Group #47983157 — subgroups

Subgroups — #47983157

Order 35456.

336 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 95 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×D₄×Dic₂₇₇	C₂	1	1	yes	maximal, characteristic
2	#31800025	C₂	1	2	yes	maximal, complemented
3	C₂×#22579797	C₂	1	1	yes	maximal, characteristic
4	#31799770	C₂	1	2	yes	maximal, complemented
5	C₂×#100×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₂₇₇	C₄	1	2	yes	—
7	C₂×#17984547	C₂²	1	2	yes	—
8	D₄×Dic₂₇₇	—	2	2	no	—
9	D₄×Dic₂₇₇	—	2	2	no	—
10	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Dic₂₇₇	C₄	1	2	yes	—
12	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
13	#22579809	—	2	2	no	—
14	#22579809	—	2	2	no	—
15	#22579797	—	2	2	no	—
16	C₂²×D₄×C₂₇₇	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₂₇₇	C₂²	1	2	yes	—
18	#100×C₂₇₇	—	2	2	no	—
19	C₂²×Dic₂₇₇	—	2	2	no	—
20	C₂²×Dic₂₇₇	—	2	2	no	—
21	C₂²×Dic₂₇₇	—	4	4	no	—
22	C₂²×Dic₂₇₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₂₇₇	C₂×C₄	1	2	yes	—
24	C₂²×Dic₂₇₇	C₂×C₄	1	1	yes	characteristic
25	#17984547	—	2	2	no	—
26	#17984547	—	2	2	no	—
27	C₂×#14555660	—	2	2	no	—
28	C₄⋊Dic₂₇₇	—	2	2	no	—
29	C₂²⋊Dic₂₇₇	—	2	4	no	—
30	C₄×Dic₂₇₇	D₄	1	2	yes	—
31	C₄×Dic₂₇₇	D₄	1	2	yes	—
32	C₂⁴×C₂₇₇	C₂×C₄	1	2	yes	—
33	C₂×D₄×C₂₇₇	—	2	2	no	—
34	C₂×D₄×C₂₇₇	—	2	2	no	—
35	C₂×D₄×C₂₇₇	D₄	1	2	yes	—
36	C₂×D₄×C₂₇₇	Q₈	1	2	yes	—
37	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
38	M₁₆×C₂₇₇	—	2	2	no	—
39	M₁₆×C₂₇₇	—	2	2	no	—
40	C₂×C₈×C₂₇₇	—	2	2	no	—
41	C₂×Dic₂₇₇	—	2	2	no	—
42	C₂×Dic₂₇₇	—	2	2	no	—
43	C₂×Dic₂₇₇	—	4	1	no	—
44	C₂×Dic₂₇₇	—	4	4	no	—
45	C₂×Dic₂₇₇	—	4	4	no	—
46	C₂×Dic₂₇₇	—	4	2	no	—
47	C₂×Dic₂₇₇	C₂²⋊C₄	1	2	yes	—
48	C₂×Dic₂₇₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Dic₂₇₇	C₂²⋊C₄	1	1	yes	characteristic
50	#14555660	—	4	2	no	—
51	C₂³×C₂₇₇	—	2	2	no	—
52	C₂³×C₂₇₇	—	2	2	no	—
53	C₂³×C₂₇₇	—	4	2	no	—
54	C₂³×C₂₇₇	—	4	2	no	—
55	C₂³×C₂₇₇	C₄⋊C₄	1	4	yes	—
56	C₂³×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
57	D₄×C₂₇₇	—	2	4	no	—
58	D₄×C₂₇₇	—	2	4	no	—
59	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₂₇₇	#41	1	2	yes	—
63	C₈×C₂₇₇	—	4	2	no	—
64	C₂²×C₂₇₇	—	2	1	no	—
65	C₂²×C₂₇₇	—	2	1	no	—
66	C₂²×C₂₇₇	—	2	4	no	—
67	C₂²×C₂₇₇	—	4	4	no	—
68	C₂²×C₂₇₇	—	4	4	no	—
69	C₂²×C₂₇₇	—	4	2	no	—
70	C₂²×C₂₇₇	—	4	2	no	—
71	C₂²×C₂₇₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₂₇₇	#117	1	1	yes	characteristic
74	C₄×C₂₇₇	#121	1	2	yes	—
75	C₄×C₂₇₇	C₂²⋊Q₈	1	2	yes	—
76	Dic₂₇₇	—	2	1	no	—
77	Dic₂₇₇	—	2	2	no	—
78	Dic₂₇₇	—	4	2	no	—
79	Dic₂₇₇	#100	1	2	yes	complemented
