Subgroups — #15683027

Order 2944.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625080	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092928	C₂	1	1	yes	maximal, characteristic
3	C₂×#10625095	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092675	C₂	1	2	yes	maximal, complemented
5	#13092663	C₂	1	2	yes	maximal, complemented
6	#13092658	C₂	1	2	yes	maximal, complemented
7	#13092646	C₂	1	2	yes	maximal, complemented
8	#13092459	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092427	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092419	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×#67288	C₂²	1	1	yes	characteristic
14	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₂₃	—	2	4	no	—
16	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
20	#10625124	—	2	1	no	—
21	C₂×C₂²⋊Dic₂₃	—	2	1	no	—
22	#10625101	C₂²	1	1	yes	characteristic
23	#10625101	C₂²	1	1	yes	characteristic
24	#10625100	—	2	1	no	—
25	C₄×C₂²⋊D₂₃	—	2	1	no	—
26	C₄×C₂²⋊D₂₃	—	2	1	no	—
27	#10625097	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
29	#10625095	—	2	1	no	—
30	#10625095	—	2	1	no	—
31	#10625095	C₂²	1	2	yes	—
32	#10625095	C₂²	1	1	yes	characteristic
33	#10625095	C₂²	1	2	yes	—
34	#10625095	C₂²	1	1	yes	characteristic
35	C₂×Dic₂₃⋊C₄	—	2	1	no	—
36	#10625080	—	2	4	no	—
37	C₂×Dic₉₂	C₂²	1	2	yes	complemented
38	C₂×#67263	C₂²	1	2	yes	complemented
39	#10625003	C₂²	1	2	yes	complemented
40	#10625000	C₂²	1	2	yes	complemented
41	#10624999	C₂²	1	2	yes	complemented
42	#10624998	C₂²	1	2	yes	complemented
43	#10624998	C₂²	1	2	yes	complemented
44	#10624998	C₂²	1	2	yes	complemented
45	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	—	2	2	no	—
47	C₂²×C₄×C₂₃	—	2	2	no	—
48	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×C₂²⋊D₂₃	—	2	1	no	—
55	C₂×C₂²⋊D₂₃	—	2	1	no	—
56	C₂×C₂²⋊D₂₃	—	2	1	no	—
57	C₂²×Dic₂₃	—	2	1	no	—
58	C₂²×Dic₂₃	—	2	1	no	—
59	#67288	—	2	2	no	—
60	#67288	—	2	2	no	—
61	C₂×D₉₂	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₂₃	—	2	1	no	—
63	C₂×Dic₄₆	—	2	2	no	—
64	C₂×Dic₄₆	—	2	2	no	—
65	C₂×Dic₄₆	—	2	1	no	—
66	C₂×Dic₄₆	C₂³	1	2	yes	—
67	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
68	M₁₆×C₂₃	—	2	2	no	—
69	M₁₆×C₂₃	—	2	4	no	—
70	M₁₆×C₂₃	D₄	1	4	yes	complemented
71	C₂×C₈×C₂₃	—	2	4	no	—
72	C₂×C₈×C₂₃	C₂³	1	2	yes	—
73	C₂×C₈×C₂₃	C₂³	1	2	yes	—
74	C₂²⋊Dic₂₃	—	2	2	no	—
75	C₂²⋊Dic₂₃	—	2	2	no	—
76	C₂²⋊Dic₂₃	—	2	1	no	—
77	C₂²⋊Dic₂₃	—	2	1	no	—
78	C₂²⋊Dic₂₃	—	2	1	no	—
79	(C₂×D₂₃)⋊C₄	—	2	1	no	—
80	(C₂×D₂₃)⋊C₄	—	2	1	no	—
81	C₄⋊Dic₂₃	—	2	1	no	—
82	C₄⋊Dic₂₃	C₂³	1	2	yes	—
83	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
85	Dic₂₃⋊C₄	—	2	2	no	—
86	Dic₂₃⋊C₄	—	2	2	no	—
87	Dic₂₃⋊C₄	—	2	1	no	—
88	Dic₂₃⋊C₄	—	2	1	no	—
89	C₄×Dic₂₃	—	2	1	no	—
90	Dic₉₂	—	2	4	no	—
91	#67263	—	2	4	no	—
92	C₂³×C₂₃	—	2	2	no	—
93	C₂³×C₂₃	—	2	2	no	—
94	C₂³×C₂₃	—	4	1	no	—
95	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×D₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₂₃	#41	1	2	yes	—
111	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
112	C₂²⋊D₂₃	—	4	2	no	—
113	C₂²⋊D₂₃	—	4	2	no	—
114	C₂²⋊D₂₃	—	4	2	no	—
115	C₂×Dic₂₃	—	2	1	no	—
116	C₂×Dic₂₃	—	2	1	no	—
117	C₂×Dic₂₃	—	2	2	no	—
