Group #31796289 — subgroups

Subgroups — D₁₃₇⋊#333 #31796289

Order 17536.

256 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578404	C₂	1	2	yes	maximal
2	#22578263	C₂	1	4	yes	maximal, complemented
3	C₄⋊Q₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555518	—	2	2	no	—
5	#17984155	C₂²	1	4	yes	—
6	C₄⋊#11764047	—	2	2	no	—
7	C₂×Q₈×D₁₃₇	C₄	1	2	yes	—
8	#17984088	C₄	1	1	yes	characteristic
9	#17984087	—	2	2	no	—
10	#17984086	C₂²	1	2	yes	—
11	C₄⋊C₄×D₁₃₇	—	2	2	no	—
12	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₁₃₇	C₄	1	1	yes	characteristic
14	#14555518	—	4	4	no	—
15	C₂×#11764047	—	2	4	no	—
16	C₂×Dic₂₇₄	—	2	2	no	—
17	C₂×Dic₂₇₄	C₂×C₄	1	2	yes	—
18	Q₈×D₁₃₇	—	4	4	no	—
19	C₂×C₄×D₁₃₇	—	2	2	no	—
20	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×D₁₃₇	D₄	1	2	yes	—
22	C₄⋊Dic₁₃₇	—	2	2	no	—
23	#14555489	—	2	4	no	—
24	C₄×Dic₁₃₇	D₄	1	2	yes	—
25	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
26	C₂×Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
27	C₄⋊C₄×C₁₃₇	—	2	2	no	—
28	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
29	#11764047	—	4	4	no	—
30	C₂²×D₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
31	Dic₂₇₄	—	4	4	no	—
32	Dic₂₇₄	—	4	2	no	—
33	C₂×Dic₁₃₇	—	2	2	no	—
34	C₂×Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₁₃₇	C₂×D₄	1	2	yes	—
36	C₄×D₁₃₇	—	2	2	no	—
37	C₄×D₁₃₇	—	4	4	no	—
38	C₄×D₁₃₇	SD₁₆	1	4	yes	—
39	C₄×D₁₃₇	Q₁₆	1	4	yes	—
40	Q₈×C₁₃₇	—	4	2	no	—
41	C₂×C₄×C₁₃₇	—	2	2	no	—
42	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×D₁₃₇	#100	1	2	yes	—
46	C₂×D₁₃₇	#103	1	4	yes	—
47	C₄×C₁₃₇	—	2	1	no	—
48	C₄×C₁₃₇	—	4	2	no	—
49	C₄×C₁₃₇	#103	1	4	yes	—
50	Dic₁₃₇	—	2	1	no	—
51	Dic₁₃₇	—	4	2	no	—
52	Dic₁₃₇	C₂×SD₁₆	1	2	yes	—
53	Dic₁₃₇	C₂×Q₁₆	1	2	yes	—
54	C₂×C₁₃₇	C₂×#100	1	1	yes	characteristic
55	C₂×C₁₃₇	C₂×#103	1	2	yes	—
56	D₁₃₇	#333	1	4	yes	complemented
57	C₁₃₇	C₂×#333	1	1	yes	characteristic, complemented, Sylow
58	C₂×#333	—	137	1	no	maximal, Sylow
59	C₂×C₄⋊Q₈	—	137	1	no	—
60	C₂×C₄⋊C₈	—	137	2	no	—
61	#333	—	137	4	no	—
62	C₂²×Q₈	—	137	2	no	—
63	C₂²×C₈	—	274	2	no	—
64	C₄⋊Q₈	—	274	2	no	—
65	C₄⋊Q₈	—	137	2	no	—
66	C₄⋊Q₈	—	137	1	no	—
67	C₄⋊Q₈	#10587739	1	1	yes	characteristic
68	C₂×C₄⋊C₄	—	274	2	no	—
69	C₂×C₄²	—	137	1	no	—
70	C₄⋊C₈	—	274	2	no	—
71	C₄⋊C₈	—	137	4	no	—
72	C₂×Q₈	—	274	2	no	—
73	C₂×Q₈	—	548	4	no	—
74	C₂×Q₈	—	137	2	no	—
75	C₂×Q₈	C₂×#10587739	1	2	yes	—
76	C₂²×C₄	—	274	2	no	—
77	C₂²×C₄	—	137	1	no	—
78	C₂²×C₄	—	137	2	no	—
79	C₂×C₈	—	274	4	no	—
80	C₂×C₈	—	548	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	274	4	no	—
83	C₄⋊C₄	—	274	2	no	—
84	C₄²	—	137	2	no	—
85	C₄²	—	137	1	no	—
86	C₄²	C₂×#10587739	1	1	yes	characteristic
87	C₂³	—	137	1	no	—
88	Q₈	—	4	2	no	—
89	Q₈	—	548	2	no	—
90	Q₈	—	548	4	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	274	2	no	—
93	C₂×C₄	—	274	2	no	—
94	C₂×C₄	—	548	4	no	—
95	C₂×C₄	—	137	1	no	—
96	C₂×C₄	—	137	2	no	—
97	C₂×C₄	—	137	4	no	—
98	C₂×C₄	—	137	4	no	—
99	C₂×C₄	C₂²⋊#10587739	1	2	yes	—
100	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
101	C₈	—	548	4	no	—
102	C₂²	—	137	2	no	—
103	C₂²	—	137	4	no	—
104	C₂²	C₂×C₂²⋊#10587739	1	1	yes	characteristic
105	C₄	—	2	1	no	—
106	C₄	—	4	2	no	—
107	C₄	—	274	1	no	—
108	C₄	—	548	2	no	—
109	C₄	—	137	2	no	—
110	C₄	—	137	2	no	—
111	C₄	#17984151	1	4	yes	—
112	C₂	—	137	4	no	—
113	C₂	D₁₃₇⋊#100	1	1	yes	characteristic
114	C₂	C₂×#17984151	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22578404

