Subgroups — #15689749

Order 3008.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13094003	C₂	1	4	yes	maximal, complemented
2	#13094001	C₂	1	8	yes	maximal, complemented
3	C₂×C₂²⋊Dic₄₇	C₂	1	2	yes	maximal
4	C₂×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×C₄₇	C₂²	1	1	yes	characteristic
6	C₂²×C₄×C₄₇	C₂²	1	2	yes	—
7	C₂²×Dic₄₇	C₂²	1	4	yes	—
8	C₂×Dic₉₄	C₂²	1	4	yes	complemented
9	C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
10	C₂²⋊Dic₄₇	C₂²	1	4	yes	—
11	C₂²⋊Dic₄₇	C₂²	1	4	yes	—
12	C₄⋊Dic₄₇	C₂²	1	4	yes	complemented
13	Dic₄₇⋊C₄	C₂²	1	8	yes	complemented
14	C₂³×C₄₇	—	4	2	no	—
15	C₂³×C₄₇	Q₈	1	4	yes	complemented
16	C₂³×C₄₇	C₂³	1	1	yes	characteristic
17	C₂³×C₄₇	C₂³	1	2	yes	—
18	C₂×C₄×C₄₇	—	4	2	no	—
19	C₂×C₄×C₄₇	C₂³	1	4	yes	—
20	C₂×Dic₄₇	—	4	4	no	—
21	C₂×Dic₄₇	C₂³	1	8	yes	—
22	Dic₉₄	—	4	4	no	—
23	C₂²×C₄₇	—	2	4	no	—
24	C₂²×C₄₇	—	2	2	no	—
25	C₂²×C₄₇	—	4	4	no	—
26	C₂²×C₄₇	C₂×Q₈	1	2	yes	—
27	C₂²×C₄₇	C₂×Q₈	1	4	yes	—
28	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
29	C₄×C₄₇	—	2	4	no	—
30	Dic₄₇	—	2	8	no	—
31	C₂×C₄₇	—	2	4	no	—
32	C₂×C₄₇	—	2	2	no	—
33	C₂×C₄₇	C₂²×Q₈	1	1	yes	characteristic
34	C₂×C₄₇	H(5,𝔽₂)	1	2	yes	—
35	#543	—	47	1	no	maximal, Sylow
36	C₄₇	#543	1	1	yes	characteristic, complemented, Sylow
37	C₂²⋊Q₈	—	47	4	no	—
38	C₂²⋊Q₈	—	47	8	no	—
39	C₂×C₂²⋊C₄	—	47	2	no	—
40	C₂×C₂²⋊C₄	D₄₇	1	1	yes	characteristic
41	C₂⁴	C₂×D₄₇	1	1	yes	characteristic
42	C₂×Q₈	—	47	4	no	—
43	C₂²×C₄	—	47	4	no	—
44	C₂²×C₄	C₂×D₄₇	1	2	yes	—
45	C₄⋊C₄	—	47	8	no	—
46	C₄⋊C₄	—	47	4	no	—
47	C₂²⋊C₄	—	47	4	no	—
48	C₂²⋊C₄	—	47	4	no	—
49	C₂²⋊C₄	C₂×D₄₇	1	4	yes	—
50	C₂³	—	4	2	no	—
51	C₂³	Dic₉₄	1	4	yes	complemented
52	C₂³	C₂²×D₄₇	1	1	yes	characteristic
53	C₂³	C₂²×D₄₇	1	2	yes	—
54	Q₈	—	188	4	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	47	8	no	—
57	C₂×C₄	—	188	4	no	—
58	C₂×C₄	C₂²×D₄₇	1	4	yes	—
59	C₂²	—	2	2	no	—
60	C₂²	—	2	4	no	—
61	C₂²	—	4	4	no	—
62	C₂²	C₂×Dic₉₄	1	2	yes	—
63	C₂²	C₂×Dic₉₄	1	4	yes	—
64	C₂²	C₂³×D₄₇	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	94	8	no	—
67	C₂	—	2	2	no	—
68	C₂	—	2	4	no	—
69	C₂	C₂²×Dic₉₄	1	1	yes	characteristic
70	C₂	#13094069	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	C₂×C₂²⋊C₄×C₄₇	C₂
prime radical	C₂⁴×C₄₇	C₂²
hypercenter	C₂×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	#543
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. #13094003	C₂	semidirect
2. #13094001	C₂	semidirect
8. C₂×Dic₉₄	C₂²	semidirect
12. C₄⋊Dic₄₇	C₂²	semidirect
13. Dic₄₇⋊C₄	C₂²	semidirect
15. C₂³×C₄₇	Q₈	semidirect
36. C₄₇	#543	semidirect
51. C₂³	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#13094003	2
#13094001	2
C₂×C₂²⋊Dic₄₇	2
C₂×C₂²⋊C₄×C₄₇	2
#543	47