Group #10309 — subgroups

Subgroups — #10309

Order 256.

309 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 93 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×#500	C₂	1	2	yes	maximal
2	C₂×#494	C₂	1	1	yes	maximal, characteristic
3	#1990	C₂	1	2	yes	maximal
4	#1956	C₂	1	1	yes	maximal, characteristic
5	#1273	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆	—	2	2	no	—
7	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	#500	—	2	4	no	—
10	#500	—	2	4	no	—
11	#494	—	2	4	no	—
12	C₂×#107	—	2	4	no	—
13	C₂×#103	—	2	2	no	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	2	yes	—
16	#384	C₂²	1	1	yes	characteristic
17	#340	—	2	4	no	—
18	C₂²×Q₈	—	2	2	no	—
19	C₂²×Q₈	—	2	1	no	—
20	C₂×Q₁₆	—	2	8	no	—
21	C₂×Q₁₆	—	4	4	no	—
22	C₂×Q₁₆	—	4	4	no	—
23	C₂²×C₈	D₄	1	2	yes	—
24	C₂²×C₈	D₄	1	4	yes	—
25	C₄⋊Q₈	—	2	2	no	—
26	C₄⋊Q₈	—	2	2	no	—
27	C₄⋊Q₈	—	2	2	no	—
28	C₄⋊Q₈	—	2	2	no	—
29	C₄⋊Q₈	—	2	1	no	—
30	C₄⋊Q₈	D₄	1	4	yes	—
31	C₄⋊Q₈	D₄	1	2	yes	—
32	C₂×C₄⋊C₄	—	2	2	no	—
33	C₂×C₄⋊C₄	—	2	2	no	—
34	C₂×C₄⋊C₄	—	2	2	no	—
35	C₂×C₄⋊C₄	—	2	2	no	—
36	C₂×C₄⋊C₄	—	2	1	no	—
37	C₂×C₄²	C₂³	1	1	yes	characteristic
38	#107	—	2	16	no	—
39	#103	—	4	4	no	—
40	C₄×C₈	—	2	2	no	—
41	C₄×C₈	—	2	2	no	—
42	C₄×C₈	—	2	2	no	—
43	C₂×Q₈	—	2	4	no	—
44	C₂×Q₈	—	2	2	no	—
45	C₂×Q₈	—	4	1	no	—
46	C₂×Q₈	—	4	2	no	—
47	C₂×Q₈	—	4	2	no	—
48	C₂×Q₈	—	4	4	no	—
49	C₂×Q₈	—	4	4	no	—
50	C₂×Q₈	—	4	2	no	—
51	C₂²×C₄	—	4	2	no	—
52	C₂²×C₄	—	4	1	no	—
53	C₂²×C₄	—	4	1	no	—
54	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄	C₂×D₄	1	2	yes	—
56	Q₁₆	—	4	16	no	—
57	C₂×C₈	—	2	4	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	2	4	no	—
61	C₂×C₈	D₈	1	4	yes	—
62	C₂×C₈	Q₁₆	1	8	yes	complemented
63	C₄⋊C₄	—	2	4	no	—
64	C₄⋊C₄	—	2	4	no	—
65	C₄⋊C₄	—	2	4	no	—
66	C₄⋊C₄	—	4	2	no	—
67	C₄⋊C₄	—	4	2	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	4	no	—
71	C₄²	C₂×D₄	1	2	yes	—
72	C₄²	C₂×D₄	1	1	yes	characteristic
73	C₄²	#41	1	1	yes	characteristic
74	C₂³	#127	1	1	yes	characteristic
75	Q₈	—	4	8	no	—
76	Q₈	—	4	4	no	—
77	Q₈	—	8	2	no	—
78	Q₈	—	8	4	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	—	4	4	no	—
81	C₂×C₄	—	8	1	no	—
82	C₂×C₄	—	8	2	no	—
83	C₂×C₄	—	8	2	no	—
84	C₂×C₄	—	8	1	no	—
85	C₂×C₄	—	8	1	no	—
86	C₂×C₄	—	8	2	no	—
87	C₂×C₄	#120	1	1	yes	characteristic
88	C₂×C₄	#120	1	2	yes	—
89	C₂×C₄	#121	1	1	yes	characteristic
90	C₂×C₄	#121	1	2	yes	—
91	C₂×C₄	C₂×D₈	1	2	yes	—
92	C₂×C₄	C₂×Q₁₆	1	4	yes	—
93	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
94	C₂×C₄	#137	1	4	yes	—
95	C₈	—	2	4	no	—
96	C₈	—	2	8	no	—
97	C₂²	#392	1	1	yes	characteristic
98	C₂²	#493	1	1	yes	characteristic
99	C₂²	#494	1	2	yes	—
100	C₂²	#496	1	1	yes	characteristic
101	C₂²	#497	1	2	yes	—
102	C₄	—	8	4	no	—
103	C₄	—	8	2	no	—
104	C₄	—	16	1	no	—
105	C₄	#447	1	2	yes	—
106	C₄	#451	1	4	yes	—
107	C₄	#459	1	2	yes	—
108	C₄	#462	1	4	yes	—
109	C₂	#1713	1	1	yes	characteristic
110	C₂	#1722	1	1	yes	characteristic
111	C₂	#1723	1	2	yes	—
112	C₂	#2011	1	1	yes	characteristic
113	C₂	#2016	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#500

C₂

yes

maximal

C₂×#494

C₂

yes

maximal, characteristic

#1990

C₂

yes

maximal

#1956

C₂

yes

maximal, characteristic

#1273

C₂

yes

maximal, characteristic

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂²

yes

—

C₂×C₄⋊Q₈

C₂²

yes

—

—

—

—

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

#384

C₂²

yes

characteristic

#340

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₄

yes

—

C₄⋊Q₈

D₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₈

yes

—

C₂×C₈

Q₁₆

yes

complemented

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄

yes

—

C₄²

C₂×D₄

yes

C₄²

yes

C₂³

yes

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#120

yes

characteristic

C₂×C₄

#120

yes

—

C₂×C₄

#121

yes

characteristic

C₂×C₄

#121

yes

—

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

C₂×Q₁₆

yes

—

C₂×C₄

AGL(1,ℤ/8)

yes

—

C₂×C₄

#137

yes

—

C₈

—

C₈

—

C₂²

yes

C₂²

yes

C₂²

yes

—

100

C₂²

#496

yes

characteristic

101

C₂²

#497

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#447

yes

—

106

C₄

#451

yes

—

107

C₄

#459

yes

—

108

C₄

#462

yes

—

109

C₂

#1713

yes

characteristic

110

C₂

#1722

yes

characteristic

111

C₂

#1723

yes

—

112

C₂

#2011

yes

characteristic

113

C₂

#2016

yes

—

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

subgroup

G/H

center, socle, prime radical

C₂³

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
62. C₂×C₈	Q₁₆	semidirect

H ≅ G/N

type

62. C₂×C₈

Q₁₆

semidirect

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

subgroup	index
C₂×#500	2
C₂×#494	2
#1990	2
#1956	2
#1273	2

subgroup

index

Subgroups — #10309

Conjugacy classes of subgroups

Complements

Maximal subgroups