Subgroups — #36342201

Order 21568.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24925033	C₂	1	2	yes	maximal, complemented
2	C₂×#19129169	C₂	1	1	yes	maximal, characteristic
3	#19129188	—	2	2	no	—
4	#19129169	—	2	2	no	—
5	C₂×#14582106	C₄	1	1	yes	characteristic
6	C₂×#14582106	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₃₃₇	C₄	1	1	yes	characteristic
8	#14582106	—	2	1	no	—
9	#14582106	D₄	1	2	yes	—
10	C₂³×C₃₃₇	C₈	1	1	yes	characteristic
11	C₂×C₄×C₃₃₇	—	2	1	no	—
12	C₂×C₄×C₃₃₇	—	2	1	no	—
13	C₂×C₄×C₃₃₇	C₈	1	1	yes	characteristic
14	C₂×C₄×C₃₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₃₃₇	—	2	1	no	—
16	C₂²×C₃₃₇	—	4	1	no	—
17	C₂²×C₃₃₇	C₂×C₈	1	1	yes	characteristic
18	C₄×C₃₃₇	—	2	1	no	—
19	C₄×C₃₃₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₄×C₃₃₇	M₁₆	1	1	yes	characteristic
21	C₂×C₃₃₇	—	2	1	no	—
22	C₂×C₃₃₇	—	4	1	no	—
23	C₂×C₃₃₇	C₂²⋊C₈	1	1	yes	characteristic
24	C₃₃₇	#349	1	1	yes	characteristic, complemented, Sylow
25	#349	—	337	1	no	maximal, Sylow
26	C₂×M₁₆	—	337	1	no	—
27	M₃₂	—	337	2	no	—
28	C₂²×C₄	#13062438	1	1	yes	characteristic
29	M₁₆	—	674	2	no	—
30	C₂×C₈	—	337	1	no	—
31	C₂×C₈	—	337	1	no	—
32	C₁₆	—	674	2	no	—
33	C₂³	#14582113	1	1	yes	characteristic
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	—	2	1	no	—
36	C₂×C₄	#14582113	1	1	yes	characteristic
37	C₂×C₄	C₂×#13062438	1	1	yes	characteristic
38	C₈	—	337	2	no	—
39	C₈	—	674	1	no	—
40	C₂²	—	2	1	no	—
41	C₂²	—	4	1	no	—
42	C₂²	C₂×#14582113	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	C₂²⋊#13062438	1	1	yes	characteristic
45	C₄	#19129194	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	4	1	no	—
48	C₂	C₂²⋊#14582113	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#13062438
commutator	C₂²×C₃₃₇	C₂×C₈
Frattini	C₂×C₄	C₂×#13062438
Fitting	C₂²×C₄×C₃₃₇	C₄
socle	C₂×C₃₃₇	C₂²⋊C₈
prime radical	C₂³×C₃₃₇	C₈
cosocle	C₂×#14582106	C₂²
hypercenter	C₂²×C₄	#13062438
hypocenter	C₃₃₇	#349
LCS	G ▷ C₂²×C₃₃₇ ▷ C₂×C₃₃₇ ▷ C₃₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. #24925033	C₂	semidirect
24. C₃₃₇	#349	semidirect

Maximal subgroups

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

subgroup	index
#24925033	2
C₂×#19129169	2
#349	337