Group #13213 — subgroups

Subgroups — #13213

Order 256.

287 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 87 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×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×#361	C₂	1	4	yes	maximal, complemented
3	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈	—	2	1	no	—
6	C₂²×D₈	C₂²	1	2	yes	—
7	C₂×#127	—	2	1	no	—
8	C₂×M₃₂	C₂²	1	2	yes	—
9	#493	—	2	1	no	—
10	#493	—	2	1	no	—
11	#493	C₄	1	2	yes	—
12	#493	C₄	1	2	yes	—
13	C₂×#108	—	2	1	no	—
14	C₂×#108	C₂²	1	2	yes	—
15	#423	—	2	1	no	—
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	#361	—	2	8	no	—
18	#346	—	2	1	no	—
19	C₄×C₁₆	—	2	1	no	—
20	C₂²×D₄	—	2	2	no	—
21	C₂²×D₄	—	4	2	no	—
22	C₂×D₈	—	2	2	no	—
23	C₂×D₈	—	2	2	no	—
24	C₂×D₈	—	2	4	no	—
25	C₂×D₈	—	2	2	no	—
26	C₂×D₈	—	2	4	no	—
27	C₂×D₈	—	4	2	no	—
28	C₂×D₈	—	4	2	no	—
29	C₂×D₈	C₂×C₄	1	4	yes	—
30	C₂×M₁₆	—	2	2	no	—
31	C₂²×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₈	C₂³	1	1	yes	characteristic
33	#127	—	2	2	no	—
34	#127	—	2	2	no	—
35	#127	—	4	2	no	—
36	C₂×C₄²	D₄	1	1	yes	characteristic
37	M₃₂	—	2	2	no	—
38	M₃₂	D₄	1	4	yes	complemented
39	C₂×C₁₆	—	2	2	no	—
40	#108	—	2	4	no	—
41	#108	—	4	2	no	—
42	C₄⋊C₈	—	4	2	no	—
43	C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈	D₄	1	1	yes	characteristic
46	C₄×C₈	D₄	1	1	yes	characteristic
47	C₂⁴	—	4	2	no	—
48	C₂×D₄	—	2	4	no	—
49	C₂×D₄	—	4	4	no	—
50	C₂×D₄	—	4	4	no	—
51	C₂×D₄	—	4	2	no	—
52	C₂×D₄	—	4	2	no	—
53	C₂×D₄	—	4	2	no	—
54	C₂×D₄	—	4	2	no	—
55	C₂×D₄	—	4	2	no	—
56	C₂×D₄	—	8	8	no	—
57	C₂²×C₄	SD₁₆	1	2	yes	—
58	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
59	D₈	—	2	8	no	—
60	D₈	—	4	4	no	—
61	D₈	—	4	4	no	—
62	D₈	—	4	4	no	—
63	M₁₆	—	4	2	no	—
64	M₁₆	—	4	2	no	—
65	C₂×C₈	—	4	2	no	—
66	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈	D₈	1	2	yes	—
69	C₂×C₈	SD₁₆	1	2	yes	—
70	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈	C₂×D₄	1	2	yes	—
72	C₂×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈	#41	1	2	yes	—
74	C₄²	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²	D₈	1	2	yes	—
77	C₁₆	—	2	4	no	—
78	C₂³	—	4	4	no	—
79	C₂³	—	8	8	no	—
80	C₂³	—	8	2	no	—
81	C₂³	—	8	2	no	—
82	C₂³	C₂×SD₁₆	1	1	yes	characteristic
83	D₄	—	4	4	no	—
84	D₄	—	4	4	no	—
85	D₄	—	8	8	no	—
86	D₄	—	8	8	no	—
87	D₄	—	8	2	no	—
88	D₄	—	8	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	#102	1	2	yes	—
91	C₂×C₄	#102	1	2	yes	—
92	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄	#124	1	1	yes	characteristic
94	C₂×C₄	#127	1	1	yes	characteristic
95	C₂×C₄	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
98	C₈	—	8	2	no	—
99	C₈	#102	1	2	yes	—
100	C₈	#102	1	2	yes	—
101	C₈	C₄×D₄	1	2	yes	—
102	C₈	#121	1	2	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	—	8	8	no	—
105	C₂²	—	8	4	no	—
106	C₂²	—	16	4	no	—
107	C₂²	—	16	4	no	—
108	C₂²	C₂×#102	1	1	yes	characteristic
109	C₂²	#486	1	1	yes	characteristic
110	C₂²	#492	1	1	yes	characteristic
111	C₄	—	2	2	no	—
112	C₄	#361	1	4	yes	complemented
113	C₄	#390	1	1	yes	characteristic
114	C₄	C₂×#102	1	1	yes	characteristic
115	C₄	#486	1	1	yes	characteristic
116	C₄	#493	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	16	4	no	—
120	C₂	#1957	1	1	yes	characteristic
121	C₂	C₂×#361	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#493

C₂

yes

maximal, characteristic

C₂×#361

C₂

yes

maximal, complemented

C₄×M₃₂

C₂

yes

maximal, characteristic, complemented

#1848

C₂

yes

maximal, characteristic, complemented

C₂²×D₈

—

C₂²×D₈

C₂²

yes

—

C₂×#127

—

C₂×M₃₂

C₂²

yes

—

#493

—

#493

—

#493

C₄

yes

—

#493

C₄

yes

—

C₂×#108

—

C₂×#108

C₂²

yes

—

#423

—

C₂×C₄×C₈

C₂²

yes

characteristic

#361

—

#346

—

C₄×C₁₆

—

C₂²×D₄

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

C₂×C₄

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

—

—

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

D₄

yes

complemented

C₂×C₁₆

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

SD₁₆

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

D₈

—

D₈

—

D₈

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

D₈

yes

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

#41

yes

—

C₄²

C₂²⋊C₄

yes

characteristic

C₄²

C₂²⋊C₄

yes

characteristic

C₄²

D₈

yes

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×SD₁₆

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

#102

yes

—

C₂×C₄

#102

yes

—

C₂×C₄

C₂×C₂²⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₈

—

C₈

#102

yes

—

100

C₈

#102

yes

—

101

C₈

C₄×D₄

yes

—

102

C₈

#121

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₂×#102

yes

characteristic

109

C₂²

#486

yes

characteristic

110

C₂²

#492

yes

characteristic

111

C₄

—

112

C₄

#361

yes

complemented

113

C₄

#390

yes

characteristic

114

C₄

C₂×#102

yes

characteristic

115

C₄

#486

yes

characteristic

116

C₄

#493

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#1957

yes

characteristic

121

C₂

C₂×#361

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#102
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#102

commutator

C₂×C₈

C₂²×C₄

Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂×#493

C₂

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. C₂×#361	C₂	semidirect
3. C₄×M₃₂	C₂	semidirect
4. #1848	C₂	semidirect
38. M₃₂	D₄	semidirect
112. C₄	#361	semidirect

H ≅ G/N

type

2. C₂×#361

C₂

semidirect

3. C₄×M₃₂

C₂

semidirect

4. #1848

C₂

semidirect

38. M₃₂

D₄

semidirect

112. C₄

#361

semidirect

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

subgroup	index
C₂×#493	2
C₂×#361	2
C₄×M₃₂	2
#1848	2

subgroup

index

C₂×#493

C₂×#361

C₄×M₃₂

#1848

Subgroups — #13213

Conjugacy classes of subgroups

Complements

Maximal subgroups