Subgroups — C₂₃×#18679 #19166305

Order 5888.

946 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3506	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3249	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3167	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3077	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1652	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1641	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1638	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1631	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1547	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#559	—	2	2	no	—
13	C₂₃×#556	C₂²	1	2	yes	—
14	C₂₃×#555	—	2	1	no	—
15	C₂₃×#553	—	2	2	no	—
16	C₂₃×#550	C₂²	1	2	yes	—
17	C₂₃×#547	—	2	1	no	—
18	C₂₃×#538	—	2	2	no	—
19	C₂₃×#535	—	2	1	no	—
20	C₂₃×#533	C₂²	1	1	yes	characteristic
21	C₂₃×#533	C₂²	1	1	yes	characteristic
22	C₂₃×#532	C₂²	1	1	yes	characteristic
23	C₂₃×#525	—	2	2	no	—
24	C₂₃×#520	—	2	1	no	—
25	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#498	C₂²	1	1	yes	characteristic, complemented
27	C₂₃×#492	C₂²	1	1	yes	characteristic, complemented
28	C₂₃×#487	C₂²	1	1	yes	characteristic, complemented
29	C₂₃×#486	C₂²	1	1	yes	characteristic, complemented
30	C₂₃×#478	—	2	2	no	—
31	C₂₃×#477	—	2	1	no	—
32	C₂₃×#477	C₂²	1	2	yes	complemented
33	C₂₃×#476	C₂²	1	2	yes	—
34	C₂₃×#474	C₂²	1	2	yes	—
35	C₂₃×#467	—	2	2	no	—
36	C₂₃×#465	—	2	2	no	—
37	C₂₃×#462	—	2	1	no	—
38	C₂₃×#462	C₂²	1	2	yes	complemented
39	C₄×SD₁₆×C₂₃	—	2	2	no	—
40	C₂₃×#428	—	2	1	no	—
41	C₂₃×#423	C₂²	1	1	yes	characteristic
42	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₂₃×#418	—	2	1	no	—
44	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
45	C₂₃×#332	C₂²	1	2	yes	complemented
46	C₂₃×#331	C₂²	1	2	yes	complemented
47	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
48	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
49	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
50	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
51	C₂×#41×C₂₃	—	2	2	no	—
52	C₂×#41×C₂₃	—	2	1	no	—
53	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₂₃	—	2	1	no	—
55	C₂×Q₁₆×C₂₃	—	2	2	no	—
56	C₂×SD₁₆×C₂₃	—	2	2	no	—
57	C₂×M₁₆×C₂₃	—	2	1	no	—
58	C₂²×C₈×C₂₃	—	2	1	no	—
59	C₂²×C₈×C₂₃	—	2	1	no	—
60	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
61	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
62	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₂₃×#127	—	2	2	no	—
66	C₂₃×#127	C₂³	1	1	yes	characteristic
67	C₂₃×#126	—	2	4	no	—
68	C₂₃×#125	—	2	1	no	—
69	C₂₃×#125	—	2	1	no	—
70	C₂₃×#124	—	2	4	no	—
71	C₂₃×#124	—	2	2	no	—
72	C₂₃×#124	—	2	1	no	—
73	C₂₃×#123	—	2	4	no	—
74	C₂₃×#123	—	2	4	no	—
75	C₂₃×C₂²⋊Q₈	—	2	2	no	—
76	C₂₃×C₂²⋊Q₈	—	2	1	no	—
77	C₂₃×C₂²⋊Q₈	—	2	1	no	—
78	C₂₃×C₂²⋊Q₈	—	2	1	no	—
79	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
80	C₂₃×#121	—	2	2	no	—
81	C₂₃×#121	—	2	2	no	—
82	C₂₃×#121	—	2	4	no	—
83	C₂₃×#121	—	2	1	no	—
84	C₂₃×#121	—	2	2	no	—
