Subgroups — #26104088

Order 12032.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688956	C₂	1	1	yes	maximal, characteristic
2	#20241599	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×D₄₇	C₄	1	1	yes	characteristic
5	#15688956	—	2	1	no	—
6	#15688956	—	2	1	no	—
7	C₂×C₄₇⋊C₃₂	C₄	1	1	yes	characteristic
8	C₄₇⋊C₆₄	—	2	1	no	—
9	C₂×C₃₂×C₄₇	C₂²	1	1	yes	characteristic
10	C₄₇×C₆₄	—	2	1	no	—
11	C₂×C₈×D₄₇	C₈	1	1	yes	characteristic
12	C₂×C₄₇⋊C₁₆	C₈	1	1	yes	characteristic
13	C₁₆×D₄₇	—	2	1	no	—
14	C₁₆×D₄₇	—	2	1	no	—
15	C₄₇⋊C₃₂	—	2	1	no	—
16	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
17	C₃₂×C₄₇	D₄	1	1	yes	characteristic
18	C₃₂×C₄₇	D₄	1	1	yes	characteristic
19	C₂×C₄×D₄₇	C₁₆	1	1	yes	characteristic
20	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
21	C₂×C₄₇⋊C₈	C₁₆	1	1	yes	characteristic
22	C₈×D₄₇	—	2	1	no	—
23	C₈×D₄₇	—	2	1	no	—
24	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₁₆×C₄₇	M₁₆	1	1	yes	characteristic
26	C₄₇⋊C₁₆	—	2	1	no	—
27	C₂²×D₄₇	C₃₂	1	1	yes	characteristic
28	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
29	C₂×Dic₄₇	C₃₂	1	1	yes	characteristic
30	C₄×D₄₇	—	2	1	no	—
31	C₄×D₄₇	—	2	1	no	—
32	C₈×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₈×C₄₇	M₃₂	1	1	yes	characteristic
34	C₄₇⋊C₈	—	2	1	no	—
35	#7513	—	47	1	no	maximal, Sylow
36	C₂²×C₄₇	C₂×C₃₂	1	1	yes	characteristic
37	C₂×D₄₇	—	2	1	no	—
38	C₂×D₄₇	—	4	1	no	—
39	C₄×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
40	C₄×C₄₇	M₆₄	1	1	yes	characteristic
41	Dic₄₇	—	2	1	no	—
42	C₂×M₆₄	—	47	1	no	—
43	M₁₂₈	—	47	1	no	—
44	M₁₂₈	D₄₇	1	1	yes	characteristic, complemented
45	C₂×C₄₇	—	2	1	no	—
46	C₂×C₄₇	C₂²⋊C₃₂	1	1	yes	characteristic
47	D₄₇	—	4	1	no	—
48	C₂²×C₁₆	—	47	1	no	—
49	M₆₄	—	94	1	no	—
50	M₆₄	—	94	1	no	—
51	C₂×C₃₂	—	47	1	no	—
52	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
53	C₆₄	—	2	1	no	—
54	C₆₄	—	94	1	no	—
55	C₄₇	#7513	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₈	—	47	1	no	—
57	C₂×C₁₆	—	47	1	no	—
58	C₂×C₁₆	—	94	1	no	—
59	C₂×C₁₆	—	94	1	no	—
60	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
61	C₃₂	—	94	1	no	—
62	C₃₂	D₁₈₈	1	1	yes	characteristic
63	C₃₂	C₂²⋊D₄₇	1	1	yes	characteristic
64	C₂²×C₄	—	47	1	no	—
65	C₂×C₈	—	47	1	no	—
66	C₂×C₈	—	94	1	no	—
67	C₂×C₈	—	94	1	no	—
68	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
69	C₁₆	—	94	1	no	—
70	C₁₆	#10625324	1	1	yes	characteristic
71	C₁₆	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
72	C₂³	—	47	1	no	—
73	C₂×C₄	—	47	1	no	—
74	C₂×C₄	—	94	1	no	—
75	C₂×C₄	—	94	1	no	—
76	C₂×C₄	C₁₆×D₄₇	1	1	yes	characteristic
77	C₈	—	94	1	no	—
78	C₈	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
79	C₈	#13093963	1	1	yes	characteristic
80	C₂²	—	94	1	no	—
81	C₂²	—	188	1	no	—
82	C₂²	C₃₂×D₄₇	1	1	yes	characteristic
83	C₄	—	94	1	no	—
84	C₄	(C₂×D₄₇)⋊C₁₆	1	1	yes	characteristic
85	C₄	#15688956	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	188	1	no	—
88	C₂	(C₂×D₄₇)⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄₇	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₄₇
Fitting	C₄₇×M₁₂₈	C₂
socle	C₂×C₄₇	C₂²⋊C₃₂
prime radical	C₂²×D₄₇	C₃₂
cosocle	C₂×C₃₂×C₄₇	C₂²
hypercenter	M₁₂₈	D₄₇
hypocenter	C₄₇	#7513
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
2. #20241599	C₂	semidirect
3. C₄₇×M₁₂₈	C₂	semidirect
44. M₁₂₈	D₄₇	semidirect
55. C₄₇	#7513	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688956	2
#20241599	2
C₄₇×M₁₂₈	2
#7513	47