Subgroups — #15682060

Order 2944.

392 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092885	C₂	1	1	yes	maximal, characteristic
2	#13092478	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092477	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092477	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092472	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092471	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092468	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092467	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092439	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#13092382	C₂	1	1	yes	maximal, characteristic
12	#13092382	C₂	1	1	yes	maximal, characteristic
13	#13092374	C₂	1	1	yes	maximal, characteristic, complemented
14	#13092371	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#67288	C₂²	1	1	yes	characteristic
17	C₂₃×#117	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
19	C₂×#67262	C₂²	1	1	yes	characteristic
20	C₂×#67262	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
23	#10625064	C₄	1	1	yes	characteristic
24	#10625064	C₄	1	1	yes	characteristic
25	#10625058	C₄	1	1	yes	characteristic
26	#10625058	C₄	1	1	yes	characteristic
27	(C₂²⋊D₂₃)⋊C₄	C₄	1	1	yes	characteristic
28	(C₂²⋊D₂₃)⋊C₄	C₄	1	1	yes	characteristic
29	(C₂²⋊D₂₃)⋊C₄	C₄	1	1	yes	characteristic
30	(C₂²⋊D₂₃)⋊C₄	C₄	1	1	yes	characteristic
31	#10625043	C₂²	1	1	yes	characteristic
32	#10625043	C₂²	1	1	yes	characteristic
33	C₄²×D₂₃	C₂²	1	1	yes	characteristic
34	C₄²×D₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
38	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₂₃×#97	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₂₃	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	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
46	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
47	#10624997	C₂²	1	1	yes	characteristic
48	#10624997	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	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
53	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
54	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
55	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
56	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
57	#10624985	C₂²	1	1	yes	characteristic
58	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₂₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₂₃	C₂×C₄	1	1	yes	characteristic
62	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
64	#67288	—	2	2	no	—
65	#67288	—	2	2	no	—
66	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
73	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
74	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
80	C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
81	C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
82	C₄²×C₂₃	C₂³	1	1	yes	characteristic
83	C₄²×C₂₃	C₂³	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	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
87	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
88	Dic₂₃⋊C₄	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	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₂₃	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	C₂×C₄	1	1	yes	characteristic
