Group #47983719 — subgroups

Subgroups — #47983719

Order 35456.

308 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 89 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	#31800573	C₂	1	1	yes	maximal, characteristic
2	#31799986	C₂	1	1	yes	maximal, characteristic
3	C₂×#22579798	C₂	1	2	yes	maximal, complemented
4	#31799774	C₂	1	2	yes	maximal, complemented
5	#568×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₅₄	C₂²	1	1	yes	characteristic
7	C₂×#17984547	C₂²	1	2	yes	—
8	#22579878	—	2	1	no	—
9	#22579867	—	2	1	no	—
10	#22579861	C₄	1	4	yes	—
11	#22579859	—	2	2	no	—
12	#22579839	C₂²	1	1	yes	characteristic
13	#22579799	—	2	1	no	—
14	#22579798	—	2	4	no	—
15	#22579797	—	2	1	no	—
16	C₂×#41×C₂₇₇	C₂²	1	1	yes	characteristic
17	#131×C₂₇₇	—	2	4	no	—
18	C₂×M₁₆×C₂₇₇	—	2	2	no	—
19	C₂×M₁₆×C₂₇₇	C₂²	1	2	yes	—
20	C₂×Dic₅₅₄	—	2	2	no	—
21	C₂×Dic₅₅₄	—	2	1	no	—
22	C₂×Dic₅₅₄	—	2	1	no	—
23	C₂×Dic₅₅₄	—	2	2	no	—
24	C₂²×Dic₂₇₇	C₂×C₄	1	2	yes	—
25	#17984547	—	2	2	no	—
26	#17984547	—	2	2	no	—
27	C₂×#14555660	—	2	2	no	—
28	C₄⋊Dic₂₇₇	—	2	1	no	—
29	#17984542	—	2	4	no	—
30	C₂²⋊Dic₂₇₇	—	2	4	no	—
31	C₂²⋊Dic₂₇₇	C₂×C₄	1	2	yes	—
32	C₄×Dic₂₇₇	—	2	1	no	—
33	#41×C₂₇₇	—	2	2	no	—
34	#41×C₂₇₇	—	2	2	no	—
35	C₂×Q₈×C₂₇₇	D₄	1	1	yes	characteristic
36	C₂×D₄×C₂₇₇	C₂×C₄	1	2	yes	—
37	C₂×D₄×C₂₇₇	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₇₇	D₄	1	2	yes	—
39	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
40	M₁₆×C₂₇₇	—	2	4	no	—
41	M₁₆×C₂₇₇	—	2	2	no	—
42	M₁₆×C₂₇₇	D₄	1	4	yes	—
43	C₂×C₈×C₂₇₇	—	2	4	no	—
44	C₂×C₈×C₂₇₇	—	2	2	no	—
45	Dic₅₅₄	—	4	1	no	—
46	Dic₅₅₄	—	4	2	no	—
47	Dic₅₅₄	—	4	1	no	—
48	C₂×Dic₂₇₇	—	2	2	no	—
49	C₂×Dic₂₇₇	—	2	2	no	—
50	C₂×Dic₂₇₇	—	2	4	no	—
51	#14555660	—	4	2	no	—
52	C₂³×C₂₇₇	C₂²⋊C₄	1	2	yes	—
53	C₂³×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
54	Q₈×C₂₇₇	—	2	2	no	—
55	D₄×C₂₇₇	—	2	4	no	—
56	D₄×C₂₇₇	—	2	2	no	—
57	C₂×C₄×C₂₇₇	—	2	2	no	—
58	C₂×C₄×C₂₇₇	—	2	2	no	—
59	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₂₇₇	#41	1	2	yes	—
64	C₈×C₂₇₇	—	2	4	no	—
65	C₂²×C₂₇₇	—	2	2	no	—
66	C₂²×C₂₇₇	—	4	1	no	—
67	C₂²×C₂₇₇	—	4	2	no	—
68	C₂²×C₂₇₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₂₇₇	C₄×D₄	1	2	yes	—
70	C₄×C₂₇₇	—	2	2	no	—
71	C₄×C₂₇₇	#120	1	1	yes	characteristic
72	C₄×C₂₇₇	#121	1	2	yes	—
73	C₄×C₂₇₇	#123	1	1	yes	characteristic
