Subgroups — #15689853

Order 3008.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13094023	C₂	1	4	yes	maximal
2	#13094022	C₂	1	4	yes	maximal
3	C₄×Dic₉₄	C₂	1	2	yes	maximal
4	#13093989	C₂	1	4	yes	maximal
5	C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₄₇	C₂²	1	2	yes	—
7	C₂×Dic₉₄	C₂²	1	4	yes	—
8	C₂×Dic₉₄	C₂²	1	2	yes	—
9	C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
10	C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	Dic₄₇⋊C₄	C₂²	1	8	yes	—
13	Dic₄₇⋊C₄	C₂²	1	4	yes	—
14	C₄×Dic₄₇	C₂²	1	4	yes	—
15	C₄×Dic₄₇	C₂²	1	4	yes	—
16	Q₈×C₄₇	—	4	2	no	—
17	C₂×C₄×C₄₇	C₂³	1	4	yes	—
18	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₄₇	C₂³	1	2	yes	—
20	C₂×Dic₄₇	C₂³	1	4	yes	—
21	C₂×Dic₄₇	C₂³	1	4	yes	—
22	Dic₉₄	—	4	4	no	—
23	Dic₉₄	Q₈	1	8	yes	complemented
24	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
25	C₄×C₄₇	—	2	4	no	—
26	C₄×C₄₇	—	2	1	no	—
27	C₄×C₄₇	C₂×Q₈	1	4	yes	—
28	Dic₄₇	—	2	4	no	—
29	Dic₄₇	C₂×Q₈	1	8	yes	—
30	C₂×C₄₇	C₂²×Q₈	1	2	yes	—
31	C₂×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
32	Q₈²	—	47	1	no	maximal, Sylow
33	C₄₇	Q₈²	1	1	yes	characteristic, complemented, Sylow
34	C₄⋊Q₈	—	47	4	no	—
35	C₄⋊Q₈	—	47	4	no	—
36	C₄⋊Q₈	D₄₇	1	1	yes	characteristic
37	C₄×Q₈	—	47	4	no	—
38	C₄×Q₈	—	47	2	no	—
39	C₂×Q₈	—	47	2	no	—
40	C₂×Q₈	—	47	4	no	—
41	C₂×Q₈	C₂×D₄₇	1	2	yes	—
42	C₄⋊C₄	—	47	8	no	—
43	C₄⋊C₄	—	47	4	no	—
44	C₄⋊C₄	—	47	2	no	—
45	C₄⋊C₄	C₂×D₄₇	1	4	yes	—
46	C₄²	—	47	4	no	—
47	C₄²	—	47	4	no	—
48	C₄²	C₂×D₄₇	1	1	yes	characteristic
49	Q₈	—	4	2	no	—
50	Q₈	—	47	8	no	—
51	Q₈	—	188	4	no	—
52	C₂×C₄	—	47	4	no	—
53	C₂×C₄	—	47	4	no	—
54	C₂×C₄	C₂²×D₄₇	1	4	yes	—
55	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₄₇	1	2	yes	—
57	C₂²	C₂³×D₄₇	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	4	no	—
60	C₄	—	47	8	no	—
61	C₄	—	94	4	no	—
62	C₄	Q₈×D₄₇	1	4	yes	—
63	C₂	C₂×Q₈×D₄₇	1	2	yes	—
64	C₂	#13094069	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	Q₈²
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
23. Dic₉₄	Q₈	semidirect
33. C₄₇	Q₈²	semidirect

Maximal subgroups

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

subgroup	index
#13094023	2
#13094022	2
C₄×Dic₉₄	2
#13093989	2
C₄⋊Q₈×C₄₇	2
Q₈²	47