Group #10672 — subgroups

Subgroups — #10672

Order 256.

255 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 85 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	#1957	C₂	1	1	yes	maximal, characteristic
2	#1957	C₂	1	1	yes	maximal, characteristic
3	#1957	C₂	1	1	yes	maximal, characteristic
4	#1845	C₂	1	1	yes	maximal, characteristic, complemented
5	#1844	C₂	1	1	yes	maximal, characteristic, complemented
6	#1834	C₂	1	1	yes	maximal, characteristic, complemented
7	#1273	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#127	C₂²	1	1	yes	characteristic
9	C₂×#107	—	2	2	no	—
10	C₂×#106	—	2	2	no	—
11	C₂×#102	—	2	2	no	—
12	C₂×#102	—	2	2	no	—
13	C₂×#102	—	2	2	no	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	#384	C₂²	1	1	yes	characteristic
18	#384	C₂²	1	1	yes	characteristic
19	#384	C₂²	1	1	yes	characteristic
20	#340	—	2	2	no	—
21	C₂²×D₄	—	2	1	no	—
22	C₂²×D₄	—	2	1	no	—
23	C₂²×D₄	—	2	1	no	—
24	C₂²×C₈	D₄	1	2	yes	—
25	C₂²×C₈	D₄	1	2	yes	—
26	C₂²×C₈	D₄	1	2	yes	—
27	#127	—	2	1	no	—
28	#127	—	2	1	no	—
29	#127	—	2	1	no	—
30	#127	Q₈	1	2	yes	—
31	C₂×C₄⋊C₄	—	2	1	no	—
32	C₂×C₄⋊C₄	—	2	1	no	—
33	C₂×C₄⋊C₄	—	2	1	no	—
34	C₂×C₄⋊C₄	—	2	1	no	—
35	C₂×C₄⋊C₄	—	2	1	no	—
36	C₂×C₄⋊C₄	—	2	1	no	—
37	C₂×C₄⋊C₄	—	2	1	no	—
38	C₂×C₄⋊C₄	—	2	1	no	—
39	C₂×C₄⋊C₄	—	2	1	no	—
40	C₂×C₄²	C₂³	1	1	yes	characteristic
41	#107	—	4	4	no	—
42	#106	—	4	4	no	—
43	#102	—	4	4	no	—
44	#102	—	4	4	no	—
45	#102	—	4	4	no	—
46	C₄×C₈	—	2	1	no	—
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	—	2	1	no	—
49	C₄×C₈	—	2	1	no	—
50	C₄×C₈	—	2	1	no	—
51	C₄×C₈	—	2	1	no	—
52	C₂⁴	—	4	1	no	—
53	C₂×D₄	—	2	2	no	—
54	C₂×D₄	—	2	2	no	—
55	C₂×D₄	—	2	2	no	—
56	C₂×D₄	—	4	1	no	—
57	C₂×D₄	—	4	1	no	—
58	C₂×D₄	—	4	1	no	—
59	C₂×D₄	—	4	2	no	—
60	C₂×D₄	—	4	2	no	—
61	C₂×D₄	—	4	2	no	—
62	C₂×D₄	—	4	2	no	—
63	C₂×D₄	—	4	2	no	—
64	C₂×D₄	—	4	2	no	—
65	C₂²×C₄	—	4	1	no	—
66	C₂²×C₄	—	4	1	no	—
67	C₂²×C₄	—	4	1	no	—
68	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	D₈	1	4	yes	—
79	C₂×C₈	SD₁₆	1	4	yes	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄⋊C₄	—	4	2	no	—
82	C₄⋊C₄	—	4	2	no	—
83	C₄⋊C₄	—	4	2	no	—
84	C₄⋊C₄	—	4	2	no	—
85	C₄⋊C₄	—	4	2	no	—
86	C₄⋊C₄	—	4	2	no	—
87	C₄⋊C₄	—	4	2	no	—
88	C₄⋊C₄	—	4	2	no	—
89	C₄²	C₂×Q₈	1	1	yes	characteristic
90	C₄²	#41	1	1	yes	characteristic
91	C₄²	#41	1	1	yes	characteristic
92	C₄²	#41	1	1	yes	characteristic
93	C₂³	—	4	2	no	—
94	C₂³	—	8	1	no	—
95	C₂³	—	8	1	no	—
96	C₂³	—	8	1	no	—
97	C₂³	—	8	1	no	—
98	C₂³	—	8	1	no	—
99	C₂³	—	8	1	no	—
100	C₂³	#127	1	1	yes	characteristic
101	D₄	—	4	4	no	—
102	D₄	—	4	4	no	—
103	D₄	—	4	4	no	—
104	D₄	—	8	2	no	—
105	D₄	—	8	2	no	—
106	D₄	—	8	2	no	—
107	C₂×C₄	—	8	1	no	—
108	C₂×C₄	—	8	1	no	—
109	C₂×C₄	—	8	1	no	—
110	C₂×C₄	—	8	1	no	—
111	C₂×C₄	—	8	1	no	—
112	C₂×C₄	—	8	1	no	—
113	C₂×C₄	—	8	1	no	—
114	C₂×C₄	—	8	1	no	—
115	C₂×C₄	—	8	1	no	—
116	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄	#123	1	1	yes	characteristic
120	C₂×C₄	#123	1	1	yes	characteristic
121	C₂×C₄	#123	1	1	yes	characteristic
122	C₂×C₄	C₂×D₈	1	2	yes	—
123	C₂×C₄	C₂×SD₁₆	1	2	yes	—
124	C₂×C₄	#135	1	2	yes	—
125	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄	#137	1	2	yes	—
128	C₈	—	2	4	no	—
129	C₈	—	2	4	no	—
130	C₈	—	4	2	no	—
131	C₂²	—	8	2	no	—
132	C₂²	—	8	2	no	—
133	C₂²	—	8	2	no	—
134	C₂²	—	16	1	no	—
135	C₂²	—	16	1	no	—
136	C₂²	—	16	1	no	—
137	C₂²	—	16	1	no	—
138	C₂²	#399	1	1	yes	characteristic
139	C₂²	#492	1	1	yes	characteristic
140	C₂²	#493	1	1	yes	characteristic
141	C₂²	#495	1	1	yes	characteristic
142	C₂²	#496	1	1	yes	characteristic
143	C₂²	#496	1	1	yes	characteristic
144	C₂²	#497	1	1	yes	characteristic
145	C₄	—	16	1	no	—
146	C₄	—	16	1	no	—
147	C₄	—	16	1	no	—
148	C₄	#474	1	2	yes	—
149	C₄	#476	1	2	yes	—
150	C₄	#478	1	2	yes	—
151	C₄	#480	1	2	yes	—
152	C₄	#481	1	2	yes	—
153	C₄	#482	1	2	yes	—
154	C₂	—	16	2	no	—
155	C₂	#1706	1	1	yes	characteristic
156	C₂	#1714	1	1	yes	characteristic
157	C₂	#1720	1	1	yes	characteristic
158	C₂	#1721	1	1	yes	characteristic
159	C₂	#2092	1	1	yes	characteristic
160	C₂	#2093	1	1	yes	characteristic
161	C₂	#2094	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1957

