Subgroups — C₂³×C₂²⋊C₄×C₁₃₇ #31778430

Order 17536.

4632 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊C₄×C₁₃₇	C₂	1	28	yes	maximal, direct product factor
4	C₂⁵×C₁₃₇	—	2	16	no	—
5	C₂⁵×C₁₃₇	C₄	1	16	yes	complemented
6	C₂⁵×C₁₃₇	C₂²	1	14	yes	—
7	C₂⁵×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₁₃₇	—	2	16	no	—
9	C₂³×C₄×C₁₃₇	C₂²	1	28	yes	complemented
10	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	112	yes	direct product factor
11	C₂⁴×C₁₃₇	—	2	128	no	—
12	C₂⁴×C₁₃₇	—	2	112	no	—
13	C₂⁴×C₁₃₇	C₂×C₄	1	8	yes	—
14	C₂⁴×C₁₃₇	C₂×C₄	1	112	yes	complemented
15	C₂⁴×C₁₃₇	D₄	1	16	yes	—
16	C₂⁴×C₁₃₇	C₂³	1	28	yes	—
17	C₂⁴×C₁₃₇	C₂³	1	7	yes	—
18	C₂²×C₄×C₁₃₇	—	2	112	no	—
19	C₂²×C₄×C₁₃₇	C₂³	1	56	yes	complemented
20	C₂²⋊C₄×C₁₃₇	C₂³	1	64	yes	direct product factor
21	C₂³×C₁₃₇	—	2	448	no	—
22	C₂³×C₁₃₇	—	2	112	no	—
23	C₂³×C₁₃₇	C₂²⋊C₄	1	64	yes	direct product factor
24	C₂³×C₁₃₇	C₂²×C₄	1	28	yes	—
25	C₂³×C₁₃₇	C₂²×C₄	1	112	yes	complemented
26	C₂³×C₁₃₇	C₂×D₄	1	56	yes	—
27	C₂³×C₁₃₇	C₂⁴	1	8	yes	—
28	C₂³×C₁₃₇	C₂⁴	1	7	yes	—
29	C₂×C₄×C₁₃₇	—	2	112	no	—
30	C₂×C₄×C₁₃₇	C₂⁴	1	16	yes	complemented
31	C₂²×C₁₃₇	—	2	224	no	—
32	C₂²×C₁₃₇	—	2	16	no	—
33	C₂²×C₁₃₇	C₂×C₂²⋊C₄	1	112	yes	direct product factor
34	C₂²×C₁₃₇	C₂³×C₄	1	14	yes	—
35	C₂²×C₁₃₇	C₂³×C₄	1	16	yes	complemented
36	C₂²×C₁₃₇	C₂²×D₄	1	28	yes	—
37	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
38	C₄×C₁₃₇	—	2	16	no	—
39	C₂×C₁₃₇	—	2	16	no	—
40	C₂×C₁₃₇	C₂²×C₂²⋊C₄	1	28	yes	direct product factor
41	C₂×C₁₃₇	C₂⁴×C₄	1	1	yes	characteristic
42	C₂×C₁₃₇	C₂³×D₄	1	2	yes	—
43	C₁₃₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
44	C₂³×C₂²⋊C₄	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂⁶	C₂×C₁₃₇	1	1	yes	characteristic
46	C₂⁴×C₄	C₂×C₁₃₇	1	2	yes	complemented
47	C₂²×C₂²⋊C₄	C₂×C₁₃₇	1	28	yes	direct product factor
48	C₂⁵	—	2	16	no	—
49	C₂⁵	C₄×C₁₃₇	1	16	yes	complemented
50	C₂⁵	C₂²×C₁₃₇	1	14	yes	—
51	C₂⁵	C₂²×C₁₃₇	1	1	yes	characteristic
52	C₂³×C₄	—	2	16	no	—
53	C₂³×C₄	C₂²×C₁₃₇	1	28	yes	complemented
54	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	112	yes	direct product factor
55	C₂⁴	—	2	128	no	—
56	C₂⁴	—	2	112	no	—
57	C₂⁴	C₂×C₄×C₁₃₇	1	8	yes	—
58	C₂⁴	C₂×C₄×C₁₃₇	1	112	yes	complemented
59	C₂⁴	D₄×C₁₃₇	1	16	yes	—
60	C₂⁴	C₂³×C₁₃₇	1	28	yes	—
