Group #31781610 — subgroups

Subgroups — #31781610

Order 17536.

308 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×#17984012	C₂	1	4	yes	maximal
2	C₂×#17983999	C₂	1	1	yes	maximal, characteristic
3	#22577269	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11764040	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
7	#17984012	—	2	8	no	—
8	#17983999	—	2	2	no	—
9	#17983995	—	2	2	no	—
10	C₂²×Q₈×C₁₃₇	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
12	C₄⋊Q₈×C₁₃₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
14	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₃₇	—	2	2	no	—
16	C₂×#11764040	—	2	2	no	—
17	C₂×#11764040	—	2	2	no	—
18	C₂×#11764040	D₄	1	4	yes	—
19	C₄⋊Dic₁₃₇	—	2	4	no	—
20	C₂×Q₈×C₁₃₇	—	2	2	no	—
21	C₂×Q₈×C₁₃₇	—	2	4	no	—
22	C₂×Q₈×C₁₃₇	—	2	4	no	—
23	C₂×Q₈×C₁₃₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₃₇	—	2	2	no	—
25	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₃₇	—	2	8	no	—
28	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
29	C₄²×C₁₃₇	D₄	1	2	yes	—
30	C₂×Dic₁₃₇	—	4	2	no	—
31	C₂×Dic₁₃₇	—	4	4	no	—
32	#11764040	—	2	4	no	—
33	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₁₃₇	—	2	8	no	—
35	Q₈×C₁₃₇	—	4	4	no	—
36	C₂×C₄×C₁₃₇	—	2	4	no	—
37	C₂×C₄×C₁₃₇	—	2	4	no	—
38	C₂×C₄×C₁₃₇	—	4	2	no	—
39	C₂×C₄×C₁₃₇	—	4	2	no	—
40	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃₇	#41	1	2	yes	—
45	C₂²×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₃₇	#124	1	1	yes	characteristic
47	C₂²×C₁₃₇	#127	1	1	yes	characteristic
48	C₂²×C₁₃₇	#137	1	4	yes	—
49	C₄×C₁₃₇	—	2	4	no	—
50	C₄×C₁₃₇	—	4	4	no	—
51	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
52	C₄×C₁₃₇	#121	1	2	yes	—
53	Dic₁₃₇	—	8	2	no	—
54	C₂×C₁₃₇	#390	1	1	yes	characteristic
55	C₂×C₁₃₇	#419	1	2	yes	—
56	C₂×C₁₃₇	#491	1	2	yes	—
57	C₂×C₁₃₇	#497	1	2	yes	—
58	C₁₃₇	#1960	1	1	yes	characteristic, complemented, Sylow
59	#1960	—	137	1	no	maximal, Sylow
60	C₂×C₄⋊Q₈	D₁₃₇	1	1	yes	characteristic
61	C₂×#103	—	137	4	no	—
62	C₂×#97	—	137	1	no	—
63	#382	—	137	1	no	—
64	C₂²×Q₈	C₂×D₁₃₇	1	2	yes	—
65	C₂²×C₈	—	137	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	Dic₁₃₇	1	4	yes	complemented
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	137	2	no	—
70	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
71	#103	—	274	8	no	—
72	#97	—	274	2	no	—
73	#95	—	274	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₁₃₇	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	274	2	no	—
80	C₂²×C₄	C₂²⋊D₁₃₇	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
82	C₂×C₈	—	274	2	no	—
83	C₂×C₈	—	274	2	no	—
84	C₂×C₈	—	137	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	274	4	no	—
87	C₄²	C₂×Dic₁₃₇	1	2	yes	—
88	C₄²	C₂²⋊D₁₃₇	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	548	2	no	—
97	C₂×C₄	—	548	4	no	—
98	C₂×C₄	C₂²⋊Dic₁₃₇	1	4	yes	—
99	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₃₇	1	2	yes	—
101	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
102	C₂×C₄	#14555509	1	2	yes	—
103	C₈	—	274	4	no	—
104	C₂²	C₂×C₂²⋊Dic₁₃₇	1	1	yes	characteristic
105	C₂²	#17984075	1	1	yes	characteristic
106	C₂²	#17984084	1	1	yes	characteristic
107	C₂²	#17984119	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	1096	2	no	—
111	C₄	D₄×Dic₁₃₇	1	2	yes	—
112	C₄	#17984059	1	2	yes	—
113	C₂	#22577306	1	1	yes	characteristic
114	C₂	#22577412	1	2	yes	—
115	C₂	#22577771	1	2	yes	—
116	C₂	#22577795	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984012

