Subgroups — #26371731

Order 12032.

858 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689561	C₂	1	2	yes	maximal, complemented
2	C₂×#15689555	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×D₇₅₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#20253874	C₂	1	2	yes	maximal, complemented
5	#20245654	C₂	1	4	yes	maximal, complemented
6	#20245649	C₂	1	2	yes	maximal, complemented
7	#20245638	C₂	1	2	yes	maximal, complemented
8	C₄₇×#2108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094052	—	2	2	no	—
10	C₂×#13094046	C₂²	1	2	yes	—
11	C₂²×D₃₇₆	C₂²	1	2	yes	—
12	#15689692	—	2	2	no	—
13	#15689561	—	2	8	no	—
14	#15689555	—	2	4	no	—
15	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic, complemented
16	C₂×#13093969	C₂²	1	4	yes	complemented
17	C₂×#13093969	C₂²	1	2	yes	complemented
18	C₂×D₇₅₂	—	2	4	no	—
19	C₂×D₇₅₂	C₂²	1	2	yes	—
20	C₂×D₇₅₂	C₂²	1	2	yes	—
21	C₂×D₇₅₂	C₂²	1	4	yes	complemented
22	C₂×D₇₅₂	C₂²	1	1	yes	characteristic, complemented
23	#15689512	—	2	4	no	—
24	#15689507	C₂²	1	4	yes	complemented
25	#15689500	C₂²	1	4	yes	complemented
26	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#405	C₂²	1	1	yes	characteristic
29	C₄₇×#346	C₂²	1	2	yes	complemented
30	C₄×C₁₆×C₄₇	C₂²	1	2	yes	complemented
31	C₂×#10625350	—	2	2	no	—
32	C₂²×D₁₈₈	—	2	2	no	—
33	#13094052	—	4	8	no	—
34	#13094046	—	2	4	no	—
35	#13094046	—	2	4	no	—
36	C₂×Dic₁₈₈	C₂³	1	2	yes	—
37	C₂×#10625325	—	2	4	no	—
38	C₂×#10625325	—	2	2	no	—
39	C₂×D₃₇₆	—	2	4	no	—
40	C₂×D₃₇₆	—	2	4	no	—
41	C₂×D₃₇₆	—	2	4	no	—
42	C₂×D₃₇₆	—	2	2	no	—
43	C₂×D₃₇₆	C₂³	1	2	yes	—
44	C₂×D₃₇₆	C₂³	1	4	yes	—
45	#13094021	—	2	4	no	—
46	#13094012	—	2	4	no	—
47	#13093999	—	4	4	no	—
48	#13093994	—	4	4	no	—
49	Dic₃₇₆	—	2	2	no	—
50	#13093969	—	2	4	no	—
51	#13093969	—	2	8	no	—
52	D₇₅₂	—	2	4	no	—
53	D₇₅₂	—	2	4	no	—
54	D₇₅₂	—	2	2	no	—
55	D₇₅₂	—	2	8	no	—
56	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
58	#117×C₄₇	D₄	1	1	yes	characteristic
59	M₃₂×C₄₇	D₄	1	4	yes	complemented
60	C₂×C₁₆×C₄₇	D₄	1	4	yes	complemented
61	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
64	#97×C₄₇	D₄	1	2	yes	—
65	C₄×C₈×C₄₇	C₂³	1	2	yes	—
66	C₂³×D₄₇	—	4	2	no	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₄₇	—	4	2	no	—
69	#10625350	—	4	4	no	—
70	#10625350	—	4	4	no	—
71	C₂×D₁₈₈	—	2	4	no	—
72	C₂×D₁₈₈	—	2	2	no	—
73	C₂×D₁₈₈	—	4	4	no	—
74	C₂×D₁₈₈	—	4	4	no	—
75	C₂×D₁₈₈	—	4	2	no	—
76	C₂×D₁₈₈	—	4	8	no	—
77	C₂×C₄×D₄₇	—	4	2	no	—
78	C₂×Dic₉₄	—	2	2	no	—
79	M₁₆×C₄₇	D₈	1	4	yes	—
80	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₄₇	D₈	1	2	yes	—
86	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
87	C₄²×C₄₇	C₂×D₄	1	2	yes	—
88	(C₂×D₄₇)⋊C₄	—	4	8	no	—
89	Dic₁₈₈	—	2	2	no	—
90	Dic₁₈₈	—	2	2	no	—
91	D₃₇₆	—	2	4	no	—
92	D₃₇₆	—	2	2	no	—
93	D₃₇₆	—	2	4	no	—
94	D₃₇₆	—	2	2	no	—
95	D₃₇₆	—	4	4	no	—
96	D₃₇₆	—	4	8	no	—
97	#10625325	—	4	4	no	—
98	#10625325	—	4	8	no	—
99	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
100	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
101	C₁₆×C₄₇	C₂×D₄	1	4	yes	—
102	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
103	C₂²×D₄₇	—	4	4	no	—
104	C₂²×D₄₇	—	4	2	no	—
105	C₂²×D₄₇	—	8	4	no	—
