Subgroups — #15682367

Order 2944.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092947	C₂	1	1	yes	maximal, characteristic
2	#13092545	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092541	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092493	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092488	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	(C₂²⋊D₂₃)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	(C₂²⋊D₂₃)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092477	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092475	C₂	1	1	yes	maximal, characteristic, complemented
11	#13092471	C₂	1	1	yes	maximal, characteristic, complemented
12	#13092375	C₂	1	1	yes	maximal, characteristic, complemented
13	#13092374	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#435	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
19	#10625110	C₄	1	1	yes	characteristic
20	#10625106	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₂₃	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₂₃	C₂²	1	1	yes	characteristic
23	#10625097	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
25	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
26	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×#67262	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
29	#10625069	C₄	1	1	yes	characteristic
30	#10625060	C₄	1	1	yes	characteristic
31	#10625057	C₄	1	1	yes	characteristic
32	#10625056	C₄	1	1	yes	characteristic
33	#10625050	C₄	1	1	yes	characteristic
34	#10625049	C₄	1	1	yes	characteristic
35	#10625043	C₂²	1	1	yes	characteristic
36	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
37	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₂₃×#97	C₂²	1	1	yes	characteristic
41	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
42	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
43	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
44	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
45	#10624997	C₂²	1	1	yes	characteristic
46	#10624997	C₂²	1	1	yes	characteristic
47	#10624997	C₂²	1	1	yes	characteristic
48	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
49	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
50	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
51	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
52	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
53	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
54	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
55	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
56	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
57	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
58	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₂₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₂₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
65	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
67	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
68	M₁₆×C₂₃	—	2	2	no	—
69	C₂×C₈×C₂₃	—	2	2	no	—
70	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
75	C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
76	C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
77	C₄²×C₂₃	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
79	C₂²⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
80	C₂²⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
81	C₂²⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
82	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
83	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
84	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
85	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
