Subgroups — C₂₃×#17600 #19165226

Order 5888.

982 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3520	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2974	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2319	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1520	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1516	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1513	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1511	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1509	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1506	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1503	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1500	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#556	C₄	1	1	yes	characteristic, complemented
13	C₂₃×#554	—	2	1	no	—
14	C₂₃×#552	C₄	1	2	yes	complemented
15	C₂₃×#551	—	2	2	no	—
16	D₄×Q₈×C₂₃	C₄	1	1	yes	characteristic, complemented
17	C₂₃×#548	C₄	1	2	yes	complemented
18	C₂₃×#547	C₄	1	1	yes	characteristic, complemented
19	C₂₃×#547	C₄	1	1	yes	characteristic, complemented
20	C₂₃×#546	—	2	1	no	—
21	C₂₃×#542	—	2	1	no	—
22	C₂₃×#539	—	2	2	no	—
23	C₂₃×#533	C₂²	1	1	yes	characteristic
24	C₂₃×#529	C₂²	1	1	yes	characteristic
25	C₂₃×#525	C₂²	1	2	yes	—
26	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₂₃×#520	—	2	1	no	—
28	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#436	—	2	1	no	—
30	C₂₃×#435	—	2	2	no	—
31	C₈×D₄×C₂₃	—	2	1	no	—
32	C₂₃×#432	C₂²	1	1	yes	characteristic
33	C₂₃×#431	C₂²	1	1	yes	characteristic
34	C₂₃×#407	C₂²	1	2	yes	—
35	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
36	C₂₃×#390	—	2	1	no	—
37	C₂₃×#389	—	2	1	no	—
38	C₂₃×#388	—	2	2	no	—
39	C₂₃×#381	C₂²	1	2	yes	—
40	C₂₃×#380	C₂²	1	2	yes	—
41	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
42	C₂₃×#376	C₂²	1	1	yes	characteristic
43	C₂₃×#328	C₂²	1	1	yes	characteristic
44	C₂₃×#328	C₂²	1	1	yes	characteristic
45	C₂₃×#328	C₂²	1	2	yes	—
46	C₂₃×#328	C₂²	1	2	yes	—
47	C₂₃×#328	C₂²	1	2	yes	—
48	C₂₃×#328	C₂²	1	2	yes	—
49	C₂₃×#328	C₂²	1	2	yes	—
50	C₂₃×#327	C₂²	1	1	yes	characteristic
51	C₂₃×#327	C₂²	1	1	yes	characteristic
52	C₂₃×#327	C₂²	1	2	yes	—
53	C₂×#41×C₂₃	—	2	1	no	—
54	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
55	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
56	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
57	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
58	C₂×M₁₆×C₂₃	—	2	2	no	—
59	C₂²×C₈×C₂₃	—	2	2	no	—
60	C₂₃×C₄⋊Q₈	—	2	1	no	—
61	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
62	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
63	C₂₃×#126	—	2	2	no	—
64	C₂₃×#126	—	2	2	no	—
65	C₂₃×#126	—	2	1	no	—
66	C₂₃×#125	—	2	2	no	—
67	C₂₃×#125	—	2	1	no	—
68	C₂₃×#125	C₂×C₄	1	1	yes	characteristic
69	C₂₃×#124	—	2	1	no	—
70	C₂₃×#124	—	2	2	no	—
71	C₂₃×#124	C₂×C₄	1	1	yes	characteristic
72	C₂₃×#123	—	2	2	no	—
73	C₂₃×#123	—	2	1	no	—
74	C₂₃×#123	—	2	2	no	—
75	C₂₃×#123	—	2	2	no	—
76	C₂₃×#123	—	2	2	no	—
77	C₂₃×C₂²⋊Q₈	—	2	2	no	—
78	C₂₃×C₂²⋊Q₈	—	2	1	no	—
79	C₂₃×C₂²⋊Q₈	—	2	2	no	—
80	C₂₃×C₂²⋊Q₈	—	2	1	no	—
81	C₂₃×C₂²⋊Q₈	—	2	2	no	—
82	C₂₃×C₂²⋊Q₈	—	2	2	no	—
83	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
