Subgroups — #15683613

Order 2944.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093029	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092897	C₂	1	1	yes	maximal, characteristic
3	#13092753	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092748	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092725	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092718	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092628	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092623	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092581	C₂	1	1	yes	maximal, characteristic
10	#13092581	C₂	1	1	yes	maximal, characteristic
11	#13092525	C₂	1	1	yes	maximal, characteristic, complemented
12	#13092519	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Dic₉₂	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#67263	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	#10625095	C₂²	1	1	yes	characteristic
20	#10625095	C₂²	1	1	yes	characteristic
21	C₂×Dic₉₂	C₂²	1	1	yes	characteristic
22	C₂×Dic₉₂	C₂²	1	1	yes	characteristic
23	C₂×#67263	C₂²	1	1	yes	characteristic
24	C₂×#67263	C₂²	1	1	yes	characteristic
25	#10625063	—	2	1	no	—
26	C₄⋊C₄×D₂₃	—	2	1	no	—
27	#10625060	—	2	1	no	—
28	#10625060	—	2	1	no	—
29	(C₂²⋊D₂₃)⋊C₄	—	2	1	no	—
30	(C₂²⋊D₂₃)⋊C₄	—	2	1	no	—
31	#10625050	—	2	1	no	—
32	#10625050	—	2	1	no	—
33	C₄×D₉₂	C₂²	1	1	yes	characteristic
34	C₄×D₉₂	C₂²	1	1	yes	characteristic
35	#10625041	C₂²	1	1	yes	characteristic
36	#10625040	C₂²	1	1	yes	characteristic
37	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
38	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
39	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
40	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
41	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	C₂₃×#97	C₂²	1	1	yes	characteristic
46	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
47	#10625003	C₂²	1	1	yes	characteristic
48	#10625003	C₂²	1	1	yes	characteristic
49	#10625000	C₂²	1	1	yes	characteristic
50	#10625000	C₂²	1	1	yes	characteristic
51	#10624999	C₂²	1	1	yes	characteristic
52	#10624999	C₂²	1	1	yes	characteristic
53	#10624998	C₂²	1	1	yes	characteristic
54	#10624998	C₂²	1	1	yes	characteristic
55	#10624998	C₂²	1	1	yes	characteristic
56	#10624998	C₂²	1	1	yes	characteristic
57	#10624998	C₂²	1	1	yes	characteristic
58	#10624998	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₂₃	—	2	1	no	—
61	C₂²×Dic₂₃	—	2	1	no	—
62	#67288	—	2	2	no	—
63	#67288	—	2	2	no	—
64	C₂×D₉₂	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₂₃	—	2	1	no	—
66	C₂×C₄×D₂₃	—	2	1	no	—
67	C₂×C₄×D₂₃	—	2	1	no	—
68	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
69	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
70	C₂×Dic₄₆	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	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
77	C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
79	C₄²×C₂₃	C₂³	1	1	yes	characteristic
80	C₄²×C₂₃	C₂³	1	1	yes	characteristic
81	C₂²⋊Dic₂₃	—	2	1	no	—
82	C₂²⋊Dic₂₃	—	2	1	no	—
83	(C₂×D₂₃)⋊C₄	—	2	1	no	—
84	(C₂×D₂₃)⋊C₄	—	2	1	no	—
85	C₄⋊Dic₂₃	—	2	1	no	—
86	C₄⋊Dic₂₃	—	2	1	no	—
87	C₄⋊Dic₂₃	—	2	1	no	—
88	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
92	Dic₂₃⋊C₄	—	2	1	no	—
93	Dic₂₃⋊C₄	—	2	1	no	—
94	Dic₂₃⋊C₄	—	2	1	no	—
95	Dic₂₃⋊C₄	—	2	1	no	—
96	Dic₂₃⋊C₄	—	2	1	no	—
97	Dic₂₃⋊C₄	—	2	1	no	—
98	Dic₂₃⋊C₄	—	2	1	no	—
