Subgroups — #26104166

Order 12032.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245697	C₂	1	1	yes	maximal, characteristic
2	#20241590	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689555	—	2	1	no	—
5	Dic₇₅₂	C₄	1	1	yes	characteristic, complemented
6	#15688962	—	2	1	no	—
7	D₁₅₀₄	C₄	1	1	yes	characteristic, complemented
8	#15688949	—	2	1	no	—
9	C₂×C₃₂×C₄₇	C₂²	1	1	yes	characteristic
10	C₄₇×C₆₄	—	2	1	no	—
11	#13094046	—	4	1	no	—
12	Dic₃₇₆	—	2	1	no	—
13	#13093969	—	4	1	no	—
14	D₇₅₂	—	2	1	no	—
15	#13093958	—	4	1	no	—
16	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
17	C₃₂×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₄₇	D₄	1	1	yes	characteristic
19	#10625350	—	8	1	no	—
20	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
21	C₄⋊Dic₄₇	—	8	1	no	—
22	Dic₁₈₈	—	4	1	no	—
23	D₃₇₆	—	4	1	no	—
24	#10625325	—	8	1	no	—
25	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₄₇	SD₁₆	1	1	yes	characteristic
27	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
28	C₂²⋊D₄₇	—	16	1	no	—
29	C₂×Dic₄₇	—	16	1	no	—
30	D₁₈₈	—	8	1	no	—
31	C₄×D₄₇	—	16	1	no	—
32	Dic₉₄	—	8	1	no	—
33	C₈×C₄₇	#102	1	1	yes	characteristic
34	C₈×C₄₇	SD₃₂	1	1	yes	characteristic
35	#7540	—	47	1	no	maximal, Sylow
36	C₂²×C₄₇	D₃₂	1	1	yes	characteristic
37	C₂×D₄₇	—	16	1	no	—
38	C₄×C₄₇	#357	1	1	yes	characteristic
39	C₄×C₄₇	SD₆₄	1	1	yes	characteristic
40	Dic₄₇	—	16	1	no	—
41	Dic₄₇	—	32	1	no	—
42	#2262	—	47	1	no	—
43	M₁₂₈	D₄₇	1	1	yes	characteristic, complemented
44	#1424	—	47	1	no	—
45	C₂×C₄₇	—	2	1	no	—
46	C₂×C₄₇	#1415	1	1	yes	characteristic
47	D₄₇	—	32	1	no	—
48	#508	—	94	1	no	—
49	Q₆₄	—	47	1	no	—
50	SD₆₄	—	94	1	no	—
51	D₃₂	—	47	1	no	—
52	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
53	#366	—	94	1	no	—
54	C₆₄	—	2	1	no	—
55	C₄₇	#7540	1	1	yes	characteristic, complemented, Sylow
56	#135	—	188	1	no	—
57	Q₃₂	—	94	1	no	—
58	SD₃₂	—	188	1	no	—
59	D₁₆	—	94	1	no	—
60	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
61	#107	—	188	1	no	—
62	C₃₂	C₄×D₄₇	1	1	yes	characteristic
63	C₃₂	C₂²⋊D₄₇	1	1	yes	characteristic
64	#41	—	376	1	no	—
65	Q₁₆	—	188	1	no	—
66	SD₁₆	—	376	1	no	—
67	D₈	—	188	1	no	—
68	C₂×C₈	D₃₇₆	1	1	yes	characteristic
69	C₄⋊C₄	—	376	1	no	—
70	C₁₆	#10625325	1	1	yes	characteristic
71	C₁₆	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
72	Q₈	—	376	1	no	—
73	D₄	—	376	1	no	—
74	D₄	—	752	1	no	—
75	C₂×C₄	—	752	1	no	—
76	C₂×C₄	—	752	1	no	—
77	C₂×C₄	D₇₅₂	1	1	yes	characteristic
78	C₈	#13093946	1	1	yes	characteristic
79	C₈	#13093969	1	1	yes	characteristic
80	C₂²	—	752	1	no	—
81	C₂²	D₁₅₀₄	1	1	yes	characteristic
82	C₄	—	1504	1	no	—
83	C₄	—	752	1	no	—
84	C₄	#15688925	1	1	yes	characteristic
85	C₄	#15688962	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	1504	1	no	—
88	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	#20245697	C₂
cosocle	C₂×C₃₂×C₄₇	C₂²
hypercenter	M₁₂₈	D₄₇
hypocenter	C₄₇	#7540
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
2. #20241590	C₂	semidirect
3. C₄₇×M₁₂₈	C₂	semidirect
5. Dic₇₅₂	C₄	semidirect
7. D₁₅₀₄	C₄	semidirect
43. M₁₂₈	D₄₇	semidirect
55. C₄₇	#7540	semidirect

Maximal subgroups

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

subgroup	index
#20245697	2
#20241590	2
C₄₇×M₁₂₈	2
#7540	47