C₂

yes

maximal, characteristic

#1957

C₂

yes

maximal, characteristic

#1957

C₂

yes

maximal, characteristic

#1845

C₂

yes

maximal, characteristic, complemented

#1844

C₂

yes

maximal, characteristic, complemented

#1834

C₂

yes

maximal, characteristic, complemented

#1273

C₂

yes

maximal, characteristic, complemented

C₂×#127

C₂²

yes

—

—

—

—

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#384

C₂²

yes

characteristic

#384

C₂²

yes

characteristic

#384

C₂²

yes

characteristic

#340

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

—

—

—

Q₈

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₈

yes

—

C₂×C₈

SD₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Q₈

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

#127

yes

characteristic

101

D₄

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂²⋊Q₈

yes

characteristic

117

C₂×C₄

C₂²⋊Q₈

yes

characteristic

118

C₂×C₄

C₂²⋊Q₈

yes

characteristic

119

C₂×C₄

#123

yes

characteristic

120

C₂×C₄

#123

yes

characteristic

121

C₂×C₄

#123

yes

characteristic

122

C₂×C₄

C₂×D₈

yes

—

123

C₂×C₄

C₂×SD₁₆

yes

—

124

C₂×C₄

#135

yes

—

125

C₂×C₄

AGL(1,ℤ/8)

yes

—

126

C₂×C₄

AGL(1,ℤ/8)

yes

—

127

C₂×C₄

#137

yes

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#399

yes

characteristic

139

C₂²

#492

yes

characteristic

140

C₂²

#493

yes

characteristic

141

C₂²

#495

yes

characteristic

142

C₂²

#496

yes

characteristic

143

C₂²

#496

yes

characteristic

144

C₂²

#497

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

#474

yes

—

149

C₄

#476

yes

—

150

C₄

#478

yes

—

151

C₄

#480

yes

—

152

C₄

#481

yes

—

153

C₄

#482

yes

—

154

C₂

—

155

C₂

#1706

yes

characteristic

156

C₂

#1714

yes

characteristic

157

C₂

#1720

yes

characteristic

158

C₂

#1721

yes

characteristic

159

C₂

#2092

yes

characteristic

160

C₂

#2093

yes

characteristic

161

C₂

#2094

yes

characteristic

	subgroup	G/H
center, socle	C₂³	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂³

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³

prime radical

C₂×#127

C₂²

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
4. #1845	C₂	semidirect
5. #1844	C₂	semidirect
6. #1834	C₂	semidirect
7. #1273	C₂	semidirect

H ≅ G/N

type

4. #1845

C₂

semidirect

5. #1844

C₂

semidirect

6. #1834

C₂

semidirect

7. #1273

C₂

semidirect

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

subgroup	index
#1957	2
#1845	2
#1844	2
#1834	2
#1273	2

subgroup

index

Subgroups — #10672

Conjugacy classes of subgroups

Complements

Maximal subgroups