Group #13524 — subgroups

Subgroups — #13524

Order 256.

144 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 51 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	#2161	C₂	1	1	yes	maximal, characteristic
2	#2155	C₂	1	1	yes	maximal, characteristic
3	C₂×#362	C₂	1	1	yes	maximal, characteristic
4	#2141	C₂	1	1	yes	maximal, characteristic
5	#1979	C₂	1	1	yes	maximal, characteristic
6	#1833	C₂	1	1	yes	maximal, characteristic
7	#1378	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×#108	C₂²	1	1	yes	characteristic
12	C₂×#108	C₂²	1	1	yes	characteristic
13	#427	C₂²	1	1	yes	characteristic
14	#419	—	2	1	no	—
15	C₂×#101	—	2	1	no	—
16	#405	C₂²	1	1	yes	characteristic
17	#368	—	2	2	no	—
18	#365	—	2	2	no	—
19	#362	—	2	2	no	—
20	#358	—	2	2	no	—
21	#344	—	2	1	no	—
22	H(3,ℤ/4)	—	2	1	no	—
23	C₂²×Q₈	—	2	1	no	—
24	C₂×Q₁₆	—	2	1	no	—
25	C₂×Q₁₆	—	2	1	no	—
26	C₂×Q₁₆	—	2	1	no	—
27	C₂×Q₁₆	—	2	1	no	—
28	C₂×Q₁₆	—	2	1	no	—
29	C₂×Q₁₆	Q₈	1	2	yes	—
30	C₂×M₁₆	—	2	1	no	—
31	C₂×M₁₆	—	2	1	no	—
32	C₂×M₁₆	D₄	1	1	yes	characteristic
33	C₂²×C₈	C₂³	1	1	yes	characteristic
34	#117	—	2	1	no	—
35	#117	D₄	1	1	yes	characteristic
36	M₃₂	—	2	1	no	—
37	M₃₂	—	2	1	no	—
38	M₃₂	—	2	1	no	—
39	M₃₂	D₄	1	2	yes	—
40	C₂×C₁₆	—	2	1	no	—
41	C₂×C₁₆	D₄	1	2	yes	—
42	#108	—	2	1	no	—
43	#108	—	2	1	no	—
44	#108	—	2	1	no	—
45	#108	—	2	1	no	—
46	#107	—	2	1	no	—
47	#106	—	2	1	no	—
48	#103	—	4	2	no	—
49	#101	—	4	2	no	—
50	#97	—	2	1	no	—
51	C₄×C₈	—	2	1	no	—
52	C₂×Q₈	—	2	2	no	—
53	C₂×Q₈	—	4	1	no	—
54	C₂×Q₈	—	4	1	no	—
55	C₂×Q₈	—	4	1	no	—
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	4	1	no	—
58	C₂²×C₄	—	4	1	no	—
59	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
60	Q₁₆	—	2	2	no	—
61	Q₁₆	—	2	2	no	—
62	Q₁₆	—	4	1	no	—
63	Q₁₆	—	4	1	no	—
64	M₁₆	—	2	1	no	—
65	M₁₆	—	2	1	no	—
66	M₁₆	—	4	1	no	—
67	M₁₆	—	4	1	no	—
68	M₁₆	—	4	1	no	—
69	M₁₆	—	4	1	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	4	1	no	—
72	C₂×C₈	—	4	1	no	—
73	C₂×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₈	#41	1	1	yes	characteristic
77	C₂×C₈	#41	1	1	yes	characteristic
78	C₂×C₈	#41	1	1	yes	characteristic
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	4	1	no	—
81	C₂²⋊C₄	—	2	1	no	—
82	C₂²⋊C₄	—	4	1	no	—
83	C₄²	—	2	1	no	—
84	C₄²	—	4	1	no	—
85	C₁₆	—	2	2	no	—
86	C₁₆	—	2	2	no	—
87	C₂³	#135	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	1	no	—
91	Q₈	—	8	1	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	—	8	1	no	—
95	C₂×C₄	—	8	1	no	—
96	C₂×C₄	—	8	1	no	—
97	C₂×C₄	C₂²⋊Q₈	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	1	yes	characteristic
101	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₈	—	4	1	no	—
104	C₈	—	8	1	no	—
105	C₈	—	8	1	no	—
106	C₈	C₂²⋊Q₈	1	1	yes	characteristic
107	C₈	C₂²⋊Q₈	1	1	yes	characteristic
108	C₈	#123	1	1	yes	characteristic
109	C₈	#123	1	1	yes	characteristic
110	C₂²	—	4	1	no	—
111	C₂²	#478	1	1	yes	characteristic
112	C₂²	#482	1	1	yes	characteristic
113	C₂²	#495	1	1	yes	characteristic
114	C₄	—	8	1	no	—
115	C₄	—	8	2	no	—
116	C₄	—	16	1	no	—
117	C₄	#399	1	1	yes	characteristic
118	C₄	#478	1	1	yes	characteristic
119	C₄	#482	1	1	yes	characteristic
120	C₄	#496	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	2	1	no	—
123	C₂	—	2	1	no	—
124	C₂	#2092	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2161

C₂

yes

maximal, characteristic

#2155

C₂

yes

maximal, characteristic

C₂×#362

C₂

yes

maximal, characteristic

#2141

C₂

yes

maximal, characteristic

#1979

C₂

yes

maximal, characteristic

#1833

C₂

yes

maximal, characteristic

#1378

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×#108

C₂²

yes

characteristic

C₂×#108

C₂²

yes

characteristic

#427

C₂²

yes

—

—

C₂²

yes

—

—

—

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Q₈

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#117

—

#117

D₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

D₄

yes

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

—

—

—

—

—

—

—

—

—

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

#135

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Q₈

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

#135

yes

characteristic

101

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

102

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

C₂²⋊Q₈

yes

characteristic

107

C₈

C₂²⋊Q₈

yes

characteristic

108

C₈

#123

yes

characteristic

109

C₈

#123

yes

characteristic

110

C₂²

—

111

C₂²

#478

yes

characteristic

112

C₂²

#482

yes

characteristic

113

C₂²

#495

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

#399

yes

characteristic

118

C₄

#478

yes

characteristic

119

C₄

#482

yes

characteristic

120

C₄

#496

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

124

C₂

#2092

yes

characteristic

	subgroup	G/H
center, socle	C₂	#2092
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂

#2092

commutator, Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂³

#135

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ 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
#2161	2
#2155	2
C₂×#362	2
#2141	2
#1979	2
#1833	2
#1378	2

subgroup

index

Subgroups — #13524

Conjugacy classes of subgroups

Maximal subgroups