Subgroups — #15689734

Order 3008.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13094019	C₂	1	4	yes	maximal
2	#13094016	C₂	1	1	yes	maximal, characteristic, complemented
3	#13094014	C₂	1	4	yes	maximal, complemented
4	#13094000	C₂	1	2	yes	maximal
5	Q₈×Dic₄₇	C₂	1	2	yes	maximal, complemented
6	#13093981	C₂	1	1	yes	maximal, characteristic
7	C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×C₄₇	C₂²	1	2	yes	—
9	C₂×C₄×D₄₇	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
11	C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
13	(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
14	C₄⋊Dic₄₇	C₂²	1	2	yes	—
15	C₄⋊Dic₄₇	C₂²	1	4	yes	—
16	Dic₄₇⋊C₄	C₂²	1	2	yes	—
17	Dic₄₇⋊C₄	C₂²	1	4	yes	—
18	Dic₄₇⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₄₇	C₂²	1	1	yes	characteristic
20	C₄×Dic₄₇	C₂²	1	4	yes	—
21	C₂²×D₄₇	C₂³	1	1	yes	characteristic
22	Q₈×C₄₇	—	2	4	no	—
23	C₂×C₄×C₄₇	C₂³	1	4	yes	—
24	C₂×C₄×C₄₇	C₂³	1	2	yes	—
25	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×Dic₄₇	C₂³	1	2	yes	—
27	C₂×Dic₄₇	C₂³	1	4	yes	—
28	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
29	C₄×D₄₇	—	2	2	no	—
30	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
31	C₂×D₄₇	—	2	1	no	—
32	C₂×D₄₇	—	2	2	no	—
33	C₄×C₄₇	—	2	4	no	—
34	C₄×C₄₇	—	2	2	no	—
35	C₄×C₄₇	#41	1	2	yes	—
36	Dic₄₇	—	2	1	no	—
37	Dic₄₇	—	2	2	no	—
38	Dic₄₇	—	2	4	no	—
39	C₂×C₄₇	C₂×#41	1	1	yes	characteristic
40	C₂×C₄₇	2₋¹⁺⁴	1	2	yes	—
41	D₄₇	—	4	1	no	—
42	#541	—	47	1	no	maximal, Sylow
43	C₄₇	#541	1	1	yes	characteristic, complemented, Sylow
44	C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
45	#126	—	47	4	no	—
46	#125	—	47	4	no	—
47	#125	—	47	1	no	—
48	C₂²⋊Q₈	—	47	2	no	—
49	C₄×Q₈	—	47	2	no	—
50	#117	—	47	1	no	—
51	C₂×Q₈	C₂×D₄₇	1	2	yes	—
52	C₂²×C₄	—	47	1	no	—
53	C₄⋊C₄	—	47	2	no	—
54	C₄⋊C₄	—	47	4	no	—
55	C₄⋊C₄	—	47	4	no	—
56	C₄⋊C₄	—	47	2	no	—
57	C₄⋊C₄	—	47	4	no	—
58	C₄⋊C₄	C₂×D₄₇	1	4	yes	—
59	C₂²⋊C₄	—	47	2	no	—
60	C₂²⋊C₄	—	47	4	no	—
61	C₄²	—	47	1	no	—
62	C₄²	—	47	4	no	—
63	C₄²	C₂×D₄₇	1	1	yes	characteristic
64	C₂³	—	47	1	no	—
65	Q₈	—	2	4	no	—
66	C₂×C₄	—	47	2	no	—
67	C₂×C₄	—	47	4	no	—
68	C₂×C₄	—	47	1	no	—
69	C₂×C₄	—	94	2	no	—
70	C₂×C₄	C₂²×D₄₇	1	4	yes	—
71	C₂×C₄	C₂²×D₄₇	1	2	yes	—
72	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
73	C₂²	—	94	1	no	—
74	C₂²	—	94	2	no	—
75	C₂²	C₂³×D₄₇	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	2	no	—
78	C₄	—	94	1	no	—
79	C₄	—	94	2	no	—
80	C₄	—	94	4	no	—
81	C₄	#10625352	1	2	yes	—
82	C₂	—	188	1	no	—
83	C₂	C₂×#10625352	1	1	yes	characteristic
84	C₂	#13094072	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	C₄⋊Q₈×C₄₇	C₂
prime radical	C₂²×D₄₇	C₂³
hypercenter	C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#541
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
2. #13094016	C₂	semidirect
3. #13094014	C₂	semidirect
5. Q₈×Dic₄₇	C₂	semidirect
7. C₄⋊Q₈×C₄₇	C₂	semidirect
43. C₄₇	#541	semidirect
44. C₄⋊Q₈	D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#13094019	2
#13094016	2
#13094014	2
#13094000	2
Q₈×Dic₄₇	2
#13093981	2
C₄⋊Q₈×C₄₇	2
#541	47