Group #49094544 — subgroups

Subgroups — #49094544

Order 35968.

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	#31824192	C₂	1	1	yes	maximal, characteristic
2	#31823866	C₂	1	1	yes	maximal, characteristic
3	#31823670	C₂	1	1	yes	maximal, characteristic
4	#31823208	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#14555886	C₂	1	1	yes	maximal, characteristic
6	#31823193	C₂	1	1	yes	maximal, characteristic
7	#475×C₂₈₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#17985128	—	2	1	no	—
9	#22582879	—	2	1	no	—
10	#22582879	C₂²	1	1	yes	characteristic
11	#22582870	—	2	2	no	—
12	#22582870	—	2	1	no	—
13	Q₈×Dic₂₈₁	—	2	1	no	—
14	C₄×Dic₅₆₂	—	2	1	no	—
15	C₄×Dic₅₆₂	C₂²	1	1	yes	characteristic
16	#22582812	—	2	2	no	—
17	C₄⋊#14555886	C₂²	1	1	yes	characteristic
18	#22582803	—	2	1	no	—
19	C₄×#14555886	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₂₈₁	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₂₈₁	C₂²	1	1	yes	characteristic
22	#106×C₂₈₁	—	2	1	no	—
23	C₄⋊C₈×C₂₈₁	C₂²	1	1	yes	characteristic
24	#103×C₂₈₁	—	2	1	no	—
25	C₂×Dic₅₆₂	—	2	1	no	—
26	C₂×Dic₅₆₂	D₄	1	1	yes	characteristic
27	#17985128	—	4	2	no	—
28	C₂×#14555886	—	2	1	no	—
29	C₂×#14555886	—	2	1	no	—
30	C₄⋊Dic₂₈₁	—	2	2	no	—
31	C₄⋊Dic₂₈₁	—	2	2	no	—
32	C₄⋊Dic₂₈₁	—	2	1	no	—
33	C₄⋊Dic₂₈₁	D₄	1	1	yes	characteristic
34	C₄⋊Dic₂₈₁	D₄	1	2	yes	—
35	#17985117	—	2	2	no	—
36	#17985117	—	2	1	no	—
37	C₄×Dic₂₈₁	—	2	2	no	—
38	C₄×Dic₂₈₁	—	2	1	no	—
39	C₂×Q₈×C₂₈₁	—	2	1	no	—
40	C₂×Q₈×C₂₈₁	D₄	1	1	yes	characteristic
41	C₂×C₈×C₂₈₁	—	2	1	no	—
42	C₄⋊C₄×C₂₈₁	—	2	1	no	—
43	C₄⋊C₄×C₂₈₁	—	2	1	no	—
44	C₄⋊C₄×C₂₈₁	—	2	1	no	—
45	C₄⋊C₄×C₂₈₁	D₄	1	1	yes	characteristic
46	C₄²×C₂₈₁	—	2	1	no	—
47	C₄²×C₂₈₁	C₂³	1	1	yes	characteristic
48	Dic₅₆₂	—	2	1	no	—
49	Dic₅₆₂	—	2	1	no	—
50	Dic₅₆₂	—	4	1	no	—
51	Dic₅₆₂	—	4	1	no	—
52	C₂×Dic₂₈₁	—	2	1	no	—
53	C₂×Dic₂₈₁	—	2	1	no	—
54	C₂×Dic₂₈₁	—	2	2	no	—
55	#14555886	—	2	2	no	—
56	#14555886	—	4	1	no	—
57	Q₈×C₂₈₁	—	2	1	no	—
58	Q₈×C₂₈₁	—	4	1	no	—
59	Q₈×C₂₈₁	—	4	1	no	—
60	Q₈×C₂₈₁	Q₁₆	1	2	yes	complemented
61	C₂×C₄×C₂₈₁	—	2	1	no	—
62	C₂×C₄×C₂₈₁	—	2	1	no	—
63	C₂×C₄×C₂₈₁	—	4	1	no	—
64	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
67	C₈×C₂₈₁	—	4	1	no	—
68	C₂²×C₂₈₁	#120	1	1	yes	characteristic
69	C₄×C₂₈₁	—	2	2	no	—
70	C₄×C₂₈₁	—	2	1	no	—
71	C₄×C₂₈₁	—	4	1	no	—
72	C₄×C₂₈₁	—	8	1	no	—
73	C₄×C₂₈₁	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₂₈₁	#135	1	1	yes	characteristic
