Subgroups — #14580577

Order 2624.

114 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	#13050143	C₂	1	2	yes	maximal
3	#13050098	C₂	1	1	yes	maximal, characteristic, complemented
4	#13050089	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×C₄₁	C₂²	1	1	yes	characteristic
7	C₂×Dic₈₂	C₂²	1	2	yes	—
8	C₂×C₈×C₄₁	—	2	2	no	—
9	C₂×C₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₄₁	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₄₁	—	2	2	no	—
12	C₄⋊Dic₄₁	C₂²	1	2	yes	—
13	#10620279	—	2	2	no	—
14	Dic₁₆₄	—	2	2	no	—
15	C₂³×C₄₁	D₄	1	1	yes	characteristic
16	C₂×C₄×C₄₁	—	2	1	no	—
17	C₂×C₄×C₄₁	—	2	1	no	—
18	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
20	C₂×Dic₄₁	—	2	2	no	—
21	C₂×Dic₄₁	—	2	2	no	—
22	Dic₈₂	—	2	2	no	—
23	Dic₈₂	—	2	2	no	—
24	C₈×C₄₁	—	2	1	no	—
25	C₈×C₄₁	D₄	1	2	yes	—
26	C₂²×C₄₁	—	2	1	no	—
27	C₂²×C₄₁	—	2	1	no	—
28	C₂²×C₄₁	Q₁₆	1	2	yes	complemented
29	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
30	C₄×C₄₁	—	2	1	no	—
31	C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
32	C₄×C₄₁	#41	1	1	yes	characteristic
33	Dic₄₁	—	4	2	no	—
34	Dic₄₁	—	4	2	no	—
35	C₂×C₄₁	—	2	2	no	—
36	C₂×C₄₁	#121	1	1	yes	characteristic
37	C₂×C₄₁	C₂×Q₁₆	1	1	yes	characteristic
38	C₂×C₄₁	#135	1	1	yes	characteristic
39	#467	—	41	1	no	maximal, Sylow
40	C₄₁	#467	1	1	yes	characteristic, complemented, Sylow
41	C₂×Q₁₆	—	41	1	no	—
42	C₂²×C₈	D₄₁	1	1	yes	characteristic
43	C₂²⋊Q₈	—	41	2	no	—
44	#107	—	41	1	no	—
45	#103	—	41	2	no	—
46	C₂×Q₈	—	41	2	no	—
47	C₂²×C₄	C₂×D₄₁	1	1	yes	characteristic
48	Q₁₆	—	82	2	no	—
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	C₂×D₄₁	1	1	yes	characteristic
51	C₂×C₈	C₂×D₄₁	1	1	yes	characteristic
52	C₄⋊C₄	—	41	2	no	—
53	C₄⋊C₄	—	82	2	no	—
54	C₂²⋊C₄	—	82	2	no	—
55	C₂³	D₁₆₄	1	1	yes	characteristic
56	Q₈	—	82	2	no	—
57	Q₈	—	82	2	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	82	2	no	—
61	C₂×C₄	—	82	2	no	—
62	C₂×C₄	D₁₆₄	1	1	yes	characteristic
63	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
64	C₈	—	2	1	no	—
65	C₈	C₂²⋊D₄₁	1	2	yes	—
66	C₂²	—	2	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	Dic₁₆₄	1	2	yes	complemented
69	C₂²	C₂×D₁₆₄	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	164	2	no	—
72	C₄	—	164	2	no	—
73	C₄	#10620306	1	1	yes	characteristic
74	C₄	C₂×C₂²⋊D₄₁	1	1	yes	characteristic
75	C₂	—	2	2	no	—
76	C₂	#13050138	1	1	yes	characteristic
77	C₂	C₂×Dic₁₆₄	1	1	yes	characteristic
78	C₂	#13050186	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₆₄
commutator, cosocle	C₂×C₄×C₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₁
Fitting	C₂²×C₈×C₄₁	C₂
socle	C₂²×C₄₁	C₂×D₄
prime radical	C₂³×C₄₁	D₄
hypercenter	C₂²×C₈	D₄₁
hypocenter	C₄₁	#467
LCS	G ▷ C₂×C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₁₆₄	C₂	semidirect
3. #13050098	C₂	semidirect
4. #13050089	C₂	semidirect
28. C₂²×C₄₁	Q₁₆	semidirect
40. C₄₁	#467	semidirect
68. C₂²	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
#13050143	2
#13050098	2
#13050089	2
C₂²×C₈×C₄₁	2
#467	41