85	C₂₃×#121	D₄	1	2	yes	—
86	C₂₃×#120	—	2	4	no	—
87	C₄×Q₈×C₂₃	—	2	2	no	—
88	C₄×Q₈×C₂₃	—	2	1	no	—
89	C₄×Q₈×C₂₃	—	2	1	no	—
90	C₄×Q₈×C₂₃	—	2	1	no	—
91	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
92	C₄×D₄×C₂₃	—	2	4	no	—
93	C₄×D₄×C₂₃	—	2	4	no	—
94	C₄×D₄×C₂₃	—	2	2	no	—
95	C₄×D₄×C₂₃	—	2	1	no	—
96	C₄×D₄×C₂₃	—	2	1	no	—
97	C₄×D₄×C₂₃	—	2	1	no	—
98	C₄×D₄×C₂₃	D₄	1	2	yes	—
99	C₄×D₄×C₂₃	D₄	1	2	yes	—
100	C₄×D₄×C₂₃	C₂³	1	2	yes	—
101	C₂₃×#117	—	2	2	no	—
102	C₂₃×#117	—	2	1	no	—
103	C₂₃×#117	—	2	2	no	—
104	C₂₃×#117	—	2	1	no	—
105	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
106	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
107	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
108	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
109	C₂₃×#107	—	2	2	no	—
110	C₂₃×#107	—	2	2	no	—
111	C₂₃×#106	—	2	2	no	—
112	C₂₃×#106	—	2	2	no	—
113	C₂₃×C₄⋊C₈	—	2	1	no	—
114	C₂₃×C₄⋊C₈	—	2	1	no	—
115	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
116	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
117	C₂₃×#103	—	2	2	no	—
118	C₂₃×#103	—	2	2	no	—
119	C₂₃×#103	—	2	2	no	—
120	C₂₃×#102	—	2	2	no	—
121	C₂₃×#102	—	2	2	no	—
122	C₂₃×#102	—	2	2	no	—
123	C₄×C₈×C₂₃	—	2	1	no	—
124	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
126	C₂⁴×C₂₃	—	2	1	no	—
127	#41×C₂₃	—	4	2	no	—
128	#41×C₂₃	—	4	2	no	—
129	#41×C₂₃	—	8	2	no	—
130	#41×C₂₃	—	8	2	no	—
131	C₂×Q₈×C₂₃	—	2	1	no	—
132	C₂×Q₈×C₂₃	—	2	1	no	—
133	C₂×Q₈×C₂₃	—	2	1	no	—
134	C₂×Q₈×C₂₃	—	2	1	no	—
135	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
136	C₂×D₄×C₂₃	—	2	2	no	—
137	C₂×D₄×C₂₃	—	2	4	no	—
138	C₂×D₄×C₂₃	—	2	4	no	—
139	C₂×D₄×C₂₃	—	2	1	no	—
140	C₂×D₄×C₂₃	—	2	2	no	—
141	C₂×D₄×C₂₃	—	2	2	no	—
142	C₂×D₄×C₂₃	—	2	1	no	—
143	C₂×D₄×C₂₃	—	2	1	no	—
144	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
145	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
146	C₂²×C₄×C₂₃	—	2	4	no	—
147	C₂²×C₄×C₂₃	—	2	1	no	—
148	C₂²×C₄×C₂₃	—	2	2	no	—
149	C₂²×C₄×C₂₃	—	2	2	no	—
150	C₂²×C₄×C₂₃	—	2	2	no	—
151	C₂²×C₄×C₂₃	—	8	1	no	—
152	C₂²×C₄×C₂₃	—	8	1	no	—
153	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
156	Q₁₆×C₂₃	—	8	2	no	—
157	SD₁₆×C₂₃	—	4	4	no	—
158	M₁₆×C₂₃	—	4	2	no	—
159	C₂×C₈×C₂₃	—	2	1	no	—
160	C₂×C₈×C₂₃	—	2	1	no	—
161	C₂×C₈×C₂₃	—	2	2	no	—
162	C₂×C₈×C₂₃	—	2	2	no	—
163	C₂×C₈×C₂₃	—	4	2	no	—
164	C₂×C₈×C₂₃	—	8	1	no	—
165	C₄⋊C₄×C₂₃	—	2	4	no	—
166	C₄⋊C₄×C₂₃	—	2	1	no	—
167	C₄⋊C₄×C₂₃	—	2	1	no	—
168	C₄⋊C₄×C₂₃	—	2	1	no	—
169	C₄⋊C₄×C₂₃	—	2	2	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	1	no	—
174	C₄⋊C₄×C₂₃	—	2	1	no	—
175	C₄⋊C₄×C₂₃	—	2	1	no	—
176	C₄⋊C₄×C₂₃	—	2	1	no	—
177	C₄⋊C₄×C₂₃	—	4	1	no	—
178	C₄⋊C₄×C₂₃	—	4	1	no	—
179	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
180	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
181	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
182	C₂²⋊C₄×C₂₃	—	2	4	no	—
183	C₂²⋊C₄×C₂₃	—	2	4	no	—
184	C₂²⋊C₄×C₂₃	—	2	4	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	1	no	—
191	C₄²×C₂₃	—	2	4	no	—
