Subgroups — #36342168

Order 21568.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14582113	C₂	1	2	yes	maximal
2	C₈×Dic₃₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582113	—	2	2	no	—
4	C₂×#14582106	C₄	1	1	yes	characteristic
5	C₄×Dic₃₃₇	C₂²	1	1	yes	characteristic
6	C₂×C₈×C₃₃₇	C₄	1	1	yes	characteristic
7	#14582113	—	4	2	no	—
8	C₂×Dic₃₃₇	D₄	1	1	yes	characteristic
9	C₂×Dic₃₃₇	Q₈	1	1	yes	characteristic
10	#14582106	—	2	1	no	—
11	C₂×C₄×C₃₃₇	C₂×C₄	1	1	yes	characteristic
12	C₈×C₃₃₇	C₈	1	2	yes	complemented
13	C₂²×C₃₃₇	C₄⋊C₄	1	1	yes	characteristic
14	C₄×C₃₃₇	C₂×C₈	1	1	yes	characteristic
15	C₄×C₃₃₇	M₁₆	1	1	yes	characteristic
16	Dic₃₃₇	—	2	1	no	—
17	Dic₃₃₇	D₈	1	1	yes	characteristic
18	Dic₃₃₇	Q₁₆	1	1	yes	characteristic
19	C₂×C₃₃₇	C₄⋊C₈	1	1	yes	characteristic
20	C₂×C₃₃₇	#107	1	1	yes	characteristic
21	C₂×C₃₃₇	#108	1	1	yes	characteristic
22	C₃₃₇	#335	1	1	yes	characteristic, complemented, Sylow
23	#335	—	337	1	no	maximal, Sylow
24	C₄⋊C₈	—	337	2	no	—
25	C₄×C₈	—	337	1	no	—
26	C₂×C₈	—	337	1	no	—
27	C₂×C₈	—	674	2	no	—
28	C₂×C₈	#13062438	1	1	yes	characteristic
29	C₄²	—	337	1	no	—
30	C₂×C₄	—	337	1	no	—
31	C₂×C₄	—	337	1	no	—
32	C₂×C₄	C₂×#13062438	1	1	yes	characteristic
33	C₈	—	1348	2	no	—
34	C₈	—	674	1	no	—
35	C₈	#14582113	1	2	yes	complemented
36	C₂²	C₄⋊#13062438	1	1	yes	characteristic
37	C₄	—	337	1	no	—
38	C₄	—	337	1	no	—
39	C₄	—	674	1	no	—
40	C₄	C₂×#14582113	1	1	yes	characteristic
41	C₄	#19129195	1	1	yes	characteristic
42	C₂	C₄⋊#14582113	1	1	yes	characteristic
43	C₂	#24925028	1	1	yes	characteristic
44	C₂	#24925030	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊#13062438
commutator	C₄×C₃₃₇	C₂×C₈
Frattini	C₂×C₄	C₂×#13062438
Fitting	C₂×C₈×C₃₃₇	C₄
socle, prime radical	C₂²×C₃₃₇	C₄⋊C₄
cosocle	C₄×Dic₃₃₇	C₂²
hypercenter	C₂×C₈	#13062438
hypocenter	C₃₃₇	#335
LCS	G ▷ C₄×C₃₃₇ ▷ C₂×C₃₃₇ ▷ C₃₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
12. C₈×C₃₃₇	C₈	semidirect
22. C₃₃₇	#335	semidirect
35. C₈	#14582113	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14582113	2
C₈×Dic₃₃₇	2
#335	337