Group #13193 — subgroups

Subgroups — #13193

Order 256.

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

normal subgroups only

This group has 223 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	#2931	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#350	C₂	1	4	yes	maximal, complemented
4	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#131	C₂²	1	2	yes	—
6	#532	C₄	1	1	yes	characteristic
7	C₂×M₃₂	C₂²	1	2	yes	—
8	C₂²×C₁₆	C₂²	1	2	yes	—
9	#436	—	2	1	no	—
10	#436	—	2	1	no	—
11	C₈×D₄	—	2	1	no	—
12	C₈×D₄	—	2	1	no	—
13	#423	C₄	1	1	yes	characteristic
14	#408	C₄	1	2	yes	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆	—	2	1	no	—
17	C₄⋊C₁₆	—	2	1	no	—
18	#350	—	2	8	no	—
19	#346	—	2	1	no	—
20	C₄×C₁₆	—	2	1	no	—
21	C₂×#41	C₂×C₄	1	2	yes	—
22	#131	—	2	4	no	—
23	#131	—	2	4	no	—
24	C₂×M₁₆	—	2	2	no	—
25	C₂×M₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₈	—	2	2	no	—
27	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈	C₈	1	1	yes	characteristic
30	#127	C₈	1	1	yes	characteristic
31	#124	C₈	1	2	yes	—
32	#121	—	2	1	no	—
33	#121	—	2	1	no	—
34	C₄×D₄	—	2	1	no	—
35	C₄×D₄	—	2	1	no	—
36	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
37	M₃₂	—	2	2	no	—
38	M₃₂	D₄	1	4	yes	complemented
39	C₂×C₁₆	—	2	2	no	—
40	C₂×C₁₆	—	2	4	no	—
41	C₂×C₁₆	—	2	2	no	—
42	C₄⋊C₈	—	2	1	no	—
43	C₄⋊C₈	—	2	1	no	—
44	C₂²⋊C₈	—	2	2	no	—
45	C₂²⋊C₈	—	2	2	no	—
46	C₄×C₈	D₄	1	1	yes	characteristic
47	C₄×C₈	D₄	1	1	yes	characteristic
48	C₄×C₈	D₄	1	1	yes	characteristic
49	C₄×C₈	D₄	1	1	yes	characteristic
50	#41	—	2	4	no	—
51	#41	—	2	4	no	—
52	C₂×Q₈	C₂×C₈	1	2	yes	—
53	C₂×D₄	—	2	2	no	—
54	C₂×D₄	—	2	1	no	—
55	C₂×D₄	—	2	1	no	—
56	C₂×D₄	C₂×C₈	1	2	yes	—
57	C₂²×C₄	—	2	2	no	—
58	C₂²×C₄	M₁₆	1	1	yes	characteristic
59	C₂²×C₄	M₁₆	1	1	yes	characteristic
60	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
61	M₁₆	—	2	4	no	—
62	M₁₆	—	4	2	no	—
63	M₁₆	—	4	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	4	4	no	—
67	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈	C₂×D₄	1	2	yes	—
70	C₂×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈	#41	1	2	yes	—
73	C₄⋊C₄	—	2	1	no	—
74	C₄⋊C₄	—	2	1	no	—
75	C₂²⋊C₄	—	2	2	no	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²	C₂×C₈	1	1	yes	characteristic
80	C₄²	C₂×C₈	1	1	yes	characteristic
81	C₁₆	—	2	4	no	—
82	C₁₆	—	4	2	no	—
83	C₂³	—	2	2	no	—
84	C₂³	C₂×M₁₆	1	1	yes	characteristic
85	Q₈	—	2	4	no	—
86	D₄	—	2	4	no	—
87	D₄	—	4	2	no	—
88	D₄	—	4	4	no	—
89	D₄	—	4	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	4	1	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
97	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄	C₄×D₄	1	2	yes	—
101	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄	#131	1	2	yes	—
103	C₈	—	2	1	no	—
104	C₈	—	2	1	no	—
105	C₈	—	4	2	no	—
106	C₈	#121	1	2	yes	—
107	C₈	#124	1	1	yes	characteristic
108	C₈	#127	1	1	yes	characteristic
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	4	2	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	2	no	—
114	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₂²	#436	1	2	yes	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	1	no	—
119	C₄	—	4	2	no	—
120	C₄	#350	1	2	yes	complemented
121	C₄	#390	1	1	yes	characteristic
122	C₄	#408	1	1	yes	characteristic
123	C₄	C₈×D₄	1	2	yes	—
124	C₂	—	2	2	no	—
125	C₂	—	8	2	no	—
126	C₂	#1954	1	1	yes	characteristic
127	C₂	C₂×#350	1	1	yes	characteristic
128	C₂	#2117	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2931

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#350

C₂

yes

maximal, complemented

C₄×M₃₂

C₂

yes

maximal, characteristic, complemented

C₂×#131

C₂²

yes

—

#532

C₄

yes

characteristic

C₂×M₃₂

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

—

#436

—

#436

—

C₈×D₄

—

C₈×D₄

—

#423

C₄

yes

characteristic

#408

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#350

—

#346

—

C₄×C₁₆

—

C₂×#41

C₂×C₄

yes

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

C₈

yes

characteristic

#127

C₈

yes

characteristic

#124

C₈

yes

—

#121

—

#121

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

C₂×C₄

yes

characteristic

M₃₂

—

M₃₂

D₄

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

#41

—

#41

—

C₂×Q₈

C₂×C₈

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

M₁₆

yes

characteristic

C₂²×C₄

M₁₆

yes

characteristic

C₂²×C₄

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊C₄

yes

characteristic

C₄²

C₂²⋊C₄

yes

characteristic

C₄²

C₂×C₈

yes

characteristic

C₄²

C₂×C₈

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

—

C₂³

C₂×M₁₆

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₈

yes

characteristic

C₂×C₄

C₂²⋊C₈

yes

characteristic

C₂×C₄

C₂²⋊C₈

yes

characteristic

C₂×C₄

C₂²⋊C₈

yes

characteristic

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

100

C₂×C₄

C₄×D₄

yes

—

101

C₂×C₄

C₂²×C₈

yes

characteristic

102

C₂×C₄

#131

yes

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

#121

yes

—

107

C₈

#124

yes

characteristic

108

C₈

#127

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×C₂²⋊C₈

yes

characteristic

115

C₂²

#436

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#350

yes

complemented

121

C₄

#390

yes

characteristic

122

C₄

#408

yes

characteristic

123

C₄

C₈×D₄

yes

—

124

C₂

—

125

C₂

—

126

C₂

#1954

yes

characteristic

127

C₂

C₂×#350

yes

characteristic

128

C₂

#2117

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×C₂²⋊C₈
prime radical	#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ 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

C₂×C₂²⋊C₈

C₄

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. C₂×C₄⋊C₁₆	C₂	semidirect
3. C₂×#350	C₂	semidirect
4. C₄×M₃₂	C₂	semidirect
38. M₃₂	D₄	semidirect
120. C₄	#350	semidirect

H ≅ G/N

type

2. C₂×C₄⋊C₁₆

C₂

semidirect

3. C₂×#350

C₂

semidirect

4. C₄×M₃₂

C₂

semidirect

38. M₃₂

D₄

semidirect

120. C₄

#350

semidirect

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

subgroup	index
#2931	2
C₂×C₄⋊C₁₆	2
C₂×#350	2
C₄×M₃₂	2

subgroup

index

#2931

C₂×C₄⋊C₁₆

C₂×#350

C₄×M₃₂

Subgroups — #13193

Conjugacy classes of subgroups

Complements

Maximal subgroups