Subgroups — #36342251

Order 21568.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#13062438	C₂	1	6	yes	maximal
2	C₄²×D₃₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13062438	C₂²	1	4	yes	—
4	C₄⋊#13062438	—	2	12	no	—
5	C₂×C₄×D₃₃₇	C₂²	1	3	yes	—
6	C₄×Dic₃₃₇	C₄	1	3	yes	complemented
7	C₄²×C₃₃₇	C₄	1	1	yes	characteristic, complemented
8	C₂×#13062438	—	2	12	no	—
9	C₂²×D₃₃₇	C₂³	1	1	yes	characteristic
10	C₂×Dic₃₃₇	C₂×C₄	1	3	yes	—
11	C₄×D₃₃₇	D₄	1	6	yes	—
12	C₄×D₃₃₇	Q₈	1	6	yes	—
13	C₂×C₄×C₃₃₇	C₂×C₄	1	3	yes	—
14	C₂²×C₃₃₇	C₂²×C₄	1	1	yes	characteristic
15	C₂×D₃₃₇	C₂×D₄	1	3	yes	—
16	C₂×D₃₃₇	C₂×Q₈	1	3	yes	—
17	#13062438	—	4	4	no	—
18	C₄×C₃₃₇	C₄⋊C₄	1	6	yes	complemented
19	Dic₃₃₇	C₄⋊C₄	1	6	yes	complemented
20	C₂×C₃₃₇	C₂×C₄⋊C₄	1	3	yes	—
21	D₃₃₇	#127	1	1	yes	characteristic
22	D₃₃₇	C₄⋊Q₈	1	3	yes	—
23	C₃₃₇	#384	1	1	yes	characteristic, complemented, Sylow
24	#384	—	337	1	no	maximal, Sylow
25	C₂×C₄⋊C₄	—	337	6	no	—
26	C₂×C₄²	—	337	1	no	—
27	C₂²×C₄	—	337	3	no	—
28	C₂²×C₄	—	337	4	no	—
29	C₄⋊C₄	—	674	12	no	—
30	C₄²	—	337	3	no	—
31	C₄²	#13062438	1	1	yes	characteristic, complemented
32	C₂³	—	337	1	no	—
33	C₂×C₄	—	337	3	no	—
34	C₂×C₄	—	337	6	no	—
35	C₂×C₄	—	337	6	no	—
36	C₂×C₄	—	674	12	no	—
37	C₂×C₄	C₂×#13062438	1	3	yes	—
38	C₂²	—	337	3	no	—
39	C₂²	—	337	3	no	—
40	C₂²	C₂²×#13062438	1	1	yes	characteristic
41	C₄	—	1348	4	no	—
42	C₄	—	337	6	no	—
43	C₄	C₄⋊#13062438	1	6	yes	complemented
44	C₂	—	337	1	no	—
45	C₂	—	337	3	no	—
46	C₂	C₂×C₄⋊#13062438	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#13062438
commutator, socle	C₂²×C₃₃₇	C₂²×C₄
Fitting	C₄²×C₃₃₇	C₄
prime radical, cosocle	C₂²×D₃₃₇	C₂³
hypercenter	C₄²	#13062438
hypocenter	C₃₃₇	#384
LCS	G ▷ C₂²×C₃₃₇ ▷ C₃₃₇
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
6. C₄×Dic₃₃₇	C₄	semidirect
7. C₄²×C₃₃₇	C₄	semidirect
18. C₄×C₃₃₇	C₄⋊C₄	semidirect
19. Dic₃₃₇	C₄⋊C₄	semidirect
23. C₃₃₇	#384	semidirect
31. C₄²	#13062438	semidirect
43. C₄	C₄⋊#13062438	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#13062438	2
C₄²×D₃₃₇	2
#384	337