Group #134870 — subgroups

Subgroups — #134870

Order 512.

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

normal subgroups only

This group has 194 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	#33475	C₂	1	1	yes	maximal, characteristic
2	#12027	C₂	1	2	yes	maximal
3	#11539	C₂	1	2	yes	maximal
4	#7240	C₂	1	2	yes	maximal
5	#2960	C₂²	1	1	yes	characteristic
6	#2183	—	2	2	no	—
7	#2183	—	2	2	no	—
8	#2166	C₂²	1	4	yes	—
9	#2151	C₄	1	1	yes	characteristic
10	#2151	C₂²	1	1	yes	characteristic
11	#2150	C₄	1	1	yes	characteristic
12	#2108	C₄	1	1	yes	characteristic
13	#2108	C₂²	1	1	yes	characteristic
14	C₂×#346	C₄	1	1	yes	characteristic
15	#1373	—	2	2	no	—
16	#1369	—	2	2	no	—
17	#1328	—	2	2	no	—
18	#1327	—	2	2	no	—
19	#533	Q₈	1	1	yes	characteristic
20	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
21	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
24	#446	—	2	2	no	—
25	C₈×Q₈	—	2	2	no	—
26	#424	D₄	1	1	yes	characteristic
27	#424	Q₈	1	1	yes	characteristic
28	#423	D₄	1	1	yes	characteristic
29	#405	C₂×C₄	1	1	yes	characteristic
30	#405	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	#364	—	2	4	no	—
33	C₄⋊C₁₆	—	2	2	no	—
34	C₄⋊C₁₆	—	2	2	no	—
35	C₄⋊C₁₆	—	2	2	no	—
36	#349	—	2	4	no	—
37	#347	—	2	4	no	—
38	#346	—	2	1	no	—
39	#346	—	2	1	no	—
40	#346	—	2	1	no	—
41	#346	D₄	1	2	yes	—
42	C₄×C₁₆	—	2	1	no	—
43	C₄×C₁₆	D₄	1	2	yes	—
44	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
45	C₂×M₁₆	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₄⋊Q₈	—	2	1	no	—
49	#125	—	2	1	no	—
50	C₂²⋊Q₈	—	2	2	no	—
51	C₄×Q₈	—	2	2	no	—
52	#117	C₄⋊C₄	1	1	yes	characteristic
53	#117	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
55	M₃₂	—	4	1	no	—
56	M₃₂	—	4	1	no	—
57	M₃₂	—	4	4	no	—
58	C₂×C₁₆	—	2	2	no	—
59	C₂×C₁₆	—	2	1	no	—
60	C₂×C₁₆	—	2	1	no	—
61	C₂×C₁₆	—	4	1	no	—
62	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
64	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	—	2	2	no	—
68	#97	—	2	1	no	—
69	#97	#41	1	2	yes	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	C₂×D₄	1	2	yes	—
74	C₂×Q₈	—	2	2	no	—
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
77	M₁₆	—	4	1	no	—
78	M₁₆	—	4	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈	#117	1	1	yes	characteristic
86	C₂×C₈	#121	1	2	yes	—
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	1	no	—
90	C₄⋊C₄	—	2	1	no	—
91	C₂²⋊C₄	—	2	1	no	—
92	C₂²⋊C₄	#108	1	2	yes	—
93	C₄²	—	2	1	no	—
94	C₄²	—	2	1	no	—
95	C₄²	#108	1	1	yes	characteristic
96	C₄²	#108	1	1	yes	characteristic
97	C₄²	C₂²⋊Q₈	1	2	yes	—
98	C₁₆	—	2	1	no	—
99	C₁₆	—	2	2	no	—
100	C₁₆	—	4	1	no	—
101	C₁₆	—	4	1	no	—
102	C₁₆	—	8	4	no	—
103	C₁₆	#101	1	2	yes	—
104	C₂³	C₂×#108	1	1	yes	characteristic
105	Q₈	—	8	2	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	8	1	no	—
113	C₂×C₄	#380	1	1	yes	characteristic
114	C₂×C₄	C₂×#108	1	1	yes	characteristic
115	C₈	—	2	1	no	—
116	C₈	—	2	2	no	—
117	C₈	—	4	1	no	—
118	C₈	—	4	1	no	—
119	C₈	C₂×#101	1	1	yes	characteristic
120	C₈	#413	1	1	yes	characteristic
121	C₂²	—	2	1	no	—
122	C₂²	—	4	1	no	—
123	C₂²	#1810	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	4	1	no	—
126	C₄	—	4	1	no	—
127	C₄	—	4	2	no	—
128	C₄	—	4	2	no	—
129	C₄	#1801	1	1	yes	characteristic
130	C₄	#1812	1	1	yes	characteristic
131	C₂	—	2	1	no	—
132	C₂	—	4	1	no	—
133	C₂	#8623	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33475

C₂

yes

maximal, characteristic

#12027

C₂

yes

maximal

#11539

C₂

yes

maximal

#7240

C₂

yes

maximal

#2960

C₂²

yes

—

—

C₂²

yes

—

#2151

C₄

yes

characteristic

#2151

C₂²

yes

characteristic

#2150

C₄

yes

characteristic

#2108

C₄

yes

characteristic

#2108

C₂²

yes

characteristic

C₂×#346

C₄

yes

—

—

—

—

Q₈

yes

characteristic

C₂×M₃₂

C₂×C₄

yes

characteristic

C₂×M₃₂

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

#446

—

C₈×Q₈

—

#424

D₄

yes

characteristic

#424

Q₈

yes

characteristic

#423

D₄

yes

characteristic

#405

C₂×C₄

yes

characteristic

#405

C₂³

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

—

—

D₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

C₂×M₁₆

C₄⋊C₄

yes

characteristic

C₂×M₁₆

C₂×D₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₄⋊C₄

yes

characteristic

#117

C₂×Q₈

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

#41

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₂²⋊C₄

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

#108

yes

—

C₄²

—

C₄²

—

C₄²

yes

C₄²

yes

C₄²

C₂²⋊Q₈

yes

—

C₁₆

—

C₁₆

—

100

C₁₆

—

101

C₁₆

—

102

C₁₆

—

103

C₁₆

#101

yes

—

104

C₂³

C₂×#108

yes

characteristic

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#380

yes

characteristic

114

C₂×C₄

C₂×#108

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

C₂×#101

yes

characteristic

120

C₈

#413

yes

characteristic

121

C₂²

—

122

C₂²

—

123

C₂²

#1810

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#1801

yes

characteristic

130

C₄

#1812

yes

characteristic

131

C₂

—

132

C₂

—

133

C₂

#8623

yes

characteristic

	subgroup	G/H
center	C₄	#1812
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#405	C₂³
socle	C₂	#8623
prime radical	C₂³	C₂×#108
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2151 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#1812

commutator

C₂²×C₈

C₂²×C₄

C₂³

C₂

C₂³

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2151 ◁ G

LECS

G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#33475	2
#12027	2
#11539	2
#7240	2

subgroup

index

Subgroups — #134870

Conjugacy classes of subgroups

Maximal subgroups