Group #17993253 — subgroups

Subgroups — #17993253

Order 4736.

212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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	#14557586	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#64436	C₂	1	1	yes	maximal, characteristic
3	#14557282	C₂	1	2	yes	maximal, complemented
4	#14557279	C₂	1	2	yes	maximal, complemented
5	C₃₇×#533	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64436	—	2	1	no	—
7	C₄⋊#64436	—	2	1	no	—
8	C₄⋊#64436	C₂²	1	2	yes	—
9	C₄⋊#64436	C₂²	1	2	yes	—
10	C₂²⋊#64436	—	2	1	no	—
11	C₄×#64436	—	2	1	no	—
12	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
14	#125×C₃₇	—	2	1	no	—
15	C₂²⋊Q₈×C₃₇	—	2	1	no	—
16	C₂²⋊Q₈×C₃₇	C₄	1	2	yes	—
17	C₄×Q₈×C₃₇	—	2	1	no	—
18	C₄×Q₈×C₃₇	C₄	1	2	yes	—
19	#117×C₃₇	—	2	1	no	—
20	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₃₇	—	2	1	no	—
22	C₂×Q₈×C₃₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
26	C₄⋊C₄×C₃₇	—	2	2	no	—
27	C₄⋊C₄×C₃₇	—	2	2	no	—
28	C₄⋊C₄×C₃₇	—	2	1	no	—
29	C₄⋊C₄×C₃₇	—	2	1	no	—
30	C₄⋊C₄×C₃₇	—	2	1	no	—
31	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
32	C₂²⋊C₄×C₃₇	—	2	2	no	—
33	C₄²×C₃₇	—	2	2	no	—
34	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₃₇	D₄	1	1	yes	characteristic
36	C₄²×C₃₇	D₄	1	1	yes	characteristic
37	C₄²×C₃₇	C₂³	1	1	yes	characteristic
38	C₂×#64436	—	2	2	no	—
39	C₂×#64436	—	2	2	no	—
40	C₂×#64436	—	4	2	no	—
41	C₂³×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈×C₃₇	—	4	2	no	—
43	Q₈×C₃₇	—	4	2	no	—
44	C₂×C₄×C₃₇	—	2	2	no	—
45	C₂×C₄×C₃₇	—	2	2	no	—
46	C₂×C₄×C₃₇	—	2	2	no	—
47	C₂×C₄×C₃₇	—	4	1	no	—
48	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₃₇	SD₁₆	1	2	yes	—
52	C₂×C₄×C₃₇	Q₁₆	1	2	yes	—
53	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
56	#64436	—	4	2	no	—
57	#64436	—	4	2	no	—
58	C₂²×C₃₇	—	2	1	no	—
59	C₂²×C₃₇	—	2	1	no	—
60	C₂²×C₃₇	#103	1	2	yes	—
61	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₃₇	—	2	1	no	—
63	C₄×C₃₇	—	2	1	no	—
64	C₄×C₃₇	—	2	1	no	—
65	C₄×C₃₇	—	4	2	no	—
66	C₄×C₃₇	—	4	2	no	—
67	C₄×C₃₇	#103	1	2	yes	—
68	C₄×C₃₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₃₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₄×C₃₇	#135	1	2	yes	—
71	#1542	—	37	1	no	maximal, Sylow
72	C₂×C₃₇	—	2	2	no	—
73	C₂×C₃₇	#413	1	1	yes	characteristic
74	C₂×C₃₇	C₂×#103	1	1	yes	characteristic
75	C₂×C₃₇	#416	1	1	yes	characteristic
76	#533	D₃₇	1	1	yes	characteristic
77	#431	—	37	1	no	—
78	C₂×C₄⋊C₈	—	37	1	no	—
79	#333	—	37	2	no	—
80	#332	—	37	2	no	—
81	C₃₇	#1542	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₈	—	74	1	no	—
83	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
84	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
85	#125	—	2	1	no	—
86	C₂²⋊Q₈	—	2	1	no	—
87	C₂²⋊Q₈	Dic₃₇	1	2	yes	—
88	C₄×Q₈	—	2	1	no	—
89	C₄×Q₈	Dic₃₇	1	2	yes	—
90	#117	—	2	1	no	—
91	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
92	C₄⋊C₈	—	37	2	no	—
93	C₄⋊C₈	—	37	2	no	—
94	C₄⋊C₈	—	74	1	no	—
95	C₂²⋊C₈	—	74	1	no	—
96	C₄×C₈	—	74	1	no	—
97	C₂×Q₈	—	2	1	no	—
98	C₂×Q₈	C₂×Dic₃₇	1	2	yes	—
99	C₂²×C₄	C₂×Dic₃₇	1	1	yes	characteristic
100	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
101	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
102	C₂×C₈	—	74	2	no	—
103	C₂×C₈	—	74	2	no	—
104	C₂×C₈	—	148	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	C₂×Dic₃₇	1	2	yes	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	C₂×Dic₃₇	1	1	yes	characteristic
114	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
115	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
116	C₄²	C₂²×D₃₇	1	1	yes	characteristic
117	C₂³	C₂²⋊Dic₃₇	1	1	yes	characteristic
118	Q₈	—	4	2	no	—
119	Q₈	—	4	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	#10596941	1	2	yes	—
125	C₂×C₄	#10596942	1	2	yes	—
126	C₂×C₄	C₂²⋊Dic₃₇	1	1	yes	characteristic
127	C₂×C₄	C₂²⋊Dic₃₇	1	1	yes	characteristic
128	C₂×C₄	C₂²⋊Dic₃₇	1	1	yes	characteristic
129	C₂×C₄	C₂²×Dic₃₇	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
132	C₈	—	148	2	no	—
133	C₈	—	148	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	#11924460	1	2	yes	—
137	C₂²	C₂×C₂²⋊Dic₃₇	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	4	2	no	—
142	C₄	—	4	2	no	—
143	C₄	#11924460	1	2	yes	—
144	C₄	C₂×#10596941	1	1	yes	characteristic
145	C₄	C₂×#10596942	1	1	yes	characteristic
146	C₄	#11924556	1	2	yes	—
147	C₂	—	2	2	no	—
148	C₂	#14557517	1	1	yes	characteristic
149	C₂	C₂×#11924460	1	1	yes	characteristic
150	C₂	#14557529	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557586

