Group #13107 — subgroups

Subgroups — #13107

Order 256.

135 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 135 conjugacy classes of subgroups — too many to draw. Only its 63 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×#364	C₂	1	1	yes	maximal, characteristic
2	C₂×#347	C₂	1	1	yes	maximal, characteristic
3	#1771	C₂	1	1	yes	maximal, characteristic
4	#1377	C₂	1	1	yes	maximal, characteristic
5	#1377	C₂	1	1	yes	maximal, characteristic
6	#1376	C₂	1	1	yes	maximal, characteristic
7	#1375	C₂	1	1	yes	maximal, characteristic
8	C₂×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈	C₄	1	1	yes	characteristic
13	C₂×#97	C₂²	1	1	yes	characteristic
14	C₂×#97	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
17	#364	—	2	2	no	—
18	#347	—	2	2	no	—
19	#336	C₄	1	1	yes	characteristic
20	#336	C₄	1	1	yes	characteristic
21	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊C₄	—	2	1	no	—
26	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
29	M₃₂	—	2	1	no	—
30	M₃₂	—	2	1	no	—
31	M₃₂	—	2	1	no	—
32	M₃₂	—	2	1	no	—
33	M₃₂	—	2	1	no	—
34	M₃₂	—	2	1	no	—
35	M₃₂	D₄	1	2	yes	—
36	M₃₂	Q₈	1	2	yes	—
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	C₄⋊C₈	—	2	2	no	—
42	#97	—	2	1	no	—
43	#97	—	2	1	no	—
44	#97	—	2	1	no	—
45	#97	—	2	1	no	—
46	C₄×C₈	—	2	1	no	—
47	C₄×C₈	—	2	1	no	—
48	#95	—	2	1	no	—
49	C₂²×C₄	—	2	1	no	—
50	C₂²×C₄	—	2	1	no	—
51	C₂²×C₄	M₁₆	1	1	yes	characteristic
52	C₂²×C₄	M₁₆	1	1	yes	characteristic
53	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	—	2	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	2	1	no	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	M₁₆	1	2	yes	—
63	C₂×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈	#41	1	1	yes	characteristic
66	C₂×C₈	#41	1	1	yes	characteristic
67	C₂×C₈	#41	1	1	yes	characteristic
68	C₂×C₈	#41	1	1	yes	characteristic
69	C₄⋊C₄	—	4	2	no	—
70	C₄²	—	2	1	no	—
71	C₄²	—	2	1	no	—
72	C₄²	—	2	1	no	—
73	C₄²	—	2	1	no	—
74	C₄²	—	2	1	no	—
75	C₄²	—	2	1	no	—
76	C₁₆	—	2	2	no	—
77	C₁₆	—	2	2	no	—
78	C₁₆	—	4	1	no	—
79	C₁₆	—	4	1	no	—
80	C₂³	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	—	4	1	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	—	4	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	#117	1	1	yes	characteristic
92	C₂×C₄	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
94	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄	#131	1	1	yes	characteristic
96	C₂×C₄	#131	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₈	—	4	1	no	—
99	C₈	—	4	1	no	—
100	C₈	C₂²⋊Q₈	1	1	yes	characteristic
101	C₈	#123	1	1	yes	characteristic
102	C₈	#125	1	1	yes	characteristic
103	C₈	#126	1	1	yes	characteristic
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	#432	1	1	yes	characteristic
107	C₂²	#436	1	1	yes	characteristic
108	C₂²	#446	1	1	yes	characteristic
109	C₄	—	4	2	no	—
110	C₄	—	4	1	no	—
111	C₄	—	4	1	no	—
112	C₄	—	8	1	no	—
113	C₄	#387	1	1	yes	characteristic
114	C₄	#433	1	1	yes	characteristic
115	C₄	#435	1	1	yes	characteristic
116	C₄	#446	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	#1917	1	1	yes	characteristic
120	C₂	#2165	1	1	yes	characteristic
121	C₂	#2166	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#364

C₂

yes

maximal, characteristic

C₂×#347

C₂

yes

maximal, characteristic

#1771

C₂

yes

maximal, characteristic

#1377

C₂

yes

maximal, characteristic

#1377

C₂

yes

maximal, characteristic

#1376

C₂

yes

maximal, characteristic

#1375

C₂

yes

maximal, characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₄

yes

characteristic

C₂×#97

C₂²

yes

characteristic

C₂×#97

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₄

yes

—

—

C₄

yes

characteristic

#336

C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

D₄

yes

—

M₃₂

Q₈

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

M₁₆

yes

characteristic

C₂²×C₄

M₁₆

yes

characteristic

C₂²×C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

M₁₆

yes

—

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

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

C₂×M₁₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

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

C₂×M₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

100

C₈

C₂²⋊Q₈

yes

characteristic

101

C₈

#123

yes

characteristic

102

C₈

#125

yes

characteristic

103

C₈

#126

yes

characteristic

104

C₂²

—

105

C₂²

—

106

C₂²

#432

yes

characteristic

107

C₂²

#436

yes

characteristic

108

C₂²

#446

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#387

yes

characteristic

114

C₄

#433

yes

characteristic

115

C₄

#435

yes

characteristic

116

C₄

#446

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

#1917

yes

characteristic

120

C₂

#2165

yes

characteristic

121

C₂

#2166

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#117
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	#432
prime radical	C₂³	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

C₂×C₄

C₂²×C₄

C₂²×C₈

C₂³

socle

C₂²

#432

prime radical

C₂³

C₂×M₁₆

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ 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
C₂×#364	2
C₂×#347	2
#1771	2
#1377	2
#1376	2
#1375	2

subgroup

index

Subgroups — #13107

Conjugacy classes of subgroups

Maximal subgroups