Subgroups — C₂₃×#17767 #19165393

Order 5888.

862 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3509	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2989	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2925	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1561	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1559	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1559	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1556	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1554	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1554	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1552	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1552	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1550	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1544	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1542	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#557	—	2	1	no	—
17	C₂₃×#557	C₄	1	2	yes	—
18	C₂₃×#556	—	2	1	no	—
19	C₂₃×#556	C₄	1	2	yes	—
20	C₂₃×#552	—	2	1	no	—
21	C₂₃×#552	—	2	1	no	—
22	C₂₃×#552	C₄	1	2	yes	—
23	C₂₃×#552	C₄	1	2	yes	—
24	C₂₃×#541	—	2	1	no	—
25	C₂₃×#541	—	2	1	no	—
26	C₂₃×#536	—	2	1	no	—
27	C₂₃×#533	C₂²	1	1	yes	characteristic
28	C₂₃×#533	C₂²	1	1	yes	characteristic
29	C₂₃×#533	C₂²	1	1	yes	characteristic
30	C₂₃×#533	C₂²	1	1	yes	characteristic
31	C₂₃×#533	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
33	C₂₃×#519	—	2	1	no	—
34	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#446	—	2	1	no	—
36	C₂₃×#436	—	2	1	no	—
37	C₂₃×#435	—	2	1	no	—
38	C₂₃×#435	—	2	1	no	—
39	C₂₃×#433	—	2	1	no	—
40	C₂₃×#433	—	2	1	no	—
41	C₂₃×#432	C₂²	1	1	yes	characteristic
42	C₂₃×#432	C₂²	1	1	yes	characteristic
43	C₂₃×#432	C₂²	1	1	yes	characteristic
44	C₂₃×#432	C₂²	1	1	yes	characteristic
45	C₂₃×#431	C₂²	1	1	yes	characteristic
46	C₂₃×#431	C₂²	1	1	yes	characteristic
47	C₂₃×#431	C₂²	1	1	yes	characteristic
48	C₂₃×#423	C₂²	1	1	yes	characteristic
49	C₂₃×#423	C₂²	1	1	yes	characteristic
50	C₂₃×#423	C₂²	1	1	yes	characteristic
51	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
52	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
53	C₂₃×#408	—	2	1	no	—
54	C₂₃×#405	—	2	1	no	—
55	C₂₃×#333	C₂²	1	1	yes	characteristic
56	C₂₃×#333	C₂²	1	1	yes	characteristic
57	C₂₃×#333	C₂²	1	1	yes	characteristic
58	C₂₃×#333	C₂²	1	1	yes	characteristic
59	C₂₃×#332	C₂²	1	1	yes	characteristic
60	C₂₃×#332	C₂²	1	1	yes	characteristic
61	C₂₃×#332	C₂²	1	1	yes	characteristic
62	C₂₃×#332	C₂²	1	1	yes	characteristic
63	C₂₃×#332	C₂²	1	1	yes	characteristic
64	C₂₃×#332	C₂²	1	1	yes	characteristic
65	C₂₃×#332	C₂²	1	1	yes	characteristic
66	C₂₃×#332	C₂²	1	1	yes	characteristic
67	C₂₃×#332	C₂²	1	1	yes	characteristic
68	C₂₃×#332	C₂²	1	1	yes	characteristic
69	C₂₃×#332	C₂²	1	1	yes	characteristic
70	C₂₃×#332	C₂²	1	1	yes	characteristic
71	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
72	C₂²×Q₈×C₂₃	—	2	1	no	—
73	C₂×M₁₆×C₂₃	—	2	1	no	—
74	C₂×M₁₆×C₂₃	—	2	1	no	—
75	C₂×M₁₆×C₂₃	—	2	1	no	—
76	C₂²×C₈×C₂₃	—	2	1	no	—
77	C₂²×C₈×C₂₃	—	2	1	no	—
78	C₂₃×C₄⋊Q₈	—	2	2	no	—
79	C₂₃×C₄⋊Q₈	—	2	1	no	—
80	C₂₃×C₄⋊Q₈	—	2	1	no	—
81	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
82	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₂₃×#126	—	2	2	no	—
