Group #13195 — subgroups

Subgroups — #13195

Order 256.

219 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 219 conjugacy classes of subgroups — too many to draw. Only its 83 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×#443	C₂	1	1	yes	maximal, characteristic
2	C₂×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#350	C₂	1	2	yes	maximal, complemented
4	#2113	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#135	C₄	1	1	yes	characteristic
7	C₂×#131	C₂²	1	2	yes	—
8	C₂×M₃₂	C₂²	1	2	yes	—
9	C₂²×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆	C₂²	1	1	yes	characteristic
11	#443	—	2	2	no	—
12	#443	—	2	2	no	—
13	C₂×#108	C₄	1	1	yes	characteristic
14	C₂×#104	C₄	1	2	yes	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	#364	—	2	2	no	—
17	#350	—	2	4	no	—
18	C₂²⋊C₁₆	—	2	4	no	—
19	C₄×C₁₆	—	2	2	no	—
20	C₂×#41	C₂×C₄	1	2	yes	—
21	#135	—	2	2	no	—
22	#135	—	2	2	no	—
23	C₂×Q₁₆	C₈	1	1	yes	characteristic
24	C₂×SD₁₆	C₈	1	2	yes	—
25	C₂×D₈	C₈	1	1	yes	characteristic
26	#131	—	2	2	no	—
27	#131	—	2	2	no	—
28	#131	—	2	2	no	—
29	#131	—	2	2	no	—
30	C₂×M₁₆	—	2	2	no	—
31	C₂×M₁₆	C₂×C₄	1	2	yes	—
32	C₂²×C₈	—	2	2	no	—
33	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈	C₂³	1	1	yes	characteristic
35	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
36	M₃₂	—	2	2	no	—
37	M₃₂	—	2	2	no	—
38	C₂×C₁₆	—	2	1	no	—
39	C₂×C₁₆	—	2	1	no	—
40	C₂×C₁₆	—	2	1	no	—
41	C₂×C₁₆	—	2	1	no	—
42	C₂×C₁₆	—	2	2	no	—
43	C₂×C₁₆	D₄	1	2	yes	—
44	C₂×C₁₆	D₄	1	2	yes	—
45	#108	—	2	2	no	—
46	#104	—	2	4	no	—
47	C₄×C₈	D₄	1	2	yes	—
48	C₄×C₈	D₄	1	2	yes	—
49	#41	—	2	2	no	—
50	#41	—	2	2	no	—
51	#41	—	2	2	no	—
52	#41	—	2	2	no	—
53	C₂×Q₈	C₂×C₈	1	2	yes	—
54	C₂×D₄	—	2	2	no	—
55	C₂×D₄	C₂×C₈	1	2	yes	—
56	C₂²×C₄	—	2	1	no	—
57	C₂²×C₄	—	2	2	no	—
58	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
59	Q₁₆	—	2	2	no	—
60	SD₁₆	—	2	4	no	—
61	D₈	—	2	2	no	—
62	M₁₆	—	2	2	no	—
63	M₁₆	—	2	2	no	—
64	M₁₆	—	4	2	no	—
65	M₁₆	—	4	2	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	4	2	no	—
68	C₂×C₈	—	4	2	no	—
69	C₂×C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈	C₂×C₈	1	1	yes	characteristic
71	C₂×C₈	C₂×C₈	1	1	yes	characteristic
72	C₂×C₈	M₁₆	1	2	yes	—
73	C₂×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈	#41	1	1	yes	characteristic
78	C₂×C₈	#41	1	1	yes	characteristic
79	C₄²	—	2	1	no	—
80	C₄²	C₂²⋊C₄	1	2	yes	—
81	C₁₆	—	2	2	no	—
82	C₁₆	—	2	2	no	—
83	C₁₆	—	4	2	no	—
84	C₂³	—	2	2	no	—
85	C₂³	#131	1	1	yes	characteristic
86	Q₈	—	2	2	no	—
87	Q₈	—	2	2	no	—
88	D₄	—	2	2	no	—
89	D₄	—	2	2	no	—
90	D₄	—	4	2	no	—
91	D₄	—	4	2	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄	#131	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₈	C₂²⋊C₈	1	2	yes	—
105	C₈	C₂²⋊C₈	1	2	yes	—
106	C₈	#121	1	1	yes	characteristic
107	C₈	#121	1	1	yes	characteristic
108	C₈	#124	1	1	yes	characteristic
109	C₈	#127	1	1	yes	characteristic
110	C₂²	—	2	2	no	—
111	C₂²	—	4	4	no	—
112	C₂²	—	4	2	no	—
113	C₂²	#408	1	1	yes	characteristic
114	C₂²	C₈×D₄	1	1	yes	characteristic
115	C₂²	#436	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	—	4	2	no	—
118	C₄	#390	1	1	yes	characteristic
119	C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₄	C₈×D₄	1	1	yes	characteristic
121	C₄	#436	1	1	yes	characteristic
122	C₂	—	2	1	no	—
123	C₂	—	2	1	no	—
124	C₂	—	8	2	no	—
125	C₂	#1954	1	1	yes	characteristic
126	C₂	#2170	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#443

