Subgroups — #26130990

Order 12032.

398 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245638	C₂	1	4	yes	maximal, complemented
2	#20243003	C₂	1	2	yes	maximal, complemented
3	C₈×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₇₅₂	—	2	8	no	—
5	#15689500	—	2	4	no	—
6	#15689500	C₂²	1	4	yes	—
7	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
8	C₈²×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×D₃₇₆	—	2	8	no	—
10	C₂×D₃₇₆	—	4	8	no	—
11	#13094021	—	2	4	no	—
12	D₇₅₂	—	4	16	no	—
13	C₂×C₁₆×C₄₇	D₄	1	4	yes	—
14	C₄×C₈×C₄₇	D₄	1	2	yes	—
15	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
16	C₂×D₁₈₈	—	4	4	no	—
17	C₂×D₁₈₈	—	4	8	no	—
18	C₂×C₈×C₄₇	D₈	1	4	yes	—
19	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
20	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
21	D₃₇₆	—	4	16	no	—
22	D₃₇₆	—	8	16	no	—
23	C₁₆×C₄₇	D₈	1	8	yes	complemented
24	C₂²×D₄₇	—	8	4	no	—
25	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
26	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
27	D₁₈₈	—	8	8	no	—
28	D₁₈₈	—	8	16	no	—
29	C₈×C₄₇	D₁₆	1	8	yes	complemented
30	C₈×C₄₇	C₂×D₈	1	4	yes	—
31	#12515	—	47	1	no	maximal, Sylow
32	C₂²×C₄₇	#493	1	1	yes	characteristic
33	C₂×D₄₇	—	16	4	no	—
34	C₂×D₄₇	—	16	8	no	—
35	C₄×C₄₇	#493	1	2	yes	—
36	C₄×C₄₇	C₂×D₁₆	1	4	yes	—
37	#2246	—	47	4	no	—
38	#1712	—	47	2	no	—
39	C₈×C₁₆	D₄₇	1	1	yes	characteristic, complemented
40	C₂×C₄₇	#1712	1	1	yes	characteristic
41	C₂×C₄₇	#2246	1	2	yes	—
42	D₄₇	—	32	4	no	—
43	C₂×D₁₆	—	94	8	no	—
44	#493	—	47	4	no	—
45	#493	—	94	4	no	—
46	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
47	C₈²	C₂×D₄₇	1	1	yes	characteristic
48	C₄₇	#12515	1	1	yes	characteristic, complemented, Sylow
49	C₂×D₈	—	94	8	no	—
50	C₂×D₈	—	188	8	no	—
51	#127	—	94	4	no	—
52	D₁₆	—	188	16	no	—
53	C₂×C₁₆	D₁₈₈	1	4	yes	—
54	C₄×C₈	D₁₈₈	1	2	yes	—
55	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
56	C₂×D₄	—	188	4	no	—
57	C₂×D₄	—	188	8	no	—
58	D₈	—	376	16	no	—
59	D₈	—	188	16	no	—
60	C₂×C₈	D₃₇₆	1	4	yes	—
61	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
62	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
63	C₁₆	D₃₇₆	1	8	yes	complemented
64	C₂³	—	376	4	no	—
65	D₄	—	376	8	no	—
66	D₄	—	376	16	no	—
67	C₂×C₄	#13094021	1	1	yes	characteristic
68	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
69	C₈	D₇₅₂	1	8	yes	complemented
70	C₈	C₂×D₃₇₆	1	4	yes	—
71	C₂²	—	752	4	no	—
72	C₂²	—	752	8	no	—
73	C₂²	#15689500	1	1	yes	characteristic
74	C₄	#15689500	1	2	yes	—
75	C₄	C₂×D₇₅₂	1	4	yes	—
76	C₂	—	1504	4	no	—
77	C₂	#20243003	1	1	yes	characteristic
78	C₂	#20245638	1	2	yes	—

	subgroup	G/H
center	C₂²	#15689500
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₈×C₁₆×C₄₇	C₂
socle	C₂²×C₄₇	#493
hypercenter	C₈×C₁₆	D₄₇
hypocenter	C₄₇	#12515
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
1. #20245638	C₂	semidirect
2. #20243003	C₂	semidirect
3. C₈×C₁₆×C₄₇	C₂	semidirect
23. C₁₆×C₄₇	D₈	semidirect
29. C₈×C₄₇	D₁₆	semidirect
39. C₈×C₁₆	D₄₇	semidirect
48. C₄₇	#12515	semidirect
63. C₁₆	D₃₇₆	semidirect
69. C₈	D₇₅₂	semidirect

Maximal subgroups

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

subgroup	index
#20245638	2
#20243003	2
C₈×C₁₆×C₄₇	2
#12515	47