Subgroups — C₂×#17984082 #22577928

Order 8768.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984082	C₂	1	8	yes	maximal, direct product factor
2	C₂×#14555489	C₂	1	3	yes	maximal, complemented
3	C₂×#14555487	C₂	1	3	yes	maximal
4	C₂×C₄²×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₁₃₇	C₂²	1	1	yes	characteristic
6	C₂²×Dic₁₃₇	C₂²	1	3	yes	—
7	#14555489	C₂²	1	12	yes	complemented
8	#14555487	C₂²	1	12	yes	—
9	C₂²×C₄×C₁₃₇	C₂²	1	3	yes	—
10	C₄²×C₁₃₇	C₂²	1	4	yes	complemented
11	C₂²×D₁₃₇	—	2	6	no	—
12	C₂²×D₁₃₇	C₂³	1	2	yes	—
13	C₂×Dic₁₃₇	—	2	6	no	—
14	C₂×Dic₁₃₇	C₂³	1	6	yes	—
15	C₂³×C₁₃₇	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₁₃₇	—	2	6	no	—
17	C₂×C₄×C₁₃₇	C₂³	1	6	yes	—
18	C₂²×C₁₃₇	#41	1	6	yes	—
19	C₂²×C₁₃₇	C₂⁴	1	1	yes	characteristic
20	C₂×D₁₃₇	—	2	6	no	—
21	C₂×D₁₃₇	—	4	4	no	—
22	C₄×C₁₃₇	—	2	6	no	—
23	Dic₁₃₇	—	2	6	no	—
24	C₂×C₁₃₇	#126	1	4	yes	complemented
25	C₂×C₁₃₇	C₂×#41	1	3	yes	—
26	D₁₃₇	—	4	2	no	—
27	C₁₃₇	C₂×#126	1	1	yes	characteristic, complemented, Sylow
28	C₂×#126	—	137	1	no	maximal, Sylow
29	#126	—	137	8	no	—
30	C₂×C₄⋊C₄	—	137	3	no	—
31	C₂×C₂²⋊C₄	—	137	3	no	—
32	C₂×C₄²	D₁₃₇	1	1	yes	characteristic, complemented
33	C₂⁴	—	137	1	no	—
34	C₂²×C₄	—	137	3	no	—
35	C₂²×C₄	C₂×D₁₃₇	1	3	yes	—
36	C₄⋊C₄	—	137	12	no	—
37	C₂²⋊C₄	—	137	12	no	—
38	C₄²	C₂×D₁₃₇	1	4	yes	complemented
39	C₂³	—	274	6	no	—
40	C₂³	—	137	2	no	—
41	C₂³	C₂²×D₁₃₇	1	1	yes	characteristic
42	C₂×C₄	—	2	6	no	—
43	C₂×C₄	—	274	6	no	—
44	C₂×C₄	—	137	6	no	—
45	C₂×C₄	C₂²×D₁₃₇	1	6	yes	—
46	C₂²	—	274	6	no	—
47	C₂²	—	548	4	no	—
48	C₂²	#14555510	1	6	yes	—
49	C₂²	C₂³×D₁₃₇	1	1	yes	characteristic
50	C₄	—	2	6	no	—
51	C₄	—	274	6	no	—
52	C₂	—	548	2	no	—
53	C₂	#17984082	1	4	yes	direct product factor
54	C₂	C₂×#14555510	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₃₇
commutator	C₂²×C₁₃₇	C₂⁴
Frattini	C₂²	C₂³×D₁₃₇
Fitting	C₂×C₄²×C₁₃₇	C₂
socle	C₂³×C₁₃₇	C₂³
hypercenter	C₂×C₄²	D₁₃₇
hypocenter	C₁₃₇	C₂×#126
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #17984082	C₂	direct (N × H)
2. C₂×#14555489	C₂	semidirect
4. C₂×C₄²×C₁₃₇	C₂	semidirect
7. #14555489	C₂²	semidirect
10. C₄²×C₁₃₇	C₂²	semidirect
24. C₂×C₁₃₇	#126	semidirect
27. C₁₃₇	C₂×#126	semidirect
32. C₂×C₄²	D₁₃₇	semidirect
38. C₄²	C₂×D₁₃₇	semidirect
53. C₂	#17984082	direct (N × H)

Subgroups — C2×#17984082 #22577928

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17984082 #22577928