Group #17996139 — subgroups

Subgroups — #17996139

Order 4736.

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

normal subgroups only

This group has 254 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×#11924467	C₂	1	2	yes	maximal
2	C₂×#11924460	C₂	1	2	yes	maximal
3	#14557403	C₂	1	1	yes	maximal, characteristic
4	#14557288	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#391	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64436	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₃₇	—	2	2	no	—
8	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
9	#11924467	—	2	4	no	—
10	#11924460	—	2	4	no	—
11	C₂²×Q₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
14	#95×C₃₇	—	2	2	no	—
15	C₂×Q₈×C₃₇	—	2	1	no	—
16	C₂×Q₈×C₃₇	—	2	2	no	—
17	C₂×Q₈×C₃₇	—	2	2	no	—
18	C₂×Q₈×C₃₇	Q₈	1	2	yes	—
19	C₂²×C₄×C₃₇	—	2	1	no	—
20	C₂²×C₄×C₃₇	—	2	1	no	—
21	C₂²×C₄×C₃₇	D₄	1	2	yes	—
22	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
23	C₂²×Dic₃₇	—	2	2	no	—
24	C₄⋊C₄×C₃₇	—	2	2	no	—
25	C₄²×C₃₇	—	2	2	no	—
26	C₄⋊Dic₃₇	—	2	2	no	—
27	C₄⋊Dic₃₇	—	2	2	no	—
28	C₄⋊Dic₃₇	—	4	4	no	—
29	C₂×#64436	—	2	2	no	—
30	C₂×#64436	—	2	2	no	—
31	C₂×#64436	D₄	1	4	yes	—
32	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₃₇	—	2	4	no	—
34	Q₈×C₃₇	—	4	2	no	—
35	C₂×C₄×C₃₇	—	2	2	no	—
36	C₂×C₄×C₃₇	—	2	2	no	—
37	C₂×C₄×C₃₇	—	2	2	no	—
38	C₂×C₄×C₃₇	—	4	1	no	—
39	C₂×C₄×C₃₇	—	4	1	no	—
40	C₂×C₄×C₃₇	—	4	1	no	—
41	C₂×C₄×C₃₇	—	4	2	no	—
42	C₂×C₄×C₃₇	SD₁₆	1	4	yes	—
43	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
46	C₂×C₄×C₃₇	#41	1	2	yes	—
47	C₂×Dic₃₇	—	4	2	no	—
48	C₂×Dic₃₇	—	4	4	no	—
49	#64436	—	2	4	no	—
50	C₂²×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₃₇	#123	1	1	yes	characteristic
52	C₂²×C₃₇	#127	1	1	yes	characteristic
53	C₂²×C₃₇	C₂×SD₁₆	1	2	yes	—
54	C₂²×C₃₇	#137	1	2	yes	—
55	C₄×C₃₇	—	2	4	no	—
56	C₄×C₃₇	—	4	2	no	—
57	C₄×C₃₇	—	8	1	no	—
58	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₃₇	#123	1	2	yes	—
60	Dic₃₇	—	8	2	no	—
61	#2096	—	37	1	no	maximal, Sylow
62	C₂×C₃₇	#399	1	1	yes	characteristic
63	C₂×C₃₇	#475	1	2	yes	—
64	C₂×C₃₇	#483	1	2	yes	—
65	C₂×C₃₇	#492	1	1	yes	characteristic
66	C₂×C₃₇	#497	1	1	yes	characteristic
67	C₂×#106	—	37	2	no	—
68	C₂×#103	—	37	2	no	—
69	#391	D₃₇	1	1	yes	characteristic
70	#384	—	37	1	no	—
71	#336	—	37	1	no	—
72	C₃₇	#2096	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	C₂×D₃₇	1	1	yes	characteristic
74	C₂²×C₈	—	37	2	no	—
75	C₂×C₄⋊C₄	—	37	2	no	—
76	C₂×C₄⋊C₄	—	74	2	no	—
77	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
78	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
79	#106	—	74	4	no	—
80	#103	—	74	4	no	—
81	#95	—	2	2	no	—
82	C₂×Q₈	—	2	1	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	Dic₇₄	1	2	yes	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	74	2	no	—
89	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
90	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
91	C₂×C₈	—	37	4	no	—
92	C₂×C₈	—	74	2	no	—
93	C₂×C₈	—	74	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	74	2	no	—
96	C₄⋊C₄	—	74	2	no	—
97	C₄⋊C₄	—	148	4	no	—
98	C₄²	—	2	2	no	—
99	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
100	Q₈	—	2	4	no	—
101	Q₈	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	148	2	no	—
110	C₂×C₄	—	148	4	no	—
111	C₂×C₄	#10596941	1	4	yes	—
112	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
113	C₂×C₄	#10596962	1	1	yes	characteristic
114	C₂×C₄	D₄×D₃₇	1	2	yes	—
115	C₂×C₄	#10596964	1	2	yes	—
116	C₈	—	74	4	no	—
117	C₂²	#11924515	1	1	yes	characteristic
118	C₂²	#11924517	1	1	yes	characteristic
119	C₂²	#11924532	1	1	yes	characteristic
120	C₂²	C₂×#10596941	1	2	yes	—
121	C₂²	#11924566	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	4	2	no	—
124	C₄	—	8	1	no	—
125	C₄	—	296	2	no	—
126	C₄	#11924513	1	2	yes	—
127	C₄	#11924518	1	2	yes	—
128	C₂	#14557469	1	1	yes	characteristic
129	C₂	#14557818	1	2	yes	—
130	C₂	#14557861	1	2	yes	—
131	C₂	#14557907	1	1	yes	characteristic
132	C₂	#14557928	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924467

