Subgroups — #15689871

Order 3008.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13094012	C₂	1	2	yes	maximal, complemented
2	#13094009	C₂	1	4	yes	maximal, complemented
3	#13094002	C₂	1	4	yes	maximal, complemented
4	#13093999	C₂	1	4	yes	maximal, complemented
5	#125×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₈₈	C₂²	1	4	yes	—
7	C₂×D₁₈₈	C₂²	1	2	yes	—
8	C₂×C₄×D₄₇	C₂²	1	4	yes	—
9	C₂×Dic₉₄	C₂²	1	2	yes	—
10	C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
11	C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
12	C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
14	(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
15	(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
16	Dic₄₇⋊C₄	C₂²	1	4	yes	—
17	C₂²×D₄₇	C₂³	1	4	yes	—
18	C₂×C₄×C₄₇	C₂³	1	4	yes	—
19	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₄₇	C₂³	1	2	yes	—
21	C₂×Dic₄₇	C₂³	1	4	yes	—
22	D₁₈₈	—	4	2	no	—
23	D₁₈₈	—	4	4	no	—
24	C₄×D₄₇	—	4	4	no	—
25	Dic₉₄	—	4	2	no	—
26	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
27	C₂×D₄₇	—	2	4	no	—
28	C₂×D₄₇	—	2	4	no	—
29	C₂×D₄₇	—	2	4	no	—
30	C₄×C₄₇	—	2	2	no	—
31	C₄×C₄₇	—	2	4	no	—
32	C₄×C₄₇	—	2	1	no	—
33	Dic₄₇	—	2	4	no	—
34	C₂×C₄₇	H(5,𝔽₂)	1	2	yes	—
35	C₂×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
36	D₄₇	—	4	4	no	—
37	#562	—	47	1	no	maximal, Sylow
38	C₄₇	#562	1	1	yes	characteristic, complemented, Sylow
39	#125	D₄₇	1	1	yes	characteristic, complemented
40	#124	—	47	2	no	—
41	#123	—	47	4	no	—
42	C₂²⋊Q₈	—	47	4	no	—
43	#121	—	47	4	no	—
44	C₂×Q₈	—	47	2	no	—
45	C₂×D₄	—	47	4	no	—
46	C₂×D₄	—	47	2	no	—
47	C₂²×C₄	—	47	4	no	—
48	C₄⋊C₄	—	47	4	no	—
49	C₄⋊C₄	C₂×D₄₇	1	4	yes	—
50	C₄⋊C₄	C₂×D₄₇	1	2	yes	—
51	C₂²⋊C₄	—	47	4	no	—
52	C₂²⋊C₄	—	47	4	no	—
53	C₂²⋊C₄	—	47	4	no	—
54	C₄²	C₂×D₄₇	1	1	yes	characteristic
55	C₂³	—	47	4	no	—
56	Q₈	—	188	2	no	—
57	D₄	—	188	4	no	—
58	D₄	—	188	2	no	—
59	C₂×C₄	—	47	4	no	—
60	C₂×C₄	—	188	4	no	—
61	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₄₇	1	2	yes	—
63	C₂×C₄	C₂²×D₄₇	1	4	yes	—
64	C₂²	—	94	4	no	—
65	C₂²	—	94	4	no	—
66	C₂²	—	94	4	no	—
67	C₂²	C₂³×D₄₇	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₄	—	94	4	no	—
72	C₂	—	188	4	no	—
73	C₂	#13094070	1	2	yes	—
74	C₂	#13094072	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	#125×C₄₇	C₂
hypercenter	#125	D₄₇
hypocenter	C₄₇	#562
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #13094012	C₂	semidirect
2. #13094009	C₂	semidirect
3. #13094002	C₂	semidirect
4. #13093999	C₂	semidirect
5. #125×C₄₇	C₂	semidirect
38. C₄₇	#562	semidirect
39. #125	D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#13094012	2
#13094009	2
#13094002	2
#13093999	2
#125×C₄₇	2
#562	47