Subgroups — #36378637

Order 22592.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#15667732	C₂	1	1	yes	maximal, characteristic
2	C₄×#15667732	C₂	1	1	yes	maximal, characteristic
3	#26014957	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667732	—	2	1	no	—
5	C₂×#15667732	—	2	1	no	—
6	C₂×Dic₇₀₆	C₄	1	1	yes	characteristic
7	#19150716	—	2	1	no	—
8	C₄×Dic₃₅₃	—	2	1	no	—
9	C₄×Dic₃₅₃	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₃₅₃	C₄	1	1	yes	characteristic
11	#15667732	—	2	2	no	—
12	#15667732	—	4	1	no	—
13	Dic₇₀₆	—	2	1	no	—
14	Dic₇₀₆	C₈	1	2	yes	complemented
15	C₂×Dic₃₅₃	—	2	1	no	—
16	C₂×Dic₃₅₃	D₄	1	1	yes	characteristic
17	C₂×Dic₃₅₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₃₅₃	—	2	1	no	—
19	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₃₅₃	—	4	1	no	—
22	C₄×C₃₅₃	C₂×C₈	1	1	yes	characteristic
23	C₄×C₃₅₃	M₁₆	1	1	yes	characteristic
24	Dic₃₅₃	—	2	2	no	—
25	Dic₃₅₃	—	2	1	no	—
26	Dic₃₅₃	SD₁₆	1	1	yes	characteristic
27	Dic₃₅₃	Q₁₆	1	1	yes	characteristic
28	C₂×C₃₅₃	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×C₃₅₃	#103	1	1	yes	characteristic
30	C₂×C₃₅₃	#104	1	1	yes	characteristic
31	C₃₅₃	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
32	Q₈⋊C₈	—	353	1	no	maximal, Sylow
33	C₄×Q₈	—	353	1	no	—
34	C₄⋊C₈	—	353	1	no	—
35	C₄×C₈	—	353	1	no	—
36	C₂×Q₈	—	353	1	no	—
37	C₂×C₈	—	706	1	no	—
38	C₂×C₈	—	706	1	no	—
39	C₄⋊C₄	—	706	1	no	—
40	C₄⋊C₄	#13083443	1	1	yes	characteristic
41	C₄²	—	353	1	no	—
42	C₄²	—	706	1	no	—
43	Q₈	—	353	2	no	—
44	Q₈	—	706	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	353	1	no	—
47	C₂×C₄	—	353	1	no	—
48	C₂×C₄	—	706	1	no	—
49	C₂×C₄	C₂×#13083443	1	1	yes	characteristic
50	C₈	—	1412	1	no	—
51	C₈	—	706	2	no	—
52	C₂²	C₂²⋊#13083443	1	1	yes	characteristic
53	C₄	—	4	1	no	—
54	C₄	—	353	1	no	—
55	C₄	—	353	1	no	—
56	C₄	—	706	2	no	—
57	C₄	—	706	1	no	—
58	C₄	#19150745	1	1	yes	characteristic
59	C₄	#19150747	1	1	yes	characteristic
60	C₂	#26015049	1	1	yes	characteristic
61	C₂	#26015055	1	1	yes	characteristic
62	C₂	#26015056	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#13083443
commutator	C₄×C₃₅₃	C₂×C₈
Frattini	C₂×C₄	C₂×#13083443
Fitting	C₄⋊C₄×C₃₅₃	C₄
socle, prime radical	C₂²×C₃₅₃	C₂²⋊C₄
cosocle	C₄×Dic₃₅₃	C₂²
hypercenter	C₄⋊C₄	#13083443
hypocenter	C₃₅₃	Q₈⋊C₈
LCS	G ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
14. Dic₇₀₆	C₈	semidirect
31. C₃₅₃	Q₈⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#15667732	2
C₄×#15667732	2
#26014957	2
Q₈⋊C₈	353