84	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
85	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
86	C₂₃×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
87	C₂₃×C₂²⋊Q₈	C₂³	1	2	yes	—
88	C₂₃×#121	—	2	1	no	—
89	C₂₃×#121	—	2	2	no	—
90	C₂₃×#121	C₂×C₄	1	2	yes	—
91	C₂₃×#121	C₂³	1	1	yes	characteristic
92	C₂₃×#120	—	2	2	no	—
93	C₄×Q₈×C₂₃	—	2	1	no	—
94	C₄×Q₈×C₂₃	—	2	1	no	—
95	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
96	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
97	C₄×D₄×C₂₃	—	2	1	no	—
98	C₄×D₄×C₂₃	—	2	2	no	—
99	C₄×D₄×C₂₃	—	2	1	no	—
100	C₄×D₄×C₂₃	—	2	2	no	—
101	C₄×D₄×C₂₃	—	2	2	no	—
102	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
103	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
104	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
105	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
106	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
107	C₄×D₄×C₂₃	D₄	1	2	yes	—
108	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
109	C₂₃×#117	—	2	1	no	—
110	C₂₃×#117	—	2	2	no	—
111	C₂₃×#117	—	2	1	no	—
112	C₂₃×#117	C₂×C₄	1	2	yes	—
113	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
114	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
115	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
116	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
117	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
118	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
119	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
120	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
121	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
122	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
123	C₂×C₄²×C₂₃	—	2	1	no	—
124	C₂×C₄²×C₂₃	—	2	1	no	—
125	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
126	C₂₃×C₄⋊C₈	—	2	1	no	—
127	C₂₃×C₄⋊C₈	—	2	1	no	—
128	C₂₃×C₂²⋊C₈	—	2	2	no	—
129	C₂₃×C₂²⋊C₈	—	2	2	no	—
130	C₂₃×C₂²⋊C₈	C₂³	1	1	yes	characteristic
131	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
132	C₂₃×C₂²⋊C₈	C₂³	1	1	yes	characteristic
133	C₂₃×#97	—	2	1	no	—
134	C₄×C₈×C₂₃	—	2	1	no	—
135	C₂₃×#95	—	2	1	no	—
136	C₂₃×#95	—	2	2	no	—
137	C₂₃×#95	C₂³	1	1	yes	characteristic
138	C₂₃×#95	C₂³	1	1	yes	characteristic
139	C₂₃×#95	C₂³	1	2	yes	—
140	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
141	#41×C₂₃	—	4	4	no	—
142	#41×C₂₃	—	8	2	no	—
143	C₂×Q₈×C₂₃	—	2	1	no	—
144	C₂×Q₈×C₂₃	—	2	2	no	—
145	C₂×Q₈×C₂₃	—	8	2	no	—
146	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
147	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
148	C₂×D₄×C₂₃	—	2	2	no	—
149	C₂×D₄×C₂₃	—	2	1	no	—
150	C₂×D₄×C₂₃	—	2	2	no	—
151	C₂×D₄×C₂₃	—	4	2	no	—
152	C₂×D₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
153	C₂×D₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
154	C₂×D₄×C₂₃	C₂²⋊C₄	1	2	yes	—
155	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	—	2	2	no	—
157	C₂²×C₄×C₂₃	—	2	2	no	—
158	C₂²×C₄×C₂₃	—	2	1	no	—
159	C₂²×C₄×C₂₃	—	2	2	no	—
160	C₂²×C₄×C₂₃	—	2	1	no	—
161	C₂²×C₄×C₂₃	—	2	2	no	—
162	C₂²×C₄×C₂₃	—	2	1	no	—
163	C₂²×C₄×C₂₃	—	8	2	no	—
164	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
