Group #145575 — subgroups

Subgroups — #145575

Order 512.

232 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 78 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×#2253	C₂	1	1	yes	maximal, characteristic
2	#13273	C₂	1	2	yes	maximal
3	#13161	C₂	1	1	yes	maximal, characteristic
4	C₂×#1366	C₂	1	2	yes	maximal
5	#7160	C₂	1	1	yes	maximal, characteristic
6	C₂×#499	C₂²	1	2	yes	—
7	#2253	—	2	4	no	—
8	C₂×#367	—	2	1	no	—
9	C₂×#366	—	2	1	no	—
10	C₂×#364	C₂²	1	1	yes	characteristic
11	C₂×#364	C₂²	1	2	yes	—
12	#2151	—	2	1	no	—
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1772	C₂²	1	1	yes	characteristic
15	#1386	—	2	2	no	—
16	#1383	—	2	2	no	—
17	#1379	—	2	1	no	—
18	#1366	—	2	4	no	—
19	C₂×#125	—	2	2	no	—
20	C₂×M₃₂	—	2	1	no	—
21	C₂×M₃₂	—	2	2	no	—
22	C₂²×C₁₆	D₄	1	1	yes	characteristic
23	C₂²×C₁₆	D₄	1	1	yes	characteristic
24	#499	—	2	2	no	—
25	#499	—	2	2	no	—
26	#499	—	2	2	no	—
27	#499	D₄	1	4	yes	—
28	C₂×#108	D₄	1	1	yes	characteristic
29	#427	D₄	1	1	yes	characteristic
30	C₂×#107	—	2	2	no	—
31	C₂×#106	—	2	2	no	—
32	#405	—	2	1	no	—
33	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
34	#367	—	2	4	no	—
35	#366	—	2	4	no	—
36	#364	—	2	2	no	—
37	#364	—	2	2	no	—
38	#364	—	2	2	no	—
39	C₄⋊C₁₆	—	4	2	no	—
40	C₄×C₁₆	—	2	2	no	—
41	#339	—	2	1	no	—
42	C₂×M₁₆	—	2	1	no	—
43	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
45	#125	—	2	4	no	—
46	#125	—	4	2	no	—
47	#125	—	8	2	no	—
48	#117	—	2	1	no	—
49	C₂×C₄⋊C₄	—	4	2	no	—
50	C₂×C₄⋊C₄	—	8	2	no	—
51	C₂×C₄²	#41	1	1	yes	characteristic
52	M₃₂	—	4	1	no	—
53	M₃₂	—	4	2	no	—
54	M₃₂	—	4	1	no	—
55	M₃₂	—	4	2	no	—
56	C₂×C₁₆	—	2	1	no	—
57	C₂×C₁₆	—	2	1	no	—
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	—	2	1	no	—
60	C₂×C₁₆	—	4	1	no	—
61	C₂×C₁₆	—	4	2	no	—
62	C₂×C₁₆	SD₁₆	1	2	yes	—
63	C₂×C₁₆	Q₁₆	1	2	yes	—
64	#108	—	2	2	no	—
65	#107	—	2	4	no	—
66	#107	—	4	2	no	—
67	#107	Q₁₆	1	2	yes	—
68	#106	—	2	1	no	—
69	#106	—	2	4	no	—
70	#106	—	4	2	no	—
71	#97	—	4	1	no	—
72	C₄×C₈	—	4	1	no	—
73	C₄×C₈	C₂×D₄	1	2	yes	—
74	C₄×C₈	#41	1	2	yes	—
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	—	8	2	no	—
77	C₂²×C₄	#121	1	1	yes	characteristic
78	M₁₆	—	4	1	no	—
79	M₁₆	—	4	1	no	—
80	C₂×C₈	—	4	1	no	—
81	C₂×C₈	#120	1	1	yes	characteristic
82	C₂×C₈	#121	1	2	yes	—
83	C₂×C₈	#124	1	1	yes	characteristic
84	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₈	#135	1	1	yes	characteristic
88	C₂×C₈	#137	1	1	yes	characteristic
89	C₂×C₈	#137	1	1	yes	characteristic
90	C₂×C₈	#137	1	2	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	4	4	no	—
93	C₄⋊C₄	—	8	4	no	—
94	C₄⋊C₄	—	8	4	no	—
95	C₄⋊C₄	—	8	2	no	—
96	C₂²⋊C₄	—	4	1	no	—
97	C₄²	—	2	1	no	—
98	C₄²	—	4	1	no	—
99	C₄²	#123	1	2	yes	—
100	C₁₆	—	2	2	no	—
101	C₁₆	—	2	2	no	—
102	C₁₆	—	8	1	no	—
103	C₁₆	—	8	2	no	—
104	C₂³	#472	1	1	yes	characteristic
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	8	4	no	—
109	C₂×C₄	—	16	4	no	—
110	C₂×C₄	#394	1	1	yes	characteristic
111	C₂×C₄	#461	1	1	yes	characteristic
112	C₂×C₄	#462	1	1	yes	characteristic
113	C₂×C₄	#467	1	1	yes	characteristic
114	C₂×C₄	#470	1	1	yes	characteristic
115	C₂×C₄	#472	1	1	yes	characteristic
116	C₈	—	8	1	no	—
117	C₈	#448	1	1	yes	characteristic
118	C₈	#451	1	1	yes	characteristic
119	C₈	#462	1	2	yes	—
120	C₈	#464	1	2	yes	—
121	C₈	#487	1	1	yes	characteristic
122	C₈	#491	1	1	yes	characteristic
123	C₂²	—	2	2	no	—
124	C₂²	#1675	1	1	yes	characteristic
125	C₂²	#1696	1	1	yes	characteristic
126	C₂²	#2045	1	1	yes	characteristic
127	C₄	—	4	2	no	—
128	C₄	—	8	2	no	—
129	C₄	—	16	4	no	—
130	C₄	#1676	1	1	yes	characteristic
131	C₄	#1695	1	1	yes	characteristic
132	C₄	#2043	1	1	yes	characteristic
133	C₄	#2056	1	1	yes	characteristic
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#10543	1	1	yes	characteristic
137	C₂	#12464	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2253

