Subgroups — #15683308

Order 2944.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092924	C₂	1	1	yes	maximal, characteristic
2	#117×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092757	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092756	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092751	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092746	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092740	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092739	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092734	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092729	C₂	1	1	yes	maximal, characteristic, complemented
11	#13092455	C₂	1	1	yes	maximal, characteristic, complemented
12	#13092453	C₂	1	1	yes	maximal, characteristic, complemented
13	#13092413	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×(C₂×D₂₃)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#416	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#117	C₂²	1	1	yes	characteristic
20	#10625117	C₂²	1	1	yes	characteristic
21	#10625117	C₂²	1	1	yes	characteristic
22	#10625109	C₂²	1	1	yes	characteristic
23	#10625109	C₂²	1	1	yes	characteristic
24	#10625108	—	2	1	no	—
25	#10625106	—	2	1	no	—
26	C₄×C₂²⋊D₂₃	—	2	1	no	—
27	C₄×C₂²⋊D₂₃	—	2	1	no	—
28	#10625097	C₂²	1	1	yes	characteristic
29	#10625097	C₂²	1	1	yes	characteristic
30	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
31	#10625063	C₂²	1	1	yes	characteristic
32	#10625063	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
35	C₂²⋊C₄×D₂₃	—	2	1	no	—
36	#10625048	—	2	1	no	—
37	#10625043	—	2	1	no	—
38	C₄²×D₂₃	—	2	1	no	—
39	C₂₃×#103	C₂²	1	1	yes	characteristic, complemented
40	C₂₃×#103	C₂²	1	1	yes	characteristic, complemented
41	C₂₃×#102	C₂²	1	1	yes	characteristic, complemented
42	C₂₃×#102	C₂²	1	1	yes	characteristic, complemented
43	#10625017	C₂²	1	1	yes	characteristic, complemented
44	#10625017	C₂²	1	1	yes	characteristic, complemented
45	#10625014	C₂²	1	1	yes	characteristic, complemented
46	#10625014	C₂²	1	1	yes	characteristic, complemented
47	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
48	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
51	#10625000	C₂²	1	1	yes	characteristic, complemented
52	#10625000	C₂²	1	1	yes	characteristic, complemented
53	#10624999	C₂²	1	1	yes	characteristic, complemented
54	#10624999	C₂²	1	1	yes	characteristic, complemented
55	#10624990	C₂²	1	1	yes	characteristic, complemented
56	#10624990	C₂²	1	1	yes	characteristic, complemented
57	#10624989	C₂²	1	1	yes	characteristic, complemented
58	#10624989	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₂₃	D₄	1	1	yes	characteristic
60	#41×C₂₃	—	2	2	no	—
61	#41×C₂₃	—	2	2	no	—
62	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₂₃	—	2	1	no	—
64	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	—	2	1	no	—
66	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₂₃	—	2	1	no	—
68	C₂×C₂²⋊D₂₃	—	2	1	no	—