165	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
166	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
170	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
172	M₁₆×C₂₃	—	8	2	no	—
173	C₂×C₈×C₂₃	—	2	1	no	—
174	C₂×C₈×C₂₃	—	2	2	no	—
175	C₂×C₈×C₂₃	—	2	1	no	—
176	C₂×C₈×C₂₃	—	8	2	no	—
177	C₄⋊C₄×C₂₃	—	2	2	no	—
178	C₄⋊C₄×C₂₃	—	2	1	no	—
179	C₄⋊C₄×C₂₃	—	2	2	no	—
180	C₄⋊C₄×C₂₃	—	2	2	no	—
181	C₄⋊C₄×C₂₃	—	2	1	no	—
182	C₄⋊C₄×C₂₃	—	2	1	no	—
183	C₄⋊C₄×C₂₃	—	2	1	no	—
184	C₄⋊C₄×C₂₃	—	2	2	no	—
185	C₄⋊C₄×C₂₃	—	2	1	no	—
186	C₄⋊C₄×C₂₃	—	2	2	no	—
187	C₄⋊C₄×C₂₃	—	2	1	no	—
188	C₄⋊C₄×C₂₃	—	2	1	no	—
189	C₄⋊C₄×C₂₃	—	2	2	no	—
190	C₄⋊C₄×C₂₃	—	4	2	no	—
191	C₄⋊C₄×C₂₃	—	8	2	no	—
192	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
193	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
194	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
195	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
196	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
197	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
198	C₂²⋊C₄×C₂₃	—	2	2	no	—
199	C₂²⋊C₄×C₂₃	—	2	2	no	—
200	C₂²⋊C₄×C₂₃	—	2	2	no	—
201	C₂²⋊C₄×C₂₃	—	2	2	no	—
202	C₂²⋊C₄×C₂₃	—	2	1	no	—
203	C₂²⋊C₄×C₂₃	—	2	2	no	—
204	C₂²⋊C₄×C₂₃	—	2	1	no	—
205	C₂²⋊C₄×C₂₃	—	2	1	no	—
206	C₂²⋊C₄×C₂₃	—	2	2	no	—
207	C₂²⋊C₄×C₂₃	—	2	1	no	—
208	C₂²⋊C₄×C₂₃	—	2	1	no	—
209	C₂²⋊C₄×C₂₃	—	8	2	no	—
210	C₂²⋊C₄×C₂₃	—	8	2	no	—
211	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
212	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
213	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
214	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
215	C₄²×C₂₃	—	2	2	no	—
216	C₄²×C₂₃	—	2	2	no	—
217	C₄²×C₂₃	—	2	1	no	—
218	C₄²×C₂₃	—	4	2	no	—
219	C₄²×C₂₃	—	8	1	no	—
220	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
221	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
222	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
223	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
224	#17600	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂³×C₂₃	—	2	2	no	—
226	C₂³×C₂₃	—	2	2	no	—
227	C₂³×C₂₃	—	2	1	no	—
228	C₂³×C₂₃	—	4	2	no	—
229	C₂³×C₂₃	—	4	2	no	—
230	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
231	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
232	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
233	Q₈×C₂₃	—	4	1	no	—
234	Q₈×C₂₃	—	4	2	no	—
235	Q₈×C₂₃	—	4	2	no	—
236	Q₈×C₂₃	—	8	2	no	—
237	D₄×C₂₃	—	2	2	no	—
238	D₄×C₂₃	—	2	2	no	—
239	D₄×C₂₃	—	4	2	no	—
240	D₄×C₂₃	—	4	1	no	—
241	D₄×C₂₃	—	4	4	no	—
242	D₄×C₂₃	—	4	1	no	—
243	D₄×C₂₃	—	4	1	no	—
244	D₄×C₂₃	—	8	2	no	—
245	C₂×C₄×C₂₃	—	2	1	no	—
246	C₂×C₄×C₂₃	—	2	1	no	—
247	C₂×C₄×C₂₃	—	2	2	no	—
248	C₂×C₄×C₂₃	—	2	1	no	—
249	C₂×C₄×C₂₃	—	2	2	no	—
250	C₂×C₄×C₂₃	—	2	1	no	—
251	C₂×C₄×C₂₃	—	2	2	no	—
252	C₂×C₄×C₂₃	—	2	1	no	—
253	C₂×C₄×C₂₃	—	4	2	no	—
254	C₂×C₄×C₂₃	—	4	2	no	—
255	C₂×C₄×C₂₃	—	4	2	no	—
256	C₂×C₄×C₂₃	—	4	1	no	—
257	C₂×C₄×C₂₃	—	4	2	no	—
258	C₂×C₄×C₂₃	—	4	2	no	—
259	C₂×C₄×C₂₃	—	4	4	no	—
260	C₂×C₄×C₂₃	—	4	2	no	—
261	C₂×C₄×C₂₃	—	4	1	no	—
262	C₂×C₄×C₂₃	—	4	1	no	—
263	C₂×C₄×C₂₃	—	4	1	no	—
264	C₂×C₄×C₂₃	—	4	2	no	—
265	C₂×C₄×C₂₃	—	4	2	no	—
