Group #13491 — subgroups

Subgroups — #13491

Order 256.

136 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 47 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×#362	C₂	1	1	yes	maximal, characteristic
2	#2141	C₂	1	1	yes	maximal, characteristic
3	#1979	C₂	1	1	yes	maximal, characteristic
4	#1833	C₂	1	1	yes	maximal, characteristic
5	#1389	C₂	1	1	yes	maximal, characteristic
6	#1388	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	#362	—	2	2	no	—
18	#358	—	2	2	no	—
19	#344	—	2	1	no	—
20	H(3,ℤ/4)	—	2	1	no	—
21	C₂²×Q₈	—	2	1	no	—
22	C₂×Q₁₆	—	2	1	no	—
23	C₂×Q₁₆	—	2	1	no	—
24	C₂×Q₁₆	—	2	1	no	—
25	C₂×Q₁₆	—	2	1	no	—
26	C₂×Q₁₆	—	2	1	no	—
27	C₂×Q₁₆	D₄	1	2	yes	—
28	C₂×M₁₆	—	2	1	no	—
29	C₂×M₁₆	—	2	1	no	—
30	C₂×M₁₆	D₄	1	1	yes	characteristic
31	C₂²×C₈	C₂³	1	1	yes	characteristic
32	#117	—	2	1	no	—
33	#117	D₄	1	1	yes	characteristic
34	M₃₂	—	2	1	no	—
35	M₃₂	—	2	1	no	—
36	M₃₂	—	2	1	no	—
37	M₃₂	—	2	1	no	—
38	C₂×C₁₆	—	2	1	no	—
39	C₂×C₁₆	—	2	1	no	—
40	#108	—	2	1	no	—
41	#108	—	2	1	no	—
42	#108	—	2	1	no	—
43	#108	—	2	1	no	—
44	#107	—	2	1	no	—
45	#106	—	2	1	no	—
46	#103	—	4	2	no	—
47	#101	—	4	2	no	—
48	#97	—	2	1	no	—
49	C₄×C₈	—	2	1	no	—
50	C₂×Q₈	—	2	2	no	—
51	C₂×Q₈	—	4	1	no	—
52	C₂×Q₈	—	4	1	no	—
53	C₂×Q₈	—	4	1	no	—
54	C₂×Q₈	—	4	1	no	—
55	C₂×Q₈	—	4	1	no	—
56	C₂²×C₄	—	4	1	no	—
57	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
58	Q₁₆	—	2	2	no	—
59	Q₁₆	—	2	2	no	—
60	Q₁₆	—	4	1	no	—
61	Q₁₆	—	4	1	no	—
62	M₁₆	—	2	1	no	—
63	M₁₆	—	2	1	no	—
64	M₁₆	—	4	1	no	—
65	M₁₆	—	4	1	no	—
66	M₁₆	—	4	1	no	—
67	M₁₆	—	4	1	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	4	1	no	—
70	C₂×C₈	—	4	1	no	—
71	C₂×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈	#41	1	1	yes	characteristic
73	C₂×C₈	#41	1	1	yes	characteristic
74	C₂×C₈	#41	1	1	yes	characteristic
75	C₂×C₈	#41	1	1	yes	characteristic
76	C₂×C₈	#41	1	1	yes	characteristic
77	C₄⋊C₄	—	2	1	no	—
78	C₄⋊C₄	—	4	1	no	—
79	C₂²⋊C₄	—	2	1	no	—
80	C₂²⋊C₄	—	4	1	no	—
81	C₄²	—	2	1	no	—
82	C₄²	—	4	1	no	—
83	C₁₆	—	4	1	no	—
84	C₁₆	—	4	1	no	—
85	C₂³	#135	1	1	yes	characteristic
86	Q₈	—	4	2	no	—
87	Q₈	—	4	2	no	—
88	Q₈	—	8	1	no	—
89	Q₈	—	8	1	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	—	8	1	no	—
93	C₂×C₄	—	8	1	no	—
94	C₂×C₄	—	8	1	no	—
95	C₂×C₄	#121	1	1	yes	characteristic
96	C₂×C₄	#123	1	1	yes	characteristic
97	C₂×C₄	#124	1	1	yes	characteristic
98	C₂×C₄	#135	1	1	yes	characteristic
99	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₈	—	4	1	no	—
102	C₈	—	8	1	no	—
103	C₈	—	8	1	no	—
104	C₈	#123	1	1	yes	characteristic
105	C₈	#123	1	1	yes	characteristic
106	C₈	#126	1	1	yes	characteristic
107	C₈	#126	1	1	yes	characteristic
108	C₂²	—	4	1	no	—
109	C₂²	#463	1	1	yes	characteristic
110	C₂²	#482	1	1	yes	characteristic
111	C₂²	#488	1	1	yes	characteristic
112	C₄	—	8	1	no	—
113	C₄	—	8	2	no	—
114	C₄	—	16	1	no	—
115	C₄	#397	1	1	yes	characteristic
116	C₄	#463	1	1	yes	characteristic
117	C₄	#482	1	1	yes	characteristic
118	C₄	#490	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#2073	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#362

C₂

yes

maximal, characteristic

#2141

C₂

yes

maximal, characteristic

#1979

C₂

yes

maximal, characteristic

#1833

C₂

yes

maximal, characteristic

#1389

C₂

yes

maximal, characteristic

#1388

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

D₄

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#117

—

#117

D₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

—

—

—

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

C₂×C₈

yes

C₂×C₈

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

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

#123

yes

characteristic

105

C₈

#123

yes

characteristic

106

C₈

#126

yes

characteristic

107

C₈

#126

yes

characteristic

108

C₂²

—

109

C₂²

#463

yes

characteristic

110

C₂²

#482

yes

characteristic

111

C₂²

#488

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#397

yes

characteristic

116

C₄

#463

yes

characteristic

117

C₄

#482

yes

characteristic

118

C₄

#490

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

#2073

yes

characteristic

	subgroup	G/H
center, socle	C₂	#2073
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

#2073

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
C₂×#362	2
#2141	2
#1979	2
#1833	2
#1389	2
#1388	2
#1378	2

subgroup

index

Subgroups — #13491

Conjugacy classes of subgroups

Maximal subgroups