Subgroups — C₂₃×#17591 #19165217

Order 5888.

1016 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3444	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1516	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1512	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1506	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1505	C₂	1	2	yes	maximal, complemented
8	C₂₃×#533	C₄	1	1	yes	characteristic, complemented
9	C₂₃×#533	C₄	1	1	yes	characteristic, complemented
10	C₂₃×#533	C₂²	1	1	yes	characteristic
11	C₂₃×#532	C₄	1	1	yes	characteristic, complemented
12	C₂₃×#532	C₄	1	1	yes	characteristic, complemented
13	C₂₃×#532	C₂²	1	1	yes	characteristic
14	C₂₃×#529	—	2	1	no	—
15	C₂₃×#529	—	2	1	no	—
16	C₂₃×#529	—	2	1	no	—
17	C₂₃×#529	—	2	1	no	—
18	C₂₃×#529	C₄	1	2	yes	complemented
19	C₂₃×#529	C₂²	1	2	yes	—
20	C₂₃×#525	—	2	1	no	—
21	C₂₃×#525	—	2	1	no	—
22	C₂₃×#525	C₂²	1	2	yes	—
23	C₄×#41×C₂₃	—	2	1	no	—
24	C₄×#41×C₂₃	—	2	1	no	—
25	C₄×#41×C₂₃	C₄	1	2	yes	complemented
26	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#432	—	2	1	no	—
28	C₂₃×#432	—	2	1	no	—
29	C₂₃×#432	C₂²	1	2	yes	—
30	C₂₃×#432	C₂²	1	2	yes	—
31	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
32	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₂×C₂₃×#97	—	2	1	no	—
34	C₂₃×#383	—	2	1	no	—
35	C₂₃×#382	C₂²	1	2	yes	—
36	C₂₃×#381	C₂²	1	2	yes	—
37	C₂₃×#380	—	2	1	no	—
38	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
39	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
40	C₂₃×#376	C₂²	1	2	yes	—
41	C₂₃×#328	C₂²	1	8	yes	—
42	C₂₃×#327	C₂²	1	8	yes	—
43	C₂×#41×C₂₃	—	2	1	no	—
44	C₂×#41×C₂₃	—	2	1	no	—
45	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
46	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₂₃	—	2	2	no	—
50	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
51	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
52	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
53	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
54	C₂₃×#126	—	2	2	no	—
55	C₂₃×#126	—	2	2	no	—
56	C₂₃×#125	—	2	1	no	—
57	C₂₃×#125	—	2	1	no	—
58	C₂₃×#125	C₂×C₄	1	2	yes	—
59	C₂₃×#124	—	2	1	no	—
60	C₂₃×#124	—	2	1	no	—
61	C₂₃×#124	C₂×C₄	1	2	yes	—
62	C₂₃×#123	—	2	2	no	—
63	C₂₃×#123	—	2	2	no	—
64	C₂₃×#123	—	2	2	no	—
65	C₂₃×C₂²⋊Q₈	—	2	1	no	—
66	C₂₃×C₂²⋊Q₈	—	2	1	no	—
67	C₂₃×C₂²⋊Q₈	—	2	1	no	—
68	C₂₃×C₂²⋊Q₈	—	2	1	no	—
69	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
70	C₂₃×C₂²⋊Q₈	C₂³	1	2	yes	—
71	C₂₃×#121	—	2	1	no	—
72	C₂₃×#121	—	2	1	no	—
73	C₂₃×#121	—	2	1	no	—
74	C₂₃×#121	—	2	1	no	—
75	C₂₃×#121	C₂×C₄	1	2	yes	—
76	C₂₃×#121	C₂³	1	2	yes	—
77	C₂₃×#120	—	2	2	no	—
78	C₄×Q₈×C₂₃	—	2	1	no	—
79	C₄×Q₈×C₂₃	—	2	1	no	—
80	C₄×Q₈×C₂₃	—	2	1	no	—
81	C₄×Q₈×C₂₃	—	2	1	no	—
82	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
83	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
84	C₄×D₄×C₂₃	—	2	2	no	—
85	C₄×D₄×C₂₃	—	2	2	no	—
86	C₄×D₄×C₂₃	—	2	2	no	—
87	C₄×D₄×C₂₃	—	2	2	no	—
88	C₄×D₄×C₂₃	—	2	2	no	—
89	C₄×D₄×C₂₃	—	2	2	no	—
90	C₄×D₄×C₂₃	—	2	1	no	—
91	C₄×D₄×C₂₃	—	2	1	no	—
92	C₄×D₄×C₂₃	—	2	1	no	—
93	C₄×D₄×C₂₃	—	2	1	no	—
94	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
95	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