61	C₂⁴	C₂³×C₁₃₇	1	7	yes	—
62	C₂²×C₄	—	2	112	no	—
63	C₂²×C₄	C₂³×C₁₃₇	1	56	yes	complemented
64	C₂²⋊C₄	C₂³×C₁₃₇	1	64	yes	direct product factor
65	C₂³	—	2	448	no	—
66	C₂³	—	2	112	no	—
67	C₂³	C₂²⋊C₄×C₁₃₇	1	64	yes	direct product factor
68	C₂³	C₂²×C₄×C₁₃₇	1	28	yes	—
69	C₂³	C₂²×C₄×C₁₃₇	1	112	yes	complemented
70	C₂³	C₂×D₄×C₁₃₇	1	56	yes	—
71	C₂³	C₂⁴×C₁₃₇	1	8	yes	—
72	C₂³	C₂⁴×C₁₃₇	1	7	yes	—
73	C₂×C₄	—	2	112	no	—
74	C₂×C₄	C₂⁴×C₁₃₇	1	16	yes	complemented
75	C₂²	—	2	224	no	—
76	C₂²	—	2	16	no	—
77	C₂²	C₂×C₂²⋊C₄×C₁₃₇	1	112	yes	direct product factor
78	C₂²	C₂³×C₄×C₁₃₇	1	14	yes	—
79	C₂²	C₂³×C₄×C₁₃₇	1	16	yes	complemented
80	C₂²	C₂²×D₄×C₁₃₇	1	28	yes	—
81	C₂²	C₂⁵×C₁₃₇	1	1	yes	characteristic
82	C₄	—	2	16	no	—
83	C₂	—	2	16	no	—
84	C₂	C₂²×C₂²⋊C₄×C₁₃₇	1	28	yes	direct product factor
85	C₂	C₂⁴×C₄×C₁₃₇	1	1	yes	characteristic
86	C₂	C₂³×D₄×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁵×C₁₃₇	C₂²
commutator	C₂	C₂⁴×C₄×C₁₃₇
Frattini, cosocle	C₂²	C₂⁵×C₁₃₇
prime radical	C₂⁶×C₁₃₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×C₁₃₇	C₂	semidirect
3. C₂²×C₂²⋊C₄×C₁₃₇	C₂	direct (N × H)
5. C₂⁵×C₁₃₇	C₄	semidirect
9. C₂³×C₄×C₁₃₇	C₂²	semidirect
10. C₂×C₂²⋊C₄×C₁₃₇	C₂²	direct (N × H)
14. C₂⁴×C₁₃₇	C₂×C₄	semidirect
19. C₂²×C₄×C₁₃₇	C₂³	semidirect
20. C₂²⋊C₄×C₁₃₇	C₂³	direct (N × H)
23. C₂³×C₁₃₇	C₂²⋊C₄	direct (N × H)
25. C₂³×C₁₃₇	C₂²×C₄	semidirect
30. C₂×C₄×C₁₃₇	C₂⁴	semidirect
33. C₂²×C₁₃₇	C₂×C₂²⋊C₄	direct (N × H)
35. C₂²×C₁₃₇	C₂³×C₄	semidirect
40. C₂×C₁₃₇	C₂²×C₂²⋊C₄	direct (N × H)
43. C₁₃₇	C₂³×C₂²⋊C₄	direct (N × H)
44. C₂³×C₂²⋊C₄	C₁₃₇	direct (N × H)
46. C₂⁴×C₄	C₂×C₁₃₇	semidirect
47. C₂²×C₂²⋊C₄	C₂×C₁₃₇	direct (N × H)
49. C₂⁵	C₄×C₁₃₇	semidirect
53. C₂³×C₄	C₂²×C₁₃₇	semidirect
54. C₂×C₂²⋊C₄	C₂²×C₁₃₇	direct (N × H)
58. C₂⁴	C₂×C₄×C₁₃₇	semidirect
63. C₂²×C₄	C₂³×C₁₃₇	semidirect
64. C₂²⋊C₄	C₂³×C₁₃₇	direct (N × H)
67. C₂³	C₂²⋊C₄×C₁₃₇	direct (N × H)
69. C₂³	C₂²×C₄×C₁₃₇	semidirect
74. C₂×C₄	C₂⁴×C₁₃₇	semidirect
77. C₂²	C₂×C₂²⋊C₄×C₁₃₇	direct (N × H)
79. C₂²	C₂³×C₄×C₁₃₇	semidirect
84. C₂	C₂²×C₂²⋊C₄×C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁶×C₁₃₇	2
C₂⁴×C₄×C₁₃₇	2
C₂²×C₂²⋊C₄×C₁₃₇	2
C₂³×C₂²⋊C₄	137

Subgroups — C23×C22⋊C4×C137 #31778430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂²⋊C₄×C₁₃₇ #31778430