69	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
70	C₂×C₄×D₂₃	—	2	2	no	—
71	C₂×C₄×D₂₃	—	2	2	no	—
72	C₂×C₄×D₂₃	—	2	1	no	—
73	C₂×C₄×D₂₃	—	2	1	no	—
74	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
75	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
76	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
77	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₂₃	—	2	1	no	—
84	C₄²×C₂₃	—	2	1	no	—
85	C₂²⋊Dic₂₃	—	2	1	no	—
86	C₂²⋊Dic₂₃	—	2	1	no	—
87	C₂²⋊Dic₂₃	—	2	1	no	—
88	C₂²⋊Dic₂₃	—	2	1	no	—
89	(C₂×D₂₃)⋊C₄	—	2	1	no	—
90	(C₂×D₂₃)⋊C₄	—	2	1	no	—
91	(C₂×D₂₃)⋊C₄	—	2	1	no	—
92	(C₂×D₂₃)⋊C₄	—	2	1	no	—
93	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
97	Dic₂₃⋊C₄	—	2	1	no	—
98	Dic₂₃⋊C₄	—	2	1	no	—
99	Dic₂₃⋊C₄	—	2	1	no	—
100	Dic₂₃⋊C₄	—	2	1	no	—
101	C₄×Dic₂₃	—	2	1	no	—
102	C₄×Dic₂₃	—	2	1	no	—
103	C₄×Dic₂₃	—	2	1	no	—
104	C₄×Dic₂₃	—	2	1	no	—
105	C₂×C₂₃⋊C₈	—	2	2	no	—
106	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
108	C₂³×C₂₃	—	2	1	no	—
109	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₂₃	—	2	2	no	—
111	C₂²×D₂₃	—	2	2	no	—
112	C₂²×D₂₃	—	4	1	no	—
113	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₂₃	—	2	1	no	—
116	Q₈×C₂₃	—	2	1	no	—
117	D₄×C₂₃	—	2	1	no	—
118	D₄×C₂₃	—	2	1	no	—
119	D₄×C₂₃	—	4	2	no	—
120	C₂×C₄×C₂₃	—	2	1	no	—
121	C₂×C₄×C₂₃	—	2	1	no	—
122	C₂×C₄×C₂₃	—	2	1	no	—
123	C₂×C₄×C₂₃	—	4	2	no	—
124	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₂₃	#41	1	2	yes	—
126	C₂×C₄×C₂₃	#41	1	2	yes	—
127	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₂₃	—	4	2	no	—
129	C₂²⋊D₂₃	—	4	2	no	—
130	C₂×Dic₂₃	—	2	2	no	—
131	C₂×Dic₂₃	—	2	1	no	—
132	C₂×Dic₂₃	—	2	1	no	—
133	C₂×Dic₂₃	—	2	1	no	—
134	C₂×Dic₂₃	—	2	1	no	—
135	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
137	C₄×D₂₃	—	2	2	no	—
138	C₄×D₂₃	—	2	2	no	—
139	C₄×D₂₃	—	2	1	no	—
140	C₄×D₂₃	—	2	1	no	—
141	C₄×D₂₃	—	2	1	no	—
142	C₄×D₂₃	—	2	1	no	—
143	C₄×D₂₃	—	4	2	no	—
144	C₄×D₂₃	—	4	2	no	—
145	C₈×C₂₃	—	2	1	no	—
146	C₈×C₂₃	—	2	1	no	—
147	C₂₃⋊C₈	—	2	1	no	—
148	C₂₃⋊C₈	—	2	1	no	—
149	#3091	—	23	1	no	maximal, Sylow
150	C₂²×C₂₃	—	2	1	no	—
151	C₂²×C₂₃	—	2	1	no	—
152	C₂²×C₂₃	—	4	1	no	—
153	C₂²×C₂₃	—	4	1	no	—
154	C₂²×C₂₃	—	4	1	no	—
155	C₂²×C₂₃	#123	1	2	yes	—
156	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×D₂₃	—	2	1	no	—
158	C₂×D₂₃	—	2	1	no	—
159	C₂×D₂₃	—	2	1	no	—
160	C₂×D₂₃	—	2	1	no	—
161	C₂×D₂₃	—	2	1	no	—
162	C₂×D₂₃	—	4	4	no	—
163	C₂×D₂₃	#135	1	2	yes	—
164	C₄×C₂₃	—	4	1	no	—
165	C₄×C₂₃	—	4	1	no	—
166	C₄×C₂₃	—	4	1	no	—
167	C₄×C₂₃	#123	1	2	yes	—
168	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
169	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
170	Dic₂₃	—	2	1	no	—
171	Dic₂₃	—	2	1	no	—
172	Dic₂₃	—	4	1	no	—
173	Dic₂₃	—	4	1	no	—
174	Dic₂₃	—	4	1	no	—
175	Dic₂₃	—	4	1	no	—
176	#525	—	23	1	no	—
177	C₂×#117	—	23	1	no	—
178	#485	—	23	1	no	—
179	#485	—	23	1	no	—
180	#484	—	23	1	no	—
181	#483	—	23	1	no	—
182	#482	—	23	1	no	—
183	#482	—	23	1	no	—
184	#481	—	23	1	no	—