C₂

yes

maximal, characteristic

#13273

C₂

yes

maximal

#13161

C₂

yes

maximal, characteristic

C₂×#1366

C₂

yes

maximal

#7160

C₂

yes

maximal, characteristic

C₂×#499

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#364

C₂²

yes

—

#2151

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1772

C₂²

yes

—

—

—

—

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

—

—

—

D₄

yes

—

C₂×#108

D₄

yes

characteristic

#427

D₄

yes

—

—

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

C₄⋊C₁₆

—

C₄×C₁₆

—

#339

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

—

—

—

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

#41

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

Q₁₆

yes

—

—

—

—

Q₁₆

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

#120

yes

characteristic

C₂×C₈

#121

yes

—

C₂×C₈

#124

yes

characteristic

C₂×C₈

C₂×SD₁₆

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

#123

yes

—

100

C₁₆

—

101

C₁₆

—

102

C₁₆

—

103

C₁₆

—

104

C₂³

#472

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

#394

yes

characteristic

111

C₂×C₄

#461

yes

characteristic

112

C₂×C₄

#462

yes

characteristic

113

C₂×C₄

#467

yes

characteristic

114

C₂×C₄

#470

yes

characteristic

115

C₂×C₄

#472

yes

characteristic

116

C₈

—

117

C₈

#448

yes

characteristic

118

C₈

#451

yes

characteristic

119

C₈

#462

yes

—

120

C₈

#464

yes

—

121

C₈

#487

yes

characteristic

122

C₈

#491

yes

characteristic

123

C₂²

—

124

C₂²

#1675

yes

characteristic

125

C₂²

#1696

yes

characteristic

126

C₂²

#2045

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#1676

yes

characteristic

131

C₄

#1695

yes

characteristic

132

C₄

#2043

yes

characteristic

133

C₄

#2056

yes

characteristic

134

C₂

—

135

C₂

—

136

C₂

#10543

yes

characteristic

137

C₂

#12464

yes

—

	subgroup	G/H
center, socle	C₂²	#1675
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1675

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#472

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#2253	2
#13273	2
#13161	2
C₂×#1366	2
#7160	2

subgroup

index

Subgroups — #145575

Conjugacy classes of subgroups

Maximal subgroups