Subgroups — Q₃₂×Dic₁₃₇ #31782648

Order 17536.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×Dic₁₃₇	C₂	1	2	yes	maximal
2	#22577229	C₂	1	1	yes	maximal, characteristic
3	#22577207	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₁₃₇	—	2	2	no	—
7	C₂×#14555485	C₂²	1	1	yes	characteristic
8	#17984022	C₂²	1	2	yes	—
9	#17984012	—	2	2	no	—
10	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
12	Q₃₂×C₁₃₇	C₄	1	4	yes	complemented
13	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#11764040	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₃₇	—	2	2	no	—
16	C₄⋊Dic₁₃₇	—	4	2	no	—
17	C₄×Dic₁₃₇	—	4	2	no	—
18	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
19	#14555485	—	2	1	no	—
20	C₂×Q₈×C₁₃₇	—	2	2	no	—
21	Q₁₆×C₁₃₇	—	2	2	no	—
22	Q₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
23	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
24	C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
25	C₂×Dic₁₃₇	—	4	2	no	—
26	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
27	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
28	#11764040	—	2	1	no	—
29	Q₈×C₁₃₇	—	2	4	no	—
30	Q₈×C₁₃₇	—	4	2	no	—
31	C₂×C₄×C₁₃₇	—	4	2	no	—
32	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
33	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₃₇	#41	1	1	yes	characteristic
35	C₂²×C₁₃₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₁₃₇	—	4	4	no	—
37	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
38	C₄×C₁₃₇	#135	1	1	yes	characteristic
39	Dic₁₃₇	—	2	1	no	—
40	Dic₁₃₇	—	8	2	no	—
41	Dic₁₃₇	Q₃₂	1	2	yes	direct product factor
42	C₂×C₁₃₇	C₄×D₈	1	1	yes	characteristic
43	C₂×C₁₃₇	C₂×Q₃₂	1	1	yes	characteristic
44	C₂×C₁₃₇	#508	1	1	yes	characteristic
45	C₁₃₇	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₃₂	—	137	1	no	maximal, Sylow
47	C₂×Q₃₂	D₁₃₇	1	1	yes	characteristic
48	C₄×Q₁₆	—	137	2	no	—
49	#366	—	137	1	no	—
50	#358	—	137	2	no	—
51	C₄×C₁₆	—	137	1	no	—
52	C₂×Q₁₆	C₂×D₁₃₇	1	2	yes	—
53	C₄×Q₈	—	274	2	no	—
54	Q₃₂	Dic₁₃₇	1	4	yes	direct product factor
55	C₂×C₁₆	—	137	1	no	—
56	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
57	#107	—	137	2	no	—
58	#103	—	274	2	no	—
59	C₄×C₈	—	137	1	no	—
60	C₂×Q₈	—	2	2	no	—
61	Q₁₆	—	2	2	no	—
62	Q₁₆	C₂×Dic₁₃₇	1	4	yes	—
63	C₂×C₈	—	137	1	no	—
64	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
65	C₄⋊C₄	—	274	2	no	—
66	C₄⋊C₄	—	548	2	no	—
67	C₄²	—	548	2	no	—
68	C₄²	—	137	1	no	—
69	C₁₆	—	274	1	no	—
70	C₁₆	C₂×Dic₁₃₇	1	2	yes	—
71	Q₈	—	2	4	no	—
72	Q₈	—	4	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	548	2	no	—
75	C₂×C₄	—	137	1	no	—
76	C₂×C₄	—	137	1	no	—
77	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
78	C₈	—	274	1	no	—
79	C₈	C₂²×Dic₁₃₇	1	1	yes	characteristic
80	C₈	#14555509	1	1	yes	characteristic
81	C₂²	D₈×D₁₃₇	1	1	yes	characteristic
82	C₄	—	4	4	no	—
83	C₄	—	274	1	no	—
84	C₄	—	1096	2	no	—
85	C₄	—	137	2	no	—
86	C₄	D₄×Dic₁₃₇	1	1	yes	characteristic
87	C₄	#17984107	1	1	yes	characteristic
88	C₂	D₈×Dic₁₃₇	1	1	yes	characteristic
89	C₂	Q₃₂×D₁₃₇	1	1	yes	characteristic
90	C₂	#22577836	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₃₇
commutator	C₈×C₁₃₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	C₂×Q₃₂×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	C₂×D₈
cosocle	C₂×C₈×C₁₃₇	C₂³
hypercenter	C₂×Q₃₂	D₁₃₇
hypocenter	C₁₃₇	C₄×Q₃₂
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
12. Q₃₂×C₁₃₇	C₄	semidirect
41. Dic₁₃₇	Q₃₂	direct (N × H)
45. C₁₃₇	C₄×Q₃₂	semidirect
54. Q₃₂	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
Q₁₆×Dic₁₃₇	2
#22577229	2
#22577207	2
C₁₆×Dic₁₃₇	2
C₂×Q₃₂×C₁₃₇	2
C₄×Q₃₂	137

Subgroups — Q32×Dic137 #31782648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₃₂×Dic₁₃₇ #31782648