Subgroups — C₂×SD₆₄×C₁₃₇ #31777271

Order 17536.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	SD₆₄×C₁₃₇	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₁₆×C₁₃₇	—	2	1	no	—
6	C₂×D₈×C₁₃₇	—	2	1	no	—
7	Q₃₂×C₁₃₇	—	2	1	no	—
8	Q₃₂×C₁₃₇	C₂²	1	2	yes	complemented
9	D₁₆×C₁₃₇	—	2	1	no	—
10	D₁₆×C₁₃₇	C₂²	1	2	yes	—
11	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₃₂×C₁₃₇	C₂²	1	2	yes	complemented
13	C₂×Q₈×C₁₃₇	—	4	1	no	—
14	C₂×D₄×C₁₃₇	—	4	1	no	—
15	Q₁₆×C₁₃₇	—	2	2	no	—
16	Q₁₆×C₁₃₇	—	4	1	no	—
17	D₈×C₁₃₇	—	2	2	no	—
18	D₈×C₁₃₇	—	4	1	no	—
19	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
20	C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
21	C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
22	C₂³×C₁₃₇	—	8	1	no	—
23	Q₈×C₁₃₇	—	4	2	no	—
24	Q₈×C₁₃₇	—	8	1	no	—
25	D₄×C₁₃₇	—	4	2	no	—
26	D₄×C₁₃₇	—	8	1	no	—
27	C₂×C₄×C₁₃₇	—	8	1	no	—
28	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
29	C₈×C₁₃₇	D₈	1	1	yes	characteristic
30	C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₁₃₇	—	8	2	no	—
32	C₂²×C₁₃₇	—	16	2	no	—
33	C₂²×C₁₃₇	D₁₆	1	1	yes	characteristic
34	C₄×C₁₃₇	—	8	2	no	—
35	C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
36	C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
37	C₂×C₁₃₇	—	16	2	no	—
38	C₂×C₁₃₇	SD₆₄	1	2	yes	direct product factor
39	C₂×C₁₃₇	C₂×D₁₆	1	1	yes	characteristic
40	C₁₃₇	C₂×SD₆₄	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×SD₆₄	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×Q₃₂	C₂×C₁₃₇	1	1	yes	characteristic, complemented
43	C₂×D₁₆	C₂×C₁₃₇	1	1	yes	characteristic
44	SD₆₄	C₂×C₁₃₇	1	4	yes	direct product factor
45	C₂×C₃₂	C₂×C₁₃₇	1	1	yes	characteristic, complemented
46	C₂×Q₁₆	—	2	1	no	—
47	C₂×D₈	—	2	1	no	—
48	Q₃₂	—	2	1	no	—
49	Q₃₂	C₂²×C₁₃₇	1	2	yes	complemented
50	D₁₆	—	2	1	no	—
51	D₁₆	C₂²×C₁₃₇	1	2	yes	—
52	C₂×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
53	C₃₂	C₂²×C₁₃₇	1	2	yes	complemented
54	C₂×Q₈	—	4	1	no	—
55	C₂×D₄	—	4	1	no	—
56	Q₁₆	—	2	2	no	—
57	Q₁₆	—	4	1	no	—
58	D₈	—	2	2	no	—
59	D₈	—	4	1	no	—
60	C₂×C₈	D₄×C₁₃₇	1	1	yes	characteristic
61	C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
62	C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
63	C₂³	—	8	1	no	—
64	Q₈	—	4	2	no	—
65	Q₈	—	8	1	no	—
66	D₄	—	4	2	no	—
67	D₄	—	8	1	no	—
68	C₂×C₄	—	8	1	no	—
69	C₂×C₄	D₈×C₁₃₇	1	1	yes	characteristic
70	C₈	D₈×C₁₃₇	1	1	yes	characteristic
71	C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
72	C₂²	—	8	2	no	—
73	C₂²	—	16	2	no	—
74	C₂²	D₁₆×C₁₃₇	1	1	yes	characteristic
75	C₄	—	8	2	no	—
76	C₄	D₁₆×C₁₃₇	1	1	yes	characteristic
77	C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
78	C₂	—	16	2	no	—
79	C₂	SD₆₄×C₁₃₇	1	2	yes	direct product factor
80	C₂	C₂×D₁₆×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂³×C₁₃₇
prime radical	C₂×D₁₆×C₁₃₇	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₈×C₁₃₇ ◁ C₂×C₁₆×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×Q₃₂×C₁₃₇	C₂	semidirect
3. SD₆₄×C₁₃₇	C₂	direct (N × H)
4. C₂×C₃₂×C₁₃₇	C₂	semidirect
8. Q₃₂×C₁₃₇	C₂²	semidirect
12. C₃₂×C₁₃₇	C₂²	semidirect
38. C₂×C₁₃₇	SD₆₄	direct (N × H)
40. C₁₃₇	C₂×SD₆₄	direct (N × H)
41. C₂×SD₆₄	C₁₃₇	direct (N × H)
42. C₂×Q₃₂	C₂×C₁₃₇	semidirect
44. SD₆₄	C₂×C₁₃₇	direct (N × H)
45. C₂×C₃₂	C₂×C₁₃₇	semidirect
49. Q₃₂	C₂²×C₁₃₇	semidirect
53. C₃₂	C₂²×C₁₃₇	semidirect
79. C₂	SD₆₄×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
C₂×Q₃₂×C₁₃₇	2
C₂×D₁₆×C₁₃₇	2
SD₆₄×C₁₃₇	2
C₂×C₃₂×C₁₃₇	2
C₂×SD₆₄	137

Subgroups — C2×SD64×C137 #31777271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×SD₆₄×C₁₃₇ #31777271