192	C₄²×C₂₃	—	2	2	no	—
193	C₄²×C₂₃	—	4	1	no	—
194	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
196	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
197	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
198	#18679	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
199	C₂³×C₂₃	—	2	2	no	—
200	C₂³×C₂₃	—	2	2	no	—
201	C₂³×C₂₃	—	4	2	no	—
202	C₂³×C₂₃	—	8	1	no	—
203	C₂³×C₂₃	—	8	1	no	—
204	C₂³×C₂₃	#120	1	1	yes	characteristic
205	Q₈×C₂₃	—	2	2	no	—
206	Q₈×C₂₃	—	2	2	no	—
207	Q₈×C₂₃	—	4	2	no	—
208	Q₈×C₂₃	—	4	1	no	—
209	Q₈×C₂₃	—	4	1	no	—
210	Q₈×C₂₃	—	4	1	no	—
211	Q₈×C₂₃	—	4	1	no	—
212	Q₈×C₂₃	—	8	1	no	—
213	D₄×C₂₃	—	2	2	no	—
214	D₄×C₂₃	—	2	2	no	—
215	D₄×C₂₃	—	4	2	no	—
216	D₄×C₂₃	—	4	4	no	—
217	D₄×C₂₃	—	4	1	no	—
218	D₄×C₂₃	—	4	1	no	—
219	D₄×C₂₃	—	4	1	no	—
220	D₄×C₂₃	—	4	1	no	—
221	D₄×C₂₃	—	4	2	no	—
222	D₄×C₂₃	—	8	4	no	—
223	D₄×C₂₃	—	8	4	no	—
224	C₂×C₄×C₂₃	—	2	1	no	—
225	C₂×C₄×C₂₃	—	2	1	no	—
226	C₂×C₄×C₂₃	—	2	2	no	—
227	C₂×C₄×C₂₃	—	2	2	no	—
228	C₂×C₄×C₂₃	—	2	2	no	—
229	C₂×C₄×C₂₃	—	2	2	no	—
230	C₂×C₄×C₂₃	—	4	1	no	—
231	C₂×C₄×C₂₃	—	4	2	no	—
232	C₂×C₄×C₂₃	—	4	2	no	—
233	C₂×C₄×C₂₃	—	4	4	no	—
234	C₂×C₄×C₂₃	—	4	1	no	—
235	C₂×C₄×C₂₃	—	4	1	no	—
236	C₂×C₄×C₂₃	—	8	2	no	—
237	C₂×C₄×C₂₃	—	8	1	no	—
238	C₂×C₄×C₂₃	—	8	4	no	—
239	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
240	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
241	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
242	C₂×C₄×C₂₃	#135	1	2	yes	—
243	C₂×C₄×C₂₃	#137	1	2	yes	—
244	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
247	C₈×C₂₃	—	2	2	no	—
248	C₈×C₂₃	—	4	1	no	—
249	C₈×C₂₃	—	4	2	no	—
250	C₈×C₂₃	—	4	2	no	—
251	#3506	C₂×C₂₃	1	1	yes	characteristic
252	#3249	C₂×C₂₃	1	1	yes	characteristic, complemented
253	#3167	C₂×C₂₃	1	1	yes	characteristic, complemented
254	#3077	C₂×C₂₃	1	1	yes	characteristic, complemented
255	#1652	C₂×C₂₃	1	2	yes	complemented
256	#1641	C₂×C₂₃	1	2	yes	complemented
257	#1638	C₂×C₂₃	1	2	yes	complemented
258	#1631	C₂×C₂₃	1	2	yes	complemented
259	#1547	C₂×C₂₃	1	1	yes	characteristic, complemented
260	#1493	C₂×C₂₃	1	1	yes	characteristic, complemented
261	C₂×D₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
262	C₂²×C₂₃	—	2	1	no	—
263	C₂²×C₂₃	—	2	1	no	—
264	C₂²×C₂₃	—	4	2	no	—
265	C₂²×C₂₃	—	4	2	no	—
266	C₂²×C₂₃	—	4	2	no	—
267	C₂²×C₂₃	—	4	2	no	—
268	C₂²×C₂₃	—	4	2	no	—
269	C₂²×C₂₃	—	4	2	no	—
270	C₂²×C₂₃	—	8	4	no	—
271	C₂²×C₂₃	#452	1	2	yes	—
272	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
273	C₄×C₂₃	—	2	1	no	—
274	C₄×C₂₃	—	2	1	no	—
275	C₄×C₂₃	—	2	1	no	—
276	C₄×C₂₃	—	4	2	no	—
277	C₄×C₂₃	—	4	2	no	—
278	C₄×C₂₃	—	4	2	no	—
279	C₄×C₂₃	—	4	2	no	—
280	C₄×C₂₃	—	8	1	no	—
281	C₄×C₂₃	—	8	1	no	—
282	C₄×C₂₃	#452	1	2	yes	—
283	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
284	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
285	C₄×C₂₃	#576	1	1	yes	characteristic
286	C₄×C₂₃	#577	1	1	yes	characteristic
287	#559	—	2	2	no	—
288	#556	C₂²×C₂₃	1	2	yes	—
289	#555	—	2	1	no	—
290	#553	—	2	2	no	—