75	C₄×C₂₈₁	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₄×C₂₈₁	#137	1	1	yes	characteristic
77	Dic₂₈₁	—	4	1	no	—
78	Dic₂₈₁	—	4	2	no	—
79	Dic₂₈₁	—	4	1	no	—
80	C₂×C₂₈₁	#449	1	1	yes	characteristic
81	C₂×C₂₈₁	#451	1	1	yes	characteristic
82	C₂×C₂₈₁	#456	1	1	yes	characteristic
83	C₂₈₁	#1648	1	1	yes	characteristic, complemented, Sylow
84	#1648	—	281	1	no	maximal, Sylow
85	#557	—	281	1	no	—
86	#500	—	281	1	no	—
87	#475	D₂₈₁	1	1	yes	characteristic
88	#462	—	281	1	no	—
89	#332	—	281	1	no	—
90	Q₈⋊C₈	—	281	1	no	—
91	Q₈⋊C₈	—	281	1	no	—
92	C₂×Q₁₆	—	562	1	no	—
93	C₄⋊Q₈	—	281	1	no	—
94	C₄⋊Q₈	—	562	1	no	—
95	C₄⋊Q₈	C₂×D₂₈₁	1	1	yes	characteristic
96	#125	—	562	2	no	—
97	#125	—	562	1	no	—
98	C₄×Q₈	—	281	1	no	—
99	C₄×Q₈	—	562	1	no	—
100	C₄×Q₈	—	562	1	no	—
101	C₄×Q₈	C₂×D₂₈₁	1	1	yes	characteristic
102	#107	—	562	2	no	—
103	#106	—	2	1	no	—
104	C₄⋊C₈	—	281	1	no	—
105	C₄⋊C₈	C₂×D₂₈₁	1	1	yes	characteristic
106	#103	—	2	1	no	—
107	#103	—	562	1	no	—
108	C₄×C₈	—	281	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	281	1	no	—
111	C₂×Q₈	—	562	1	no	—
112	C₂×Q₈	C₂²⋊D₂₈₁	1	1	yes	characteristic
113	Q₁₆	—	1124	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	562	1	no	—
116	C₂×C₈	—	562	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	281	1	no	—
121	C₄⋊C₄	—	281	2	no	—
122	C₄⋊C₄	—	562	2	no	—
123	C₄⋊C₄	—	562	2	no	—
124	C₄⋊C₄	—	562	2	no	—
125	C₄⋊C₄	—	562	1	no	—
126	C₄⋊C₄	—	562	1	no	—
127	C₄⋊C₄	C₂²⋊D₂₈₁	1	1	yes	characteristic
128	C₄²	—	2	1	no	—
129	C₄²	—	562	2	no	—
130	C₄²	—	562	1	no	—
131	C₄²	C₂²×D₂₈₁	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	562	1	no	—
136	Q₈	—	562	1	no	—
137	Q₈	—	1124	1	no	—
138	Q₈	—	1124	1	no	—
139	Q₈	#17985128	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	—	562	1	no	—
144	C₂×C₄	—	562	2	no	—
145	C₂×C₄	—	562	1	no	—
146	C₂×C₄	D₄×D₂₈₁	1	1	yes	characteristic
147	C₂×C₄	D₄×D₂₈₁	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	562	2	no	—
151	C₈	—	1124	1	no	—
152	C₂²	#22582847	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	—	1124	1	no	—
158	C₄	—	1124	1	no	—
159	C₄	—	1124	2	no	—
160	C₄	C₂×#17985128	1	1	yes	characteristic
161	C₄	#22582900	1	1	yes	characteristic
162	C₄	#22582906	1	1	yes	characteristic
163	C₄	#22582909	1	1	yes	characteristic
164	C₂	#31823607	1	1	yes	characteristic
165	C₂	#31823618	1	1	yes	characteristic
166	C₂	#31823644	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31824192