80	C₂×C₂₇₇	—	2	1	no	—
81	C₂×C₂₇₇	—	2	1	no	—
82	C₂×C₂₇₇	—	4	4	no	—
83	C₂×C₂₇₇	#380	1	1	yes	characteristic
84	C₂×C₂₇₇	C₂×#100	1	1	yes	characteristic
85	C₂×C₂₇₇	#413	1	1	yes	characteristic
86	C₂₇₇	#1800	1	1	yes	characteristic, complemented, Sylow
87	#1800	—	277	1	no	maximal, Sylow
88	C₂×C₄×D₄	—	277	1	no	—
89	#423	—	277	2	no	—
90	C₂×#100	—	277	1	no	—
91	C₂×#100	D₂₇₇	1	1	yes	characteristic
92	#343	—	277	2	no	—
93	C₂²×D₄	C₂×D₂₇₇	1	1	yes	characteristic
94	C₂³×C₄	—	277	2	no	—
95	C₂×M₁₆	—	277	2	no	—
96	C₂×M₁₆	C₂×D₂₇₇	1	2	yes	—
97	C₄×D₄	—	554	2	no	—
98	C₄×D₄	—	554	2	no	—
99	C₂×C₄⋊C₄	—	277	1	no	—
100	C₂×C₂²⋊C₄	—	277	2	no	—
101	C₂×C₄²	—	277	1	no	—
102	C₄⋊C₈	—	554	2	no	—
103	C₄⋊C₈	—	554	2	no	—
104	#100	—	2	2	no	—
105	#100	—	554	2	no	—
106	C₂⁴	C₄×D₂₇₇	1	2	yes	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	C₂²⋊D₂₇₇	1	2	yes	—
110	C₂×D₄	Dic₅₅₄	1	2	yes	—
111	C₂²×C₄	—	277	1	no	—
112	C₂²×C₄	—	277	1	no	—
113	C₂²×C₄	—	277	2	no	—
114	C₂²×C₄	—	554	2	no	—
115	C₂²×C₄	—	554	2	no	—
116	C₂²×C₄	—	1108	4	no	—
117	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
118	M₁₆	—	2	2	no	—
119	M₁₆	—	2	2	no	—
120	M₁₆	—	554	2	no	—
121	M₁₆	—	554	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	554	2	no	—
124	C₄⋊C₄	—	554	2	no	—
125	C₂²⋊C₄	—	554	4	no	—
126	C₄²	—	277	2	no	—
127	C₄²	—	277	2	no	—
128	C₂³	—	2	2	no	—
129	C₂³	—	2	2	no	—
130	C₂³	—	4	2	no	—
131	C₂³	—	4	2	no	—
132	C₂³	#17984542	1	4	yes	—
133	C₂³	C₂×C₄×D₂₇₇	1	1	yes	characteristic
134	D₄	—	2	4	no	—
135	D₄	—	2	4	no	—
136	C₂×C₄	—	277	1	no	—
137	C₂×C₄	—	277	2	no	—
138	C₂×C₄	—	277	1	no	—
139	C₂×C₄	—	554	2	no	—
140	C₂×C₄	—	554	2	no	—
141	C₂×C₄	—	1108	4	no	—
142	C₂×C₄	—	1108	1	no	—
143	C₂×C₄	—	1108	4	no	—
144	C₂×C₄	—	1108	2	no	—
145	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
147	C₂×C₄	C₂×Dic₅₅₄	1	1	yes	characteristic
148	C₂×C₄	#17984562	1	2	yes	—
149	C₈	—	4	2	no	—
150	C₈	—	1108	2	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	4	4	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	4	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	C₂²⋊C₄×D₂₇₇	1	1	yes	characteristic
159	C₂²	C₂×#17984542	1	1	yes	characteristic
160	C₂²	#22579836	1	1	yes	characteristic
161	C₄	—	277	2	no	—
162	C₄	—	554	1	no	—
163	C₄	—	554	2	no	—
164	C₄	—	1108	2	no	—
165	C₄	#22579852	1	2	yes	—
166	C₄	#22579857	1	2	yes	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	4	4	no	—
170	C₂	#31799860	1	1	yes	characteristic
171	C₂	#100×D₂₇₇	1	1	yes	characteristic
172	C₂	#31799983	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×Dic₂₇₇