266	C₂×C₄×C₂₃	—	4	1	no	—
267	C₂×C₄×C₂₃	—	4	2	no	—
268	C₂×C₄×C₂₃	—	4	1	no	—
269	C₂×C₄×C₂₃	—	4	1	no	—
270	C₂×C₄×C₂₃	—	8	2	no	—
271	C₂×C₄×C₂₃	—	8	2	no	—
272	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
274	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
277	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
278	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
279	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
280	C₈×C₂₃	—	4	1	no	—
281	C₈×C₂₃	—	4	2	no	—
282	C₈×C₂₃	—	4	1	no	—
283	#3520	C₂×C₂₃	1	1	yes	characteristic
284	#2974	C₂×C₂₃	1	1	yes	characteristic, complemented
285	#2319	C₂×C₂₃	1	1	yes	characteristic, complemented
286	#1520	C₂×C₂₃	1	2	yes	complemented
287	#1516	C₂×C₂₃	1	2	yes	complemented
288	#1513	C₂×C₂₃	1	2	yes	complemented
289	#1511	C₂×C₂₃	1	2	yes	complemented
290	#1509	C₂×C₂₃	1	1	yes	characteristic, complemented
291	#1506	C₂×C₂₃	1	1	yes	characteristic, complemented
292	#1503	C₂×C₂₃	1	1	yes	characteristic, complemented
293	#1500	C₂×C₂₃	1	1	yes	characteristic, complemented
294	C₂²×C₂₃	—	2	1	no	—
295	C₂²×C₂₃	—	2	1	no	—
296	C₂²×C₂₃	—	2	2	no	—
297	C₂²×C₂₃	—	2	2	no	—
298	C₂²×C₂₃	—	2	2	no	—
299	C₂²×C₂₃	—	2	2	no	—
300	C₂²×C₂₃	—	2	2	no	—
301	C₂²×C₂₃	—	2	2	no	—
302	C₂²×C₂₃	—	4	1	no	—
303	C₂²×C₂₃	—	4	1	no	—
304	C₂²×C₂₃	—	4	1	no	—
305	C₂²×C₂₃	—	8	2	no	—
306	C₂²×C₂₃	—	8	2	no	—
307	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
308	C₂²×C₂₃	#577	1	1	yes	characteristic
309	C₂²×C₂₃	#578	1	1	yes	characteristic
310	C₄×C₂₃	—	2	1	no	—
311	C₄×C₂₃	—	2	1	no	—
312	C₄×C₂₃	—	2	2	no	—
313	C₄×C₂₃	—	2	2	no	—
314	C₄×C₂₃	—	2	1	no	—
315	C₄×C₂₃	—	2	1	no	—
316	C₄×C₂₃	—	2	1	no	—
317	C₄×C₂₃	—	4	2	no	—
318	C₄×C₂₃	—	4	1	no	—
319	C₄×C₂₃	—	4	1	no	—
320	C₄×C₂₃	—	4	1	no	—
321	C₄×C₂₃	—	4	2	no	—
322	C₄×C₂₃	—	8	1	no	—
323	C₄×C₂₃	—	8	2	no	—
324	C₄×C₂₃	—	8	1	no	—
325	C₄×C₂₃	#421	1	2	yes	—
326	#556	C₄×C₂₃	1	1	yes	characteristic, complemented
327	#554	—	2	1	no	—
328	#552	C₄×C₂₃	1	2	yes	complemented
329	#551	—	2	2	no	—
330	D₄×Q₈	C₄×C₂₃	1	1	yes	characteristic, complemented
331	#548	C₄×C₂₃	1	2	yes	complemented
332	#547	C₄×C₂₃	1	1	yes	characteristic, complemented
333	#547	C₄×C₂₃	1	1	yes	characteristic, complemented
334	#546	—	2	1	no	—
335	#542	—	2	1	no	—
336	#539	—	2	2	no	—
337	#533	C₂²×C₂₃	1	1	yes	characteristic
338	#529	C₂²×C₂₃	1	1	yes	characteristic
339	#525	C₂²×C₂₃	1	2	yes	—
340	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
341	#520	—	2	1	no	—
342	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
343	#436	—	2	1	no	—
344	#435	—	2	2	no	—
345	C₈×D₄	—	2	1	no	—
346	#432	C₂²×C₂₃	1	1	yes	characteristic
347	#431	C₂²×C₂₃	1	1	yes	characteristic
348	#407	C₂²×C₂₃	1	2	yes	—
349	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
350	#390	—	2	1	no	—
351	#389	—	2	1	no	—
352	#388	—	2	2	no	—
353	#381	C₂²×C₂₃	1	2	yes	—
354	#380	C₂²×C₂₃	1	2	yes	—
355	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
356	#376	C₂²×C₂₃	1	1	yes	characteristic
357	#328	C₂²×C₂₃	1	1	yes	characteristic
358	#328	C₂²×C₂₃	1	1	yes	characteristic
359	#328	C₂²×C₂₃	1	2	yes	—
360	#328	C₂²×C₂₃	1	2	yes	—
361	#328	C₂²×C₂₃	1	2	yes	—
362	#328	C₂²×C₂₃	1	2	yes	—
363	#328	C₂²×C₂₃	1	2	yes	—
