Group #13479 — subgroups

Subgroups — #13479

Order 256.

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

normal subgroups only

This group has 159 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	#2159	C₂	1	1	yes	maximal, characteristic
2	C₂×#364	C₂	1	1	yes	maximal, characteristic
3	C₂×#362	C₂	1	1	yes	maximal, characteristic
4	#2141	C₂	1	1	yes	maximal, characteristic
5	#1969	C₂	1	1	yes	maximal, characteristic
6	#1945	C₂	1	1	yes	maximal, characteristic
7	#1387	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	#427	C₂²	1	1	yes	characteristic
13	C₂×#107	C₂²	1	1	yes	characteristic
14	#424	—	2	1	no	—
15	C₂×#103	—	2	1	no	—
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	#391	—	2	1	no	—
18	#367	—	2	1	no	—
19	#366	—	2	1	no	—
20	#364	—	2	2	no	—
21	#362	—	2	2	no	—
22	#358	—	2	1	no	—
23	#358	—	2	1	no	—
24	#339	—	2	1	no	—
25	C₂²×Q₈	—	2	1	no	—
26	C₂×Q₁₆	—	2	1	no	—
27	C₂×Q₁₆	—	2	1	no	—
28	C₂×Q₁₆	—	2	1	no	—
29	C₂×Q₁₆	—	2	1	no	—
30	C₂×Q₁₆	—	2	1	no	—
31	C₂×Q₁₆	Q₈	1	2	yes	—
32	C₂×M₁₆	—	2	1	no	—
33	C₂²×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₈	C₂³	1	1	yes	characteristic
35	#117	—	2	1	no	—
36	C₂×C₄⋊C₄	—	2	1	no	—
37	C₂×C₄²	D₄	1	1	yes	characteristic
38	M₃₂	—	2	1	no	—
39	M₃₂	—	2	1	no	—
40	M₃₂	—	2	1	no	—
41	M₃₂	D₄	1	2	yes	—
42	C₂×C₁₆	—	2	1	no	—
43	C₂×C₁₆	—	2	1	no	—
44	#108	—	2	1	no	—
45	#108	—	2	1	no	—
46	#107	—	2	1	no	—
47	#107	—	2	1	no	—
48	#107	D₄	1	1	yes	characteristic
49	#107	D₄	1	1	yes	characteristic
50	#106	—	2	1	no	—
51	C₄⋊C₈	—	4	2	no	—
52	#103	—	4	2	no	—
53	C₄×C₈	—	2	1	no	—
54	C₄×C₈	—	2	1	no	—
55	#95	—	4	2	no	—
56	C₂×Q₈	—	2	2	no	—
57	C₂×Q₈	—	4	1	no	—
58	C₂×Q₈	—	4	1	no	—
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	4	1	no	—
61	C₂×Q₈	—	4	1	no	—
62	C₂²×C₄	—	2	1	no	—
63	C₂²×C₄	—	4	1	no	—
64	C₂²×C₄	—	4	1	no	—
65	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
66	Q₁₆	—	2	2	no	—
67	Q₁₆	—	2	2	no	—
68	Q₁₆	—	4	1	no	—
69	Q₁₆	—	4	1	no	—
70	M₁₆	—	4	1	no	—
71	M₁₆	—	4	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	4	1	no	—
75	C₂×C₈	D₈	1	2	yes	—
76	C₂×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₈	#41	1	1	yes	characteristic
80	C₂×C₈	#41	1	1	yes	characteristic
81	C₂×C₈	#41	1	1	yes	characteristic
82	C₄⋊C₄	—	2	1	no	—
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	4	1	no	—
85	C₄⋊C₄	—	4	1	no	—
86	C₂²⋊C₄	—	4	1	no	—
87	C₄²	—	2	1	no	—
88	C₄²	—	2	1	no	—
89	C₄²	—	4	1	no	—
90	C₁₆	—	2	2	no	—
91	C₁₆	—	4	1	no	—
92	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
93	Q₈	—	4	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	8	1	no	—
96	Q₈	—	8	1	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	8	1	no	—
104	C₂×C₄	—	8	1	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	—	8	1	no	—
107	C₂×C₄	#121	1	1	yes	characteristic
108	C₂×C₄	#121	1	1	yes	characteristic
109	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄	#135	1	1	yes	characteristic
112	C₂×C₄	#137	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	8	1	no	—
115	C₈	#121	1	1	yes	characteristic
116	C₈	C₂²⋊Q₈	1	1	yes	characteristic
117	C₈	C₂²⋊Q₈	1	1	yes	characteristic
118	C₈	#126	1	1	yes	characteristic
119	C₂²	—	2	1	no	—
120	C₂²	—	2	1	no	—
121	C₂²	#459	1	1	yes	characteristic
122	C₂²	#468	1	1	yes	characteristic
123	C₂²	#478	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	—	8	2	no	—
126	C₄	—	8	1	no	—
127	C₄	—	8	1	no	—
128	C₄	—	16	1	no	—
129	C₄	#396	1	1	yes	characteristic
130	C₄	#464	1	1	yes	characteristic
131	C₄	#466	1	1	yes	characteristic
132	C₄	#474	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	2	1	no	—
135	C₂	#2067	1	1	yes	characteristic
136	C₂	#2212	1	1	yes	characteristic
137	C₂	#2214	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2159

C₂

yes

maximal, characteristic

C₂×#364

C₂

yes

maximal, characteristic

C₂×#362

C₂

yes

maximal, characteristic

#2141

C₂

yes

maximal, characteristic

#1969

C₂

yes

maximal, characteristic

#1945

C₂

yes

maximal, characteristic

#1387

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×#108

C₂²

yes

characteristic

#427

C₂²

yes

characteristic

C₂×#107

C₂²

yes

characteristic

#424

—

C₂×#103

—

C₂×C₄×C₈

C₂²

yes

—

—

—

—

—

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Q₈

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

D₄

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

D₄

yes

characteristic

#107

D₄

yes

characteristic

#106

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₈

yes

—

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

AGL(1,ℤ/8)

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#121

yes

characteristic

108

C₂×C₄

#121

yes

characteristic

109

C₂×C₄

C₂²⋊Q₈

yes

characteristic

110

C₂×C₄

C₂×D₈

yes

characteristic

111

C₂×C₄

#135

yes

characteristic

112

C₂×C₄

#137

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

#121

yes

characteristic

116

C₈

C₂²⋊Q₈

yes

characteristic

117

C₈

C₂²⋊Q₈

yes

characteristic

118

C₈

#126

yes

characteristic

119

C₂²

—

120

C₂²

—

121

C₂²

#459

yes

characteristic

122

C₂²

#468

yes

characteristic

123

C₂²

#478

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#396

yes

characteristic

130

C₄

#464

yes

characteristic

131

C₄

#466

yes

characteristic

132

C₄

#474

yes

characteristic

133

C₂

—

134

C₂

—

135

C₂

#2067

yes

characteristic

136

C₂

#2212

yes

characteristic

137

C₂

#2214

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#459
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

#459

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
#2159	2
C₂×#364	2
C₂×#362	2
#2141	2
#1969	2
#1945	2
#1387	2

subgroup

index

Subgroups — #13479

Conjugacy classes of subgroups

Maximal subgroups