C₂

yes

maximal, characteristic

#31800025

C₂

yes

maximal, complemented

C₂×#22579797

C₂

yes

maximal, characteristic

#31799770

C₂

yes

maximal, complemented

C₂×#100×C₂₇₇

C₂

yes

maximal, characteristic

C₂³×Dic₂₇₇

C₄

yes

—

C₂×#17984547

C₂²

yes

—

D₄×Dic₂₇₇

—

D₄×Dic₂₇₇

—

C₂×C₄⋊Dic₂₇₇

C₂²

yes

characteristic

C₂×C₂²⋊Dic₂₇₇

C₄

yes

—

C₂×C₄×Dic₂₇₇

C₂²

yes

—

—

—

C₂²×D₄×C₂₇₇

C₂²

yes

characteristic

C₂×M₁₆×C₂₇₇

C₂²

yes

—

#100×C₂₇₇

—

C₂²×Dic₂₇₇

—

C₂²×Dic₂₇₇

—

C₂²×Dic₂₇₇

—

C₂²×Dic₂₇₇

C₂×C₄

yes

characteristic

C₂²×Dic₂₇₇

C₂×C₄

yes

—

C₂²×Dic₂₇₇

C₂×C₄

yes

—

—

—

C₄⋊Dic₂₇₇

—

C₂²⋊Dic₂₇₇

—

C₄×Dic₂₇₇

D₄

yes

—

C₄×Dic₂₇₇

D₄

yes

—

C₂⁴×C₂₇₇

C₂×C₄

yes

—

C₂×D₄×C₂₇₇

—

C₂×D₄×C₂₇₇

—

C₂×D₄×C₂₇₇

D₄

yes

—

C₂×D₄×C₂₇₇

Q₈

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

M₁₆×C₂₇₇

—

M₁₆×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

C₂²⋊C₄

yes

—

C₂×Dic₂₇₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₇₇

C₂²⋊C₄

yes

characteristic

#14555660

—

C₂³×C₂₇₇

—

C₂³×C₂₇₇

—

C₂³×C₂₇₇

—

C₂³×C₂₇₇

—

C₂³×C₂₇₇

C₄⋊C₄

yes

—

C₂³×C₂₇₇

C₂²×C₄

yes

characteristic

D₄×C₂₇₇

—

D₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₇

#41

yes

—

C₈×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₇₇

C₂×C₄⋊C₄

yes

C₂²×C₂₇₇

yes

C₄×C₂₇₇

yes

—

C₄×C₂₇₇

C₂²⋊Q₈

yes

—

Dic₂₇₇

—

Dic₂₇₇

—

Dic₂₇₇

—

Dic₂₇₇

#100

yes

complemented

C₂×C₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂₇₇

yes

characteristic, complemented, Sylow

#1800

—

277

maximal, Sylow

C₂×C₄×D₄

—

277

—

#423

—

277

—

C₂×#100

—

277

—

C₂×#100

D₂₇₇

yes

characteristic

#343

—

277

—

C₂²×D₄

C₂×D₂₇₇

yes

characteristic

C₂³×C₄

—

277

—

C₂×M₁₆

—

277

—

C₂×M₁₆

C₂×D₂₇₇

yes

—

C₄×D₄

—

554

—

C₄×D₄

—

554

—

C₂×C₄⋊C₄

—

277

—

100

C₂×C₂²⋊C₄

—

277

—

101

C₂×C₄²

—

277

—

102

C₄⋊C₈

—

554

—

103

C₄⋊C₈

—

554

—

104

#100

—

105

#100

—

554

—

106

C₂⁴

C₄×D₂₇₇

yes

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

C₂²⋊D₂₇₇

yes

—

110

C₂×D₄

Dic₅₅₄

yes

—

111

C₂²×C₄

—

277

—

112

C₂²×C₄

—

277

—

113

C₂²×C₄

—

277

—

114

C₂²×C₄

—

554

—

115

C₂²×C₄

—

554

—

116

C₂²×C₄

—

1108

—

117

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

118

M₁₆

—

119

M₁₆

—

120

M₁₆

—

554

—

