Subgroups — #15689849

Order 3008.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13094023	C₂	1	3	yes	maximal
2	#13094016	C₂	1	3	yes	maximal
3	#13094015	C₂	1	3	yes	maximal
4	Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
5	Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₉₄	C₂	1	3	yes	maximal
7	C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₉₄	C₂²	1	3	yes	—
10	C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
11	C₄²×C₄₇	C₂²	1	3	yes	—
12	C₄⋊Dic₄₇	C₂²	1	3	yes	—
13	C₄⋊Dic₄₇	C₂²	1	3	yes	—
14	C₄⋊Dic₄₇	C₂²	1	3	yes	—
15	Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
16	Dic₄₇⋊C₄	C₂²	1	3	yes	—
17	Dic₄₇⋊C₄	C₂²	1	6	yes	—
18	C₄×Dic₄₇	C₂²	1	3	yes	—
19	C₄×Dic₄₇	C₂²	1	3	yes	—
20	Q₈×C₄₇	Q₈	1	4	yes	complemented
21	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₄₇	C₂³	1	3	yes	—
23	C₂×C₄×C₄₇	C₂³	1	3	yes	—
24	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
25	C₂×Dic₄₇	C₂³	1	3	yes	—
26	C₂×Dic₄₇	C₂³	1	3	yes	—
27	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
28	Dic₉₄	—	2	6	no	—
29	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
30	C₄×C₄₇	—	2	3	no	—
31	C₄×C₄₇	—	2	1	no	—
32	C₄×C₄₇	C₂×Q₈	1	6	yes	—
33	Dic₄₇	—	2	3	no	—
34	Dic₄₇	—	2	1	no	—
35	Dic₄₇	—	2	3	no	—
36	Dic₄₇	#41	1	2	yes	—
37	C₂×C₄₇	C₂²×Q₈	1	1	yes	characteristic
38	C₂×C₄₇	C₂×#41	1	1	yes	characteristic
39	C₂×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
40	#557	—	47	1	no	maximal, Sylow
41	C₄₇	#557	1	1	yes	characteristic, complemented, Sylow
42	C₄⋊Q₈	—	47	3	no	—
43	#125	—	47	3	no	—
44	#125	—	47	3	no	—
45	C₄×Q₈	—	47	1	no	—
46	C₄×Q₈	—	47	1	no	—
47	C₄×Q₈	—	47	3	no	—
48	C₄×Q₈	D₄₇	1	1	yes	characteristic
49	C₂×Q₈	—	47	3	no	—
50	C₂×Q₈	C₂×D₄₇	1	1	yes	characteristic
51	C₄⋊C₄	—	47	1	no	—
52	C₄⋊C₄	—	47	3	no	—
53	C₄⋊C₄	—	47	6	no	—
54	C₄⋊C₄	—	47	3	no	—
55	C₄⋊C₄	—	47	3	no	—
56	C₄⋊C₄	—	47	3	no	—
57	C₄⋊C₄	C₂×D₄₇	1	3	yes	—
58	C₄²	—	47	3	no	—
59	C₄²	—	47	3	no	—
60	C₄²	C₂×D₄₇	1	3	yes	—
61	Q₈	—	94	6	no	—
62	Q₈	Dic₉₄	1	4	yes	complemented
63	C₂×C₄	—	47	1	no	—
64	C₂×C₄	—	47	1	no	—
65	C₂×C₄	—	47	3	no	—
66	C₂×C₄	—	47	3	no	—
67	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
68	C₂×C₄	C₂²×D₄₇	1	3	yes	—
69	C₂×C₄	C₂²×D₄₇	1	3	yes	—
70	C₂²	C₂³×D₄₇	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	2	1	no	—
73	C₄	—	47	2	no	—
74	C₄	—	94	3	no	—
75	C₄	—	94	1	no	—
76	C₄	—	94	3	no	—
77	C₄	C₂×Dic₉₄	1	6	yes	—
78	C₂	C₂²×Dic₉₄	1	1	yes	characteristic
79	C₂	#41×D₄₇	1	1	yes	characteristic
80	C₂	#13094072	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, prime radical, cosocle	C₂²×C₄₇	C₂⁴
Fitting	C₄×Q₈×C₄₇	C₂
hypercenter	C₄×Q₈	D₄₇
hypocenter	C₄₇	#557
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
20. Q₈×C₄₇	Q₈	semidirect
41. C₄₇	#557	semidirect
62. Q₈	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
#13094016	2
#13094015	2
Q₈×Dic₄₇	2
C₄×Dic₉₄	2
C₄×Q₈×C₄₇	2
#557	47