Subgroups — #36342224

Order 21568.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24925034	C₂	1	1	yes	maximal, characteristic
2	#24925029	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×D₃₃₇	C₂	1	1	yes	maximal, characteristic
4	#19129195	—	2	1	no	—
5	#19129188	—	2	1	no	—
6	Q₈×D₃₃₇	—	2	1	no	—
7	Dic₁₃₄₈	C₄	1	1	yes	characteristic
8	#19129175	—	2	1	no	—
9	C₈×D₃₃₇	C₂²	1	1	yes	characteristic
10	Q₁₆×C₃₃₇	C₄	1	1	yes	characteristic
11	#14582113	—	4	1	no	—
12	Dic₆₇₄	—	2	1	no	—
13	Dic₆₇₄	—	4	1	no	—
14	C₄×D₃₃₇	—	4	1	no	—
15	C₄×D₃₃₇	D₄	1	1	yes	characteristic
16	#14582106	D₄	1	1	yes	characteristic
17	Q₈×C₃₃₇	—	2	1	no	—
18	C₈×C₃₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂×D₃₃₇	SD₁₆	1	1	yes	characteristic
20	C₄×C₃₃₇	—	4	1	no	—
21	C₄×C₃₃₇	C₂²⋊C₄	1	1	yes	characteristic
22	Dic₃₃₇	—	4	1	no	—
23	Dic₃₃₇	D₈	1	1	yes	characteristic
24	C₂×C₃₃₇	#102	1	1	yes	characteristic
25	D₃₃₇	—	2	1	no	—
26	C₃₃₇	#362	1	1	yes	characteristic, complemented, Sylow
27	#362	—	337	1	no	maximal, Sylow
28	C₂×Q₁₆	—	337	1	no	—
29	M₃₂	—	337	1	no	—
30	#108	—	337	1	no	—
31	C₂×Q₈	—	674	1	no	—
32	Q₁₆	—	337	1	no	—
33	Q₁₆	—	674	1	no	—
34	Q₁₆	#13062438	1	1	yes	characteristic
35	M₁₆	—	674	1	no	—
36	C₂×C₈	—	337	1	no	—
37	C₁₆	—	674	1	no	—
38	Q₈	—	2	1	no	—
39	Q₈	—	1348	1	no	—
40	Q₈	—	674	1	no	—
41	C₂×C₄	—	1348	1	no	—
42	C₂×C₄	—	337	1	no	—
43	C₈	—	1348	1	no	—
44	C₈	—	337	1	no	—
45	C₈	C₂×#13062438	1	1	yes	characteristic
46	C₂²	—	337	1	no	—
47	C₄	—	4	1	no	—
48	C₄	—	1348	1	no	—
49	C₄	—	337	1	no	—
50	C₄	C₂²⋊#13062438	1	1	yes	characteristic
51	C₂	—	674	1	no	—
52	C₂	#24925021	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#24925021
commutator	C₈×C₃₃₇	C₂×C₄
Frattini	C₈	C₂×#13062438
Fitting	Q₁₆×C₃₃₇	C₄
socle	C₂×C₃₃₇	#102
prime radical	C₂×D₃₃₇	SD₁₆
cosocle	C₈×D₃₃₇	C₂²
hypercenter	Q₁₆	#13062438
hypocenter	C₃₃₇	#362
LCS	G ▷ C₈×C₃₃₇ ▷ C₄×C₃₃₇ ▷ C₂×C₃₃₇ ▷ C₃₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
26. C₃₃₇	#362	semidirect

Maximal subgroups

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

subgroup	index
#24925034	2
#24925029	2
Q₁₆×D₃₃₇	2
#362	337