Subgroups — #36342333

Order 21568.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19129193	C₂	1	2	yes	maximal, complemented
2	C₄⋊#14582113	C₂	1	1	yes	maximal, characteristic, complemented
3	#24925016	C₂	1	2	yes	maximal, complemented
4	C₄×#14582113	C₂	1	1	yes	maximal, characteristic, complemented
5	#24924921	C₂	1	1	yes	maximal, characteristic
6	#19129193	—	2	4	no	—
7	C₂×#14582113	C₂²	1	1	yes	characteristic
8	C₂×#14582113	C₂²	1	2	yes	—
9	C₂×#14582113	C₂²	1	1	yes	characteristic
10	C₂×D₁₃₄₈	C₄	1	1	yes	characteristic
11	C₂×C₄×D₃₃₇	C₂²	1	2	yes	—
12	#19129162	C₄	1	2	yes	—
13	C₄×Dic₃₃₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₃₃₇	C₄	1	1	yes	characteristic
15	#14582113	—	2	2	no	—
16	#14582113	—	2	1	no	—
17	#14582113	D₄	1	2	yes	complemented
18	C₂²×D₃₃₇	C₂×C₄	1	2	yes	—
19	D₁₃₄₈	C₈	1	4	yes	complemented
20	C₂×Dic₃₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₃₃₇	C₂³	1	1	yes	characteristic
22	C₄×D₃₃₇	—	2	2	no	—
23	C₄×D₃₃₇	—	2	2	no	—
24	C₂×C₄×C₃₃₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₃₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃₃₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₃₃₇	—	2	2	no	—
28	C₂×D₃₃₇	—	2	2	no	—
29	C₂×D₃₃₇	C₂×C₈	1	4	yes	complemented
30	C₄×C₃₃₇	—	2	2	no	—
31	C₄×C₃₃₇	C₂×C₈	1	2	yes	complemented
32	Dic₃₃₇	—	2	1	no	—
33	Dic₃₃₇	C₂×D₄	1	1	yes	characteristic
34	Dic₃₃₇	#41	1	1	yes	characteristic
35	C₂×C₃₃₇	C₄×D₄	1	1	yes	characteristic
36	C₂×C₃₃₇	C₂²×C₈	1	1	yes	characteristic
37	C₂×C₃₃₇	#131	1	1	yes	characteristic
38	D₃₃₇	—	2	4	no	—
39	C₃₃₇	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
40	C₈×D₄	—	337	1	no	maximal, Sylow
41	C₂²×C₈	—	337	2	no	—
42	C₄×D₄	—	337	1	no	—
43	C₄⋊C₈	—	337	1	no	—
44	C₂²⋊C₈	—	337	2	no	—
45	C₄×C₈	—	337	1	no	—
46	C₂×D₄	—	337	1	no	—
47	C₂²×C₄	—	337	2	no	—
48	C₂×C₈	—	337	1	no	—
49	C₂×C₈	—	337	1	no	—
50	C₂×C₈	—	337	2	no	—
51	C₂×C₈	—	674	4	no	—
52	C₄⋊C₄	#13062438	1	1	yes	characteristic
53	C₂²⋊C₄	—	337	2	no	—
54	C₄²	—	337	1	no	—
55	C₂³	—	337	2	no	—
56	D₄	—	337	4	no	—
57	C₂×C₄	—	337	1	no	—
58	C₂×C₄	—	337	1	no	—
59	C₂×C₄	—	674	2	no	—
60	C₂×C₄	—	674	2	no	—
61	C₂×C₄	C₂×#13062438	1	2	yes	—
62	C₂×C₄	C₂×#13062438	1	1	yes	characteristic
63	C₈	—	337	2	no	—
64	C₈	—	674	2	no	—
65	C₈	—	674	1	no	—
66	C₂²	—	337	4	no	—
67	C₂²	—	674	2	no	—
68	C₂²	—	674	2	no	—
69	C₂²	C₂²×#13062438	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	337	1	no	—
72	C₄	—	337	1	no	—
73	C₄	—	674	1	no	—
74	C₄	#19129193	1	2	yes	complemented
75	C₂	—	674	4	no	—
76	C₂	D₄×#13062438	1	1	yes	characteristic
77	C₂	C₂×#19129193	1	1	yes	characteristic
78	C₂	#24925047	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#13062438
commutator	C₂×C₃₃₇	C₂²×C₈
Fitting	C₄⋊C₄×C₃₃₇	C₄
socle	C₂²×C₃₃₇	C₂²×C₄
prime radical	C₂×D₁₃₄₈	C₄
cosocle	C₂×Dic₃₃₇	C₂³
hypercenter	C₄⋊C₄	#13062438
hypocenter	C₃₃₇	C₈×D₄
LCS	G ▷ C₂×C₃₃₇ ▷ C₃₃₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#19129193	C₂	semidirect
2. C₄⋊#14582113	C₂	semidirect
3. #24925016	C₂	semidirect
4. C₄×#14582113	C₂	semidirect
17. #14582113	D₄	semidirect
19. D₁₃₄₈	C₈	semidirect
29. C₂×D₃₃₇	C₂×C₈	semidirect
31. C₄×C₃₃₇	C₂×C₈	semidirect
39. C₃₃₇	C₈×D₄	semidirect
74. C₄	#19129193	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19129193	2
C₄⋊#14582113	2
#24925016	2
C₄×#14582113	2
#24924921	2
C₈×D₄	337