364	#327	C₂²×C₂₃	1	1	yes	characteristic
365	#327	C₂²×C₂₃	1	1	yes	characteristic
366	#327	C₂²×C₂₃	1	2	yes	—
367	C₂×C₂₃	—	2	1	no	—
368	C₂×C₂₃	—	2	1	no	—
369	C₂×C₂₃	—	4	2	no	—
370	C₂×C₂₃	—	4	2	no	—
371	C₂×C₂₃	—	8	1	no	—
372	C₂×C₂₃	#2883	1	1	yes	characteristic
373	C₂×C₂₃	#2898	1	1	yes	characteristic
374	C₂×C₂₃	C₂×#421	1	1	yes	characteristic
375	C₂×#41	—	2	1	no	—
376	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
377	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
378	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
379	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
380	C₂×M₁₆	—	2	2	no	—
381	C₂²×C₈	—	2	2	no	—
382	C₄⋊Q₈	—	2	1	no	—
383	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
384	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
385	#126	—	2	2	no	—
386	#126	—	2	2	no	—
387	#126	—	2	1	no	—
388	#125	—	2	2	no	—
389	#125	—	2	1	no	—
390	#125	C₂×C₄×C₂₃	1	1	yes	characteristic
391	#124	—	2	1	no	—
392	#124	—	2	2	no	—
393	#124	C₂×C₄×C₂₃	1	1	yes	characteristic
394	#123	—	2	2	no	—
395	#123	—	2	2	no	—
396	#123	—	2	1	no	—
397	#123	—	2	2	no	—
398	#123	—	2	2	no	—
399	C₂²⋊Q₈	—	2	2	no	—
400	C₂²⋊Q₈	—	2	2	no	—
401	C₂²⋊Q₈	—	2	1	no	—
402	C₂²⋊Q₈	—	2	2	no	—
403	C₂²⋊Q₈	—	2	1	no	—
404	C₂²⋊Q₈	—	2	2	no	—
405	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
406	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
407	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
408	C₂²⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
409	C₂²⋊Q₈	C₂³×C₂₃	1	2	yes	—
410	#121	—	2	2	no	—
411	#121	—	2	1	no	—
412	#121	C₂×C₄×C₂₃	1	2	yes	—
413	#121	C₂³×C₂₃	1	1	yes	characteristic
414	#120	—	2	2	no	—
415	C₄×Q₈	—	2	1	no	—
416	C₄×Q₈	—	2	1	no	—
417	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
418	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
419	C₄×D₄	—	2	1	no	—
420	C₄×D₄	—	2	2	no	—
421	C₄×D₄	—	2	2	no	—
422	C₄×D₄	—	2	1	no	—
423	C₄×D₄	—	2	2	no	—
424	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
425	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
426	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
427	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
428	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
429	C₄×D₄	D₄×C₂₃	1	2	yes	—
430	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
431	#117	—	2	1	no	—
432	#117	—	2	2	no	—
433	#117	—	2	1	no	—
434	#117	C₂×C₄×C₂₃	1	2	yes	—
435	C₂×C₄⋊C₄	—	2	2	no	—
436	C₂×C₄⋊C₄	—	2	1	no	—
437	C₂×C₄⋊C₄	—	2	1	no	—
438	C₂×C₄⋊C₄	—	2	2	no	—
439	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
440	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
441	C₂×C₂²⋊C₄	—	2	2	no	—
442	C₂×C₂²⋊C₄	—	2	2	no	—
443	C₂×C₂²⋊C₄	—	2	2	no	—
444	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
445	C₂×C₄²	—	2	1	no	—
446	C₂×C₄²	—	2	1	no	—
447	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
448	C₄⋊C₈	—	2	1	no	—
449	C₄⋊C₈	—	2	1	no	—
450	C₂²⋊C₈	—	2	2	no	—
451	C₂²⋊C₈	—	2	2	no	—
452	C₂²⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
453	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
454	C₂²⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
455	#97	—	2	1	no	—
456	C₄×C₈	—	2	1	no	—
457	#95	—	2	1	no	—
