Subgroups — #26137332

Order 12032.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245697	C₂	1	2	yes	maximal, complemented
2	Dic₁₅₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241604	C₂	1	2	yes	maximal, complemented
4	D₃₀₀₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₇×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689555	—	2	2	no	—
7	Dic₇₅₂	C₂²	1	2	yes	—
8	#15688962	—	2	2	no	—
9	D₁₅₀₄	C₂²	1	2	yes	—
10	C₂×C₃₂×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×C₆₄	C₂²	1	1	yes	characteristic
12	C₄₇×C₆₄	C₂²	1	1	yes	characteristic
13	#13094046	—	4	2	no	—
14	Dic₃₇₆	—	2	2	no	—
15	#13093969	—	4	2	no	—
16	D₇₅₂	—	2	2	no	—
17	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
18	C₃₂×C₄₇	D₄	1	1	yes	characteristic
19	C₃₂×C₄₇	C₂³	1	1	yes	characteristic
20	#10625350	—	8	2	no	—
21	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
22	Dic₁₈₈	—	4	2	no	—
23	D₃₇₆	—	4	2	no	—
24	#10625325	—	8	2	no	—
25	C₁₆×C₄₇	D₈	1	1	yes	characteristic
26	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
28	C₂²⋊D₄₇	—	16	2	no	—
29	D₁₈₈	—	8	2	no	—
30	C₄×D₄₇	—	16	2	no	—
31	Dic₉₄	—	8	2	no	—
32	C₈×C₄₇	D₁₆	1	1	yes	characteristic
33	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
34	#13741	—	47	1	no	maximal, Sylow
35	C₂²×C₄₇	D₃₂	1	1	yes	characteristic
36	C₂×D₄₇	—	16	2	no	—
37	C₄×C₄₇	D₃₂	1	1	yes	characteristic
38	C₄×C₄₇	C₂×D₁₆	1	1	yes	characteristic
39	Dic₄₇	—	16	2	no	—
40	#2262	—	47	2	no	—
41	Q₁₂₈	—	47	1	no	—
42	SD₁₂₈	—	47	2	no	—
43	D₆₄	—	47	1	no	—
44	C₂×C₆₄	D₄₇	1	1	yes	characteristic, complemented
45	C₂×C₄₇	—	2	1	no	—
46	C₂×C₄₇	C₂×D₃₂	1	1	yes	characteristic
47	D₄₇	—	32	2	no	—
48	#508	—	94	2	no	—
49	Q₆₄	—	47	2	no	—
50	SD₆₄	—	94	2	no	—
51	D₃₂	—	47	2	no	—
52	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
53	C₆₄	C₂×D₄₇	1	1	yes	characteristic
54	C₆₄	C₂×D₄₇	1	1	yes	characteristic
55	C₄₇	#13741	1	1	yes	characteristic, complemented, Sylow
56	#135	—	188	2	no	—
57	Q₃₂	—	94	2	no	—
58	SD₃₂	—	188	2	no	—
59	D₁₆	—	94	2	no	—
60	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
61	C₃₂	D₁₈₈	1	1	yes	characteristic
62	C₃₂	C₂²×D₄₇	1	1	yes	characteristic
63	#41	—	376	2	no	—
64	Q₁₆	—	188	2	no	—
65	SD₁₆	—	376	2	no	—
66	D₈	—	188	2	no	—
67	C₂×C₈	D₃₇₆	1	1	yes	characteristic
68	C₁₆	D₃₇₆	1	1	yes	characteristic
69	C₁₆	C₂×D₁₈₈	1	1	yes	characteristic
70	Q₈	—	376	2	no	—
71	D₄	—	376	2	no	—
72	D₄	—	752	2	no	—
73	C₂×C₄	—	752	2	no	—
74	C₂×C₄	D₇₅₂	1	1	yes	characteristic
75	C₈	D₇₅₂	1	1	yes	characteristic
76	C₈	C₂×D₃₇₆	1	1	yes	characteristic
77	C₂²	—	752	2	no	—
78	C₂²	D₁₅₀₄	1	1	yes	characteristic
79	C₄	—	752	2	no	—
80	C₄	D₁₅₀₄	1	1	yes	characteristic
81	C₄	C₂×D₇₅₂	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	1504	2	no	—
84	C₂	C₂×D₁₅₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₅₀₄
commutator, cosocle	C₃₂×C₄₇	C₂³
Frattini	C₃₂	C₂²×D₄₇
Fitting	C₂×C₄₇×C₆₄	C₂
socle	C₂×C₄₇	C₂×D₃₂
hypercenter	C₂×C₆₄	D₄₇
hypocenter	C₄₇	#13741
LCS	G ▷ C₃₂×C₄₇ ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
1. #20245697	C₂	semidirect
2. Dic₁₅₀₄	C₂	semidirect
3. #20241604	C₂	semidirect
4. D₃₀₀₈	C₂	semidirect
5. C₂×C₄₇×C₆₄	C₂	semidirect
44. C₂×C₆₄	D₄₇	semidirect
55. C₄₇	#13741	semidirect

Maximal subgroups

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

subgroup	index
#20245697	2
Dic₁₅₀₄	2
#20241604	2
D₃₀₀₈	2
C₂×C₄₇×C₆₄	2
#13741	47