96	C₂₃×#117	—	2	1	no	—
97	C₂₃×#117	—	2	1	no	—
98	C₂₃×#117	—	2	2	no	—
99	C₂₃×#117	—	2	2	no	—
100	C₂₃×#117	—	2	2	no	—
101	C₂₃×#117	C₂×C₄	1	2	yes	—
102	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
103	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
104	C₂×C₄²×C₂₃	—	2	2	no	—
105	C₂×C₄²×C₂₃	—	2	2	no	—
106	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
107	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
108	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
109	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
110	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
111	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
112	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
113	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
114	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
118	C₂₃×C₄⋊C₈	—	2	2	no	—
119	C₂₃×C₄⋊C₈	—	2	2	no	—
120	C₂₃×C₂²⋊C₈	—	2	2	no	—
121	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
122	C₂₃×#97	—	2	2	no	—
123	C₂₃×#97	—	2	2	no	—
124	C₂₃×#95	—	2	2	no	—
125	C₂₃×#95	—	2	2	no	—
126	C₂₃×#95	—	2	2	no	—
127	C₂₃×#95	C₂³	1	4	yes	—
128	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
129	#41×C₂₃	—	4	4	no	—
130	#41×C₂₃	—	4	4	no	—
131	#41×C₂₃	—	4	4	no	—
132	C₂×Q₈×C₂₃	—	2	1	no	—
133	C₂×Q₈×C₂₃	—	2	1	no	—
134	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
135	C₂×D₄×C₂₃	—	2	2	no	—
136	C₂×D₄×C₂₃	—	2	2	no	—
137	C₂×D₄×C₂₃	—	2	2	no	—
138	C₂×D₄×C₂₃	—	2	1	no	—
139	C₂×D₄×C₂₃	—	2	1	no	—
140	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
141	C₂²×C₄×C₂₃	—	2	2	no	—
142	C₂²×C₄×C₂₃	—	2	2	no	—
143	C₂²×C₄×C₂₃	—	2	2	no	—
144	C₂²×C₄×C₂₃	—	2	4	no	—
145	C₂²×C₄×C₂₃	—	4	2	no	—
146	C₂²×C₄×C₂₃	—	4	2	no	—
147	C₂²×C₄×C₂₃	—	4	2	no	—
148	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
166	C₂×C₈×C₂₃	—	2	4	no	—
167	C₂×C₈×C₂₃	—	4	4	no	—
168	C₄⋊C₄×C₂₃	—	2	1	no	—
169	C₄⋊C₄×C₂₃	—	2	1	no	—
170	C₄⋊C₄×C₂₃	—	2	1	no	—
171	C₄⋊C₄×C₂₃	—	2	1	no	—
172	C₄⋊C₄×C₂₃	—	2	2	no	—
173	C₄⋊C₄×C₂₃	—	2	2	no	—
174	C₄⋊C₄×C₂₃	—	2	2	no	—
175	C₄⋊C₄×C₂₃	—	2	2	no	—
176	C₄⋊C₄×C₂₃	—	2	2	no	—
177	C₄⋊C₄×C₂₃	—	2	2	no	—
178	C₄⋊C₄×C₂₃	—	4	4	no	—
179	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
180	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
181	C₂²⋊C₄×C₂₃	—	2	2	no	—
182	C₂²⋊C₄×C₂₃	—	2	2	no	—
183	C₂²⋊C₄×C₂₃	—	2	2	no	—
184	C₂²⋊C₄×C₂₃	—	2	2	no	—
185	C₂²⋊C₄×C₂₃	—	2	2	no	—
186	C₂²⋊C₄×C₂₃	—	2	2	no	—
187	C₂²⋊C₄×C₂₃	—	2	2	no	—
188	C₂²⋊C₄×C₂₃	—	2	2	no	—
189	C₂²⋊C₄×C₂₃	—	2	2	no	—
190	C₂²⋊C₄×C₂₃	—	4	4	no	—
191	C₄²×C₂₃	—	2	2	no	—
192	C₄²×C₂₃	—	2	2	no	—
193	C₄²×C₂₃	—	2	2	no	—
194	C₄²×C₂₃	—	4	4	no	—
195	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
196	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
197	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
198	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
199	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
200	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
201	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
202	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
203	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
204	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
