Subgroups — #36378517

Order 22592.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26014991	C₂	1	4	yes	maximal
2	#26014990	C₂	1	4	yes	maximal
3	C₄×Dic₇₀₆	C₂	1	2	yes	maximal
4	#26014957	C₂	1	4	yes	maximal
5	C₄⋊Q₈×C₃₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₀₆	C₂²	1	4	yes	—
7	C₂×Dic₇₀₆	C₂²	1	2	yes	—
8	C₄⋊Dic₃₅₃	C₂²	1	2	yes	—
9	#19150716	C₂²	1	4	yes	—
10	#19150716	C₂²	1	8	yes	—
11	C₄×Dic₃₅₃	C₂²	1	4	yes	—
12	C₄×Dic₃₅₃	C₂²	1	4	yes	—
13	C₂×Q₈×C₃₅₃	C₂²	1	2	yes	—
14	C₄⋊C₄×C₃₅₃	C₂²	1	4	yes	—
15	C₄²×C₃₅₃	C₂²	1	1	yes	characteristic
16	Dic₇₀₆	—	4	4	no	—
17	Dic₇₀₆	Q₈	1	8	yes	complemented
18	C₂×Dic₃₅₃	C₂³	1	4	yes	—
19	C₂×Dic₃₅₃	C₂³	1	4	yes	—
20	Q₈×C₃₅₃	—	4	2	no	—
21	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₃₅₃	C₂³	1	4	yes	—
23	C₂×C₄×C₃₅₃	C₂³	1	2	yes	—
24	C₂²×C₃₅₃	C₂⁴	1	1	yes	characteristic
25	C₄×C₃₅₃	—	2	1	no	—
26	C₄×C₃₅₃	—	2	4	no	—
27	C₄×C₃₅₃	C₂×Q₈	1	4	yes	—
28	Dic₃₅₃	—	2	4	no	—
29	Dic₃₅₃	C₂×Q₈	1	8	yes	—
30	C₂×C₃₅₃	C₂²×Q₈	1	2	yes	—
31	C₂×C₃₅₃	H(5,𝔽₂)	1	1	yes	characteristic
32	C₃₅₃	Q₈²	1	1	yes	characteristic, complemented, Sylow
33	Q₈²	—	353	1	no	maximal, Sylow
34	C₄⋊Q₈	—	353	4	no	—
35	C₄⋊Q₈	—	353	4	no	—
36	C₄⋊Q₈	D₃₅₃	1	1	yes	characteristic
37	C₄×Q₈	—	353	4	no	—
38	C₄×Q₈	—	353	2	no	—
39	C₂×Q₈	—	353	2	no	—
40	C₂×Q₈	—	353	4	no	—
41	C₂×Q₈	C₂×D₃₅₃	1	2	yes	—
42	C₄⋊C₄	—	353	8	no	—
43	C₄⋊C₄	—	353	4	no	—
44	C₄⋊C₄	—	353	2	no	—
45	C₄⋊C₄	C₂×D₃₅₃	1	4	yes	—
46	C₄²	—	353	4	no	—
47	C₄²	—	353	4	no	—
48	C₄²	C₂×D₃₅₃	1	1	yes	characteristic
49	Q₈	—	4	2	no	—
50	Q₈	—	353	8	no	—
51	Q₈	—	1412	4	no	—
52	C₂×C₄	—	353	4	no	—
53	C₂×C₄	—	353	4	no	—
54	C₂×C₄	C₂²×D₃₅₃	1	4	yes	—
55	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₃₅₃	1	2	yes	—
57	C₂²	C₂³×D₃₅₃	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	4	no	—
60	C₄	—	353	8	no	—
61	C₄	—	706	4	no	—
62	C₄	Q₈×D₃₅₃	1	4	yes	—
63	C₂	C₂×Q₈×D₃₅₃	1	2	yes	—
64	C₂	#26015037	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₅₃
commutator, socle, prime radical, cosocle	C₂²×C₃₅₃	C₂⁴
Fitting	C₄⋊Q₈×C₃₅₃	C₂
hypercenter	C₄⋊Q₈	D₃₅₃
hypocenter	C₃₅₃	Q₈²
LCS	G ▷ C₂²×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
17. Dic₇₀₆	Q₈	semidirect
32. C₃₅₃	Q₈²	semidirect

Maximal subgroups

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

subgroup	index
#26014991	2
#26014990	2
C₄×Dic₇₀₆	2
#26014957	2
C₄⋊Q₈×C₃₅₃	2
Q₈²	353