87	C₂₃×#126	—	2	2	no	—
88	C₂₃×#126	—	2	2	no	—
89	C₂₃×#126	—	2	2	no	—
90	C₂₃×#125	—	2	2	no	—
91	C₂₃×#125	—	2	2	no	—
92	C₂₃×#125	—	2	2	no	—
93	C₂₃×#125	—	2	2	no	—
94	C₂₃×#125	—	2	1	no	—
95	C₂₃×#125	—	2	1	no	—
96	C₂₃×#125	—	2	1	no	—
97	C₂₃×#125	C₂×C₄	1	2	yes	—
98	C₂₃×#125	C₂×C₄	1	2	yes	—
99	C₂₃×#124	—	2	2	no	—
100	C₂₃×#123	—	2	2	no	—
101	C₂₃×#123	—	2	2	no	—
102	C₂₃×C₂²⋊Q₈	—	2	1	no	—
103	C₂₃×C₂²⋊Q₈	—	2	1	no	—
104	C₂₃×C₂²⋊Q₈	—	2	1	no	—
105	C₂₃×C₂²⋊Q₈	—	2	1	no	—
106	C₂₃×C₂²⋊Q₈	—	2	1	no	—
107	C₂₃×C₂²⋊Q₈	—	2	1	no	—
108	C₂₃×C₂²⋊Q₈	—	2	1	no	—
109	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
110	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
111	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
112	C₄×Q₈×C₂₃	—	2	2	no	—
113	C₄×Q₈×C₂₃	—	2	2	no	—
114	C₄×Q₈×C₂₃	—	2	1	no	—
115	C₄×Q₈×C₂₃	—	2	1	no	—
116	C₄×Q₈×C₂₃	—	2	1	no	—
117	C₄×Q₈×C₂₃	—	2	1	no	—
118	C₄×Q₈×C₂₃	—	2	1	no	—
119	C₄×Q₈×C₂₃	—	2	1	no	—
120	C₄×Q₈×C₂₃	—	2	1	no	—
121	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
122	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
123	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
124	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
125	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
126	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
127	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
128	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
129	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
130	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
131	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
132	C₂₃×#117	—	2	1	no	—
133	C₂₃×#117	—	2	1	no	—
134	C₂₃×#117	—	2	1	no	—
135	C₂₃×#117	—	2	2	no	—
136	C₂₃×#117	—	2	2	no	—
137	C₂₃×#117	C₂×C₄	1	2	yes	—
138	C₂₃×#117	C₂×C₄	1	2	yes	—
139	C₂₃×#117	D₄	1	1	yes	characteristic
140	C₂₃×#117	D₄	1	1	yes	characteristic
141	C₂₃×#117	D₄	1	1	yes	characteristic
142	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
143	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
144	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
145	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
146	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
147	C₂₃×C₄⋊C₈	—	2	1	no	—
148	C₂₃×C₄⋊C₈	—	2	1	no	—
149	C₂₃×C₄⋊C₈	—	2	1	no	—
150	C₂₃×C₄⋊C₈	—	2	1	no	—
151	C₂₃×C₄⋊C₈	—	2	1	no	—
152	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
153	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
154	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
155	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
156	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
157	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
158	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
159	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
160	C₂₃×C₂²⋊C₈	—	2	1	no	—
161	C₂₃×C₂²⋊C₈	—	2	1	no	—
162	C₂₃×C₂²⋊C₈	—	2	1	no	—
163	C₂₃×C₂²⋊C₈	—	2	1	no	—
164	C₂₃×C₂²⋊C₈	—	2	1	no	—
165	C₂₃×C₂²⋊C₈	—	2	1	no	—
166	C₂₃×C₂²⋊C₈	—	2	1	no	—
167	C₂₃×#97	—	2	1	no	—
168	C₂₃×#97	—	2	1	no	—
169	C₂₃×#97	—	2	1	no	—
170	C₂₃×#97	—	2	1	no	—
171	C₄×C₈×C₂₃	—	2	1	no	—
172	C₄×C₈×C₂₃	—	2	1	no	—
173	C₄×C₈×C₂₃	—	2	1	no	—
174	#41×C₂₃	—	4	1	no	—
175	#41×C₂₃	—	4	1	no	—
176	C₂×Q₈×C₂₃	—	2	2	no	—
