Subgroups — #36377551

Order 22592.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150721	C₂	1	1	yes	maximal, characteristic
2	C₂²×#15667725	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×C₃₅₃	C₂	1	1	yes	maximal, characteristic
4	#19150721	—	2	1	no	—
5	#19150721	C₄	1	2	yes	—
6	C₂×#15667725	—	2	1	no	—
7	C₂×#15667725	—	2	1	no	—
8	C₂×#15667725	—	2	1	no	—
9	C₂×#15667725	C₄	1	2	yes	—
10	C₂²×C₄×C₃₅₃	C₂²	1	1	yes	characteristic
11	M₁₆×C₃₅₃	—	2	1	no	—
12	M₁₆×C₃₅₃	C₄	1	2	yes	—
13	C₂×C₈×C₃₅₃	—	2	1	no	—
14	#15667725	—	2	2	no	—
15	#15667725	—	2	2	no	—
16	C₂³×C₃₅₃	Q₈	1	1	yes	characteristic
17	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₃₅₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₃₅₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₃₅₃	D₄	1	1	yes	characteristic
23	C₈×C₃₅₃	—	2	2	no	—
24	C₂²×C₃₅₃	—	2	2	no	—
25	C₂²×C₃₅₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₃₅₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₃₅₃	C₄⋊C₄	1	1	yes	characteristic
28	C₄×C₃₅₃	C₄²	1	1	yes	characteristic
29	C₄×C₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₄×C₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₄×C₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₃₅₃	—	2	1	no	—
33	C₂×C₃₅₃	—	2	1	no	—
34	C₂×C₃₅₃	#95	1	1	yes	characteristic
35	C₂×C₃₅₃	#108	1	2	yes	—
36	C₃₅₃	#341	1	1	yes	characteristic, complemented, Sylow
37	#341	—	353	1	no	maximal, Sylow
38	C₂×M₁₆	—	353	1	no	—
39	C₂×M₁₆	D₃₅₃	1	1	yes	characteristic
40	C₂²×C₈	—	353	1	no	—
41	C₂²×C₄	C₂×D₃₅₃	1	1	yes	characteristic
42	M₁₆	—	2	1	no	—
43	M₁₆	—	353	2	no	—
44	M₁₆	—	706	1	no	—
45	M₁₆	Dic₃₅₃	1	2	yes	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	353	2	no	—
48	C₂×C₈	—	706	1	no	—
49	C₂×C₈	—	706	1	no	—
50	C₂×C₈	—	706	1	no	—
51	C₂³	Dic₇₀₆	1	1	yes	characteristic
52	C₂×C₄	C₄×D₃₅₃	1	1	yes	characteristic
53	C₂×C₄	C₄×D₃₅₃	1	1	yes	characteristic
54	C₂×C₄	C₂×Dic₃₅₃	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₃₅₃	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₃₅₃	1	1	yes	characteristic
57	C₂×C₄	D₁₄₁₂	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	706	2	no	—
60	C₈	—	706	2	no	—
61	C₂²	—	2	2	no	—
62	C₂²	#19150716	1	1	yes	characteristic
63	C₂²	#19150716	1	1	yes	characteristic
64	C₂²	C₄⋊Dic₃₅₃	1	1	yes	characteristic
65	C₄	C₄×Dic₃₅₃	1	1	yes	characteristic
66	C₄	#19150714	1	1	yes	characteristic
67	C₄	#19150714	1	1	yes	characteristic
68	C₄	C₂²⋊Dic₃₅₃	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	2	1	no	—
71	C₂	#26014899	1	1	yes	characteristic
72	C₂	#26014927	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₃₅₃
commutator	C₄×C₃₅₃	C₄²
Frattini	C₂²×C₄	C₂×D₃₅₃
Fitting	C₂×M₁₆×C₃₅₃	C₂
socle	C₂²×C₃₅₃	C₄⋊C₄
prime radical	C₂³×C₃₅₃	Q₈
cosocle	C₂²×C₄×C₃₅₃	C₂²
hypercenter	C₂×M₁₆	D₃₅₃
hypocenter	C₃₅₃	#341
LCS	G ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
36. C₃₅₃	#341	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19150721	2
C₂²×#15667725	2
C₂×M₁₆×C₃₅₃	2
#341	353