98	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
99	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
104	#67262	—	2	2	no	—
105	#67262	—	2	2	no	—
106	C₈×D₂₃	—	2	2	no	—
107	C₈×D₂₃	—	2	2	no	—
108	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	—	2	1	no	—
113	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₂₃	—	2	2	no	—
120	C₂²⋊D₂₃	—	2	2	no	—
121	C₂×Dic₂₃	—	2	2	no	—
122	C₂×Dic₂₃	—	4	1	no	—
123	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
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	D₉₂	—	2	2	no	—
130	C₄×D₂₃	—	2	1	no	—
131	C₄×D₂₃	—	2	1	no	—
132	C₄×D₂₃	—	2	2	no	—
133	C₄×D₂₃	—	2	2	no	—
134	C₄×D₂₃	—	4	1	no	—
135	C₄×D₂₃	—	4	1	no	—
136	C₄×D₂₃	#41	1	2	yes	—
137	C₄×D₂₃	#41	1	2	yes	—
138	Dic₄₆	—	2	2	no	—
139	C₈×C₂₃	—	2	1	no	—
140	C₈×C₂₃	—	2	1	no	—
141	C₈×C₂₃	—	2	1	no	—
142	C₈×C₂₃	—	2	1	no	—
143	C₂₃⋊C₈	—	2	1	no	—
144	C₂₃⋊C₈	—	2	1	no	—
145	C₂₃⋊C₈	—	2	1	no	—
146	C₂₃⋊C₈	—	2	1	no	—
147	#2925	—	23	1	no	maximal, Sylow
148	C₂²×C₂₃	—	2	1	no	—
149	C₂²×C₂₃	—	2	1	no	—
150	C₂²×C₂₃	—	2	1	no	—
151	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
152	C₂×D₂₃	—	2	1	no	—
153	C₂×D₂₃	—	2	1	no	—
154	C₂×D₂₃	—	2	1	no	—
155	C₂×D₂₃	—	2	1	no	—
156	C₂×D₂₃	—	2	1	no	—
157	C₂×D₂₃	#117	1	2	yes	—
158	C₄×C₂₃	—	2	1	no	—
159	C₄×C₂₃	—	2	1	no	—
160	C₄×C₂₃	—	2	1	no	—
161	C₄×C₂₃	—	2	1	no	—
162	C₄×C₂₃	—	2	1	no	—
163	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
164	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
165	Dic₂₃	—	2	1	no	—
166	Dic₂₃	—	2	1	no	—
167	Dic₂₃	—	2	1	no	—
168	Dic₂₃	—	2	1	no	—
169	Dic₂₃	#117	1	2	yes	—
170	Dic₂₃	#131	1	2	yes	—
171	C₄×#41	—	23	1	no	—
172	#433	—	23	1	no	—
173	#433	—	23	1	no	—
174	#433	—	23	1	no	—
175	#433	D₂₃	1	1	yes	characteristic, complemented
176	#432	—	23	1	no	—
177	#432	—	23	1	no	—
178	#431	—	23	1	no	—
179	#431	—	23	1	no	—
180	#423	—	23	1	no	—
181	C₂×C₄⋊C₈	—	23	1	no	—
182	#408	—	23	1	no	—
183	#408	—	23	1	no	—
184	#405	—	23	1	no	—
185	#405	—	23	1	no	—
186	C₂×C₂₃	—	4	1	no	—
187	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
188	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
189	C₂×C₂₃	#568	1	1	yes	characteristic
190	D₂₃	—	2	2	no	—
191	D₂₃	—	4	1	no	—
192	C₂×#41	—	23	1	no	—
193	C₂×M₁₆	—	23	1	no	—
194	C₂×M₁₆	—	23	1	no	—
195	C₂²×C₈	—	23	1	no	—
196	C₂²×C₈	—	23	1	no	—
197	C₄×Q₈	—	23	1	no	—
198	C₄×Q₈	—	23	1	no	—
199	C₄×D₄	—	23	1	no	—
200	C₄×D₄	—	23	1	no	—
201	C₄×D₄	—	23	1	no	—
202	C₄×D₄	—	23	1	no	—
203	C₄×D₄	—	23	1	no	—
204	C₄×D₄	—	23	1	no	—
205	#117	—	23	1	no	—
206	#117	—	23	1	no	—
207	#117	C₂×D₂₃	1	1	yes	characteristic
208	C₂×C₄²	—	23	1	no	—
209	C₂×C₄²	—	23	1	no	—
210	C₂×C₄²	—	23	1	no	—
211	C₄⋊C₈	—	23	1	no	—
212	C₄⋊C₈	—	23	1	no	—
213	C₄⋊C₈	—	23	1	no	—
214	C₄⋊C₈	—	23	1	no	—
215	C₄⋊C₈	—	23	1	no	—
216	C₄⋊C₈	—	23	1	no	—
217	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
218	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
219	C₂²⋊C₈	—	23	1	no	—
220	C₂²⋊C₈	—	23	1	no	—
221	C₂²⋊C₈	—	23	1	no	—
222	C₂²⋊C₈	—	23	1	no	—
223	C₂²⋊C₈	—	23	1	no	—
224	C₂²⋊C₈	—	23	1	no	—
225	C₂²⋊C₈	C₂×D₂₃	1	1	yes	characteristic
226	C₂²⋊C₈	C₂×D₂₃	1	1	yes	characteristic
227	#97	—	23	1	no	—
228	#97	—	23	1	no	—
229	#97	—	23	1	no	—
230	#97	C₂×D₂₃	1	1	yes	characteristic
231	C₄×C₈	—	23	1	no	—
232	C₄×C₈	—	23	1	no	—
233	C₄×C₈	—	23	1	no	—
234	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
