Group #47983570 — subgroups

Subgroups — #47983570

Order 35456.

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

normal subgroups only

This group has 344 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×#22579869	C₂	1	1	yes	maximal, characteristic
2	#31800006	C₂	1	2	yes	maximal
3	C₂×#22579799	C₂	1	1	yes	maximal, characteristic, complemented
4	#31799761	C₂	1	2	yes	maximal, complemented
5	#519×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₁₀₈	C₂²	1	1	yes	characteristic
7	C₂×#17984547	C₂²	1	2	yes	—
8	#22579869	—	2	2	no	—
9	#22579869	C₄	1	4	yes	complemented
10	C₂×#17984540	—	2	2	no	—
11	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
12	#22579800	—	2	4	no	—
13	#22579799	—	2	2	no	—
14	C₂²×Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₂₇₇	—	2	4	no	—
16	#117×C₂₇₇	—	2	2	no	—
17	#117×C₂₇₇	C₂²	1	2	yes	—
18	C₂³×D₂₇₇	—	2	1	no	—
19	C₂×D₁₁₀₈	—	2	2	no	—
20	C₂×D₁₁₀₈	—	2	1	no	—
21	C₂×D₁₁₀₈	—	2	2	no	—
22	C₂×D₁₁₀₈	C₂×C₄	1	2	yes	—
23	C₂²×Dic₂₇₇	—	2	1	no	—
24	#17984547	—	2	2	no	—
25	#17984547	—	2	2	no	—
26	C₂×#14555660	—	2	2	no	—
27	#17984540	—	2	8	no	—
28	C₄×Dic₂₇₇	—	2	1	no	—
29	C₄×Dic₂₇₇	C₂×C₄	1	2	yes	—
30	C₂×Q₈×C₂₇₇	—	2	2	no	—
31	C₂×Q₈×C₂₇₇	—	2	2	no	—
32	C₂×Q₈×C₂₇₇	C₂×C₄	1	2	yes	—
33	C₂×Q₈×C₂₇₇	D₄	1	2	yes	—
34	C₂²×C₄×C₂₇₇	D₄	1	2	yes	—
35	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₂₇₇	—	2	4	no	—
37	C₄⋊C₄×C₂₇₇	D₄	1	4	yes	—
38	C₂²⋊C₄×C₂₇₇	—	2	2	no	—
39	C₄²×C₂₇₇	—	2	4	no	—
40	C₄²×C₂₇₇	—	2	2	no	—
41	C₂²×D₂₇₇	—	2	2	no	—
42	C₂²×D₂₇₇	—	4	4	no	—
43	C₂²×D₂₇₇	—	4	1	no	—
44	C₂²×D₂₇₇	—	4	1	no	—
45	D₁₁₀₈	—	2	4	no	—
46	D₁₁₀₈	—	4	2	no	—
47	C₂×Dic₂₇₇	—	2	2	no	—
48	C₂×Dic₂₇₇	—	4	2	no	—
49	#14555660	—	4	2	no	—
50	C₂³×C₂₇₇	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₂₇₇	—	2	4	no	—
52	Q₈×C₂₇₇	—	2	4	no	—
53	C₂×C₄×C₂₇₇	—	2	2	no	—
54	C₂×C₄×C₂₇₇	—	2	2	no	—
55	C₂×C₄×C₂₇₇	—	2	4	no	—
56	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	4	yes	—
57	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₂₇₇	#41	1	2	yes	—
61	C₂²×C₂₇₇	—	2	2	no	—
62	C₂²×C₂₇₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₂₇₇	#120	1	1	yes	characteristic
64	C₂²×C₂₇₇	#123	1	1	yes	characteristic
65	C₂×D₂₇₇	—	4	4	no	—
66	C₂×D₂₇₇	—	4	2	no	—
67	C₂×D₂₇₇	—	8	2	no	—
68	C₂×D₂₇₇	—	8	2	no	—
69	C₄×C₂₇₇	—	2	4	no	—
70	C₄×C₂₇₇	—	4	4	no	—
71	C₄×C₂₇₇	C₄×D₄	1	2	yes	—
