Subgroups — #36341027

Order 21568.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14582106	C₂	1	1	yes	maximal, characteristic
2	C₄×#14582106	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×C₃₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582106	C₄	1	2	yes	—
5	C₂×#14582106	C₄	1	2	yes	—
6	C₂×C₈×C₃₃₇	C₄	1	2	yes	—
7	C₄²×C₃₃₇	C₂²	1	1	yes	characteristic
8	#14582106	—	2	2	no	—
9	#14582106	C₈	1	4	yes	complemented
10	C₂×C₄×C₃₃₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄×C₃₃₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₃₃₇	C₂×C₄	1	1	yes	characteristic
13	C₈×C₃₃₇	—	2	2	no	—
14	C₂²×C₃₃₇	C₄²	1	1	yes	characteristic
15	C₄×C₃₃₇	C₂×C₈	1	2	yes	—
16	C₄×C₃₃₇	M₁₆	1	2	yes	—
17	C₄×C₃₃₇	M₁₆	1	2	yes	—
18	C₂×C₃₃₇	C₄×C₈	1	1	yes	characteristic
19	C₂×C₃₃₇	#97	1	1	yes	characteristic
20	C₂×C₃₃₇	#97	1	1	yes	characteristic
21	C₃₃₇	#322	1	1	yes	characteristic, complemented, Sylow
22	#322	—	337	1	no	maximal, Sylow
23	C₄×C₈	—	337	1	no	—
24	C₄×C₈	—	337	1	no	—
25	C₄×C₈	D₃₃₇	1	1	yes	characteristic
26	C₂×C₈	—	337	2	no	—
27	C₂×C₈	—	337	2	no	—
28	C₂×C₈	Dic₃₃₇	1	2	yes	—
29	C₄²	C₂×D₃₃₇	1	1	yes	characteristic
30	C₂×C₄	C₄×D₃₃₇	1	1	yes	characteristic
31	C₂×C₄	C₄×D₃₃₇	1	1	yes	characteristic
32	C₂×C₄	C₂×Dic₃₃₇	1	1	yes	characteristic
33	C₈	—	2	2	no	—
34	C₈	—	337	4	no	—
35	C₈	—	674	2	no	—
36	C₂²	C₄×Dic₃₃₇	1	1	yes	characteristic
37	C₄	C₈×D₃₃₇	1	2	yes	—
38	C₄	#19129168	1	2	yes	—
39	C₄	#19129169	1	2	yes	—
40	C₂	C₈×Dic₃₃₇	1	1	yes	characteristic
41	C₂	#24924869	1	1	yes	characteristic
42	C₂	#24924870	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂×D₃₃₇
commutator	C₂×C₃₃₇	C₄×C₈
Fitting	C₄×C₈×C₃₃₇	C₂
socle, prime radical	C₂²×C₃₃₇	C₄²
cosocle	C₄²×C₃₃₇	C₂²
hypercenter	C₄×C₈	D₃₃₇
hypocenter	C₃₃₇	#322
LCS	G ▷ C₂×C₃₃₇ ▷ C₃₃₇
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
9. #14582106	C₈	semidirect
21. C₃₃₇	#322	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14582106	2
C₄×C₈×C₃₃₇	2
#322	337