Subgroups — #36378708

Order 22592.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150740	C₂	1	2	yes	maximal
2	C₈×Dic₃₅₃	C₂	1	1	yes	maximal, characteristic
3	#19150740	—	2	2	no	—
4	C₂×#15667725	C₂²	1	1	yes	characteristic
5	C₄×Dic₃₅₃	C₄	1	1	yes	characteristic
6	C₂×C₈×C₃₅₃	C₄	1	1	yes	characteristic
7	C₂×Dic₃₅₃	C₈	1	1	yes	characteristic
8	C₂×Dic₃₅₃	C₈	1	1	yes	characteristic
9	#15667725	D₄	1	1	yes	characteristic
10	#15667725	Q₈	1	1	yes	characteristic
11	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
12	C₈×C₃₅₃	—	2	1	no	—
13	C₂²×C₃₅₃	C₂×C₈	1	1	yes	characteristic
14	C₄×C₃₅₃	C₄⋊C₄	1	1	yes	characteristic
15	C₄×C₃₅₃	M₁₆	1	1	yes	characteristic
16	Dic₃₅₃	—	2	1	no	—
17	Dic₃₅₃	C₁₆	1	2	yes	complemented
18	C₂×C₃₅₃	C₄⋊C₈	1	1	yes	characteristic
19	C₂×C₃₅₃	C₂×C₁₆	1	1	yes	characteristic
20	C₂×C₃₅₃	M₃₂	1	1	yes	characteristic
21	C₃₅₃	C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
22	C₄⋊C₁₆	—	353	1	no	maximal, Sylow
23	C₂×C₁₆	—	353	2	no	—
24	C₄×C₈	—	353	1	no	—
25	C₂×C₈	—	353	1	no	—
26	C₂×C₈	#13083443	1	1	yes	characteristic
27	C₄²	—	353	1	no	—
28	C₁₆	—	706	2	no	—
29	C₂×C₄	—	353	1	no	—
30	C₂×C₄	—	353	1	no	—
31	C₂×C₄	C₂×#13083443	1	1	yes	characteristic
32	C₈	—	2	1	no	—
33	C₈	—	353	1	no	—
34	C₈	—	353	1	no	—
35	C₂²	#19150745	1	1	yes	characteristic
36	C₄	—	353	2	no	—
37	C₄	—	706	1	no	—
38	C₄	C₄⋊#13083443	1	1	yes	characteristic
39	C₄	#19150746	1	1	yes	characteristic
40	C₂	#26015059	1	1	yes	characteristic
41	C₂	#26015065	1	1	yes	characteristic
42	C₂	#26015067	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×#13083443
commutator	C₂×C₃₅₃	C₂×C₁₆
Fitting	C₂×C₈×C₃₅₃	C₄
socle, prime radical	C₂²×C₃₅₃	C₂×C₈
cosocle	C₂×#15667725	C₂²
hypercenter	C₂×C₈	#13083443
hypocenter	C₃₅₃	C₄⋊C₁₆
LCS	G ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
17. Dic₃₅₃	C₁₆	semidirect
21. C₃₅₃	C₄⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19150740	2
C₈×Dic₃₅₃	2
C₄⋊C₁₆	353