Group #17996173 — subgroups

Subgroups — #17996173

Order 4736.

158 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 57 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	#14557581	C₂	1	1	yes	maximal, characteristic
2	#14557556	C₂	1	1	yes	maximal, characteristic
3	#14557305	C₂	1	1	yes	maximal, characteristic
4	#14557294	C₂	1	1	yes	maximal, characteristic
5	#14557292	C₂	1	1	yes	maximal, characteristic
6	#14557289	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#344	C₂	1	1	yes	maximal, characteristic
8	C₂×#10596934	C₂²	1	1	yes	characteristic
9	C₂×#10596934	C₂²	1	1	yes	characteristic
10	#11924495	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
12	#11924471	—	2	2	no	—
13	#11924465	—	2	2	no	—
14	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
16	#117×C₃₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₃₇	—	2	1	no	—
19	M₁₆×C₃₇	—	2	1	no	—
20	M₁₆×C₃₇	—	2	1	no	—
21	M₁₆×C₃₇	—	2	1	no	—
22	M₁₆×C₃₇	—	2	1	no	—
23	C₂×C₈×C₃₇	—	2	1	no	—
24	C₂×C₈×C₃₇	—	2	1	no	—
25	C₄⋊C₄×C₃₇	—	2	1	no	—
26	C₂²⋊C₄×C₃₇	—	2	1	no	—
27	C₄²×C₃₇	—	2	1	no	—
28	C₂²⋊Dic₃₇	—	2	1	no	—
29	C₄⋊Dic₃₇	—	2	1	no	—
30	C₄×Dic₃₇	—	2	1	no	—
31	C₄×Dic₃₇	—	2	1	no	—
32	C₄×Dic₃₇	Q₈	1	1	yes	characteristic
33	C₄×Dic₃₇	Q₈	1	1	yes	characteristic
34	#10596934	—	2	1	no	—
35	#10596934	—	2	1	no	—
36	#10596934	—	2	1	no	—
37	#10596934	—	2	1	no	—
38	C₂×#64436	—	2	1	no	—
39	C₂×#64436	D₄	1	2	yes	—
40	C₂³×C₃₇	#41	1	1	yes	characteristic
41	C₂×C₄×C₃₇	—	4	1	no	—
42	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
45	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
46	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
47	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
48	C₂×Dic₃₇	—	2	1	no	—
49	C₂×Dic₃₇	—	2	1	no	—
50	C₂×Dic₃₇	—	4	2	no	—
51	C₂×Dic₃₇	—	4	1	no	—
52	C₈×C₃₇	—	4	1	no	—
53	C₈×C₃₇	—	4	1	no	—
54	#64436	—	2	2	no	—
55	#64436	—	4	1	no	—
56	C₂²×C₃₇	—	4	1	no	—
57	C₂²×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂²×C₃₇	#126	1	1	yes	characteristic
59	C₂²×C₃₇	#126	1	1	yes	characteristic
60	C₄×C₃₇	—	8	1	no	—
61	C₄×C₃₇	#123	1	1	yes	characteristic
62	C₄×C₃₇	#124	1	1	yes	characteristic
63	C₄×C₃₇	#125	1	1	yes	characteristic
64	C₄×C₃₇	#125	1	1	yes	characteristic
65	Dic₃₇	—	2	2	no	—
66	Dic₃₇	—	4	1	no	—
67	Dic₃₇	—	8	1	no	—
68	#2102	—	37	1	no	maximal, Sylow
69	C₂×C₃₇	—	2	1	no	—
70	C₂×C₃₇	—	2	1	no	—
71	C₂×C₃₇	—	2	1	no	—
72	C₂×C₃₇	#400	1	1	yes	characteristic
73	#430	—	37	1	no	—
74	#423	—	37	1	no	—
75	#344	—	37	1	no	—
76	#344	D₃₇	1	1	yes	characteristic
77	#339	—	37	1	no	—
78	#339	—	37	1	no	—
79	H(3,ℤ/4)	—	37	1	no	—
80	C₃₇	#2102	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₁₆	—	37	1	no	—
82	C₂×M₁₆	—	37	1	no	—
83	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
84	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
85	#117	—	37	1	no	—
86	#117	C₂×D₃₇	1	1	yes	characteristic
87	C₂×C₄²	—	37	1	no	—
88	#108	—	74	2	no	—
89	C₄⋊C₈	—	74	2	no	—
90	C₂²×C₄	—	74	1	no	—
91	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
92	M₁₆	—	2	1	no	—
93	M₁₆	—	2	1	no	—
94	M₁₆	—	2	1	no	—
95	M₁₆	—	2	1	no	—
96	M₁₆	—	74	1	no	—
97	M₁₆	—	74	1	no	—
98	M₁₆	—	74	1	no	—
99	M₁₆	—	74	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	37	2	no	—
103	C₂×C₈	—	74	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	74	1	no	—
106	C₂²⋊C₄	—	2	1	no	—
107	C₂²⋊C₄	—	74	1	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	37	1	no	—
110	C₄²	—	37	1	no	—
111	C₄²	—	74	1	no	—
112	C₄²	—	74	1	no	—
113	C₂³	#10596962	1	1	yes	characteristic
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	74	1	no	—
116	C₂×C₄	—	74	1	no	—
117	C₂×C₄	—	148	1	no	—
118	C₂×C₄	—	148	2	no	—
119	C₂×C₄	#10596962	1	1	yes	characteristic
120	C₂×C₄	#10596962	1	1	yes	characteristic
121	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
122	C₂×C₄	#10596964	1	1	yes	characteristic
123	C₂×C₄	#10596964	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₃₇	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	4	1	no	—
127	C₈	—	74	2	no	—
128	C₈	—	148	1	no	—
129	C₂²	—	4	1	no	—
130	C₂²	#11924514	1	1	yes	characteristic
131	C₂²	#11924529	1	1	yes	characteristic
132	C₂²	#11924530	1	1	yes	characteristic
133	C₄	—	8	1	no	—
134	C₄	—	296	1	no	—
135	C₄	—	74	2	no	—
136	C₄	—	148	1	no	—
137	C₄	#11924518	1	1	yes	characteristic
138	C₄	#11924522	1	1	yes	characteristic
139	C₄	#11924526	1	1	yes	characteristic
140	C₄	#11924526	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#14557474	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557581

