Subgroups — #14580547

Order 2624.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₆₄	C₂	1	2	yes	maximal
2	#13050164	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₈₂	C₂	1	1	yes	maximal, characteristic
4	#13050089	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₂	C₂²	1	2	yes	—
7	C₂×Dic₈₂	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₄₁	C₂²	1	2	yes	—
9	C₄²×C₄₁	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₄₁	—	2	1	no	—
11	C₄⋊Dic₄₁	—	2	1	no	—
12	C₄⋊Dic₄₁	C₂²	1	1	yes	characteristic
13	#10620279	—	2	1	no	—
14	C₄×Dic₄₁	—	2	1	no	—
15	Dic₁₆₄	—	2	4	no	—
16	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
17	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
18	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
19	C₂×Dic₄₁	—	2	1	no	—
20	C₂×Dic₄₁	—	2	1	no	—
21	C₂×Dic₄₁	—	2	2	no	—
22	Dic₈₂	—	2	1	no	—
23	Dic₈₂	—	2	2	no	—
24	Dic₈₂	—	2	2	no	—
25	Dic₈₂	D₄	1	2	yes	—
26	C₈×C₄₁	—	2	2	no	—
27	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
28	C₄×C₄₁	—	2	1	no	—
29	C₄×C₄₁	Q₁₆	1	2	yes	complemented
30	C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
31	C₄×C₄₁	#41	1	1	yes	characteristic
32	Dic₄₁	—	2	2	no	—
33	Dic₄₁	—	4	2	no	—
34	Dic₄₁	—	4	1	no	—
35	C₂×C₄₁	#121	1	1	yes	characteristic
36	C₂×C₄₁	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₄₁	#137	1	1	yes	characteristic
38	#462	—	41	1	no	maximal, Sylow
39	C₄₁	#462	1	1	yes	characteristic, complemented, Sylow
40	C₂×Q₁₆	—	41	2	no	—
41	C₄⋊Q₈	—	41	1	no	—
42	C₄×Q₈	—	41	1	no	—
43	C₄⋊C₈	D₄₁	1	1	yes	characteristic
44	#103	—	41	2	no	—
45	C₂×Q₈	—	41	1	no	—
46	C₂×Q₈	—	41	2	no	—
47	Q₁₆	—	82	4	no	—
48	C₂×C₈	C₂×D₄₁	1	2	yes	—
49	C₄⋊C₄	—	41	1	no	—
50	C₄⋊C₄	—	82	1	no	—
51	C₄⋊C₄	—	82	1	no	—
52	C₄⋊C₄	—	82	1	no	—
53	C₄²	—	82	1	no	—
54	C₄²	C₂×D₄₁	1	1	yes	characteristic
55	Q₈	—	41	2	no	—
56	Q₈	—	82	1	no	—
57	Q₈	—	82	2	no	—
58	Q₈	—	82	2	no	—
59	C₂×C₄	—	82	1	no	—
60	C₂×C₄	—	82	1	no	—
61	C₂×C₄	—	82	2	no	—
62	C₂×C₄	D₁₆₄	1	1	yes	characteristic
63	C₂×C₄	D₁₆₄	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₂²	C₂×D₁₆₄	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	82	2	no	—
69	C₄	—	164	2	no	—
70	C₄	—	164	1	no	—
71	C₄	Dic₁₆₄	1	2	yes	complemented
72	C₄	D₄×D₄₁	1	1	yes	characteristic
73	C₄	#10620310	1	1	yes	characteristic
74	C₂	#13050139	1	1	yes	characteristic
75	C₂	C₂×Dic₁₆₄	1	1	yes	characteristic
76	C₂	#13050197	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₆₄
commutator, cosocle	C₂×C₄×C₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₁
Fitting	C₄⋊C₈×C₄₁	C₂
socle, prime radical	C₂²×C₄₁	C₂×D₄
hypercenter	C₄⋊C₈	D₄₁
hypocenter	C₄₁	#462
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
29. C₄×C₄₁	Q₁₆	semidirect
39. C₄₁	#462	semidirect
71. C₄	Dic₁₆₄	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
#13050164	2
C₄×Dic₈₂	2
#13050089	2
C₄⋊C₈×C₄₁	2
#462	41