Subgroups — #36342174

Order 21568.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₃₇⋊M₁₆	C₂	1	2	yes	maximal
2	C₂×#19129168	C₂	1	1	yes	maximal, characteristic
3	#19129195	—	2	2	no	—
4	#19129194	—	2	2	no	—
5	#19129193	C₄	1	4	yes	—
6	C₂×C₄×D₃₃₇	C₂²	1	1	yes	characteristic
7	#19129168	—	2	2	no	—
8	C₂×#14582106	C₄	1	1	yes	characteristic
9	C₂×C₈×C₃₃₇	C₄	1	1	yes	characteristic
10	#14582113	—	2	4	no	—
11	C₂²×D₃₃₇	Q₈	1	1	yes	characteristic
12	C₂×Dic₃₃₇	D₄	1	1	yes	characteristic
13	C₄×D₃₃₇	C₂×C₄	1	2	yes	—
14	C₄×D₃₃₇	D₄	1	2	yes	—
15	#14582106	—	2	1	no	—
16	C₂×C₄×C₃₃₇	C₂×C₄	1	1	yes	characteristic
17	C₈×C₃₃₇	—	2	1	no	—
18	C₂²×C₃₃₇	C₄⋊C₄	1	1	yes	characteristic
19	C₂×D₃₃₇	—	4	1	no	—
20	C₂×D₃₃₇	C₄⋊C₄	1	2	yes	—
21	C₄×C₃₃₇	C₄²	1	1	yes	characteristic
22	C₄×C₃₃₇	C₂²⋊C₄	1	1	yes	characteristic
23	Dic₃₃₇	C₂²⋊C₄	1	2	yes	—
24	C₂×C₃₃₇	—	2	1	no	—
25	C₂×C₃₃₇	#95	1	1	yes	characteristic
26	D₃₃₇	—	2	2	no	—
27	C₃₃₇	#338	1	1	yes	characteristic, complemented, Sylow
28	#338	—	337	1	no	maximal, Sylow
29	C₂×M₁₆	—	337	2	no	—
30	C₂×M₁₆	—	337	1	no	—
31	C₂²×C₄	—	337	1	no	—
32	M₁₆	—	674	2	no	—
33	M₁₆	—	674	2	no	—
34	M₁₆	—	674	2	no	—
35	C₂×C₈	—	337	4	no	—
36	C₂×C₈	—	337	1	no	—
37	C₂×C₈	#13062438	1	1	yes	characteristic
38	C₂³	—	337	1	no	—
39	C₂×C₄	—	337	1	no	—
40	C₂×C₄	—	337	2	no	—
41	C₂×C₄	—	337	2	no	—
42	C₂×C₄	C₂×#13062438	1	1	yes	characteristic
43	C₈	—	2	1	no	—
44	C₈	—	674	4	no	—
45	C₈	—	674	1	no	—
46	C₂²	—	1348	1	no	—
47	C₂²	—	337	2	no	—
48	C₂²	C₄⋊#13062438	1	1	yes	characteristic
49	C₄	—	337	2	no	—
50	C₄	C₄×#13062438	1	1	yes	characteristic
51	C₄	C₂²⋊#13062438	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	674	2	no	—
54	C₂	(C₂×#13062438)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×#13062438
commutator	C₄×C₃₃₇	C₄²
Frattini	C₂×C₄	C₂×#13062438
Fitting	C₂×C₈×C₃₃₇	C₄
socle	C₂×C₃₃₇	#95
prime radical	C₂²×D₃₃₇	Q₈
cosocle	C₂×C₄×D₃₃₇	C₂²
hypercenter	C₂×C₈	#13062438
hypocenter	C₃₃₇	#338
LCS	G ▷ C₄×C₃₃₇ ▷ C₃₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
27. C₃₃₇	#338	semidirect

Maximal subgroups

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

subgroup	index
D₃₃₇⋊M₁₆	2
C₂×#19129168	2
#338	337