121

M₁₆

—

554

—

122

C₂×C₈

—

123

C₂×C₈

—

554

—

124

C₄⋊C₄

—

554

—

125

C₂²⋊C₄

—

554

—

126

C₄²

—

277

—

127

C₄²

—

277

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

#17984542

yes

—

133

C₂³

C₂×C₄×D₂₇₇

yes

characteristic

134

D₄

—

135

D₄

—

136

C₂×C₄

—

277

—

137

C₂×C₄

—

277

—

138

C₂×C₄

—

277

—

139

C₂×C₄

—

554

—

140

C₂×C₄

—

554

—

141

C₂×C₄

—

1108

—

142

C₂×C₄

—

1108

—

143

C₂×C₄

—

1108

—

144

C₂×C₄

—

1108

—

145

C₂×C₄

D₄×D₂₇₇

yes

—

146

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

147

C₂×C₄

C₂×Dic₅₅₄

yes

characteristic

148

C₂×C₄

#17984562

yes

—

149

C₈

—

150

C₈

—

1108

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

C₂²⋊C₄×D₂₇₇

yes

characteristic

159

C₂²

C₂×#17984542

yes

characteristic

160

C₂²

#22579836

yes

characteristic

161

C₄

—

277

—

162

C₄

—

554

—

163

C₄

—

554

—

164

C₄

—

1108

—

165

C₄

#22579852

yes

—

166

C₄

#22579857

yes

—

167

C₂

—

168

C₂

—

169

C₂

—

170

C₂

#31799860

yes

characteristic

171

C₂

#100×D₂₇₇

yes

characteristic

172

C₂

#31799983

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₂₇₇
commutator	C₂³×C₂₇₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	C₂×#100×C₂₇₇	C₂
socle	C₂²×C₂₇₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₂₇₇	C₂²
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	C₂×#100	D₂₇₇
hypocenter	C₂₇₇	#1800
LCS	G ▷ C₂³×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₂₇₇

commutator

C₂³×C₂₇₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

C₂×#100×C₂₇₇

C₂

socle

C₂²×C₂₇₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₄×C₂₇₇

C₂²

cosocle

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₂³×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

N	H ≅ G/N	type
2. #31800025	C₂	semidirect
4. #31799770	C₂	semidirect
79. Dic₂₇₇	#100	semidirect
86. C₂₇₇	#1800	semidirect

H ≅ G/N

type

2. #31800025

C₂

semidirect

4. #31799770

C₂

semidirect

79. Dic₂₇₇

#100

semidirect

86. C₂₇₇

#1800

semidirect

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

subgroup	index
C₂×D₄×Dic₂₇₇	2
#31800025	2
C₂×#22579797	2
#31799770	2
C₂×#100×C₂₇₇	2
#1800	277

subgroup

index

C₂×D₄×Dic₂₇₇

C₂×#100×C₂₇₇

277

Subgroups — #47983157

Conjugacy classes of subgroups

Complements

Maximal subgroups