106	C₂²×D₄₇	—	16	4	no	—
107	C₂×C₄×C₄₇	#127	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
109	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	C₂²⋊D₄₇	—	8	4	no	—
113	C₂×Dic₄₇	—	4	2	no	—
114	D₁₈₈	—	4	2	no	—
115	D₁₈₈	—	4	4	no	—
116	D₁₈₈	—	4	4	no	—
117	D₁₈₈	—	4	2	no	—
118	D₁₈₈	—	8	4	no	—
119	D₁₈₈	—	16	8	no	—
120	C₄×D₄₇	—	8	4	no	—
121	Dic₉₄	—	4	2	no	—
122	Dic₉₄	—	4	2	no	—
123	C₈×C₄₇	#127	1	2	yes	—
124	C₈×C₄₇	C₂×D₈	1	4	yes	—
125	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
127	#33720	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	#493	1	2	yes	—
130	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
131	C₂×D₄₇	—	8	2	no	—
132	C₂×D₄₇	—	8	4	no	—
133	C₂×D₄₇	—	8	4	no	—
134	C₂×D₄₇	—	8	8	no	—
135	C₂×D₄₇	—	16	8	no	—
136	C₄×C₄₇	—	2	4	no	—
137	C₄×C₄₇	#493	1	2	yes	—
138	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
139	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
140	Dic₄₇	—	8	2	no	—
141	C₂×#509	—	47	2	no	—
142	C₂×#508	—	47	1	no	—
143	C₂²×D₁₆	—	47	1	no	—
144	#3155	—	47	2	no	—
145	#2250	—	47	4	no	—
146	#2249	—	47	2	no	—
147	#2246	—	47	2	no	—
148	#2108	D₄₇	1	1	yes	characteristic, complemented
149	C₂×C₄₇	—	2	2	no	—
150	C₂×C₄₇	C₂×#493	1	1	yes	characteristic
151	C₂×C₄₇	#3415	1	2	yes	—
152	D₄₇	—	16	2	no	—
153	D₄₇	—	16	4	no	—
154	C₂×AGL(1,ℤ/8)	—	94	2	no	—
155	C₂×#135	—	47	2	no	—
156	C₂²×D₈	—	47	2	no	—
157	#535	—	94	2	no	—
158	#509	—	94	8	no	—
159	#508	—	94	4	no	—
160	C₂×Q₃₂	—	47	1	no	—
161	C₂×SD₃₂	—	47	2	no	—
162	C₂×SD₃₂	—	47	4	no	—
163	C₂×D₁₆	—	47	2	no	—
164	C₂×D₁₆	—	47	2	no	—
165	C₂×D₁₆	—	47	1	no	—
166	C₂×D₁₆	—	47	4	no	—
167	C₂×D₁₆	—	94	4	no	—
168	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
169	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
170	#496	—	94	4	no	—
171	#495	—	47	4	no	—
172	#493	—	47	4	no	—
173	#405	C₂×D₄₇	1	1	yes	characteristic
174	#346	C₂×D₄₇	1	2	yes	complemented
175	C₄×C₁₆	C₂×D₄₇	1	2	yes	complemented
176	C₄₇	#33720	1	1	yes	characteristic, complemented, Sylow
177	C₂×#41	—	94	2	no	—
178	C₂²×D₄	—	94	2	no	—
179	AGL(1,ℤ/8)	—	188	8	no	—
180	#135	—	94	4	no	—
181	#135	—	94	4	no	—
182	C₂×Q₁₆	—	47	2	no	—
183	C₂×SD₁₆	—	94	2	no	—
184	C₂×SD₁₆	—	94	4	no	—
185	C₂×D₈	—	47	4	no	—
186	C₂×D₈	—	47	2	no	—
187	C₂×D₈	—	94	4	no	—
188	C₂×D₈	—	94	4	no	—
189	C₂×D₈	—	94	2	no	—
190	C₂×D₈	—	94	4	no	—
191	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
193	#127	—	94	4	no	—
194	#124	—	94	4	no	—
195	#121	—	188	4	no	—
196	#120	—	188	4	no	—
197	#117	D₁₈₈	1	1	yes	characteristic
198	Q₃₂	—	94	2	no	—
199	SD₃₂	—	94	4	no	—
200	SD₃₂	—	94	8	no	—
201	D₁₆	—	94	4	no	—
202	D₁₆	—	94	4	no	—
203	D₁₆	—	94	2	no	—
204	D₁₆	—	94	8	no	—
205	M₃₂	D₁₈₈	1	4	yes	complemented
206	C₂×C₁₆	D₁₈₈	1	4	yes	complemented
207	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
208	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
209	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
210	#97	D₁₈₈	1	2	yes	—
211	C₄×C₈	C₂²×D₄₇	1	2	yes	—
212	C₂⁴	—	188	2	no	—
213	#41	—	188	4	no	—
214	#41	—	188	4	no	—
215	C₂×Q₈	—	94	2	no	—
216	C₂×D₄	—	94	4	no	—
217	C₂×D₄	—	94	2	no	—
218	C₂×D₄	—	188	2	no	—
219	C₂×D₄	—	188	4	no	—
220	C₂×D₄	—	188	2	no	—
