Subgroups — #15689420

Order 3008.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₈₈	C₂	1	1	yes	maximal, characteristic
2	#13093957	C₂	1	1	yes	maximal, characteristic, complemented
3	#13093955	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#13093944	C₂	1	1	yes	maximal, characteristic
6	#13093944	C₂	1	1	yes	maximal, characteristic
7	#125×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₈₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄₇	—	2	1	no	—
10	C₄⋊C₄×C₄₇	—	2	1	no	—
11	C₄⋊C₄×C₄₇	—	2	1	no	—
12	C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	(C₂×D₄₇)⋊C₄	—	2	1	no	—
16	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×D₄₇	—	2	1	no	—
20	C₂×C₄×C₄₇	—	2	1	no	—
21	C₂×C₄×C₄₇	—	2	1	no	—
22	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂×Dic₄₇	—	2	1	no	—
26	D₁₈₈	—	2	1	no	—
27	D₁₈₈	Q₈	1	2	yes	—
28	C₄×D₄₇	—	4	2	no	—
29	C₄₇⋊C₈	—	2	1	no	—
30	C₄₇⋊C₈	—	2	1	no	—
31	C₂²×C₄₇	C₂×D₄	1	1	yes	characteristic
32	C₂×D₄₇	—	2	2	no	—
33	C₂×D₄₇	—	4	1	no	—
34	C₂×D₄₇	—	4	1	no	—
35	C₄×C₄₇	—	2	1	no	—
36	C₄×C₄₇	—	2	1	no	—
37	C₄×C₄₇	—	4	1	no	—
38	C₄×C₄₇	—	4	1	no	—
39	C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
40	C₄×C₄₇	#41	1	1	yes	characteristic
41	Dic₄₇	—	4	1	no	—
42	C₂×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
43	C₂×C₄₇	#135	1	1	yes	characteristic
44	C₂×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
45	D₄₇	—	4	2	no	—
46	#478	—	47	1	no	maximal, Sylow
47	C₄₇	#478	1	1	yes	characteristic, complemented, Sylow
48	#125	D₄₇	1	1	yes	characteristic, complemented
49	C₄×D₄	—	47	1	no	—
50	#107	—	47	1	no	—
51	#106	—	47	1	no	—
52	C₄⋊C₈	—	47	1	no	—
53	#102	—	47	1	no	—
54	#102	—	47	1	no	—
55	C₂×D₄	—	47	1	no	—
56	C₂²×C₄	—	94	1	no	—
57	C₂×C₈	—	47	1	no	—
58	C₂×C₈	—	47	1	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	2	1	no	—
61	C₄⋊C₄	—	47	1	no	—
62	C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
63	C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
64	C₂²⋊C₄	—	94	1	no	—
65	C₄²	C₂×D₄₇	1	1	yes	characteristic
66	C₂³	—	94	1	no	—
67	D₄	—	47	2	no	—
68	D₄	—	94	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	94	1	no	—
72	C₂×C₄	—	188	2	no	—
73	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
74	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
75	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
76	C₈	—	94	1	no	—
77	C₈	—	94	1	no	—
78	C₂²	—	94	2	no	—
79	C₂²	—	188	1	no	—
80	C₂²	—	188	1	no	—
81	C₂²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	188	1	no	—
87	C₄	Q₈×D₄₇	1	1	yes	characteristic
88	C₄	#10625354	1	1	yes	characteristic
89	C₂	—	188	2	no	—
90	C₂	#13094000	1	1	yes	characteristic
91	C₂	#13094044	1	1	yes	characteristic
92	C₂	#13094049	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂³
Frattini	C₂×C₄	C₂²×D₄₇
Fitting	#125×C₄₇	C₂
socle	C₂²×C₄₇	C₂×D₄
prime radical	C₂×D₁₈₈	C₂²
hypercenter	#125	D₄₇
hypocenter	C₄₇	#478
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #13093957	C₂	semidirect
3. #13093955	C₂	semidirect
4. C₄⋊(C₄₇⋊C₈)	C₂	semidirect
7. #125×C₄₇	C₂	semidirect
47. C₄₇	#478	semidirect
48. #125	D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₈₈	2
#13093957	2
#13093955	2
C₄⋊(C₄₇⋊C₈)	2
#13093944	2
#125×C₄₇	2
#478	47