Group #47984309 — subgroups

Subgroups — #47984309

Order 35456.

290 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 91 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×#22579880	C₂	1	1	yes	maximal, characteristic
2	C₂×#22579805	C₂	1	2	yes	maximal
3	#31799908	C₂	1	1	yes	maximal, characteristic
4	#31799765	C₂	1	2	yes	maximal
5	#336×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₅₄	—	2	1	no	—
7	C₂²×Dic₅₅₄	C₂²	1	1	yes	characteristic
8	#22579880	—	2	4	no	—
9	C₂×C₄⋊Dic₂₇₇	—	2	1	no	—
10	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₂₇₇	C₂²	1	2	yes	—
12	#22579805	—	2	4	no	—
13	#22579787	—	2	1	no	—
14	#22579787	—	2	1	no	—
15	C₂²×C₈×C₂₇₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₅₄	—	2	4	no	—
18	C₂×Dic₅₅₄	—	2	2	no	—
19	C₂×Dic₅₅₄	—	2	1	no	—
20	C₂×Dic₅₅₄	—	2	2	no	—
21	C₂×Dic₅₅₄	—	4	2	no	—
22	C₂×Dic₅₅₄	—	4	2	no	—
23	C₂×Dic₅₅₄	Q₈	1	2	yes	complemented
24	C₂²×Dic₂₇₇	—	2	1	no	—
25	C₂²×Dic₂₇₇	—	2	2	no	—
26	C₂²×Dic₂₇₇	—	2	1	no	—
27	C₄⋊Dic₂₇₇	—	2	4	no	—
28	C₄⋊Dic₂₇₇	—	2	2	no	—
29	C₄⋊Dic₂₇₇	—	2	2	no	—
30	C₄⋊Dic₂₇₇	D₄	1	4	yes	—
31	C₂²×C₄×C₂₇₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₇₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₂₇₇	—	2	2	no	—
35	C₂×C₈×C₂₇₇	—	2	2	no	—
36	C₂×C₈×C₂₇₇	—	2	2	no	—
37	C₄²×C₂₇₇	—	2	2	no	—
38	Dic₅₅₄	—	2	4	no	—
39	Dic₅₅₄	—	4	8	no	—
40	Dic₅₅₄	—	4	2	no	—
41	C₂×Dic₂₇₇	—	2	4	no	—
42	C₂×Dic₂₇₇	—	2	2	no	—
43	C₂×Dic₂₇₇	—	4	1	no	—
44	C₂×Dic₂₇₇	—	4	1	no	—
45	C₂×Dic₂₇₇	—	4	2	no	—
46	C₂×Dic₂₇₇	—	4	2	no	—
47	C₂×Dic₂₇₇	—	4	1	no	—
48	C₂×Dic₂₇₇	—	4	1	no	—
49	C₂×Dic₂₇₇	—	4	1	no	—
50	C₂³×C₂₇₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₇₇	—	2	1	no	—
52	C₂×C₄×C₂₇₇	—	2	1	no	—
53	C₂×C₄×C₂₇₇	—	2	1	no	—
54	C₂×C₄×C₂₇₇	—	2	1	no	—
55	C₂×C₄×C₂₇₇	SD₁₆	1	2	yes	—
56	C₂×C₄×C₂₇₇	Q₁₆	1	2	yes	—
57	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₂₇₇	#41	1	1	yes	characteristic
60	C₂×C₄×C₂₇₇	#41	1	2	yes	—
61	C₈×C₂₇₇	—	4	2	no	—
62	C₂²×C₂₇₇	#120	1	1	yes	characteristic
63	C₂²×C₂₇₇	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₂₇₇	#124	1	1	yes	characteristic
65	C₂²×C₂₇₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₂₇₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂²×C₂₇₇	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂²×C₂₇₇	#137	1	1	yes	characteristic
69	C₄×C₂₇₇	—	2	2	no	—
70	C₄×C₂₇₇	—	2	2	no	—
71	C₄×C₂₇₇	C₂²⋊Q₈	1	2	yes	—
72	C₄×C₂₇₇	#124	1	2	yes	—
73	Dic₂₇₇	—	4	2	no	—
74	Dic₂₇₇	—	4	4	no	—
75	Dic₂₇₇	—	8	1	no	—
76	C₂×C₂₇₇	#393	1	1	yes	characteristic
77	C₂×C₂₇₇	#450	1	1	yes	characteristic