C₂

yes

maximal, characteristic

#14557556

C₂

yes

maximal, characteristic

#14557305

C₂

yes

maximal, characteristic

#14557294

C₂

yes

maximal, characteristic

#14557292

C₂

yes

maximal, characteristic

#14557289

C₂

yes

maximal, characteristic

C₃₇×#344

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

characteristic

C₂×#10596934

C₂²

yes

characteristic

#11924495

C₂²

yes

characteristic

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924471

—

#11924465

—

C₂×M₁₆×C₃₇

C₂²

yes

characteristic

C₂×M₁₆×C₃₇

C₂²

yes

characteristic

#117×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

M₁₆×C₃₇

—

M₁₆×C₃₇

—

M₁₆×C₃₇

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₄⋊C₄×C₃₇

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

Q₈

yes

characteristic

C₄×Dic₃₇

Q₈

yes

—

—

—

—

—

D₄

yes

—

C₂³×C₃₇

#41

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

C₂×C₄×C₃₇

yes

C₂×C₄×C₃₇

yes

C₂×C₄×C₃₇

yes

C₂×C₄×C₃₇

yes

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₈×C₃₇

—

C₈×C₃₇

—

#64436

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂²⋊Q₈

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₄×C₃₇

—

C₄×C₃₇

yes

C₄×C₃₇

yes

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

#2102

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

yes

—

—

—

D₃₇

yes

—

—

—

C₃₇

#2102

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

characteristic

C₂×M₁₆

C₂×D₃₇

yes

characteristic

#117

—

#117

C₂×D₃₇

yes

characteristic

C₂×C₄²

—

#108

—

C₄⋊C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₂³

#10596962

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

148

—

118

C₂×C₄

—

148

—

119

C₂×C₄

#10596962

yes

characteristic

120

C₂×C₄

#10596962

yes

characteristic

121

C₂×C₄

D₄×D₃₇

yes

characteristic

122

C₂×C₄

#10596964

yes

characteristic

123

C₂×C₄

#10596964

yes

characteristic

124

C₂×C₄

Q₈×D₃₇

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

148

—

129

C₂²

—

130

C₂²

#11924514

yes

characteristic

131

C₂²

#11924529

yes

characteristic

132

C₂²

#11924530

yes

characteristic

133

C₄

—

134

C₄

—

296

—

135

C₄

—

136

C₄

—

148

—

137

C₄

#11924518

yes

characteristic

138

C₄

#11924522

yes

characteristic

139

C₄

#11924526

yes

characteristic

140

C₄

#11924526

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

#14557474

yes

characteristic

	subgroup	G/H
center	C₄	#11924522
commutator, cosocle	C₂²×C₄×C₃₇	C₂³
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#344	C₂
socle	C₂×C₃₇	#400
prime radical	C₂³×C₃₇	#41
hypercenter	#344	D₃₇
hypocenter	C₃₇	#2102
LCS	G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#11924522

commutator, cosocle

C₂²×C₄×C₃₇

C₂³

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#344

C₂

C₂×C₃₇

C₂³×C₃₇

D₃₇

C₃₇

G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
80. C₃₇	#2102	semidirect

H ≅ G/N

type

80. C₃₇

#2102

semidirect

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

subgroup	index
#14557581	2
#14557556	2
#14557305	2
#14557294	2
#14557292	2
#14557289	2
C₃₇×#344	2
#2102	37

subgroup

index

C₃₇×#344

Subgroups — #17996173

Conjugacy classes of subgroups

Complements

Maximal subgroups