C₂

yes

maximal

C₂×#11924460

C₂

yes

maximal

#14557403

C₂

yes

maximal, characteristic

#14557288

C₂

yes

maximal, characteristic

C₃₇×#391

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

—

C₂×C₄⋊Dic₃₇

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

#11924467

—

#11924460

—

C₂²×Q₈×C₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₇

C₂²

yes

characteristic

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#95×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

Q₈

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₂×#64436

—

C₂×#64436

—

C₂×#64436

D₄

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

SD₁₆

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

C₂×C₄×C₃₇

yes

C₂×C₄×C₃₇

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

#64436

—

C₂²×C₃₇

C₂²⋊Q₈

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₂²×C₃₇

C₂×SD₁₆

yes

—

C₂²×C₃₇

#137

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

#123

yes

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

#483

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

—

D₃₇

yes

characteristic

#384

—

#336

—

C₃₇

#2096

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₃₇

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

characteristic

C₂×C₄²

C₂×D₃₇

yes

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₇₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

148

—

C₄²

—

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

100

Q₈

—

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

148

—

110

C₂×C₄

—

148

—

111

C₂×C₄

#10596941

yes

—

112

C₂×C₄

C₂×Dic₇₄

yes

characteristic

113

C₂×C₄

#10596962

yes

characteristic

114

C₂×C₄

D₄×D₃₇

yes

—

115

C₂×C₄

#10596964

yes

—

116

C₈

—

117

C₂²

#11924515

yes

characteristic

118

C₂²

#11924517

yes

characteristic

119

C₂²

#11924532

yes

characteristic

120

C₂²

C₂×#10596941

yes

—

121

C₂²

#11924566

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

296

—

126

C₄

#11924513

yes

—

127

C₄

#11924518

yes

—

128

C₂

#14557469

yes

characteristic

129

C₂

#14557818

yes

—

130

C₂

#14557861

yes

—

131

C₂

#14557907

yes

characteristic

132

C₂

#14557928

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₇
commutator, cosocle	C₂²×C₄×C₃₇	C₂³
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#391	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
hypercenter	#391	D₃₇
hypocenter	C₃₇	#2096
LCS	G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₇

commutator, cosocle

C₂²×C₄×C₃₇

C₂³

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#391

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

D₃₇

C₃₇

G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
72. C₃₇	#2096	semidirect

H ≅ G/N

type

72. C₃₇

#2096

semidirect

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

subgroup	index
C₂×#11924467	2
C₂×#11924460	2
#14557403	2
#14557288	2
C₃₇×#391	2
#2096	37

subgroup

index

C₃₇×#391

Subgroups — #17996139

Conjugacy classes of subgroups

Complements

Maximal subgroups