Subgroups — #20246565

Order 6016.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689003	C₂	1	12	yes	maximal, complemented
2	#15688991	C₂	1	2	yes	maximal, complemented
3	C₂×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₄₇	C₂²	1	4	yes	—
5	C₂×C₂²⋊Dic₄₇	C₂²	1	12	yes	—
6	C₄₇⋊#95	C₂²	1	12	yes	—
7	C₂²×D₄×C₄₇	C₂²	1	6	yes	—
8	#127×C₄₇	C₄	1	8	yes	complemented
9	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄₇	C₂³	1	4	yes	—
11	C₂×D₄×C₄₇	—	4	6	no	—
12	C₂×D₄×C₄₇	—	4	6	no	—
13	C₂×D₄×C₄₇	C₂×C₄	1	24	yes	complemented
14	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
15	C₂²×Dic₄₇	—	4	12	no	—
16	C₂²×Dic₄₇	C₂³	1	8	yes	—
17	C₄²×C₄₇	C₂×C₄	1	4	yes	complemented
18	C₂²⋊Dic₄₇	—	4	12	no	—
19	C₂³×C₄₇	—	2	12	no	—
20	C₂³×C₄₇	—	2	12	no	—
21	C₂³×C₄₇	C₂²×C₄	1	8	yes	—
22	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
23	D₄×C₄₇	—	4	24	no	—
24	C₂×C₄×C₄₇	—	2	6	no	—
25	C₂×C₄×C₄₇	C₂²×C₄	1	6	yes	—
26	C₂×Dic₄₇	—	2	24	no	—
27	C₂×Dic₄₇	—	4	16	no	—
28	C₂²×C₄₇	—	2	24	no	—
29	C₂²×C₄₇	—	4	4	no	—
30	C₂²×C₄₇	—	4	12	no	—
31	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
32	C₂²×C₄₇	H(5,𝔽₂)	1	6	yes	—
33	C₄×C₄₇	—	2	6	no	—
34	Dic₄₇	—	4	8	no	—
35	#2380	—	47	1	no	maximal, Sylow
36	C₂×C₄₇	—	4	8	no	—
37	C₂×C₄₇	#518	1	3	yes	—
38	C₂×C₄₇	#560	1	4	yes	—
39	C₂×#127	D₄₇	1	1	yes	characteristic
40	#386	—	47	12	no	—
41	#383	—	47	2	no	—
42	C₄₇	#2380	1	1	yes	characteristic, complemented, Sylow
43	C₂²×D₄	C₂×D₄₇	1	6	yes	—
44	C₂³×C₄	—	47	4	no	—
45	#127	Dic₄₇	1	8	yes	complemented
46	C₂×C₂²⋊C₄	—	47	12	no	—
47	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
48	#95	—	47	12	no	—
49	C₂⁴	C₂²×D₄₇	1	4	yes	—
50	C₂×D₄	—	4	6	no	—
51	C₂×D₄	—	4	6	no	—
52	C₂×D₄	C₂×Dic₄₇	1	24	yes	complemented
53	C₂²×C₄	—	47	8	no	—
54	C₂²×C₄	—	188	12	no	—
55	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
56	C₂²⋊C₄	—	188	12	no	—
57	C₄²	C₂×Dic₄₇	1	4	yes	complemented
58	C₂³	—	2	12	no	—
59	C₂³	—	2	12	no	—
60	C₂³	C₂²×Dic₄₇	1	8	yes	—
61	C₂³	C₂³×D₄₇	1	1	yes	characteristic
62	D₄	—	4	24	no	—
63	C₂×C₄	—	2	6	no	—
64	C₂×C₄	—	94	24	no	—
65	C₂×C₄	—	188	16	no	—
66	C₂×C₄	C₂²×Dic₄₇	1	6	yes	—
67	C₂²	—	2	24	no	—
68	C₂²	—	4	4	no	—
69	C₂²	—	4	12	no	—
70	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
71	C₂²	#13094069	1	6	yes	—
72	C₄	—	2	6	no	—
73	C₄	—	188	8	no	—
74	C₂	—	4	8	no	—
75	C₂	#15689598	1	3	yes	—
76	C₂	#15689860	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₂×#127×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×#127	D₄₇
hypocenter	C₄₇	#2380
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. #15689003	C₂	semidirect
2. #15688991	C₂	semidirect
8. #127×C₄₇	C₄	semidirect
13. C₂×D₄×C₄₇	C₂×C₄	semidirect
17. C₄²×C₄₇	C₂×C₄	semidirect
42. C₄₇	#2380	semidirect
45. #127	Dic₄₇	semidirect
52. C₂×D₄	C₂×Dic₄₇	semidirect
57. C₄²	C₂×Dic₄₇	semidirect