291	#550	C₂²×C₂₃	1	2	yes	—
292	#547	—	2	1	no	—
293	#538	—	2	2	no	—
294	#535	—	2	1	no	—
295	#533	C₂²×C₂₃	1	1	yes	characteristic
296	#533	C₂²×C₂₃	1	1	yes	characteristic
297	#532	C₂²×C₂₃	1	1	yes	characteristic
298	#525	—	2	2	no	—
299	#520	—	2	1	no	—
300	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
301	#498	C₂²×C₂₃	1	1	yes	characteristic, complemented
302	#492	C₂²×C₂₃	1	1	yes	characteristic, complemented
303	#487	C₂²×C₂₃	1	1	yes	characteristic, complemented
304	#486	C₂²×C₂₃	1	1	yes	characteristic, complemented
305	#478	—	2	2	no	—
306	#477	—	2	1	no	—
307	#477	C₂²×C₂₃	1	2	yes	complemented
308	#476	C₂²×C₂₃	1	2	yes	—
309	#474	C₂²×C₂₃	1	2	yes	—
310	#467	—	2	2	no	—
311	#465	—	2	2	no	—
312	#462	—	2	1	no	—
313	#462	C₂²×C₂₃	1	2	yes	complemented
314	C₄×SD₁₆	—	2	2	no	—
315	#428	—	2	1	no	—
316	#423	C₂²×C₂₃	1	1	yes	characteristic
317	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
318	#418	—	2	1	no	—
319	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
320	#332	C₂²×C₂₃	1	2	yes	complemented
321	#331	C₂²×C₂₃	1	2	yes	complemented
322	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	complemented
323	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	complemented
324	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
325	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
326	C₂×C₂₃	—	2	2	no	—
327	C₂×C₂₃	—	8	2	no	—
328	C₂×C₂₃	—	8	2	no	—
329	C₂×C₂₃	C₂×#452	1	1	yes	characteristic
330	C₂×C₂₃	#3012	1	1	yes	characteristic
331	C₂×C₂₃	#3018	1	1	yes	characteristic
332	C₂×#41	—	2	2	no	—
333	C₂×#41	—	2	1	no	—
334	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
335	C₂³×C₄	—	2	1	no	—
336	C₂×Q₁₆	—	2	2	no	—
337	C₂×SD₁₆	—	2	2	no	—
338	C₂×M₁₆	—	2	1	no	—
339	C₂²×C₈	—	2	1	no	—
340	C₂²×C₈	—	2	1	no	—
341	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
342	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
343	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
344	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
345	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
346	#127	—	2	2	no	—
347	#127	C₂³×C₂₃	1	1	yes	characteristic
348	#126	—	2	4	no	—
349	#125	—	2	1	no	—
350	#125	—	2	1	no	—
351	#124	—	2	4	no	—
352	#124	—	2	2	no	—
353	#124	—	2	1	no	—
354	#123	—	2	4	no	—
355	#123	—	2	4	no	—
356	C₂²⋊Q₈	—	2	2	no	—
357	C₂²⋊Q₈	—	2	1	no	—
358	C₂²⋊Q₈	—	2	1	no	—
359	C₂²⋊Q₈	—	2	1	no	—
360	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
361	#121	—	2	2	no	—
362	#121	—	2	2	no	—
363	#121	—	2	4	no	—
364	#121	—	2	1	no	—
365	#121	—	2	2	no	—
366	#121	D₄×C₂₃	1	2	yes	—
367	#120	—	2	4	no	—
368	C₄×Q₈	—	2	2	no	—
369	C₄×Q₈	—	2	1	no	—
370	C₄×Q₈	—	2	1	no	—
371	C₄×Q₈	—	2	1	no	—
372	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
373	C₄×D₄	—	2	4	no	—
374	C₄×D₄	—	2	4	no	—
375	C₄×D₄	—	2	2	no	—
376	C₄×D₄	—	2	1	no	—
377	C₄×D₄	—	2	1	no	—
378	C₄×D₄	—	2	1	no	—
379	C₄×D₄	D₄×C₂₃	1	2	yes	—
380	C₄×D₄	D₄×C₂₃	1	2	yes	—
381	C₄×D₄	C₂³×C₂₃	1	2	yes	—
382	#117	—	2	2	no	—
383	#117	—	2	1	no	—
