Subgroups — #14580616

Order 2624.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13050164	C₂	1	1	yes	maximal, characteristic, complemented
2	#13050097	C₂	1	2	yes	maximal, complemented
3	C₄⋊#66491	C₂	1	1	yes	maximal, characteristic, complemented
4	#13050085	C₂	1	2	yes	maximal
5	C₄×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic
6	C₂×D₄×C₄₁	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₄₁	—	2	1	no	—
8	C₂×Dic₈₂	—	2	1	no	—
9	C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
10	C₂²⋊C₄×C₄₁	—	2	1	no	—
11	C₄²×C₄₁	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₄₁	—	2	1	no	—
13	C₄⋊Dic₄₁	C₂²	1	2	yes	—
14	C₂×#66491	C₂²	1	2	yes	—
15	C₂³×C₄₁	—	2	1	no	—
16	D₄×C₄₁	—	2	1	no	—
17	D₄×C₄₁	Q₈	1	2	yes	complemented
18	C₂×C₄×C₄₁	—	2	1	no	—
19	C₂×C₄×C₄₁	—	4	2	no	—
20	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
21	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
22	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
23	C₂×Dic₄₁	—	2	2	no	—
24	Dic₈₂	—	4	2	no	—
25	#66491	—	2	2	no	—
26	C₂²×C₄₁	—	2	2	no	—
27	C₂²×C₄₁	—	4	1	no	—
28	C₂²×C₄₁	—	4	1	no	—
29	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
30	C₄×C₄₁	—	2	1	no	—
31	C₄×C₄₁	—	4	1	no	—
32	C₄×C₄₁	D₈	1	2	yes	—
33	C₄×C₄₁	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₄₁	#41	1	1	yes	characteristic
35	Dic₄₁	—	4	2	no	—
36	C₂×C₄₁	—	4	2	no	—
37	C₂×C₄₁	C₂²⋊Q₈	1	1	yes	characteristic
38	C₂×C₄₁	C₂×D₈	1	1	yes	characteristic
39	C₂×C₄₁	#137	1	1	yes	characteristic
40	#474	—	41	1	no	maximal, Sylow
41	C₄₁	#474	1	1	yes	characteristic, complemented, Sylow
42	C₄⋊Q₈	—	41	1	no	—
43	C₄×D₄	D₄₁	1	1	yes	characteristic
44	#107	—	41	2	no	—
45	C₄⋊C₈	—	41	1	no	—
46	#102	—	41	2	no	—
47	C₂×Q₈	—	82	1	no	—
48	C₂×D₄	C₂×D₄₁	1	1	yes	characteristic
49	C₂²×C₄	—	2	1	no	—
50	C₂×C₈	—	41	2	no	—
51	C₄⋊C₄	—	41	2	no	—
52	C₄⋊C₄	—	82	1	no	—
53	C₄⋊C₄	C₂×D₄₁	1	1	yes	characteristic
54	C₂²⋊C₄	—	2	1	no	—
55	C₄²	C₂×D₄₁	1	1	yes	characteristic
56	C₂³	—	2	1	no	—
57	Q₈	—	164	2	no	—
58	D₄	—	2	1	no	—
59	D₄	Dic₈₂	1	2	yes	complemented
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	4	2	no	—
62	C₂×C₄	—	82	2	no	—
63	C₂×C₄	C₂²⋊D₄₁	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₄₁	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
66	C₈	—	82	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	C₂×C₂²⋊D₄₁	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	164	2	no	—
74	C₄	#10620282	1	2	yes	—
75	C₄	C₂×Dic₈₂	1	1	yes	characteristic
76	C₄	#10620306	1	1	yes	characteristic
77	C₂	—	4	2	no	—
78	C₂	#13050143	1	1	yes	characteristic
79	C₂	C₂×#10620282	1	1	yes	characteristic
80	C₂	#13050194	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×D₄×C₄₁	C₂
socle	C₂²×C₄₁	C₂×D₄
prime radical	C₂×D₄×C₄₁	C₂²
hypercenter	C₄×D₄	D₄₁
hypocenter	C₄₁	#474
LCS	G ▷ C₂×C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #13050164	C₂	semidirect
2. #13050097	C₂	semidirect
3. C₄⋊#66491	C₂	semidirect
17. D₄×C₄₁	Q₈	semidirect
41. C₄₁	#474	semidirect
59. D₄	Dic₈₂	semidirect

Maximal subgroups

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

subgroup	index
#13050164	2
#13050097	2
C₄⋊#66491	2
#13050085	2
C₄×D₄×C₄₁	2
#474	41