118	C₂×Dic₂₃	—	2	2	no	—
119	C₂×Dic₂₃	—	2	1	no	—
120	C₂×Dic₂₃	—	4	2	no	—
121	C₂×Dic₂₃	—	4	2	no	—
122	D₉₂	—	2	1	no	—
123	D₉₂	—	2	1	no	—
124	C₄×D₂₃	—	4	2	no	—
125	Dic₄₆	—	2	2	no	—
126	Dic₄₆	—	2	1	no	—
127	Dic₄₆	—	2	2	no	—
128	Dic₄₆	—	2	1	no	—
129	Dic₄₆	—	4	2	no	—
130	C₈×C₂₃	—	2	2	no	—
131	C₈×C₂₃	C₂×D₄	1	4	yes	—
132	#3056	—	23	1	no	maximal, Sylow
133	C₂²×C₂₃	—	2	1	no	—
134	C₂²×C₂₃	—	2	1	no	—
135	C₂²×C₂₃	—	2	1	no	—
136	C₂²×C₂₃	—	2	1	no	—
137	C₂²×C₂₃	—	2	1	no	—
138	C₂²×C₂₃	—	2	1	no	—
139	C₂²×C₂₃	—	2	1	no	—
140	C₂²×C₂₃	—	4	4	no	—
141	C₂²×C₂₃	#121	1	2	yes	—
142	C₂²×C₂₃	#137	1	2	yes	complemented
143	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₂₃	—	4	1	no	—
145	C₂×D₂₃	—	4	1	no	—
146	C₂×D₂₃	—	4	1	no	—
147	C₄×C₂₃	—	2	1	no	—
148	C₄×C₂₃	—	2	1	no	—
149	C₄×C₂₃	#121	1	2	yes	—
150	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
151	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
152	Dic₂₃	—	4	2	no	—
153	Dic₂₃	—	4	1	no	—
154	Dic₂₃	—	4	1	no	—
155	Dic₂₃	—	4	2	no	—
156	Dic₂₃	—	4	1	no	—
157	C₂×#137	—	23	1	no	—
158	C₂²×M₁₆	D₂₃	1	1	yes	characteristic, complemented
159	#525	—	23	1	no	—
160	C₂×C₂²⋊Q₈	—	23	1	no	—
161	#470	—	23	2	no	—
162	#468	—	23	2	no	—
163	#467	—	23	2	no	—
164	#465	—	23	2	no	—
165	#428	—	23	1	no	—
166	#419	—	23	1	no	—
167	#417	—	23	1	no	—
168	C₂×C₂₃	—	2	2	no	—
169	C₂×C₂₃	—	2	2	no	—
170	C₂×C₂₃	—	4	1	no	—
171	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
172	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
173	C₂×C₂₃	#575	1	1	yes	characteristic
174	D₂₃	—	8	1	no	—
175	C₂×#41	—	23	1	no	—
176	C₂²×Q₈	—	23	1	no	—
177	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
178	#137	—	46	4	no	—
179	C₂×Q₁₆	—	23	2	no	—
180	C₂×SD₁₆	—	23	2	no	—
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
183	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
184	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
185	C₂²×C₈	C₂×D₂₃	1	2	yes	—
186	#123	—	46	1	no	—
187	C₂²⋊Q₈	—	23	2	no	—
188	C₂²⋊Q₈	—	23	2	no	—
189	C₂²⋊Q₈	—	23	1	no	—
190	C₂²⋊Q₈	—	23	1	no	—
191	C₂²⋊Q₈	—	46	1	no	—
192	C₂²⋊Q₈	—	46	1	no	—
193	#121	—	23	1	no	—
194	#121	—	23	1	no	—
195	#120	—	46	1	no	—
196	C₄×D₄	—	46	1	no	—
197	C₄×D₄	—	46	1	no	—
198	#117	—	23	1	no	—
199	C₂×C₄⋊C₄	—	23	1	no	—
200	C₂×C₄⋊C₄	—	46	1	no	—
201	C₂×C₂²⋊C₄	—	46	1	no	—
202	#107	—	23	2	no	—
203	#106	—	23	2	no	—
204	#103	—	23	2	no	—
205	#103	—	23	2	no	—
206	#103	—	23	2	no	—
207	#102	—	23	2	no	—
208	C₂₃	#3056	1	1	yes	characteristic, complemented, Sylow
209	C₂⁴	D₉₂	1	1	yes	characteristic
210	#41	—	46	2	no	—
211	#41	—	46	2	no	—
212	C₂×Q₈	—	23	2	no	—
213	C₂×Q₈	—	23	1	no	—
214	C₂×Q₈	—	46	1	no	—
215	C₂×Q₈	—	46	2	no	—
216	C₂×Q₈	—	46	2	no	—
217	C₂×D₄	—	23	1	no	—
218	C₂×D₄	—	46	1	no	—
219	C₂×D₄	—	46	1	no	—
220	C₂×D₄	—	46	1	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	46	1	no	—
224	C₂²×C₄	—	46	1	no	—
225	C₂²×C₄	—	46	1	no	—
226	C₂²×C₄	D₉₂	1	1	yes	characteristic
227	C₂²×C₄	D₉₂	1	1	yes	characteristic
228	C₂²×C₄	D₉₂	1	1	yes	characteristic
229	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
232	Q₁₆	—	46	4	no	—
233	SD₁₆	—	46	4	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	C₂²⋊D₂₃	1	4	yes	complemented
