Subgroups — C₂×#24925055 #36342423

Order 21568.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24925055	C₂	1	4	yes	maximal, direct product factor
2	C₂×#19129197	C₂	1	2	yes	maximal, complemented
3	C₂²×#14582113	C₂	1	1	yes	maximal, characteristic
4	#19129197	C₂²	1	4	yes	complemented
5	C₂×#14582113	C₄	1	2	yes	—
6	C₂×#14582113	C₄	1	1	yes	characteristic
7	C₂×#14582113	C₂²	1	1	yes	characteristic
8	C₂×#14582113	C₂²	1	2	yes	—
9	C₂²×Dic₃₃₇	C₄	1	1	yes	characteristic
10	#14582113	C₂×C₄	1	1	yes	characteristic
11	#14582113	C₂×C₄	1	2	yes	—
12	#14582113	C₂³	1	1	yes	characteristic
13	C₂×Dic₃₃₇	C₈	1	2	yes	—
14	C₂×Dic₃₃₇	C₈	1	1	yes	characteristic
15	C₂×Dic₃₃₇	C₂×C₄	1	2	yes	—
16	C₂×Dic₃₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₃₃₇	C₈	1	1	yes	characteristic
18	C₂²×C₃₃₇	—	2	2	no	—
19	C₂²×C₃₃₇	C₂×C₈	1	2	yes	—
20	C₂²×C₃₃₇	C₂×C₈	1	1	yes	characteristic
21	Dic₃₃₇	C₂×C₈	1	2	yes	—
22	Dic₃₃₇	C₂×C₈	1	1	yes	characteristic
23	Dic₃₃₇	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₃₃₇	—	2	2	no	—
25	C₂×C₃₃₇	M₃₂	1	2	yes	complemented
26	C₂×C₃₃₇	C₂²×C₈	1	1	yes	characteristic
27	C₃₃₇	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
28	C₂×M₃₂	—	337	1	no	maximal, Sylow
29	C₂²×C₈	—	337	1	no	—
30	M₃₂	—	337	4	no	—
31	C₂×C₁₆	—	337	2	no	—
32	C₂²×C₄	—	337	1	no	—
33	C₂×C₈	—	337	2	no	—
34	C₂×C₈	—	337	1	no	—
35	C₂×C₈	—	337	1	no	—
36	C₂×C₈	—	337	2	no	—
37	C₁₆	—	337	4	no	—
38	C₂³	#14582115	1	1	yes	characteristic
39	C₂×C₄	—	337	2	no	—
40	C₂×C₄	—	337	2	no	—
41	C₂×C₄	—	337	1	no	—
42	C₂×C₄	—	337	1	no	—
43	C₈	—	337	1	no	—
44	C₈	—	337	1	no	—
45	C₈	—	337	2	no	—
46	C₂²	—	2	2	no	—
47	C₂²	C₂×#14582115	1	2	yes	—
48	C₂²	C₂×#14582115	1	1	yes	characteristic
49	C₄	—	337	1	no	—
50	C₄	—	337	2	no	—
51	C₄	—	337	1	no	—
52	C₂	—	2	2	no	—
53	C₂	#24925055	1	2	yes	direct product factor
54	C₂	C₂²×#14582115	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14582115
commutator	C₂×C₃₃₇	C₂²×C₈
Frattini	C₂	C₂²×#14582115
Fitting, prime radical	C₂³×C₃₃₇	C₈
socle	C₂²×C₃₃₇	C₂×C₈
cosocle	#14582113	C₂³
hypercenter	C₂³	#14582115
hypocenter	C₃₃₇	C₂×M₃₂
LCS	G ▷ C₂×C₃₃₇ ▷ C₃₃₇
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. #24925055	C₂	direct (N × H)
2. C₂×#19129197	C₂	semidirect
4. #19129197	C₂²	semidirect
25. C₂×C₃₃₇	M₃₂	semidirect
27. C₃₃₇	C₂×M₃₂	semidirect
53. C₂	#24925055	direct (N × H)

Maximal subgroups

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

subgroup	index
#24925055	2
C₂×#19129197	2
C₂²×#14582113	2
C₂×M₃₂	337

Subgroups — C2×#24925055 #36342423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24925055 #36342423