99	Dic₂₃⋊C₄	—	2	1	no	—
100	C₄×Dic₂₃	—	2	1	no	—
101	C₄×Dic₂₃	—	2	1	no	—
102	C₄×Dic₂₃	—	2	1	no	—
103	C₄×Dic₂₃	—	2	1	no	—
104	Dic₉₂	—	2	2	no	—
105	Dic₉₂	—	2	2	no	—
106	#67263	—	2	2	no	—
107	#67263	—	2	2	no	—
108	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	2	1	no	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₂₃	—	2	4	no	—
119	C₂×Dic₂₃	—	2	1	no	—
120	C₂×Dic₂₃	—	2	1	no	—
121	C₂×Dic₂₃	—	2	1	no	—
122	C₂×Dic₂₃	—	2	1	no	—
123	C₂×Dic₂₃	—	2	1	no	—
124	C₂×Dic₂₃	—	2	1	no	—
125	C₂×Dic₂₃	—	2	1	no	—
126	C₂×Dic₂₃	—	8	1	no	—
127	D₉₂	—	2	1	no	—
128	D₉₂	#41	1	2	yes	—
129	C₄×D₂₃	—	2	4	no	—
130	C₄×D₂₃	—	8	1	no	—
131	C₄×D₂₃	—	8	1	no	—
132	Dic₄₆	—	2	1	no	—
133	Dic₄₆	—	2	1	no	—
134	Dic₄₆	—	2	1	no	—
135	Dic₄₆	—	2	1	no	—
136	Dic₄₆	—	2	1	no	—
137	Dic₄₆	#41	1	2	yes	—
138	C₈×C₂₃	—	2	1	no	—
139	C₈×C₂₃	—	2	1	no	—
140	C₈×C₂₃	—	2	1	no	—
141	C₈×C₂₃	—	2	1	no	—
142	#3120	—	23	1	no	maximal, Sylow
143	C₂²×C₂₃	—	2	1	no	—
144	C₂²×C₂₃	—	2	1	no	—
145	C₂²×C₂₃	—	2	1	no	—
146	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₂₃	—	2	2	no	—
148	C₂×D₂₃	—	4	1	no	—
149	C₂×D₂₃	—	4	1	no	—
150	C₄×C₂₃	—	2	1	no	—
151	C₄×C₂₃	—	2	1	no	—
152	C₄×C₂₃	—	2	1	no	—
153	C₄×C₂₃	—	2	1	no	—
154	C₄×C₂₃	—	2	1	no	—
155	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
156	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
157	Dic₂₃	—	2	2	no	—
158	Dic₂₃	—	4	1	no	—
159	Dic₂₃	—	4	1	no	—
160	Dic₂₃	—	4	1	no	—
161	Dic₂₃	—	4	1	no	—
162	Dic₂₃	—	4	1	no	—
163	Dic₂₃	—	4	1	no	—
164	#541	—	23	1	no	—
165	#520	—	23	1	no	—
166	#484	—	23	1	no	—
167	#483	—	23	1	no	—
168	#479	—	23	1	no	—
169	#478	—	23	1	no	—
170	#462	—	23	1	no	—
171	#461	—	23	1	no	—
172	#452	—	23	1	no	—
173	#452	—	23	1	no	—
174	#441	—	23	1	no	—
175	#440	—	23	1	no	—
176	C₄×Q₁₆	—	23	1	no	—
177	C₄×SD₁₆	—	23	1	no	—
178	#433	D₂₃	1	1	yes	characteristic, complemented
179	C₂×C₂₃	—	4	1	no	—
180	C₂×C₂₃	#525	1	1	yes	characteristic
181	C₂×C₂₃	#577	1	1	yes	characteristic
182	C₂×C₂₃	#578	1	1	yes	characteristic
183	D₂₃	—	4	2	no	—
184	C₂×#41	—	23	1	no	—
185	C₂×Q₁₆	—	23	1	no	—
186	C₂×Q₁₆	—	23	1	no	—
187	C₂×SD₁₆	—	23	1	no	—
188	C₂×SD₁₆	—	23	1	no	—
189	C₄⋊Q₈	—	23	1	no	—
190	#126	—	46	1	no	—
191	#126	—	46	1	no	—
192	#125	—	23	1	no	—
193	#125	—	46	1	no	—
194	#125	—	46	1	no	—
195	C₂²⋊Q₈	—	23	1	no	—
196	C₂²⋊Q₈	—	23	1	no	—
197	C₄×Q₈	—	23	1	no	—
198	C₄×Q₈	—	23	1	no	—
199	C₄×Q₈	—	23	1	no	—
200	C₄×Q₈	—	23	1	no	—
201	C₄×D₄	—	23	1	no	—
202	C₄×D₄	—	23	1	no	—
203	C₄×D₄	—	46	1	no	—
204	C₄×D₄	—	46	1	no	—
205	#117	—	46	1	no	—
206	#117	C₂×D₂₃	1	1	yes	characteristic
207	C₂×C₄⋊C₄	—	23	1	no	—
208	C₂×C₄⋊C₄	—	46	1	no	—
209	#107	—	23	1	no	—
210	#107	—	23	1	no	—
211	#106	—	23	1	no	—
212	#106	—	23	1	no	—
213	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
214	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
215	#103	—	23	1	no	—
216	#103	—	23	1	no	—
217	#103	—	23	1	no	—
218	#103	—	23	1	no	—
219	#103	—	23	1	no	—
220	#103	—	23	1	no	—
221	#102	—	23	1	no	—
222	#102	—	23	1	no	—
223	C₂²⋊C₈	C₂×D₂₃	1	1	yes	characteristic
224	C₂²⋊C₈	C₂×D₂₃	1	1	yes	characteristic
225	#97	C₂×D₂₃	1	1	yes	characteristic
226	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
227	C₂₃	#3120	1	1	yes	characteristic, complemented, Sylow
228	#41	—	46	2	no	—
229	#41	—	46	2	no	—
230	C₂×Q₈	—	23	1	no	—
231	C₂×Q₈	—	23	1	no	—
232	C₂×Q₈	—	23	1	no	—
233	C₂×D₄	—	23	1	no	—
234	C₂×D₄	—	46	1	no	—
235	C₂²×C₄	—	46	1	no	—
