Subgroups — C₂³×Q₈×D₁₃₇ #31795583

Order 17536.

6264 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₇₄	C₂	1	3	yes	maximal, complemented
2	C₂²×Q₈×D₁₃₇	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄×D₁₃₇	C₂	1	3	yes	maximal
4	C₂³×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₁₃₇	C₂²	1	1	yes	characteristic
6	C₂²×Dic₂₇₄	C₂²	1	84	yes	complemented
7	C₂×Q₈×D₁₃₇	C₂²	1	448	yes	direct product factor
8	C₂³×Dic₁₃₇	C₂²	1	3	yes	—
9	C₂²×C₄×D₁₃₇	C₂²	1	84	yes	—
10	C₂²×Q₈×C₁₃₇	C₂²	1	28	yes	complemented
11	C₂³×C₄×C₁₃₇	C₂²	1	3	yes	—
12	C₂³×D₁₃₇	Q₈	1	16	yes	direct product factor
13	C₂³×D₁₃₇	C₂³	1	14	yes	—
14	C₂×Dic₂₇₄	C₂³	1	336	yes	complemented
15	Q₈×D₁₃₇	C₂³	1	512	yes	direct product factor
16	C₂²×Dic₁₃₇	C₂³	1	42	yes	—
17	C₂×C₄×D₁₃₇	C₂³	1	336	yes	—
18	C₂⁴×C₁₃₇	C₂³	1	1	yes	characteristic
19	C₂×Q₈×C₁₃₇	C₂³	1	112	yes	complemented
20	C₂²×C₄×C₁₃₇	C₂³	1	42	yes	—
21	C₂²×D₁₃₇	C₂×Q₈	1	112	yes	direct product factor
22	C₂²×D₁₃₇	C₂⁴	1	28	yes	—
23	Dic₂₇₄	C₂⁴	1	192	yes	complemented
24	C₂×Dic₁₃₇	C₂⁴	1	84	yes	—
25	C₄×D₁₃₇	C₂⁴	1	192	yes	—
26	C₂³×C₁₃₇	C₂×Q₈	1	8	yes	complemented
27	C₂³×C₁₃₇	C₂⁴	1	7	yes	—
28	Q₈×C₁₃₇	C₂⁴	1	64	yes	complemented
29	C₂×C₄×C₁₃₇	C₂⁴	1	84	yes	—
30	C₂²×C₁₃₇	C₂²×Q₈	1	28	yes	complemented
31	C₂²×C₁₃₇	C₂⁵	1	7	yes	—
32	C₂×D₁₃₇	C₂²×Q₈	1	112	yes	direct product factor
33	C₂×D₁₃₇	C₂⁵	1	8	yes	—
34	C₄×C₁₃₇	C₂⁵	1	24	yes	—
35	Dic₁₃₇	C₂⁵	1	24	yes	—
36	C₂×C₁₃₇	C₂³×Q₈	1	14	yes	complemented
37	C₂×C₁₃₇	C₂⁶	1	1	yes	characteristic
38	D₁₃₇	C₂³×Q₈	1	16	yes	direct product factor
39	C₁₃₇	C₂⁴×Q₈	1	1	yes	characteristic, complemented, Sylow
40	C₂⁴×Q₈	—	137	1	no	maximal, Sylow
41	C₂³×Q₈	—	137	56	no	—
42	C₂³×Q₈	—	137	3	no	—
43	C₂³×Q₈	D₁₃₇	1	1	yes	characteristic, direct product factor
44	C₂⁴×C₄	—	137	3	no	—
45	C₂⁵	—	137	1	no	—
46	C₂²×Q₈	—	137	448	no	—
47	C₂²×Q₈	—	137	84	no	—
48	C₂²×Q₈	C₂×D₁₃₇	1	28	yes	direct product factor
49	C₂³×C₄	—	137	3	no	—
50	C₂³×C₄	—	137	84	no	—
51	C₂³×C₄	C₂×D₁₃₇	1	3	yes	—
52	C₂⁴	—	137	16	no	—
53	C₂⁴	—	137	14	no	—
54	C₂⁴	C₂²×D₁₃₇	1	1	yes	characteristic
