Subgroups — #15689795

Order 3008.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625354	C₂	1	2	yes	maximal, complemented
2	#13094021	C₂	1	3	yes	maximal, complemented
3	#13093999	C₂	1	6	yes	maximal, complemented
4	C₄×D₁₈₈	C₂	1	3	yes	maximal, complemented
5	C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₄₇	C₂²	1	1	yes	characteristic
7	#10625354	—	2	8	no	—
8	C₂×D₁₈₈	C₂²	1	3	yes	—
9	C₂×D₁₈₈	C₂²	1	6	yes	—
10	C₂×D₁₈₈	C₂²	1	6	yes	—
11	C₂×C₄×D₄₇	C₂²	1	6	yes	—
12	C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
13	C₄²×C₄₇	C₂²	1	3	yes	—
14	(C₂×D₄₇)⋊C₄	C₂²	1	6	yes	—
15	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂²×D₄₇	C₂³	1	6	yes	—
17	Q₈×C₄₇	D₄	1	4	yes	complemented
18	C₂×C₄×C₄₇	C₂³	1	3	yes	—
19	C₂×C₄×C₄₇	C₂³	1	3	yes	—
20	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
21	C₂×Dic₄₇	C₂³	1	2	yes	—
22	D₁₈₈	—	2	12	no	—
23	D₁₈₈	—	2	6	no	—
24	D₁₈₈	—	4	6	no	—
25	C₄×D₄₇	—	2	12	no	—
26	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
27	C₂×D₄₇	—	2	6	no	—
28	C₂×D₄₇	—	2	6	no	—
29	C₂×D₄₇	—	2	6	no	—
30	C₄×C₄₇	—	2	3	no	—
31	C₄×C₄₇	C₂×D₄	1	6	yes	—
32	C₄×C₄₇	#41	1	2	yes	complemented
33	Dic₄₇	—	2	2	no	—
34	C₂×C₄₇	C₂²×D₄	1	1	yes	characteristic
35	C₂×C₄₇	C₂×#41	1	1	yes	characteristic
36	C₂×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
37	D₄₇	—	4	6	no	—
38	#550	—	47	1	no	maximal, Sylow
39	C₄₇	#550	1	1	yes	characteristic, complemented, Sylow
40	C₂×#41	—	47	2	no	—
41	#127	—	47	3	no	—
42	#121	—	47	6	no	—
43	C₄×Q₈	D₄₇	1	1	yes	characteristic, complemented
44	C₄×D₄	—	47	3	no	—
45	#41	—	94	8	no	—
46	C₂×Q₈	C₂×D₄₇	1	1	yes	characteristic
47	C₂×D₄	—	47	3	no	—
48	C₂×D₄	—	47	6	no	—
49	C₂×D₄	—	47	6	no	—
50	C₂²×C₄	—	47	6	no	—
51	C₄⋊C₄	—	47	1	no	—
52	C₄⋊C₄	C₂×D₄₇	1	3	yes	—
53	C₂²⋊C₄	—	47	6	no	—
54	C₄²	C₂×D₄₇	1	3	yes	—
55	C₂³	—	47	6	no	—
56	Q₈	D₁₈₈	1	4	yes	complemented
57	D₄	—	94	12	no	—
58	D₄	—	94	6	no	—
59	D₄	—	188	6	no	—
60	C₂×C₄	—	47	2	no	—
61	C₂×C₄	—	94	12	no	—
62	C₂×C₄	C₂²×D₄₇	1	3	yes	—
63	C₂×C₄	C₂²×D₄₇	1	3	yes	—
64	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
65	C₂²	—	94	6	no	—
66	C₂²	—	94	6	no	—
67	C₂²	—	94	6	no	—
68	C₂²	C₂³×D₄₇	1	1	yes	characteristic
69	C₄	—	2	3	no	—
70	C₄	—	94	2	no	—
71	C₄	C₂×D₁₈₈	1	6	yes	—
72	C₄	#10625354	1	2	yes	complemented
73	C₂	—	188	6	no	—
74	C₂	C₂²×D₁₈₈	1	1	yes	characteristic
75	C₂	C₂×#10625354	1	1	yes	characteristic
76	C₂	#13094070	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	#550
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. C₂×#10625354	C₂	semidirect
2. #13094021	C₂	semidirect
3. #13093999	C₂	semidirect
4. C₄×D₁₈₈	C₂	semidirect
5. C₄×Q₈×C₄₇	C₂	semidirect
17. Q₈×C₄₇	D₄	semidirect
32. C₄×C₄₇	#41	semidirect
39. C₄₇	#550	semidirect
43. C₄×Q₈	D₄₇	semidirect
56. Q₈	D₁₈₈	semidirect
72. C₄	#10625354	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625354	2
#13094021	2
#13093999	2
C₄×D₁₈₈	2
C₄×Q₈×C₄₇	2
#550	47