205	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
206	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
207	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
209	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
211	#17591	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂³×C₂₃	—	2	2	no	—
213	C₂³×C₂₃	—	2	2	no	—
214	C₂³×C₂₃	—	2	2	no	—
215	C₂³×C₂₃	—	4	1	no	—
216	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
218	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
219	Q₈×C₂₃	—	4	2	no	—
220	Q₈×C₂₃	—	4	2	no	—
221	Q₈×C₂₃	—	4	2	no	—
222	D₄×C₂₃	—	4	2	no	—
223	D₄×C₂₃	—	4	2	no	—
224	D₄×C₂₃	—	4	4	no	—
225	D₄×C₂₃	—	4	4	no	—
226	D₄×C₂₃	—	4	4	no	—
227	D₄×C₂₃	—	4	2	no	—
228	C₂×C₄×C₂₃	—	2	2	no	—
229	C₂×C₄×C₂₃	—	2	2	no	—
230	C₂×C₄×C₂₃	—	2	2	no	—
231	C₂×C₄×C₂₃	—	2	2	no	—
232	C₂×C₄×C₂₃	—	2	2	no	—
233	C₂×C₄×C₂₃	—	2	2	no	—
234	C₂×C₄×C₂₃	—	2	2	no	—
235	C₂×C₄×C₂₃	—	2	2	no	—
236	C₂×C₄×C₂₃	—	2	2	no	—
237	C₂×C₄×C₂₃	—	2	2	no	—
238	C₂×C₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	4	1	no	—
240	C₂×C₄×C₂₃	—	4	1	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	4	1	no	—
244	C₂×C₄×C₂₃	—	4	1	no	—
245	C₂×C₄×C₂₃	—	4	1	no	—
246	C₂×C₄×C₂₃	—	4	1	no	—
247	C₂×C₄×C₂₃	—	4	1	no	—
248	C₂×C₄×C₂₃	—	4	1	no	—
249	C₂×C₄×C₂₃	—	4	4	no	—
250	C₂×C₄×C₂₃	—	4	4	no	—
251	C₂×C₄×C₂₃	—	4	4	no	—
252	C₂×C₄×C₂₃	—	4	4	no	—
253	C₂×C₄×C₂₃	—	4	4	no	—
254	C₂×C₄×C₂₃	—	4	4	no	—
255	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
266	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
267	C₈×C₂₃	—	4	4	no	—
268	#3444	C₂×C₂₃	1	1	yes	characteristic
269	C₂×#432	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#2283	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#1516	C₂×C₂₃	1	4	yes	complemented
272	#1512	C₂×C₂₃	1	4	yes	complemented
273	#1506	C₂×C₂₃	1	2	yes	complemented
274	#1505	C₂×C₂₃	1	2	yes	complemented
275	C₂²×C₂₃	—	2	1	no	—
276	C₂²×C₂₃	—	2	1	no	—
277	C₂²×C₂₃	—	2	1	no	—
278	C₂²×C₂₃	—	2	1	no	—
279	C₂²×C₂₃	—	2	1	no	—
280	C₂²×C₂₃	—	2	1	no	—
281	C₂²×C₂₃	—	2	1	no	—
282	C₂²×C₂₃	—	4	4	no	—
283	C₂²×C₂₃	—	4	2	no	—
284	C₂²×C₂₃	—	4	2	no	—
285	C₂²×C₂₃	—	4	2	no	—
286	C₂²×C₂₃	#410	1	2	yes	—
287	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
288	C₂²×C₂₃	#575	1	2	yes	—
289	C₄×C₂₃	—	2	1	no	—
290	C₄×C₂₃	—	2	1	no	—
291	C₄×C₂₃	—	2	1	no	—
292	C₄×C₂₃	—	2	1	no	—
293	C₄×C₂₃	—	2	1	no	—
294	C₄×C₂₃	—	2	1	no	—
295	C₄×C₂₃	—	2	1	no	—
296	C₄×C₂₃	—	2	1	no	—
297	C₄×C₂₃	—	2	1	no	—
298	C₄×C₂₃	—	2	1	no	—
299	C₄×C₂₃	—	2	1	no	—
300	C₄×C₂₃	—	4	2	no	—
301	C₄×C₂₃	—	4	2	no	—
302	C₄×C₂₃	—	4	2	no	—
303	C₄×C₂₃	—	8	4	no	—
304	C₄×C₂₃	#421	1	2	yes	—
305	#533	C₄×C₂₃	1	1	yes	characteristic, complemented
306	#533	C₄×C₂₃	1	1	yes	characteristic, complemented
307	#533	C₂²×C₂₃	1	1	yes	characteristic
308	#532	C₄×C₂₃	1	1	yes	characteristic, complemented
309	#532	C₄×C₂₃	1	1	yes	characteristic, complemented
310	#532	C₂²×C₂₃	1	1	yes	characteristic
311	#529	—	2	1	no	—
312	#529	—	2	1	no	—
313	#529	—	2	1	no	—
314	#529	—	2	1	no	—
315	#529	C₄×C₂₃	1	2	yes	complemented
316	#529	C₂²×C₂₃	1	2	yes	—
317	#525	—	2	1	no	—
318	#525	—	2	1	no	—
319	#525	C₂²×C₂₃	1	2	yes	—
320	C₄×#41	—	2	1	no	—
321	C₄×#41	—	2	1	no	—
322	C₄×#41	C₄×C₂₃	1	2	yes	complemented
