Subgroups — #26104172

Order 12032.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₅₂	C₂	1	1	yes	maximal, characteristic
2	#20241592	C₂	1	1	yes	maximal, characteristic
3	C₄₇×M₁₂₈	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₇₆	—	2	1	no	—
5	Dic₇₅₂	—	2	1	no	—
6	Dic₇₅₂	C₄	1	2	yes	—
7	#15688953	—	2	1	no	—
8	C₂×C₃₂×C₄₇	C₂²	1	1	yes	characteristic
9	C₄₇×C₆₄	—	2	1	no	—
10	C₂×Dic₁₈₈	—	4	1	no	—
11	Dic₃₇₆	—	2	2	no	—
12	Dic₃₇₆	—	4	1	no	—
13	#13093960	—	4	1	no	—
14	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
15	C₃₂×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₄₇	D₄	1	1	yes	characteristic
17	C₂×Dic₉₄	—	8	1	no	—
18	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
19	#10625329	—	8	1	no	—
20	Dic₁₈₈	—	4	2	no	—
21	Dic₁₈₈	—	8	1	no	—
22	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₄₇	SD₁₆	1	1	yes	characteristic
24	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
25	C₂×Dic₄₇	—	16	1	no	—
26	Dic₉₄	—	8	2	no	—
27	Dic₉₄	—	16	1	no	—
28	C₈×C₄₇	#102	1	1	yes	characteristic
29	C₈×C₄₇	SD₃₂	1	1	yes	characteristic
30	C₄₇⋊C₈	—	16	1	no	—
31	#7542	—	47	1	no	maximal, Sylow
32	C₂²×C₄₇	SD₆₄	1	1	yes	characteristic
33	C₄×C₄₇	#357	1	1	yes	characteristic
34	C₄×C₄₇	D₃₂	1	1	yes	characteristic
35	Dic₄₇	—	16	2	no	—
36	C₂×Q₆₄	—	47	1	no	—
37	M₁₂₈	D₄₇	1	1	yes	characteristic
38	#1425	—	47	1	no	—
39	C₂×C₄₇	—	2	1	no	—
40	C₂×C₄₇	#1415	1	1	yes	characteristic
41	C₂×Q₃₂	—	94	1	no	—
42	Q₆₄	—	47	2	no	—
43	Q₆₄	—	94	1	no	—
44	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
45	#368	—	94	1	no	—
46	C₆₄	—	2	1	no	—
47	C₄₇	#7542	1	1	yes	characteristic, complemented, Sylow
48	C₂×Q₁₆	—	188	1	no	—
49	Q₃₂	—	94	2	no	—
50	Q₃₂	—	188	1	no	—
51	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
52	#108	—	188	1	no	—
53	C₃₂	C₄×D₄₇	1	1	yes	characteristic
54	C₃₂	C₂²⋊D₄₇	1	1	yes	characteristic
55	C₂×Q₈	—	376	1	no	—
56	Q₁₆	—	376	1	no	—
57	Q₁₆	—	188	2	no	—
58	M₁₆	—	376	1	no	—
59	C₂×C₈	D₃₇₆	1	1	yes	characteristic
60	C₁₆	#10625325	1	1	yes	characteristic
61	C₁₆	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
62	Q₈	—	376	2	no	—
63	Q₈	—	752	1	no	—
64	C₂×C₄	—	752	1	no	—
65	C₂×C₄	D₇₅₂	1	1	yes	characteristic
66	C₈	—	752	1	no	—
67	C₈	#13093946	1	1	yes	characteristic
68	C₈	#13093969	1	1	yes	characteristic
69	C₂²	#15688962	1	1	yes	characteristic
70	C₄	—	752	2	no	—
71	C₄	#15688925	1	1	yes	characteristic
72	C₄	D₁₅₀₄	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	#20241567	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20241567
commutator	C₃₂×C₄₇	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₄₇
Fitting	C₄₇×M₁₂₈	C₂
socle	C₂×C₄₇	#1415
prime radical	C₂²×C₄₇	SD₆₄
cosocle	C₂×C₃₂×C₄₇	C₂²
hypercenter	M₁₂₈	D₄₇
hypocenter	C₄₇	#7542
LCS	G ▷ C₃₂×C₄₇ ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
47. C₄₇	#7542	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₅₂	2
#20241592	2
C₄₇×M₁₂₈	2
#7542	47