Group #124919 — subgroups

Subgroups — #124919

Order 512.

278 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 72 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	#21736	C₂	1	1	yes	maximal, characteristic
2	#12595	C₂	1	1	yes	maximal, characteristic
3	#8134	C₂	1	1	yes	maximal, characteristic
4	#7119	C₂	1	4	yes	maximal
5	#3162	C₄	1	1	yes	characteristic
6	#2961	—	2	1	no	—
7	C₂×#347	—	2	1	no	—
8	#1994	—	2	1	no	—
9	#1946	—	2	1	no	—
10	#1945	C₂²	1	2	yes	—
11	#1849	C₄	1	1	yes	characteristic
12	#1773	C₄	1	2	yes	—
13	#1772	—	2	1	no	—
14	#1744	—	2	1	no	—
15	C₄×#97	C₂²	1	1	yes	characteristic
16	#1377	—	2	1	no	—
17	#1376	C₂²	1	2	yes	—
18	#1273	—	2	1	no	—
19	#1273	C₂²	1	2	yes	—
20	#533	—	2	1	no	—
21	C₂×M₃₂	—	2	2	no	—
22	#501	—	2	4	no	—
23	#446	—	4	4	no	—
24	C₂×#108	—	2	1	no	—
25	C₂×#108	C₂×C₄	1	2	yes	—
26	#427	—	2	1	no	—
27	#427	C₂×C₄	1	2	yes	—
28	#424	—	2	2	no	—
29	#423	—	2	1	no	—
30	#405	—	2	2	no	—
31	C₂×#97	C₂×C₄	1	1	yes	characteristic
32	C₂×#97	C₂×C₄	1	1	yes	characteristic
33	C₂×#97	D₄	1	1	yes	characteristic
34	C₂×#97	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	—	2	2	no	—
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
38	C₄³	D₄	1	1	yes	characteristic
39	#347	—	2	4	no	—
40	#339	—	2	2	no	—
41	#339	—	2	2	no	—
42	#336	—	2	4	no	—
43	C₂×M₁₆	—	2	2	no	—
44	C₂²×C₈	—	2	4	no	—
45	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
49	C₄⋊Q₈	—	2	2	no	—
50	#125	—	2	2	no	—
51	C₂²⋊Q₈	—	4	2	no	—
52	C₄×Q₈	—	4	2	no	—
53	#117	—	2	2	no	—
54	C₂×C₄²	—	2	2	no	—
55	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
58	M₃₂	—	2	4	no	—
59	M₃₂	—	4	2	no	—
60	C₂×C₁₆	—	4	2	no	—
61	#108	—	4	2	no	—
62	#108	—	4	2	no	—
63	#107	—	2	2	no	—
64	#107	M₁₆	1	4	yes	—
65	#106	—	2	2	no	—
66	#106	—	2	2	no	—
67	C₄⋊C₈	—	4	2	no	—
68	C₄⋊C₈	—	4	4	no	—
69	#97	—	2	2	no	—
70	#97	—	2	2	no	—
71	#97	—	2	2	no	—
72	#97	—	2	2	no	—
73	#97	—	4	2	no	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	—	4	1	no	—
76	C₄×C₈	—	4	2	no	—
77	C₄×C₈	—	4	2	no	—
78	C₄×C₈	—	4	2	no	—
79	C₂×Q₈	—	4	2	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	#104	1	2	yes	—
83	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	M₁₆	—	4	2	no	—
85	M₁₆	—	4	2	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	4	4	no	—
95	C₂×C₈	—	4	8	no	—
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	#101	1	2	yes	—
98	C₂×C₈	#104	1	2	yes	—
99	C₂×C₈	C₂×M₁₆	1	2	yes	—
100	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	4	4	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₂²⋊C₄	—	4	2	no	—
105	C₄²	—	2	1	no	—
106	C₄²	—	2	1	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	2	1	no	—
110	C₄²	—	4	4	no	—
111	C₄²	—	4	2	no	—
112	C₄²	#137	1	2	yes	—
113	C₁₆	—	4	4	no	—
114	C₂³	C₂×#104	1	1	yes	characteristic
115	Q₈	—	16	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	4	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	#407	1	1	yes	characteristic
124	C₂×C₄	#410	1	1	yes	characteristic
125	C₂×C₄	C₂×#101	1	1	yes	characteristic
126	C₂×C₄	#418	1	1	yes	characteristic
127	C₂×C₄	#419	1	1	yes	characteristic
128	C₂×C₄	C₂×#104	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	2	2	no	—
131	C₈	—	2	2	no	—
132	C₈	—	2	2	no	—
133	C₈	—	8	4	no	—
134	C₈	—	8	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	#1490	1	1	yes	characteristic
137	C₂²	#1506	1	1	yes	characteristic
138	C₂²	#1526	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	4	2	no	—
141	C₄	—	8	1	no	—
142	C₄	—	8	4	no	—
143	C₄	#1467	1	1	yes	characteristic
144	C₄	#1491	1	1	yes	characteristic
145	C₄	#1505	1	1	yes	characteristic
146	C₄	#1556	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#7638	1	1	yes	characteristic
150	C₂	#11670	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21736

C₂

yes

maximal, characteristic

#12595

C₂

yes

maximal, characteristic

#8134

C₂

yes

maximal, characteristic

#7119

C₂

yes

maximal

#3162

C₄

yes

—

—

—

—

C₂²

yes

—

#1849

C₄

yes

characteristic

#1773

C₄

yes

—

#1772

—

#1744

—

C₄×#97

C₂²

yes

characteristic

#1377

—

#1376

C₂²

yes

—

#1273

—

#1273

C₂²

yes

—

#533

—

C₂×M₃₂

—

—

—

—

C₂×C₄

yes

—

#427

—

#427

C₂×C₄

yes

—

#424

—

#423

—

#405

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

—

—

—

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

M₁₆

yes

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#104

yes

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#101

yes

—

C₂×C₈

#104

yes

—

C₂×C₈

C₂×M₁₆

yes

—

100

C₂×C₈

AGL(1,ℤ/8)

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₄²

#137

yes

—

113

C₁₆

—

114

C₂³

C₂×#104

yes

characteristic

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

#407

yes

characteristic

124

C₂×C₄

#410

yes

characteristic

125

C₂×C₄

C₂×#101

yes

characteristic

126

C₂×C₄

#418

yes

characteristic

127

C₂×C₄

#419

yes

characteristic

128

C₂×C₄

C₂×#104

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₂²

—

136

C₂²

#1490

yes

characteristic

137

C₂²

#1506

yes

characteristic

138

C₂²

#1526

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#1467

yes

characteristic

144

C₄

#1491

yes

characteristic

145

C₄

#1505

yes

characteristic

146

C₄

#1556

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#7638

yes

characteristic

150

C₂

#11670

yes

—

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1490
prime radical	C₂³	C₂×#104
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#418

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₄×#97 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#21736	2
#12595	2
#8134	2
#7119	2

subgroup

index

Subgroups — #124919

Conjugacy classes of subgroups

Maximal subgroups