Subgroups — #15689839

Order 3008.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13094013	C₂	1	4	yes	maximal, complemented
2	#13093999	C₂	1	4	yes	maximal, complemented
3	C₄×D₁₈₈	C₂	1	2	yes	maximal, complemented
4	#13093982	C₂	1	4	yes	maximal, complemented
5	C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₄₇	C₂²	1	2	yes	—
7	C₂×D₁₈₈	C₂²	1	2	yes	—
8	C₂×D₁₈₈	C₂²	1	4	yes	—
9	C₂×C₄×D₄₇	C₂²	1	4	yes	—
10	C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
11	C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	(C₂×D₄₇)⋊C₄	C₂²	1	8	yes	—
13	(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
14	C₄⋊Dic₄₇	C₂²	1	2	yes	—
15	C₄×Dic₄₇	C₂²	1	4	yes	—
16	C₂²×D₄₇	C₂³	1	4	yes	—
17	Q₈×C₄₇	—	4	2	no	—
18	C₂×C₄×C₄₇	C₂³	1	4	yes	—
19	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₄₇	C₂³	1	2	yes	—
21	C₂×Dic₄₇	C₂³	1	4	yes	—
22	D₁₈₈	—	2	4	no	—
23	D₁₈₈	—	4	4	no	—
24	C₄×D₄₇	—	2	8	no	—
25	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
26	C₂×D₄₇	—	2	4	no	—
27	C₂×D₄₇	—	2	4	no	—
28	C₂×D₄₇	—	2	4	no	—
29	C₄×C₄₇	—	2	4	no	—
30	C₄×C₄₇	—	2	1	no	—
31	C₄×C₄₇	#41	1	4	yes	—
32	Dic₄₇	—	2	4	no	—
33	C₂×C₄₇	C₂×#41	1	2	yes	—
34	C₂×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
35	D₄₇	—	4	4	no	—
36	#555	—	47	1	no	maximal, Sylow
37	C₄₇	#555	1	1	yes	characteristic, complemented, Sylow
38	C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
39	#124	—	47	4	no	—
40	#121	—	47	4	no	—
41	C₄×D₄	—	47	2	no	—
42	#117	—	47	4	no	—
43	C₂×Q₈	C₂×D₄₇	1	2	yes	—
44	C₂×D₄	—	47	2	no	—
45	C₂×D₄	—	47	4	no	—
46	C₂²×C₄	—	47	4	no	—
47	C₄⋊C₄	—	47	2	no	—
48	C₄⋊C₄	C₂×D₄₇	1	4	yes	—
49	C₂²⋊C₄	—	47	8	no	—
50	C₂²⋊C₄	—	47	4	no	—
51	C₄²	—	47	4	no	—
52	C₄²	C₂×D₄₇	1	1	yes	characteristic
53	C₂³	—	47	4	no	—
54	Q₈	—	4	2	no	—
55	D₄	—	94	4	no	—
56	D₄	—	188	4	no	—
57	C₂×C₄	—	47	4	no	—
58	C₂×C₄	—	94	8	no	—
59	C₂×C₄	C₂²×D₄₇	1	4	yes	—
60	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₄₇	1	2	yes	—
62	C₂²	—	94	4	no	—
63	C₂²	—	94	4	no	—
64	C₂²	—	94	4	no	—
65	C₂²	C₂³×D₄₇	1	1	yes	characteristic
66	C₄	—	2	4	no	—
67	C₄	—	2	1	no	—
68	C₄	—	94	4	no	—
69	C₄	#10625354	1	4	yes	—
70	C₂	—	188	4	no	—
71	C₂	C₂×#10625354	1	2	yes	—
72	C₂	#13094069	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	C₄⋊Q₈×C₄₇	C₂
hypercenter	C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#555
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
1. #13094013	C₂	semidirect
2. #13093999	C₂	semidirect
3. C₄×D₁₈₈	C₂	semidirect
4. #13093982	C₂	semidirect
5. C₄⋊Q₈×C₄₇	C₂	semidirect
37. C₄₇	#555	semidirect
38. 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
#13094013	2
#13093999	2
C₄×D₁₈₈	2
#13093982	2
C₄⋊Q₈×C₄₇	2
#555	47