C₂

yes

maximal, characteristic

C₂×C₄⋊#64436

C₂

yes

maximal, characteristic

#14557282

C₂

yes

maximal, complemented

#14557279

C₂

yes

maximal, complemented

C₃₇×#533

C₂

yes

maximal, characteristic

C₂²×#64436

—

C₄⋊#64436

—

C₄⋊#64436

C₂²

yes

—

C₄⋊#64436

C₂²

yes

—

C₂²⋊#64436

—

C₄×#64436

—

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

#125×C₃₇

—

C₂²⋊Q₈×C₃₇

—

C₂²⋊Q₈×C₃₇

C₄

yes

—

C₄×Q₈×C₃₇

—

C₄×Q₈×C₃₇

C₄

yes

—

#117×C₃₇

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

D₄

yes

characteristic

C₄²×C₃₇

D₄

yes

characteristic

C₄²×C₃₇

C₂³

yes

—

—

—

C₂³×C₃₇

C₂²⋊C₄

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₇

SD₁₆

yes

—

C₂×C₄×C₃₇

Q₁₆

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

#64436

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#103

yes

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#103

yes

—

C₄×C₃₇

C₂×SD₁₆

yes

characteristic

C₄×C₃₇

C₂×Q₁₆

yes

characteristic

C₄×C₃₇

#135

yes

—

#1542

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

yes

characteristic

#431

—

C₂×C₄⋊C₈

—

#333

—

#332

—

C₃₇

#1542

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄⋊Q₈

C₂×D₃₇

yes

characteristic

C₄⋊Q₈

C₂×D₃₇

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

Dic₃₇

yes

—

C₄×Q₈

—

C₄×Q₈

Dic₃₇

yes

—

#117

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₃₇

yes

—

C₂²×C₄

C₂×Dic₃₇

yes

characteristic

100

C₂²×C₄

C₂²⋊D₃₇

yes

characteristic

101

C₂²×C₄

C₂²⋊D₃₇

yes

characteristic

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

148

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

C₂×Dic₃₇

yes

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

C₂×Dic₃₇

yes

characteristic

114

C₄²

C₂²⋊D₃₇

yes

characteristic

115

C₄²

C₂²⋊D₃₇

yes

characteristic

116

C₄²

C₂²×D₃₇

yes

characteristic

117

C₂³

C₂²⋊Dic₃₇

yes

characteristic

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#10596941

yes

—

125

C₂×C₄

#10596942

yes

—

126

C₂×C₄

C₂²⋊Dic₃₇

yes

characteristic

127

C₂×C₄

C₂²⋊Dic₃₇

yes

characteristic

128

C₂×C₄

C₂²⋊Dic₃₇

yes

characteristic

129

C₂×C₄

C₂²×Dic₃₇

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

132

C₈

—

148

—

133

C₈

—

148

—

134

C₂²

—

135

C₂²

—

136

C₂²

#11924460

yes

—

137

C₂²

C₂×C₂²⋊Dic₃₇

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#11924460

yes

—

144

C₄

C₂×#10596941

yes

characteristic

145

C₄

C₂×#10596942

yes

characteristic

146

C₄

#11924556

yes

—

147

C₂

—

148

C₂

#14557517

yes

characteristic

149

C₂

C₂×#11924460

yes

characteristic

150

C₂

#14557529

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#533	C₂
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₃₇	C₂²⋊C₄
cosocle	C₄²×C₃₇	C₂³
hypercenter	#533	D₃₇
hypocenter	C₃₇	#1542
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#533

C₂

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₃₇

C₂²⋊C₄

cosocle

C₄²×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. #14557282	C₂	semidirect
4. #14557279	C₂	semidirect
81. C₃₇	#1542	semidirect

H ≅ G/N

type

3. #14557282

C₂

semidirect

4. #14557279

C₂

semidirect

81. C₃₇

#1542

semidirect

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

subgroup	index
#14557586	2
C₂×C₄⋊#64436	2
#14557282	2
#14557279	2
C₃₇×#533	2
#1542	37

subgroup

index

C₂×C₄⋊#64436

C₃₇×#533

Subgroups — #17993253

Conjugacy classes of subgroups

Complements

Maximal subgroups