Group #12820 — subgroups

Subgroups — #12820

Order 256.

207 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 79 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	#2139	C₂	1	2	yes	maximal
4	#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	#1387	C₂	1	2	yes	maximal, complemented
6	C₂×#135	C₂²	1	1	yes	characteristic
7	C₂×#131	—	2	1	no	—
8	C₂×M₃₂	C₂²	1	2	yes	—
9	#443	—	2	2	no	—
10	#443	C₄	1	4	yes	complemented
11	C₂×#108	C₂²	1	1	yes	characteristic
12	C₂×#107	C₂²	1	2	yes	—
13	C₂×#106	—	2	1	no	—
14	C₂×#104	—	2	1	no	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	#382	—	2	1	no	—
17	#364	—	2	2	no	—
18	#358	—	2	2	no	—
19	#357	—	2	2	no	—
20	C₂×#41	—	2	1	no	—
21	#135	—	2	1	no	—
22	#135	—	2	1	no	—
23	#135	C₂×C₄	1	2	yes	—
24	#135	Q₈	1	2	yes	—
25	C₂×Q₁₆	D₄	1	1	yes	characteristic
26	C₂×SD₁₆	—	2	1	no	—
27	C₂×D₈	D₄	1	1	yes	characteristic
28	#131	—	2	4	no	—
29	#131	—	4	2	no	—
30	C₂×M₁₆	—	2	1	no	—
31	C₂×M₁₆	—	4	1	no	—
32	C₂²×C₈	—	4	1	no	—
33	C₂²×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₈	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄	—	2	2	no	—
36	C₂×C₄²	D₄	1	1	yes	characteristic
37	M₃₂	—	2	2	no	—
38	M₃₂	—	2	2	no	—
39	C₂×C₁₆	—	2	2	no	—
40	#108	—	2	1	no	—
41	#108	C₂×C₄	1	2	yes	—
42	#107	—	2	2	no	—
43	#107	—	2	2	no	—
44	#106	—	4	1	no	—
45	#106	—	4	1	no	—
46	#104	—	2	4	no	—
47	C₄×C₈	C₂×C₄	1	2	yes	—
48	C₄×C₈	D₄	1	2	yes	—
49	#95	—	4	2	no	—
50	#41	—	2	2	no	—
51	#41	—	2	2	no	—
52	#41	—	4	1	no	—
53	#41	—	4	1	no	—
54	C₂×Q₈	—	2	1	no	—
55	C₂×D₄	—	2	1	no	—
56	C₂×D₄	—	4	1	no	—
57	C₂²×C₄	—	2	1	no	—
58	C₂²×C₄	—	4	1	no	—
59	C₂²×C₄	—	4	2	no	—
60	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
61	Q₁₆	—	2	1	no	—
62	Q₁₆	C₄⋊C₄	1	2	yes	—
63	SD₁₆	—	2	2	no	—
64	SD₁₆	—	2	2	no	—
65	D₈	—	2	1	no	—
66	D₈	C₄⋊C₄	1	2	yes	—
67	M₁₆	—	2	2	no	—
68	M₁₆	—	4	1	no	—
69	M₁₆	—	4	4	no	—
70	C₂×C₈	—	4	1	no	—
71	C₂×C₈	—	4	4	no	—
72	C₂×C₈	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈	D₈	1	1	yes	characteristic
74	C₂×C₈	D₈	1	1	yes	characteristic
75	C₂×C₈	SD₁₆	1	2	yes	—
76	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₈	#41	1	1	yes	characteristic
81	C₂×C₈	#41	1	1	yes	characteristic
82	C₄⋊C₄	—	4	2	no	—
83	C₄⋊C₄	—	4	2	no	—
84	C₄²	—	2	1	no	—
85	C₄²	C₂²⋊C₄	1	2	yes	—
86	C₁₆	—	4	2	no	—
87	C₂³	—	4	1	no	—
88	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
89	Q₈	—	2	2	no	—
90	Q₈	—	4	1	no	—
91	D₄	—	2	2	no	—
92	D₄	—	4	2	no	—
93	D₄	—	4	2	no	—
94	D₄	—	4	1	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	8	4	no	—
101	C₂×C₄	—	8	2	no	—
102	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄	#121	1	1	yes	characteristic
104	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄	#137	1	1	yes	characteristic
108	C₈	—	4	2	no	—
109	C₈	#102	1	2	yes	—
110	C₈	#102	1	2	yes	—
111	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₈	#117	1	1	yes	characteristic
113	C₈	#121	1	1	yes	characteristic
114	C₈	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂²	—	2	2	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	8	2	no	—
118	C₂²	#417	1	1	yes	characteristic
119	C₂²	#459	1	1	yes	characteristic
120	C₂²	#476	1	1	yes	characteristic
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₄	—	16	2	no	—
124	C₄	#380	1	1	yes	characteristic
125	C₄	C₂×#102	1	1	yes	characteristic
126	C₄	#461	1	1	yes	characteristic
127	C₄	#474	1	1	yes	characteristic
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	—	8	2	no	—
131	C₂	#1802	1	1	yes	characteristic
132	C₂	#2212	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#443

