Subgroups — #15689754

Order 3008.

206 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13094023	C₂	1	4	yes	maximal, complemented
2	#13094015	C₂	1	4	yes	maximal, complemented
3	#13094003	C₂	1	4	yes	maximal
4	#13093983	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×C₄₇	C₂²	1	2	yes	—
7	C₂²×C₄×C₄₇	C₂²	1	1	yes	characteristic
8	C₂×Dic₉₄	C₂²	1	4	yes	—
9	C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
10	C₂²⋊Dic₄₇	C₂²	1	4	yes	—
11	C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₄⋊Dic₄₇	C₂²	1	4	yes	—
14	Dic₄₇⋊C₄	C₂²	1	8	yes	—
15	C₄×Dic₄₇	C₂²	1	4	yes	—
16	C₂³×C₄₇	C₂³	1	1	yes	characteristic
17	C₂×C₄×C₄₇	—	4	2	no	—
18	C₂×C₄×C₄₇	Q₈	1	4	yes	complemented
19	C₂×C₄×C₄₇	C₂³	1	4	yes	—
20	C₂×C₄×C₄₇	C₂³	1	2	yes	—
21	C₂×Dic₄₇	C₂³	1	4	yes	—
22	C₂×Dic₄₇	C₂³	1	4	yes	—
23	Dic₉₄	—	4	4	no	—
24	C₂²×C₄₇	—	2	1	no	—
25	C₂²×C₄₇	—	2	1	no	—
26	C₂²×C₄₇	C₂×Q₈	1	2	yes	—
27	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
28	C₄×C₄₇	—	2	4	no	—
29	C₄×C₄₇	C₂×Q₈	1	4	yes	—
30	Dic₄₇	—	2	4	no	—
31	Dic₄₇	—	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	1	yes	characteristic
35	C₂×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
36	#544	—	47	1	no	maximal, Sylow
37	C₄₇	#544	1	1	yes	characteristic, complemented, Sylow
38	C₄⋊Q₈	—	47	4	no	—
39	#125	—	47	4	no	—
40	C₂²⋊Q₈	—	47	4	no	—
41	#117	—	47	2	no	—
42	C₂×C₄⋊C₄	D₄₇	1	1	yes	characteristic
43	C₂×Q₈	—	47	4	no	—
44	C₂²×C₄	C₂×D₄₇	1	2	yes	—
45	C₂²×C₄	C₂×D₄₇	1	1	yes	characteristic
46	C₄⋊C₄	—	47	8	no	—
47	C₄⋊C₄	—	47	2	no	—
48	C₄⋊C₄	—	47	2	no	—
49	C₄⋊C₄	—	47	4	no	—
50	C₄⋊C₄	C₂×D₄₇	1	4	yes	—
51	C₂²⋊C₄	—	47	4	no	—
52	C₄²	—	47	4	no	—
53	C₂³	C₂²×D₄₇	1	1	yes	characteristic
54	Q₈	—	188	4	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	47	4	no	—
57	C₂×C₄	—	47	4	no	—
58	C₂×C₄	Dic₉₄	1	4	yes	complemented
59	C₂×C₄	C₂²×D₄₇	1	4	yes	—
60	C₂×C₄	C₂²×D₄₇	1	2	yes	—
61	C₂²	—	2	1	no	—
62	C₂²	—	2	1	no	—
63	C₂²	C₂×Dic₉₄	1	2	yes	—
64	C₂²	C₂³×D₄₇	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	94	4	no	—
67	C₄	—	94	4	no	—
68	C₄	C₂×Dic₉₄	1	4	yes	—
69	C₂	—	2	2	no	—
70	C₂	C₂²×Dic₉₄	1	1	yes	characteristic
71	C₂	#13094069	1	1	yes	characteristic
72	C₂	#13094072	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	C₂×C₄⋊C₄×C₄₇	C₂
prime radical	C₂³×C₄₇	C₂³
hypercenter	C₂×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#544
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #13094023	C₂	semidirect
2. #13094015	C₂	semidirect
18. C₂×C₄×C₄₇	Q₈	semidirect
37. C₄₇	#544	semidirect
58. C₂×C₄	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#13094023	2
#13094015	2
#13094003	2
#13093983	2
C₂×C₄⋊C₄×C₄₇	2
#544	47