Group #49159926 — subgroups

Subgroups — #49159926

Order 37504.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 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×#22627624	C₂	1	1	yes	maximal, characteristic
2	C₂×#22627548	C₂	1	2	yes	maximal
3	#34035891	C₂	1	2	yes	maximal
4	#34035884	C₂	1	1	yes	maximal, characteristic
5	#391×C₂₉₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₈₆	—	2	1	no	—
7	C₂²×#14556873	C₂²	1	2	yes	—
8	#22627624	—	2	4	no	—
9	C₂×C₄⋊Dic₂₉₃	—	2	1	no	—
10	C₂×C₄⋊Dic₂₉₃	C₂²	1	2	yes	—
11	#22627548	—	2	4	no	—
12	C₂²×Q₈×C₂₉₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₉₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₉₃	C₂²	1	1	yes	characteristic
15	#95×C₂₉₃	—	2	1	no	—
16	#95×C₂₉₃	—	2	1	no	—
17	C₂×Dic₅₈₆	—	2	4	no	—
18	C₂×Dic₅₈₆	—	4	2	no	—
19	C₂×Dic₅₈₆	—	4	2	no	—
20	C₂²×Dic₂₉₃	—	2	2	no	—
21	C₂×#14556873	—	2	2	no	—
22	C₂×#14556873	—	2	2	no	—
23	C₂×#14556873	—	2	2	no	—
24	C₄⋊Dic₂₉₃	—	2	4	no	—
25	C₄⋊Dic₂₉₃	—	2	2	no	—
26	C₄⋊Dic₂₉₃	D₄	1	4	yes	—
27	C₂×Q₈×C₂₉₃	—	2	1	no	—
28	C₂×Q₈×C₂₉₃	—	2	2	no	—
29	C₂×Q₈×C₂₉₃	—	2	2	no	—
30	C₂×Q₈×C₂₉₃	Q₈	1	2	yes	complemented
31	C₂²×C₄×C₂₉₃	—	2	1	no	—
32	C₂²×C₄×C₂₉₃	—	2	1	no	—
33	C₂²×C₄×C₂₉₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₉₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₂₉₃	—	2	2	no	—
37	C₄²×C₂₉₃	—	2	2	no	—
38	Dic₅₈₆	—	4	8	no	—
39	C₂×Dic₂₉₃	—	2	4	no	—
40	C₂×Dic₂₉₃	—	4	2	no	—
41	C₂×Dic₂₉₃	—	4	2	no	—
42	#14556873	—	4	2	no	—
43	C₂³×C₂₉₃	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₂₉₃	—	2	4	no	—
45	Q₈×C₂₉₃	—	4	2	no	—
46	C₂×C₄×C₂₉₃	—	2	1	no	—
47	C₂×C₄×C₂₉₃	—	2	1	no	—
48	C₂×C₄×C₂₉₃	—	2	1	no	—
49	C₂×C₄×C₂₉₃	—	2	1	no	—
50	C₂×C₄×C₂₉₃	—	2	2	no	—
51	C₂×C₄×C₂₉₃	—	4	1	no	—
52	C₂×C₄×C₂₉₃	—	4	1	no	—
53	C₂×C₄×C₂₉₃	—	4	1	no	—
54	C₂×C₄×C₂₉₃	—	4	1	no	—
55	C₂×C₄×C₂₉₃	—	4	1	no	—
56	C₂×C₄×C₂₉₃	SD₁₆	1	2	yes	—
57	C₂×C₄×C₂₉₃	Q₁₆	1	2	yes	—
58	C₂×C₄×C₂₉₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₂₉₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₂₉₃	#41	1	2	yes	—
61	C₂×C₄×C₂₉₃	#41	1	1	yes	characteristic
62	C₂²×C₂₉₃	#120	1	1	yes	characteristic
63	C₂²×C₂₉₃	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₂₉₃	#124	1	1	yes	characteristic
65	C₂²×C₂₉₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₂₉₃	C₂×Q₁₆	1	1	yes	characteristic
67	C₂²×C₂₉₃	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂²×C₂₉₃	#137	1	1	yes	characteristic
69	C₄×C₂₉₃	—	2	2	no	—
70	C₄×C₂₉₃	—	2	2	no	—
71	C₄×C₂₉₃	—	4	2	no	—
72	C₄×C₂₉₃	—	8	1	no	—
73	C₄×C₂₉₃	C₂²⋊Q₈	1	2	yes	—
74	C₄×C₂₉₃	#124	1	2	yes	—
75	Dic₂₉₃	—	4	4	no	—
