Group #10669 — subgroups

Subgroups — #10669

Order 256.

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

normal subgroups only

This group has 148 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	#1995	C₂	1	1	yes	maximal, characteristic
2	#1832	C₂	1	1	yes	maximal, characteristic
3	#1571	C₂	1	1	yes	maximal, characteristic
4	#1570	C₂	1	1	yes	maximal, characteristic
5	#1481	C₂	1	1	yes	maximal, characteristic
6	#1481	C₂	1	1	yes	maximal, characteristic
7	#1274	C₂	1	1	yes	maximal, characteristic
8	#533	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	#376	C₂²	1	1	yes	characteristic
15	#339	—	2	2	no	—
16	#336	—	2	2	no	—
17	#335	—	2	2	no	—
18	#334	—	2	2	no	—
19	Q₈⋊C₈	—	2	2	no	—
20	Q₈⋊C₈	—	2	2	no	—
21	C₂×M₁₆	—	2	1	no	—
22	C₂²×C₈	—	2	1	no	—
23	C₂²×C₈	—	2	1	no	—
24	C₄⋊Q₈	Q₈	1	1	yes	characteristic
25	C₄⋊Q₈	Q₈	1	1	yes	characteristic
26	#125	—	2	1	no	—
27	C₂²⋊Q₈	—	2	1	no	—
28	C₂²⋊Q₈	—	2	1	no	—
29	C₄×Q₈	—	2	1	no	—
30	C₄×Q₈	—	2	1	no	—
31	#117	—	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	C₂²⋊C₈	—	2	1	no	—
43	C₂²⋊C₈	—	2	1	no	—
44	C₄×C₈	D₄	1	2	yes	—
45	C₄×C₈	D₄	1	2	yes	—
46	#95	—	2	1	no	—
47	#95	—	2	1	no	—
48	C₂×Q₈	—	2	1	no	—
49	C₂×Q₈	—	2	1	no	—
50	C₂²×C₄	—	2	2	no	—
51	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄	#41	1	1	yes	characteristic
53	C₂²×C₄	#41	1	1	yes	characteristic
54	M₁₆	—	4	2	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	4	1	no	—
58	C₂×C₈	—	4	1	no	—
59	C₂×C₈	—	4	1	no	—
60	C₂×C₈	—	4	2	no	—
61	C₂×C₈	—	4	2	no	—
62	C₄⋊C₄	—	2	1	no	—
63	C₄⋊C₄	—	2	1	no	—
64	C₄⋊C₄	—	2	1	no	—
65	C₄⋊C₄	—	2	1	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	4	1	no	—
68	C₂²⋊C₄	—	4	1	no	—
69	C₄²	—	2	2	no	—
70	C₄²	—	2	2	no	—
71	C₄²	—	4	1	no	—
72	C₄²	C₂×D₄	1	1	yes	characteristic
73	C₄²	C₂×D₄	1	1	yes	characteristic
74	C₄²	C₂×Q₈	1	1	yes	characteristic
75	C₄²	#41	1	1	yes	characteristic
76	C₂³	#123	1	1	yes	characteristic
77	Q₈	—	4	2	no	—
78	Q₈	—	4	2	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	—	4	1	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
89	C₂×C₄	#123	1	1	yes	characteristic
90	C₂×C₄	#123	1	1	yes	characteristic
91	C₂×C₄	#127	1	1	yes	characteristic
92	C₂×C₄	#135	1	2	yes	—
93	C₂×C₄	#137	1	2	yes	—
94	C₈	—	4	2	no	—
95	C₈	—	4	2	no	—
96	C₈	—	8	1	no	—
97	C₈	—	8	1	no	—
98	C₈	—	8	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	#399	1	1	yes	characteristic
102	C₂²	#485	1	2	yes	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	—	8	2	no	—
108	C₄	—	8	1	no	—
109	C₄	—	8	1	no	—
110	C₄	#479	1	2	yes	—
111	C₄	#495	1	1	yes	characteristic
112	C₄	#497	1	1	yes	characteristic
113	C₂	—	2	2	no	—
114	C₂	#2097	1	1	yes	characteristic
115	C₂	#2098	1	1	yes	characteristic
116	C₂	#2098	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1995

C₂

yes

maximal, characteristic

#1832

C₂

yes

maximal, characteristic

#1571

C₂

yes

maximal, characteristic

#1570

C₂

yes

maximal, characteristic

#1481

C₂

yes

maximal, characteristic

#1481

C₂

yes

maximal, characteristic

#1274

C₂

yes

maximal, characteristic

#533

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

#376

C₂²

yes

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Q₈

yes

characteristic

C₄⋊Q₈

Q₈

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

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

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

C₄²

yes

C₂³

yes

Q₈

—

Q₈

—

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

C₂×C₄

yes

—

C₂×C₄

#137

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

100

C₂²

—

101

C₂²

#399

yes

characteristic

102

C₂²

#485

yes

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#479

yes

—

111

C₄

#495

yes

characteristic

112

C₄

#497

yes

characteristic

113

C₂

—

114

C₂

#2097

yes

characteristic

115

C₂

#2098

yes

characteristic

116

C₂

#2098

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#399
prime radical	C₂³	#123
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

C₂³

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#1995	2
#1832	2
#1571	2
#1570	2
#1481	2
#1274	2

subgroup

index

Subgroups — #10669

Conjugacy classes of subgroups

Maximal subgroups