185	#480	—	23	1	no	—
186	#427	—	23	1	no	—
187	#427	—	23	1	no	—
188	#416	—	23	1	no	—
189	#416	D₂₃	1	1	yes	characteristic, complemented
190	C₂×C₂²⋊C₈	—	23	1	no	—
191	C₂×C₂₃	—	2	2	no	—
192	C₂×C₂₃	—	8	1	no	—
193	C₂×C₂₃	C₂×#123	1	1	yes	characteristic
194	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
195	C₂×C₂₃	#575	1	1	yes	characteristic
196	D₂₃	—	2	2	no	—
197	D₂₃	—	4	1	no	—
198	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
199	C₂³×C₄	—	23	1	no	—
200	C₂²×C₈	—	23	1	no	—
201	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
202	#123	—	46	1	no	—
203	C₂²⋊Q₈	—	23	1	no	—
204	C₂²⋊Q₈	—	23	1	no	—
205	#121	—	23	1	no	—
206	#121	—	23	1	no	—
207	#120	—	46	1	no	—
208	C₄×D₄	—	46	1	no	—
209	C₄×D₄	—	46	1	no	—
210	#117	—	23	1	no	—
211	#117	—	23	1	no	—
212	#117	—	23	1	no	—
213	#117	—	23	1	no	—
214	#117	—	46	1	no	—
215	#117	—	46	1	no	—
216	#117	C₂×D₂₃	1	1	yes	characteristic
217	C₂×C₄⋊C₄	—	23	1	no	—
218	C₂×C₄⋊C₄	—	23	1	no	—
219	C₂×C₂²⋊C₄	—	46	1	no	—
220	C₂×C₄²	—	46	1	no	—
221	#107	—	23	1	no	—
222	#107	—	23	1	no	—
223	#107	—	23	1	no	—
224	#107	—	23	1	no	—
225	#106	—	23	1	no	—
226	#106	—	23	1	no	—
227	#106	—	23	1	no	—
228	#106	—	23	1	no	—
229	#103	—	23	1	no	—
230	#103	—	23	1	no	—
231	#103	C₂×D₂₃	1	1	yes	characteristic, complemented
232	#103	C₂×D₂₃	1	1	yes	characteristic, complemented
233	#102	—	23	1	no	—
234	#102	—	23	1	no	—
235	#102	C₂×D₂₃	1	1	yes	characteristic, complemented
236	#102	C₂×D₂₃	1	1	yes	characteristic, complemented
237	C₂²⋊C₈	—	23	1	no	—
238	C₂²⋊C₈	—	23	1	no	—
239	C₂²⋊C₈	—	23	1	no	—
240	C₂²⋊C₈	—	23	1	no	—
241	C₂₃	#3091	1	1	yes	characteristic, complemented, Sylow
242	C₂⁴	—	23	1	no	—
243	#41	—	2	2	no	—
244	#41	—	2	2	no	—
245	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	46	1	no	—
248	C₂×D₄	—	46	1	no	—
249	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	23	1	no	—
252	C₂²×C₄	—	23	1	no	—
253	C₂²×C₄	—	23	1	no	—
254	C₂²×C₄	—	23	1	no	—
255	C₂²×C₄	—	23	1	no	—
256	C₂²×C₄	—	46	1	no	—
257	C₂²×C₄	—	46	1	no	—
258	C₂²×C₄	—	46	2	no	—
259	C₂²×C₄	—	46	2	no	—
260	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	23	1	no	—
263	C₂×C₈	—	23	1	no	—
264	C₂×C₈	—	46	2	no	—
265	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
266	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
267	C₄⋊C₄	—	23	1	no	—
268	C₄⋊C₄	—	23	1	no	—
269	C₄⋊C₄	—	23	1	no	—
270	C₄⋊C₄	—	23	1	no	—
271	C₄⋊C₄	—	46	1	no	—
272	C₄⋊C₄	—	46	1	no	—
273	C₄⋊C₄	—	46	1	no	—
274	C₄⋊C₄	—	46	1	no	—
275	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	46	1	no	—
279	C₂²⋊C₄	—	46	1	no	—
280	C₂²⋊C₄	—	46	1	no	—
281	C₂²⋊C₄	—	46	1	no	—
282	C₂²⋊C₄	—	46	1	no	—
283	C₂²⋊C₄	—	46	1	no	—
284	C₂²⋊C₄	—	46	1	no	—
285	C₂²⋊C₄	—	46	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	46	1	no	—
288	C₄²	—	46	1	no	—
289	C₄²	—	46	1	no	—
290	C₄²	—	46	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	23	1	no	—
293	C₂³	—	23	1	no	—
294	C₂³	—	46	2	no	—
295	C₂³	—	46	2	no	—
296	C₂³	—	92	1	no	—
