Subgroups — #106×D₃₅₃ #36377816

Order 22592.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₃₅₃	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×D₃₅₃	C₂	1	2	yes	maximal
3	#26014924	C₂	1	1	yes	maximal, characteristic, complemented
4	#26014923	C₂	1	2	yes	maximal, complemented
5	#106×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₃₅₃	—	2	2	no	—
7	C₂×C₄×D₃₅₃	C₂²	1	1	yes	characteristic
8	C₂×#15667725	C₂²	1	1	yes	characteristic
9	C₈×D₃₅₃	C₄	1	4	yes	complemented
10	C₄⋊Dic₃₅₃	C₂²	1	2	yes	—
11	#19150716	—	2	2	no	—
12	C₂×C₈×C₃₅₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₃₅₃	C₂²	1	2	yes	—
14	C₂²×D₃₅₃	D₄	1	1	yes	characteristic
15	C₂×Dic₃₅₃	—	2	2	no	—
16	C₂×Dic₃₅₃	D₄	1	1	yes	characteristic
17	C₄×D₃₅₃	—	4	4	no	—
18	C₄×D₃₅₃	C₂×C₄	1	2	yes	—
19	C₄×D₃₅₃	Q₈	1	2	yes	—
20	#15667725	C₂×C₄	1	2	yes	complemented
21	C₂×C₄×C₃₅₃	—	2	2	no	—
22	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
23	C₈×C₃₅₃	C₂×C₄	1	2	yes	complemented
24	C₂²×C₃₅₃	C₂×D₄	1	1	yes	characteristic
25	C₂×D₃₅₃	C₄⋊C₄	1	2	yes	—
26	C₂×D₃₅₃	SD₁₆	1	4	yes	—
27	C₄×C₃₅₃	—	4	2	no	—
28	C₄×C₃₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₃₅₃	C₂×Q₈	1	1	yes	characteristic
30	Dic₃₅₃	—	4	2	no	—
31	Dic₃₅₃	C₄⋊C₄	1	2	yes	—
32	C₂×C₃₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₃₅₃	C₂×SD₁₆	1	2	yes	—
34	D₃₅₃	#106	1	4	yes	direct product factor
35	C₃₅₃	C₂×#106	1	1	yes	characteristic, complemented, Sylow
36	C₂×#106	—	353	1	no	maximal, Sylow
37	C₂²×C₈	—	353	1	no	—
38	C₂×C₄⋊C₄	—	353	2	no	—
39	#106	—	353	1	no	—
40	#106	—	353	2	no	—
41	#106	D₃₅₃	1	1	yes	characteristic, direct product factor
42	C₂²×C₄	—	353	1	no	—
43	C₂²×C₄	—	706	2	no	—
44	C₂×C₈	—	353	1	no	—
45	C₂×C₈	—	353	4	no	—
46	C₂×C₈	C₂×D₃₅₃	1	1	yes	characteristic
47	C₄⋊C₄	—	353	2	no	—
48	C₄⋊C₄	—	706	2	no	—
49	C₄⋊C₄	C₂×D₃₅₃	1	2	yes	—
50	C₂³	—	353	1	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	353	1	no	—
53	C₂×C₄	—	353	2	no	—
54	C₂×C₄	—	353	2	no	—
55	C₂×C₄	—	1412	4	no	—
56	C₂×C₄	—	706	2	no	—
57	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
58	C₈	—	353	2	no	—
59	C₈	C₄×D₃₅₃	1	2	yes	complemented
60	C₂²	—	353	4	no	—
61	C₂²	—	353	2	no	—
62	C₂²	D₄×D₃₅₃	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	—	353	2	no	—
65	C₄	—	1412	2	no	—
66	C₄	C₂×C₄×D₃₅₃	1	1	yes	characteristic
67	C₄	Q₈×D₃₅₃	1	1	yes	characteristic
68	C₂	—	353	4	no	—
69	C₂	C₄⋊C₄×D₃₅₃	1	1	yes	characteristic
70	C₂	SD₁₆×D₃₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₃₅₃
commutator	C₄×C₃₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₅₃
Fitting	#106×C₃₅₃	C₂
socle	C₂²×C₃₅₃	C₂×D₄
prime radical	C₂²×D₃₅₃	D₄
cosocle	C₂×C₄×C₃₅₃	C₂³
hypercenter	#106	D₃₅₃
hypocenter	C₃₅₃	C₂×#106
LCS	G ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
3. #26014924	C₂	semidirect
4. #26014923	C₂	semidirect
5. #106×C₃₅₃	C₂	semidirect
9. C₈×D₃₅₃	C₄	semidirect
20. #15667725	C₂×C₄	semidirect
23. C₈×C₃₅₃	C₂×C₄	semidirect
34. D₃₅₃	#106	direct (N × H)
35. C₃₅₃	C₂×#106	semidirect
41. #106	D₃₅₃	direct (N × H)
59. C₈	C₄×D₃₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₃₅₃	2
C₄⋊C₄×D₃₅₃	2
#26014924	2
#26014923	2
#106×C₃₅₃	2
C₂×#106	353

Subgroups — #106×D353 #36377816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #106×D₃₅₃ #36377816