236	C₂²×C₄	—	46	1	no	—
237	C₂²×C₄	—	46	1	no	—
238	C₂²×C₄	—	46	1	no	—
239	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
240	Q₁₆	—	46	2	no	—
241	Q₁₆	—	46	2	no	—
242	SD₁₆	—	46	2	no	—
243	SD₁₆	—	46	2	no	—
244	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
245	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
248	C₄⋊C₄	—	23	1	no	—
249	C₄⋊C₄	—	23	1	no	—
250	C₄⋊C₄	—	23	1	no	—
251	C₄⋊C₄	—	23	1	no	—
252	C₄⋊C₄	—	46	1	no	—
253	C₄⋊C₄	—	46	1	no	—
254	C₄⋊C₄	—	46	1	no	—
255	C₄⋊C₄	—	46	1	no	—
256	C₄⋊C₄	—	46	1	no	—
257	C₄⋊C₄	—	46	1	no	—
258	C₄⋊C₄	—	46	1	no	—
259	C₄⋊C₄	—	46	1	no	—
260	C₄⋊C₄	—	46	1	no	—
261	C₄⋊C₄	—	46	1	no	—
262	C₄⋊C₄	—	46	1	no	—
263	C₄⋊C₄	D₉₂	1	1	yes	characteristic
264	C₄⋊C₄	D₉₂	1	1	yes	characteristic
265	C₂²⋊C₄	—	46	1	no	—
266	C₂²⋊C₄	—	46	1	no	—
267	C₂²⋊C₄	—	46	1	no	—
268	C₂²⋊C₄	—	46	1	no	—
269	C₂²⋊C₄	D₉₂	1	1	yes	characteristic
270	C₂²⋊C₄	D₉₂	1	1	yes	characteristic
271	C₄²	—	46	1	no	—
272	C₄²	—	46	1	no	—
273	C₄²	—	46	1	no	—
274	C₄²	—	46	1	no	—
275	C₄²	C₂²×D₂₃	1	1	yes	characteristic
276	C₄²	C₂²×D₂₃	1	1	yes	characteristic
277	C₂³	—	46	1	no	—
278	C₂³	C₂×D₉₂	1	1	yes	characteristic
279	Q₈	—	23	2	no	—
280	Q₈	—	46	1	no	—
281	Q₈	—	46	1	no	—
282	Q₈	—	46	1	no	—
283	Q₈	—	46	1	no	—
284	Q₈	—	46	1	no	—
285	D₄	—	23	2	no	—
286	D₄	—	46	4	no	—
287	D₄	—	46	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	46	4	no	—
291	C₂×C₄	—	46	1	no	—
292	C₂×C₄	—	46	1	no	—
293	C₂×C₄	—	46	1	no	—
294	C₂×C₄	—	46	1	no	—
295	C₂×C₄	—	46	1	no	—
296	C₂×C₄	—	46	1	no	—
297	C₂×C₄	—	46	1	no	—
298	C₂×C₄	—	184	1	no	—
299	C₂×C₄	—	184	1	no	—
300	C₂×C₄	—	184	1	no	—
301	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
302	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
303	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
304	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
305	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	46	2	no	—
315	C₂²	—	92	1	no	—
316	C₂²	—	92	1	no	—
317	C₂²	C₂²×D₉₂	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	46	2	no	—
324	C₄	—	92	1	no	—
325	C₄	—	92	1	no	—
326	C₄	—	92	1	no	—
327	C₄	—	92	1	no	—
328	C₄	—	92	1	no	—
329	C₄	—	92	1	no	—
330	C₄	#41×D₂₃	1	1	yes	characteristic
331	C₄	#41×D₂₃	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	92	2	no	—
334	C₂	#13092929	1	1	yes	characteristic
335	C₂	#13093229	1	1	yes	characteristic
336	C₂	#13093234	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₉₂
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#433	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#67288	C₂²
hypercenter	#433	D₂₃
hypocenter	C₂₃	#3120
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

Complements

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

N	H ≅ G/N	type
1. #13093029	C₂	semidirect
3. #13092753	C₂	semidirect
4. #13092748	C₂	semidirect
5. #13092725	C₂	semidirect
6. #13092718	C₂	semidirect
7. #13092628	C₂	semidirect
8. #13092623	C₂	semidirect
11. #13092525	C₂	semidirect
12. #13092519	C₂	semidirect
13. C₄×Dic₉₂	C₂	semidirect
14. C₄×#67263	C₂	semidirect
15. C₂₃×#433	C₂	semidirect
178. #433	D₂₃	semidirect
227. C₂₃	#3120	semidirect

Maximal subgroups

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

subgroup	index
#13093029	2
#13092897	2
#13092753	2
#13092748	2
#13092725	2
#13092718	2
#13092628	2
#13092623	2
#13092581	2
#13092525	2
#13092519	2
C₄×Dic₉₂	2
C₄×#67263	2
C₂₃×#433	2
#3120	23