C₂

yes

maximal

#22578263

C₂

yes

maximal, complemented

C₄⋊Q₈×D₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555518

—

#17984155

C₂²

yes

—

C₄⋊#11764047

—

C₂×Q₈×D₁₃₇

C₄

yes

—

#17984088

C₄

yes

characteristic

#17984087

—

#17984086

C₂²

yes

—

C₄⋊C₄×D₁₃₇

—

C₄²×D₁₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₇

C₄

yes

characteristic

#14555518

—

C₂×#11764047

—

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

C₂×C₄

yes

—

Q₈×D₁₃₇

—

C₂×C₄×D₁₃₇

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃₇

D₄

yes

—

C₄⋊Dic₁₃₇

—

#14555489

—

C₄×Dic₁₃₇

D₄

yes

—

C₄×Dic₁₃₇

C₂³

yes

characteristic

C₂×Q₈×C₁₃₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

#11764047

—

C₂²×D₁₃₇

C₂²⋊C₄

yes

characteristic

Dic₂₇₄

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃₇

C₂×D₄

yes

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

SD₁₆

yes

—

C₄×D₁₃₇

Q₁₆

yes

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂²×C₁₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₃₇

#100

yes

—

C₂×D₁₃₇

#103

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#103

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

C₂×SD₁₆

yes

—

Dic₁₃₇

C₂×Q₁₆

yes

—

C₂×C₁₃₇

C₂×#100

yes

characteristic

C₂×C₁₃₇

C₂×#103

yes

—

D₁₃₇

#333

yes

complemented

C₁₃₇

C₂×#333

yes

characteristic, complemented, Sylow

C₂×#333

—

137

maximal, Sylow

C₂×C₄⋊Q₈

—

137

—

C₂×C₄⋊C₈

—

137

—

#333

—

137

—

C₂²×Q₈

—

137

—

C₂²×C₈

—

274

—

C₄⋊Q₈

—

274

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

#10587739

yes

characteristic

C₂×C₄⋊C₄

—

274

—

C₂×C₄²

—

137

—

C₄⋊C₈

—

274

—

C₄⋊C₈

—

137

—

C₂×Q₈

—

274

—

C₂×Q₈

—

548

—

C₂×Q₈

—

137

—

C₂×Q₈

C₂×#10587739

yes

—

C₂²×C₄

—

274

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂×C₈

—

274

—

C₂×C₈

—

548

—

C₄⋊C₄

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

C₂×#10587739

yes

characteristic

C₂³

—

137

—

Q₈

—

Q₈

—

548

—

Q₈

—

548

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

548

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂²⋊#10587739

yes

—

100

C₂×C₄

C₂²×#10587739

yes

characteristic

101

C₈

—

548

—

102

C₂²

—

137

—

103

C₂²

—

137

—

104

C₂²

C₂×C₂²⋊#10587739

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

274

—

108

C₄

—

548

—

109

C₄

—

137

—

110

C₄

—

137

—

111

C₄

#17984151

yes

—

112

C₂

—

137

—

113

C₂

D₁₃₇⋊#100

yes

characteristic

114

C₂

C₂×#17984151

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#10587739
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10587739
Fitting	C₄⋊Q₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₃₇	C₂²⋊C₄
cosocle	C₄×Dic₁₃₇	C₂³
hypercenter	C₄⋊Q₈	#10587739
hypocenter	C₁₃₇	C₂×#333
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#10587739

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#10587739

Fitting

C₄⋊Q₈×C₁₃₇

C₄

socle

C₂²×C₁₃₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₃₇

C₂²⋊C₄

cosocle

C₄×Dic₁₃₇

C₂³

C₄⋊Q₈

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. #22578263	C₂	semidirect
56. D₁₃₇	#333	semidirect
57. C₁₃₇	C₂×#333	semidirect

H ≅ G/N

type

2. #22578263

C₂

semidirect

56. D₁₃₇

#333

semidirect

57. C₁₃₇

C₂×#333

semidirect

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

subgroup	index
#22578404	2
#22578263	2
C₄⋊Q₈×D₁₃₇	2
C₂×#333	137

subgroup

index

#22578404

#22578263

C₄⋊Q₈×D₁₃₇

C₂×#333

137

Subgroups — D₁₃₇⋊#333 #31796289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D137⋊#333 #31796289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₃₇⋊#333 #31796289