177	C₂×Q₈×C₂₃	—	2	1	no	—
178	C₂×Q₈×C₂₃	—	2	1	no	—
179	C₂×Q₈×C₂₃	—	2	1	no	—
180	C₂×Q₈×C₂₃	—	4	2	no	—
181	C₂×Q₈×C₂₃	—	4	2	no	—
182	C₂×Q₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
183	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
184	C₂×D₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
185	C₂×D₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
186	C₂×D₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
187	C₂²×C₄×C₂₃	—	2	2	no	—
188	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
189	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
190	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
191	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
193	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
194	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
195	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
196	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
197	M₁₆×C₂₃	—	4	2	no	—
198	M₁₆×C₂₃	—	8	1	no	—
199	M₁₆×C₂₃	—	8	1	no	—
200	C₂×C₈×C₂₃	—	2	1	no	—
201	C₂×C₈×C₂₃	—	2	1	no	—
202	C₂×C₈×C₂₃	—	2	1	no	—
203	C₂×C₈×C₂₃	—	2	1	no	—
204	C₂×C₈×C₂₃	—	2	1	no	—
205	C₂×C₈×C₂₃	—	2	1	no	—
206	C₂×C₈×C₂₃	—	2	1	no	—
207	C₂×C₈×C₂₃	—	2	1	no	—
208	C₂×C₈×C₂₃	—	4	2	no	—
209	C₂×C₈×C₂₃	—	8	1	no	—
210	C₄⋊C₄×C₂₃	—	2	2	no	—
211	C₄⋊C₄×C₂₃	—	2	2	no	—
212	C₄⋊C₄×C₂₃	—	2	2	no	—
213	C₄⋊C₄×C₂₃	—	2	2	no	—
214	C₄⋊C₄×C₂₃	—	2	2	no	—
215	C₄⋊C₄×C₂₃	—	2	2	no	—
216	C₄⋊C₄×C₂₃	—	2	2	no	—
217	C₄⋊C₄×C₂₃	—	2	2	no	—
218	C₄⋊C₄×C₂₃	—	2	2	no	—
219	C₄⋊C₄×C₂₃	—	2	2	no	—
220	C₄⋊C₄×C₂₃	—	2	2	no	—
221	C₄⋊C₄×C₂₃	—	2	2	no	—
222	C₄⋊C₄×C₂₃	—	2	1	no	—
223	C₄⋊C₄×C₂₃	—	2	1	no	—
224	C₄⋊C₄×C₂₃	—	2	1	no	—
225	C₄⋊C₄×C₂₃	—	2	1	no	—
226	C₄⋊C₄×C₂₃	—	2	1	no	—
227	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
228	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
229	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
230	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
231	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
232	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
233	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
234	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
235	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
236	C₂²⋊C₄×C₂₃	—	2	2	no	—
237	C₂²⋊C₄×C₂₃	—	2	2	no	—
238	C₂²⋊C₄×C₂₃	—	2	2	no	—
239	C₂²⋊C₄×C₂₃	—	2	2	no	—
240	C₂²⋊C₄×C₂₃	—	2	2	no	—
241	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
242	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
243	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
244	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
245	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
246	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
247	C₄²×C₂₃	—	2	2	no	—
248	C₄²×C₂₃	—	2	2	no	—
249	C₄²×C₂₃	—	2	2	no	—
250	C₄²×C₂₃	—	2	2	no	—
251	C₄²×C₂₃	—	2	2	no	—
252	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
253	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
254	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
255	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
256	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
257	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
258	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
