Subgroups — #14581184

Order 2624.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#66493	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#66493	C₂	1	1	yes	maximal, characteristic, complemented
3	#13050126	C₂	1	1	yes	maximal, characteristic
4	C₂×D₁₆₄	C₄	1	1	yes	characteristic
5	C₂×C₄×D₄₁	—	2	1	no	—
6	C₂×#66493	—	2	1	no	—
7	C₂×#66493	—	2	1	no	—
8	C₄⋊C₄×C₄₁	C₄	1	1	yes	characteristic
9	#10620281	—	2	1	no	—
10	C₄×Dic₄₁	C₂²	1	1	yes	characteristic
11	C₂²×D₄₁	—	2	1	no	—
12	C₂×C₄×C₄₁	—	2	1	no	—
13	C₂×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₄₁	D₄	1	1	yes	characteristic
15	C₂×Dic₄₁	D₄	1	1	yes	characteristic
16	D₁₆₄	—	2	1	no	—
17	D₁₆₄	C₈	1	2	yes	complemented
18	C₄×D₄₁	—	4	1	no	—
19	C₄×D₄₁	—	4	1	no	—
20	#66493	—	2	2	no	—
21	#66493	—	4	1	no	—
22	C₂²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₄₁	—	2	2	no	—
24	C₂×D₄₁	—	4	1	no	—
25	C₂×D₄₁	—	4	1	no	—
26	C₄×C₄₁	—	4	1	no	—
27	C₄×C₄₁	C₂×C₈	1	1	yes	characteristic
28	C₄×C₄₁	M₁₆	1	1	yes	characteristic
29	Dic₄₁	—	2	1	no	—
30	Dic₄₁	D₈	1	1	yes	characteristic
31	Dic₄₁	SD₁₆	1	1	yes	characteristic
32	C₂×C₄₁	C₂²⋊C₈	1	1	yes	characteristic
33	C₂×C₄₁	#102	1	1	yes	characteristic
34	C₂×C₄₁	#104	1	1	yes	characteristic
35	D₄₁	—	4	2	no	—
36	D₄⋊C₈	—	41	1	no	maximal, Sylow
37	C₄₁	D₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
38	C₄×D₄	—	41	1	no	—
39	C₄⋊C₈	—	41	1	no	—
40	C₄×C₈	—	41	1	no	—
41	C₂×D₄	—	41	1	no	—
42	C₂²×C₄	—	82	1	no	—
43	C₂×C₈	—	82	1	no	—
44	C₂×C₈	—	82	1	no	—
45	C₄⋊C₄	#4223	1	1	yes	characteristic
46	C₂²⋊C₄	—	82	1	no	—
47	C₄²	—	41	1	no	—
48	C₂³	—	82	1	no	—
49	D₄	—	41	2	no	—
50	D₄	—	82	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	41	1	no	—
53	C₂×C₄	—	41	1	no	—
54	C₂×C₄	—	164	1	no	—
55	C₂×C₄	—	164	1	no	—
56	C₂×C₄	C₂×#4223	1	1	yes	characteristic
57	C₈	—	82	2	no	—
58	C₈	—	164	1	no	—
59	C₂²	—	82	2	no	—
60	C₂²	—	164	1	no	—
61	C₂²	—	164	1	no	—
62	C₂²	C₂²⋊#4223	1	1	yes	characteristic
63	C₄	—	4	1	no	—
64	C₄	—	41	1	no	—
65	C₄	—	41	1	no	—
66	C₄	—	82	1	no	—
67	C₄	#10620296	1	1	yes	characteristic
68	C₄	#10620297	1	1	yes	characteristic
69	C₂	—	164	2	no	—
70	C₂	#13050221	1	1	yes	characteristic
71	C₂	#13050226	1	1	yes	characteristic
72	C₂	#13050229	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#4223
commutator	C₄×C₄₁	C₂×C₈
Frattini	C₂×C₄	C₂×#4223
Fitting	C₄⋊C₄×C₄₁	C₄
socle	C₂²×C₄₁	C₂²⋊C₄
prime radical	C₂×D₁₆₄	C₄
cosocle	C₄×Dic₄₁	C₂²
hypercenter	C₄⋊C₄	#4223
hypocenter	C₄₁	D₄⋊C₈
LCS	G ▷ C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊#66493	C₂	semidirect
2. C₄×#66493	C₂	semidirect
17. D₁₆₄	C₈	semidirect
37. C₄₁	D₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#66493	2
C₄×#66493	2
#13050126	2
D₄⋊C₈	41