Subgroups — C₂×#346×C₁₃₇ #31777117

Order 17536.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#346×C₁₃₇	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₈×C₁₃₇	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₁₃₇	C₄	1	8	yes	complemented
8	C₂×C₁₆×C₁₃₇	C₂²	1	4	yes	—
9	C₄×C₈×C₁₃₇	C₄	1	2	yes	—
10	C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
11	C₂²×C₄×C₁₃₇	C₈	1	2	yes	—
12	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₁₃₇	C₈	1	4	yes	—
14	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
16	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
18	C₄²×C₁₃₇	C₈	1	2	yes	—
19	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
20	C₁₆×C₁₃₇	C₂×C₄	1	8	yes	complemented
21	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
22	C₂×C₄×C₁₃₇	—	2	4	no	—
23	C₂×C₄×C₁₃₇	C₄²	1	2	yes	—
24	C₂×C₄×C₁₃₇	C₂×C₈	1	4	yes	—
25	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
28	C₈×C₁₃₇	C₄²	1	2	yes	—
29	C₈×C₁₃₇	C₂×C₈	1	4	yes	—
30	C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
31	C₂²×C₁₃₇	C₄×C₈	1	2	yes	—
32	C₂²×C₁₃₇	M₃₂	1	4	yes	—
33	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
34	C₄×C₁₃₇	—	2	4	no	—
35	C₄×C₁₃₇	C₄×C₈	1	2	yes	—
36	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
37	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
38	C₂×C₁₃₇	#346	1	4	yes	direct product factor
39	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
40	C₂×C₁₃₇	C₂×M₃₂	1	2	yes	—
41	C₁₃₇	C₂×#346	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×#346	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₁₆	C₂×C₁₃₇	1	2	yes	—
44	C₂×C₄×C₈	C₂×C₁₃₇	1	1	yes	characteristic
45	#346	C₂×C₁₃₇	1	4	yes	direct product factor
46	C₂²×C₈	C₄×C₁₃₇	1	1	yes	characteristic
47	C₂²×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
48	C₂×C₄²	C₄×C₁₃₇	1	1	yes	characteristic
49	C₂×C₁₆	C₄×C₁₃₇	1	8	yes	complemented
50	C₂×C₁₆	C₂²×C₁₃₇	1	4	yes	—
51	C₄×C₈	C₄×C₁₃₇	1	2	yes	—
52	C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
53	C₂²×C₄	C₈×C₁₃₇	1	2	yes	—
54	C₂²×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
55	C₂×C₈	C₈×C₁₃₇	1	4	yes	—
56	C₂×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
57	C₂×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
58	C₂×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
59	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
60	C₄²	C₈×C₁₃₇	1	2	yes	—
61	C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
62	C₁₆	C₂×C₄×C₁₃₇	1	8	yes	complemented
63	C₂³	C₂×C₈×C₁₃₇	1	1	yes	characteristic
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	C₄²×C₁₃₇	1	2	yes	—
66	C₂×C₄	C₂×C₈×C₁₃₇	1	4	yes	—
67	C₂×C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
68	C₂×C₄	C₂×C₈×C₁₃₇	1	2	yes	—
69	C₂×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
70	C₈	C₄²×C₁₃₇	1	2	yes	—
71	C₈	C₂×C₈×C₁₃₇	1	4	yes	—
72	C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
73	C₂²	C₄×C₈×C₁₃₇	1	2	yes	—
74	C₂²	M₃₂×C₁₃₇	1	4	yes	—
75	C₂²	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	C₄×C₈×C₁₃₇	1	2	yes	—
78	C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
79	C₄	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
80	C₂	#346×C₁₃₇	1	4	yes	direct product factor
81	C₂	C₂×C₄×C₈×C₁₃₇	1	1	yes	characteristic
82	C₂	C₂×M₃₂×C₁₃₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #346×C₁₃₇	C₂	direct (N × H)
7. C₂×C₁₆×C₁₃₇	C₄	semidirect
20. C₁₆×C₁₃₇	C₂×C₄	semidirect
38. C₂×C₁₃₇	#346	direct (N × H)
41. C₁₃₇	C₂×#346	direct (N × H)
42. C₂×#346	C₁₃₇	direct (N × H)
45. #346	C₂×C₁₃₇	direct (N × H)
49. C₂×C₁₆	C₄×C₁₃₇	semidirect
62. C₁₆	C₂×C₄×C₁₃₇	semidirect
80. C₂	#346×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
C₂²×C₁₆×C₁₃₇	2
C₂×C₄×C₈×C₁₃₇	2
#346×C₁₃₇	2
C₂×#346	137

Subgroups — C2×#346×C137 #31777117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#346×C₁₃₇ #31777117