Group #10627 — subgroups

Subgroups — #10627

Order 256.

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

normal subgroups only

This group has 165 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	#1843	C₂	1	1	yes	maximal, characteristic, complemented
3	#1540	C₂	1	2	yes	maximal
4	#1301	C₂	1	2	yes	maximal, complemented
5	#1274	C₂	1	1	yes	maximal, characteristic, complemented
6	#532	C₂²	1	1	yes	characteristic
7	#431	C₂²	1	2	yes	—
8	#423	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈	—	2	1	no	—
10	C₂×C₄⋊C₈	—	2	1	no	—
11	C₂×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×#104	—	2	1	no	—
13	C₂×#104	—	2	1	no	—
14	#376	C₂²	1	1	yes	characteristic
15	#332	—	2	2	no	—
16	#331	—	2	2	no	—
17	C₂×#41	—	2	1	no	—
18	C₂×M₁₆	—	2	1	no	—
19	C₂²×C₈	—	2	2	no	—
20	C₄⋊Q₈	D₄	1	1	yes	characteristic
21	#127	D₄	1	1	yes	characteristic
22	#124	—	2	1	no	—
23	#121	—	2	2	no	—
24	C₄×D₄	—	2	2	no	—
25	C₂×C₄²	D₄	1	1	yes	characteristic
26	C₂×C₄²	D₄	1	1	yes	characteristic
27	C₂×C₄²	C₂³	1	1	yes	characteristic
28	C₄⋊C₈	—	2	2	no	—
29	C₄⋊C₈	—	2	2	no	—
30	C₄⋊C₈	—	2	2	no	—
31	C₄⋊C₈	—	2	2	no	—
32	C₄⋊C₈	—	4	2	no	—
33	C₄⋊C₈	—	4	2	no	—
34	#104	—	4	2	no	—
35	#104	—	4	2	no	—
36	C₂²⋊C₈	—	2	2	no	—
37	C₄×C₈	—	2	2	no	—
38	#95	—	2	2	no	—
39	#41	—	4	1	no	—
40	#41	—	4	1	no	—
41	#41	—	4	1	no	—
42	#41	—	4	1	no	—
43	C₂×Q₈	—	2	1	no	—
44	C₂×D₄	—	2	1	no	—
45	C₂×D₄	—	2	2	no	—
46	C₂×D₄	—	4	1	no	—
47	C₂²×C₄	—	2	1	no	—
48	C₂²×C₄	—	2	1	no	—
49	C₂²×C₄	—	4	1	no	—
50	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄	#41	1	2	yes	—
52	M₁₆	—	4	1	no	—
53	M₁₆	—	4	1	no	—
54	C₂×C₈	—	4	2	no	—
55	C₂×C₈	—	4	2	no	—
56	C₂×C₈	—	4	1	no	—
57	C₂×C₈	—	4	2	no	—
58	C₂×C₈	—	4	2	no	—
59	C₄⋊C₄	—	2	2	no	—
60	C₂²⋊C₄	—	4	2	no	—
61	C₄²	—	2	1	no	—
62	C₄²	—	2	1	no	—
63	C₄²	D₈	1	2	yes	—
64	C₄²	SD₁₆	1	2	yes	—
65	C₄²	C₂×D₄	1	1	yes	characteristic
66	C₄²	#41	1	1	yes	characteristic
67	C₄²	#41	1	2	yes	—
68	C₂³	—	4	1	no	—
69	C₂³	#123	1	1	yes	characteristic
70	Q₈	—	4	1	no	—
71	Q₈	—	4	1	no	—
72	D₄	—	4	1	no	—
73	D₄	—	4	1	no	—
74	D₄	—	8	2	no	—
75	D₄	—	8	1	no	—
76	D₄	—	8	1	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	1	no	—
83	C₂×C₄	—	8	1	no	—
84	C₂×C₄	—	8	1	no	—
85	C₂×C₄	#121	1	1	yes	characteristic
86	C₂×C₄	#123	1	2	yes	—
87	C₂×C₄	#124	1	1	yes	characteristic
88	C₂×C₄	#126	1	2	yes	—
89	C₂×C₄	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
92	C₂×C₄	#137	1	1	yes	characteristic
93	C₈	—	8	2	no	—
94	C₈	—	8	2	no	—
95	C₈	—	8	1	no	—
96	C₂²	—	2	2	no	—
97	C₂²	—	8	1	no	—
98	C₂²	—	8	2	no	—
99	C₂²	#397	1	1	yes	characteristic
100	C₂²	#480	1	1	yes	characteristic
101	C₂²	#481	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	8	1	no	—
107	C₄	#459	1	1	yes	characteristic
108	C₄	#461	1	1	yes	characteristic
109	C₄	#486	1	1	yes	characteristic
110	C₄	#489	1	1	yes	characteristic
111	C₂	—	2	1	no	—
112	C₂	—	2	1	no	—
113	C₂	—	16	1	no	—
114	C₂	#2071	1	1	yes	characteristic
115	C₂	#2080	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#1963

C₂

yes

maximal, characteristic

#1843

C₂

yes

maximal, characteristic, complemented

#1540

C₂

yes

maximal

#1301

C₂

yes

maximal, complemented

#1274

C₂

yes

maximal, characteristic, complemented

#532

C₂²

yes

characteristic

#431

C₂²

yes

—

#423

C₂²

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×#104

—

C₂×#104

—

#376

C₂²

yes

characteristic

#332

—

#331

—

C₂×#41

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

D₄

yes

characteristic

#127

D₄

yes

characteristic

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#104

—

#104

—

C₂²⋊C₈

—

C₄×C₈

—

—

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

D₈

yes

—

C₄²

SD₁₆

yes

—

C₄²

C₂×D₄

yes

C₄²

yes

C₄²

yes

—

C₂³

—

C₂³

#123

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#121

yes

characteristic

C₂×C₄

#123

yes

—

C₂×C₄

#124

yes

characteristic

C₂×C₄

#126

yes

—

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#397

yes

characteristic

100

C₂²

#480

yes

characteristic

101

C₂²

#481

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

#459

yes

characteristic

108

C₄

#461

yes

characteristic

109

C₄

#486

yes

characteristic

110

C₄

#489

yes

characteristic

111

C₂

—

112

C₂

—

113

C₂

—

114

C₂

#2071

yes

characteristic

115

C₂

#2080

yes

—

	subgroup	G/H
center	C₂×C₄	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#397
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

#124

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. #1843	C₂	semidirect
4. #1301	C₂	semidirect
5. #1274	C₂	semidirect

H ≅ G/N

type

2. #1843

C₂

semidirect

4. #1301

C₂

semidirect

5. #1274

C₂

semidirect

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

subgroup	index
#1963	2
#1843	2
#1540	2
#1301	2
#1274	2

subgroup

index

Subgroups — #10627

Conjugacy classes of subgroups

Complements

Maximal subgroups