Group #31780918 — subgroups

Subgroups — #31780918

Order 17536.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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×C₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#22577268	C₂	1	2	yes	maximal
3	#22577166	C₂	1	1	yes	maximal, characteristic
4	#22577165	C₂	1	2	yes	maximal
5	#340×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11764040	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₃₇	C₄	1	4	yes	complemented
10	#17983995	—	2	2	no	—
11	#17983995	—	2	2	no	—
12	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
14	C₂²×Dic₁₃₇	—	2	2	no	—
15	C₂²×Dic₁₃₇	D₄	1	1	yes	characteristic
16	C₂²×Dic₁₃₇	D₄	1	1	yes	characteristic
17	C₂×#11764040	—	2	2	no	—
18	C₂×#11764040	C₂×C₄	1	2	yes	—
19	C₄⋊Dic₁₃₇	—	2	2	no	—
20	C₄⋊Dic₁₃₇	—	2	2	no	—
21	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₃₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₃₇	—	2	2	no	—
24	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₃₇	—	2	2	no	—
26	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
27	C₄⋊C₄×C₁₃₇	—	2	2	no	—
28	C₄⋊C₄×C₁₃₇	—	2	2	no	—
29	C₂×Dic₁₃₇	—	2	1	no	—
30	C₂×Dic₁₃₇	—	2	2	no	—
31	C₂×Dic₁₃₇	—	2	1	no	—
32	C₂×Dic₁₃₇	—	4	2	no	—
33	C₂×Dic₁₃₇	—	4	2	no	—
34	C₂×Dic₁₃₇	—	4	2	no	—
35	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
37	#11764040	—	2	2	no	—
38	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₇	—	4	2	no	—
40	C₂×C₄×C₁₃₇	—	4	2	no	—
41	C₂×C₄×C₁₃₇	—	4	2	no	—
42	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₃₇	#41	1	2	yes	—
45	C₂×C₄×C₁₃₇	#41	1	2	yes	—
46	C₈×C₁₃₇	—	2	2	no	—
47	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₃₇	#124	1	2	yes	—
49	C₂²×C₁₃₇	#135	1	1	yes	characteristic
50	C₂²×C₁₃₇	#135	1	2	yes	—
51	C₂²×C₁₃₇	#135	1	1	yes	characteristic
52	C₄×C₁₃₇	—	8	2	no	—
53	C₄×C₁₃₇	#117	1	1	yes	characteristic
54	C₄×C₁₃₇	#117	1	1	yes	characteristic
55	C₄×C₁₃₇	#125	1	2	yes	—
56	Dic₁₃₇	—	2	2	no	—
57	Dic₁₃₇	—	2	2	no	—
58	Dic₁₃₇	—	8	2	no	—
59	C₂×C₁₃₇	#382	1	1	yes	characteristic
60	C₂×C₁₃₇	#427	1	1	yes	characteristic
61	C₂×C₁₃₇	#427	1	1	yes	characteristic
62	C₂×C₁₃₇	#488	1	2	yes	—
63	C₂×C₁₃₇	#488	1	2	yes	—
64	C₁₃₇	#1835	1	1	yes	characteristic, complemented, Sylow
65	#1835	—	137	1	no	maximal, Sylow
66	C₂×C₄×C₈	—	137	1	no	—
67	#382	—	137	2	no	—
68	#340	—	137	1	no	—
69	#340	—	137	2	no	—
70	#340	D₁₃₇	1	1	yes	characteristic
71	C₂²×C₈	—	137	1	no	—
72	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	137	2	no	—
74	C₂×C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
75	C₂×C₄²	—	137	1	no	—
76	C₄×C₈	—	137	4	no	—
77	#95	—	274	2	no	—
78	#95	—	274	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	274	2	no	—
81	C₂²×C₄	—	137	1	no	—
82	C₂²×C₄	—	137	1	no	—
83	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	274	2	no	—
86	C₂×C₈	—	137	2	no	—
87	C₂×C₈	C₄×D₁₃₇	1	2	yes	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	274	2	no	—
91	C₄⋊C₄	—	274	2	no	—
92	C₄²	—	137	2	no	—
93	C₄²	—	137	2	no	—
94	C₂³	D₄×D₁₃₇	1	1	yes	characteristic
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	274	1	no	—
99	C₂×C₄	—	274	2	no	—
100	C₂×C₄	—	274	1	no	—
101	C₂×C₄	—	548	2	no	—
102	C₂×C₄	—	548	2	no	—
103	C₂×C₄	—	548	2	no	—
104	C₂×C₄	—	137	2	no	—
105	C₂×C₄	—	137	2	no	—
106	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₁₃₇	1	1	yes	characteristic
108	C₂×C₄	#14555509	1	2	yes	—
109	C₂×C₄	#14555510	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	274	2	no	—
112	C₂²	C₄⋊C₄×D₁₃₇	1	1	yes	characteristic
113	C₂²	#17984074	1	2	yes	—
114	C₂²	#17984107	1	1	yes	characteristic
115	C₂²	#17984109	1	2	yes	—
116	C₂²	#17984111	1	1	yes	characteristic
117	C₄	—	8	2	no	—
118	C₄	—	274	2	no	—
119	C₄	—	274	2	no	—
120	C₄	—	1096	2	no	—
121	C₄	#17984044	1	1	yes	characteristic
122	C₄	#17984045	1	1	yes	characteristic
123	C₄	#17984078	1	2	yes	—
124	C₂	#22577267	1	1	yes	characteristic
125	C₂	#22577438	1	1	yes	characteristic
126	C₂	#22577440	1	1	yes	characteristic
127	C₂	#22577756	1	2	yes	—
128	C₂	#22577759	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₃₇

