Subgroups — #15689175

Order 3008.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₄₇	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₄₇)⋊C₈	C₂	1	2	yes	maximal
4	C₈×Dic₄₇	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄₇	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄₇	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄₇	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₄₇	C₂²	1	2	yes	—
10	C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄×Dic₄₇	C₄	1	1	yes	characteristic
12	C₄×Dic₄₇	C₄	1	1	yes	characteristic
13	C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂²×D₄₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₂×Dic₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₄₇	C₂×C₄	1	1	yes	characteristic
22	C₄×D₄₇	—	2	1	no	—
23	C₄×D₄₇	—	2	1	no	—
24	C₄×D₄₇	—	2	2	no	—
25	C₄×D₄₇	C₈	1	4	yes	complemented
26	C₈×C₄₇	—	2	2	no	—
27	C₄₇⋊C₈	—	2	2	no	—
28	C₂²×C₄₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×D₄₇	—	2	1	no	—
30	C₂×D₄₇	—	2	1	no	—
31	C₂×D₄₇	C₂×C₈	1	2	yes	—
32	C₄×C₄₇	—	2	1	no	—
33	C₄×C₄₇	C₂×C₈	1	2	yes	—
34	C₄×C₄₇	#41	1	1	yes	characteristic
35	C₄×C₄₇	#41	1	1	yes	characteristic
36	Dic₄₇	—	2	1	no	—
37	Dic₄₇	C₂×C₈	1	2	yes	—
38	Dic₄₇	M₁₆	1	2	yes	—
39	C₂×C₄₇	#117	1	1	yes	characteristic
40	C₂×C₄₇	C₂²×C₈	1	1	yes	characteristic
41	C₂×C₄₇	C₂×M₁₆	1	1	yes	characteristic
42	D₄₇	—	2	2	no	—
43	#431	—	47	1	no	maximal, Sylow
44	C₄₇	#431	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₄²	—	47	1	no	—
46	C₄⋊C₈	—	47	1	no	—
47	C₄⋊C₈	D₄₇	1	1	yes	characteristic, complemented
48	C₂²⋊C₈	—	47	2	no	—
49	C₄×C₈	—	47	2	no	—
50	C₂²×C₄	—	47	1	no	—
51	C₂²×C₄	—	47	1	no	—
52	C₂²×C₄	—	47	1	no	—
53	C₂×C₈	—	47	2	no	—
54	C₂×C₈	C₂×D₄₇	1	2	yes	—
55	C₄²	—	47	1	no	—
56	C₄²	—	47	1	no	—
57	C₄²	—	47	1	no	—
58	C₄²	C₂×D₄₇	1	1	yes	characteristic
59	C₂³	—	47	1	no	—
60	C₂×C₄	—	47	1	no	—
61	C₂×C₄	—	47	1	no	—
62	C₂×C₄	—	47	4	no	—
63	C₂×C₄	—	47	1	no	—
64	C₂×C₄	—	94	2	no	—
65	C₂×C₄	—	94	1	no	—
66	C₂×C₄	—	94	1	no	—
67	C₂×C₄	C₄×D₄₇	1	1	yes	characteristic
68	C₂×C₄	C₄×D₄₇	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	94	2	no	—
72	C₂²	—	47	2	no	—
73	C₂²	—	94	1	no	—
74	C₂²	—	94	1	no	—
75	C₂²	C₂×C₄×D₄₇	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	47	2	no	—
78	C₄	—	47	2	no	—
79	C₄	—	94	1	no	—
80	C₄	C₈×D₄₇	1	2	yes	—
81	C₄	#10625352	1	1	yes	characteristic
82	C₄	#10625354	1	1	yes	characteristic
83	C₂	—	94	2	no	—
84	C₂	#13093982	1	1	yes	characteristic
85	C₂	C₂×C₈×D₄₇	1	1	yes	characteristic
86	C₂	M₁₆×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₇
commutator	C₂×C₄₇	C₂²×C₈
Fitting	C₄⋊C₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₄
prime radical	C₂²×D₄₇	C₂×C₄
cosocle	C₂×C₄×C₄₇	C₂³
hypercenter	C₄⋊C₈	D₄₇
hypocenter	C₄₇	#431
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₄₇⋊C₈)	C₂	semidirect
4. C₈×Dic₄₇	C₂	semidirect
5. C₄⋊C₈×C₄₇	C₂	semidirect
25. C₄×D₄₇	C₈	semidirect
44. C₄₇	#431	semidirect
47. C₄⋊C₈	D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₄₇	2
C₄⋊(C₄₇⋊C₈)	2
(C₂×D₄₇)⋊C₈	2
C₈×Dic₄₇	2
C₄⋊C₈×C₄₇	2
#431	47