78	C₂×C₂₇₇	#451	1	1	yes	characteristic
79	C₂×C₂₇₇	#475	1	1	yes	characteristic
80	C₂×C₂₇₇	#477	1	1	yes	characteristic
81	C₂×C₂₇₇	#487	1	1	yes	characteristic
82	C₂×C₂₇₇	#489	1	1	yes	characteristic
83	C₂₇₇	#2024	1	1	yes	characteristic, complemented, Sylow
84	#2024	—	277	1	no	maximal, Sylow
85	C₂×C₄⋊Q₈	—	277	1	no	—
86	C₂×#103	—	277	2	no	—
87	#391	—	277	1	no	—
88	#340	—	277	2	no	—
89	#336	D₂₇₇	1	1	yes	characteristic
90	C₂²×Q₈	—	277	1	no	—
91	C₂²×Q₈	—	554	1	no	—
92	C₂²×C₈	C₂×D₂₇₇	1	2	yes	—
93	C₄⋊Q₈	—	554	4	no	—
94	C₂×C₄⋊C₄	—	277	2	no	—
95	C₂×C₄⋊C₄	—	277	1	no	—
96	C₂×C₄⋊C₄	—	554	1	no	—
97	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
98	#103	—	554	4	no	—
99	#95	—	554	1	no	—
100	#95	—	554	1	no	—
101	C₂×Q₈	—	277	2	no	—
102	C₂×Q₈	—	554	4	no	—
103	C₂×Q₈	—	554	1	no	—
104	C₂×Q₈	—	554	2	no	—
105	C₂×Q₈	—	554	2	no	—
106	C₂×Q₈	—	1108	2	no	—
107	C₂×Q₈	—	1108	2	no	—
108	C₂²×C₄	—	554	1	no	—
109	C₂²×C₄	—	554	2	no	—
110	C₂²×C₄	—	554	1	no	—
111	C₂²×C₄	D₁₁₀₈	1	1	yes	characteristic
112	C₂²×C₄	D₁₁₀₈	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₄⋊C₄	—	277	4	no	—
118	C₄⋊C₄	—	554	4	no	—
119	C₄⋊C₄	—	554	2	no	—
120	C₄⋊C₄	—	554	2	no	—
121	C₄²	—	2	2	no	—
122	C₂³	C₂×D₁₁₀₈	1	1	yes	characteristic
123	Q₈	—	554	4	no	—
124	Q₈	—	1108	8	no	—
125	Q₈	—	1108	2	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	—	554	4	no	—
131	C₂×C₄	—	554	2	no	—
132	C₂×C₄	—	1108	1	no	—
133	C₂×C₄	—	1108	1	no	—
134	C₂×C₄	—	1108	2	no	—
135	C₂×C₄	—	1108	2	no	—
136	C₂×C₄	—	1108	1	no	—
137	C₂×C₄	—	1108	1	no	—
138	C₂×C₄	—	1108	1	no	—
139	C₂×C₄	#17984552	1	2	yes	—
140	C₂×C₄	Dic₁₁₀₈	1	2	yes	—
141	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
142	C₂×C₄	Q₈×D₂₇₇	1	1	yes	characteristic
143	C₂×C₄	#17984562	1	1	yes	characteristic
144	C₂×C₄	#17984563	1	2	yes	—
145	C₈	—	4	2	no	—
146	C₂²	#22579850	1	1	yes	characteristic
147	C₂²	#22579860	1	1	yes	characteristic
148	C₂²	#22579868	1	1	yes	characteristic
149	C₂²	C₂×#17984552	1	1	yes	characteristic
150	C₂²	C₂×Dic₁₁₀₈	1	1	yes	characteristic
151	C₂²	#22579908	1	1	yes	characteristic
152	C₂²	#22579913	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	1108	4	no	—
156	C₄	—	1108	2	no	—
157	C₄	—	2216	1	no	—
158	C₄	#22579858	1	2	yes	—
159	C₄	#22579866	1	2	yes	—
160	C₂	#31799915	1	1	yes	characteristic
161	C₂	#31800155	1	1	yes	characteristic
162	C₂	#31800160	1	1	yes	characteristic
163	C₂	#31800288	1	1	yes	characteristic
164	C₂	#31800298	1	1	yes	characteristic
165	C₂	#31800353	1	1	yes	characteristic
166	C₂	#31800362	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22579880