458	#95	—	2	2	no	—
459	#95	C₂³×C₂₃	1	1	yes	characteristic
460	#95	C₂³×C₂₃	1	1	yes	characteristic
461	#95	C₂³×C₂₃	1	2	yes	—
462	C₂₃	#17600	1	1	yes	characteristic, direct product factor, Sylow
463	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
464	#41	—	4	4	no	—
465	#41	—	8	2	no	—
466	C₂×Q₈	—	2	1	no	—
467	C₂×Q₈	—	2	2	no	—
468	C₂×Q₈	—	8	2	no	—
469	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
470	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
471	C₂×D₄	—	2	2	no	—
472	C₂×D₄	—	2	1	no	—
473	C₂×D₄	—	2	2	no	—
474	C₂×D₄	—	4	2	no	—
475	C₂×D₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
476	C₂×D₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
477	C₂×D₄	C₂²⋊C₄×C₂₃	1	2	yes	—
478	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	8	2	no	—
487	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
488	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
489	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
490	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
491	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
492	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
493	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
494	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
495	M₁₆	—	8	2	no	—
496	C₂×C₈	—	2	1	no	—
497	C₂×C₈	—	2	1	no	—
498	C₂×C₈	—	2	2	no	—
499	C₂×C₈	—	8	2	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	2	1	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	4	2	no	—
514	C₄⋊C₄	—	8	2	no	—
515	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
516	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
517	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
518	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
519	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
520	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	2	2	no	—
523	C₂²⋊C₄	—	2	2	no	—
524	C₂²⋊C₄	—	2	2	no	—
525	C₂²⋊C₄	—	2	1	no	—
526	C₂²⋊C₄	—	2	2	no	—
527	C₂²⋊C₄	—	2	1	no	—
528	C₂²⋊C₄	—	2	1	no	—
529	C₂²⋊C₄	—	2	2	no	—
530	C₂²⋊C₄	—	2	1	no	—
531	C₂²⋊C₄	—	2	1	no	—
532	C₂²⋊C₄	—	8	2	no	—
533	C₂²⋊C₄	—	8	2	no	—
534	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
535	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
536	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
537	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
538	C₄²	—	2	2	no	—
539	C₄²	—	2	2	no	—
540	C₄²	—	2	1	no	—
541	C₄²	—	4	2	no	—
542	C₄²	—	8	1	no	—
543	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
544	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
545	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
546	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
547	C₂³	—	2	2	no	—
548	C₂³	—	2	2	no	—
549	C₂³	—	2	1	no	—
550	C₂³	—	4	2	no	—
551	C₂³	—	4	2	no	—
552	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
553	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
554	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
555	Q₈	—	4	1	no	—
556	Q₈	—	4	2	no	—
557	Q₈	—	4	2	no	—
558	Q₈	—	8	2	no	—
559	D₄	—	2	2	no	—
560	D₄	—	2	2	no	—
561	D₄	—	4	2	no	—
562	D₄	—	4	1	no	—
563	D₄	—	4	4	no	—
564	D₄	—	4	1	no	—