297	C₂³	D₄×D₂₃	1	1	yes	characteristic
298	Q₈	—	2	1	no	—
299	Q₈	—	2	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	4	2	no	—
303	D₄	—	92	2	no	—
304	D₄	—	92	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	23	1	no	—
310	C₂×C₄	—	23	1	no	—
311	C₂×C₄	—	46	2	no	—
312	C₂×C₄	—	46	2	no	—
313	C₂×C₄	—	46	2	no	—
314	C₂×C₄	—	46	1	no	—
315	C₂×C₄	—	46	1	no	—
316	C₂×C₄	—	46	1	no	—
317	C₂×C₄	—	46	1	no	—
318	C₂×C₄	—	46	1	no	—
319	C₂×C₄	—	46	1	no	—
320	C₂×C₄	—	46	1	no	—
321	C₂×C₄	—	46	1	no	—
322	C₂×C₄	—	92	2	no	—
323	C₂×C₄	—	92	2	no	—
324	C₂×C₄	#67288	1	2	yes	—
325	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
326	C₂×C₄	#67290	1	2	yes	—
327	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	46	1	no	—
331	C₈	—	46	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	23	2	no	—
338	C₂²	—	46	1	no	—
339	C₂²	—	46	1	no	—
340	C₂²	—	46	1	no	—
341	C₂²	—	46	1	no	—
342	C₂²	—	46	1	no	—
343	C₂²	—	92	4	no	—
344	C₂²	#10625054	1	2	yes	—
345	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	46	1	no	—
350	C₄	—	46	1	no	—
351	C₄	—	92	1	no	—
352	C₄	—	92	1	no	—
353	C₄	—	92	1	no	—
354	C₄	—	92	1	no	—
355	C₄	#10625054	1	2	yes	—
356	C₄	C₂×#67288	1	1	yes	characteristic
357	C₄	C₂×#67290	1	1	yes	characteristic
358	C₂	—	2	2	no	—
359	C₂	—	8	1	no	—
360	C₂	—	46	2	no	—
361	C₂	—	92	1	no	—
362	C₂	C₂×#10625054	1	1	yes	characteristic
363	C₂	#135×D₂₃	1	1	yes	characteristic
364	C₂	#13093214	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#416	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	#13092924	C₂
hypercenter	#416	D₂₃
hypocenter	C₂₃	#3091
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. #117×D₂₃	C₂	semidirect
3. #13092757	C₂	semidirect
4. #13092756	C₂	semidirect
5. #13092751	C₂	semidirect
6. #13092746	C₂	semidirect
7. #13092740	C₂	semidirect
8. #13092739	C₂	semidirect
9. #13092734	C₂	semidirect
10. #13092729	C₂	semidirect
11. #13092455	C₂	semidirect
12. #13092453	C₂	semidirect
13. #13092413	C₂	semidirect
14. C₂×(C₂×D₂₃)⋊C₈	C₂	semidirect
15. C₂₃×#416	C₂	semidirect
39. C₂₃×#103	C₂²	semidirect
40. C₂₃×#103	C₂²	semidirect
41. C₂₃×#102	C₂²	semidirect
42. C₂₃×#102	C₂²	semidirect
43. #10625017	C₂²	semidirect
44. #10625017	C₂²	semidirect
45. #10625014	C₂²	semidirect
46. #10625014	C₂²	semidirect
51. #10625000	C₂²	semidirect
52. #10625000	C₂²	semidirect
53. #10624999	C₂²	semidirect
54. #10624999	C₂²	semidirect
55. #10624990	C₂²	semidirect
56. #10624990	C₂²	semidirect
57. #10624989	C₂²	semidirect
58. #10624989	C₂²	semidirect
189. #416	D₂₃	semidirect
231. #103	C₂×D₂₃	semidirect
232. #103	C₂×D₂₃	semidirect
235. #102	C₂×D₂₃	semidirect
236. #102	C₂×D₂₃	semidirect
241. C₂₃	#3091	semidirect

Maximal subgroups

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

subgroup	index
#13092924	2
#117×D₂₃	2
#13092757	2
#13092756	2
#13092751	2
#13092746	2
#13092740	2
#13092739	2
#13092734	2
#13092729	2
#13092455	2
#13092453	2
#13092413	2
C₂×(C₂×D₂₃)⋊C₈	2
C₂₃×#416	2
#3091	23