Group #47983268 — subgroups

Subgroups — #47983268

Order 35456.

262 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 81 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	#41×Dic₂₇₇	C₂	1	1	yes	maximal, characteristic
2	#31799986	C₂	1	1	yes	maximal, characteristic
3	#31799761	C₂	1	2	yes	maximal, complemented
4	#31799758	C₂	1	2	yes	maximal, complemented
5	#410×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#17984547	C₂²	1	2	yes	—
7	Q₈×Dic₂₇₇	—	2	1	no	—
8	D₄×Dic₂₇₇	—	2	2	no	—
9	D₄×Dic₂₇₇	—	2	1	no	—
10	#22579839	C₄	1	2	yes	complemented
11	#22579839	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₂₇₇	C₄	1	2	yes	complemented
13	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
14	#22579799	—	2	1	no	—
15	#22579797	—	2	1	no	—
16	C₂×#41×C₂₇₇	C₂²	1	1	yes	characteristic
17	#117×C₂₇₇	C₂²	1	2	yes	—
18	#99×C₂₇₇	—	2	2	no	—
19	C₂²×Dic₂₇₇	—	2	2	no	—
20	C₂²×Dic₂₇₇	—	2	1	no	—
21	#17984547	—	2	2	no	—
22	#17984547	—	2	2	no	—
23	C₂×#14555660	—	2	2	no	—
24	C₄⋊Dic₂₇₇	—	2	2	no	—
25	C₄⋊Dic₂₇₇	—	2	1	no	—
26	C₂²⋊Dic₂₇₇	—	2	2	no	—
27	C₂²⋊Dic₂₇₇	—	2	1	no	—
28	C₄×Dic₂₇₇	—	2	2	no	—
29	C₄×Dic₂₇₇	—	2	1	no	—
30	C₄×Dic₂₇₇	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₂₇₇	C₂×C₄	1	2	yes	—
32	C₄×Dic₂₇₇	C₂×C₄	1	1	yes	characteristic
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	D₄	1	2	yes	—
37	C₂×D₄×C₂₇₇	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₇₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₂₇₇	—	2	2	no	—
41	C₂²⋊C₄×C₂₇₇	—	2	2	no	—
42	C₄²×C₂₇₇	—	2	2	no	—
43	C₂×Dic₂₇₇	—	2	1	no	—
44	C₂×Dic₂₇₇	—	2	2	no	—
45	C₂×Dic₂₇₇	—	2	1	no	—
46	C₂×Dic₂₇₇	—	4	2	no	—
47	C₂×Dic₂₇₇	—	4	4	no	—
48	#14555660	—	4	2	no	—
49	C₂³×C₂₇₇	C₂²⋊C₄	1	2	yes	—
50	C₂³×C₂₇₇	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₂₇₇	—	2	2	no	—
52	D₄×C₂₇₇	—	2	4	no	—
53	D₄×C₂₇₇	—	2	2	no	—
54	C₂×C₄×C₂₇₇	—	2	2	no	—
55	C₂×C₄×C₂₇₇	—	2	2	no	—
56	C₂×C₄×C₂₇₇	—	4	2	no	—
57	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₂₇₇	#41	1	2	yes	—
61	C₂×C₄×C₂₇₇	#41	1	2	yes	—
62	C₂²×C₂₇₇	—	2	2	no	—
63	C₂²×C₂₇₇	—	4	1	no	—
64	C₂²×C₂₇₇	—	4	2	no	—
65	C₂²×C₂₇₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₂₇₇	#123	1	2	yes	—
67	C₄×C₂₇₇	—	2	2	no	—
68	C₄×C₂₇₇	—	8	2	no	—
69	C₄×C₂₇₇	#117	1	1	yes	characteristic
70	C₄×C₂₇₇	#117	1	1	yes	characteristic