86	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
87	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
88	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
89	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
90	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
91	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
92	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
93	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
94	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
98	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
99	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₂₃⋊C₈	—	2	2	no	—
101	C₂×C₂₃⋊C₈	—	2	2	no	—
102	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
106	#67262	—	2	2	no	—
107	C₈×D₂₃	—	2	2	no	—
108	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
109	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
111	D₄×C₂₃	—	2	2	no	—
112	C₂×C₄×C₂₃	—	2	1	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₂₃	—	2	2	no	—
122	C₂²⋊D₂₃	—	2	2	no	—
123	C₂×Dic₂₃	—	2	2	no	—
124	C₂×Dic₂₃	—	2	2	no	—
125	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
132	C₄×D₂₃	—	2	1	no	—
133	C₄×D₂₃	—	2	1	no	—
134	Dic₄₆	—	2	2	no	—
135	C₈×C₂₃	—	2	1	no	—
136	C₈×C₂₃	—	2	1	no	—
137	C₈×C₂₃	—	2	1	no	—
138	C₈×C₂₃	#41	1	2	yes	—
139	C₂₃⋊C₈	—	2	1	no	—
140	C₂₃⋊C₈	—	2	1	no	—
141	C₂₃⋊C₈	—	2	1	no	—
142	C₂₃⋊C₈	#41	1	2	yes	—
143	#2966	—	23	1	no	maximal, Sylow
144	C₂²×C₂₃	—	2	1	no	—
145	C₂²×C₂₃	—	2	1	no	—
146	C₂²×C₂₃	—	2	1	no	—
147	C₂²×C₂₃	—	2	1	no	—
148	C₂²×C₂₃	—	2	1	no	—
149	C₂²×C₂₃	#131	1	2	yes	—
150	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
151	C₂×D₂₃	—	2	1	no	—
152	C₂×D₂₃	—	2	1	no	—
153	C₂×D₂₃	—	2	1	no	—
154	C₄×C₂₃	—	2	1	no	—
155	C₄×C₂₃	—	2	1	no	—
156	C₄×C₂₃	—	2	1	no	—
157	C₄×C₂₃	—	2	1	no	—
158	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
159	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
160	Dic₂₃	—	2	1	no	—
161	Dic₂₃	—	2	1	no	—
162	Dic₂₃	—	2	1	no	—
163	Dic₂₃	—	2	1	no	—
164	Dic₂₃	—	2	1	no	—
165	Dic₂₃	—	2	1	no	—
166	Dic₂₃	#131	1	2	yes	—
167	#529	—	23	1	no	—
168	#446	—	23	1	no	—
169	C₈×Q₈	—	23	1	no	—
170	#436	—	23	1	no	—
171	#435	—	23	1	no	—
172	#435	D₂₃	1	1	yes	characteristic, complemented
173	C₈×D₄	—	23	1	no	—
174	C₈×D₄	—	23	1	no	—
175	C₈×D₄	—	23	1	no	—
176	#433	—	23	1	no	—
177	#433	—	23	1	no	—
178	#432	—	23	1	no	—
179	#405	—	23	1	no	—
180	#405	—	23	1	no	—
181	C₂×C₄×C₈	—	23	1	no	—
182	C₂×C₂₃	—	2	2	no	—
183	C₂×C₂₃	—	4	1	no	—
184	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
185	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
186	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
187	D₂₃	—	4	1	no	—
188	C₂×M₁₆	—	23	1	no	—
189	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
190	C₂²×C₈	—	23	1	no	—
191	C₂²×C₈	—	23	1	no	—
192	C₂²×C₈	—	23	1	no	—
193	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
194	#126	—	23	1	no	—
195	#126	—	23	1	no	—
196	#125	—	23	1	no	—
197	#124	—	23	1	no	—
198	#123	—	23	1	no	—
199	#123	—	23	1	no	—
200	C₂²⋊Q₈	—	23	1	no	—
201	#121	—	23	1	no	—
202	C₄×Q₈	—	23	1	no	—
203	C₄×D₄	—	23	1	no	—
204	C₄×D₄	—	23	1	no	—
205	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic
206	#117	—	23	1	no	—
207	#117	—	23	1	no	—
208	C₂×C₄²	—	23	1	no	—
209	C₄⋊C₈	—	23	1	no	—
210	C₄⋊C₈	—	23	1	no	—
211	C₄⋊C₈	—	23	1	no	—
212	C₄⋊C₈	—	23	1	no	—
213	C₄⋊C₈	—	23	1	no	—
214	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
215	C₂²⋊C₈	—	23	1	no	—
216	C₂²⋊C₈	—	23	1	no	—
217	C₂²⋊C₈	—	23	1	no	—
218	C₂²⋊C₈	—	23	1	no	—
219	C₂²⋊C₈	C₂×D₂₃	1	1	yes	characteristic
220	C₂²⋊C₈	C₂×D₂₃	1	1	yes	characteristic
221	#97	—	23	1	no	—