C₂

yes

maximal, characteristic

#22577268

C₂

yes

maximal

#22577166

C₂

yes

maximal, characteristic

#22577165

C₂

yes

maximal

#340×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×#11764040

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃₇

C₂²

yes

—

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

C₈×Dic₁₃₇

C₄

yes

complemented

#17983995

—

#17983995

—

C₂²×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₃₇

C₂²

yes

—

C₂²×Dic₁₃₇

—

C₂²×Dic₁₃₇

D₄

yes

characteristic

C₂²×Dic₁₃₇

D₄

yes

characteristic

C₂×#11764040

—

C₂×#11764040

C₂×C₄

yes

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

Q₈

yes

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₄⋊C₄

yes

—

C₂×Dic₁₃₇

C₄⋊C₄

yes

—

#11764040

—

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

#41

yes

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₇

#124

yes

—

C₂²×C₁₃₇

#135

yes

characteristic

C₂²×C₁₃₇

#135

yes

—

C₂²×C₁₃₇

#135

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

—

C₂×C₁₃₇

#488

yes

—

C₁₃₇

#1835

yes

characteristic, complemented, Sylow

#1835

—

137

maximal, Sylow

C₂×C₄×C₈

—

137

—

#382

—

137

—

#340

—

137

—

#340

—

137

—

#340

D₁₃₇

yes

characteristic

C₂²×C₈

—

137

—

C₂²×C₈

C₂×D₁₃₇

yes

characteristic

C₂×C₄⋊C₄

—

137

—

C₂×C₄⋊C₄

C₂×D₁₃₇

yes

—

C₂×C₄²

—

137

—

C₄×C₈

—

137

—

#95

—

274

—

#95

—

274

—

C₂²×C₄

—

C₂²×C₄

—

274

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

C₄×D₁₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄²

—

137

—

C₄²

—

137

—

C₂³

D₄×D₁₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

100

C₂×C₄

—

274

—

101

C₂×C₄

—

548

—

102

C₂×C₄

—

548

—

103

C₂×C₄

—

548

—

104

C₂×C₄

—

137

—

105

C₂×C₄

—

137

—

106

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

107

C₂×C₄

Q₈×D₁₃₇

yes

characteristic

108

C₂×C₄

#14555509

yes

—

109

C₂×C₄

#14555510

yes

—

110

C₈

—

111

C₈

—

274

—

112

C₂²

C₄⋊C₄×D₁₃₇

yes

characteristic

113

C₂²

#17984074

yes

—

114

C₂²

#17984107

yes

characteristic

115

C₂²

#17984109

yes

—

116

C₂²

#17984111

yes

characteristic

117

C₄

—

118

C₄

—

274

—

119

C₄

—

274

—

120

C₄

—

1096

—

121

C₄

#17984044

yes

characteristic

122

C₄

#17984045

yes

characteristic

123

C₄

#17984078

yes

—

124

C₂

#22577267

yes

characteristic

125

C₂

#22577438

yes

characteristic

126

C₂

#22577440

yes

characteristic

127

C₂

#22577756

yes

—

128

C₂

#22577759

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	#340×C₁₃₇	C₂
socle, prime radical	C₂³×C₁₃₇	C₂×D₄
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	#340	D₁₃₇
hypocenter	C₁₃₇	#1835
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₁₃₇

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₇

Fitting

#340×C₁₃₇

C₂

socle, prime radical

C₂³×C₁₃₇

C₂×D₄

cosocle

C₂²×C₄×C₁₃₇

C₂³

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
9. C₈×Dic₁₃₇	C₄	semidirect
64. C₁₃₇	#1835	semidirect

H ≅ G/N

type

9. C₈×Dic₁₃₇

C₄

semidirect

64. C₁₃₇

#1835

semidirect

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

subgroup	index
C₂×C₈×Dic₁₃₇	2
#22577268	2
#22577166	2
#22577165	2
#340×C₁₃₇	2
#1835	137

subgroup

index

C₂×C₈×Dic₁₃₇

#340×C₁₃₇

137

Subgroups — #31780918

Conjugacy classes of subgroups

Complements

Maximal subgroups