Group #49158026 — subgroups

Subgroups — #49158026

Order 37504.

194 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 57 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	#34036858	C₂	1	1	yes	maximal, characteristic
2	#34036532	C₂	1	1	yes	maximal, characteristic
3	#34036336	C₂	1	1	yes	maximal, characteristic
4	#34035874	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#14556873	C₂	1	1	yes	maximal, characteristic
6	#34035859	C₂	1	1	yes	maximal, characteristic
7	#475×C₂₉₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#17989792	—	2	1	no	—
9	#22627624	—	2	1	no	—
10	#22627624	C₂²	1	1	yes	characteristic
11	#22627615	—	2	2	no	—
12	#22627615	—	2	1	no	—
13	Q₈×Dic₂₉₃	—	2	1	no	—
14	C₄×Dic₅₈₆	—	2	1	no	—
15	C₄×Dic₅₈₆	C₂²	1	1	yes	characteristic
16	#22627557	—	2	2	no	—
17	C₄⋊#14556873	C₂²	1	1	yes	characteristic
18	#22627548	—	2	1	no	—
19	C₄×#14556873	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₂₉₃	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₂₉₃	C₂²	1	1	yes	characteristic
22	#106×C₂₉₃	—	2	1	no	—
23	C₄⋊C₈×C₂₉₃	C₂²	1	1	yes	characteristic
24	#103×C₂₉₃	—	2	1	no	—
25	C₂×Dic₅₈₆	—	2	1	no	—
26	C₂×Dic₅₈₆	D₄	1	1	yes	characteristic
27	#17989792	—	4	2	no	—
28	C₂×#14556873	—	2	1	no	—
29	C₂×#14556873	—	2	1	no	—
30	C₄⋊Dic₂₉₃	—	2	2	no	—
31	C₄⋊Dic₂₉₃	—	2	2	no	—
32	C₄⋊Dic₂₉₃	—	2	1	no	—
33	C₄⋊Dic₂₉₃	D₄	1	1	yes	characteristic
34	C₄⋊Dic₂₉₃	D₄	1	2	yes	—
35	#17989781	—	2	2	no	—
36	#17989781	—	2	1	no	—
37	C₄×Dic₂₉₃	—	2	2	no	—
38	C₄×Dic₂₉₃	—	2	1	no	—
39	C₂×Q₈×C₂₉₃	—	2	1	no	—
40	C₂×Q₈×C₂₉₃	D₄	1	1	yes	characteristic
41	C₂×C₈×C₂₉₃	—	2	1	no	—
42	C₄⋊C₄×C₂₉₃	—	2	1	no	—
43	C₄⋊C₄×C₂₉₃	—	2	1	no	—
44	C₄⋊C₄×C₂₉₃	—	2	1	no	—
45	C₄⋊C₄×C₂₉₃	D₄	1	1	yes	characteristic
46	C₄²×C₂₉₃	—	2	1	no	—
47	C₄²×C₂₉₃	C₂³	1	1	yes	characteristic
48	Dic₅₈₆	—	2	1	no	—
49	Dic₅₈₆	—	2	1	no	—
50	Dic₅₈₆	—	4	1	no	—
51	Dic₅₈₆	—	4	1	no	—
52	C₂×Dic₂₉₃	—	2	1	no	—
53	C₂×Dic₂₉₃	—	2	1	no	—
54	C₂×Dic₂₉₃	—	2	2	no	—
55	#14556873	—	2	2	no	—
56	#14556873	—	4	1	no	—
57	Q₈×C₂₉₃	—	2	1	no	—
58	Q₈×C₂₉₃	—	4	1	no	—
59	Q₈×C₂₉₃	—	4	1	no	—
60	Q₈×C₂₉₃	Q₁₆	1	2	yes	complemented
61	C₂×C₄×C₂₉₃	—	2	1	no	—
62	C₂×C₄×C₂₉₃	—	2	1	no	—
63	C₂×C₄×C₂₉₃	—	4	1	no	—
64	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
67	C₈×C₂₉₃	—	4	1	no	—
68	C₂²×C₂₉₃	#120	1	1	yes	characteristic
69	C₄×C₂₉₃	—	2	2	no	—
70	C₄×C₂₉₃	—	2	1	no	—
71	C₄×C₂₉₃	—	4	1	no	—
72	C₄×C₂₉₃	—	8	1	no	—
73	C₄×C₂₉₃	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₂₉₃	#135	1	1	yes	characteristic