222	#97	—	23	1	no	—
223	#97	—	23	1	no	—
224	#97	C₂×D₂₃	1	1	yes	characteristic
225	C₄×C₈	—	23	1	no	—
226	C₄×C₈	—	23	1	no	—
227	C₄×C₈	—	23	1	no	—
228	C₄×C₈	—	23	1	no	—
229	C₄×C₈	—	23	1	no	—
230	C₄×C₈	—	23	1	no	—
231	C₂₃	#2966	1	1	yes	characteristic, complemented, Sylow
232	C₂×Q₈	—	23	1	no	—
233	C₂×D₄	—	23	1	no	—
234	C₂×D₄	—	23	1	no	—
235	C₂×D₄	C₄×D₂₃	1	1	yes	characteristic
236	C₂²×C₄	—	23	1	no	—
237	C₂²×C₄	—	23	1	no	—
238	C₂²×C₄	—	23	1	no	—
239	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
241	M₁₆	—	2	2	no	—
242	M₁₆	—	46	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	23	1	no	—
245	C₂×C₈	—	23	1	no	—
246	C₂×C₈	—	23	1	no	—
247	C₂×C₈	—	23	1	no	—
248	C₂×C₈	—	46	2	no	—
249	C₂×C₈	—	46	2	no	—
250	C₂×C₈	—	46	2	no	—
251	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
254	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
255	C₄⋊C₄	—	23	1	no	—
256	C₄⋊C₄	—	23	1	no	—
257	C₄⋊C₄	—	23	1	no	—
258	C₄⋊C₄	—	23	1	no	—
259	C₄⋊C₄	—	23	1	no	—
260	C₄⋊C₄	—	23	1	no	—
261	C₄⋊C₄	—	23	1	no	—
262	C₄⋊C₄	—	23	1	no	—
263	C₄⋊C₄	—	23	1	no	—
264	C₄⋊C₄	C₄×D₂₃	1	1	yes	characteristic
265	C₂²⋊C₄	—	23	1	no	—
266	C₂²⋊C₄	—	23	1	no	—
267	C₂²⋊C₄	—	23	1	no	—
268	C₂²⋊C₄	—	23	1	no	—
269	C₂²⋊C₄	—	23	1	no	—
270	C₂²⋊C₄	—	23	1	no	—
271	C₂²⋊C₄	—	23	1	no	—
272	C₂²⋊C₄	—	23	1	no	—
273	C₂²⋊C₄	C₄×D₂₃	1	1	yes	characteristic
274	C₂²⋊C₄	C₄×D₂₃	1	1	yes	characteristic
275	C₄²	—	23	1	no	—
276	C₄²	—	23	1	no	—
277	C₄²	—	23	1	no	—
278	C₄²	—	23	1	no	—
279	C₄²	—	23	1	no	—
280	C₄²	C₂²×D₂₃	1	1	yes	characteristic
281	C₂³	—	23	1	no	—
282	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
283	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
284	Q₈	—	46	2	no	—
285	D₄	—	2	2	no	—
286	D₄	—	46	2	no	—
287	D₄	—	46	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	23	1	no	—
293	C₂×C₄	—	23	1	no	—
294	C₂×C₄	—	23	1	no	—
295	C₂×C₄	—	23	1	no	—
296	C₂×C₄	—	23	1	no	—
297	C₂×C₄	—	23	1	no	—
298	C₂×C₄	—	23	1	no	—
299	C₂×C₄	—	46	2	no	—
300	C₂×C₄	—	46	2	no	—
301	C₂×C₄	—	46	1	no	—
302	C₂×C₄	—	46	1	no	—
303	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	23	2	no	—
312	C₈	—	46	1	no	—
313	C₈	—	46	1	no	—
314	C₈	—	46	1	no	—
315	C₈	#67290	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	46	1	no	—
322	C₂²	—	46	1	no	—
323	C₂²	—	46	1	no	—
324	C₂²	#10625078	1	2	yes	—
325	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	23	2	no	—
331	C₄	—	46	1	no	—
332	C₄	—	46	1	no	—
333	C₄	—	46	1	no	—
334	C₄	—	46	1	no	—
335	C₄	—	46	1	no	—
336	C₄	—	46	1	no	—
337	C₄	C₂×#67290	1	1	yes	characteristic
338	C₄	#41×D₂₃	1	1	yes	characteristic
339	C₂	—	2	2	no	—
340	C₂	—	4	1	no	—
341	C₂	—	92	1	no	—
342	C₂	C₄×#67290	1	1	yes	characteristic
343	C₂	#131×D₂₃	1	1	yes	characteristic
344	C₂	C₂×#10625078	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#435	C₂
prime radical	#10625110	C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#435	D₂₃
hypocenter	C₂₃	#2966
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. #13092545	C₂	semidirect
3. #13092541	C₂	semidirect
4. #13092493	C₂	semidirect
5. #13092488	C₂	semidirect
6. D₄×C₂₃⋊C₈	C₂	semidirect
7. (C₂²⋊D₂₃)⋊C₈	C₂	semidirect
8. (C₂²⋊D₂₃)⋊C₈	C₂	semidirect
9. #13092477	C₂	semidirect
10. #13092475	C₂	semidirect
11. #13092471	C₂	semidirect
12. #13092375	C₂	semidirect
13. #13092374	C₂	semidirect
14. C₂×C₈×Dic₂₃	C₂	semidirect
15. C₂₃×#435	C₂	semidirect
172. #435	D₂₃	semidirect
231. C₂₃	#2966	semidirect

Maximal subgroups

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

subgroup	index
#13092947	2
#13092545	2
#13092541	2
#13092493	2
#13092488	2
D₄×C₂₃⋊C₈	2
(C₂²⋊D₂₃)⋊C₈	2
#13092477	2
#13092475	2
#13092471	2
#13092375	2
#13092374	2
C₂×C₈×Dic₂₃	2
C₂₃×#435	2
#2966	23