259	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
260	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
261	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
262	#17767	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
263	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
265	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
266	Q₈×C₂₃	—	4	1	no	—
267	Q₈×C₂₃	—	4	2	no	—
268	Q₈×C₂₃	—	4	4	no	—
269	Q₈×C₂₃	—	4	1	no	—
270	Q₈×C₂₃	—	4	1	no	—
271	Q₈×C₂₃	—	4	1	no	—
272	Q₈×C₂₃	—	4	1	no	—
273	Q₈×C₂₃	—	4	1	no	—
274	Q₈×C₂₃	—	4	1	no	—
275	D₄×C₂₃	—	4	1	no	—
276	D₄×C₂₃	—	4	1	no	—
277	D₄×C₂₃	—	4	1	no	—
278	C₂×C₄×C₂₃	—	2	1	no	—
279	C₂×C₄×C₂₃	—	2	1	no	—
280	C₂×C₄×C₂₃	—	2	1	no	—
281	C₂×C₄×C₂₃	—	2	1	no	—
282	C₂×C₄×C₂₃	—	2	1	no	—
283	C₂×C₄×C₂₃	—	2	1	no	—
284	C₂×C₄×C₂₃	—	2	1	no	—
285	C₂×C₄×C₂₃	—	2	1	no	—
286	C₂×C₄×C₂₃	—	2	1	no	—
287	C₂×C₄×C₂₃	—	2	1	no	—
288	C₂×C₄×C₂₃	—	2	1	no	—
289	C₂×C₄×C₂₃	—	2	1	no	—
290	C₂×C₄×C₂₃	—	2	2	no	—
291	C₂×C₄×C₂₃	—	2	2	no	—
292	C₂×C₄×C₂₃	—	2	2	no	—
293	C₂×C₄×C₂₃	—	2	2	no	—
294	C₂×C₄×C₂₃	—	4	1	no	—
295	C₂×C₄×C₂₃	—	4	1	no	—
296	C₂×C₄×C₂₃	—	4	1	no	—
297	C₂×C₄×C₂₃	—	4	4	no	—
298	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
299	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
300	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
301	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
302	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
303	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
304	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
305	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
306	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
307	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
308	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
309	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
310	C₈×C₂₃	—	4	1	no	—
311	C₈×C₂₃	—	4	1	no	—
312	C₈×C₂₃	—	4	1	no	—
313	C₈×C₂₃	—	4	1	no	—
314	C₈×C₂₃	—	4	1	no	—
315	C₈×C₂₃	—	4	1	no	—
316	C₈×C₂₃	—	4	1	no	—
317	C₈×C₂₃	—	4	1	no	—
318	#3509	C₂×C₂₃	1	1	yes	characteristic
319	#2989	C₂×C₂₃	1	1	yes	characteristic
320	#2925	C₂×C₂₃	1	1	yes	characteristic
321	#1561	C₂×C₂₃	1	1	yes	characteristic, complemented
322	#1559	C₂×C₂₃	1	1	yes	characteristic, complemented
323	#1559	C₂×C₂₃	1	1	yes	characteristic, complemented
324	#1556	C₂×C₂₃	1	1	yes	characteristic, complemented
325	#1554	C₂×C₂₃	1	1	yes	characteristic, complemented
326	#1554	C₂×C₂₃	1	1	yes	characteristic, complemented
327	#1552	C₂×C₂₃	1	1	yes	characteristic, complemented
328	#1552	C₂×C₂₃	1	1	yes	characteristic, complemented
329	#1550	C₂×C₂₃	1	1	yes	characteristic, complemented
330	#1549	C₂×C₂₃	1	1	yes	characteristic, complemented
331	#1544	C₂×C₂₃	1	1	yes	characteristic, complemented
332	#1542	C₂×C₂₃	1	1	yes	characteristic, complemented
333	C₂²×C₂₃	—	2	1	no	—
334	C₂²×C₂₃	—	2	1	no	—
335	C₂²×C₂₃	—	2	1	no	—
336	C₂²×C₂₃	—	2	1	no	—
337	C₂²×C₂₃	—	2	1	no	—
338	C₂²×C₂₃	—	2	1	no	—
339	C₂²×C₂₃	—	2	1	no	—
340	C₂²×C₂₃	—	2	1	no	—
341	C₂²×C₂₃	—	2	1	no	—
342	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
343	C₄×C₂₃	—	2	1	no	—
344	C₄×C₂₃	—	2	1	no	—
345	C₄×C₂₃	—	2	1	no	—
346	C₄×C₂₃	—	2	1	no	—
347	C₄×C₂₃	—	2	1	no	—
348	C₄×C₂₃	—	2	1	no	—
349	C₄×C₂₃	—	2	1	no	—