71	C₄×C₂₇₇	#123	1	2	yes	—
72	Dic₂₇₇	—	2	2	no	—
73	Dic₂₇₇	—	4	2	no	—
74	Dic₂₇₇	—	4	1	no	—
75	C₂×C₂₇₇	—	2	1	no	—
76	C₂×C₂₇₇	—	2	2	no	—
77	C₂×C₂₇₇	—	4	2	no	—
78	C₂×C₂₇₇	#381	1	1	yes	characteristic
79	C₂₇₇	#1827	1	1	yes	characteristic, complemented, Sylow
80	#1827	—	277	1	no	maximal, Sylow
81	C₄×#41	—	277	1	no	—
82	#413	—	277	1	no	—
83	#410	D₂₇₇	1	1	yes	characteristic
84	#339	—	277	2	no	—
85	H(3,ℤ/4)	—	277	2	no	—
86	C₂×#41	C₂×D₂₇₇	1	1	yes	characteristic
87	C₂×M₁₆	—	277	2	no	—
88	C₄×Q₈	—	554	1	no	—
89	C₄×D₄	—	554	2	no	—
90	C₄×D₄	—	554	1	no	—
91	#117	—	277	2	no	—
92	#117	—	277	1	no	—
93	#117	C₂×D₂₇₇	1	2	yes	—
94	C₂×C₄²	—	277	2	no	—
95	C₂×C₄²	—	277	1	no	—
96	#101	—	554	1	no	—
97	#100	—	554	1	no	—
98	#99	—	2	2	no	—
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	C₂×Q₈	D₁₁₀₈	1	1	yes	characteristic
102	C₂×D₄	C₂²⋊D₂₇₇	1	2	yes	—
103	C₂×D₄	D₁₁₀₈	1	1	yes	characteristic
104	C₂²×C₄	—	554	2	no	—
105	C₂²×C₄	—	554	1	no	—
106	C₂²×C₄	C₄×D₂₇₇	1	2	yes	—
107	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
108	M₁₆	—	554	2	no	—
109	M₁₆	—	554	2	no	—
110	C₂×C₈	—	554	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	554	2	no	—
113	C₄⋊C₄	—	554	1	no	—
114	C₂²⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	554	2	no	—
116	C₂²⋊C₄	—	554	1	no	—
117	C₄²	—	2	2	no	—
118	C₄²	—	277	1	no	—
119	C₄²	—	277	2	no	—
120	C₄²	—	277	1	no	—
121	C₄²	—	554	2	no	—
122	C₄²	—	554	1	no	—
123	C₂³	#17984540	1	2	yes	—
124	C₂³	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	554	1	no	—
132	C₂×C₄	—	554	2	no	—
133	C₂×C₄	—	554	1	no	—
134	C₂×C₄	—	1108	2	no	—
135	C₂×C₄	—	1108	4	no	—
136	C₂×C₄	#17984540	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
138	C₂×C₄	C₂×D₁₁₀₈	1	1	yes	characteristic
139	C₂×C₄	#17984562	1	2	yes	—
140	C₂×C₄	#17984562	1	2	yes	—
141	C₈	—	1108	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	4	2	no	—
145	C₂²	C₂×#17984540	1	1	yes	characteristic
146	C₂²	#22579861	1	2	yes	—
147	C₄	—	2	2	no	—
148	C₄	—	8	2	no	—
149	C₄	—	554	2	no	—
150	C₄	—	1108	2	no	—
151	C₄	—	1108	1	no	—
152	C₄	#22579836	1	1	yes	characteristic
153	C₄	#22579836	1	1	yes	characteristic
154	C₄	#22579864	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	—	2	1	no	—
157	C₂	—	4	2	no	—
158	C₂	#31799863	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#41×Dic₂₇₇

