Subgroups — #36378135

Order 22592.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26014989	C₂	1	1	yes	maximal, characteristic
2	#26014912	C₂	1	4	yes	maximal, complemented
3	C₄×#15667725	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊Q₈×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₄₁₂	—	2	1	no	—
6	C₂×D₁₄₁₂	—	2	1	no	—
7	C₂×D₁₄₁₂	C₂²	1	2	yes	—
8	C₂×#15667725	C₂²	1	2	yes	—
9	C₂×Q₈×C₃₅₃	—	2	1	no	—
10	C₄⋊C₄×C₃₅₃	—	2	1	no	—
11	C₄⋊C₄×C₃₅₃	C₂²	1	2	yes	—
12	C₄²×C₃₅₃	C₂²	1	1	yes	characteristic
13	C₂²×D₃₅₃	—	2	2	no	—
14	D₁₄₁₂	—	2	4	no	—
15	D₁₄₁₂	—	4	2	no	—
16	D₁₄₁₂	—	4	2	no	—
17	#15667725	—	2	2	no	—
18	Q₈×C₃₅₃	—	4	2	no	—
19	C₂×C₄×C₃₅₃	—	2	2	no	—
20	C₂×C₄×C₃₅₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₃₅₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
23	C₂²×C₃₅₃	C₂×D₄	1	1	yes	characteristic
24	C₂×D₃₅₃	—	4	2	no	—
25	C₂×D₃₅₃	—	4	2	no	—
26	C₂×D₃₅₃	—	4	2	no	—
27	C₄×C₃₅₃	—	4	2	no	—
28	C₄×C₃₅₃	D₈	1	2	yes	—
29	C₄×C₃₅₃	SD₁₆	1	2	yes	—
30	C₄×C₃₅₃	#41	1	2	yes	—
31	C₂×C₃₅₃	#124	1	1	yes	characteristic
32	C₂×C₃₅₃	C₂×D₈	1	1	yes	characteristic
33	C₂×C₃₅₃	C₂×SD₁₆	1	1	yes	characteristic
34	D₃₅₃	—	8	2	no	—
35	C₃₅₃	#486	1	1	yes	characteristic, complemented, Sylow
36	#486	—	353	1	no	maximal, Sylow
37	C₄⋊Q₈	D₃₅₃	1	1	yes	characteristic, complemented
38	#127	—	353	1	no	—
39	#102	—	353	4	no	—
40	C₄×C₈	—	353	1	no	—
41	C₂×Q₈	—	2	1	no	—
42	C₂×D₄	—	353	2	no	—
43	C₂×D₄	—	706	1	no	—
44	C₂×D₄	—	706	1	no	—
45	C₂×C₈	—	353	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	C₂×D₃₅₃	1	2	yes	—
48	C₄²	C₂×D₃₅₃	1	1	yes	characteristic
49	C₂³	—	706	2	no	—
50	Q₈	—	4	2	no	—
51	D₄	—	1412	2	no	—
52	D₄	—	1412	2	no	—
53	D₄	—	706	4	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₂²⋊D₃₅₃	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₃₅₃	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
58	C₈	—	706	2	no	—
59	C₂²	—	1412	2	no	—
60	C₂²	—	1412	2	no	—
61	C₂²	—	1412	2	no	—
62	C₂²	C₂×C₂²⋊D₃₅₃	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	#19150722	1	2	yes	—
65	C₄	#19150724	1	2	yes	—
66	C₄	#19150738	1	2	yes	—
67	C₂	—	2824	2	no	—
68	C₂	#26014981	1	1	yes	characteristic
69	C₂	C₂×#19150722	1	1	yes	characteristic
70	C₂	C₂×#19150724	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₅₃
commutator, cosocle	C₂×C₄×C₃₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₅₃
Fitting	C₄⋊Q₈×C₃₅₃	C₂
socle	C₂²×C₃₅₃	C₂×D₄
prime radical	#26014989	C₂
hypercenter	C₄⋊Q₈	D₃₅₃
hypocenter	C₃₅₃	#486
LCS	G ▷ C₂×C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #26014912	C₂	semidirect
3. C₄×#15667725	C₂	semidirect
4. C₄⋊Q₈×C₃₅₃	C₂	semidirect
35. C₃₅₃	#486	semidirect
37. C₄⋊Q₈	D₃₅₃	semidirect

Maximal subgroups

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

subgroup	index
#26014989	2
#26014912	2
C₄×#15667725	2
C₄⋊Q₈×C₃₅₃	2
#486	353