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	C₂²×D₂₃	1	2	yes	—
239	C₂×C₈	C₂²×D₂₃	1	2	yes	—
240	C₄⋊C₄	—	23	2	no	—
241	C₄⋊C₄	—	23	1	no	—
242	C₄⋊C₄	—	23	1	no	—
243	C₄⋊C₄	—	46	1	no	—
244	C₄⋊C₄	—	46	2	no	—
245	C₄⋊C₄	—	46	2	no	—
246	C₄⋊C₄	—	46	1	no	—
247	C₄⋊C₄	—	46	1	no	—
248	C₂²⋊C₄	—	46	2	no	—
249	C₂²⋊C₄	—	46	2	no	—
250	C₂²⋊C₄	—	46	1	no	—
251	C₂²⋊C₄	—	46	1	no	—
252	C₂²⋊C₄	—	46	1	no	—
253	C₂²⋊C₄	—	46	1	no	—
254	C₂²⋊C₄	—	46	1	no	—
255	C₄²	—	46	1	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	4	1	no	—
259	C₂³	—	46	1	no	—
260	C₂³	C₂×D₉₂	1	1	yes	characteristic
261	C₂³	C₂×D₉₂	1	1	yes	characteristic
262	C₂³	C₂×D₉₂	1	1	yes	characteristic
263	Q₈	—	46	2	no	—
264	Q₈	—	46	1	no	—
265	Q₈	—	46	2	no	—
266	Q₈	—	46	1	no	—
267	Q₈	—	92	2	no	—
268	D₄	—	46	1	no	—
269	D₄	—	46	1	no	—
270	D₄	—	92	2	no	—
271	D₄	—	92	2	no	—
272	D₄	—	92	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	46	1	no	—
281	C₂×C₄	—	46	1	no	—
282	C₂×C₄	—	46	2	no	—
283	C₂×C₄	—	46	1	no	—
284	C₂×C₄	—	46	2	no	—
285	C₂×C₄	—	92	2	no	—
286	C₂×C₄	—	92	2	no	—
287	C₂×C₄	—	92	2	no	—
288	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
289	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
290	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
291	C₂×C₄	#67288	1	2	yes	—
292	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
293	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	C₂×C₂²⋊D₂₃	1	4	yes	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	92	1	no	—
305	C₂²	—	92	1	no	—
306	C₂²	—	92	1	no	—
307	C₂²	#10625080	1	2	yes	complemented
308	C₂²	#10625101	1	2	yes	—
309	C₂²	C₂²×D₉₂	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	92	1	no	—
313	C₄	—	92	2	no	—
314	C₄	—	92	1	no	—
315	C₄	—	92	2	no	—
316	C₄	—	92	1	no	—
317	C₄	#10625101	1	2	yes	—
318	C₄	C₂×#67288	1	1	yes	characteristic
319	C₄	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	2	2	no	—
322	C₂	—	4	1	no	—
323	C₂	—	184	1	no	—
324	C₂	C₂×#10625101	1	1	yes	characteristic
325	C₂	C₂×#10625080	1	1	yes	characteristic
326	C₂	#13093217	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₉₂
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂²×M₁₆×C₂₃	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	#13092928	C₂
hypercenter	C₂²×M₁₆	D₂₃
hypocenter	C₂₃	#3056
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#10625080	C₂	semidirect
3. C₂×#10625095	C₂	semidirect
4. #13092675	C₂	semidirect
5. #13092663	C₂	semidirect
6. #13092658	C₂	semidirect
7. #13092646	C₂	semidirect
8. #13092459	C₂	semidirect
9. #13092427	C₂	semidirect
10. #13092419	C₂	semidirect
11. C₂²×M₁₆×C₂₃	C₂	semidirect
37. C₂×Dic₉₂	C₂²	semidirect
38. C₂×#67263	C₂²	semidirect
39. #10625003	C₂²	semidirect
40. #10625000	C₂²	semidirect
41. #10624999	C₂²	semidirect
42. #10624998	C₂²	semidirect
43. #10624998	C₂²	semidirect
44. #10624998	C₂²	semidirect
70. M₁₆×C₂₃	D₄	semidirect
142. C₂²×C₂₃	#137	semidirect
158. C₂²×M₁₆	D₂₃	semidirect
208. C₂₃	#3056	semidirect
236. M₁₆	C₂²⋊D₂₃	semidirect
307. C₂²	#10625080	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625080	2
#13092928	2
C₂×#10625095	2
#13092675	2
#13092663	2
#13092658	2
#13092646	2
#13092459	2
#13092427	2
#13092419	2
C₂²×M₁₆×C₂₃	2
#3056	23