Subgroups — #26137305

Order 12032.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245663	C₂	1	2	yes	maximal
2	#20241590	C₂	1	4	yes	maximal
3	C₄×C₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
4	#15689528	—	2	2	no	—
5	#15688949	—	2	2	no	—
6	#15688949	C₂²	1	4	yes	—
7	C₂×C₃₂×C₄₇	C₂²	1	2	yes	—
8	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
9	#13094024	—	4	2	no	—
10	#13093958	—	2	4	no	—
11	#13093958	—	4	2	no	—
12	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
14	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
15	C₃₂×C₄₇	Q₈	1	4	yes	complemented
16	C₂×Dic₉₄	—	8	2	no	—
17	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
18	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
19	C₄²×C₄₇	D₈	1	1	yes	characteristic
20	C₄⋊Dic₄₇	—	4	4	no	—
21	C₄⋊Dic₄₇	—	8	2	no	—
22	C₁₆×C₄₇	Q₁₆	1	2	yes	—
23	C₁₆×C₄₇	C₂×Q₈	1	2	yes	—
24	C₂×C₄×C₄₇	D₁₆	1	2	yes	—
25	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
26	C₂×Dic₄₇	—	8	4	no	—
27	Dic₉₄	—	16	4	no	—
28	C₈×C₄₇	Q₃₂	1	2	yes	—
29	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
30	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
31	#13733	—	47	1	no	maximal, Sylow
32	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
33	C₄×C₄₇	SD₆₄	1	4	yes	—
34	C₄×C₄₇	#500	1	1	yes	characteristic
35	C₄×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
36	Dic₄₇	—	16	4	no	—
37	#2252	—	47	2	no	—
38	#1424	—	47	4	no	—
39	C₄×C₃₂	D₄₇	1	1	yes	characteristic
40	C₂×C₄₇	#2252	1	1	yes	characteristic
41	C₂×C₄₇	C₂×SD₆₄	1	2	yes	—
42	#500	—	94	2	no	—
43	C₂×C₃₂	C₂×D₄₇	1	2	yes	—
44	#366	—	47	4	no	—
45	#366	—	94	2	no	—
46	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
47	C₄₇	#13733	1	1	yes	characteristic, complemented, Sylow
48	C₄⋊Q₈	—	188	2	no	—
49	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
50	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
51	#107	—	94	4	no	—
52	#107	—	188	2	no	—
53	C₄×C₈	D₁₈₈	1	1	yes	characteristic
54	C₃₂	Dic₉₄	1	4	yes	complemented
55	C₂×Q₈	—	376	2	no	—
56	C₂×C₈	D₃₇₆	1	1	yes	characteristic
57	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
58	C₄⋊C₄	—	376	2	no	—
59	C₄⋊C₄	—	188	4	no	—
60	C₄²	D₃₇₆	1	1	yes	characteristic
61	C₁₆	Dic₁₈₈	1	2	yes	—
62	C₁₆	C₂×Dic₉₄	1	2	yes	—
63	Q₈	—	752	4	no	—
64	C₂×C₄	—	376	4	no	—
65	C₂×C₄	D₇₅₂	1	2	yes	—
66	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
67	C₈	Dic₃₇₆	1	2	yes	—
68	C₈	#13094024	1	1	yes	characteristic
69	C₈	C₂×Dic₁₈₈	1	1	yes	characteristic
70	C₂²	C₂×D₇₅₂	1	1	yes	characteristic
71	C₄	—	752	4	no	—
72	C₄	#15688962	1	4	yes	—
73	C₄	#15689528	1	1	yes	characteristic
74	C₄	C₂×Dic₃₇₆	1	1	yes	characteristic
75	C₂	#20245663	1	1	yes	characteristic
76	C₂	C₂×#15688962	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₇₅₂
commutator, cosocle	C₂×C₁₆×C₄₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₄×C₃₂×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	C₂×D₁₆
hypercenter	C₄×C₃₂	D₄₇
hypocenter	C₄₇	#13733
LCS	G ▷ C₂×C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₄₇	Q₈	semidirect
47. C₄₇	#13733	semidirect
54. C₃₂	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20245663	2
#20241590	2
C₄×C₃₂×C₄₇	2
#13733	47