221	C₂×D₄	—	188	4	no	—
222	C₂×D₄	—	188	8	no	—
223	C₂²×C₄	—	188	2	no	—
224	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
225	Q₁₆	—	94	2	no	—
226	Q₁₆	—	94	2	no	—
227	SD₁₆	—	188	4	no	—
228	SD₁₆	—	188	8	no	—
229	D₈	—	94	4	no	—
230	D₈	—	94	2	no	—
231	D₈	—	94	4	no	—
232	D₈	—	94	2	no	—
233	D₈	—	188	4	no	—
234	D₈	—	188	8	no	—
235	M₁₆	D₃₇₆	1	4	yes	—
236	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
237	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
238	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
239	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
240	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
241	C₄⋊C₄	D₃₇₆	1	2	yes	—
242	C₂²⋊C₄	—	188	8	no	—
243	C₂²⋊C₄	D₃₇₆	1	2	yes	—
244	C₄²	C₂×D₁₈₈	1	2	yes	—
245	C₁₆	C₂×D₁₈₈	1	2	yes	—
246	C₁₆	C₂×D₁₈₈	1	2	yes	—
247	C₁₆	C₂×D₁₈₈	1	4	yes	—
248	C₂³	—	376	4	no	—
249	C₂³	—	188	4	no	—
250	C₂³	—	188	2	no	—
251	C₂³	—	752	4	no	—
252	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
253	Q₈	—	188	2	no	—
254	Q₈	—	188	2	no	—
255	D₄	—	376	4	no	—
256	D₄	—	376	4	no	—
257	D₄	—	188	2	no	—
258	D₄	—	188	4	no	—
259	D₄	—	188	4	no	—
260	D₄	—	188	2	no	—
261	D₄	—	752	8	no	—
262	C₂×C₄	—	376	4	no	—
263	C₂×C₄	—	188	2	no	—
264	C₂×C₄	#13094021	1	2	yes	—
265	C₂×C₄	C₂×D₃₇₆	1	4	yes	—
266	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
267	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
268	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
269	C₈	#13094021	1	2	yes	—
270	C₈	C₂×D₃₇₆	1	4	yes	—
271	C₈	C₂²×D₁₈₈	1	1	yes	characteristic
272	C₈	C₂²×D₁₈₈	1	1	yes	characteristic
273	C₂²	—	2	2	no	—
274	C₂²	—	376	4	no	—
275	C₂²	—	376	8	no	—
276	C₂²	—	376	2	no	—
277	C₂²	—	376	4	no	—
278	C₂²	—	752	8	no	—
279	C₂²	#15689500	1	2	yes	—
280	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
281	C₄	—	2	4	no	—
282	C₄	—	376	2	no	—
283	C₄	#15689500	1	2	yes	—
284	C₄	C₂×#13094021	1	1	yes	characteristic
285	C₄	C₂²×D₃₇₆	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	752	4	no	—
288	C₂	—	752	2	no	—
289	C₂	C₂×#15689500	1	1	yes	characteristic
290	C₂	#20256529	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2108	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#2108	D₄₇
hypocenter	C₄₇	#33720
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
1. C₂×#15689561	C₂	semidirect
2. C₂×#15689555	C₂	semidirect
3. C₂²×D₇₅₂	C₂	semidirect
4. #20253874	C₂	semidirect
5. #20245654	C₂	semidirect
6. #20245649	C₂	semidirect
7. #20245638	C₂	semidirect
8. C₄₇×#2108	C₂	semidirect
15. C₂×Dic₃₇₆	C₂²	semidirect
16. C₂×#13093969	C₂²	semidirect
17. C₂×#13093969	C₂²	semidirect
21. C₂×D₇₅₂	C₂²	semidirect
22. C₂×D₇₅₂	C₂²	semidirect
24. #15689507	C₂²	semidirect
25. #15689500	C₂²	semidirect
29. C₄₇×#346	C₂²	semidirect
30. C₄×C₁₆×C₄₇	C₂²	semidirect
59. M₃₂×C₄₇	D₄	semidirect
60. C₂×C₁₆×C₄₇	D₄	semidirect
148. #2108	D₄₇	semidirect
174. #346	C₂×D₄₇	semidirect
175. C₄×C₁₆	C₂×D₄₇	semidirect
176. C₄₇	#33720	semidirect
205. M₃₂	D₁₈₈	semidirect
206. C₂×C₁₆	D₁₈₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689561	2
C₂×#15689555	2
C₂²×D₇₅₂	2
#20253874	2
#20245654	2
#20245649	2
#20245638	2
C₄₇×#2108	2
#33720	47