76	C₂×C₂₉₃	#393	1	1	yes	characteristic
77	C₂×C₂₉₃	#450	1	1	yes	characteristic
78	C₂×C₂₉₃	#451	1	1	yes	characteristic
79	C₂×C₂₉₃	#475	1	1	yes	characteristic
80	C₂×C₂₉₃	#477	1	1	yes	characteristic
81	C₂×C₂₉₃	#487	1	1	yes	characteristic
82	C₂×C₂₉₃	#489	1	1	yes	characteristic
83	C₂₉₃	#2024	1	1	yes	characteristic, complemented, Sylow
84	#2024	—	293	1	no	maximal, Sylow
85	C₂×C₄⋊Q₈	—	293	1	no	—
86	C₂×#103	—	293	2	no	—
87	#391	D₂₉₃	1	1	yes	characteristic
88	#340	—	293	2	no	—
89	#336	—	293	1	no	—
90	C₂²×Q₈	—	586	1	no	—
91	C₂²×Q₈	C₂×D₂₉₃	1	1	yes	characteristic
92	C₂²×C₈	—	293	2	no	—
93	C₄⋊Q₈	—	586	4	no	—
94	C₂×C₄⋊C₄	—	293	2	no	—
95	C₂×C₄⋊C₄	—	586	1	no	—
96	C₂×C₄⋊C₄	C₂×D₂₉₃	1	1	yes	characteristic
97	C₂×C₄²	C₂×D₂₉₃	1	1	yes	characteristic
98	#103	—	586	4	no	—
99	#95	—	2	1	no	—
100	#95	—	2	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	586	4	no	—
105	C₂×Q₈	—	1172	2	no	—
106	C₂×Q₈	—	1172	2	no	—
107	C₂×Q₈	Dic₅₈₆	1	2	yes	complemented
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	586	2	no	—
111	C₂²×C₄	C₂²⋊D₂₉₃	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊D₂₉₃	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
114	C₂×C₈	—	586	2	no	—
115	C₂×C₈	—	586	2	no	—
116	C₂×C₈	—	586	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	293	4	no	—
119	C₄⋊C₄	—	586	4	no	—
120	C₄⋊C₄	—	586	2	no	—
121	C₄²	—	2	2	no	—
122	C₂³	C₂×C₂²⋊D₂₉₃	1	1	yes	characteristic
123	Q₈	—	2	4	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	1172	8	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	586	4	no	—
137	C₂×C₄	—	1172	2	no	—
138	C₂×C₄	—	1172	2	no	—
139	C₂×C₄	#17989789	1	2	yes	—
140	C₂×C₄	#17989792	1	2	yes	—
141	C₂×C₄	D₄×D₂₉₃	1	2	yes	—
142	C₂×C₄	C₂×Dic₅₈₆	1	1	yes	characteristic
143	C₂×C₄	#17989801	1	2	yes	—
144	C₂×C₄	#17989802	1	1	yes	characteristic
145	C₈	—	1172	2	no	—
146	C₂²	#22627592	1	1	yes	characteristic
147	C₂²	#22627603	1	1	yes	characteristic
148	C₂²	#22627611	1	1	yes	characteristic
149	C₂²	C₂×#17989789	1	1	yes	characteristic
150	C₂²	C₂×#17989792	1	1	yes	characteristic
151	C₂²	#22627649	1	1	yes	characteristic
152	C₂²	#22627654	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	—	8	1	no	—
157	C₄	—	1172	4	no	—
158	C₄	#22627601	1	2	yes	—
159	C₄	#22627610	1	2	yes	—
160	C₂	#34036041	1	1	yes	characteristic
161	C₂	#34036279	1	1	yes	characteristic
162	C₂	#34036284	1	1	yes	characteristic
163	C₂	#34036414	1	1	yes	characteristic
164	C₂	#34036424	1	1	yes	characteristic
165	C₂	#34036479	1	1	yes	characteristic
166	C₂	#34036488	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22627624

