Subgroups — C₂²×#19150738 #36378615

Order 22592.

890 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150738	C₂	1	24	yes	maximal, direct product factor
2	C₂²×D₁₄₁₂	C₂	1	3	yes	maximal, complemented
3	C₂²×C₄×D₃₅₃	C₂	1	3	yes	maximal, complemented
4	C₂²×Q₈×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₃₅₃	C₂²	1	3	yes	—
6	#19150738	C₂²	1	64	yes	direct product factor
7	C₂×D₁₄₁₂	C₂²	1	36	yes	complemented
8	C₂²×Dic₃₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₃₅₃	C₂²	1	36	yes	complemented
10	C₂×Q₈×C₃₅₃	C₂²	1	12	yes	complemented
11	C₂²×C₄×C₃₅₃	C₂²	1	3	yes	—
12	C₂²×D₃₅₃	—	2	12	no	—
13	C₂²×D₃₅₃	C₂³	1	18	yes	—
14	D₁₄₁₂	C₂³	1	48	yes	complemented
15	C₂×Dic₃₅₃	C₂³	1	6	yes	—
16	C₄×D₃₅₃	C₂³	1	48	yes	complemented
17	C₂³×C₃₅₃	C₂³	1	1	yes	characteristic
18	Q₈×C₃₅₃	C₂³	1	16	yes	complemented
19	C₂×C₄×C₃₅₃	C₂³	1	18	yes	—
20	C₂²×C₃₅₃	#41	1	4	yes	complemented
21	C₂²×C₃₅₃	C₂⁴	1	3	yes	—
22	C₂×D₃₅₃	—	2	36	no	—
23	C₂×D₃₅₃	C₂⁴	1	12	yes	—
24	C₄×C₃₅₃	C₂⁴	1	12	yes	—
25	Dic₃₅₃	C₂⁴	1	4	yes	—
26	C₂×C₃₅₃	C₂×#41	1	6	yes	complemented
27	C₂×C₃₅₃	C₂⁵	1	1	yes	characteristic
28	D₃₅₃	—	2	12	no	—
29	C₃₅₃	C₂²×#41	1	1	yes	characteristic, complemented, Sylow
30	C₂²×#41	—	353	1	no	maximal, Sylow
31	C₂×#41	—	353	24	no	—
32	C₂²×Q₈	D₃₅₃	1	1	yes	characteristic, complemented
33	C₂²×D₄	—	353	3	no	—
34	C₂³×C₄	—	353	3	no	—
35	C₂⁴	—	353	3	no	—
36	#41	—	353	64	no	—
37	C₂×Q₈	C₂×D₃₅₃	1	12	yes	complemented
38	C₂×D₄	—	353	36	no	—
39	C₂²×C₄	—	353	1	no	—
40	C₂²×C₄	—	353	36	no	—
41	C₂²×C₄	C₂×D₃₅₃	1	3	yes	—
42	C₂³	—	353	18	no	—
43	C₂³	—	706	12	no	—
44	C₂³	C₂²×D₃₅₃	1	1	yes	characteristic
45	Q₈	C₂²×D₃₅₃	1	16	yes	complemented
46	D₄	—	353	48	no	—
47	C₂×C₄	—	353	6	no	—
48	C₂×C₄	—	353	48	no	—
49	C₂×C₄	C₂²×D₃₅₃	1	18	yes	—
50	C₂²	—	353	12	no	—
51	C₂²	—	706	36	no	—
52	C₂²	#19150738	1	4	yes	direct product factor
53	C₂²	C₂³×D₃₅₃	1	3	yes	—
54	C₄	—	353	4	no	—
55	C₄	C₂³×D₃₅₃	1	12	yes	—
56	C₂	—	706	12	no	—
57	C₂	C₂×#19150738	1	6	yes	direct product factor
58	C₂	C₂⁴×D₃₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃₅₃
commutator, cosocle	C₂×C₃₅₃	C₂⁵
Frattini	C₂	C₂⁴×D₃₅₃
Fitting	C₂²×Q₈×C₃₅₃	C₂
socle	C₂³×C₃₅₃	C₂³
hypercenter	C₂²×Q₈	D₃₅₃
hypocenter	C₃₅₃	C₂²×#41
LCS	G ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#19150738	C₂	direct (N × H)
2. C₂²×D₁₄₁₂	C₂	semidirect
3. C₂²×C₄×D₃₅₃	C₂	semidirect
4. C₂²×Q₈×C₃₅₃	C₂	semidirect
6. #19150738	C₂²	direct (N × H)
7. C₂×D₁₄₁₂	C₂²	semidirect
9. C₂×C₄×D₃₅₃	C₂²	semidirect
10. C₂×Q₈×C₃₅₃	C₂²	semidirect
14. D₁₄₁₂	C₂³	semidirect
16. C₄×D₃₅₃	C₂³	semidirect
18. Q₈×C₃₅₃	C₂³	semidirect
20. C₂²×C₃₅₃	#41	semidirect
26. C₂×C₃₅₃	C₂×#41	semidirect
29. C₃₅₃	C₂²×#41	semidirect
32. C₂²×Q₈	D₃₅₃	semidirect
37. C₂×Q₈	C₂×D₃₅₃	semidirect
45. Q₈	C₂²×D₃₅₃	semidirect
52. C₂²	#19150738	direct (N × H)
57. C₂	C₂×#19150738	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19150738	2
C₂²×D₁₄₁₂	2
C₂²×C₄×D₃₅₃	2
C₂²×Q₈×C₃₅₃	2
C₂²×#41	353

Subgroups — C22×#19150738 #36378615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19150738 #36378615