384	#117	—	2	2	no	—
385	#117	—	2	1	no	—
386	C₂×C₄⋊C₄	—	2	2	no	—
387	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
388	C₂×C₂²⋊C₄	—	2	1	no	—
389	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
390	#107	—	2	2	no	—
391	#107	—	2	2	no	—
392	#106	—	2	2	no	—
393	#106	—	2	2	no	—
394	C₄⋊C₈	—	2	1	no	—
395	C₄⋊C₈	—	2	1	no	—
396	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
397	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
398	#103	—	2	2	no	—
399	#103	—	2	2	no	—
400	#103	—	2	2	no	—
401	#102	—	2	2	no	—
402	#102	—	2	2	no	—
403	#102	—	2	2	no	—
404	C₄×C₈	—	2	1	no	—
405	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
406	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
407	C₂₃	#18679	1	1	yes	characteristic, direct product factor, Sylow
408	C₂⁴	—	2	1	no	—
409	#41	—	4	2	no	—
410	#41	—	4	2	no	—
411	#41	—	8	2	no	—
412	#41	—	8	2	no	—
413	C₂×Q₈	—	2	1	no	—
414	C₂×Q₈	—	2	1	no	—
415	C₂×Q₈	—	2	1	no	—
416	C₂×Q₈	—	2	1	no	—
417	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	2	1	no	—
425	C₂×D₄	—	2	1	no	—
426	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
427	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	8	1	no	—
434	C₂²×C₄	—	8	1	no	—
435	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
436	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
437	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
438	Q₁₆	—	8	2	no	—
439	SD₁₆	—	4	4	no	—
440	M₁₆	—	4	2	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	2	no	—
444	C₂×C₈	—	2	2	no	—
445	C₂×C₈	—	4	2	no	—
446	C₂×C₈	—	8	1	no	—
447	C₄⋊C₄	—	2	4	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	2	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	2	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	4	1	no	—
460	C₄⋊C₄	—	4	1	no	—
461	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
462	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
463	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	4	no	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	2	2	no	—
468	C₂²⋊C₄	—	2	2	no	—
469	C₂²⋊C₄	—	2	2	no	—
470	C₂²⋊C₄	—	2	2	no	—
471	C₂²⋊C₄	—	2	2	no	—
472	C₂²⋊C₄	—	4	1	no	—
473	C₄²	—	2	4	no	—
474	C₄²	—	2	2	no	—
475	C₄²	—	4	1	no	—
476	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
477	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
479	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
480	C₂³	—	2	2	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	4	2	no	—
483	C₂³	—	8	1	no	—
484	C₂³	—	8	1	no	—
485	C₂³	C₂₃×#120	1	1	yes	characteristic
486	Q₈	—	2	2	no	—
487	Q₈	—	2	2	no	—
488	Q₈	—	4	2	no	—
489	Q₈	—	4	1	no	—
490	Q₈	—	4	1	no	—
491	Q₈	—	4	1	no	—
492	Q₈	—	4	1	no	—
493	Q₈	—	8	1	no	—
494	D₄	—	2	2	no	—
495	D₄	—	2	2	no	—
496	D₄	—	4	2	no	—
497	D₄	—	4	4	no	—
498	D₄	—	4	1	no	—
499	D₄	—	4	1	no	—
500	D₄	—	4	1	no	—
501	D₄	—	4	1	no	—
502	D₄	—	4	2	no	—
503	D₄	—	8	4	no	—
504	D₄	—	8	4	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	2	no	—
513	C₂×C₄	—	4	2	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	—	4	1	no	—