75	C₄×C₂₉₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₄×C₂₉₃	#137	1	1	yes	characteristic
77	Dic₂₉₃	—	4	1	no	—
78	Dic₂₉₃	—	4	2	no	—
79	Dic₂₉₃	—	4	1	no	—
80	C₂×C₂₉₃	#449	1	1	yes	characteristic
81	C₂×C₂₉₃	#451	1	1	yes	characteristic
82	C₂×C₂₉₃	#456	1	1	yes	characteristic
83	C₂₉₃	#1648	1	1	yes	characteristic, complemented, Sylow
84	#1648	—	293	1	no	maximal, Sylow
85	#557	—	293	1	no	—
86	#500	—	293	1	no	—
87	#475	D₂₉₃	1	1	yes	characteristic
88	#462	—	293	1	no	—
89	#332	—	293	1	no	—
90	Q₈⋊C₈	—	293	1	no	—
91	Q₈⋊C₈	—	293	1	no	—
92	C₂×Q₁₆	—	586	1	no	—
93	C₄⋊Q₈	—	293	1	no	—
94	C₄⋊Q₈	—	586	1	no	—
95	C₄⋊Q₈	C₂×D₂₉₃	1	1	yes	characteristic
96	#125	—	586	2	no	—
97	#125	—	586	1	no	—
98	C₄×Q₈	—	293	1	no	—
99	C₄×Q₈	—	586	1	no	—
100	C₄×Q₈	—	586	1	no	—
101	C₄×Q₈	C₂×D₂₉₃	1	1	yes	characteristic
102	#107	—	586	2	no	—
103	#106	—	2	1	no	—
104	C₄⋊C₈	—	293	1	no	—
105	C₄⋊C₈	C₂×D₂₉₃	1	1	yes	characteristic
106	#103	—	2	1	no	—
107	#103	—	586	1	no	—
108	C₄×C₈	—	293	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	293	1	no	—
111	C₂×Q₈	—	586	1	no	—
112	C₂×Q₈	C₂²⋊D₂₉₃	1	1	yes	characteristic
113	Q₁₆	—	1172	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	586	1	no	—
116	C₂×C₈	—	586	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	293	1	no	—
121	C₄⋊C₄	—	293	2	no	—
122	C₄⋊C₄	—	586	2	no	—
123	C₄⋊C₄	—	586	2	no	—
124	C₄⋊C₄	—	586	2	no	—
125	C₄⋊C₄	—	586	1	no	—
126	C₄⋊C₄	—	586	1	no	—
127	C₄⋊C₄	C₂²⋊D₂₉₃	1	1	yes	characteristic
128	C₄²	—	2	1	no	—
129	C₄²	—	586	2	no	—
130	C₄²	—	586	1	no	—
131	C₄²	C₂²×D₂₉₃	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	586	1	no	—
136	Q₈	—	586	1	no	—
137	Q₈	—	1172	1	no	—
138	Q₈	—	1172	1	no	—
139	Q₈	#17989792	1	2	yes	complemented
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	586	1	no	—
144	C₂×C₄	—	586	2	no	—
145	C₂×C₄	—	586	1	no	—
146	C₂×C₄	D₄×D₂₉₃	1	1	yes	characteristic
147	C₂×C₄	D₄×D₂₉₃	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	586	2	no	—
151	C₈	—	1172	1	no	—
152	C₂²	#22627592	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	8	1	no	—
157	C₄	—	1172	1	no	—
158	C₄	—	1172	2	no	—
159	C₄	—	1172	1	no	—
160	C₄	C₂×#17989792	1	1	yes	characteristic
161	C₄	#22627645	1	1	yes	characteristic
162	C₄	#22627651	1	1	yes	characteristic
163	C₄	#22627654	1	1	yes	characteristic
164	C₂	#34036273	1	1	yes	characteristic
165	C₂	#34036284	1	1	yes	characteristic
166	C₂	#34036310	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#34036858

