Subgroups — #15689873

Order 3008.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13094022	C₂	1	2	yes	maximal, complemented
2	#13094017	C₂	1	4	yes	maximal, complemented
3	#13094016	C₂	1	2	yes	maximal, complemented
4	#13094005	C₂	1	2	yes	maximal
5	#13094001	C₂	1	4	yes	maximal, complemented
6	#124×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈×C₄₇	C₂²	1	1	yes	characteristic
8	C₂×D₄×C₄₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₄₇	C₂²	1	2	yes	—
10	C₂×Dic₉₄	C₂²	1	2	yes	—
11	C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
12	C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₄₇	C₂²	1	4	yes	—
14	C₂²⋊Dic₄₇	C₂²	1	2	yes	—
15	C₄⋊Dic₄₇	C₂²	1	2	yes	—
16	C₄⋊Dic₄₇	C₂²	1	2	yes	—
17	Dic₄₇⋊C₄	C₂²	1	4	yes	—
18	Dic₄₇⋊C₄	C₂²	1	4	yes	—
19	Dic₄₇⋊C₄	C₂²	1	4	yes	—
20	C₄×Dic₄₇	C₂²	1	2	yes	—
21	C₂³×C₄₇	C₂³	1	2	yes	—
22	Q₈×C₄₇	—	4	1	no	—
23	D₄×C₄₇	—	4	1	no	—
24	C₂×C₄×C₄₇	C₂³	1	2	yes	—
25	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₄₇	C₂³	1	2	yes	—
27	C₂×Dic₄₇	—	4	2	no	—
28	C₂×Dic₄₇	C₂³	1	4	yes	—
29	C₂×Dic₄₇	C₂³	1	4	yes	—
30	Dic₉₄	—	4	2	no	—
31	C₂²×C₄₇	—	2	2	no	—
32	C₂²×C₄₇	—	2	4	no	—
33	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
34	C₄×C₄₇	—	2	2	no	—
35	C₄×C₄₇	—	2	1	no	—
36	C₄×C₄₇	—	2	2	no	—
37	Dic₄₇	—	2	4	no	—
38	Dic₄₇	—	2	4	no	—
39	C₂×C₄₇	—	4	2	no	—
40	C₂×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
41	C₂×C₄₇	2₋¹⁺⁴	1	2	yes	—
42	#563	—	47	1	no	maximal, Sylow
43	C₄₇	#563	1	1	yes	characteristic, complemented, Sylow
44	C₄⋊Q₈	—	47	2	no	—
45	#126	—	47	4	no	—
46	#125	—	47	2	no	—
47	#124	D₄₇	1	1	yes	characteristic
48	#123	—	47	2	no	—
49	C₂²⋊Q₈	—	47	4	no	—
50	C₂×Q₈	—	47	2	no	—
51	C₂×Q₈	C₂×D₄₇	1	1	yes	characteristic
52	C₂×D₄	C₂×D₄₇	1	1	yes	characteristic
53	C₂²×C₄	—	47	2	no	—
54	C₄⋊C₄	—	47	4	no	—
55	C₄⋊C₄	—	47	4	no	—
56	C₄⋊C₄	—	47	4	no	—
57	C₄⋊C₄	—	47	2	no	—
58	C₄⋊C₄	—	47	2	no	—
59	C₂²⋊C₄	—	47	4	no	—
60	C₂²⋊C₄	—	47	2	no	—
61	C₂²⋊C₄	C₂×D₄₇	1	4	yes	—
62	C₄²	—	47	2	no	—
63	C₄²	C₂×D₄₇	1	1	yes	characteristic
64	C₂³	C₂²×D₄₇	1	2	yes	—
65	Q₈	—	4	1	no	—
66	Q₈	—	188	2	no	—
67	D₄	—	4	1	no	—
68	C₂×C₄	—	47	4	no	—
69	C₂×C₄	—	47	4	no	—
70	C₂×C₄	—	188	2	no	—
71	C₂×C₄	C₂²×D₄₇	1	2	yes	—
72	C₂×C₄	C₂²×D₄₇	1	2	yes	—
73	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
74	C₂²	—	2	2	no	—
75	C₂²	—	2	4	no	—
76	C₂²	C₂³×D₄₇	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	2	2	no	—
79	C₄	—	2	2	no	—
80	C₄	—	94	4	no	—
81	C₄	—	94	4	no	—
82	C₂	—	4	2	no	—
83	C₂	#13094069	1	1	yes	characteristic
84	C₂	#13094071	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	#124×C₄₇	C₂
prime radical	C₂×D₄×C₄₇	C₂²
hypercenter	#124	D₄₇
hypocenter	C₄₇	#563
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
1. #13094022	C₂	semidirect
2. #13094017	C₂	semidirect
3. #13094016	C₂	semidirect
5. #13094001	C₂	semidirect
43. C₄₇	#563	semidirect

Maximal subgroups

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

subgroup	index
#13094022	2
#13094017	2
#13094016	2
#13094005	2
#13094001	2
#124×C₄₇	2
#563	47