Subgroups — #15689852

Order 3008.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13094024	C₂	1	3	yes	maximal
2	#13094015	C₂	1	6	yes	maximal
3	Q₈×Dic₄₇	C₂	1	2	yes	maximal
4	C₄×Dic₉₄	C₂	1	3	yes	maximal
5	C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₄₇	C₂²	1	1	yes	characteristic
7	C₂×Dic₉₄	C₂²	1	3	yes	—
8	C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
9	C₄²×C₄₇	C₂²	1	3	yes	—
10	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₄₇	C₂²	1	6	yes	—
12	C₄⋊Dic₄₇	C₂²	1	6	yes	—
13	Dic₄₇⋊C₄	C₂²	1	6	yes	—
14	C₄×Dic₄₇	C₂²	1	6	yes	—
15	Q₈×C₄₇	Q₈	1	4	yes	complemented
16	C₂×C₄×C₄₇	C₂³	1	3	yes	—
17	C₂×C₄×C₄₇	C₂³	1	3	yes	—
18	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₂×Dic₄₇	C₂³	1	2	yes	—
20	C₂×Dic₄₇	C₂³	1	6	yes	—
21	Dic₉₄	—	2	6	no	—
22	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
23	C₄×C₄₇	—	2	3	no	—
24	C₄×C₄₇	C₂×Q₈	1	6	yes	—
25	C₄×C₄₇	#41	1	2	yes	—
26	Dic₄₇	—	2	6	no	—
27	Dic₄₇	—	2	2	no	—
28	C₂×C₄₇	C₂²×Q₈	1	1	yes	characteristic
29	C₂×C₄₇	C₂×#41	1	1	yes	characteristic
30	C₂×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
31	#557	—	47	1	no	maximal, Sylow
32	C₄₇	#557	1	1	yes	characteristic, complemented, Sylow
33	C₄⋊Q₈	—	47	3	no	—
34	#125	—	47	6	no	—
35	C₄×Q₈	—	47	2	no	—
36	C₄×Q₈	—	47	3	no	—
37	C₄×Q₈	D₄₇	1	1	yes	characteristic
38	C₂×Q₈	—	47	3	no	—
39	C₂×Q₈	C₂×D₄₇	1	1	yes	characteristic
40	C₄⋊C₄	—	47	6	no	—
41	C₄⋊C₄	—	47	1	no	—
42	C₄⋊C₄	—	47	6	no	—
43	C₄⋊C₄	—	47	6	no	—
44	C₄⋊C₄	C₂×D₄₇	1	3	yes	—
45	C₄²	—	47	6	no	—
46	C₄²	C₂×D₄₇	1	3	yes	—
47	Q₈	—	94	6	no	—
48	Q₈	Dic₉₄	1	4	yes	complemented
49	C₂×C₄	—	47	2	no	—
50	C₂×C₄	—	47	6	no	—
51	C₂×C₄	C₂²×D₄₇	1	3	yes	—
52	C₂×C₄	C₂²×D₄₇	1	3	yes	—
53	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
54	C₂²	C₂³×D₄₇	1	1	yes	characteristic
55	C₄	—	2	3	no	—
56	C₄	—	94	6	no	—
57	C₄	—	94	2	no	—
58	C₄	C₂×Dic₉₄	1	6	yes	—
59	C₄	#10625354	1	2	yes	—
60	C₂	C₂²×Dic₉₄	1	1	yes	characteristic
61	C₂	C₂×#10625354	1	1	yes	characteristic
62	C₂	#13094071	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
15. Q₈×C₄₇	Q₈	semidirect
32. C₄₇	#557	semidirect
48. Q₈	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#13094024	2
#13094015	2
Q₈×Dic₄₇	2
C₄×Dic₉₄	2
C₄×Q₈×C₄₇	2
#557	47