Subgroups — #36342426

Order 21568.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24925056	C₂	1	1	yes	maximal, characteristic, complemented
2	#24925055	C₂	1	2	yes	maximal, complemented
3	#24925054	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19129197	C₂	1	2	yes	maximal, complemented
5	#24925046	C₂	1	1	yes	maximal, characteristic
6	#19129197	C₂²	1	1	yes	characteristic
7	#19129197	C₂²	1	2	yes	—
8	#19129197	C₂²	1	1	yes	characteristic
9	#19129195	C₄	1	1	yes	characteristic
10	#19129194	C₄	1	2	yes	—
11	#19129193	C₂²	1	1	yes	characteristic
12	C₂×#14582113	C₂²	1	2	yes	—
13	#19129184	C₄	1	1	yes	characteristic
14	#14582113	C₂×C₄	1	2	yes	—
15	#14582113	C₂×C₄	1	1	yes	characteristic
16	#14582113	C₂³	1	1	yes	characteristic
17	Dic₆₇₄	C₈	1	1	yes	characteristic
18	C₂²⋊D₃₃₇	C₈	1	2	yes	—
19	C₂×Dic₃₃₇	C₂×C₄	1	2	yes	—
20	C₄×D₃₃₇	C₂×C₄	1	1	yes	characteristic
21	D₄×C₃₃₇	C₈	1	1	yes	characteristic
22	C₂²×C₃₃₇	C₂×C₈	1	2	yes	—
23	C₂×D₃₃₇	C₂×C₈	1	1	yes	characteristic
24	C₄×C₃₃₇	C₂×C₈	1	1	yes	characteristic
25	Dic₃₃₇	C₂×C₈	1	2	yes	—
26	Dic₃₃₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₃₃₇	—	2	2	no	—
28	C₂×C₃₃₇	C₂²×C₈	1	1	yes	characteristic
29	D₃₃₇	—	2	1	no	—
30	C₃₃₇	#504	1	1	yes	characteristic, complemented, Sylow
31	#504	—	337	1	no	maximal, Sylow
32	#131	—	337	1	no	—
33	M₃₂	—	337	2	no	—
34	M₃₂	—	337	1	no	—
35	C₂×C₁₆	—	337	2	no	—
36	C₂×C₁₆	—	337	1	no	—
37	#41	—	337	1	no	—
38	M₁₆	—	337	2	no	—
39	M₁₆	—	337	1	no	—
40	C₂×C₈	—	337	2	no	—
41	C₂×C₈	—	337	1	no	—
42	C₁₆	—	337	1	no	—
43	C₁₆	—	337	2	no	—
44	C₁₆	—	337	1	no	—
45	Q₈	—	337	1	no	—
46	D₄	—	337	2	no	—
47	D₄	#14582115	1	1	yes	characteristic
48	C₂×C₄	—	337	2	no	—
49	C₂×C₄	—	337	1	no	—
50	C₈	—	337	1	no	—
51	C₈	—	337	2	no	—
52	C₈	—	337	1	no	—
53	C₂²	—	337	1	no	—
54	C₂²	C₂×#14582115	1	2	yes	—
55	C₄	—	337	1	no	—
56	C₄	—	337	2	no	—
57	C₄	C₂×#14582115	1	1	yes	characteristic
58	C₂	—	2	2	no	—
59	C₂	—	674	1	no	—
60	C₂	C₂²×#14582115	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×#14582115
commutator, socle	C₂×C₃₃₇	C₂²×C₈
Fitting	D₄×C₃₃₇	C₈
prime radical	#19129184	C₄
cosocle	#14582113	C₂³
hypercenter	D₄	#14582115
hypocenter	C₃₃₇	#504
LCS	G ▷ C₂×C₃₃₇ ▷ C₃₃₇
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. #24925056	C₂	semidirect
2. #24925055	C₂	semidirect
3. #24925054	C₂	semidirect
4. C₂×#19129197	C₂	semidirect
30. C₃₃₇	#504	semidirect

Maximal subgroups

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

subgroup	index
#24925056	2
#24925055	2
#24925054	2
C₂×#19129197	2
#24925046	2
#504	337