Subgroups — #36378788

Order 22592.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150745	C₂	1	2	yes	maximal
2	#26015059	C₂	1	2	yes	maximal, complemented
3	C₄⋊#15667732	C₂	1	2	yes	maximal, complemented
4	C₄²×D₃₅₃	C₂	1	1	yes	maximal, characteristic
5	#19150745	—	2	4	no	—
6	C₂×#15667732	C₂²	1	4	yes	—
7	C₂×C₄×D₃₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄×D₃₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×D₃₅₃	C₂²	1	1	yes	characteristic
10	C₄×Dic₃₅₃	C₄	1	1	yes	characteristic
11	C₄×Dic₃₅₃	C₂²	1	1	yes	characteristic
12	C₄×Dic₃₅₃	C₂²	1	1	yes	characteristic
13	C₄²×C₃₅₃	C₄	1	1	yes	characteristic
14	#15667732	—	2	4	no	—
15	C₂²×D₃₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₃₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₃₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₃₅₃	C₂³	1	1	yes	characteristic
19	C₄×D₃₅₃	—	2	2	no	—
20	C₄×D₃₅₃	C₈	1	4	yes	complemented
21	C₄×D₃₅₃	D₄	1	2	yes	—
22	C₄×D₃₅₃	Q₈	1	2	yes	—
23	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₃₅₃	C₄⋊C₄	1	2	yes	—
28	C₂×D₃₅₃	C₂×C₈	1	2	yes	—
29	C₂×D₃₅₃	M₁₆	1	2	yes	—
30	C₄×C₃₅₃	—	2	1	no	—
31	C₄×C₃₅₃	C₄⋊C₄	1	2	yes	—
32	C₄×C₃₅₃	C₂×C₈	1	2	yes	complemented
33	Dic₃₅₃	—	2	1	no	—
34	Dic₃₅₃	C₂×C₈	1	2	yes	complemented
35	Dic₃₅₃	C₂×D₄	1	1	yes	characteristic
36	Dic₃₅₃	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₃₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₃₅₃	C₂²×C₈	1	1	yes	characteristic
39	C₂×C₃₅₃	C₂×M₁₆	1	1	yes	characteristic
40	D₃₅₃	C₄⋊C₈	1	4	yes	complemented
41	C₃₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₄⋊C₈	—	353	1	no	maximal, Sylow
43	C₂²×C₈	—	353	2	no	—
44	C₂×C₄²	—	353	1	no	—
45	C₄⋊C₈	—	353	2	no	—
46	C₄⋊C₈	—	353	2	no	—
47	C₂²×C₄	—	353	1	no	—
48	C₂²×C₄	—	353	1	no	—
49	C₂²×C₄	—	353	1	no	—
50	C₂×C₈	—	353	4	no	—
51	C₂×C₈	—	706	4	no	—
52	C₄²	—	353	1	no	—
53	C₄²	—	353	1	no	—
54	C₄²	—	353	1	no	—
55	C₄²	#13083443	1	1	yes	characteristic
56	C₂³	—	353	1	no	—
57	C₂×C₄	—	353	1	no	—
58	C₂×C₄	—	353	1	no	—
59	C₂×C₄	—	353	2	no	—
60	C₂×C₄	—	353	2	no	—
61	C₂×C₄	—	353	4	no	—
62	C₂×C₄	—	353	1	no	—
63	C₂×C₄	—	706	2	no	—
64	C₂×C₄	C₂×#13083443	1	1	yes	characteristic
65	C₂×C₄	C₂×#13083443	1	1	yes	characteristic
66	C₂×C₄	C₂×#13083443	1	1	yes	characteristic
67	C₈	—	706	4	no	—
68	C₂²	—	353	2	no	—
69	C₂²	—	353	2	no	—
70	C₂²	—	353	2	no	—
71	C₂²	C₂²×#13083443	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	—	353	1	no	—
74	C₄	—	353	1	no	—
75	C₄	—	353	2	no	—
76	C₄	—	706	1	no	—
77	C₄	C₄⋊#13083443	1	2	yes	—
78	C₄	#19150745	1	2	yes	complemented
79	C₂	—	353	4	no	—
80	C₂	C₂×C₄⋊#13083443	1	1	yes	characteristic
81	C₂	C₂×#19150745	1	1	yes	characteristic
82	C₂	D₃₅₃⋊M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#13083443
commutator	C₂×C₃₅₃	C₂²×C₈
Frattini	C₂²	C₂²×#13083443
Fitting	C₄²×C₃₅₃	C₄
socle	C₂²×C₃₅₃	C₂²×C₄
prime radical	C₂²×D₃₅₃	C₂×C₄
cosocle	C₂×Dic₃₅₃	C₂³
hypercenter	C₄²	#13083443
hypocenter	C₃₅₃	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. #26015059	C₂	semidirect
3. C₄⋊#15667732	C₂	semidirect
20. C₄×D₃₅₃	C₈	semidirect
32. C₄×C₃₅₃	C₂×C₈	semidirect
34. Dic₃₅₃	C₂×C₈	semidirect
40. D₃₅₃	C₄⋊C₈	semidirect
41. C₃₅₃	C₂×C₄⋊C₈	semidirect
78. C₄	#19150745	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19150745	2
#26015059	2
C₄⋊#15667732	2
C₄²×D₃₅₃	2
C₂×C₄⋊C₈	353