C₂

yes

maximal

C₂×#17983999

C₂

yes

maximal, characteristic

#22577269

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×#11764040

C₂²

yes

—

C₂×C₄⋊Dic₁₃₇

C₂²

yes

—

#17984012

—

#17983999

—

#17983995

—

C₂²×Q₈×C₁₃₇

C₂²

yes

—

C₄⋊Q₈×C₁₃₇

—

C₄⋊Q₈×C₁₃₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₃₇

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

C₂²×Dic₁₃₇

—

C₂×#11764040

—

C₂×#11764040

—

C₂×#11764040

D₄

yes

—

C₄⋊Dic₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

D₄

yes

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

—

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

—

C₂²×C₁₃₇

C₂×C₂²⋊C₄

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₄×D₄

yes

—

C₄×C₁₃₇

#121

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

#390

yes

characteristic

C₂×C₁₃₇

#419

yes

—

C₂×C₁₃₇

#491

yes

—

C₂×C₁₃₇

#497

yes

—

C₁₃₇

#1960

yes

characteristic, complemented, Sylow

#1960

—

137

maximal, Sylow

C₂×C₄⋊Q₈

D₁₃₇

yes

characteristic

C₂×#103

—

137

—

C₂×#97

—

137

—

#382

—

137

—

C₂²×Q₈

C₂×D₁₃₇

yes

—

C₂²×C₈

—

137

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₁₃₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

#103

—

274

—

#97

—

274

—

#95

—

274

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₃₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

274

—

C₂²×C₄

C₂²⋊D₁₃₇

yes

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₄⋊C₄

—

C₄⋊C₄

—

274

—

C₄²

C₂×Dic₁₃₇

yes

—

C₄²

C₂²⋊D₁₃₇

yes

—

C₂³

C₂×C₂²⋊D₁₃₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

548

—

C₂×C₄

—

548

—

C₂×C₄

C₂²⋊Dic₁₃₇

yes

—

C₂×C₄

C₂²×Dic₁₃₇

yes

characteristic

100

C₂×C₄

D₄×D₁₃₇

yes

—

101

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

102

C₂×C₄

#14555509

yes

—

103

C₈

—

274

—

104

C₂²

C₂×C₂²⋊Dic₁₃₇

yes

characteristic

105

C₂²

#17984075

yes

characteristic

106

C₂²

#17984084

yes

characteristic

107

C₂²

#17984119

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

1096

—

111

C₄

D₄×Dic₁₃₇

yes

—

112

C₄

#17984059

yes

—

113

C₂

#22577306

yes

characteristic

114

C₂

#22577412

yes

—

115

C₂

#22577771

yes

—

116

C₂

#22577795

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	C₂×C₄⋊Q₈×C₁₃₇	C₂
socle, prime radical	C₂³×C₁₃₇	C₂×D₄
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₃₇
hypocenter	C₁₃₇	#1960
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₇

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₇

Fitting

C₂×C₄⋊Q₈×C₁₃₇

C₂

socle, prime radical

C₂³×C₁₃₇

C₂×D₄

cosocle

C₂²×C₄×C₁₃₇

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₃₇

hypocenter

C₁₃₇

#1960

LCS

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₁₃₇	C₄	semidirect
58. C₁₃₇	#1960	semidirect
67. C₄⋊Q₈	Dic₁₃₇	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₁₃₇

C₄

semidirect

58. C₁₃₇

#1960

semidirect

67. C₄⋊Q₈

Dic₁₃₇

semidirect

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

subgroup	index
C₂×#17984012	2
C₂×#17983999	2
#22577269	2
C₂×C₄⋊Q₈×C₁₃₇	2
#1960	137

subgroup

index

C₂×C₄⋊Q₈×C₁₃₇

137

Subgroups — #31781610

Conjugacy classes of subgroups

Complements

Maximal subgroups