C₂

yes

maximal, characteristic

C₂×#364

C₂

yes

maximal, characteristic, complemented

#2139

C₂

yes

maximal

#1846

C₂

yes

maximal, characteristic, complemented

#1387

C₂

yes

maximal, complemented

C₂×#135

C₂²

yes

characteristic

C₂×#131

—

C₂×M₃₂

C₂²

yes

—

#443

—

#443

C₄

yes

complemented

C₂×#108

C₂²

yes

characteristic

C₂×#107

C₂²

yes

—

C₂×#106

—

C₂×#104

—

C₂×C₄×C₈

C₂²

yes

—

—

—

—

C₂×#41

—

#135

—

#135

—

#135

C₂×C₄

yes

—

#135

Q₈

yes

—

C₂×Q₁₆

D₄

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

D₄

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#108

—

#108

C₂×C₄

yes

—

—

—

—

—

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

—

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

C₄⋊C₄

yes

—

SD₁₆

—

SD₁₆

—

D₈

—

D₈

C₄⋊C₄

yes

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

D₈

yes

characteristic

C₂×C₈

D₈

yes

characteristic

C₂×C₈

SD₁₆

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂²⋊C₄

yes

—

C₁₆

—

C₂³

—

C₂³

AGL(1,ℤ/8)

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

103

C₂×C₄

#121

yes

characteristic

104

C₂×C₄

C₂²⋊Q₈

yes

characteristic

105

C₂×C₄

C₂×D₈

yes

characteristic

106

C₂×C₄

C₂×SD₁₆

yes

characteristic

107

C₂×C₄

#137

yes

characteristic

108

C₈

—

109

C₈

#102

yes

—

110

C₈

#102

yes

—

111

C₈

C₂×C₄⋊C₄

yes

characteristic

112

C₈

#117

yes

characteristic

113

C₈

#121

yes

characteristic

114

C₈

C₂²⋊Q₈

yes

characteristic

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

#417

yes

characteristic

119

C₂²

#459

yes

characteristic

120

C₂²

#476

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#380

yes

characteristic

125

C₄

C₂×#102

yes

characteristic

126

C₄

#461

yes

characteristic

127

C₄

#474

yes

characteristic

128

C₂

—

129

C₂

—

130

C₂

—

131

C₂

#1802

yes

characteristic

132

C₂

#2212

yes

—

	subgroup	G/H
center, socle	C₂²	#459
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#459

commutator

C₂×C₈

C₂²×C₄

Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂×#135

C₂²

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. C₂×#364	C₂	semidirect
4. #1846	C₂	semidirect
5. #1387	C₂	semidirect
10. #443	C₄	semidirect

H ≅ G/N

type

2. C₂×#364

C₂

semidirect

4. #1846

C₂

semidirect

5. #1387

C₂

semidirect

10. #443

C₄

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
#2139	2
#1846	2
#1387	2

subgroup

index

Subgroups — #12820

Conjugacy classes of subgroups

Complements

Maximal subgroups