350	C₄×C₂₃	—	2	1	no	—
351	C₄×C₂₃	—	2	1	no	—
352	C₄×C₂₃	—	2	1	no	—
353	C₄×C₂₃	—	4	2	no	—
354	C₄×C₂₃	—	4	2	no	—
355	C₄×C₂₃	—	4	2	no	—
356	C₄×C₂₃	—	4	2	no	—
357	C₄×C₂₃	#575	1	1	yes	characteristic
358	C₄×C₂₃	#575	1	1	yes	characteristic
359	C₄×C₂₃	#577	1	1	yes	characteristic
360	C₄×C₂₃	#578	1	1	yes	characteristic
361	#557	—	2	1	no	—
362	#557	C₄×C₂₃	1	2	yes	—
363	#556	—	2	1	no	—
364	#556	C₄×C₂₃	1	2	yes	—
365	#552	—	2	1	no	—
366	#552	—	2	1	no	—
367	#552	C₄×C₂₃	1	2	yes	—
368	#552	C₄×C₂₃	1	2	yes	—
369	#541	—	2	1	no	—
370	#541	—	2	1	no	—
371	#536	—	2	1	no	—
372	#533	C₂²×C₂₃	1	1	yes	characteristic
373	#533	C₂²×C₂₃	1	1	yes	characteristic
374	#533	C₂²×C₂₃	1	1	yes	characteristic
375	#533	C₂²×C₂₃	1	1	yes	characteristic
376	#533	C₂²×C₂₃	1	1	yes	characteristic
377	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
378	#519	—	2	1	no	—
379	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
380	#446	—	2	1	no	—
381	#436	—	2	1	no	—
382	#435	—	2	1	no	—
383	#435	—	2	1	no	—
384	#433	—	2	1	no	—
385	#433	—	2	1	no	—
386	#432	C₂²×C₂₃	1	1	yes	characteristic
387	#432	C₂²×C₂₃	1	1	yes	characteristic
388	#432	C₂²×C₂₃	1	1	yes	characteristic
389	#432	C₂²×C₂₃	1	1	yes	characteristic
390	#431	C₂²×C₂₃	1	1	yes	characteristic
391	#431	C₂²×C₂₃	1	1	yes	characteristic
392	#431	C₂²×C₂₃	1	1	yes	characteristic
393	#423	C₂²×C₂₃	1	1	yes	characteristic
394	#423	C₂²×C₂₃	1	1	yes	characteristic
395	#423	C₂²×C₂₃	1	1	yes	characteristic
396	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
397	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
398	#408	—	2	1	no	—
399	#405	—	2	1	no	—
400	#333	C₂²×C₂₃	1	1	yes	characteristic
401	#333	C₂²×C₂₃	1	1	yes	characteristic
402	#333	C₂²×C₂₃	1	1	yes	characteristic
403	#333	C₂²×C₂₃	1	1	yes	characteristic
404	#332	C₂²×C₂₃	1	1	yes	characteristic
405	#332	C₂²×C₂₃	1	1	yes	characteristic
406	#332	C₂²×C₂₃	1	1	yes	characteristic
407	#332	C₂²×C₂₃	1	1	yes	characteristic
408	#332	C₂²×C₂₃	1	1	yes	characteristic
409	#332	C₂²×C₂₃	1	1	yes	characteristic
410	#332	C₂²×C₂₃	1	1	yes	characteristic
411	#332	C₂²×C₂₃	1	1	yes	characteristic
412	#332	C₂²×C₂₃	1	1	yes	characteristic
413	#332	C₂²×C₂₃	1	1	yes	characteristic
414	#332	C₂²×C₂₃	1	1	yes	characteristic
415	#332	C₂²×C₂₃	1	1	yes	characteristic
416	C₂×C₂₃	—	4	1	no	—
417	C₂×C₂₃	—	4	1	no	—
418	C₂×C₂₃	—	4	1	no	—
419	C₂×C₂₃	#2889	1	1	yes	characteristic
420	C₂×C₂₃	#2896	1	1	yes	characteristic
421	C₂×C₂₃	#2898	1	1	yes	characteristic
422	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
423	C₂²×Q₈	—	2	1	no	—
424	C₂×M₁₆	—	2	1	no	—
425	C₂×M₁₆	—	2	1	no	—
426	C₂×M₁₆	—	2	1	no	—
427	C₂²×C₈	—	2	1	no	—
428	C₂²×C₈	—	2	1	no	—
429	C₄⋊Q₈	—	2	2	no	—
430	C₄⋊Q₈	—	2	1	no	—
431	C₄⋊Q₈	—	2	1	no	—
432	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
433	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
434	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
435	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
436	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
437	#126	—	2	2	no	—
438	#126	—	2	2	no	—
439	#126	—	2	2	no	—
440	#126	—	2	2	no	—
441	#125	—	2	2	no	—
442	#125	—	2	2	no	—
443	#125	—	2	2	no	—
444	#125	—	2	2	no	—
445	#125	—	2	1	no	—
446	#125	—	2	1	no	—
447	#125	—	2	1	no	—
448	#125	C₂×C₄×C₂₃	1	2	yes	—