323	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
324	#432	—	2	1	no	—
325	#432	—	2	1	no	—
326	#432	C₂²×C₂₃	1	2	yes	—
327	#432	C₂²×C₂₃	1	2	yes	—
328	C₂×C₄⋊C₈	—	2	1	no	—
329	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
330	C₂×#97	—	2	1	no	—
331	#383	—	2	1	no	—
332	#382	C₂²×C₂₃	1	2	yes	—
333	#381	C₂²×C₂₃	1	2	yes	—
334	#380	—	2	1	no	—
335	C₄×C₄⋊C₄	—	2	1	no	—
336	C₄×C₂²⋊C₄	—	2	1	no	—
337	#376	C₂²×C₂₃	1	2	yes	—
338	#328	C₂²×C₂₃	1	8	yes	—
339	#327	C₂²×C₂₃	1	8	yes	—
340	C₂×C₂₃	—	2	2	no	—
341	C₂×C₂₃	—	2	2	no	—
342	C₂×C₂₃	—	4	1	no	—
343	C₂×C₂₃	—	8	2	no	—
344	C₂×C₂₃	C₂×#410	1	1	yes	characteristic
345	C₂×C₂₃	#2896	1	1	yes	characteristic
346	C₂×C₂₃	C₂×#421	1	1	yes	characteristic
347	C₂×#41	—	2	1	no	—
348	C₂×#41	—	2	1	no	—
349	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
350	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
351	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
352	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
353	C₂²×C₈	—	2	2	no	—
354	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
355	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
356	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
357	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
358	#126	—	2	2	no	—
359	#126	—	2	2	no	—
360	#125	—	2	1	no	—
361	#125	—	2	1	no	—
362	#125	C₂×C₄×C₂₃	1	2	yes	—
363	#124	—	2	1	no	—
364	#124	—	2	1	no	—
365	#124	C₂×C₄×C₂₃	1	2	yes	—
366	#123	—	2	2	no	—
367	#123	—	2	2	no	—
368	#123	—	2	2	no	—
369	C₂²⋊Q₈	—	2	1	no	—
370	C₂²⋊Q₈	—	2	1	no	—
371	C₂²⋊Q₈	—	2	1	no	—
372	C₂²⋊Q₈	—	2	1	no	—
373	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
374	C₂²⋊Q₈	C₂³×C₂₃	1	2	yes	—
375	#121	—	2	1	no	—
376	#121	—	2	1	no	—
377	#121	—	2	1	no	—
378	#121	—	2	1	no	—
379	#121	C₂×C₄×C₂₃	1	2	yes	—
380	#121	C₂³×C₂₃	1	2	yes	—
381	#120	—	2	2	no	—
382	C₄×Q₈	—	2	1	no	—
383	C₄×Q₈	—	2	1	no	—
384	C₄×Q₈	—	2	1	no	—
385	C₄×Q₈	—	2	1	no	—
386	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
387	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
388	C₄×D₄	—	2	2	no	—
389	C₄×D₄	—	2	2	no	—
390	C₄×D₄	—	2	2	no	—
391	C₄×D₄	—	2	2	no	—
392	C₄×D₄	—	2	2	no	—
393	C₄×D₄	—	2	2	no	—
394	C₄×D₄	—	2	1	no	—
395	C₄×D₄	—	2	1	no	—
396	C₄×D₄	—	2	1	no	—
397	C₄×D₄	—	2	1	no	—
398	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
399	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
400	#117	—	2	1	no	—
401	#117	—	2	1	no	—
402	#117	—	2	2	no	—
403	#117	—	2	2	no	—
404	#117	—	2	2	no	—
405	#117	C₂×C₄×C₂₃	1	2	yes	—
406	C₂×C₄⋊C₄	—	2	2	no	—
407	C₂×C₂²⋊C₄	—	2	2	no	—
408	C₂×C₄²	—	2	2	no	—
409	C₂×C₄²	—	2	2	no	—
410	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
411	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
412	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
413	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
414	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
415	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
416	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
417	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
418	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
419	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