235	C₂₃	#2925	1	1	yes	characteristic, complemented, Sylow
236	#41	—	46	2	no	—
237	#41	—	46	2	no	—
238	C₂×Q₈	—	23	1	no	—
239	C₂×D₄	—	23	1	no	—
240	C₂×D₄	—	23	1	no	—
241	C₂×D₄	—	23	1	no	—
242	C₂²×C₄	—	23	1	no	—
243	C₂²×C₄	—	23	1	no	—
244	C₂²×C₄	—	23	1	no	—
245	C₂²×C₄	—	23	1	no	—
246	C₂²×C₄	—	23	1	no	—
247	C₂²×C₄	—	23	1	no	—
248	C₂²×C₄	—	23	1	no	—
249	C₂²×C₄	—	23	1	no	—
250	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
251	M₁₆	—	46	2	no	—
252	M₁₆	—	46	2	no	—
253	C₂×C₈	—	23	1	no	—
254	C₂×C₈	—	23	1	no	—
255	C₂×C₈	—	23	1	no	—
256	C₂×C₈	—	23	1	no	—
257	C₂×C₈	—	46	2	no	—
258	C₂×C₈	—	46	2	no	—
259	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
260	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
263	C₄⋊C₄	—	23	1	no	—
264	C₄⋊C₄	—	23	1	no	—
265	C₄⋊C₄	—	23	1	no	—
266	C₄⋊C₄	—	23	1	no	—
267	C₄⋊C₄	C₄×D₂₃	1	1	yes	characteristic
268	C₄⋊C₄	C₄×D₂₃	1	1	yes	characteristic
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	—	23	1	no	—
281	C₄²	—	23	1	no	—
282	C₄²	—	23	1	no	—
283	C₄²	C₂²×D₂₃	1	1	yes	characteristic
284	C₄²	C₂²×D₂₃	1	1	yes	characteristic
285	C₂³	—	23	1	no	—
286	C₂³	—	23	1	no	—
287	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
288	Q₈	—	46	2	no	—
289	D₄	—	46	2	no	—
290	D₄	—	46	2	no	—
291	D₄	—	46	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	23	1	no	—
295	C₂×C₄	—	23	1	no	—
296	C₂×C₄	—	23	2	no	—
297	C₂×C₄	—	23	2	no	—
298	C₂×C₄	—	23	1	no	—
299	C₂×C₄	—	23	1	no	—
300	C₂×C₄	—	23	1	no	—
301	C₂×C₄	—	23	1	no	—
302	C₂×C₄	—	46	2	no	—
303	C₂×C₄	—	46	2	no	—
304	C₂×C₄	—	46	2	no	—
305	C₂×C₄	—	46	1	no	—
306	C₂×C₄	—	46	1	no	—
307	C₂×C₄	—	92	1	no	—
308	C₂×C₄	—	92	1	no	—
309	C₂×C₄	—	92	1	no	—
310	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
311	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	46	1	no	—
321	C₈	—	46	1	no	—
322	C₈	—	46	1	no	—
323	C₈	—	46	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	23	2	no	—
328	C₂²	—	46	1	no	—
329	C₂²	—	46	1	no	—
330	C₂²	—	46	1	no	—
331	C₂²	—	46	1	no	—
332	C₂²	—	46	1	no	—
333	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	23	2	no	—
340	C₄	—	23	2	no	—
341	C₄	—	46	1	no	—
342	C₄	—	46	1	no	—
343	C₄	—	46	1	no	—
344	C₄	—	46	1	no	—
345	C₄	#41×D₂₃	1	1	yes	characteristic
346	C₄	#41×D₂₃	1	1	yes	characteristic
347	C₂	—	4	1	no	—
348	C₂	—	46	2	no	—
349	C₂	—	92	1	no	—
350	C₂	#117×D₂₃	1	1	yes	characteristic
351	C₂	#131×D₂₃	1	1	yes	characteristic
352	C₂	#13093182	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#433	C₂
prime radical	C₂×#67288	C₂²
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#433	D₂₃
hypocenter	C₂₃	#2925
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #13092478	C₂	semidirect
3. #13092477	C₂	semidirect
4. #13092477	C₂	semidirect
5. #13092472	C₂	semidirect
6. #13092471	C₂	semidirect
7. #13092468	C₂	semidirect
8. #13092467	C₂	semidirect
9. #13092439	C₂	semidirect
10. C₄⋊C₈×D₂₃	C₂	semidirect
13. #13092374	C₂	semidirect
14. #13092371	C₂	semidirect
15. C₂₃×#433	C₂	semidirect
175. #433	D₂₃	semidirect
235. C₂₃	#2925	semidirect

Maximal subgroups

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

subgroup	index
#13092885	2
#13092478	2
#13092477	2
#13092472	2
#13092471	2
#13092468	2
#13092467	2
#13092439	2
C₄⋊C₈×D₂₃	2
#13092382	2
#13092374	2
#13092371	2
C₂₃×#433	2
#2925	23