Group #153396 — subgroups

Subgroups — #153396

Order 512.

157 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 157 conjugacy classes of subgroups — too many to draw. Only its 66 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×#1426	C₂	1	1	yes	maximal, characteristic
2	#13630	C₂	1	1	yes	maximal, characteristic
3	#13173	C₂	1	1	yes	maximal, characteristic
4	#13172	C₂	1	1	yes	maximal, characteristic
5	#7479	C₂	1	1	yes	maximal, characteristic
6	#7479	C₂	1	1	yes	maximal, characteristic
7	#7474	C₂	1	1	yes	maximal, characteristic
8	C₂×M₆₄	C₂²	1	1	yes	characteristic
9	C₂×M₆₄	C₂²	1	1	yes	characteristic
10	C₂×#368	C₂²	1	1	yes	characteristic
11	C₂×#368	C₂²	1	1	yes	characteristic
12	C₂×#368	C₂²	1	1	yes	characteristic
13	C₂×#367	C₂²	1	1	yes	characteristic
14	#2108	C₂²	1	1	yes	characteristic
15	#1953	—	2	1	no	—
16	#1952	—	2	1	no	—
17	#1426	—	2	2	no	—
18	#1422	—	2	2	no	—
19	#1388	—	2	1	no	—
20	#1387	—	2	1	no	—
21	C₂×M₃₂	D₄	1	1	yes	characteristic
22	C₂²×C₁₆	C₂³	1	1	yes	characteristic
23	#430	—	4	1	no	—
24	C₂×#108	—	2	1	no	—
25	C₂×#108	—	2	1	no	—
26	C₂×#108	—	2	1	no	—
27	#428	—	4	1	no	—
28	C₂×#107	—	2	1	no	—
29	#405	D₄	1	1	yes	characteristic
30	M₆₄	—	2	1	no	—
31	M₆₄	—	2	1	no	—
32	M₆₄	—	2	1	no	—
33	M₆₄	D₄	1	2	yes	—
34	C₂×C₃₂	—	2	1	no	—
35	C₂×C₃₂	—	2	1	no	—
36	#368	—	2	1	no	—
37	#368	—	2	1	no	—
38	#368	—	2	1	no	—
39	#368	—	2	1	no	—
40	#368	Q₈	1	2	yes	—
41	#368	Q₈	1	2	yes	—
42	#367	—	2	1	no	—
43	#367	—	2	1	no	—
44	#346	—	2	1	no	—
45	C₄×C₁₆	—	2	1	no	—
46	#344	—	4	1	no	—
47	#343	—	4	1	no	—
48	C₂×M₁₆	—	4	1	no	—
49	C₂×M₁₆	—	4	1	no	—
50	C₂×M₁₆	—	4	1	no	—
51	C₂×M₁₆	D₈	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
53	#117	D₈	1	1	yes	characteristic
54	C₂×C₄⋊C₄	—	4	1	no	—
55	M₃₂	—	2	1	no	—
56	M₃₂	—	2	1	no	—
57	C₂×C₁₆	—	2	1	no	—
58	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
59	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₁₆	#41	1	1	yes	characteristic
62	C₂×C₁₆	#41	1	1	yes	characteristic
63	C₂×C₁₆	#41	1	1	yes	characteristic
64	#108	—	2	2	no	—
65	#108	—	2	2	no	—
66	#108	—	4	1	no	—
67	#108	—	4	1	no	—
68	#108	—	4	1	no	—
69	#108	—	4	1	no	—
70	#108	—	8	2	no	—
71	#107	—	4	1	no	—
72	#107	—	4	1	no	—
73	#107	—	8	1	no	—
74	#106	—	8	1	no	—
75	#97	—	2	1	no	—
76	C₄×C₈	—	2	1	no	—
77	C₃₂	—	2	2	no	—
78	C₃₂	—	4	1	no	—
79	C₂²×C₄	—	8	1	no	—
80	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
81	M₁₆	—	2	1	no	—
82	M₁₆	—	4	2	no	—
83	M₁₆	—	4	2	no	—
84	M₁₆	—	8	1	no	—
85	M₁₆	—	8	1	no	—
86	M₁₆	—	8	1	no	—
87	M₁₆	—	8	1	no	—
88	M₁₆	SD₃₂	1	2	yes	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	8	1	no	—
91	C₂×C₈	—	8	1	no	—
92	C₂×C₈	—	8	1	no	—
93	C₂×C₈	#121	1	1	yes	characteristic
94	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₈	#135	1	1	yes	characteristic
97	C₂×C₈	#137	1	1	yes	characteristic
98	C₂×C₈	#137	1	1	yes	characteristic
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	8	1	no	—
101	C₄⋊C₄	—	8	1	no	—
102	C₂²⋊C₄	—	2	1	no	—
103	C₄²	—	2	1	no	—
104	C₁₆	—	4	1	no	—
105	C₁₆	#123	1	1	yes	characteristic
106	C₁₆	#125	1	1	yes	characteristic
107	C₁₆	#125	1	1	yes	characteristic
108	C₁₆	C₄⋊Q₈	1	1	yes	characteristic
109	C₂³	#510	1	1	yes	characteristic
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	16	1	no	—
112	C₂×C₄	—	32	1	no	—
113	C₂×C₄	#466	1	1	yes	characteristic
114	C₂×C₄	#474	1	1	yes	characteristic
115	C₂×C₄	#474	1	1	yes	characteristic
116	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
117	C₂×C₄	#508	1	1	yes	characteristic
118	C₂×C₄	#509	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	8	2	no	—
121	C₈	—	8	2	no	—
122	C₈	—	16	1	no	—
123	C₈	#398	1	1	yes	characteristic
124	C₈	#470	1	1	yes	characteristic
125	C₈	#479	1	1	yes	characteristic
126	C₈	#479	1	1	yes	characteristic
127	C₂²	—	4	1	no	—
128	C₂²	#2220	1	1	yes	characteristic
129	C₂²	#2225	1	1	yes	characteristic
130	C₂²	#2229	1	1	yes	characteristic
131	C₄	—	8	1	no	—
132	C₄	—	32	1	no	—
133	C₄	#2086	1	1	yes	characteristic
134	C₄	#2217	1	1	yes	characteristic
135	C₄	#2226	1	1	yes	characteristic
136	C₄	#2228	1	1	yes	characteristic
137	C₂	—	2	1	no	—
138	C₂	—	2	1	no	—
139	C₂	—	2	1	no	—
140	C₂	#13516	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1426