74	Dic₂₇₇	—	4	2	no	—
75	Dic₂₇₇	—	4	2	no	—
76	C₂×C₂₇₇	—	2	1	no	—
77	C₂×C₂₇₇	—	2	2	no	—
78	C₂×C₂₇₇	—	4	2	no	—
79	C₂×C₂₇₇	#386	1	1	yes	characteristic
80	C₂₇₇	#1915	1	1	yes	characteristic, complemented, Sylow
81	#1915	—	277	1	no	maximal, Sylow
82	#568	D₂₇₇	1	1	yes	characteristic
83	#536	—	277	1	no	—
84	#413	—	277	1	no	—
85	C₂×#101	—	277	2	no	—
86	#344	—	277	2	no	—
87	C₂×#41	C₂×D₂₇₇	1	1	yes	characteristic
88	C₂²×Q₈	—	277	1	no	—
89	#131	—	2	4	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	—	277	2	no	—
92	C₂×M₁₆	C₂×D₂₇₇	1	2	yes	—
93	C₄⋊Q₈	—	554	1	no	—
94	#124	—	554	1	no	—
95	#123	—	277	4	no	—
96	C₂²⋊Q₈	—	554	2	no	—
97	#117	—	277	1	no	—
98	#101	—	554	1	no	—
99	#101	—	554	4	no	—
100	#100	—	554	1	no	—
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	C₂×Q₈	—	554	2	no	—
104	C₂×Q₈	—	554	1	no	—
105	C₂×Q₈	—	554	1	no	—
106	C₂×Q₈	—	554	2	no	—
107	C₂×Q₈	D₁₁₀₈	1	1	yes	characteristic
108	C₂×D₄	C₄×D₂₇₇	1	2	yes	—
109	C₂×D₄	D₁₁₀₈	1	1	yes	characteristic
110	C₂²×C₄	—	277	2	no	—
111	C₂²×C₄	C₂²⋊D₂₇₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
113	M₁₆	—	2	4	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	554	2	no	—
116	M₁₆	—	554	2	no	—
117	M₁₆	C₂²⋊D₂₇₇	1	4	yes	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	554	2	no	—
121	C₄⋊C₄	—	554	4	no	—
122	C₄⋊C₄	—	554	1	no	—
123	C₂²⋊C₄	—	277	2	no	—
124	C₂²⋊C₄	—	554	4	no	—
125	C₄²	—	554	1	no	—
126	C₂³	#17984540	1	2	yes	—
127	C₂³	C₂×C₄×D₂₇₇	1	1	yes	characteristic
128	Q₈	—	2	2	no	—
129	Q₈	—	1108	1	no	—
130	Q₈	—	1108	2	no	—
131	Q₈	—	1108	1	no	—
132	D₄	—	2	4	no	—
133	D₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	554	2	no	—
137	C₂×C₄	—	554	2	no	—
138	C₂×C₄	—	554	4	no	—
139	C₂×C₄	#17984540	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
142	C₂×C₄	C₂×D₁₁₀₈	1	1	yes	characteristic
143	C₂×C₄	#17984563	1	2	yes	—
144	C₈	—	2	4	no	—
145	C₈	—	1108	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	2	no	—
149	C₂²	C₂×#17984540	1	1	yes	characteristic
150	C₂²	C₄×C₂²⋊D₂₇₇	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	1108	2	no	—
153	C₄	—	1108	2	no	—
154	C₄	#22579849	1	1	yes	characteristic
155	C₄	#22579854	1	2	yes	—
156	C₄	#22579863	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#31799881	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31800573

