Subgroups — #35229700

Order 20896.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₆₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₆₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊D₆₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₆₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#23782584	C₂	1	1	yes	maximal, characteristic, complemented
6	#23782582	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₄×C₆₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₆₅₃	C₂²	1	1	yes	characteristic
9	C₂²×D₆₅₃	C₂²	1	1	yes	characteristic
10	D₂₆₁₂	—	2	2	no	—
11	C₂²⋊D₆₅₃	—	2	2	no	—
12	C₂²⋊D₆₅₃	—	2	2	no	—
13	C₂×Dic₆₅₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₆₅₃	C₂²	1	1	yes	characteristic
15	C₄×D₆₅₃	—	2	2	no	—
16	C₂³×C₆₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₆₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₆₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₆₅₃	—	2	1	no	—
20	C₂²×C₆₅₃	—	2	1	no	—
21	C₂²×C₆₅₃	—	2	1	no	—
22	C₂²×C₆₅₃	C₂³	1	1	yes	characteristic
23	C₂×D₆₅₃	—	2	1	no	—
24	C₂×D₆₅₃	—	2	1	no	—
25	C₂×D₆₅₃	—	2	1	no	—
26	C₂×D₆₅₃	—	2	1	no	—
27	C₂×D₆₅₃	—	2	1	no	—
28	C₂×D₆₅₃	D₄	1	2	yes	complemented
29	C₄×C₆₅₃	—	2	1	no	—
30	C₄×C₆₅₃	—	2	1	no	—
31	Dic₆₅₃	—	2	1	no	—
32	Dic₆₅₃	D₄	1	2	yes	complemented
33	C₂×C₆₅₃	—	4	1	no	—
34	C₂×C₆₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₆₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₆₅₃	#41	1	1	yes	characteristic
37	D₆₅₃	—	2	2	no	—
38	D₆₅₃	—	4	1	no	—
39	C₆₅₃	#121	1	1	yes	characteristic, complemented, Sylow
40	#121	—	653	1	no	maximal, Sylow
41	C₂×D₄	—	653	1	no	—
42	C₂×D₄	—	653	1	no	—
43	C₂×D₄	—	653	1	no	—
44	C₂²×C₄	—	653	1	no	—
45	C₄⋊C₄	—	653	1	no	—
46	C₂²⋊C₄	—	653	1	no	—
47	C₂²⋊C₄	D₆₅₃	1	1	yes	characteristic, complemented
48	C₂³	—	653	1	no	—
49	C₂³	—	653	1	no	—
50	C₂³	C₂×D₆₅₃	1	1	yes	characteristic
51	D₄	—	1306	2	no	—
52	D₄	—	1306	2	no	—
53	D₄	—	1306	2	no	—
54	C₂×C₄	—	1306	2	no	—
55	C₂×C₄	—	653	1	no	—
56	C₂×C₄	—	653	1	no	—
57	C₂×C₄	C₂×D₆₅₃	1	1	yes	characteristic
58	C₂×C₄	C₂×D₆₅₃	1	1	yes	characteristic
59	C₂²	—	2	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	—	1306	1	no	—
63	C₂²	—	1306	1	no	—
64	C₂²	—	1306	1	no	—
65	C₂²	—	1306	1	no	—
66	C₂²	—	1306	1	no	—
67	C₂²	—	653	2	no	—
68	C₂²	C₂²×D₆₅₃	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	2	1	no	—
71	C₄	—	1306	1	no	—
72	C₄	—	653	2	no	—
73	C₂	—	4	1	no	—
74	C₂	—	1306	2	no	—
75	C₂	—	2612	1	no	—
76	C₂	D₄×D₆₅₃	1	1	yes	characteristic
77	C₂	D₄×D₆₅₃	1	1	yes	characteristic
78	C₂	#23782605	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₆₅₃
commutator, socle, cosocle	C₂²×C₆₅₃	C₂³
Fitting	C₂²⋊C₄×C₆₅₃	C₂
hypercenter	C₂²⋊C₄	D₆₅₃
hypocenter	C₆₅₃	#121
LCS	G ▷ C₂²×C₆₅₃ ▷ C₆₅₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₆₁₂	C₂	semidirect
2. C₂×C₂²⋊D₆₅₃	C₂	semidirect
3. C₂×C₂²⋊D₆₅₃	C₂	semidirect
4. C₂×C₄×D₆₅₃	C₂	semidirect
5. #23782584	C₂	semidirect
6. #23782582	C₂	semidirect
7. C₂²⋊C₄×C₆₅₃	C₂	semidirect
28. C₂×D₆₅₃	D₄	semidirect
32. Dic₆₅₃	D₄	semidirect
39. C₆₅₃	#121	semidirect
47. C₂²⋊C₄	D₆₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₆₁₂	2
C₂×C₂²⋊D₆₅₃	2
C₂×C₄×D₆₅₃	2
#23782584	2
#23782582	2
C₂²⋊C₄×C₆₅₃	2
#121	653