C₂

yes

maximal, characteristic

#13630

C₂

yes

maximal, characteristic

#13173

C₂

yes

maximal, characteristic

#13172

C₂

yes

maximal, characteristic

#7479

C₂

yes

maximal, characteristic

#7479

C₂

yes

maximal, characteristic

#7474

C₂

yes

maximal, characteristic

C₂×M₆₄

C₂²

yes

characteristic

C₂×M₆₄

C₂²

yes

characteristic

C₂×#368

C₂²

yes

characteristic

C₂×#368

C₂²

yes

characteristic

C₂×#368

C₂²

yes

characteristic

C₂×#367

C₂²

yes

characteristic

#2108

C₂²

yes

—

—

—

—

—

—

C₂×M₃₂

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

—

D₄

yes

characteristic

M₆₄

—

M₆₄

—

M₆₄

—

M₆₄

D₄

yes

—

C₂×C₃₂

—

C₂×C₃₂

—

—

—

—

—

Q₈

yes

—

#368

Q₈

yes

—

#367

—

#367

—

#346

—

C₄×C₁₆

—

#344

—

#343

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

D₈

yes

characteristic

C₂×C₄⋊C₄

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

—

—

—

—

—

—

—

—

—

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

SD₃₂

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#121

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₁₆

—

105

C₁₆

#123

yes

characteristic

106

C₁₆

#125

yes

characteristic

107

C₁₆

#125

yes

characteristic

108

C₁₆

C₄⋊Q₈

yes

characteristic

109

C₂³

#510

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#466

yes

characteristic

114

C₂×C₄

#474

yes

characteristic

115

C₂×C₄

#474

yes

characteristic

116

C₂×C₄

C₂×SD₃₂

yes

characteristic

117

C₂×C₄

#508

yes

characteristic

118

C₂×C₄

#509

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

#398

yes

characteristic

124

C₈

#470

yes

characteristic

125

C₈

#479

yes

characteristic

126

C₈

#479

yes

characteristic

127

C₂²

—

128

C₂²

#2220

yes

characteristic

129

C₂²

#2225

yes

characteristic

130

C₂²

#2229

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

#2086

yes

characteristic

134

C₄

#2217

yes

characteristic

135

C₄

#2226

yes

characteristic

136

C₄

#2228

yes

characteristic

137

C₂

—

138

C₂

—

139

C₂

—

140

C₂

#13516

yes

characteristic

	subgroup	G/H
center, socle	C₂	#13516
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#510
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂

#13516

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂³

#510

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#1426	2
#13630	2
#13173	2
#13172	2
#7479	2
#7474	2

subgroup

index

Subgroups — #153396

Conjugacy classes of subgroups

Maximal subgroups