C₂

yes

maximal, characteristic

C₂×#22627548

C₂

yes

maximal

#34035891

C₂

yes

maximal

#34035884

C₂

yes

maximal, characteristic

#391×C₂₉₃

C₂

yes

maximal, characteristic

C₂²×Dic₅₈₆

—

C₂²×#14556873

C₂²

yes

—

#22627624

—

C₂×C₄⋊Dic₂₉₃

—

C₂×C₄⋊Dic₂₉₃

C₂²

yes

—

#22627548

—

C₂²×Q₈×C₂₉₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₉₃

C₂²

yes

characteristic

C₂×C₄²×C₂₉₃

C₂²

yes

characteristic

#95×C₂₉₃

—

#95×C₂₉₃

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂²×Dic₂₉₃

—

C₂×#14556873

—

C₂×#14556873

—

C₂×#14556873

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

D₄

yes

—

C₂×Q₈×C₂₉₃

—

C₂×Q₈×C₂₉₃

—

C₂×Q₈×C₂₉₃

—

C₂×Q₈×C₂₉₃

Q₈

yes

complemented

C₂²×C₄×C₂₉₃

—

C₂²×C₄×C₂₉₃

—

C₂²×C₄×C₂₉₃

D₄

yes

characteristic

C₂²×C₄×C₂₉₃

D₄

yes

characteristic

C₂²×C₄×C₂₉₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₉₃

—

C₄²×C₂₉₃

—

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

#14556873

—

C₂³×C₂₉₃

C₂×D₄

yes

characteristic

Q₈×C₂₉₃

—

Q₈×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

SD₁₆

yes

—

C₂×C₄×C₂₉₃

Q₁₆

yes

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

—

C₂×C₄×C₂₉₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₉₃

#41

yes

—

C₂×C₄×C₂₉₃

yes

C₂²×C₂₉₃

yes

C₂²×C₂₉₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₉₃

#124

yes

characteristic

C₂²×C₂₉₃

C₂×SD₁₆

yes

characteristic

C₂²×C₂₉₃

C₂×Q₁₆

yes

C₂²×C₂₉₃

yes

C₂²×C₂₉₃

yes

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

C₂²⋊Q₈

yes

—

C₄×C₂₉₃

#124

yes

—

Dic₂₉₃

—

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂₉₃

yes

characteristic, complemented, Sylow

#2024

—

293

maximal, Sylow

C₂×C₄⋊Q₈

—

293

—

C₂×#103

—

293

—

#391

D₂₉₃

yes

characteristic

#340

—

293

—

#336

—

293

—

C₂²×Q₈

—

586

—

C₂²×Q₈

C₂×D₂₉₃

yes

characteristic

C₂²×C₈

—

293

—

C₄⋊Q₈

—

586

—

C₂×C₄⋊C₄

—

293

—

C₂×C₄⋊C₄

—

586

—

C₂×C₄⋊C₄

C₂×D₂₉₃

yes

characteristic

C₂×C₄²

C₂×D₂₉₃

yes

characteristic

#103

—

586

—

#95

—

100

#95

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

586

—

105

C₂×Q₈

—

1172

—

106

C₂×Q₈

—

1172

—

107

C₂×Q₈

Dic₅₈₆

yes

complemented

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

586

—

111

C₂²×C₄

C₂²⋊D₂₉₃

yes

characteristic

112

C₂²×C₄

C₂²⋊D₂₉₃

yes

characteristic

113

C₂²×C₄

C₂²×D₂₉₃

yes

characteristic

114

C₂×C₈

—

586

—

115

C₂×C₈

—

586

—

116

C₂×C₈

—

586

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

293

—

119

C₄⋊C₄

—

586

—

120

C₄⋊C₄

—

586

—

121

C₄²

—

122

C₂³

C₂×C₂²⋊D₂₉₃

yes

characteristic

123

Q₈

—

124

Q₈

—

125

Q₈

—

1172

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

586

—

137

C₂×C₄

—

1172

—

138

C₂×C₄

—

1172

—

139

C₂×C₄

#17989789

yes

—

140

C₂×C₄

#17989792

yes

—

141

C₂×C₄

D₄×D₂₉₃

yes

—

142

C₂×C₄

C₂×Dic₅₈₆

yes

characteristic

143

C₂×C₄

#17989801

yes

—

144

C₂×C₄

#17989802

yes

characteristic

145

C₈

—

1172

—

146

C₂²

#22627592

yes

characteristic

147

C₂²

#22627603

yes

characteristic

148

C₂²

#22627611

yes

characteristic

149

C₂²

C₂×#17989789

yes

characteristic

150

C₂²

C₂×#17989792

yes

characteristic

151

C₂²

#22627649

yes

characteristic

152

C₂²

#22627654

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

1172

—

158

C₄

#22627601

yes

—

159

C₄

#22627610

yes

—

160

C₂

#34036041

yes

characteristic

161

C₂

#34036279

yes

characteristic

162

C₂

#34036284

yes

characteristic

163

C₂

#34036414

yes

characteristic

164

C₂

#34036424

yes

characteristic

165

C₂

#34036479

yes

characteristic

166

C₂

#34036488

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₉₃
commutator, cosocle	C₂²×C₄×C₂₉₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₉₃
Fitting	#391×C₂₉₃	C₂
socle, prime radical	C₂³×C₂₉₃	C₂×D₄
hypercenter	#391	D₂₉₃
hypocenter	C₂₉₃	#2024
LCS	G ▷ C₂²×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₉₃

commutator, cosocle

C₂²×C₄×C₂₉₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₉₃

Fitting

#391×C₂₉₃

C₂

socle, prime radical

C₂³×C₂₉₃

C₂×D₄

D₂₉₃

C₂₉₃

G ▷ C₂²×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
30. C₂×Q₈×C₂₉₃	Q₈	semidirect
83. C₂₉₃	#2024	semidirect
107. C₂×Q₈	Dic₅₈₆	semidirect

H ≅ G/N

type

30. C₂×Q₈×C₂₉₃

Q₈

semidirect

83. C₂₉₃

#2024

semidirect

107. C₂×Q₈

Dic₅₈₆

semidirect

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

subgroup	index
C₂×#22627624	2
C₂×#22627548	2
#34035891	2
#34035884	2
#391×C₂₉₃	2
#2024	293

subgroup

index

#391×C₂₉₃

293

Subgroups — #49159926

Conjugacy classes of subgroups

Complements

Maximal subgroups