Subgroups — #15689471

Order 3008.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13094021	C₂	1	1	yes	maximal, characteristic
2	#13093944	C₂	1	4	yes	maximal, complemented
3	C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₈×C₄₇	—	2	1	no	—
6	C₂×D₁₈₈	—	2	1	no	—
7	C₂×D₁₈₈	—	2	1	no	—
8	C₂×D₁₈₈	C₂²	1	2	yes	—
9	C₄⋊C₄×C₄₇	—	2	1	no	—
10	C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
11	C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₄₇⋊C₈	C₂²	1	2	yes	—
13	C₂²×D₄₇	—	2	2	no	—
14	Q₈×C₄₇	—	4	2	no	—
15	C₂×C₄×C₄₇	—	2	2	no	—
16	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
17	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	D₁₈₈	—	2	4	no	—
20	D₁₈₈	—	4	2	no	—
21	D₁₈₈	—	4	2	no	—
22	C₄₇⋊C₈	—	2	2	no	—
23	C₂²×C₄₇	C₂×D₄	1	1	yes	characteristic
24	C₂×D₄₇	—	4	2	no	—
25	C₂×D₄₇	—	4	2	no	—
26	C₂×D₄₇	—	4	2	no	—
27	C₄×C₄₇	—	4	2	no	—
28	C₄×C₄₇	D₈	1	2	yes	—
29	C₄×C₄₇	SD₁₆	1	2	yes	—
30	C₄×C₄₇	#41	1	2	yes	—
31	C₂×C₄₇	#124	1	1	yes	characteristic
32	C₂×C₄₇	C₂×D₈	1	1	yes	characteristic
33	C₂×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
34	D₄₇	—	8	2	no	—
35	#486	—	47	1	no	maximal, Sylow
36	C₄₇	#486	1	1	yes	characteristic, complemented, Sylow
37	C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
38	#127	—	47	1	no	—
39	#102	—	47	4	no	—
40	C₄×C₈	—	47	1	no	—
41	C₂×Q₈	—	2	1	no	—
42	C₂×D₄	—	47	2	no	—
43	C₂×D₄	—	94	1	no	—
44	C₂×D₄	—	94	1	no	—
45	C₂×C₈	—	47	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	C₂×D₄₇	1	2	yes	—
48	C₄²	C₂×D₄₇	1	1	yes	characteristic
49	C₂³	—	94	2	no	—
50	Q₈	—	4	2	no	—
51	D₄	—	94	4	no	—
52	D₄	—	188	2	no	—
53	D₄	—	188	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
58	C₈	—	94	2	no	—
59	C₂²	—	188	2	no	—
60	C₂²	—	188	2	no	—
61	C₂²	—	188	2	no	—
62	C₂²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	#10625334	1	2	yes	—
65	C₄	#10625336	1	2	yes	—
66	C₄	#10625354	1	2	yes	—
67	C₂	—	376	2	no	—
68	C₂	#13094013	1	1	yes	characteristic
69	C₂	C₂×#10625334	1	1	yes	characteristic
70	C₂	C₂×#10625336	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂³
Frattini	C₂×C₄	C₂²×D₄₇
Fitting	C₄⋊Q₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂×D₄
prime radical	#13094021	C₂
hypercenter	C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#486
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #13093944	C₂	semidirect
3. C₄×C₄₇⋊C₈	C₂	semidirect
4. C₄⋊Q₈×C₄₇	C₂	semidirect
36. C₄₇	#486	semidirect
37. 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
#13094021	2
#13093944	2
C₄×C₄₇⋊C₈	2
C₄⋊Q₈×C₄₇	2
#486	47