449	#125	C₂×C₄×C₂₃	1	2	yes	—
450	#124	—	2	2	no	—
451	#123	—	2	2	no	—
452	#123	—	2	2	no	—
453	C₂²⋊Q₈	—	2	1	no	—
454	C₂²⋊Q₈	—	2	1	no	—
455	C₂²⋊Q₈	—	2	1	no	—
456	C₂²⋊Q₈	—	2	1	no	—
457	C₂²⋊Q₈	—	2	1	no	—
458	C₂²⋊Q₈	—	2	1	no	—
459	C₂²⋊Q₈	—	2	1	no	—
460	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
461	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
462	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
463	C₄×Q₈	—	2	2	no	—
464	C₄×Q₈	—	2	2	no	—
465	C₄×Q₈	—	2	1	no	—
466	C₄×Q₈	—	2	1	no	—
467	C₄×Q₈	—	2	1	no	—
468	C₄×Q₈	—	2	1	no	—
469	C₄×Q₈	—	2	1	no	—
470	C₄×Q₈	—	2	1	no	—
471	C₄×Q₈	—	2	1	no	—
472	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
473	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
474	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
475	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
476	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
477	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
478	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
479	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
480	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
481	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
482	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
483	#117	—	2	1	no	—
484	#117	—	2	1	no	—
485	#117	—	2	1	no	—
486	#117	—	2	2	no	—
487	#117	—	2	2	no	—
488	#117	C₂×C₄×C₂₃	1	2	yes	—
489	#117	C₂×C₄×C₂₃	1	2	yes	—
490	#117	D₄×C₂₃	1	1	yes	characteristic
491	#117	D₄×C₂₃	1	1	yes	characteristic
492	#117	D₄×C₂₃	1	1	yes	characteristic
493	C₂×C₄⋊C₄	—	2	1	no	—
494	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
495	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
496	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
497	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
498	C₄⋊C₈	—	2	1	no	—
499	C₄⋊C₈	—	2	1	no	—
500	C₄⋊C₈	—	2	1	no	—
501	C₄⋊C₈	—	2	1	no	—
502	C₄⋊C₈	—	2	1	no	—
503	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
504	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
505	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
506	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
507	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
508	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
509	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
510	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
511	C₂²⋊C₈	—	2	1	no	—
512	C₂²⋊C₈	—	2	1	no	—
513	C₂²⋊C₈	—	2	1	no	—
514	C₂²⋊C₈	—	2	1	no	—
515	C₂²⋊C₈	—	2	1	no	—
516	C₂²⋊C₈	—	2	1	no	—
517	C₂²⋊C₈	—	2	1	no	—
518	#97	—	2	1	no	—
519	#97	—	2	1	no	—
520	#97	—	2	1	no	—
521	#97	—	2	1	no	—
522	C₄×C₈	—	2	1	no	—
523	C₄×C₈	—	2	1	no	—
524	C₄×C₈	—	2	1	no	—
525	C₂₃	#17767	1	1	yes	characteristic, direct product factor, Sylow
526	#41	—	4	1	no	—
527	#41	—	4	1	no	—
528	C₂×Q₈	—	2	2	no	—
529	C₂×Q₈	—	2	1	no	—
530	C₂×Q₈	—	2	1	no	—
531	C₂×Q₈	—	2	1	no	—
532	C₂×Q₈	—	4	2	no	—
533	C₂×Q₈	—	4	2	no	—
534	C₂×Q₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
535	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
536	C₂×D₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
537	C₂×D₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
538	C₂×D₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
