Group #17995341 — subgroups

Subgroups — #17995341

Order 4736.

208 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 83 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×#11924471	C₂	1	1	yes	maximal, characteristic
2	#14557569	C₂	1	1	yes	maximal, characteristic
3	#14557559	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic
5	#14557294	C₂	1	1	yes	maximal, characteristic
6	#14557292	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#424	C₂	1	1	yes	maximal, characteristic
8	C₂×#10596934	C₂²	1	1	yes	characteristic
9	C₂²×#64436	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	#11924469	C₄	1	2	yes	complemented
14	#11924467	C₄	1	2	yes	complemented
15	#11924465	—	2	2	no	—
16	C₈×Dic₃₇	—	2	2	no	—
17	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
19	#117×C₃₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₃₇	—	2	2	no	—
21	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃₇	—	2	1	no	—
23	M₁₆×C₃₇	—	2	1	no	—
24	M₁₆×C₃₇	—	2	1	no	—
25	C₂×C₈×C₃₇	—	2	1	no	—
26	C₂×C₈×C₃₇	—	2	1	no	—
27	C₂×C₈×C₃₇	—	2	1	no	—
28	C₂×C₈×C₃₇	D₄	1	2	yes	—
29	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
30	C₂²⋊C₄×C₃₇	—	2	1	no	—
31	C₄²×C₃₇	—	2	1	no	—
32	C₂²⋊Dic₃₇	—	2	1	no	—
33	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
34	C₄×Dic₃₇	—	2	1	no	—
35	C₄×Dic₃₇	—	2	1	no	—
36	C₄×Dic₃₇	Q₈	1	2	yes	—
37	#10596934	—	2	1	no	—
38	#10596934	—	2	1	no	—
39	C₂×#64436	—	2	1	no	—
40	C₂×#64436	C₂×C₄	1	1	yes	characteristic
41	C₂×#64436	C₂×C₄	1	1	yes	characteristic
42	C₂×#64436	D₄	1	2	yes	—
43	C₂×#64436	Q₈	1	2	yes	—
44	C₂³×C₃₇	#41	1	1	yes	characteristic
45	C₂×C₄×C₃₇	—	2	2	no	—
46	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
52	C₂×Dic₃₇	—	2	4	no	—
53	C₂×Dic₃₇	—	2	2	no	—
54	C₂×Dic₃₇	—	2	2	no	—
55	C₈×C₃₇	—	2	2	no	—
56	C₈×C₃₇	—	4	1	no	—
57	#64436	—	4	1	no	—
58	#64436	C₄⋊C₄	1	2	yes	—
59	#64436	C₄⋊C₄	1	2	yes	—
60	C₂²×C₃₇	—	2	2	no	—
61	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₃₇	#125	1	1	yes	characteristic
64	C₄×C₃₇	—	4	2	no	—
65	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₃₇	C₄×Q₈	1	1	yes	characteristic
67	C₄×C₃₇	#121	1	1	yes	characteristic
68	C₄×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
69	Dic₃₇	—	2	4	no	—
70	Dic₃₇	—	4	2	no	—
71	#1945	—	37	1	no	maximal, Sylow
72	C₂×C₃₇	—	2	1	no	—
73	C₂×C₃₇	—	2	1	no	—
74	C₂×C₃₇	#389	1	1	yes	characteristic
75	C₂×C₃₇	#443	1	2	yes	—
76	C₂×#108	—	37	1	no	—
77	#427	—	37	1	no	—
78	#424	—	37	1	no	—
79	#424	D₃₇	1	1	yes	characteristic
80	C₂×C₄×C₈	—	37	1	no	—
81	#339	—	37	1	no	—
82	#339	—	37	1	no	—
83	C₃₇	#1945	1	1	yes	characteristic, complemented, Sylow
84	C₂×M₁₆	—	37	1	no	—
85	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
86	C₂²×C₈	—	37	1	no	—
87	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
88	#117	—	37	1	no	—
89	#117	C₂×D₃₇	1	1	yes	characteristic
90	C₂×C₄²	—	37	1	no	—
91	#108	—	74	2	no	—
92	#107	—	37	2	no	—
93	#106	—	37	2	no	—
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	74	2	no	—
96	C₄×C₈	—	74	2	no	—
97	C₂²×C₄	—	74	1	no	—
98	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	74	1	no	—
102	M₁₆	—	74	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	37	1	no	—
107	C₂×C₈	—	37	1	no	—
108	C₂×C₈	—	37	2	no	—
109	C₂×C₈	—	37	2	no	—
110	C₂×C₈	—	74	1	no	—
111	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
112	C₄⋊C₄	—	37	2	no	—
113	C₄⋊C₄	C₄×D₃₇	1	2	yes	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	74	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	37	2	no	—
118	C₄²	—	74	1	no	—
119	C₄²	—	74	1	no	—
120	C₂³	#10596962	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	74	4	no	—
123	C₂×C₄	—	74	2	no	—
124	C₂×C₄	—	74	2	no	—
125	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
126	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
127	C₂×C₄	#10596964	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₃₇	1	1	yes	characteristic
129	C₂×C₄	Q₈×D₃₇	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	4	1	no	—
133	C₈	—	37	2	no	—
134	C₈	—	37	2	no	—
135	C₈	—	148	1	no	—
136	C₂²	—	2	2	no	—
137	C₂²	C₄×C₂²⋊D₃₇	1	1	yes	characteristic
138	C₂²	#11924514	1	1	yes	characteristic
139	C₂²	#11924526	1	1	yes	characteristic
140	C₄	—	4	2	no	—
141	C₄	—	74	4	no	—
142	C₄	—	148	2	no	—
143	C₄	C₄⋊C₄×D₃₇	1	1	yes	characteristic
144	C₄	#11924501	1	1	yes	characteristic
145	C₄	#11924508	1	1	yes	characteristic
146	C₄	#11924534	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#14557428	1	1	yes	characteristic
150	C₂	#14557648	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924471

