Subgroups — #24924941

Order 10784.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊D₃₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₃₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊Dic₃₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#19129164	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊Dic₃₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#19129162	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₄×C₃₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₃₇	C₂²	1	1	yes	characteristic
9	C₂²⋊D₃₃₇	—	2	2	no	—
10	C₂×Dic₃₃₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₃₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₃₃₇	C₂²	1	1	yes	characteristic
13	C₄×D₃₃₇	—	2	2	no	—
14	C₂³×C₃₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₃₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₃₃₇	C₂²	1	1	yes	characteristic
17	C₂²×C₃₃₇	—	2	1	no	—
18	C₂²×C₃₃₇	—	2	1	no	—
19	C₂²×C₃₃₇	—	2	1	no	—
20	C₂²×C₃₃₇	C₂³	1	1	yes	characteristic
21	C₂×D₃₃₇	—	2	1	no	—
22	C₂×D₃₃₇	—	2	1	no	—
23	C₂×D₃₃₇	D₄	1	2	yes	—
24	C₄×C₃₃₇	—	2	1	no	—
25	C₄×C₃₃₇	—	2	1	no	—
26	Dic₃₃₇	—	2	1	no	—
27	Dic₃₃₇	—	2	1	no	—
28	Dic₃₃₇	—	2	1	no	—
29	C₂×C₃₃₇	—	4	1	no	—
30	C₂×C₃₃₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₃₃₇	#41	1	1	yes	characteristic
32	C₂×C₃₃₇	#41	1	1	yes	characteristic
33	D₃₃₇	—	2	2	no	—
34	C₃₃₇	#123	1	1	yes	characteristic, complemented, Sylow
35	#123	—	337	1	no	maximal, Sylow
36	C₂×D₄	—	337	1	no	—
37	C₂²×C₄	—	337	1	no	—
38	C₄⋊C₄	—	337	1	no	—
39	C₄⋊C₄	—	337	1	no	—
40	C₂²⋊C₄	—	337	1	no	—
41	C₂²⋊C₄	—	337	1	no	—
42	C₂²⋊C₄	D₃₃₇	1	1	yes	characteristic, complemented
43	C₂³	—	337	1	no	—
44	C₂³	C₂×D₃₃₇	1	1	yes	characteristic
45	D₄	—	674	2	no	—
46	C₂×C₄	—	337	1	no	—
47	C₂×C₄	—	337	1	no	—
48	C₂×C₄	—	337	1	no	—
49	C₂×C₄	—	674	2	no	—
50	C₂×C₄	C₂×D₃₃₇	1	1	yes	characteristic
51	C₂×C₄	C₂×D₃₃₇	1	1	yes	characteristic
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	337	2	no	—
56	C₂²	—	674	1	no	—
57	C₂²	—	674	1	no	—
58	C₂²	C₂²×D₃₃₇	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	2	1	no	—
61	C₄	—	674	1	no	—
62	C₄	—	674	1	no	—
63	C₄	—	674	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	674	2	no	—
66	C₂	D₄×D₃₃₇	1	1	yes	characteristic
67	C₂	#19129184	1	1	yes	characteristic
68	C₂	#19129185	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₃₇
commutator, socle, cosocle	C₂²×C₃₃₇	C₂³
Fitting	C₂²⋊C₄×C₃₃₇	C₂
prime radical	C₂×C₂²⋊D₃₃₇	C₂
hypercenter	C₂²⋊C₄	D₃₃₇
hypocenter	C₃₃₇	#123
LCS	G ▷ C₂²×C₃₃₇ ▷ C₃₃₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₃₃₇	C₂	semidirect
3. C₄⋊Dic₃₃₇	C₂	semidirect
4. #19129164	C₂	semidirect
5. C₂²⋊Dic₃₃₇	C₂	semidirect
6. #19129162	C₂	semidirect
7. C₂²⋊C₄×C₃₃₇	C₂	semidirect
34. C₃₃₇	#123	semidirect
42. C₂²⋊C₄	D₃₃₇	semidirect