Subgroups — #36378730

Order 22592.

218 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#13083443	C₂	1	2	yes	maximal
2	(C₂×#13083443)⋊C₄	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×D₃₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13083443	C₂²	1	4	yes	—
5	C₂²⋊#13083443	—	2	4	no	—
6	C₂³×D₃₅₃	C₂²	1	1	yes	characteristic
7	C₂²×Dic₃₅₃	C₄	1	1	yes	characteristic, complemented
8	C₂×C₄×D₃₅₃	—	2	4	no	—
9	C₂×C₄×D₃₅₃	C₄	1	2	yes	complemented
10	C₂×C₄×D₃₅₃	C₂²	1	2	yes	—
11	C₂²×C₄×C₃₅₃	C₄	1	1	yes	characteristic, complemented
12	C₂×#13083443	—	2	4	no	—
13	C₂×#13083443	—	2	8	no	—
14	C₂²×D₃₅₃	—	2	4	no	—
15	C₂²×D₃₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×D₃₅₃	D₄	1	4	yes	—
17	C₂²×D₃₅₃	C₂³	1	1	yes	characteristic
18	C₂×Dic₃₅₃	—	2	2	no	—
19	C₂×Dic₃₅₃	C₂×C₄	1	2	yes	—
20	C₄×D₃₅₃	—	2	4	no	—
21	C₄×D₃₅₃	—	2	4	no	—
22	C₂³×C₃₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₃₅₃	—	2	2	no	—
24	C₂×C₄×C₃₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₃₅₃	—	2	2	no	—
26	C₂²×C₃₅₃	C₂²⋊C₄	1	2	yes	—
27	C₂²×C₃₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×D₃₅₃	—	2	2	no	—
29	C₂×D₃₅₃	—	2	8	no	—
30	C₂×D₃₅₃	C₂²⋊C₄	1	2	yes	—
31	C₂×D₃₅₃	C₂×D₄	1	2	yes	—
32	C₂×D₃₅₃	#41	1	4	yes	—
33	#13083443	—	4	4	no	—
34	C₄×C₃₅₃	—	2	2	no	—
35	Dic₃₅₃	—	2	2	no	—
36	C₂×C₃₅₃	—	2	2	no	—
37	C₂×C₃₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₃₅₃	#117	1	2	yes	—
39	D₃₅₃	—	2	2	no	—
40	D₃₅₃	#123	1	4	yes	—
41	C₃₅₃	#381	1	1	yes	characteristic, complemented, Sylow
42	#381	—	353	1	no	maximal, Sylow
43	C₂³×C₄	—	353	1	no	—
44	C₂×C₂²⋊C₄	—	353	2	no	—
45	#95	—	353	4	no	—
46	C₂⁴	—	353	1	no	—
47	C₂²×C₄	—	353	1	no	—
48	C₂²×C₄	—	353	2	no	—
49	C₂²×C₄	—	353	2	no	—
50	C₂²×C₄	—	353	4	no	—
51	C₂²×C₄	—	706	4	no	—
52	C₂²×C₄	#13083443	1	1	yes	characteristic, complemented
53	C₂²⋊C₄	—	706	4	no	—
54	C₂³	—	353	1	no	—
55	C₂³	—	353	4	no	—
56	C₂³	—	353	1	no	—
57	C₂³	—	706	4	no	—
58	C₂³	C₂×#13083443	1	1	yes	characteristic
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	353	2	no	—
61	C₂×C₄	—	706	4	no	—
62	C₂×C₄	—	706	4	no	—
63	C₂×C₄	—	706	4	no	—
64	C₂×C₄	—	706	8	no	—
65	C₂×C₄	—	706	2	no	—
66	C₂×C₄	C₂×#13083443	1	2	yes	—
67	C₂²	—	2	2	no	—
68	C₂²	—	353	2	no	—
69	C₂²	—	353	4	no	—
70	C₂²	—	353	2	no	—
71	C₂²	—	706	8	no	—
72	C₂²	—	706	2	no	—
73	C₂²	C₂²⋊#13083443	1	2	yes	—
74	C₂²	C₂²×#13083443	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	1412	4	no	—
77	C₄	—	706	2	no	—
78	C₂	—	2	2	no	—
79	C₂	—	353	4	no	—
80	C₂	—	706	2	no	—
81	C₂	C₂×C₂²⋊#13083443	1	1	yes	characteristic
82	C₂	#26015071	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#13083443
commutator, socle	C₂²×C₃₅₃	C₂²×C₄
Fitting	C₂²×C₄×C₃₅₃	C₄
prime radical	C₂³×D₃₅₃	C₂²
cosocle	C₂²×D₃₅₃	C₂³
hypercenter	C₂²×C₄	#13083443
hypocenter	C₃₅₃	#381
LCS	G ▷ C₂²×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. (C₂×#13083443)⋊C₄	C₂	semidirect
7. C₂²×Dic₃₅₃	C₄	semidirect
9. C₂×C₄×D₃₅₃	C₄	semidirect
11. C₂²×C₄×C₃₅₃	C₄	semidirect
41. C₃₅₃	#381	semidirect
52. C₂²×C₄	#13083443	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#13083443	2
(C₂×#13083443)⋊C₄	2
C₂²×C₄×D₃₅₃	2
#381	353