55	C₂×Q₈	—	137	512	no	—
56	C₂×Q₈	—	137	336	no	—
57	C₂×Q₈	C₂²×D₁₃₇	1	112	yes	direct product factor
58	C₂²×C₄	—	137	42	no	—
59	C₂²×C₄	—	137	336	no	—
60	C₂²×C₄	C₂²×D₁₃₇	1	42	yes	—
61	C₂³	—	137	112	no	—
62	C₂³	—	137	28	no	—
63	C₂³	Q₈×D₁₃₇	1	8	yes	direct product factor
64	C₂³	C₂³×D₁₃₇	1	7	yes	—
65	Q₈	—	137	192	no	—
66	Q₈	C₂³×D₁₃₇	1	64	yes	direct product factor
67	C₂×C₄	—	137	84	no	—
68	C₂×C₄	—	137	192	no	—
69	C₂×C₄	C₂³×D₁₃₇	1	84	yes	—
70	C₂²	—	137	112	no	—
71	C₂²	—	137	8	no	—
72	C₂²	C₂×Q₈×D₁₃₇	1	28	yes	direct product factor
73	C₂²	C₂⁴×D₁₃₇	1	7	yes	—
74	C₄	—	137	24	no	—
75	C₄	C₂⁴×D₁₃₇	1	24	yes	—
76	C₂	—	137	16	no	—
77	C₂	C₂²×Q₈×D₁₃₇	1	14	yes	direct product factor
78	C₂	C₂⁵×D₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₃₇
commutator, cosocle	C₂×C₁₃₇	C₂⁶
Frattini	C₂	C₂⁵×D₁₃₇
Fitting	C₂³×Q₈×C₁₃₇	C₂
socle	C₂⁴×C₁₃₇	C₂³
prime radical	C₂⁴×D₁₃₇	C₂²
hypercenter	C₂³×Q₈	D₁₃₇
hypocenter	C₁₃₇	C₂⁴×Q₈
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₂₇₄	C₂	semidirect
2. C₂²×Q₈×D₁₃₇	C₂	direct (N × H)
4. C₂³×Q₈×C₁₃₇	C₂	semidirect
6. C₂²×Dic₂₇₄	C₂²	semidirect
7. C₂×Q₈×D₁₃₇	C₂²	direct (N × H)
10. C₂²×Q₈×C₁₃₇	C₂²	semidirect
12. C₂³×D₁₃₇	Q₈	direct (N × H)
14. C₂×Dic₂₇₄	C₂³	semidirect
15. Q₈×D₁₃₇	C₂³	direct (N × H)
19. C₂×Q₈×C₁₃₇	C₂³	semidirect
21. C₂²×D₁₃₇	C₂×Q₈	direct (N × H)
23. Dic₂₇₄	C₂⁴	semidirect
26. C₂³×C₁₃₇	C₂×Q₈	semidirect
28. Q₈×C₁₃₇	C₂⁴	semidirect
30. C₂²×C₁₃₇	C₂²×Q₈	semidirect
32. C₂×D₁₃₇	C₂²×Q₈	direct (N × H)
36. C₂×C₁₃₇	C₂³×Q₈	semidirect
38. D₁₃₇	C₂³×Q₈	direct (N × H)
39. C₁₃₇	C₂⁴×Q₈	semidirect
43. C₂³×Q₈	D₁₃₇	direct (N × H)
48. C₂²×Q₈	C₂×D₁₃₇	direct (N × H)
57. C₂×Q₈	C₂²×D₁₃₇	direct (N × H)
63. C₂³	Q₈×D₁₃₇	direct (N × H)
66. Q₈	C₂³×D₁₃₇	direct (N × H)
72. C₂²	C₂×Q₈×D₁₃₇	direct (N × H)
77. C₂	C₂²×Q₈×D₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₂₇₄	2
C₂²×Q₈×D₁₃₇	2
C₂³×C₄×D₁₃₇	2
C₂³×Q₈×C₁₃₇	2
C₂⁴×Q₈	137

Subgroups — C23×Q8×D137 #31795583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×Q₈×D₁₃₇ #31795583