72	C₄×C₂₇₇	#121	1	2	yes	—
73	Dic₂₇₇	—	4	2	no	—
74	C₂×C₂₇₇	—	2	1	no	—
75	C₂×C₂₇₇	—	2	1	no	—
76	C₂×C₂₇₇	#386	1	1	yes	characteristic
77	C₂×C₂₇₇	#454	1	2	yes	—
78	D₂₇₇	—	8	2	no	—
79	C₂₇₇	#1888	1	1	yes	characteristic, complemented, Sylow
80	#1888	—	277	1	no	maximal, Sylow
81	C₂×#124	—	277	1	no	—
82	#519	D₂₇₇	1	1	yes	characteristic, complemented
83	#418	—	277	2	no	—
84	C₂×#101	—	277	1	no	—
85	#339	—	277	2	no	—
86	C₂²×Q₈	C₂×D₂₇₇	1	1	yes	characteristic
87	C₂²×D₄	—	277	1	no	—
88	C₂×M₁₆	—	277	2	no	—
89	#124	—	277	4	no	—
90	#124	—	554	2	no	—
91	C₄×Q₈	—	2	4	no	—
92	#117	—	2	2	no	—
93	#117	C₂×D₂₇₇	1	2	yes	—
94	C₂×C₂²⋊C₄	—	554	2	no	—
95	C₂×C₄²	—	277	1	no	—
96	#102	—	554	4	no	—
97	#101	—	554	2	no	—
98	C₂⁴	—	554	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₄×D₂₇₇	1	2	yes	—
102	C₂×Q₈	D₁₁₀₈	1	2	yes	—
103	C₂×D₄	—	277	2	no	—
104	C₂×D₄	—	554	2	no	—
105	C₂×D₄	—	554	1	no	—
106	C₂×D₄	—	554	2	no	—
107	C₂²×C₄	—	554	1	no	—
108	C₂²×C₄	C₂²⋊D₂₇₇	1	2	yes	—
109	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
110	M₁₆	—	554	2	no	—
111	M₁₆	—	554	2	no	—
112	C₂×C₈	—	554	2	no	—
113	C₄⋊C₄	—	2	4	no	—
114	C₄⋊C₄	C₂²⋊D₂₇₇	1	4	yes	—
115	C₂²⋊C₄	—	2	2	no	—
116	C₂²⋊C₄	—	554	8	no	—
117	C₄²	—	2	4	no	—
118	C₄²	—	2	2	no	—
119	C₄²	—	277	2	no	—
120	C₄²	—	554	1	no	—
121	C₂³	—	554	2	no	—
122	C₂³	—	1108	4	no	—
123	C₂³	—	1108	1	no	—
124	C₂³	—	1108	1	no	—
125	C₂³	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
126	Q₈	—	2	4	no	—
127	Q₈	—	2	4	no	—
128	D₄	—	554	4	no	—
129	D₄	—	1108	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	554	2	no	—
134	C₂×C₄	—	1108	2	no	—
135	C₂×C₄	#17984540	1	4	yes	—
136	C₂×C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	2	yes	—
138	C₂×C₄	C₂×D₁₁₀₈	1	1	yes	characteristic
139	C₂×C₄	#17984563	1	2	yes	—
140	C₈	—	1108	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	1108	4	no	—
143	C₂²	—	1108	2	no	—
144	C₂²	—	2216	2	no	—
145	C₂²	—	2216	2	no	—
146	C₂²	C₂×#17984540	1	1	yes	characteristic
147	C₂²	#22579849	1	1	yes	characteristic
148	C₂²	#22579863	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	4	4	no	—
151	C₄	—	1108	2	no	—
152	C₄	C₄×C₂²⋊D₂₇₇	1	2	yes	—
153	C₄	#22579854	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	—	2216	2	no	—
157	C₂	#31799881	1	1	yes	characteristic
158	C₂	#31800175	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22579869