C₂

yes

maximal, characteristic

#14557569

C₂

yes

maximal, characteristic

#14557559

C₂

yes

maximal, characteristic

C₂×C₈×Dic₃₇

C₂

yes

maximal, characteristic

#14557294

C₂

yes

maximal, characteristic

#14557292

C₂

yes

maximal, characteristic

C₃₇×#424

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

characteristic

C₂²×#64436

C₂²

yes

characteristic

#11924495

C₂²

yes

characteristic

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924471

—

#11924469

C₄

yes

complemented

#11924467

C₄

yes

complemented

#11924465

—

C₈×Dic₃₇

—

C₂×M₁₆×C₃₇

C₂²

yes

characteristic

C₂²×C₈×C₃₇

C₂²

yes

characteristic

#117×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

M₁₆×C₃₇

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

D₄

yes

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

Q₈

yes

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

D₄

yes

—

C₂×#64436

Q₈

yes

—

C₂³×C₃₇

#41

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

#41

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₈×C₃₇

—

C₈×C₃₇

—

#64436

—

#64436

C₄⋊C₄

yes

—

#64436

C₄⋊C₄

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

C₄×D₄

yes

characteristic

C₂²×C₃₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₃₇

#125

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₇

C₄×Q₈

yes

characteristic

C₄×C₃₇

#121

yes

characteristic

C₄×C₃₇

C₄⋊Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

—

#1945

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#389

yes

characteristic

C₂×C₃₇

#443

yes

—

—

—

—

D₃₇

yes

characteristic

C₂×C₄×C₈

—

#339

—

#339

—

C₃₇

#1945

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

#117

—

#117

C₂×D₃₇

yes

characteristic

C₂×C₄²

—

#108

—

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

C₂²⋊D₃₇

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₄×D₃₇

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₂³

#10596962

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

126

C₂×C₄

D₄×D₃₇

yes

characteristic

127

C₂×C₄

#10596964

yes

characteristic

128

C₂×C₄

Q₈×D₃₇

yes

characteristic

129

C₂×C₄

Q₈×D₃₇

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

148

—

136

C₂²

—

137

C₂²

C₄×C₂²⋊D₃₇

yes

characteristic

138

C₂²

#11924514

yes

characteristic

139

C₂²

#11924526

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

148

—

143

C₄

C₄⋊C₄×D₃₇

yes

characteristic

144

C₄

#11924501

yes

characteristic

145

C₄

#11924508

yes

characteristic

146

C₄

#11924534

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#14557428

yes

characteristic

150

C₂

#14557648

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#424	C₂
socle	C₂²×C₃₇	C₄×D₄
prime radical	C₂³×C₃₇	#41
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#424	D₃₇
hypocenter	C₃₇	#1945
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #424

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#424

C₂

socle

C₂²×C₃₇

C₄×D₄

prime radical

C₂³×C₃₇

#41

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #424

N	H ≅ G/N	type
13. #11924469	C₄	semidirect
14. #11924467	C₄	semidirect
83. C₃₇	#1945	semidirect

H ≅ G/N

type

13. #11924469

C₄

semidirect

14. #11924467

C₄

semidirect

83. C₃₇

#1945

semidirect

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

subgroup	index
C₂×#11924471	2
#14557569	2
#14557559	2
C₂×C₈×Dic₃₇	2
#14557294	2
#14557292	2
C₃₇×#424	2
#1945	37

subgroup

index

C₂×C₈×Dic₃₇

C₃₇×#424

Subgroups — #17995341

Conjugacy classes of subgroups

Complements

Maximal subgroups