Subgroups — #36378767

Order 22592.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26015071	C₂	1	2	yes	maximal, complemented
2	#26015050	C₂	1	4	yes	maximal, complemented
3	C₂×#19150736	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#13083443	—	2	2	no	—
5	C₂²⋊#13083443	C₂²	1	4	yes	—
6	C₄×#13083443	—	2	2	no	—
7	#19150736	—	2	2	no	—
8	#19150736	—	2	2	no	—
9	C₂×Dic₇₀₆	C₄	1	1	yes	characteristic, complemented
10	C₂×C₂²⋊D₃₅₃	C₂²	1	2	yes	—
11	C₂²×Dic₃₅₃	C₄	1	2	yes	complemented
12	C₂×C₄×D₃₅₃	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₃₅₃	C₄	1	1	yes	characteristic, complemented
14	C₂×#13083443	—	2	4	no	—
15	C₂²×D₃₅₃	C₂³	1	1	yes	characteristic
16	Dic₇₀₆	—	2	2	no	—
17	C₂²⋊D₃₅₃	—	2	4	no	—
18	C₂×Dic₃₅₃	—	2	2	no	—
19	C₂×Dic₃₅₃	—	2	2	no	—
20	C₂×Dic₃₅₃	C₂×C₄	1	2	yes	—
21	C₂×Dic₃₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×D₃₅₃	D₄	1	2	yes	—
23	C₄×D₃₅₃	D₄	1	2	yes	—
24	C₂³×C₃₅₃	C₂×C₄	1	2	yes	—
25	D₄×C₃₅₃	—	2	2	no	—
26	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃₅₃	—	2	2	no	—
28	C₂²×C₃₅₃	—	4	2	no	—
29	C₂²×C₃₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×D₃₅₃	—	4	1	no	—
31	C₂×D₃₅₃	C₂×D₄	1	2	yes	—
32	#13083443	—	4	4	no	—
33	C₄×C₃₅₃	C₂²⋊C₄	1	2	yes	—
34	Dic₃₅₃	—	2	2	no	—
35	Dic₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₃₅₃	—	2	1	no	—
38	C₂×C₃₅₃	—	4	2	no	—
39	C₂×C₃₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	D₃₅₃	—	2	2	no	—
41	C₃₅₃	#410	1	1	yes	characteristic, complemented, Sylow
42	#410	—	353	1	no	maximal, Sylow
43	C₂×#41	—	353	1	no	—
44	#117	—	353	2	no	—
45	#99	—	353	4	no	—
46	#41	—	706	2	no	—
47	#41	—	706	2	no	—
48	C₂×Q₈	—	353	1	no	—
49	C₂×D₄	—	353	2	no	—
50	C₂×D₄	#13083443	1	1	yes	characteristic, complemented
51	C₂²×C₄	—	353	1	no	—
52	C₂²×C₄	—	353	2	no	—
53	C₄⋊C₄	—	706	2	no	—
54	C₂²⋊C₄	—	353	4	no	—
55	C₄²	—	706	2	no	—
56	C₂³	—	353	1	no	—
57	C₂³	C₂×#13083443	1	2	yes	—
58	Q₈	—	706	2	no	—
59	D₄	—	2	2	no	—
60	D₄	—	706	4	no	—
61	C₂×C₄	—	353	1	no	—
62	C₂×C₄	—	353	2	no	—
63	C₂×C₄	—	353	2	no	—
64	C₂×C₄	—	353	2	no	—
65	C₂×C₄	—	706	2	no	—
66	C₂×C₄	—	706	2	no	—
67	C₂×C₄	—	706	4	no	—
68	C₂×C₄	C₂×#13083443	1	1	yes	characteristic
69	C₂²	—	2	2	no	—
70	C₂²	—	4	2	no	—
71	C₂²	—	353	2	no	—
72	C₂²	—	1412	1	no	—
73	C₂²	C₂²×#13083443	1	1	yes	characteristic
74	C₄	—	353	1	no	—
75	C₄	—	353	1	no	—
76	C₄	—	1412	4	no	—
77	C₄	—	706	2	no	—
78	C₄	C₂²⋊#13083443	1	2	yes	—
79	C₂	—	2	1	no	—
80	C₂	—	4	2	no	—
81	C₂	—	706	2	no	—
82	C₂	C₂×C₂²⋊#13083443	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#13083443
commutator	C₂²×C₃₅₃	C₂²×C₄
Frattini	C₂²	C₂²×#13083443
Fitting	C₂×D₄×C₃₅₃	C₄
socle	C₂×C₃₅₃	C₂×C₂²⋊C₄
prime radical	C₂×#19150736	C₂
cosocle	C₂²×D₃₅₃	C₂³
hypercenter	C₂×D₄	#13083443
hypocenter	C₃₅₃	#410
LCS	G ▷ C₂²×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #26015071	C₂	semidirect
2. #26015050	C₂	semidirect
9. C₂×Dic₇₀₆	C₄	semidirect
11. C₂²×Dic₃₅₃	C₄	semidirect
13. C₂×D₄×C₃₅₃	C₄	semidirect
41. C₃₅₃	#410	semidirect
50. C₂×D₄	#13083443	semidirect

Maximal subgroups

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

subgroup	index
#26015071	2
#26015050	2
C₂×#19150736	2
#410	353