Subgroups — #14580698

Order 2624.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13050152	C₂	1	1	yes	maximal, characteristic
2	#13050087	C₂	1	2	yes	maximal, complemented
3	#13050084	C₂	1	2	yes	maximal
4	#13050075	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₄₁	—	2	1	no	—
7	C₂×D₁₆₄	C₂²	1	1	yes	characteristic
8	C₂×Dic₈₂	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₄₁	—	2	1	no	—
10	C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
11	C₄²×C₄₁	C₂²	1	1	yes	characteristic
12	#10620281	—	2	1	no	—
13	#10620281	—	2	1	no	—
14	C₂×#66491	C₂²	1	2	yes	—
15	C₂²×D₄₁	—	2	1	no	—
16	Q₈×C₄₁	—	4	2	no	—
17	C₂×C₄×C₄₁	—	2	2	no	—
18	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
20	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
21	C₂×Dic₄₁	—	2	1	no	—
22	D₁₆₄	—	2	2	no	—
23	Dic₈₂	—	2	2	no	—
24	#66491	—	2	2	no	—
25	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
26	C₂×D₄₁	—	4	1	no	—
27	C₂×D₄₁	—	4	1	no	—
28	C₂×D₄₁	—	4	1	no	—
29	C₄×C₄₁	—	2	1	no	—
30	C₄×C₄₁	—	2	1	no	—
31	C₄×C₄₁	—	4	2	no	—
32	C₄×C₄₁	#41	1	2	yes	—
33	Dic₄₁	—	4	1	no	—
34	C₂×C₄₁	#124	1	1	yes	characteristic
35	C₂×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₂×C₄₁	#137	1	1	yes	characteristic
37	D₄₁	—	8	1	no	—
38	#489	—	41	1	no	maximal, Sylow
39	C₄₁	#489	1	1	yes	characteristic, complemented, Sylow
40	C₄⋊Q₈	D₄₁	1	1	yes	characteristic, complemented
41	#124	—	41	1	no	—
42	#103	—	41	2	no	—
43	#102	—	41	2	no	—
44	#97	—	41	1	no	—
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	—	41	1	no	—
47	C₂×D₄	—	41	1	no	—
48	C₂×C₈	—	41	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	C₂×D₄₁	1	2	yes	—
51	C₂²⋊C₄	—	82	1	no	—
52	C₂²⋊C₄	—	82	1	no	—
53	C₄²	C₂×D₄₁	1	1	yes	characteristic
54	C₂³	—	82	1	no	—
55	Q₈	—	4	2	no	—
56	Q₈	—	82	2	no	—
57	D₄	—	82	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	82	1	no	—
60	C₂×C₄	C₂²⋊D₄₁	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₄₁	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
63	C₈	—	82	2	no	—
64	C₂²	—	164	1	no	—
65	C₂²	—	164	1	no	—
66	C₂²	—	164	1	no	—
67	C₂²	C₂×C₂²⋊D₄₁	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	2	no	—
71	C₄	—	164	1	no	—
72	C₄	#10620310	1	2	yes	—
73	C₂	—	328	1	no	—
74	C₂	#13050153	1	1	yes	characteristic
75	C₂	#13050190	1	1	yes	characteristic
76	C₂	#13050195	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₁
commutator, cosocle	C₂×C₄×C₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₁
Fitting	C₄⋊Q₈×C₄₁	C₂
socle	C₂²×C₄₁	C₂×D₄
prime radical	C₂×D₁₆₄	C₂²
hypercenter	C₄⋊Q₈	D₄₁
hypocenter	C₄₁	#489
LCS	G ▷ C₂×C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #13050087	C₂	semidirect
4. #13050075	C₂	semidirect
5. C₄⋊Q₈×C₄₁	C₂	semidirect
39. C₄₁	#489	semidirect
40. C₄⋊Q₈	D₄₁	semidirect

Maximal subgroups

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

subgroup	index
#13050152	2
#13050087	2
#13050084	2
#13050075	2
C₄⋊Q₈×C₄₁	2
#489	41