Subgroups — C₂×C₂²⋊#10587739 #17984165

Order 4384.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587739	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#10587739	C₂	1	4	yes	maximal, direct product factor
3	C₂³×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587739	—	2	4	no	—
5	C₂×#10587739	C₂²	1	4	yes	complemented
6	C₂²×D₁₃₇	—	2	4	no	—
7	C₂²×D₁₃₇	C₄	1	2	yes	complemented
8	C₂²×D₁₃₇	C₄	1	1	yes	characteristic, complemented
9	C₂²×D₁₃₇	C₂²	1	2	yes	—
10	C₂²×D₁₃₇	C₂²	1	1	yes	characteristic
11	C₂³×C₁₃₇	C₄	1	1	yes	characteristic, complemented
12	C₂²×C₁₃₇	—	2	2	no	—
13	C₂²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₁₃₇	C₂×C₄	1	2	yes	complemented
15	C₂×D₁₃₇	—	2	2	no	—
16	C₂×D₁₃₇	—	2	4	no	—
17	C₂×D₁₃₇	—	2	4	no	—
18	C₂×D₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂×D₁₃₇	C₂×C₄	1	2	yes	complemented
20	C₂×D₁₃₇	D₄	1	4	yes	—
21	C₂×D₁₃₇	C₂³	1	1	yes	characteristic
22	#10587739	—	2	4	no	—
23	C₂×C₁₃₇	—	2	2	no	—
24	C₂×C₁₃₇	C₂²⋊C₄	1	2	yes	complemented
25	C₂×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
26	D₁₃₇	—	2	2	no	—
27	D₁₃₇	C₂²⋊C₄	1	2	yes	complemented
28	D₁₃₇	C₂×D₄	1	2	yes	—
29	C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₂²⋊C₄	—	137	1	no	maximal, Sylow
31	C₂⁴	—	137	1	no	—
32	C₂²×C₄	—	137	2	no	—
33	C₂²⋊C₄	—	137	4	no	—
34	C₂³	—	274	4	no	—
35	C₂³	—	137	1	no	—
36	C₂³	—	137	2	no	—
37	C₂³	—	137	2	no	—
38	C₂³	—	137	1	no	—
39	C₂³	#10587739	1	1	yes	characteristic, complemented
40	C₂×C₄	—	274	4	no	—
41	C₂×C₄	—	137	4	no	—
42	C₂²	—	2	2	no	—
43	C₂²	—	274	4	no	—
44	C₂²	—	274	4	no	—
45	C₂²	—	274	2	no	—
46	C₂²	—	137	2	no	—
47	C₂²	—	137	4	no	—
48	C₂²	—	137	1	no	—
49	C₂²	—	137	1	no	—
50	C₂²	C₂×#10587739	1	1	yes	characteristic
51	C₂²	C₂×#10587739	1	2	yes	complemented
52	C₄	—	274	4	no	—
53	C₂	—	2	2	no	—
54	C₂	—	274	2	no	—
55	C₂	—	137	2	no	—
56	C₂	—	137	2	no	—
57	C₂	C₂²⋊#10587739	1	2	yes	direct product factor
58	C₂	C₂²×#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#10587739
commutator	C₂×C₁₃₇	C₂²×C₄
Frattini	C₂	C₂²×#10587739
Fitting	C₂³×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₄
prime radical	C₂³×D₁₃₇	C₂
cosocle	C₂×D₁₃₇	C₂³
hypercenter	C₂³	#10587739
hypocenter	C₁₃₇	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂²×#10587739	C₂	semidirect
2. C₂²⋊#10587739	C₂	direct (N × H)
5. C₂×#10587739	C₂²	semidirect
7. C₂²×D₁₃₇	C₄	semidirect
8. C₂²×D₁₃₇	C₄	semidirect
11. C₂³×C₁₃₇	C₄	semidirect
14. C₂²×C₁₃₇	C₂×C₄	semidirect
19. C₂×D₁₃₇	C₂×C₄	semidirect
24. C₂×C₁₃₇	C₂²⋊C₄	semidirect
27. D₁₃₇	C₂²⋊C₄	semidirect
29. C₁₃₇	C₂×C₂²⋊C₄	semidirect
39. C₂³	#10587739	semidirect
51. C₂²	C₂×#10587739	semidirect
57. C₂	C₂²⋊#10587739	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#10587739	2
C₂²⋊#10587739	2
C₂³×D₁₃₇	2
C₂×C₂²⋊C₄	137

Subgroups — C2×C22⋊#10587739 #17984165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊#10587739 #17984165