Group #17994697 — subgroups

Subgroups — #17994697

Order 4736.

262 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 81 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	#41×Dic₃₇	C₂	1	1	yes	maximal, characteristic
2	#14557517	C₂	1	1	yes	maximal, characteristic
3	#14557292	C₂	1	2	yes	maximal, complemented
4	#14557289	C₂	1	2	yes	maximal, complemented
5	C₃₇×#410	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596934	C₂²	1	2	yes	—
7	Q₈×Dic₃₇	—	2	1	no	—
8	D₄×Dic₃₇	—	2	2	no	—
9	D₄×Dic₃₇	—	2	1	no	—
10	#11924495	C₄	1	2	yes	complemented
11	#11924495	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₃₇	C₄	1	2	yes	complemented
13	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
14	#11924455	—	2	1	no	—
15	#11924453	—	2	1	no	—
16	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
17	#117×C₃₇	C₂²	1	2	yes	—
18	#99×C₃₇	—	2	2	no	—
19	#41×C₃₇	—	2	2	no	—
20	#41×C₃₇	—	2	2	no	—
21	C₂×Q₈×C₃₇	D₄	1	1	yes	characteristic
22	C₂×D₄×C₃₇	D₄	1	1	yes	characteristic
23	C₂×D₄×C₃₇	D₄	1	2	yes	—
24	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₇	—	2	2	no	—
27	C₂²×Dic₃₇	—	2	1	no	—
28	C₄⋊C₄×C₃₇	—	2	2	no	—
29	C₂²⋊C₄×C₃₇	—	2	2	no	—
30	C₄²×C₃₇	—	2	2	no	—
31	C₂²⋊Dic₃₇	—	2	2	no	—
32	C₂²⋊Dic₃₇	—	2	1	no	—
33	C₄⋊Dic₃₇	—	2	2	no	—
34	C₄⋊Dic₃₇	—	2	1	no	—
35	C₄×Dic₃₇	—	2	2	no	—
36	C₄×Dic₃₇	—	2	1	no	—
37	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
39	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
40	#10596934	—	2	2	no	—
41	#10596934	—	2	2	no	—
42	C₂×#64436	—	2	2	no	—
43	C₂³×C₃₇	C₂²⋊C₄	1	2	yes	—
44	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₃₇	—	2	2	no	—
46	D₄×C₃₇	—	2	4	no	—
47	D₄×C₃₇	—	2	2	no	—
48	C₂×C₄×C₃₇	—	2	2	no	—
49	C₂×C₄×C₃₇	—	2	2	no	—
50	C₂×C₄×C₃₇	—	4	2	no	—
51	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₃₇	#41	1	2	yes	—
55	C₂×C₄×C₃₇	#41	1	2	yes	—
56	C₂×Dic₃₇	—	2	1	no	—
57	C₂×Dic₃₇	—	2	2	no	—
58	C₂×Dic₃₇	—	2	1	no	—
59	C₂×Dic₃₇	—	4	2	no	—
60	C₂×Dic₃₇	—	4	4	no	—
61	#64436	—	4	2	no	—
62	C₂²×C₃₇	—	2	2	no	—
63	C₂²×C₃₇	—	4	1	no	—
64	C₂²×C₃₇	—	4	2	no	—
65	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₃₇	#123	1	2	yes	—
67	C₄×C₃₇	—	2	2	no	—
68	C₄×C₃₇	—	8	2	no	—
69	C₄×C₃₇	#117	1	1	yes	characteristic
70	C₄×C₃₇	#117	1	1	yes	characteristic
71	C₄×C₃₇	#123	1	2	yes	—
72	Dic₃₇	—	2	2	no	—
73	Dic₃₇	—	4	2	no	—
74	Dic₃₇	—	4	1	no	—
75	#1827	—	37	1	no	maximal, Sylow
76	C₂×C₃₇	—	2	1	no	—
77	C₂×C₃₇	—	2	2	no	—
78	C₂×C₃₇	—	4	2	no	—
79	C₂×C₃₇	#381	1	1	yes	characteristic
80	C₄×#41	—	37	1	no	—
81	#413	—	37	1	no	—
82	#410	D₃₇	1	1	yes	characteristic
83	#339	—	37	2	no	—
84	H(3,ℤ/4)	—	37	2	no	—
85	C₃₇	#1827	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
87	C₂×M₁₆	—	37	2	no	—
88	C₄×Q₈	—	74	1	no	—
89	C₄×D₄	—	74	1	no	—
90	C₄×D₄	—	74	2	no	—
91	#117	—	37	2	no	—
92	#117	—	37	1	no	—
93	#117	C₂×D₃₇	1	2	yes	—
94	C₂×C₄²	—	37	2	no	—
95	C₂×C₄²	—	37	1	no	—
96	#101	—	74	1	no	—
97	#100	—	74	1	no	—
98	#99	—	2	2	no	—
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	C₂×Q₈	D₁₄₈	1	1	yes	characteristic
102	C₂×D₄	D₁₄₈	1	1	yes	characteristic
103	C₂×D₄	C₂²⋊D₃₇	1	2	yes	—
104	C₂²×C₄	—	74	2	no	—
105	C₂²×C₄	—	74	1	no	—
106	C₂²×C₄	C₄×D₃₇	1	2	yes	—
107	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
108	M₁₆	—	74	2	no	—
109	M₁₆	—	74	2	no	—
110	C₂×C₈	—	74	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	74	2	no	—
113	C₄⋊C₄	—	74	1	no	—
114	C₂²⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	74	2	no	—
116	C₂²⋊C₄	—	74	1	no	—
117	C₄²	—	2	2	no	—
118	C₄²	—	37	1	no	—
119	C₄²	—	37	2	no	—
120	C₄²	—	37	1	no	—
121	C₄²	—	74	2	no	—
122	C₄²	—	74	1	no	—
123	C₂³	#10596938	1	2	yes	—
124	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	74	1	no	—
132	C₂×C₄	—	74	2	no	—
133	C₂×C₄	—	74	1	no	—
134	C₂×C₄	—	148	2	no	—
135	C₂×C₄	—	148	4	no	—
136	C₂×C₄	#10596938	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
138	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
139	C₂×C₄	#10596964	1	2	yes	—
140	C₂×C₄	#10596964	1	2	yes	—
141	C₈	—	148	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	4	2	no	—
145	C₂²	C₂×#10596938	1	1	yes	characteristic
146	C₂²	#11924517	1	2	yes	—
147	C₄	—	2	2	no	—
148	C₄	—	8	2	no	—
149	C₄	—	74	2	no	—
150	C₄	—	148	2	no	—
151	C₄	—	148	1	no	—
152	C₄	#11924492	1	1	yes	characteristic
153	C₄	#11924492	1	1	yes	characteristic
154	C₄	#11924520	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	—	2	1	no	—
157	C₂	—	4	2	no	—
158	C₂	#14557394	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#41×Dic₃₇

