Subgroups — #26104169

Order 12032.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₀₄	C₂	1	1	yes	maximal, characteristic
2	#20241592	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₇₅₂	—	2	1	no	—
5	D₁₅₀₄	—	2	1	no	—
6	D₁₅₀₄	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×D₃₇₆	—	4	1	no	—
11	D₇₅₂	—	2	2	no	—
12	D₇₅₂	—	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×D₁₈₈	—	8	1	no	—
18	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
19	#10625329	—	8	1	no	—
20	D₃₇₆	—	4	2	no	—
21	D₃₇₆	—	8	1	no	—
22	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₄₇	SD₁₆	1	1	yes	characteristic
24	C₂²×D₄₇	—	16	1	no	—
25	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
26	D₁₈₈	—	8	2	no	—
27	D₁₈₈	—	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	#7541	—	47	1	no	maximal, Sylow
32	C₂²×C₄₇	SD₆₄	1	1	yes	characteristic
33	C₂×D₄₇	—	16	2	no	—
34	C₂×D₄₇	—	32	2	no	—
35	C₄×C₄₇	#357	1	1	yes	characteristic
36	C₄×C₄₇	D₃₂	1	1	yes	characteristic
37	C₂×D₃₂	—	47	1	no	—
38	M₁₂₈	D₄₇	1	1	yes	characteristic, complemented
39	#1425	—	47	1	no	—
40	C₂×C₄₇	—	2	1	no	—
41	C₂×C₄₇	#1415	1	1	yes	characteristic
42	D₄₇	—	32	2	no	—
43	C₂×D₁₆	—	94	1	no	—
44	D₃₂	—	47	2	no	—
45	D₃₂	—	94	1	no	—
46	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
47	#368	—	94	1	no	—
48	C₆₄	—	2	1	no	—
49	C₄₇	#7541	1	1	yes	characteristic, complemented, Sylow
50	C₂×D₈	—	188	1	no	—
51	D₁₆	—	94	2	no	—
52	D₁₆	—	188	1	no	—
53	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
54	#108	—	188	1	no	—
55	C₃₂	C₄×D₄₇	1	1	yes	characteristic
56	C₃₂	C₂²⋊D₄₇	1	1	yes	characteristic
57	C₂×D₄	—	376	1	no	—
58	D₈	—	376	1	no	—
59	D₈	—	188	2	no	—
60	M₁₆	—	376	1	no	—
61	C₂×C₈	D₃₇₆	1	1	yes	characteristic
62	C₁₆	#10625325	1	1	yes	characteristic
63	C₁₆	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
64	C₂³	—	752	1	no	—
65	D₄	—	376	2	no	—
66	D₄	—	752	1	no	—
67	C₂×C₄	D₇₅₂	1	1	yes	characteristic
68	C₈	—	752	1	no	—
69	C₈	#13093946	1	1	yes	characteristic
70	C₈	#13093969	1	1	yes	characteristic
71	C₂²	—	1504	2	no	—
72	C₂²	—	752	2	no	—
73	C₂²	#15688962	1	1	yes	characteristic
74	C₄	#15688925	1	1	yes	characteristic
75	C₄	D₁₅₀₄	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	1504	2	no	—
78	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	C₂×D₁₅₀₄	C₂
cosocle	C₂×C₃₂×C₄₇	C₂²
hypercenter	M₁₂₈	D₄₇
hypocenter	C₄₇	#7541
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. #20241592	C₂	semidirect
3. C₄₇×M₁₂₈	C₂	semidirect
38. M₁₂₈	D₄₇	semidirect
49. C₄₇	#7541	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₅₀₄	2
#20241592	2
C₄₇×M₁₂₈	2
#7541	47