Group #135663 — subgroups

Subgroups — #135663

Order 512.

216 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 106 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	#12539	C₂	1	1	yes	maximal, characteristic
2	#8780	C₂	1	2	yes	maximal
3	#7157	C₂	1	2	yes	maximal
4	#7155	C₂	1	2	yes	maximal
5	C₂×#346	C₄	1	1	yes	characteristic
6	C₂×#346	C₄	1	1	yes	characteristic
7	C₂×#346	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
9	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	#1726	C₂²	1	1	yes	characteristic
12	#1564	C₂²	1	4	yes	—
13	#1281	—	2	2	no	—
14	#1274	—	2	2	no	—
15	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
19	#431	—	2	4	no	—
20	C₂×#97	D₄	1	1	yes	characteristic
21	C₂×#97	Q₈	1	1	yes	characteristic
22	C₂×#97	Q₈	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
26	#376	D₄	1	1	yes	characteristic
27	#346	—	2	2	no	—
28	#346	—	2	2	no	—
29	#346	C₈	1	4	yes	complemented
30	C₄×C₁₆	—	2	2	no	—
31	C₄×C₁₆	—	2	2	no	—
32	C₄×C₁₆	—	2	2	no	—
33	#335	—	2	4	no	—
34	#334	—	2	4	no	—
35	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
42	C₂×C₁₆	—	2	1	no	—
43	C₂×C₁₆	—	2	1	no	—
44	C₂×C₁₆	—	2	1	no	—
45	C₂×C₁₆	—	2	1	no	—
46	C₂×C₁₆	—	2	1	no	—
47	C₂×C₁₆	—	2	1	no	—
48	C₂×C₁₆	—	2	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	C₂×C₈	1	2	yes	—
51	C₂×C₁₆	C₂×C₈	1	2	yes	—
52	C₂×C₁₆	M₁₆	1	2	yes	—
53	C₂×C₁₆	M₁₆	1	2	yes	—
54	C₄⋊C₈	—	4	4	no	—
55	C₂²⋊C₈	—	4	4	no	—
56	#97	—	2	2	no	—
57	#97	—	2	2	no	—
58	#97	—	2	2	no	—
59	C₄×C₈	—	2	1	no	—
60	C₄×C₈	—	2	1	no	—
61	C₄×C₈	—	4	4	no	—
62	C₄×C₈	C₂×C₈	1	2	yes	—
63	C₄×C₈	M₁₆	1	2	yes	—
64	C₄×C₈	#41	1	2	yes	—
65	C₄×C₈	#41	1	2	yes	—
66	#95	—	2	2	no	—
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄	#135	1	2	yes	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	8	4	no	—
78	C₂×C₈	C₄⋊C₈	1	2	yes	—
79	C₂×C₈	#108	1	2	yes	—
80	C₂×C₈	#117	1	1	yes	characteristic
81	C₂×C₈	#117	1	1	yes	characteristic
82	C₂×C₈	#125	1	2	yes	—
83	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
87	C₄²	—	2	1	no	—
88	C₄²	—	2	1	no	—
89	C₄²	C₄⋊C₈	1	2	yes	—
90	C₄²	#108	1	2	yes	—
91	C₄²	#124	1	2	yes	—
92	C₄²	#135	1	2	yes	—
93	C₁₆	—	2	2	no	—
94	C₁₆	—	2	2	no	—
95	C₁₆	—	2	2	no	—
96	C₁₆	—	2	2	no	—
97	C₂³	#427	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	#382	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
105	C₂×C₄	#423	1	1	yes	characteristic
106	C₂×C₄	#427	1	1	yes	characteristic
107	C₂×C₄	C₂×#108	1	1	yes	characteristic
108	C₂×C₄	C₂×#108	1	1	yes	characteristic
109	C₂×C₄	#488	1	4	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	—	2	2	no	—
113	C₈	—	16	4	no	—
114	C₈	#431	1	1	yes	characteristic
115	C₈	#431	1	1	yes	characteristic
116	C₈	#432	1	1	yes	characteristic
117	C₈	#432	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	#1564	1	1	yes	characteristic
120	C₂²	#1568	1	1	yes	characteristic
121	C₂²	#1835	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	4	2	no	—
124	C₄	—	4	2	no	—
125	C₄	#1564	1	1	yes	characteristic
126	C₄	#1568	1	1	yes	characteristic
127	C₄	#1831	1	1	yes	characteristic
128	C₄	#1837	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	2	1	no	—
131	C₂	#8785	1	1	yes	characteristic
132	C₂	#12893	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12539

C₂

yes

maximal, characteristic

#8780

C₂

yes

maximal

#7157

C₂

yes

maximal

#7155

C₂

yes

maximal

C₂×#346

C₄

yes

characteristic

C₂×#346

C₄

yes

characteristic

C₂×#346

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₄

yes

characteristic

C₂×C₄×C₁₆

C₄

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1726

C₂²

yes

characteristic

#1564

C₂²

yes

—

#1281

—

#1274

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

#431

—

C₂×#97

D₄

yes

characteristic

C₂×#97

Q₈

yes

characteristic

C₂×#97

Q₈

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#376

D₄

yes

—

—

C₈

yes

complemented

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

#334

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

M₁₆

yes

—

C₂×C₁₆

M₁₆

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

#41

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

#135

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

#108

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

C₂²×C₈

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

characteristic

C₄²

—

C₄²

—

C₄²

C₄⋊C₈

yes

—

C₄²

#108

yes

—

C₄²

#124

yes

—

C₄²

#135

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#427

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#382

yes

characteristic

104

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

105

C₂×C₄

#423

yes

characteristic

106

C₂×C₄

#427

yes

characteristic

107

C₂×C₄

C₂×#108

yes

characteristic

108

C₂×C₄

C₂×#108

yes

characteristic

109

C₂×C₄

#488

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

#431

yes

characteristic

115

C₈

#431

yes

characteristic

116

C₈

#432

yes

characteristic

117

C₈

#432

yes

characteristic

118

C₂²

—

119

C₂²

#1564

yes

characteristic

120

C₂²

#1568

yes

characteristic

121

C₂²

#1835

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#1564

yes

characteristic

126

C₄

#1568

yes

characteristic

127

C₄

#1831

yes

characteristic

128

C₄

#1837

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

#8785

yes

characteristic

132

C₂

#12893

yes

—

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1835
prime radical	C₂³	#427
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#427

commutator

C₂×C₈

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ #376 ◁ #1726 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
29. #346	C₈	semidirect

H ≅ G/N

type

29. #346

C₈

semidirect

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

subgroup	index
#12539	2
#8780	2
#7157	2
#7155	2

subgroup

index

Subgroups — #135663

Conjugacy classes of subgroups

Complements

Maximal subgroups