C₂

yes

maximal, characteristic

#31800006

C₂

yes

maximal

C₂×#22579799

C₂

yes

maximal, characteristic, complemented

#31799761

C₂

yes

maximal, complemented

#519×C₂₇₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₁₀₈

C₂²

yes

characteristic

C₂×#17984547

C₂²

yes

—

#22579869

—

#22579869

C₄

yes

complemented

C₂×#17984540

—

C₂×C₄×Dic₂₇₇

C₂²

yes

characteristic

#22579800

—

#22579799

—

C₂²×Q₈×C₂₇₇

C₂²

yes

characteristic

C₄×Q₈×C₂₇₇

—

#117×C₂₇₇

—

#117×C₂₇₇

C₂²

yes

—

C₂³×D₂₇₇

—

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

C₂×C₄

yes

—

C₂²×Dic₂₇₇

—

—

—

—

—

C₄×Dic₂₇₇

—

C₄×Dic₂₇₇

C₂×C₄

yes

—

C₂×Q₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

C₂×C₄

yes

—

C₂×Q₈×C₂₇₇

D₄

yes

—

C₂²×C₄×C₂₇₇

D₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

D₄

yes

—

C₂²⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

—

C₄²×C₂₇₇

—

C₂²×D₂₇₇

—

C₂²×D₂₇₇

—

C₂²×D₂₇₇

—

C₂²×D₂₇₇

—

D₁₁₀₈

—

D₁₁₀₈

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

#14555660

—

C₂³×C₂₇₇

C₂×D₄

yes

characteristic

Q₈×C₂₇₇

—

Q₈×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

#41

yes

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

C₂×C₂²⋊C₄

yes

C₂²×C₂₇₇

yes

C₂²×C₂₇₇

yes

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₄×D₄

yes

—

C₄×C₂₇₇

#121

yes

—

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#386

yes

characteristic

C₂×C₂₇₇

#454

yes

—

D₂₇₇

—

C₂₇₇

#1888

yes

characteristic, complemented, Sylow

#1888

—

277

maximal, Sylow

C₂×#124

—

277

—

#519

D₂₇₇

yes

characteristic, complemented

#418

—

277

—

C₂×#101

—

277

—

#339

—

277

—

C₂²×Q₈

C₂×D₂₇₇

yes

characteristic

C₂²×D₄

—

277

—

C₂×M₁₆

—

277

—

#124

—

277

—

#124

—

554

—

C₄×Q₈

—

#117

—

#117

C₂×D₂₇₇

yes

—

C₂×C₂²⋊C₄

—

554

—

C₂×C₄²

—

277

—

#102

—

554

—

#101

—

554

—

C₂⁴

—

554

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₄×D₂₇₇

yes

—

102

C₂×Q₈

D₁₁₀₈

yes

—

103

C₂×D₄

—

277

—

104

C₂×D₄

—

554

—

105

C₂×D₄

—

554

—

106

C₂×D₄

—

554

—

107

C₂²×C₄

—

554

—

108

C₂²×C₄

C₂²⋊D₂₇₇

yes

—

109

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

110

M₁₆

—

554

—

111

M₁₆

—

554

—

112

C₂×C₈

—

554

—

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂²⋊D₂₇₇

yes

—

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

554

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

277

—

120

C₄²

—

554

—

121

C₂³

—

554

—

122

C₂³

—

1108

—

123

C₂³

—

1108

—

124

C₂³

—

1108

—

125

C₂³

C₂×C₂²⋊D₂₇₇

yes

characteristic

126

Q₈

—

127

Q₈

—

128

D₄

—

554

—

129

D₄

—

1108

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

554

—

134

C₂×C₄

—

1108

—

135

C₂×C₄

#17984540

yes

—

136

C₂×C₄

C₂×C₄×D₂₇₇

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

—

138

C₂×C₄

C₂×D₁₁₀₈

yes

characteristic

139

C₂×C₄

#17984563

yes

—

140

C₈

—

1108

—

141

C₂²

—

142

C₂²

—

1108

—

143

C₂²

—

1108

—

144

C₂²

—

2216

—

145

C₂²

—

2216

—

146

C₂²

C₂×#17984540

yes

characteristic

147

C₂²

#22579849

yes

characteristic

148

C₂²

#22579863

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

1108

—

152

C₄

C₄×C₂²⋊D₂₇₇

yes

—

153

C₄

#22579854

yes

—

154

C₂

—

155

C₂

—

156

C₂

—

2216

—

157

C₂

#31799881

yes

characteristic

158

C₂

#31800175

yes

—

	subgroup	G/H
center	C₂²	#22579849
commutator	C₂×C₄×C₂₇₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	#519×C₂₇₇	C₂
socle	C₂²×C₂₇₇	#120
prime radical	C₂²×D₁₁₀₈	C₂²
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	#519	D₂₇₇
hypocenter	C₂₇₇	#1888
LCS	G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #519

subgroup

G/H

center

C₂²

#22579849

commutator

C₂×C₄×C₂₇₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

#519×C₂₇₇

C₂

socle

C₂²×C₂₇₇

#120

prime radical

C₂²×D₁₁₀₈

C₂²

cosocle

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #519

N	H ≅ G/N	type
3. C₂×#22579799	C₂	semidirect
4. #31799761	C₂	semidirect
5. #519×C₂₇₇	C₂	semidirect
9. #22579869	C₄	semidirect
79. C₂₇₇	#1888	semidirect
82. #519	D₂₇₇	semidirect

H ≅ G/N

type

3. C₂×#22579799

C₂

semidirect

4. #31799761

C₂

semidirect

5. #519×C₂₇₇

C₂

semidirect

9. #22579869

C₄

semidirect

79. C₂₇₇

#1888

semidirect

82. #519

D₂₇₇

semidirect

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

subgroup	index
C₂×#22579869	2
#31800006	2
C₂×#22579799	2
#31799761	2
#519×C₂₇₇	2
#1888	277

subgroup

index

#519×C₂₇₇

277

Subgroups — #47983570

Conjugacy classes of subgroups

Complements

Maximal subgroups