Group #10665 — subgroups

Subgroups — #10665

Order 256.

167 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 167 conjugacy classes of subgroups — too many to draw. Only its 53 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	#1963	C₂	1	1	yes	maximal, characteristic
2	#1832	C₂	1	1	yes	maximal, characteristic, complemented
3	#1571	C₂	1	1	yes	maximal, characteristic, complemented
4	#1570	C₂	1	1	yes	maximal, characteristic, complemented
5	#1480	C₂	1	1	yes	maximal, characteristic
6	#1480	C₂	1	1	yes	maximal, characteristic
7	#1274	C₂	1	1	yes	maximal, characteristic, complemented
8	#532	C₂²	1	1	yes	characteristic
9	#431	C₂²	1	1	yes	characteristic
10	#431	C₂²	1	1	yes	characteristic
11	#423	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×#104	—	2	2	no	—
15	#376	C₂²	1	1	yes	characteristic
16	#336	—	2	2	no	—
17	#335	—	2	2	no	—
18	#334	—	2	2	no	—
19	D₄⋊C₈	—	2	2	no	—
20	D₄⋊C₈	—	2	2	no	—
21	C₂×#41	—	2	1	no	—
22	C₂×M₁₆	—	2	1	no	—
23	C₂²×C₈	—	2	1	no	—
24	C₂²×C₈	—	2	1	no	—
25	C₄⋊Q₈	Q₈	1	1	yes	characteristic
26	#127	Q₈	1	1	yes	characteristic
27	#124	—	2	1	no	—
28	#121	—	2	1	no	—
29	#121	—	2	1	no	—
30	C₄×D₄	—	2	1	no	—
31	C₄×D₄	—	2	1	no	—
32	C₂×C₄²	D₄	1	2	yes	—
33	C₂×C₄²	C₂³	1	1	yes	characteristic
34	C₄⋊C₈	—	2	1	no	—
35	C₄⋊C₈	—	2	1	no	—
36	C₄⋊C₈	—	2	1	no	—
37	C₄⋊C₈	—	2	1	no	—
38	C₄⋊C₈	—	2	1	no	—
39	C₄⋊C₈	—	2	1	no	—
40	C₄⋊C₈	—	2	1	no	—
41	C₄⋊C₈	—	2	1	no	—
42	#104	—	4	4	no	—
43	C₂²⋊C₈	—	2	1	no	—
44	C₂²⋊C₈	—	2	1	no	—
45	C₄×C₈	D₄	1	2	yes	—
46	C₄×C₈	D₄	1	2	yes	—
47	#95	—	2	1	no	—
48	#95	—	2	1	no	—
49	#41	—	4	2	no	—
50	#41	—	4	2	no	—
51	C₂×Q₈	—	2	1	no	—
52	C₂×D₄	—	2	1	no	—
53	C₂×D₄	—	2	1	no	—
54	C₂×D₄	—	2	1	no	—
55	C₂×D₄	—	4	1	no	—
56	C₂²×C₄	—	2	2	no	—
57	C₂²×C₄	—	4	1	no	—
58	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄	#41	1	1	yes	characteristic
60	C₂²×C₄	#41	1	1	yes	characteristic
61	M₁₆	—	4	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	4	1	no	—
65	C₂×C₈	—	4	1	no	—
66	C₂×C₈	—	4	1	no	—
67	C₂×C₈	—	4	2	no	—
68	C₂×C₈	—	4	2	no	—
69	C₄⋊C₄	—	2	1	no	—
70	C₄⋊C₄	—	2	1	no	—
71	C₂²⋊C₄	—	4	1	no	—
72	C₂²⋊C₄	—	4	1	no	—
73	C₄²	—	2	2	no	—
74	C₄²	—	2	2	no	—
75	C₄²	C₂×D₄	1	1	yes	characteristic
76	C₄²	C₂×D₄	1	1	yes	characteristic
77	C₄²	C₂×Q₈	1	1	yes	characteristic
78	C₄²	#41	1	1	yes	characteristic
79	C₂³	—	4	1	no	—
80	C₂³	#123	1	1	yes	characteristic
81	Q₈	—	4	2	no	—
82	D₄	—	4	2	no	—
83	D₄	—	4	2	no	—
84	D₄	—	4	2	no	—
85	D₄	—	8	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	8	1	no	—
93	C₂×C₄	—	8	1	no	—
94	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄	#123	1	1	yes	characteristic
98	C₂×C₄	#123	1	1	yes	characteristic
99	C₂×C₄	#127	1	1	yes	characteristic
100	C₂×C₄	#135	1	2	yes	—
101	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
102	C₈	—	4	2	no	—
103	C₈	—	4	2	no	—
104	C₈	—	8	1	no	—
105	C₈	—	8	1	no	—
106	C₈	—	8	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	—	8	1	no	—
110	C₂²	—	8	1	no	—
111	C₂²	—	8	1	no	—
112	C₂²	#399	1	1	yes	characteristic
113	C₂²	#482	1	2	yes	—
114	C₄	—	2	1	no	—
115	C₄	—	2	1	no	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	8	2	no	—
119	C₄	—	8	1	no	—
120	C₄	#478	1	2	yes	—
121	C₄	#495	1	1	yes	characteristic
122	C₄	#496	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	16	1	no	—
125	C₂	#2092	1	1	yes	characteristic
126	C₂	#2098	1	1	yes	characteristic
127	C₂	#2098	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1963

C₂

yes

maximal, characteristic

#1832

C₂

yes

maximal, characteristic, complemented

#1571

C₂

yes

maximal, characteristic, complemented

#1570

C₂

yes

maximal, characteristic, complemented

#1480

C₂

yes

maximal, characteristic

#1480

C₂

yes

maximal, characteristic

#1274

C₂

yes

maximal, characteristic, complemented

#532

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#104

—

#376

C₂²

yes

—

—

—

D₄⋊C₈

—

D₄⋊C₈

—

C₂×#41

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Q₈

yes

characteristic

#127

Q₈

yes

—

—

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

D₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#104

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

—

C₂³

#123

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

#135

yes

—

101

C₂×C₄

AGL(1,ℤ/8)

yes

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

#399

yes

characteristic

113

C₂²

#482

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#478

yes

—

121

C₄

#495

yes

characteristic

122

C₄

#496

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#2092

yes

characteristic

126

C₂

#2098

yes

characteristic

127

C₂

#2098

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#399
prime radical	#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³

C₂²

C₂²

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. #1832	C₂	semidirect
3. #1571	C₂	semidirect
4. #1570	C₂	semidirect
7. #1274	C₂	semidirect

H ≅ G/N

type

2. #1832

C₂

semidirect

3. #1571

C₂

semidirect

4. #1570

C₂

semidirect

7. #1274

C₂

semidirect

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

subgroup	index
#1963	2
#1832	2
#1571	2
#1570	2
#1480	2
#1274	2

subgroup

index

Subgroups — #10665

Conjugacy classes of subgroups

Complements

Maximal subgroups