539	C₂²×C₄	—	2	2	no	—
540	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
541	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
542	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
543	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
544	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
545	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
546	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
547	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
548	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
549	M₁₆	—	4	2	no	—
550	M₁₆	—	8	1	no	—
551	M₁₆	—	8	1	no	—
552	C₂×C₈	—	2	1	no	—
553	C₂×C₈	—	2	1	no	—
554	C₂×C₈	—	2	1	no	—
555	C₂×C₈	—	2	1	no	—
556	C₂×C₈	—	2	1	no	—
557	C₂×C₈	—	2	1	no	—
558	C₂×C₈	—	2	1	no	—
559	C₂×C₈	—	2	1	no	—
560	C₂×C₈	—	4	2	no	—
561	C₂×C₈	—	8	1	no	—
562	C₄⋊C₄	—	2	2	no	—
563	C₄⋊C₄	—	2	2	no	—
564	C₄⋊C₄	—	2	2	no	—
565	C₄⋊C₄	—	2	2	no	—
566	C₄⋊C₄	—	2	2	no	—
567	C₄⋊C₄	—	2	2	no	—
568	C₄⋊C₄	—	2	2	no	—
569	C₄⋊C₄	—	2	2	no	—
570	C₄⋊C₄	—	2	2	no	—
571	C₄⋊C₄	—	2	2	no	—
572	C₄⋊C₄	—	2	2	no	—
573	C₄⋊C₄	—	2	2	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	2	1	no	—
576	C₄⋊C₄	—	2	1	no	—
577	C₄⋊C₄	—	2	1	no	—
578	C₄⋊C₄	—	2	1	no	—
579	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
580	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
581	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
582	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
583	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
584	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
585	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
586	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
587	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
588	C₂²⋊C₄	—	2	2	no	—
589	C₂²⋊C₄	—	2	2	no	—
590	C₂²⋊C₄	—	2	2	no	—
591	C₂²⋊C₄	—	2	2	no	—
592	C₂²⋊C₄	—	2	2	no	—
593	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
594	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
595	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
596	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
597	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
598	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
599	C₄²	—	2	2	no	—
600	C₄²	—	2	2	no	—
601	C₄²	—	2	2	no	—
602	C₄²	—	2	2	no	—
603	C₄²	—	2	2	no	—
604	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
605	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
606	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
607	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
608	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
609	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
610	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
611	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
612	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
613	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
614	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
615	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
616	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
617	Q₈	—	4	1	no	—
618	Q₈	—	4	2	no	—
619	Q₈	—	4	4	no	—
620	Q₈	—	4	1	no	—
621	Q₈	—	4	1	no	—
622	Q₈	—	4	1	no	—
623	Q₈	—	4	1	no	—
624	Q₈	—	4	1	no	—
625	Q₈	—	4	1	no	—
626	D₄	—	4	1	no	—