C₂

yes

maximal, characteristic

C₂×#364

C₂

yes

maximal, characteristic, complemented

C₂×#350

C₂

yes

maximal, complemented

#2113

C₂

yes

maximal, complemented

C₂×C₄×C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#135

C₄

yes

characteristic

C₂×#131

C₂²

yes

—

C₂×M₃₂

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

—

—

C₄

yes

characteristic

C₂×#104

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#364

—

#350

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₂×#41

C₂×C₄

yes

—

#135

—

#135

—

C₂×Q₁₆

C₈

yes

characteristic

C₂×SD₁₆

C₈

yes

—

C₂×D₈

C₈

yes

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

—

C₂×C₁₆

D₄

yes

—

#108

—

#104

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

—

—

—

—

C₂×Q₈

C₂×C₈

yes

—

C₂×D₄

—

C₂×D₄

C₂×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

yes

characteristic

C₂×C₈

yes

characteristic

C₂×C₈

M₁₆

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

C₂²⋊C₄

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

—

C₂³

#131

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

100

C₂×C₄

C₂²×C₈

yes

characteristic

101

C₂×C₄

C₂×M₁₆

yes

characteristic

102

C₂×C₄

#131

yes

characteristic

103

C₈

—

104

C₈

C₂²⋊C₈

yes

—

105

C₈

C₂²⋊C₈

yes

—

106

C₈

#121

yes

characteristic

107

C₈

#121

yes

characteristic

108

C₈

#124

yes

characteristic

109

C₈

#127

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

#408

yes

characteristic

114

C₂²

C₈×D₄

yes

characteristic

115

C₂²

#436

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

#390

yes

characteristic

119

C₄

C₂×C₂²⋊C₈

yes

characteristic

120

C₄

C₈×D₄

yes

characteristic

121

C₄

#436

yes

characteristic

122

C₂

—

123

C₂

—

124

C₂

—

125

C₂

#1954

yes

characteristic

126

C₂

#2170

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₈×D₄
prime radical	C₂×#135	C₄
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

C₄

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. C₂×#364	C₂	semidirect
3. C₂×#350	C₂	semidirect
4. #2113	C₂	semidirect
5. C₂×C₄×C₁₆	C₂	semidirect

H ≅ G/N

type

2. C₂×#364

C₂

semidirect

3. C₂×#350

C₂

semidirect

4. #2113

C₂

semidirect

5. C₂×C₄×C₁₆

C₂

semidirect

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

subgroup	index
C₂×#443	2
C₂×#364	2
C₂×#350	2
#2113	2
C₂×C₄×C₁₆	2

subgroup

index

C₂×C₄×C₁₆

Subgroups — #13195

Conjugacy classes of subgroups

Complements

Maximal subgroups