C₂

yes

maximal, characteristic

#34036532

C₂

yes

maximal, characteristic

#34036336

C₂

yes

maximal, characteristic

#34035874

C₂

yes

maximal, characteristic

Q₈⋊#14556873

C₂

yes

maximal, characteristic

#34035859

C₂

yes

maximal, characteristic

#475×C₂₉₃

C₂

yes

maximal, characteristic

C₂×#17989792

—

#22627624

—

#22627624

C₂²

yes

characteristic

#22627615

—

#22627615

—

Q₈×Dic₂₉₃

—

C₄×Dic₅₈₆

—

C₄×Dic₅₈₆

C₂²

yes

characteristic

#22627557

—

C₄⋊#14556873

C₂²

yes

characteristic

#22627548

—

C₄×#14556873

C₂²

yes

characteristic

C₄⋊Q₈×C₂₉₃

C₂²

yes

characteristic

C₄×Q₈×C₂₉₃

C₂²

yes

characteristic

#106×C₂₉₃

—

C₄⋊C₈×C₂₉₃

C₂²

yes

characteristic

#103×C₂₉₃

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

D₄

yes

—

—

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

D₄

yes

characteristic

C₄⋊Dic₂₉₃

D₄

yes

—

#17989781

—

#17989781

—

C₄×Dic₂₉₃

—

C₄×Dic₂₉₃

—

C₂×Q₈×C₂₉₃

—

C₂×Q₈×C₂₉₃

D₄

yes

characteristic

C₂×C₈×C₂₉₃

—

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

D₄

yes

characteristic

C₄²×C₂₉₃

—

C₄²×C₂₉₃

C₂³

yes

characteristic

Dic₅₈₆

—

Dic₅₈₆

—

Dic₅₈₆

—

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

#14556873

—

#14556873

—

Q₈×C₂₉₃

—

Q₈×C₂₉₃

—

Q₈×C₂₉₃

—

Q₈×C₂₉₃

Q₁₆

yes

complemented

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₈×C₂₉₃

—

C₂²×C₂₉₃

#120

yes

characteristic

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

C₂×Q₁₆

yes

C₄×C₂₉₃

yes

C₄×C₂₉₃

yes

C₄×C₂₉₃

yes

Dic₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

—

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂₉₃

yes

characteristic, complemented, Sylow

#1648

—

293

maximal, Sylow

#557

—

293

—

#500

—

293

—

#475

D₂₉₃

yes

characteristic

#462

—

293

—

#332

—

293

—

Q₈⋊C₈

—

293

—

Q₈⋊C₈

—

293

—

C₂×Q₁₆

—

586

—

C₄⋊Q₈

—

293

—

C₄⋊Q₈

—

586

—

C₄⋊Q₈

C₂×D₂₉₃

yes

characteristic

#125

—

586

—

#125

—

586

—

C₄×Q₈

—

293

—

C₄×Q₈

—

586

—

100

C₄×Q₈

—

586

—

101

C₄×Q₈

C₂×D₂₉₃

yes

characteristic

102

#107

—

586

—

103

#106

—

104

C₄⋊C₈

—

293

—

105

C₄⋊C₈

C₂×D₂₉₃

yes

characteristic

106

#103

—

107

#103

—

586

—

108

C₄×C₈

—

293

—

109

C₂×Q₈

—

110

C₂×Q₈

—

293

—

111

C₂×Q₈

—

586

—

112

C₂×Q₈

C₂²⋊D₂₉₃

yes

characteristic

113

Q₁₆

—

1172

—

114

C₂×C₈

—

115

C₂×C₈

—

586

—

116

C₂×C₈

—

586

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

293

—

121

C₄⋊C₄

—

293

—

122

C₄⋊C₄

—

586

—

123

C₄⋊C₄

—

586

—

124

C₄⋊C₄

—

586

—

125

C₄⋊C₄

—

586

—

126

C₄⋊C₄

—

586

—

127

C₄⋊C₄

C₂²⋊D₂₉₃

yes

characteristic

128

C₄²

—

129

C₄²

—

586

—

130

C₄²

—

586

—

131

C₄²

C₂²×D₂₉₃

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

586

—

136

Q₈

—

586

—

137

Q₈

—

1172

—

138

Q₈

—

1172

—

139

Q₈

#17989792

yes

complemented

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

586

—

144

C₂×C₄

—

586

—

145

C₂×C₄

—

586

—

146

C₂×C₄

D₄×D₂₉₃

yes

characteristic

147

C₂×C₄

D₄×D₂₉₃

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

characteristic

149

C₈

—

150

C₈

—

586

—

151

C₈

—

1172

—

152

C₂²

#22627592

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

1172

—

158

C₄

—

1172

—

159

C₄

—

1172

—

160

C₄

C₂×#17989792

yes

characteristic

161

C₄

#22627645

yes

characteristic

162

C₄

#22627651

yes

characteristic

163

C₄

#22627654

yes

characteristic

164

C₂

#34036273

yes

characteristic

165

C₂

#34036284

yes

characteristic

166

C₂

#34036310

yes

characteristic

	subgroup	G/H
center	C₂²	#22627592
commutator, cosocle	C₄²×C₂₉₃	C₂³
Frattini	C₄²	C₂²×D₂₉₃
Fitting	#475×C₂₉₃	C₂
socle, prime radical	C₂²×C₂₉₃	#120
hypercenter	#475	D₂₉₃
hypocenter	C₂₉₃	#1648
LCS	G ▷ C₄²×C₂₉₃ ▷ C₂²×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#22627592

commutator, cosocle

C₄²×C₂₉₃

C₂³

Frattini

C₄²

C₂²×D₂₉₃

Fitting

#475×C₂₉₃

C₂

C₂²×C₂₉₃

D₂₉₃

C₂₉₃

G ▷ C₄²×C₂₉₃ ▷ C₂²×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
60. Q₈×C₂₉₃	Q₁₆	semidirect
83. C₂₉₃	#1648	semidirect
139. Q₈	#17989792	semidirect

H ≅ G/N

type

60. Q₈×C₂₉₃

Q₁₆

semidirect

83. C₂₉₃

#1648

semidirect

139. Q₈

#17989792

semidirect

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

subgroup	index
#34036858	2
#34036532	2
#34036336	2
#34035874	2
Q₈⋊#14556873	2
#34035859	2
#475×C₂₉₃	2
#1648	293

subgroup

index

#475×C₂₉₃

293

Subgroups — #49158026

Conjugacy classes of subgroups

Complements

Maximal subgroups