C₂

yes

maximal, characteristic

C₂×#22579805

C₂

yes

maximal

#31799908

C₂

yes

maximal, characteristic

#31799765

C₂

yes

maximal

#336×C₂₇₇

C₂

yes

maximal, characteristic

C₂²×Dic₅₅₄

—

C₂²×Dic₅₅₄

C₂²

yes

characteristic

#22579880

—

C₂×C₄⋊Dic₂₇₇

—

C₂×C₄⋊Dic₂₇₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₇₇

C₂²

yes

—

#22579805

—

#22579787

—

#22579787

—

C₂²×C₈×C₂₇₇

C₂²

yes

—

C₂×C₄²×C₂₇₇

C₂²

yes

characteristic

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

Q₈

yes

complemented

C₂²×Dic₂₇₇

—

C₂²×Dic₂₇₇

—

C₂²×Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

D₄

yes

—

C₂²×C₄×C₂₇₇

D₄

yes

characteristic

C₂²×C₄×C₂₇₇

D₄

yes

characteristic

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₄²×C₂₇₇

—

Dic₅₅₄

—

Dic₅₅₄

—

Dic₅₅₄

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂³×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

SD₁₆

yes

—

C₂×C₄×C₂₇₇

Q₁₆

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×Q₈

yes

C₂×C₄×C₂₇₇

yes

C₂×C₄×C₂₇₇

yes

—

C₈×C₂₇₇

—

C₂²×C₂₇₇

#120

yes

characteristic

C₂²×C₂₇₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₇₇

#124

yes

characteristic

C₂²×C₂₇₇

C₂×SD₁₆

yes

characteristic

C₂²×C₂₇₇

C₂×Q₁₆

yes

C₂²×C₂₇₇

yes

C₂²×C₂₇₇

yes

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₂²⋊Q₈

yes

—

C₄×C₂₇₇

#124

yes

—

Dic₂₇₇

—

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂₇₇

yes

characteristic, complemented, Sylow

#2024

—

277

maximal, Sylow

C₂×C₄⋊Q₈

—

277

—

C₂×#103

—

277

—

#391

—

277

—

#340

—

277

—

#336

D₂₇₇

yes

characteristic

C₂²×Q₈

—

277

—

C₂²×Q₈

—

554

—

C₂²×C₈

C₂×D₂₇₇

yes

—

C₄⋊Q₈

—

554

—

C₂×C₄⋊C₄

—

277

—

C₂×C₄⋊C₄

—

277

—

C₂×C₄⋊C₄

—

554

—

C₂×C₄²

C₂×D₂₇₇

yes

characteristic

#103

—

554

—

#95

—

554

—

100

#95

—

554

—

101

C₂×Q₈

—

277

—

102

C₂×Q₈

—

554

—

103

C₂×Q₈

—

554

—

104

C₂×Q₈

—

554

—

105

C₂×Q₈

—

554

—

106

C₂×Q₈

—

1108

—

107

C₂×Q₈

—

1108

—

108

C₂²×C₄

—

554

—

109

C₂²×C₄

—

554

—

110

C₂²×C₄

—

554

—

111

C₂²×C₄

D₁₁₀₈

yes

characteristic

112

C₂²×C₄

D₁₁₀₈

yes

characteristic

113

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

277

—

118

C₄⋊C₄

—

554

—

119

C₄⋊C₄

—

554

—

120

C₄⋊C₄

—

554

—

121

C₄²

—

122

C₂³

C₂×D₁₁₀₈

yes

characteristic

123

Q₈

—

554

—

124

Q₈

—

1108

—

125

Q₈

—

1108

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

554

—

131

C₂×C₄

—

554

—

132

C₂×C₄

—

1108

—

133

C₂×C₄

—

1108

—

134

C₂×C₄

—

1108

—

135

C₂×C₄

—

1108

—

136

C₂×C₄

—

1108

—

137

C₂×C₄

—

1108

—

138

C₂×C₄

—

1108

—

139

C₂×C₄

#17984552

yes

—

140

C₂×C₄

Dic₁₁₀₈

yes

—

141

C₂×C₄

D₄×D₂₇₇

yes

—

142

C₂×C₄

Q₈×D₂₇₇

yes

characteristic

143

C₂×C₄

#17984562

yes

characteristic

144

C₂×C₄

#17984563

yes

—

145

C₈

—

146

C₂²

#22579850

yes

characteristic

147

C₂²

#22579860

yes

characteristic

148

C₂²

#22579868

yes

characteristic

149

C₂²

C₂×#17984552

yes

characteristic

150

C₂²

C₂×Dic₁₁₀₈

yes

characteristic

151

C₂²

#22579908

yes

characteristic

152

C₂²

#22579913

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

1108

—

156

C₄

—

1108

—

157

C₄

—

2216

—

158

C₄

#22579858

yes

—

159

C₄

#22579866

yes

—

160

C₂

#31799915

yes

characteristic

161

C₂

#31800155

yes

characteristic

162

C₂

#31800160

yes

characteristic

163

C₂

#31800288

yes

characteristic

164

C₂

#31800298

yes

characteristic

165

C₂

#31800353

yes

characteristic

166

C₂

#31800362

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₁₀₈
commutator, cosocle	C₂²×C₄×C₂₇₇	C₂³
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	#336×C₂₇₇	C₂
socle, prime radical	C₂³×C₂₇₇	C₂×D₄
hypercenter	#336	D₂₇₇
hypocenter	C₂₇₇	#2024
LCS	G ▷ C₂²×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₁₁₀₈

commutator, cosocle

C₂²×C₄×C₂₇₇

C₂³

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

#336×C₂₇₇

C₂

socle, prime radical

C₂³×C₂₇₇

C₂×D₄

D₂₇₇

C₂₇₇

G ▷ C₂²×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
23. C₂×Dic₅₅₄	Q₈	semidirect
83. C₂₇₇	#2024	semidirect

H ≅ G/N

type

23. C₂×Dic₅₅₄

Q₈

semidirect

83. C₂₇₇

#2024

semidirect

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

subgroup	index
C₂×#22579880	2
C₂×#22579805	2
#31799908	2
#31799765	2
#336×C₂₇₇	2
#2024	277

subgroup

index

#336×C₂₇₇

277

Subgroups — #47984309

Conjugacy classes of subgroups

Complements

Maximal subgroups