C₂

yes

maximal, characteristic

#31799986

C₂

yes

maximal, characteristic

#31799761

C₂

yes

maximal, complemented

#31799758

C₂

yes

maximal, complemented

#410×C₂₇₇

C₂

yes

maximal, characteristic

C₂×#17984547

C₂²

yes

—

Q₈×Dic₂₇₇

—

D₄×Dic₂₇₇

—

D₄×Dic₂₇₇

—

#22579839

C₄

yes

complemented

#22579839

C₂²

yes

characteristic

C₂×C₄×Dic₂₇₇

C₄

yes

complemented

C₂×C₄×Dic₂₇₇

C₂²

yes

characteristic

#22579799

—

#22579797

—

C₂×#41×C₂₇₇

C₂²

yes

characteristic

#117×C₂₇₇

C₂²

yes

—

#99×C₂₇₇

—

C₂²×Dic₂₇₇

—

C₂²×Dic₂₇₇

—

#17984547

—

#17984547

—

C₂×#14555660

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₂²⋊Dic₂₇₇

—

C₂²⋊Dic₂₇₇

—

C₄×Dic₂₇₇

—

C₄×Dic₂₇₇

—

C₄×Dic₂₇₇

C₂×C₄

yes

characteristic

C₄×Dic₂₇₇

C₂×C₄

yes

—

C₄×Dic₂₇₇

C₂×C₄

yes

characteristic

#41×C₂₇₇

—

#41×C₂₇₇

—

C₂×Q₈×C₂₇₇

D₄

yes

characteristic

C₂×D₄×C₂₇₇

D₄

yes

—

C₂×D₄×C₂₇₇

D₄

yes

characteristic

C₂²×C₄×C₂₇₇

C₂×C₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₇₇

—

C₂²⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

#14555660

—

C₂³×C₂₇₇

C₂²⋊C₄

yes

—

C₂³×C₂₇₇

C₂×D₄

yes

characteristic

Q₈×C₂₇₇

—

D₄×C₂₇₇

—

D₄×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

characteristic

C₂×C₄×C₂₇₇

#41

yes

—

C₂×C₄×C₂₇₇

#41

yes

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₇₇

#123

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

yes

C₄×C₂₇₇

yes

C₄×C₂₇₇

yes

—

Dic₂₇₇

—

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#381

yes

characteristic

C₂₇₇

#1827

yes

characteristic, complemented, Sylow

#1827

—

277

maximal, Sylow

C₄×#41

—

277

—

#413

—

277

—

#410

D₂₇₇

yes

characteristic

#339

—

277

—

H(3,ℤ/4)

—

277

—

C₂×#41

C₂×D₂₇₇

yes

characteristic

C₂×M₁₆

—

277

—

C₄×Q₈

—

554

—

C₄×D₄

—

554

—

C₄×D₄

—

554

—

#117

—

277

—

#117

—

277

—

#117

C₂×D₂₇₇

yes

—

C₂×C₄²

—

277

—

C₂×C₄²

—

277

—

#101

—

554

—

#100

—

554

—

#99

—

#41

—

100

#41

—

101

C₂×Q₈

D₁₁₀₈

yes

characteristic

102

C₂×D₄

C₂²⋊D₂₇₇

yes

—

103

C₂×D₄

D₁₁₀₈

yes

characteristic

104

C₂²×C₄

—

554

—

105

C₂²×C₄

—

554

—

106

C₂²×C₄

C₄×D₂₇₇

yes

—

107

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

108

M₁₆

—

554

—

109

M₁₆

—

554

—

110

C₂×C₈

—

554

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

554

—

113

C₄⋊C₄

—

554

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

554

—

116

C₂²⋊C₄

—

554

—

117

C₄²

—

118

C₄²

—

277

—

119

C₄²

—

277

—

120

C₄²

—

277

—

121

C₄²

—

554

—

122

C₄²

—

554

—

123

C₂³

#17984540

yes

—

124

C₂³

C₂×C₂²⋊D₂₇₇

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

554

—

132

C₂×C₄

—

554

—

133

C₂×C₄

—

554

—

134

C₂×C₄

—

1108

—

135

C₂×C₄

—

1108

—

136

C₂×C₄

#17984540

yes

—

137

C₂×C₄

C₂×C₄×D₂₇₇

yes

characteristic

138

C₂×C₄

C₂×D₁₁₀₈

yes

characteristic

139

C₂×C₄

#17984562

yes

—

140

C₂×C₄

#17984562

yes

—

141

C₈

—

1108

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₂×#17984540

yes

characteristic

146

C₂²

#22579861

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

554

—

150

C₄

—

1108

—

151

C₄

—

1108

—

152

C₄

#22579836

yes

characteristic

153

C₄

#22579836

yes

characteristic

154

C₄

#22579864

yes

—

155

C₂

—

156

C₂

—

157

C₂

—

158

C₂

#31799863

yes

characteristic

	subgroup	G/H
center	C₄	#22579836
commutator	C₂×C₄×C₂₇₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	#410×C₂₇₇	C₂
socle	C₂×C₂₇₇	#381
prime radical	C₂×#41×C₂₇₇	C₂²
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	#410	D₂₇₇
hypocenter	C₂₇₇	#1827
LCS	G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#22579836

commutator

C₂×C₄×C₂₇₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

#410×C₂₇₇

C₂

socle

C₂×C₂₇₇

#381

prime radical

C₂×#41×C₂₇₇

C₂²

cosocle

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
3. #31799761	C₂	semidirect
4. #31799758	C₂	semidirect
10. #22579839	C₄	semidirect
12. C₂×C₄×Dic₂₇₇	C₄	semidirect
79. C₂₇₇	#1827	semidirect

H ≅ G/N

type

3. #31799761

C₂

semidirect

4. #31799758

C₂

semidirect

10. #22579839

C₄

semidirect

12. C₂×C₄×Dic₂₇₇

C₄

semidirect

79. C₂₇₇

#1827

semidirect

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

subgroup	index
#41×Dic₂₇₇	2
#31799986	2
#31799761	2
#31799758	2
#410×C₂₇₇	2
#1827	277

subgroup

index

#41×Dic₂₇₇

#410×C₂₇₇

277

Subgroups — #47983268

Conjugacy classes of subgroups

Complements

Maximal subgroups