Subgroups — #14580720

Order 2624.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10620273	C₂	1	2	yes	maximal, complemented
3	C₂×D₃₂₈	C₂	1	1	yes	maximal, characteristic
4	#13050152	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆₄	C₂²	1	2	yes	—
7	C₂×Dic₈₂	C₂²	1	2	yes	—
8	C₂×C₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₄²×C₄₁	C₂²	1	1	yes	characteristic
11	#10620281	—	2	4	no	—
12	Dic₁₆₄	—	2	2	no	—
13	D₃₂₈	—	2	2	no	—
14	#10620273	—	2	4	no	—
15	C₂²×D₄₁	—	2	2	no	—
16	C₂×C₄×C₄₁	D₄	1	2	yes	—
17	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
18	C₂×Dic₄₁	—	2	2	no	—
19	D₁₆₄	—	2	2	no	—
20	D₁₆₄	—	2	2	no	—
21	Dic₈₂	—	2	2	no	—
22	Dic₈₂	—	2	2	no	—
23	C₈×C₄₁	D₄	1	2	yes	—
24	C₈×C₄₁	D₄	1	2	yes	—
25	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
26	C₂×D₄₁	—	4	2	no	—
27	C₂×D₄₁	—	4	4	no	—
28	C₄×C₄₁	—	2	2	no	—
29	C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
30	C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
31	Dic₄₁	—	4	2	no	—
32	C₂×C₄₁	#127	1	1	yes	characteristic
33	C₂×C₄₁	#135	1	2	yes	—
34	D₄₁	—	8	2	no	—
35	#495	—	41	1	no	maximal, Sylow
36	C₄₁	#495	1	1	yes	characteristic, complemented, Sylow
37	C₂×Q₁₆	—	41	1	no	—
38	C₂×SD₁₆	—	41	2	no	—
39	C₂×D₈	—	41	1	no	—
40	#124	—	41	2	no	—
41	C₄×C₈	D₄₁	1	1	yes	characteristic, complemented
42	C₂×Q₈	—	41	2	no	—
43	C₂×D₄	—	41	2	no	—
44	Q₁₆	—	82	2	no	—
45	SD₁₆	—	82	4	no	—
46	D₈	—	82	2	no	—
47	C₂×C₈	C₂×D₄₁	1	1	yes	characteristic
48	C₂×C₈	C₂×D₄₁	1	1	yes	characteristic
49	C₂²⋊C₄	—	82	4	no	—
50	C₄²	C₂×D₄₁	1	1	yes	characteristic
51	C₂³	—	82	2	no	—
52	Q₈	—	82	2	no	—
53	Q₈	—	82	2	no	—
54	D₄	—	82	2	no	—
55	D₄	—	82	2	no	—
56	C₂×C₄	—	82	2	no	—
57	C₂×C₄	D₁₆₄	1	2	yes	—
58	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
59	C₈	D₁₆₄	1	2	yes	—
60	C₈	D₁₆₄	1	2	yes	—
61	C₂²	—	164	2	no	—
62	C₂²	—	164	4	no	—
63	C₂²	C₂×D₁₆₄	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	164	2	no	—
66	C₄	C₂×D₁₆₄	1	1	yes	characteristic
67	C₄	C₂×D₁₆₄	1	1	yes	characteristic
68	C₂	—	328	2	no	—
69	C₂	#13050161	1	1	yes	characteristic
70	C₂	#13050186	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₆₄
commutator, cosocle	C₂×C₄×C₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₁
Fitting	C₄×C₈×C₄₁	C₂
socle	C₂²×C₄₁	C₂×D₄
prime radical	C₂×D₃₂₈	C₂
hypercenter	C₄×C₈	D₄₁
hypocenter	C₄₁	#495
LCS	G ▷ C₂×C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₁₆₄	C₂	semidirect
2. C₂×#10620273	C₂	semidirect
4. #13050152	C₂	semidirect
5. C₄×C₈×C₄₁	C₂	semidirect
36. C₄₁	#495	semidirect
41. C₄×C₈	D₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₆₄	2
C₂×#10620273	2
C₂×D₃₂₈	2
#13050152	2
C₄×C₈×C₄₁	2
#495	41