Subgroups — #36378700

Order 22592.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26015065	C₂	1	1	yes	maximal, characteristic
2	#26015061	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×D₃₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#13083443	—	2	1	no	—
5	#19150740	—	2	1	no	—
6	Q₈×D₃₅₃	—	2	1	no	—
7	Dic₁₄₁₂	C₄	1	1	yes	characteristic, complemented
8	#19150727	—	2	1	no	—
9	C₈×D₃₅₃	C₂²	1	1	yes	characteristic
10	Q₁₆×C₃₅₃	C₄	1	1	yes	characteristic, complemented
11	C₂×#13083443	—	4	1	no	—
12	Dic₇₀₆	—	2	1	no	—
13	Dic₇₀₆	—	4	1	no	—
14	C₄×D₃₅₃	—	4	1	no	—
15	C₄×D₃₅₃	D₄	1	1	yes	characteristic
16	#15667725	D₄	1	1	yes	characteristic
17	Q₈×C₃₅₃	—	2	1	no	—
18	C₈×C₃₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×D₃₅₃	D₈	1	1	yes	characteristic
20	#13083443	—	8	1	no	—
21	C₄×C₃₅₃	—	4	1	no	—
22	C₄×C₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	Dic₃₅₃	—	4	1	no	—
24	Dic₃₅₃	SD₁₆	1	1	yes	characteristic
25	C₂×C₃₅₃	#102	1	1	yes	characteristic
26	D₃₅₃	SD₃₂	1	1	yes	characteristic
27	D₃₅₃	Q₃₂	1	1	yes	characteristic
28	C₃₅₃	#358	1	1	yes	characteristic, complemented, Sylow
29	#358	—	353	1	no	maximal, Sylow
30	C₂×Q₁₆	—	353	1	no	—
31	C₂×C₁₆	—	353	1	no	—
32	#107	—	353	1	no	—
33	C₂×Q₈	—	706	1	no	—
34	Q₁₆	—	353	1	no	—
35	Q₁₆	—	706	1	no	—
36	Q₁₆	#13083443	1	1	yes	characteristic, complemented
37	C₂×C₈	—	353	1	no	—
38	C₄⋊C₄	—	706	1	no	—
39	C₁₆	—	706	1	no	—
40	Q₈	—	2	1	no	—
41	Q₈	—	1412	1	no	—
42	Q₈	—	706	1	no	—
43	C₂×C₄	—	353	1	no	—
44	C₂×C₄	—	1412	1	no	—
45	C₂×C₄	—	1412	1	no	—
46	C₈	—	353	1	no	—
47	C₈	C₂×#13083443	1	1	yes	characteristic
48	C₂²	—	353	1	no	—
49	C₄	—	4	1	no	—
50	C₄	—	2824	1	no	—
51	C₄	—	353	1	no	—
52	C₄	—	1412	1	no	—
53	C₄	C₂²⋊#13083443	1	1	yes	characteristic
54	C₂	—	353	1	no	—
55	C₂	—	353	1	no	—
56	C₂	#26015054	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26015054
commutator	C₈×C₃₅₃	C₂×C₄
Frattini	C₈	C₂×#13083443
Fitting	Q₁₆×C₃₅₃	C₄
socle	C₂×C₃₅₃	#102
prime radical	C₂×D₃₅₃	D₈
cosocle	C₈×D₃₅₃	C₂²
hypercenter	Q₁₆	#13083443
hypocenter	C₃₅₃	#358
LCS	G ▷ C₈×C₃₅₃ ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
7. Dic₁₄₁₂	C₄	semidirect
10. Q₁₆×C₃₅₃	C₄	semidirect
28. C₃₅₃	#358	semidirect
36. Q₁₆	#13083443	semidirect

Maximal subgroups

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

subgroup	index
#26015065	2
#26015061	2
Q₁₆×D₃₅₃	2
#358	353