C₂

yes

maximal, characteristic

#31799986

C₂

yes

maximal, characteristic

C₂×#22579798

C₂

yes

maximal, complemented

#31799774

C₂

yes

maximal, complemented

#568×C₂₇₇

C₂

yes

maximal, characteristic

C₂²×Dic₅₅₄

C₂²

yes

characteristic

C₂×#17984547

C₂²

yes

—

#22579878

—

#22579867

—

#22579861

C₄

yes

—

#22579859

—

#22579839

C₂²

yes

—

—

—

C₂×#41×C₂₇₇

C₂²

yes

characteristic

#131×C₂₇₇

—

C₂×M₁₆×C₂₇₇

—

C₂×M₁₆×C₂₇₇

C₂²

yes

—

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

—

C₂²×Dic₂₇₇

C₂×C₄

yes

—

#17984547

—

#17984547

—

C₂×#14555660

—

C₄⋊Dic₂₇₇

—

#17984542

—

C₂²⋊Dic₂₇₇

—

C₂²⋊Dic₂₇₇

C₂×C₄

yes

—

C₄×Dic₂₇₇

—

#41×C₂₇₇

—

#41×C₂₇₇

—

C₂×Q₈×C₂₇₇

D₄

yes

characteristic

C₂×D₄×C₂₇₇

C₂×C₄

yes

—

C₂×D₄×C₂₇₇

D₄

yes

characteristic

C₂²×C₄×C₂₇₇

D₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

M₁₆×C₂₇₇

—

M₁₆×C₂₇₇

—

M₁₆×C₂₇₇

D₄

yes

—

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

—

Dic₅₅₄

—

Dic₅₅₄

—

Dic₅₅₄

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

#14555660

—

C₂³×C₂₇₇

C₂²⋊C₄

yes

—

C₂³×C₂₇₇

C₂²×C₄

yes

characteristic

Q₈×C₂₇₇

—

D₄×C₂₇₇

—

D₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

#41

yes

—

C₈×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₇₇

C₄×D₄

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

#120

yes

characteristic

C₄×C₂₇₇

#121

yes

—

C₄×C₂₇₇

#123

yes

characteristic

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#386

yes

characteristic

C₂₇₇

#1915

yes

characteristic, complemented, Sylow

#1915

—

277

maximal, Sylow

#568

D₂₇₇

yes

characteristic

#536

—

277

—

#413

—

277

—

C₂×#101

—

277

—

#344

—

277

—

C₂×#41

C₂×D₂₇₇

yes

characteristic

C₂²×Q₈

—

277

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

277

—

C₂×M₁₆

C₂×D₂₇₇

yes

—

C₄⋊Q₈

—

554

—

#124

—

554

—

#123

—

277

—

C₂²⋊Q₈

—

554

—

#117

—

277

—

#101

—

554

—

#101

—

554

—

100

#100

—

554

—

101

#41

—

102

#41

—

103

C₂×Q₈

—

554

—

104

C₂×Q₈

—

554

—

105

C₂×Q₈

—

554

—

106

C₂×Q₈

—

554

—

107

C₂×Q₈

D₁₁₀₈

yes

characteristic

108

C₂×D₄

C₄×D₂₇₇

yes

—

109

C₂×D₄

D₁₁₀₈

yes

characteristic

110

C₂²×C₄

—

277

—

111

C₂²×C₄

C₂²⋊D₂₇₇

yes

—

112

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

554

—

116

M₁₆

—

554

—

117

M₁₆

C₂²⋊D₂₇₇

yes

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

554

—

121

C₄⋊C₄

—

554

—

122

C₄⋊C₄

—

554

—

123

C₂²⋊C₄

—

277

—

124

C₂²⋊C₄

—

554

—

125

C₄²

—

554

—

126

C₂³

#17984540

yes

—

127

C₂³

C₂×C₄×D₂₇₇

yes

characteristic

128

Q₈

—

129

Q₈

—

1108

—

130

Q₈

—

1108

—

131

Q₈

—

1108

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

554

—

137

C₂×C₄

—

554

—

138

C₂×C₄

—

554

—

139

C₂×C₄

#17984540

yes

—

140

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

—

141

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

142

C₂×C₄

C₂×D₁₁₀₈

yes

characteristic

143

C₂×C₄

#17984563

yes

—

144

C₈

—

145

C₈

—

1108

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×#17984540

yes

characteristic

150

C₂²

C₄×C₂²⋊D₂₇₇

yes

—

151

C₄

—

152

C₄

—

1108

—

153

C₄

—

1108

—

154

C₄

#22579849

yes

characteristic

155

C₄

#22579854

yes

—

156

C₄

#22579863

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#31799881

yes

characteristic

	subgroup	G/H
center	C₂	#31799881
commutator	C₂³×C₂₇₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	#568×C₂₇₇	C₂
socle	C₂×C₂₇₇	#386
prime radical	C₂×#41×C₂₇₇	C₂²
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	#568	D₂₇₇
hypocenter	C₂₇₇	#1915
LCS	G ▷ C₂³×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₂

#31799881

commutator

C₂³×C₂₇₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

#568×C₂₇₇

C₂

socle

C₂×C₂₇₇

#386

prime radical

C₂×#41×C₂₇₇

C₂²

cosocle

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₂³×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
3. C₂×#22579798	C₂	semidirect
4. #31799774	C₂	semidirect
80. C₂₇₇	#1915	semidirect

H ≅ G/N

type

3. C₂×#22579798

C₂

semidirect

4. #31799774

C₂

semidirect

80. C₂₇₇

#1915

semidirect

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

subgroup	index
#31800573	2
#31799986	2
C₂×#22579798	2
#31799774	2
#568×C₂₇₇	2
#1915	277

subgroup

index

#568×C₂₇₇

277

Subgroups — #47983719

Conjugacy classes of subgroups

Complements

Maximal subgroups