Subgroups — C₂×C₁₆×Dic₁₃₇ #31782375

Order 17536.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555485	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₁₆×Dic₁₃₇	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11764040	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₃₇	C₄	1	1	yes	characteristic
7	C₂×#14555485	C₄	1	4	yes	complemented
8	C₂×#14555485	C₂²	1	2	yes	—
9	C₈×Dic₁₃₇	C₄	1	2	yes	—
10	C₈×Dic₁₃₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₃₇	C₄	1	4	yes	complemented
13	C₂×C₁₆×C₁₃₇	C₂²	1	2	yes	—
14	C₂²×Dic₁₃₇	C₈	1	2	yes	—
15	C₂×#11764040	C₈	1	4	yes	—
16	C₂×#11764040	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₃₇	C₈	1	2	yes	—
18	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
19	#14555485	C₂×C₄	1	4	yes	complemented
20	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
25	C₁₆×C₁₃₇	C₂×C₄	1	4	yes	complemented
26	C₂×Dic₁₃₇	C₁₆	1	8	yes	direct product factor
27	C₂×Dic₁₃₇	C₂×C₈	1	4	yes	—
28	#11764040	C₂×C₈	1	4	yes	—
29	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₃₇	C₄²	1	2	yes	—
31	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₃₇	C₄²	1	2	yes	—
35	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₁₃₇	C₄×C₈	1	2	yes	—
38	C₂²×C₁₃₇	C₂×C₁₆	1	4	yes	—
39	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
40	C₄×C₁₃₇	C₄×C₈	1	2	yes	—
41	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
42	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
43	Dic₁₃₇	C₂×C₁₆	1	8	yes	direct product factor
44	C₂×C₁₃₇	C₄×C₁₆	1	4	yes	complemented
45	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
46	C₂×C₁₃₇	C₂²×C₁₆	1	2	yes	—
47	C₁₃₇	C₂×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₄×C₁₆	—	137	1	no	maximal, Sylow
49	C₂²×C₁₆	—	137	1	no	—
50	C₂²×C₁₆	D₁₃₇	1	1	yes	characteristic
51	C₂×C₄×C₈	—	137	1	no	—
52	C₄×C₁₆	—	137	4	no	—
53	C₂²×C₈	—	137	1	no	—
54	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
55	C₂×C₄²	—	137	1	no	—
56	C₂×C₁₆	—	137	4	no	—
57	C₂×C₁₆	—	137	2	no	—
58	C₂×C₁₆	Dic₁₃₇	1	4	yes	direct product factor
59	C₂×C₁₆	C₂×D₁₃₇	1	2	yes	—
60	C₄×C₈	—	137	2	no	—
61	C₄×C₈	—	137	2	no	—
62	C₂²×C₄	—	137	2	no	—
63	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
64	C₂×C₈	—	137	4	no	—
65	C₂×C₈	—	137	2	no	—
66	C₂×C₈	C₄×D₁₃₇	1	1	yes	characteristic
67	C₂×C₈	C₄×D₁₃₇	1	2	yes	—
68	C₂×C₈	C₂×Dic₁₃₇	1	2	yes	—
69	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
70	C₄²	—	137	2	no	—
71	C₄²	—	137	2	no	—
72	C₁₆	—	137	4	no	—
73	C₁₆	C₂×Dic₁₃₇	1	4	yes	direct product factor
74	C₂³	C₈×D₁₃₇	1	1	yes	characteristic
75	C₂×C₄	—	137	8	no	—
76	C₂×C₄	—	137	4	no	—
77	C₂×C₄	C₄×Dic₁₃₇	1	2	yes	—
78	C₂×C₄	C₈×D₁₃₇	1	2	yes	—
79	C₂×C₄	C₈×D₁₃₇	1	1	yes	characteristic
80	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
81	C₈	—	137	4	no	—
82	C₈	C₄×Dic₁₃₇	1	2	yes	—
83	C₈	C₂×C₄×D₁₃₇	1	1	yes	characteristic
84	C₈	C₂²×Dic₁₃₇	1	1	yes	characteristic
85	C₂²	C₈×Dic₁₃₇	1	2	yes	—
86	C₂²	C₁₆×D₁₃₇	1	4	yes	—
87	C₂²	C₂×C₈×D₁₃₇	1	1	yes	characteristic
88	C₄	—	137	8	no	—
89	C₄	C₈×Dic₁₃₇	1	2	yes	—
90	C₄	C₂×C₄×Dic₁₃₇	1	1	yes	characteristic
91	C₄	C₂×C₈×D₁₃₇	1	1	yes	characteristic
92	C₂	C₁₆×Dic₁₃₇	1	4	yes	direct product factor
93	C₂	C₂×C₈×Dic₁₃₇	1	1	yes	characteristic
94	C₂	C₂×C₁₆×D₁₃₇	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₁₆	D₁₃₇
commutator, hypocenter	C₁₃₇	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	C₂²×C₁₆×C₁₃₇	C₂
socle, prime radical	C₂³×C₁₃₇	C₂×C₈
cosocle	C₂×C₈×C₁₃₇	C₂³
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₁₆×Dic₁₃₇	C₂	direct (N × H)
7. C₂×#14555485	C₄	semidirect
12. C₂×C₁₆×C₁₃₇	C₄	semidirect
19. #14555485	C₂×C₄	semidirect
25. C₁₆×C₁₃₇	C₂×C₄	semidirect
26. C₂×Dic₁₃₇	C₁₆	direct (N × H)
43. Dic₁₃₇	C₂×C₁₆	direct (N × H)
44. C₂×C₁₃₇	C₄×C₁₆	semidirect
47. C₁₃₇	C₂×C₄×C₁₆	semidirect
58. C₂×C₁₆	Dic₁₃₇	direct (N × H)
73. C₁₆	C₂×Dic₁₃₇	direct (N × H)
92. C₂	C₁₆×Dic₁₃₇	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×#14555485	2
C₂×C₈×Dic₁₃₇	2
C₁₆×Dic₁₃₇	2
C₂²×C₁₆×C₁₃₇	2
C₂×C₄×C₁₆	137

Subgroups — C2×C16×Dic137 #31782375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₆×Dic₁₃₇ #31782375