420	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
421	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
422	C₄⋊C₈	—	2	2	no	—
423	C₄⋊C₈	—	2	2	no	—
424	C₂²⋊C₈	—	2	2	no	—
425	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
426	#97	—	2	2	no	—
427	#97	—	2	2	no	—
428	#95	—	2	2	no	—
429	#95	—	2	2	no	—
430	#95	—	2	2	no	—
431	#95	C₂³×C₂₃	1	4	yes	—
432	C₂₃	#17591	1	1	yes	characteristic, direct product factor, Sylow
433	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
434	#41	—	4	4	no	—
435	#41	—	4	4	no	—
436	#41	—	4	4	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	2	1	no	—
444	C₂×D₄	—	2	1	no	—
445	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	2	4	no	—
450	C₂²×C₄	—	4	2	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	—	4	2	no	—
453	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
454	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
455	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
456	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
457	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
458	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
459	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
460	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
461	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
462	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
465	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
466	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
467	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
468	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
469	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
470	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
471	C₂×C₈	—	2	4	no	—
472	C₂×C₈	—	4	4	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	2	no	—
478	C₄⋊C₄	—	2	2	no	—
479	C₄⋊C₄	—	2	2	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	4	4	no	—
484	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
485	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
486	C₂²⋊C₄	—	2	2	no	—
487	C₂²⋊C₄	—	2	2	no	—
488	C₂²⋊C₄	—	2	2	no	—
489	C₂²⋊C₄	—	2	2	no	—
490	C₂²⋊C₄	—	2	2	no	—
491	C₂²⋊C₄	—	2	2	no	—
492	C₂²⋊C₄	—	2	2	no	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	4	4	no	—
496	C₄²	—	2	2	no	—
497	C₄²	—	2	2	no	—
498	C₄²	—	2	2	no	—
499	C₄²	—	4	4	no	—
500	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
501	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
502	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
503	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
504	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
505	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
506	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
507	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
508	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
509	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
510	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
511	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
512	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
513	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
514	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