565	D₄	—	4	1	no	—
566	D₄	—	8	2	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	4	2	no	—
576	C₂×C₄	—	4	2	no	—
577	C₂×C₄	—	4	2	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	2	no	—
580	C₂×C₄	—	4	2	no	—
581	C₂×C₄	—	4	4	no	—
582	C₂×C₄	—	4	2	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	—	4	2	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	8	2	no	—
593	C₂×C₄	—	8	2	no	—
594	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
595	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
596	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
597	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
598	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
599	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
600	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
601	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
602	C₈	—	4	1	no	—
603	C₈	—	4	2	no	—
604	C₈	—	4	1	no	—
605	C₂²	—	2	1	no	—
606	C₂²	—	2	1	no	—
607	C₂²	—	2	2	no	—
608	C₂²	—	2	2	no	—
609	C₂²	—	2	2	no	—
610	C₂²	—	2	2	no	—
611	C₂²	—	2	2	no	—
612	C₂²	—	2	2	no	—
613	C₂²	—	4	1	no	—
614	C₂²	—	4	1	no	—
615	C₂²	—	4	1	no	—
616	C₂²	—	8	2	no	—
617	C₂²	—	8	2	no	—
618	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
619	C₂²	C₂₃×#577	1	1	yes	characteristic
620	C₂²	C₂₃×#578	1	1	yes	characteristic
621	C₄	—	2	1	no	—
622	C₄	—	2	1	no	—
623	C₄	—	2	2	no	—
624	C₄	—	2	2	no	—
625	C₄	—	2	1	no	—
626	C₄	—	2	1	no	—
627	C₄	—	2	1	no	—
628	C₄	—	4	2	no	—
629	C₄	—	4	1	no	—
630	C₄	—	4	1	no	—
631	C₄	—	4	1	no	—
632	C₄	—	4	2	no	—
633	C₄	—	8	1	no	—
634	C₄	—	8	2	no	—
635	C₄	—	8	1	no	—
636	C₄	C₂₃×#421	1	2	yes	—
637	C₂	—	2	1	no	—
638	C₂	—	2	1	no	—
639	C₂	—	4	2	no	—
640	C₂	—	4	2	no	—
641	C₂	—	8	1	no	—
642	C₂	C₂₃×#2883	1	1	yes	characteristic
643	C₂	C₂₃×#2898	1	1	yes	characteristic
644	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×#3520	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2974	C₂	semidirect
3. C₂₃×#2319	C₂	semidirect
4. C₂₃×#1520	C₂	semidirect
5. C₂₃×#1516	C₂	semidirect
6. C₂₃×#1513	C₂	semidirect
7. C₂₃×#1511	C₂	semidirect
8. C₂₃×#1509	C₂	semidirect
9. C₂₃×#1506	C₂	semidirect
10. C₂₃×#1503	C₂	semidirect
11. C₂₃×#1500	C₂	semidirect
12. C₂₃×#556	C₄	semidirect
14. C₂₃×#552	C₄	semidirect
16. D₄×Q₈×C₂₃	C₄	semidirect
17. C₂₃×#548	C₄	semidirect
18. C₂₃×#547	C₄	semidirect
19. C₂₃×#547	C₄	semidirect
224. #17600	C₂₃	direct (N × H)
284. #2974	C₂×C₂₃	semidirect
285. #2319	C₂×C₂₃	semidirect
286. #1520	C₂×C₂₃	semidirect
287. #1516	C₂×C₂₃	semidirect
288. #1513	C₂×C₂₃	semidirect
289. #1511	C₂×C₂₃	semidirect
290. #1509	C₂×C₂₃	semidirect
291. #1506	C₂×C₂₃	semidirect
292. #1503	C₂×C₂₃	semidirect
293. #1500	C₂×C₂₃	semidirect
326. #556	C₄×C₂₃	semidirect
328. #552	C₄×C₂₃	semidirect
330. D₄×Q₈	C₄×C₂₃	semidirect
331. #548	C₄×C₂₃	semidirect
332. #547	C₄×C₂₃	semidirect
333. #547	C₄×C₂₃	semidirect
462. C₂₃	#17600	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×#3520	2
C₂₃×#2974	2
C₂₃×#2319	2
C₂₃×#1520	2
C₂₃×#1516	2
C₂₃×#1513	2
C₂₃×#1511	2
C₂₃×#1509	2
C₂₃×#1506	2
C₂₃×#1503	2
C₂₃×#1500	2
#17600	23

Subgroups — C23×#17600 #19165226

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17600 #19165226