Group #13106 — subgroups

Subgroups — #13106

Order 256.

136 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 71 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	#2151	C₂	1	1	yes	maximal, characteristic
2	#2108	C₂	1	1	yes	maximal, characteristic
3	#1772	C₂	1	1	yes	maximal, characteristic
4	#1379	C₂	1	1	yes	maximal, characteristic
5	#1379	C₂	1	1	yes	maximal, characteristic
6	#1376	C₂	1	1	yes	maximal, characteristic
7	#1374	C₂	1	1	yes	maximal, characteristic
8	C₂×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×#108	C₄	1	1	yes	characteristic
13	#427	C₄	1	1	yes	characteristic
14	#405	C₂²	1	1	yes	characteristic
15	#405	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆	—	2	1	no	—
18	C₄⋊C₁₆	—	2	1	no	—
19	#346	—	2	1	no	—
20	C₄×C₁₆	—	2	1	no	—
21	#339	C₄	1	1	yes	characteristic
22	#339	C₄	1	1	yes	characteristic
23	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈	C₂³	1	1	yes	characteristic
27	#117	C₂×C₄	1	1	yes	characteristic
28	#117	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
30	M₃₂	—	2	1	no	—
31	M₃₂	—	2	1	no	—
32	M₃₂	—	2	1	no	—
33	M₃₂	Q₈	1	2	yes	—
34	C₂×C₁₆	—	2	1	no	—
35	C₂×C₁₆	—	2	1	no	—
36	C₂×C₁₆	—	2	1	no	—
37	C₂×C₁₆	—	2	1	no	—
38	C₂×C₁₆	—	2	1	no	—
39	C₂×C₁₆	—	2	1	no	—
40	C₂×C₁₆	—	2	1	no	—
41	C₂×C₁₆	D₄	1	1	yes	characteristic
42	C₂×C₁₆	D₄	1	1	yes	characteristic
43	#108	—	2	2	no	—
44	#107	C₈	1	1	yes	characteristic
45	#107	C₈	1	1	yes	characteristic
46	#106	C₈	1	1	yes	characteristic
47	#106	C₈	1	1	yes	characteristic
48	#97	—	2	1	no	—
49	#97	—	2	1	no	—
50	C₄×C₈	—	2	1	no	—
51	C₄×C₈	—	2	1	no	—
52	C₄×C₈	—	2	1	no	—
53	C₄×C₈	—	2	1	no	—
54	C₂²×C₄	—	2	1	no	—
55	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
56	M₁₆	—	2	1	no	—
57	M₁₆	—	2	1	no	—
58	M₁₆	—	2	1	no	—
59	M₁₆	M₁₆	1	2	yes	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	C₂×C₈	1	1	yes	characteristic
65	C₂×C₈	C₂×C₈	1	1	yes	characteristic
66	C₂×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈	#41	1	1	yes	characteristic
69	C₂×C₈	#41	1	1	yes	characteristic
70	C₂×C₈	#41	1	1	yes	characteristic
71	C₂×C₈	#41	1	1	yes	characteristic
72	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
73	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
74	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
75	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
76	C₂²⋊C₄	—	2	1	no	—
77	C₂²⋊C₄	—	2	1	no	—
78	C₄²	—	2	1	no	—
79	C₄²	—	2	1	no	—
80	C₄²	—	2	1	no	—
81	C₄²	—	2	1	no	—
82	C₁₆	—	2	2	no	—
83	C₁₆	—	2	1	no	—
84	C₁₆	—	2	1	no	—
85	C₁₆	—	4	1	no	—
86	C₁₆	—	4	1	no	—
87	C₂³	#131	1	1	yes	characteristic
88	C₂×C₄	—	2	2	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	—	4	1	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	#117	1	1	yes	characteristic
96	C₂×C₄	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
98	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄	#131	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	1	no	—
103	C₈	—	2	2	no	—
104	C₈	—	4	1	no	—
105	C₈	C₂²⋊Q₈	1	1	yes	characteristic
106	C₈	#123	1	1	yes	characteristic
107	C₈	#125	1	1	yes	characteristic
108	C₈	#126	1	1	yes	characteristic
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	#433	1	1	yes	characteristic
112	C₂²	C₈×D₄	1	1	yes	characteristic
113	C₂²	#446	1	1	yes	characteristic
114	C₄	—	4	2	no	—
115	C₄	—	4	1	no	—
116	C₄	—	4	1	no	—
117	C₄	—	4	1	no	—
118	C₄	—	4	1	no	—
119	C₄	#387	1	1	yes	characteristic
120	C₄	#431	1	1	yes	characteristic
121	C₄	#435	1	1	yes	characteristic
122	C₄	C₈×Q₈	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#1916	1	1	yes	characteristic
126	C₂	#2170	1	1	yes	characteristic
127	C₂	#2171	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2151

C₂

yes

maximal, characteristic

#2108

C₂

yes

maximal, characteristic

#1772

C₂

yes

maximal, characteristic

#1379

C₂

yes

maximal, characteristic

#1379

C₂

yes

maximal, characteristic

#1376

C₂

yes

maximal, characteristic

#1374

C₂

yes

maximal, characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

characteristic

C₂×#108

C₄

yes

characteristic

#427

C₄

yes

characteristic

#405

C₂²

yes

characteristic

#405

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

—

#339

C₄

yes

characteristic

#339

C₄

yes

characteristic

C₂×M₁₆

C₂×C₄

yes

characteristic

C₂×M₁₆

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#117

C₂×C₄

yes

characteristic

#117

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

Q₈

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

characteristic

C₂×C₁₆

D₄

yes

characteristic

#108

—

#107

C₈

yes

characteristic

#107

C₈

yes

characteristic

#106

C₈

yes

characteristic

#106

C₈

yes

characteristic

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

characteristic

C₂×C₈

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

C₂×C₈

yes

characteristic

C₄⋊C₄

C₂×C₈

yes

characteristic

C₄⋊C₄

C₂×C₈

yes

characteristic

C₄⋊C₄

C₂×C₈

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#131

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#117

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×Q₈

yes

characteristic

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

100

C₂×C₄

#131

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

C₂²⋊Q₈

yes

characteristic

106

C₈

#123

yes

characteristic

107

C₈

#125

yes

characteristic

108

C₈

#126

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

#433

yes

characteristic

112

C₂²

C₈×D₄

yes

characteristic

113

C₂²

#446

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#387

yes

characteristic

120

C₄

#431

yes

characteristic

121

C₄

#435

yes

characteristic

122

C₄

C₈×Q₈

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#1916

yes

characteristic

126

C₂

#2170

yes

characteristic

127

C₂

#2171

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₈×D₄
prime radical	C₂³	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

C₂×D₄

commutator

C₂×C₄

C₂²×C₈

Frattini, cosocle

C₂²×C₈

C₂³

socle

C₂²

C₈×D₄

C₂³

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#2151	2
#2108	2
#1772	2
#1379	2
#1376	2
#1374	2

subgroup

index

Subgroups — #13106

Conjugacy classes of subgroups

Maximal subgroups