C₂

yes

maximal, characteristic

#31823866

C₂

yes

maximal, characteristic

#31823670

C₂

yes

maximal, characteristic

#31823208

C₂

yes

maximal, characteristic

Q₈⋊#14555886

C₂

yes

maximal, characteristic

#31823193

C₂

yes

maximal, characteristic

#475×C₂₈₁

C₂

yes

maximal, characteristic

C₂×#17985128

—

#22582879

—

#22582879

C₂²

yes

characteristic

#22582870

—

#22582870

—

Q₈×Dic₂₈₁

—

C₄×Dic₅₆₂

—

C₄×Dic₅₆₂

C₂²

yes

characteristic

#22582812

—

C₄⋊#14555886

C₂²

yes

characteristic

#22582803

—

C₄×#14555886

C₂²

yes

characteristic

C₄⋊Q₈×C₂₈₁

C₂²

yes

characteristic

C₄×Q₈×C₂₈₁

C₂²

yes

characteristic

#106×C₂₈₁

—

C₄⋊C₈×C₂₈₁

C₂²

yes

characteristic

#103×C₂₈₁

—

C₂×Dic₅₆₂

—

C₂×Dic₅₆₂

D₄

yes

—

—

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

D₄

yes

characteristic

C₄⋊Dic₂₈₁

D₄

yes

—

#17985117

—

#17985117

—

C₄×Dic₂₈₁

—

C₄×Dic₂₈₁

—

C₂×Q₈×C₂₈₁

—

C₂×Q₈×C₂₈₁

D₄

yes

characteristic

C₂×C₈×C₂₈₁

—

C₄⋊C₄×C₂₈₁

—

C₄⋊C₄×C₂₈₁

—

C₄⋊C₄×C₂₈₁

—

C₄⋊C₄×C₂₈₁

D₄

yes

characteristic

C₄²×C₂₈₁

—

C₄²×C₂₈₁

C₂³

yes

characteristic

Dic₅₆₂

—

Dic₅₆₂

—

Dic₅₆₂

—

Dic₅₆₂

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

#14555886

—

#14555886

—

Q₈×C₂₈₁

—

Q₈×C₂₈₁

—

Q₈×C₂₈₁

—

Q₈×C₂₈₁

Q₁₆

yes

complemented

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₈×C₂₈₁

—

C₂²×C₂₈₁

#120

yes

characteristic

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₂×Q₁₆

yes

C₄×C₂₈₁

yes

C₄×C₂₈₁

yes

C₄×C₂₈₁

yes

Dic₂₈₁

—

Dic₂₈₁

—

Dic₂₈₁

—

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂₈₁

yes

characteristic, complemented, Sylow

#1648

—

281

maximal, Sylow

#557

—

281

—

#500

—

281

—

#475

D₂₈₁

yes

characteristic

#462

—

281

—

#332

—

281

—

Q₈⋊C₈

—

281

—

Q₈⋊C₈

—

281

—

C₂×Q₁₆

—

562

—

C₄⋊Q₈

—

281

—

C₄⋊Q₈

—

562

—

C₄⋊Q₈

C₂×D₂₈₁

yes

characteristic

#125

—

562

—

#125

—

562

—

C₄×Q₈

—

281

—

C₄×Q₈

—

562

—

100

C₄×Q₈

—

562

—

101

C₄×Q₈

C₂×D₂₈₁

yes

characteristic

102

#107

—

562

—

103

#106

—

104

C₄⋊C₈

—

281

—

105

C₄⋊C₈

C₂×D₂₈₁

yes

characteristic

106

#103

—

107

#103

—

562

—

108

C₄×C₈

—

281

—

109

C₂×Q₈

—

110

C₂×Q₈

—

281

—

111

C₂×Q₈

—

562

—

112

C₂×Q₈

C₂²⋊D₂₈₁

yes

characteristic

113

Q₁₆

—

1124

—

114

C₂×C₈

—

115

C₂×C₈

—

562

—

116

C₂×C₈

—

562

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

281

—

121

C₄⋊C₄

—

281

—

122

C₄⋊C₄

—

562

—

123

C₄⋊C₄

—

562

—

124

C₄⋊C₄

—

562

—

125

C₄⋊C₄

—

562

—

126

C₄⋊C₄

—

562

—

127

C₄⋊C₄

C₂²⋊D₂₈₁

yes

characteristic

128

C₄²

—

129

C₄²

—

562

—

130

C₄²

—

562

—

131

C₄²

C₂²×D₂₈₁

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

562

—

136

Q₈

—

562

—

137

Q₈

—

1124

—

138

Q₈

—

1124

—

139

Q₈

#17985128

yes

complemented

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

562

—

144

C₂×C₄

—

562

—

145

C₂×C₄

—

562

—

146

C₂×C₄

D₄×D₂₈₁

yes

characteristic

147

C₂×C₄

D₄×D₂₈₁

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

characteristic

149

C₈

—

150

C₈

—

562

—

151

C₈

—

1124

—

152

C₂²

#22582847

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

1124

—

158

C₄

—

1124

—

159

C₄

—

1124

—

160

C₄

C₂×#17985128

yes

characteristic

161

C₄

#22582900

yes

characteristic

162

C₄

#22582906

yes

characteristic

163

C₄

#22582909

yes

characteristic

164

C₂

#31823607

yes

characteristic

165

C₂

#31823618

yes

characteristic

166

C₂

#31823644

yes

characteristic

	subgroup	G/H
center	C₂²	#22582847
commutator, cosocle	C₄²×C₂₈₁	C₂³
Frattini	C₄²	C₂²×D₂₈₁
Fitting	#475×C₂₈₁	C₂
socle, prime radical	C₂²×C₂₈₁	#120
hypercenter	#475	D₂₈₁
hypocenter	C₂₈₁	#1648
LCS	G ▷ C₄²×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#22582847

commutator, cosocle

C₄²×C₂₈₁

C₂³

Frattini

C₄²

C₂²×D₂₈₁

Fitting

#475×C₂₈₁

C₂

C₂²×C₂₈₁

D₂₈₁

C₂₈₁

G ▷ C₄²×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
60. Q₈×C₂₈₁	Q₁₆	semidirect
83. C₂₈₁	#1648	semidirect
139. Q₈	#17985128	semidirect

H ≅ G/N

type

60. Q₈×C₂₈₁

Q₁₆

semidirect

83. C₂₈₁

#1648

semidirect

139. Q₈

#17985128

semidirect

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

subgroup	index
#31824192	2
#31823866	2
#31823670	2
#31823208	2
Q₈⋊#14555886	2
#31823193	2
#475×C₂₈₁	2
#1648	281

subgroup

index

#475×C₂₈₁

281

Subgroups — #49094544

Conjugacy classes of subgroups

Complements

Maximal subgroups