Subgroups — C₂²×#19150716 #36378237

Order 22592.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₃₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×Dic₃₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#19150716	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₄×C₃₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₅₃	—	2	4	no	—
6	C₂²×Dic₃₅₃	C₄	1	8	yes	complemented
7	C₂²×Dic₃₅₃	C₂²	1	6	yes	—
8	C₂²×Dic₃₅₃	C₂²	1	6	yes	—
9	#19150716	C₂²	1	16	yes	direct product factor
10	C₂⁴×C₃₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₃₅₃	—	2	4	no	—
12	C₂²×C₄×C₃₅₃	C₂²	1	6	yes	—
13	C₂×Dic₃₅₃	—	2	12	no	—
14	C₂×Dic₃₅₃	C₂×C₄	1	24	yes	complemented
15	C₂×Dic₃₅₃	C₂³	1	4	yes	—
16	C₂×Dic₃₅₃	C₂³	1	4	yes	—
17	C₂³×C₃₅₃	C₂×C₄	1	4	yes	—
18	C₂³×C₃₅₃	D₄	1	4	yes	—
19	C₂³×C₃₅₃	Q₈	1	4	yes	—
20	C₂³×C₃₅₃	C₂³	1	3	yes	—
21	C₂×C₄×C₃₅₃	—	2	12	no	—
22	C₂×C₄×C₃₅₃	C₂³	1	4	yes	—
23	C₂²×C₃₅₃	C₄⋊C₄	1	16	yes	complemented
24	C₂²×C₃₅₃	C₂²×C₄	1	6	yes	—
25	C₂²×C₃₅₃	C₂×D₄	1	6	yes	—
26	C₂²×C₃₅₃	C₂×Q₈	1	6	yes	—
27	C₂²×C₃₅₃	C₂⁴	1	1	yes	characteristic
28	C₄×C₃₅₃	—	2	4	no	—
29	Dic₃₅₃	—	2	4	no	—
30	Dic₃₅₃	C₂²×C₄	1	8	yes	complemented
31	C₂×C₃₅₃	C₂×C₄⋊C₄	1	12	yes	complemented
32	C₂×C₃₅₃	C₂³×C₄	1	1	yes	characteristic
33	C₂×C₃₅₃	C₂²×D₄	1	1	yes	characteristic
34	C₂×C₃₅₃	C₂²×Q₈	1	1	yes	characteristic
35	C₃₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
36	C₂²×C₄⋊C₄	—	353	1	no	maximal, Sylow
37	C₂³×C₄	—	353	1	no	—
38	C₂³×C₄	—	353	1	no	—
39	C₂³×C₄	D₃₅₃	1	1	yes	characteristic
40	C₂×C₄⋊C₄	—	353	12	no	—
41	C₂⁴	C₂×D₃₅₃	1	1	yes	characteristic
42	C₂²×C₄	—	2	4	no	—
43	C₂²×C₄	—	353	8	no	—
44	C₂²×C₄	—	353	6	no	—
45	C₂²×C₄	—	353	6	no	—
46	C₂²×C₄	—	706	4	no	—
47	C₂²×C₄	C₂×D₃₅₃	1	6	yes	—
48	C₄⋊C₄	—	353	16	no	—
49	C₂³	C₄×D₃₅₃	1	4	yes	—
50	C₂³	C₂²⋊D₃₅₃	1	4	yes	—
51	C₂³	Dic₇₀₆	1	4	yes	—
52	C₂³	C₂²×D₃₅₃	1	3	yes	—
53	C₂×C₄	—	2	12	no	—
54	C₂×C₄	—	353	24	no	—
55	C₂×C₄	—	353	4	no	—
56	C₂×C₄	—	353	4	no	—
57	C₂×C₄	—	706	12	no	—
58	C₂×C₄	C₂²×D₃₅₃	1	4	yes	—
59	C₂²	#19150716	1	16	yes	direct product factor
60	C₂²	C₂×C₄×D₃₅₃	1	6	yes	—
61	C₂²	C₂×C₂²⋊D₃₅₃	1	6	yes	—
62	C₂²	C₂×Dic₇₀₆	1	6	yes	—
63	C₂²	C₂³×D₃₅₃	1	1	yes	characteristic
64	C₄	—	2	4	no	—
65	C₄	—	353	8	no	—
66	C₄	—	706	4	no	—
67	C₂	C₂×#19150716	1	12	yes	direct product factor
68	C₂	C₂²×C₄×D₃₅₃	1	1	yes	characteristic
69	C₂	C₂²×C₂²⋊D₃₅₃	1	1	yes	characteristic
70	C₂	C₂²×Dic₇₀₆	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂×#19150716	C₂	direct (N × H)
6. C₂²×Dic₃₅₃	C₄	semidirect
9. #19150716	C₂²	direct (N × H)
14. C₂×Dic₃₅₃	C₂×C₄	semidirect
23. C₂²×C₃₅₃	C₄⋊C₄	semidirect
30. Dic₃₅₃	C₂²×C₄	semidirect
31. C₂×C₃₅₃	C₂×C₄⋊C₄	semidirect
35. C₃₅₃	C₂²×C₄⋊C₄	semidirect
59. C₂²	#19150716	direct (N × H)
67. C₂	C₂×#19150716	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₃₅₃	2
C₂×#19150716	2
C₂³×C₄×C₃₅₃	2
C₂²×C₄⋊C₄	353

Subgroups — C22×#19150716 #36378237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19150716 #36378237