516	C₂×C₄	—	4	1	no	—
517	C₂×C₄	—	8	2	no	—
518	C₂×C₄	—	8	1	no	—
519	C₂×C₄	—	8	4	no	—
520	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
521	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
522	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
523	C₂×C₄	C₂₃×#135	1	2	yes	—
524	C₂×C₄	C₂₃×#137	1	2	yes	—
525	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
526	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
528	C₈	—	2	2	no	—
529	C₈	—	4	1	no	—
530	C₈	—	4	2	no	—
531	C₈	—	4	2	no	—
532	C₂²	—	2	1	no	—
533	C₂²	—	2	1	no	—
534	C₂²	—	4	2	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	2	no	—
538	C₂²	—	4	2	no	—
539	C₂²	—	4	2	no	—
540	C₂²	—	8	4	no	—
541	C₂²	C₂₃×#452	1	2	yes	—
542	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
543	C₄	—	2	1	no	—
544	C₄	—	2	1	no	—
545	C₄	—	2	1	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	2	no	—
548	C₄	—	4	2	no	—
549	C₄	—	4	2	no	—
550	C₄	—	8	1	no	—
551	C₄	—	8	1	no	—
552	C₄	C₂₃×#452	1	2	yes	—
553	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
554	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
555	C₄	C₂₃×#576	1	1	yes	characteristic
556	C₄	C₂₃×#577	1	1	yes	characteristic
557	C₂	—	2	2	no	—
558	C₂	—	8	2	no	—
559	C₂	—	8	2	no	—
560	C₂	C₂×C₂₃×#452	1	1	yes	characteristic
561	C₂	C₂₃×#3012	1	1	yes	characteristic
562	C₂	C₂₃×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3506	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#3249	C₂	semidirect
3. C₂₃×#3167	C₂	semidirect
4. C₂₃×#3077	C₂	semidirect
5. C₂₃×#1652	C₂	semidirect
6. C₂₃×#1641	C₂	semidirect
7. C₂₃×#1638	C₂	semidirect
8. C₂₃×#1631	C₂	semidirect
9. C₂₃×#1547	C₂	semidirect
10. C₂₃×#1493	C₂	semidirect
11. C₂×C₂₃×D₄⋊C₈	C₂	semidirect
26. C₂₃×#498	C₂²	semidirect
27. C₂₃×#492	C₂²	semidirect
28. C₂₃×#487	C₂²	semidirect
29. C₂₃×#486	C₂²	semidirect
32. C₂₃×#477	C₂²	semidirect
38. C₂₃×#462	C₂²	semidirect
45. C₂₃×#332	C₂²	semidirect
46. C₂₃×#331	C₂²	semidirect
47. C₂₃×Q₈⋊C₈	C₂²	semidirect
48. C₂₃×Q₈⋊C₈	C₂²	semidirect
198. #18679	C₂₃	direct (N × H)
252. #3249	C₂×C₂₃	semidirect
253. #3167	C₂×C₂₃	semidirect
254. #3077	C₂×C₂₃	semidirect
255. #1652	C₂×C₂₃	semidirect
256. #1641	C₂×C₂₃	semidirect
257. #1638	C₂×C₂₃	semidirect
258. #1631	C₂×C₂₃	semidirect
259. #1547	C₂×C₂₃	semidirect
260. #1493	C₂×C₂₃	semidirect
261. C₂×D₄⋊C₈	C₂×C₂₃	semidirect
301. #498	C₂²×C₂₃	semidirect
302. #492	C₂²×C₂₃	semidirect
303. #487	C₂²×C₂₃	semidirect
304. #486	C₂²×C₂₃	semidirect
307. #477	C₂²×C₂₃	semidirect
313. #462	C₂²×C₂₃	semidirect
320. #332	C₂²×C₂₃	semidirect
321. #331	C₂²×C₂₃	semidirect
322. Q₈⋊C₈	C₂²×C₂₃	semidirect
323. Q₈⋊C₈	C₂²×C₂₃	semidirect
407. C₂₃	#18679	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×#3506	2
C₂₃×#3249	2
C₂₃×#3167	2
C₂₃×#3077	2
C₂₃×#1652	2
C₂₃×#1641	2
C₂₃×#1638	2
C₂₃×#1631	2
C₂₃×#1547	2
C₂₃×#1493	2
C₂×C₂₃×D₄⋊C₈	2
#18679	23

Subgroups — C23×#18679 #19166305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18679 #19166305