C₂

yes

maximal, characteristic

#14557517

C₂

yes

maximal, characteristic

#14557292

C₂

yes

maximal, complemented

#14557289

C₂

yes

maximal, complemented

C₃₇×#410

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

Q₈×Dic₃₇

—

D₄×Dic₃₇

—

D₄×Dic₃₇

—

#11924495

C₄

yes

complemented

#11924495

C₂²

yes

characteristic

C₂×C₄×Dic₃₇

C₄

yes

complemented

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924455

—

#11924453

—

C₂×#41×C₃₇

C₂²

yes

characteristic

#117×C₃₇

C₂²

yes

—

#99×C₃₇

—

#41×C₃₇

—

#41×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

characteristic

C₂×D₄×C₃₇

D₄

yes

characteristic

C₂×D₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

—

C₄⋊C₄×C₃₇

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₂²⋊Dic₃₇

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

—

—

—

C₂³×C₃₇

C₂²⋊C₄

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

#123

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

yes

C₄×C₃₇

yes

C₄×C₃₇

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

#1827

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#381

yes

characteristic

C₄×#41

—

#413

—

#410

D₃₇

yes

characteristic

#339

—

H(3,ℤ/4)

—

C₃₇

#1827

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₃₇

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

#117

C₂×D₃₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

—

—

—

—

100

#41

—

101

C₂×Q₈

D₁₄₈

yes

characteristic

102

C₂×D₄

D₁₄₈

yes

characteristic

103

C₂×D₄

C₂²⋊D₃₇

yes

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₄×D₃₇

yes

—

107

C₂²×C₄

C₂²×D₃₇

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₂³

#10596938

yes

—

124

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

148

—

135

C₂×C₄

—

148

—

136

C₂×C₄

#10596938

yes

—

137

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

138

C₂×C₄

C₂×D₁₄₈

yes

characteristic

139

C₂×C₄

#10596964

yes

—

140

C₂×C₄

#10596964

yes

—

141

C₈

—

148

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₂×#10596938

yes

characteristic

146

C₂²

#11924517

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

148

—

151

C₄

—

148

—

152

C₄

#11924492

yes

characteristic

153

C₄

#11924492

yes

characteristic

154

C₄

#11924520

yes

—

155

C₂

—

156

C₂

—

157

C₂

—

158

C₂

#14557394

yes

characteristic

	subgroup	G/H
center	C₄	#11924492
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#410	C₂
socle	C₂×C₃₇	#381
prime radical	C₂×#41×C₃₇	C₂²
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#410	D₃₇
hypocenter	C₃₇	#1827
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#11924492

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#410

C₂

socle

C₂×C₃₇

#381

prime radical

C₂×#41×C₃₇

C₂²

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
3. #14557292	C₂	semidirect
4. #14557289	C₂	semidirect
10. #11924495	C₄	semidirect
12. C₂×C₄×Dic₃₇	C₄	semidirect
85. C₃₇	#1827	semidirect

H ≅ G/N

type

3. #14557292

C₂

semidirect

4. #14557289

C₂

semidirect

10. #11924495

C₄

semidirect

12. C₂×C₄×Dic₃₇

C₄

semidirect

85. C₃₇

#1827

semidirect

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

subgroup	index
#41×Dic₃₇	2
#14557517	2
#14557292	2
#14557289	2
C₃₇×#410	2
#1827	37

subgroup

index

#41×Dic₃₇

C₃₇×#410

Subgroups — #17994697

Conjugacy classes of subgroups

Complements

Maximal subgroups