Subgroups — #36378827

Order 22592.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#13083443	C₂	1	2	yes	maximal
2	#26015060	C₂	1	1	yes	maximal, characteristic
3	#26015055	C₂	1	2	yes	maximal
4	#26015046	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×D₃₅₃	C₂	1	1	yes	maximal, characteristic
6	#19150745	C₂²	1	1	yes	characteristic
7	C₄⋊#13083443	—	2	2	no	—
8	C₄⋊#13083443	C₂²	1	2	yes	—
9	C₄×#13083443	—	2	2	no	—
10	C₄×#13083443	C₂²	1	1	yes	characteristic
11	Q₈×D₃₅₃	C₂²	1	2	yes	—
12	Dic₁₄₁₂	C₄	1	1	yes	characteristic, complemented
13	#19150727	C₄	1	2	yes	complemented
14	C₈×D₃₅₃	C₂²	1	1	yes	characteristic
15	Q₁₆×C₃₅₃	C₄	1	1	yes	characteristic, complemented
16	C₂×#13083443	—	2	2	no	—
17	C₂×#13083443	D₄	1	2	yes	—
18	#15667732	—	2	1	no	—
19	Dic₇₀₆	—	2	2	no	—
20	Dic₇₀₆	C₂×C₄	1	2	yes	—
21	C₄×D₃₅₃	—	2	2	no	—
22	C₄×D₃₅₃	C₂³	1	1	yes	characteristic
23	#15667725	C₂×C₄	1	1	yes	characteristic
24	Q₈×C₃₅₃	C₂×C₄	1	2	yes	—
25	C₈×C₃₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×D₃₅₃	C₂×D₄	1	1	yes	characteristic
27	#13083443	—	2	2	no	—
28	#13083443	—	4	2	no	—
29	C₄×C₃₅₃	—	2	2	no	—
30	C₄×C₃₅₃	C₂²×C₄	1	1	yes	characteristic
31	Dic₃₅₃	—	2	2	no	—
32	Dic₃₅₃	#41	1	1	yes	characteristic
33	C₂×C₃₅₃	C₄×D₄	1	1	yes	characteristic
34	D₃₅₃	#137	1	2	yes	—
35	C₃₅₃	#441	1	1	yes	characteristic, complemented, Sylow
36	#441	—	353	1	no	maximal, Sylow
37	C₂×Q₁₆	—	353	1	no	—
38	C₄×Q₈	—	353	2	no	—
39	#106	—	353	1	no	—
40	#103	—	353	2	no	—
41	#97	—	353	1	no	—
42	C₂×Q₈	—	353	2	no	—
43	Q₁₆	—	353	2	no	—
44	Q₁₆	—	353	1	no	—
45	Q₁₆	#13083443	1	1	yes	characteristic, complemented
46	C₂×C₈	—	353	1	no	—
47	C₂×C₈	—	353	1	no	—
48	C₄⋊C₄	—	353	2	no	—
49	C₄⋊C₄	—	706	2	no	—
50	C₄²	—	353	1	no	—
51	C₄²	—	706	2	no	—
52	Q₈	—	353	2	no	—
53	Q₈	—	706	2	no	—
54	Q₈	C₂×#13083443	1	2	yes	—
55	C₂×C₄	—	353	1	no	—
56	C₂×C₄	—	353	2	no	—
57	C₂×C₄	—	706	2	no	—
58	C₂×C₄	—	706	2	no	—
59	C₈	—	353	1	no	—
60	C₈	—	706	1	no	—
61	C₈	C₂×#13083443	1	1	yes	characteristic
62	C₂²	—	353	1	no	—
63	C₄	—	2	2	no	—
64	C₄	—	353	1	no	—
65	C₄	—	1412	2	no	—
66	C₄	—	706	2	no	—
67	C₄	—	706	2	no	—
68	C₄	C₂²×#13083443	1	1	yes	characteristic
69	C₂	—	353	2	no	—
70	C₂	D₄×#13083443	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#13083443
commutator	C₄×C₃₅₃	C₂²×C₄
Frattini	C₄	C₂²×#13083443
Fitting	Q₁₆×C₃₅₃	C₄
socle	C₂×C₃₅₃	C₄×D₄
prime radical	C₂×D₃₅₃	C₂×D₄
cosocle	C₄×D₃₅₃	C₂³
hypercenter	Q₁₆	#13083443
hypocenter	C₃₅₃	#441
LCS	G ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
12. Dic₁₄₁₂	C₄	semidirect
13. #19150727	C₄	semidirect
15. Q₁₆×C₃₅₃	C₄	semidirect
35. C₃₅₃	#441	semidirect
45. Q₁₆	#13083443	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#13083443	2
#26015060	2
#26015055	2
#26015046	2
Q₁₆×D₃₅₃	2
#441	353