515	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
516	C₂³	—	2	2	no	—
517	C₂³	—	2	2	no	—
518	C₂³	—	2	2	no	—
519	C₂³	—	4	1	no	—
520	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
521	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
522	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
523	Q₈	—	4	2	no	—
524	Q₈	—	4	2	no	—
525	Q₈	—	4	2	no	—
526	D₄	—	4	2	no	—
527	D₄	—	4	2	no	—
528	D₄	—	4	4	no	—
529	D₄	—	4	4	no	—
530	D₄	—	4	4	no	—
531	D₄	—	4	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	4	no	—
554	C₂×C₄	—	4	4	no	—
555	C₂×C₄	—	4	4	no	—
556	C₂×C₄	—	4	4	no	—
557	C₂×C₄	—	4	4	no	—
558	C₂×C₄	—	4	4	no	—
559	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
560	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
561	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
562	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
563	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
564	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
565	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
566	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
567	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
568	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
569	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
570	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
571	C₈	—	4	4	no	—
572	C₂²	—	2	1	no	—
573	C₂²	—	2	1	no	—
574	C₂²	—	2	1	no	—
575	C₂²	—	2	1	no	—
576	C₂²	—	2	1	no	—
577	C₂²	—	2	1	no	—
578	C₂²	—	2	1	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	4	2	no	—
583	C₂²	C₂₃×#410	1	2	yes	—
584	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
585	C₂²	C₂₃×#575	1	2	yes	—
586	C₄	—	2	1	no	—
587	C₄	—	2	1	no	—
588	C₄	—	2	1	no	—
589	C₄	—	2	1	no	—
590	C₄	—	2	1	no	—
591	C₄	—	2	1	no	—
592	C₄	—	2	1	no	—
593	C₄	—	2	1	no	—
594	C₄	—	2	1	no	—
595	C₄	—	2	1	no	—
596	C₄	—	2	1	no	—
597	C₄	—	4	2	no	—
598	C₄	—	4	2	no	—
599	C₄	—	4	2	no	—
600	C₄	—	8	4	no	—
601	C₄	C₂₃×#421	1	2	yes	—
602	C₂	—	2	2	no	—
603	C₂	—	2	2	no	—
604	C₂	—	4	1	no	—
605	C₂	—	8	2	no	—
606	C₂	C₂×C₂₃×#410	1	1	yes	characteristic
607	C₂	C₂₃×#2896	1	1	yes	characteristic
608	C₂	C₂×C₂₃×#421	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#432	C₂	semidirect
3. C₂₃×#2283	C₂	semidirect
4. C₂₃×#1516	C₂	semidirect
5. C₂₃×#1512	C₂	semidirect
6. C₂₃×#1506	C₂	semidirect
7. C₂₃×#1505	C₂	semidirect
8. C₂₃×#533	C₄	semidirect
9. C₂₃×#533	C₄	semidirect
11. C₂₃×#532	C₄	semidirect
12. C₂₃×#532	C₄	semidirect
18. C₂₃×#529	C₄	semidirect
25. C₄×#41×C₂₃	C₄	semidirect
211. #17591	C₂₃	direct (N × H)
269. C₂×#432	C₂×C₂₃	semidirect
270. #2283	C₂×C₂₃	semidirect
271. #1516	C₂×C₂₃	semidirect
272. #1512	C₂×C₂₃	semidirect
273. #1506	C₂×C₂₃	semidirect
274. #1505	C₂×C₂₃	semidirect
305. #533	C₄×C₂₃	semidirect
306. #533	C₄×C₂₃	semidirect
308. #532	C₄×C₂₃	semidirect
309. #532	C₄×C₂₃	semidirect
315. #529	C₄×C₂₃	semidirect
322. C₄×#41	C₄×C₂₃	semidirect
432. C₂₃	#17591	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×#3444	2
C₂×C₂₃×#432	2
C₂₃×#2283	2
C₂₃×#1516	2
C₂₃×#1512	2
C₂₃×#1506	2
C₂₃×#1505	2
#17591	23

Subgroups — C23×#17591 #19165217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17591 #19165217