Group #13521 — subgroups

Subgroups — #13521

Order 256.

135 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 135 conjugacy classes of subgroups — too many to draw. Only its 55 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×#365	C₂	1	1	yes	maximal, characteristic
2	C₂×#364	C₂	1	1	yes	maximal, characteristic
3	#1945	C₂	1	1	yes	maximal, characteristic
4	#1941	C₂	1	1	yes	maximal, characteristic
5	#1386	C₂	1	1	yes	maximal, characteristic
6	#1386	C₂	1	1	yes	maximal, characteristic
7	#1383	C₂	1	1	yes	maximal, characteristic
8	C₂×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×#108	C₂²	1	1	yes	characteristic
11	#427	C₂²	1	1	yes	characteristic
12	C₂×#106	C₂²	1	1	yes	characteristic
13	C₂×#106	C₂²	1	1	yes	characteristic
14	#424	—	2	1	no	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	#389	—	2	1	no	—
17	#365	—	2	2	no	—
18	#364	—	2	2	no	—
19	#340	—	2	1	no	—
20	#339	—	2	1	no	—
21	C₂×M₁₆	—	2	1	no	—
22	C₂²×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₈	C₂³	1	1	yes	characteristic
24	#117	—	2	1	no	—
25	C₂×C₄⋊C₄	—	2	1	no	—
26	C₂×C₄⋊C₄	—	2	1	no	—
27	C₂×C₄⋊C₄	—	4	1	no	—
28	C₂×C₄²	D₄	1	1	yes	characteristic
29	M₃₂	—	2	1	no	—
30	M₃₂	—	2	1	no	—
31	M₃₂	—	2	1	no	—
32	M₃₂	D₄	1	2	yes	—
33	C₂×C₁₆	—	2	1	no	—
34	C₂×C₁₆	—	2	1	no	—
35	#108	—	2	1	no	—
36	#108	—	2	1	no	—
37	#107	Q₈	1	1	yes	characteristic
38	#107	Q₈	1	1	yes	characteristic
39	#106	—	2	1	no	—
40	#106	—	2	1	no	—
41	#106	—	2	1	no	—
42	#106	—	2	1	no	—
43	#106	Q₈	1	2	yes	—
44	C₄⋊C₈	—	4	2	no	—
45	C₄×C₈	—	2	1	no	—
46	C₄×C₈	—	2	1	no	—
47	#95	—	4	1	no	—
48	C₂²×C₄	—	2	1	no	—
49	C₂²×C₄	—	4	1	no	—
50	C₂²×C₄	—	4	1	no	—
51	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
52	M₁₆	—	4	1	no	—
53	M₁₆	—	4	1	no	—
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	—	2	1	no	—
56	C₂×C₈	—	4	1	no	—
57	C₂×C₈	SD₁₆	1	2	yes	—
58	C₂×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈	#41	1	1	yes	characteristic
62	C₂×C₈	#41	1	1	yes	characteristic
63	C₂×C₈	#41	1	1	yes	characteristic
64	C₄⋊C₄	—	2	2	no	—
65	C₄⋊C₄	—	2	1	no	—
66	C₄⋊C₄	—	2	1	no	—
67	C₄⋊C₄	—	4	1	no	—
68	C₄⋊C₄	—	4	1	no	—
69	C₄⋊C₄	—	4	1	no	—
70	C₄⋊C₄	—	8	2	no	—
71	C₂²⋊C₄	—	4	1	no	—
72	C₄²	—	2	1	no	—
73	C₄²	—	2	1	no	—
74	C₄²	—	4	1	no	—
75	C₁₆	—	2	2	no	—
76	C₁₆	—	4	1	no	—
77	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	—	4	1	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	4	1	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	8	1	no	—
85	C₂×C₄	—	8	1	no	—
86	C₂×C₄	—	8	1	no	—
87	C₂×C₄	—	8	1	no	—
88	C₂×C₄	—	8	1	no	—
89	C₂×C₄	#121	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄	#135	1	1	yes	characteristic
94	C₂×C₄	#137	1	1	yes	characteristic
95	C₈	—	2	2	no	—
96	C₈	—	8	1	no	—
97	C₈	#123	1	1	yes	characteristic
98	C₈	#125	1	1	yes	characteristic
99	C₈	#125	1	1	yes	characteristic
100	C₈	C₄⋊Q₈	1	1	yes	characteristic
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	#468	1	1	yes	characteristic
104	C₂²	#476	1	1	yes	characteristic
105	C₂²	#478	1	1	yes	characteristic
106	C₄	—	4	2	no	—
107	C₄	—	8	2	no	—
108	C₄	—	8	1	no	—
109	C₄	—	8	1	no	—
110	C₄	—	16	1	no	—
111	C₄	#398	1	1	yes	characteristic
112	C₄	#465	1	1	yes	characteristic
113	C₄	#475	1	1	yes	characteristic
114	C₄	#479	1	1	yes	characteristic
115	C₂	—	2	1	no	—
116	C₂	—	2	1	no	—
117	C₂	#2088	1	1	yes	characteristic
118	C₂	#2230	1	1	yes	characteristic
119	C₂	#2231	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#365

C₂

yes

maximal, characteristic

C₂×#364

C₂

yes

maximal, characteristic

#1945

C₂

yes

maximal, characteristic

#1941

C₂

yes

maximal, characteristic

#1386

C₂

yes

maximal, characteristic

#1386

C₂

yes

maximal, characteristic

#1383

C₂

yes

maximal, characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×#108

C₂²

yes

characteristic

#427

C₂²

yes

characteristic

C₂×#106

C₂²

yes

characteristic

C₂×#106

C₂²

yes

characteristic

#424

—

C₂×C₄×C₈

C₂²

yes

—

—

—

—

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

D₄

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

#108

—

#107

Q₈

yes

characteristic

#107

Q₈

yes

—

—

—

—

Q₈

yes

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#121

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

yes

C₈

yes

C₈

yes

100

C₈

C₄⋊Q₈

yes

characteristic

101

C₂²

—

102

C₂²

—

103

C₂²

#468

yes

characteristic

104

C₂²

#476

yes

characteristic

105

C₂²

#478

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#398

yes

characteristic

112

C₄

#465

yes

characteristic

113

C₄

#475

yes

characteristic

114

C₄

#479

yes

characteristic

115

C₂

—

116

C₂

—

117

C₂

#2088

yes

characteristic

118

C₂

#2230

yes

characteristic

119

C₂

#2231

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	AGL(1,ℤ/8)
LCS	G ▷ C₂²×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

#476

commutator, Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂³

AGL(1,ℤ/8)

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#365	2
C₂×#364	2
#1945	2
#1941	2
#1386	2
#1383	2

subgroup

index

Subgroups — #13521

Conjugacy classes of subgroups

Maximal subgroups