Group #17995319 — subgroups

Subgroups — #17995319

Order 4736.

248 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 113 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×C₄⋊#64436	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924465	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₃₇	C₂	1	1	yes	maximal, characteristic
4	#14557287	C₂	1	2	yes	maximal
5	C₂×#97×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64436	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₃₇	C₄	1	2	yes	—
8	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
9	C₄⋊#64436	—	2	2	no	—
10	#11924465	—	2	4	no	—
11	#11924443	C₄	1	1	yes	characteristic
12	#11924443	C₄	1	1	yes	characteristic
13	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
16	#97×C₃₇	—	2	2	no	—
17	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
20	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
21	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₃₇	—	2	1	no	—
23	C₂×C₈×C₃₇	—	2	1	no	—
24	C₂×C₈×C₃₇	—	2	1	no	—
25	C₂×C₈×C₃₇	—	2	1	no	—
26	C₂×C₈×C₃₇	D₄	1	2	yes	—
27	C₂×C₈×C₃₇	Q₈	1	2	yes	—
28	C₄²×C₃₇	D₄	1	2	yes	—
29	C₄²×C₃₇	Q₈	1	2	yes	—
30	C₄⋊Dic₃₇	—	2	4	no	—
31	C₄×Dic₃₇	—	2	2	no	—
32	C₄×Dic₃₇	—	2	2	no	—
33	C₂×#64436	—	2	2	no	—
34	C₂×#64436	—	2	2	no	—
35	C₂×#64436	—	2	2	no	—
36	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₃₇	—	2	2	no	—
38	C₂×C₄×C₃₇	—	2	2	no	—
39	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
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	1	yes	characteristic
47	C₂×Dic₃₇	—	2	2	no	—
48	C₂×Dic₃₇	—	2	2	no	—
49	C₂×Dic₃₇	—	2	2	no	—
50	C₂×Dic₃₇	—	2	2	no	—
51	C₂×Dic₃₇	—	2	2	no	—
52	C₂×Dic₃₇	M₁₆	1	4	yes	—
53	C₈×C₃₇	—	2	2	no	—
54	C₈×C₃₇	—	2	2	no	—
55	#64436	—	4	2	no	—
56	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₃₇	C₄×Q₈	1	1	yes	characteristic
59	C₂²×C₃₇	C₂×M₁₆	1	2	yes	—
60	C₂²×C₃₇	#131	1	2	yes	—
61	C₄×C₃₇	—	2	2	no	—
62	C₄×C₃₇	—	2	2	no	—
63	C₄×C₃₇	#121	1	1	yes	characteristic
64	C₄×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
65	C₄×C₃₇	#125	1	1	yes	characteristic
66	C₄×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₃₇	—	2	4	no	—
68	Dic₃₇	—	4	2	no	—
69	#1940	—	37	1	no	maximal, Sylow
70	C₂×C₃₇	#389	1	1	yes	characteristic
71	C₂×C₃₇	#423	1	1	yes	characteristic
72	C₂×C₃₇	#424	1	1	yes	characteristic
73	C₂×C₃₇	#435	1	2	yes	—
74	C₂×C₃₇	#446	1	2	yes	—
75	C₂×C₄⋊C₈	—	37	2	no	—
76	C₂×C₄⋊C₈	—	37	1	no	—
77	C₂×#97	D₃₇	1	1	yes	characteristic
78	C₄×C₄⋊C₄	—	37	1	no	—
79	#336	—	37	2	no	—
80	C₃₇	#1940	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	37	2	no	—
82	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
84	C₂×C₄⋊C₄	—	37	2	no	—
85	C₂×C₄²	—	37	2	no	—
86	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
87	C₄⋊C₈	—	74	4	no	—
88	C₄⋊C₈	—	74	2	no	—
89	#97	—	2	2	no	—
90	#95	—	37	1	no	—
91	#95	—	37	1	no	—
92	C₂²×C₄	—	37	2	no	—
93	C₂²×C₄	—	37	2	no	—
94	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	74	2	no	—
102	C₂×C₈	—	74	2	no	—
103	C₂×C₈	—	74	2	no	—
104	C₂×C₈	Dic₇₄	1	2	yes	—
105	C₂×C₈	D₁₄₈	1	2	yes	—
106	C₄⋊C₄	—	74	4	no	—
107	C₄²	—	74	2	no	—
108	C₄²	—	74	2	no	—
109	C₄²	Dic₇₄	1	2	yes	—
110	C₄²	C₂²⋊D₃₇	1	2	yes	—
111	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	37	4	no	—
115	C₂×C₄	—	74	2	no	—
116	C₂×C₄	—	74	2	no	—
117	C₂×C₄	—	74	2	no	—
118	C₂×C₄	—	74	2	no	—
119	C₂×C₄	—	74	2	no	—
120	C₂×C₄	#10596936	1	2	yes	—
121	C₂×C₄	#10596936	1	2	yes	—
122	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
123	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
124	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
125	C₂×C₄	#10596962	1	1	yes	characteristic
126	C₂×C₄	#10596962	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	2	2	no	—
130	C₈	—	148	2	no	—
131	C₂²	C₂×#10596936	1	1	yes	characteristic
132	C₂²	C₄×D₁₄₈	1	1	yes	characteristic
133	C₂²	C₄×Dic₇₄	1	1	yes	characteristic
134	C₂²	M₁₆×D₃₇	1	2	yes	—
135	C₂²	#11924544	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	74	4	no	—
139	C₄	—	148	2	no	—
140	C₄	#11924510	1	1	yes	characteristic
141	C₄	#11924515	1	1	yes	characteristic
142	C₄	#11924528	1	1	yes	characteristic
143	C₄	#11924536	1	1	yes	characteristic
144	C₂	#14557431	1	1	yes	characteristic
145	C₂	#14557556	1	1	yes	characteristic
146	C₂	#14557561	1	1	yes	characteristic
147	C₂	#14557606	1	2	yes	—
148	C₂	#14557661	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#64436

