Group #7257 — subgroups

Subgroups — #7257

Order 256.

151 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 151 conjugacy classes of subgroups — too many to draw. Only its 49 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×#349	C₂	1	1	yes	maximal, characteristic
2	#1811	C₂	1	1	yes	maximal, characteristic
3	C₂×#338	C₂	1	1	yes	maximal, characteristic
4	C₂²×M₁₆	—	2	1	no	—
5	C₂²×M₁₆	C₂²	1	1	yes	characteristic
6	C₂×M₃₂	—	2	1	no	—
7	C₂×#108	C₄	1	1	yes	characteristic
8	C₂×#108	C₄	1	1	yes	characteristic
9	#407	—	2	1	no	—
10	#349	—	2	1	no	—
11	#349	C₄	1	2	yes	—
12	#341	—	2	1	no	—
13	#338	—	2	1	no	—
14	#338	C₄	1	2	yes	—
15	C₂³×C₄	D₄	1	1	yes	characteristic
16	C₂×M₁₆	—	2	2	no	—
17	C₂×M₁₆	—	2	2	no	—
18	C₂×M₁₆	—	2	1	no	—
19	C₂×M₁₆	—	2	1	no	—
20	C₂×M₁₆	—	4	1	no	—
21	C₂×M₁₆	—	4	1	no	—
22	C₂×M₁₆	—	4	1	no	—
23	C₂×M₁₆	—	4	1	no	—
24	C₂×M₁₆	—	4	1	no	—
25	C₂×M₁₆	—	4	1	no	—
26	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆	D₄	1	1	yes	characteristic
29	C₂×M₁₆	Q₈	1	1	yes	characteristic
30	C₂²×C₈	—	2	2	no	—
31	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈	D₄	1	1	yes	characteristic
33	M₃₂	—	2	2	no	—
34	M₃₂	—	4	1	no	—
35	C₂×C₁₆	—	4	1	no	—
36	#108	—	4	1	no	—
37	#108	—	4	1	no	—
38	C₂²⋊C₈	—	4	1	no	—
39	C₂²⋊C₈	—	4	1	no	—
40	C₂⁴	Q₁₆	1	1	yes	characteristic
41	C₂²×C₄	—	2	1	no	—
42	C₂²×C₄	—	2	1	no	—
43	C₂²×C₄	—	2	1	no	—
44	C₂²×C₄	—	2	1	no	—
45	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄	D₈	1	1	yes	characteristic
49	C₂²×C₄	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄	SD₁₆	1	1	yes	characteristic
51	M₁₆	—	4	1	no	—
52	M₁₆	—	4	1	no	—
53	M₁₆	—	4	1	no	—
54	M₁₆	—	4	1	no	—
55	M₁₆	—	4	2	no	—
56	M₁₆	—	4	2	no	—
57	M₁₆	—	4	2	no	—
58	M₁₆	—	4	2	no	—
59	M₁₆	—	4	1	no	—
60	M₁₆	—	4	1	no	—
61	M₁₆	—	4	1	no	—
62	M₁₆	—	4	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	—	2	4	no	—
68	C₂×C₈	—	4	2	no	—
69	C₂×C₈	—	4	2	no	—
70	C₂×C₈	—	4	1	no	—
71	C₂×C₈	—	4	1	no	—
72	C₂×C₈	C₄²	1	1	yes	characteristic
73	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
76	C₁₆	—	4	2	no	—
77	C₂³	—	2	1	no	—
78	C₂³	—	2	1	no	—
79	C₂³	—	8	1	no	—
80	C₂³	#103	1	1	yes	characteristic
81	C₂³	#103	1	1	yes	characteristic
82	C₂³	#107	1	1	yes	characteristic
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	#95	1	1	yes	characteristic
94	C₂×C₄	#99	1	1	yes	characteristic
95	C₂×C₄	#99	1	1	yes	characteristic
96	C₂×C₄	#102	1	1	yes	characteristic
97	C₂×C₄	#102	1	1	yes	characteristic
98	C₂×C₄	#104	1	1	yes	characteristic
99	C₂×C₄	#104	1	1	yes	characteristic
100	C₂×C₄	#106	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	1	no	—
103	C₈	—	2	1	no	—
104	C₈	—	2	1	no	—
105	C₈	—	4	4	no	—
106	C₈	—	8	1	no	—
107	C₈	—	8	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	—	2	1	no	—
110	C₂²	—	4	1	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	1	no	—
113	C₂²	—	4	2	no	—
114	C₂²	—	8	1	no	—
115	C₂²	#328	1	1	yes	characteristic
116	C₂²	#328	1	1	yes	characteristic
117	C₂²	#340	1	1	yes	characteristic
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	#327	1	1	yes	characteristic
121	C₄	#327	1	1	yes	characteristic
122	C₄	#339	1	1	yes	characteristic
123	C₄	#342	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	—	4	1	no	—
128	C₂	—	4	1	no	—
129	C₂	#1294	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#349

C₂

yes

maximal, characteristic

#1811

C₂

yes

maximal, characteristic

C₂×#338

C₂

yes

maximal, characteristic

C₂²×M₁₆

—

C₂²×M₁₆

C₂²

yes

characteristic

C₂×M₃₂

—

C₂×#108

C₄

yes

characteristic

C₂×#108

C₄

yes

—

—

C₄

yes

—

#341

—

#338

—

#338

C₄

yes

—

C₂³×C₄

D₄

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄

yes

characteristic

C₂×M₁₆

C₂×C₄

yes

characteristic

C₂×M₁₆

D₄

yes

characteristic

C₂×M₁₆

Q₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#108

—

#108

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂⁴

Q₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

D₈

yes

characteristic

C₂²×C₄

SD₁₆

yes

characteristic

C₂²×C₄

SD₁₆

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

characteristic

C₁₆

—

C₂³

—

C₂³

—

C₂³

—

C₂³

yes

C₂³

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

#106

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

#328

yes

characteristic

116

C₂²

#328

yes

characteristic

117

C₂²

#340

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

#327

yes

characteristic

121

C₄

#327

yes

characteristic

122

C₄

#339

yes

characteristic

123

C₄

#342

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

—

127

C₂

—

128

C₂

—

129

C₂

#1294

yes

characteristic

	subgroup	G/H
center	C₄	#327
commutator	C₂×C₈	C₄²
Frattini, cosocle	C₂²×M₁₆	C₂²
socle	C₂	#1294
prime radical	C₂⁴	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#327

commutator

C₂×C₈

C₄²

Frattini, cosocle

C₂²×M₁₆

C₂²

socle

C₂

#1294

prime radical

C₂⁴

Q₁₆

LCS

G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G

LECS

G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#349	2
#1811	2
C₂×#338	2

subgroup

index

C₂×#349

#1811

C₂×#338

Subgroups — #7257

Conjugacy classes of subgroups

Maximal subgroups