Subgroups — C₂²×#14555518 #22578474

Order 8768.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14555518	C₂	1	12	yes	maximal, direct product factor
2	C₂²×#11764047	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#14555518	C₂²	1	16	yes	direct product factor
5	C₂×#11764047	C₂²	1	12	yes	complemented
6	C₂³×D₁₃₇	C₄	1	1	yes	characteristic
7	C₂²×Dic₁₃₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₁₃₇	C₄	1	6	yes	—
9	C₂×C₄×D₁₃₇	C₂²	1	6	yes	—
10	C₂²×C₄×C₁₃₇	C₄	1	1	yes	characteristic
11	#11764047	C₂³	1	8	yes	complemented
12	C₂²×D₁₃₇	C₈	1	8	yes	complemented
13	C₂²×D₁₃₇	C₂×C₄	1	6	yes	—
14	C₂×Dic₁₃₇	C₂×C₄	1	3	yes	—
15	C₂×Dic₁₃₇	C₂³	1	3	yes	—
16	C₄×D₁₃₇	C₂×C₄	1	12	yes	—
17	C₄×D₁₃₇	C₂³	1	4	yes	—
18	C₂³×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₃₇	C₂×C₄	1	6	yes	—
20	C₂²×C₁₃₇	C₂×C₈	1	4	yes	complemented
21	C₂²×C₁₃₇	C₂²×C₄	1	3	yes	—
22	C₂×D₁₃₇	C₂×C₈	1	24	yes	complemented
23	C₂×D₁₃₇	C₂²×C₄	1	4	yes	—
24	C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
25	Dic₁₃₇	C₂²×C₄	1	3	yes	—
26	Dic₁₃₇	C₂⁴	1	1	yes	characteristic
27	C₂×C₁₃₇	C₂²×C₈	1	6	yes	complemented
28	C₂×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
29	D₁₃₇	C₂²×C₈	1	8	yes	complemented
30	C₁₃₇	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
31	C₂³×C₈	—	137	1	no	maximal, Sylow
32	C₂³×C₄	—	137	1	no	—
33	C₂²×C₈	—	137	2	no	—
34	C₂²×C₈	—	137	12	no	—
35	C₂⁴	—	137	1	no	—
36	C₂²×C₄	—	137	1	no	—
37	C₂²×C₄	—	137	6	no	—
38	C₂²×C₄	—	137	6	no	—
39	C₂²×C₄	#10587739	1	1	yes	characteristic
40	C₂×C₈	—	137	12	no	—
41	C₂×C₈	—	137	16	no	—
42	C₂³	—	137	8	no	—
43	C₂³	—	137	6	no	—
44	C₂³	C₂×#10587739	1	1	yes	characteristic
45	C₂×C₄	—	137	3	no	—
46	C₂×C₄	—	137	3	no	—
47	C₂×C₄	—	137	4	no	—
48	C₂×C₄	—	137	12	no	—
49	C₂×C₄	C₂×#10587739	1	6	yes	—
50	C₈	—	137	8	no	—
51	C₂²	—	137	24	no	—
52	C₂²	—	137	4	no	—
53	C₂²	#14555518	1	4	yes	direct product factor
54	C₂²	C₂²×#10587739	1	3	yes	—
55	C₄	—	137	1	no	—
56	C₄	—	137	3	no	—
57	C₄	C₂²×#10587739	1	4	yes	—
58	C₂	—	137	8	no	—
59	C₂	C₂×#14555518	1	6	yes	direct product factor
60	C₂	C₂³×#10587739	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	#10587739
commutator, hypocenter	C₁₃₇	C₂³×C₈
Frattini	C₂	C₂³×#10587739
Fitting	C₂²×C₄×C₁₃₇	C₄
socle	C₂³×C₁₃₇	C₂×C₄
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#14555518	C₂	direct (N × H)
2. C₂²×#11764047	C₂	semidirect
4. #14555518	C₂²	direct (N × H)
5. C₂×#11764047	C₂²	semidirect
11. #11764047	C₂³	semidirect
12. C₂²×D₁₃₇	C₈	semidirect
20. C₂²×C₁₃₇	C₂×C₈	semidirect
22. C₂×D₁₃₇	C₂×C₈	semidirect
27. C₂×C₁₃₇	C₂²×C₈	semidirect
29. D₁₃₇	C₂²×C₈	semidirect
30. C₁₃₇	C₂³×C₈	semidirect
53. C₂²	#14555518	direct (N × H)
59. C₂	C₂×#14555518	direct (N × H)

Subgroups — C22×#14555518 #22578474

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#14555518 #22578474