C₂

yes

maximal, characteristic

C₂×#11924465

C₂

yes

maximal

C₄×C₄⋊Dic₃₇

C₂

yes

maximal, characteristic

#14557287

C₂

yes

maximal

C₂×#97×C₃₇

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₄

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

—

C₄⋊#64436

—

#11924465

—

#11924443

C₄

yes

characteristic

#11924443

C₄

yes

characteristic

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#97×C₃₇

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

D₄

yes

—

C₂×C₈×C₃₇

Q₈

yes

—

C₄²×C₃₇

D₄

yes

—

C₄²×C₃₇

Q₈

yes

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₂×#64436

—

C₂×#64436

—

C₂×#64436

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

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

C₂×C₄×C₃₇

yes

C₂×C₄×C₃₇

yes

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

M₁₆

yes

—

C₈×C₃₇

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

C₄×D₄

yes

characteristic

C₂²×C₃₇

C₄×Q₈

yes

characteristic

C₂²×C₃₇

C₂×M₁₆

yes

—

C₂²×C₃₇

#131

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#121

yes

characteristic

C₄×C₃₇

C₂²⋊Q₈

yes

characteristic

C₄×C₃₇

#125

yes

characteristic

C₄×C₃₇

C₄⋊Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

#446

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

D₃₇

yes

characteristic

C₄×C₄⋊C₄

—

#336

—

C₃₇

#1940

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃₇

yes

characteristic

C₂²×C₄

C₄×D₃₇

yes

characteristic

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

Dic₇₄

yes

—

105

C₂×C₈

D₁₄₈

yes

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

Dic₇₄

yes

—

110

C₄²

C₂²⋊D₃₇

yes

—

111

C₂³

C₂×C₄×D₃₇

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

#10596936

yes

—

121

C₂×C₄

#10596936

yes

—

122

C₂×C₄

C₂×Dic₇₄

yes

characteristic

123

C₂×C₄

C₂×Dic₇₄

yes

characteristic

124

C₂×C₄

C₂×D₁₄₈

yes

characteristic

125

C₂×C₄

#10596962

yes

characteristic

126

C₂×C₄

#10596962

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

148

—

131

C₂²

C₂×#10596936

yes

characteristic

132

C₂²

C₄×D₁₄₈

yes

characteristic

133

C₂²

C₄×Dic₇₄

yes

characteristic

134

C₂²

M₁₆×D₃₇

yes

—

135

C₂²

#11924544

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

148

—

140

C₄

#11924510

yes

characteristic

141

C₄

#11924515

yes

characteristic

142

C₄

#11924528

yes

characteristic

143

C₄

#11924536

yes

characteristic

144

C₂

#14557431

yes

characteristic

145

C₂

#14557556

yes

characteristic

146

C₂

#14557561

yes

characteristic

147

C₂

#14557606

yes

—

148

C₂

#14557661

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃₇
commutator, socle, prime radical	C₂³×C₃₇	C₂²×C₄
Fitting	C₂×#97×C₃₇	C₂
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×#97	D₃₇
hypocenter	C₃₇	#1940
LCS	G ▷ C₂³×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃₇

commutator, socle, prime radical

C₂³×C₃₇

C₂²×C₄

Fitting

C₂×#97×C₃₇

C₂

cosocle

C₂²×C₄×C₃₇

C₂³

hypercenter

C₂×#97

D₃₇

C₃₇

G ▷ C₂³×C₃₇ ▷ C₃₇

C₁ ◁ C₂²×C₄ ◁ C₂×#97

N	H ≅ G/N	type
80. C₃₇	#1940	semidirect

H ≅ G/N

type

80. C₃₇

#1940

semidirect

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

subgroup	index
C₂×C₄⋊#64436	2
C₂×#11924465	2
C₄×C₄⋊Dic₃₇	2
#14557287	2
C₂×#97×C₃₇	2
#1940	37

subgroup

index

C₂×C₄⋊#64436

C₂×#11924465

C₄×C₄⋊Dic₃₇

#14557287

C₂×#97×C₃₇

#1940

Subgroups — #17995319

Conjugacy classes of subgroups

Complements

Maximal subgroups