627	D₄	—	4	1	no	—
628	D₄	—	4	1	no	—
629	C₂×C₄	—	2	1	no	—
630	C₂×C₄	—	2	1	no	—
631	C₂×C₄	—	2	1	no	—
632	C₂×C₄	—	2	1	no	—
633	C₂×C₄	—	2	1	no	—
634	C₂×C₄	—	2	1	no	—
635	C₂×C₄	—	2	1	no	—
636	C₂×C₄	—	2	1	no	—
637	C₂×C₄	—	2	1	no	—
638	C₂×C₄	—	2	1	no	—
639	C₂×C₄	—	2	1	no	—
640	C₂×C₄	—	2	1	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	4	1	no	—
646	C₂×C₄	—	4	1	no	—
647	C₂×C₄	—	4	1	no	—
648	C₂×C₄	—	4	4	no	—
649	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
650	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
651	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
652	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
653	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
654	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
655	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
656	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
657	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
658	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
659	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
660	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
661	C₈	—	4	1	no	—
662	C₈	—	4	1	no	—
663	C₈	—	4	1	no	—
664	C₈	—	4	1	no	—
665	C₈	—	4	1	no	—
666	C₈	—	4	1	no	—
667	C₈	—	4	1	no	—
668	C₈	—	4	1	no	—
669	C₂²	—	2	1	no	—
670	C₂²	—	2	1	no	—
671	C₂²	—	2	1	no	—
672	C₂²	—	2	1	no	—
673	C₂²	—	2	1	no	—
674	C₂²	—	2	1	no	—
675	C₂²	—	2	1	no	—
676	C₂²	—	2	1	no	—
677	C₂²	—	2	1	no	—
678	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
679	C₄	—	2	1	no	—
680	C₄	—	2	1	no	—
681	C₄	—	2	1	no	—
682	C₄	—	2	1	no	—
683	C₄	—	2	1	no	—
684	C₄	—	2	1	no	—
685	C₄	—	2	1	no	—
686	C₄	—	2	1	no	—
687	C₄	—	2	1	no	—
688	C₄	—	2	1	no	—
689	C₄	—	4	2	no	—
690	C₄	—	4	2	no	—
691	C₄	—	4	2	no	—
692	C₄	—	4	2	no	—
693	C₄	C₂₃×#575	1	1	yes	characteristic
694	C₄	C₂₃×#575	1	1	yes	characteristic
695	C₄	C₂₃×#577	1	1	yes	characteristic
696	C₄	C₂₃×#578	1	1	yes	characteristic
697	C₂	—	4	1	no	—
698	C₂	—	4	1	no	—
699	C₂	—	4	1	no	—
700	C₂	C₂₃×#2889	1	1	yes	characteristic
701	C₂	C₂₃×#2896	1	1	yes	characteristic
702	C₂	C₂₃×#2898	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×#41×C₂₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₂₃×#1561	C₂	semidirect
5. C₂₃×#1559	C₂	semidirect
6. C₂₃×#1559	C₂	semidirect
7. C₂₃×#1556	C₂	semidirect
8. C₂₃×#1554	C₂	semidirect
9. C₂₃×#1554	C₂	semidirect
10. C₂₃×#1552	C₂	semidirect
11. C₂₃×#1552	C₂	semidirect
12. C₂₃×#1550	C₂	semidirect
13. C₂₃×#1549	C₂	semidirect
14. C₂₃×#1544	C₂	semidirect
15. C₂₃×#1542	C₂	semidirect
262. #17767	C₂₃	direct (N × H)
321. #1561	C₂×C₂₃	semidirect
322. #1559	C₂×C₂₃	semidirect
323. #1559	C₂×C₂₃	semidirect
324. #1556	C₂×C₂₃	semidirect
325. #1554	C₂×C₂₃	semidirect
326. #1554	C₂×C₂₃	semidirect
327. #1552	C₂×C₂₃	semidirect
328. #1552	C₂×C₂₃	semidirect
329. #1550	C₂×C₂₃	semidirect
330. #1549	C₂×C₂₃	semidirect
331. #1544	C₂×C₂₃	semidirect
332. #1542	C₂×C₂₃	semidirect
525. C₂₃	#17767	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3509	2
C₂₃×#2989	2
C₂₃×#2925	2
C₂₃×#1561	2
C₂₃×#1559	2
C₂₃×#1556	2
C₂₃×#1554	2
C₂₃×#1552	2
C₂₃×#1550	2
C₂₃×#1549	2
C₂₃×#1544	2
C₂₃×#1542	2
#17767	23

Subgroups — C23×#17767 #19165393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17767 #19165393