Group #70531 — subgroups

Subgroups — #70531

Order 384.

266 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 65 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	#5748	C₂	1	1	yes	maximal, characteristic
2	C₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	#5132	C₂	1	2	yes	maximal, complemented
4	C₂×#805	C₂	1	2	yes	maximal, complemented
5	C₃⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
6	#2050	—	3	1	no	maximal, Sylow
7	C₂×#235	C₂²	1	2	yes	—
8	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₂×#208	C₂²	1	2	yes	—
12	#909	—	2	2	no	—
13	#908	—	2	2	no	—
14	#904	—	2	2	no	—
15	C₃×C₄⋊C₈	—	2	2	no	—
16	#824	—	2	2	no	—
17	#823	—	2	2	no	—
18	#805	—	2	2	no	—
19	#805	—	2	2	no	—
20	#575	—	3	1	no	—
21	#423	D₃	1	1	yes	characteristic, complemented
22	C₂×#104	—	3	2	no	—
23	#413	—	3	2	no	—
24	#338	—	3	1	no	—
25	C₂²×C₃×C₄	—	2	1	no	—
26	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₂×#67	—	2	2	no	—
28	#235	—	2	2	no	—
29	#235	—	2	2	no	—
30	#235	—	2	2	no	—
31	#235	—	2	2	no	—
32	C₂×D₁₂	D₄	1	2	yes	—
33	C₂×C₄×D₃	—	2	2	no	—
34	C₂×Dic₆	D₄	1	2	yes	—
35	C₃×M₁₆	—	2	2	no	—
36	C₃×M₁₆	—	2	2	no	—
37	C₂×C₃×C₈	—	2	1	no	—
38	C₂×C₃×C₈	D₄	1	2	yes	—
39	C₃×C₄²	—	2	1	no	—
40	C₃×C₄²	D₄	1	1	yes	characteristic
41	C₃×C₄²	D₄	1	1	yes	characteristic
42	#208	—	2	2	no	—
43	#208	—	2	2	no	—
44	C₂×C₃⋊C₈	—	2	2	no	—
45	Dic₁₂	—	2	2	no	—
46	D₂₄	—	2	2	no	—
47	#204	—	2	4	no	—
48	C₂×#41	—	3	2	no	—
49	#137	—	6	2	no	—
50	AGL(1,ℤ/8)	—	6	2	no	—
51	#135	—	6	2	no	—
52	C₂×M₁₆	—	3	2	no	—
53	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
54	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
55	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
56	C₄⋊C₈	—	2	2	no	—
57	#104	—	6	2	no	—
58	#104	—	6	2	no	—
59	#101	—	6	2	no	—
60	#100	—	6	2	no	—
61	C₂³×C₃	#41	1	1	yes	characteristic
62	C₂²×D₃	—	2	2	no	—
63	C₂×C₃×C₄	—	2	1	no	—
64	C₂×C₃×C₄	—	2	1	no	—
65	C₂×C₃×C₄	—	4	2	no	—
66	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
68	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄	#41	1	2	yes	—
70	#67	—	4	2	no	—
71	#67	—	4	2	no	—
72	C₂×Dic₃	—	2	2	no	—
73	D₁₂	—	2	2	no	—
74	D₁₂	—	2	2	no	—
75	C₄×D₃	—	4	2	no	—
76	C₄×D₃	—	4	2	no	—
77	Dic₆	—	2	2	no	—
78	Dic₆	—	2	2	no	—
79	C₃×C₈	—	2	2	no	—
80	C₃×C₈	—	4	1	no	—
81	C₃⋊C₈	—	4	2	no	—
82	#41	—	6	2	no	—
83	#41	—	6	2	no	—
84	#41	—	6	2	no	—
85	#41	—	6	2	no	—
86	C₂×Q₈	—	3	2	no	—
87	C₂×D₄	—	3	2	no	—
88	C₂×D₄	—	6	2	no	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	—	6	2	no	—
91	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
92	Q₁₆	—	6	2	no	—
93	SD₁₆	—	6	4	no	—
94	D₈	—	6	2	no	—
95	M₁₆	—	2	2	no	—
96	M₁₆	—	2	2	no	—
97	M₁₆	—	6	2	no	—
98	M₁₆	—	6	2	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	6	2	no	—
101	C₂×C₈	#67	1	2	yes	—
102	C₄²	—	2	1	no	—
103	C₄²	D₁₂	1	1	yes	characteristic
104	C₄²	D₁₂	1	1	yes	characteristic
105	C₂²×C₃	—	4	1	no	—
106	C₂²×C₃	#121	1	1	yes	characteristic
107	C₂²×C₃	#123	1	2	yes	—
108	C₂×D₃	—	4	2	no	—
109	C₂×D₃	—	8	2	no	—
110	C₃×C₄	—	2	2	no	—
111	C₃×C₄	—	4	1	no	—
112	C₃×C₄	#120	1	1	yes	characteristic
113	C₃×C₄	#121	1	2	yes	—
114	C₃×C₄	#124	1	1	yes	characteristic
115	Dic₃	—	4	2	no	—
116	C₂³	—	6	2	no	—
117	C₂³	#235	1	1	yes	characteristic
118	Q₈	—	6	2	no	—
119	Q₈	—	6	2	no	—
120	D₄	—	6	2	no	—
121	D₄	—	6	2	no	—
122	D₄	—	12	2	no	—
123	D₄	—	12	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	6	2	no	—
128	C₂×C₄	—	12	2	no	—
129	C₂×C₄	—	12	2	no	—
130	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
131	C₂×C₄	D₃×D₄	1	2	yes	—
132	C₂×C₄	#239	1	2	yes	—
133	C₂×C₄	C₂×#67	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	—	4	1	no	—
136	C₈	—	12	2	no	—
137	C₂×C₃	—	2	1	no	—
138	C₂×C₃	—	2	2	no	—
139	C₂×C₃	#394	1	1	yes	characteristic
140	D₃	—	8	2	no	—
141	C₂²	—	4	1	no	—
142	C₂²	—	12	2	no	—
143	C₂²	—	24	2	no	—
144	C₂²	#894	1	2	yes	—
145	C₂²	#930	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	4	1	no	—
148	C₄	—	12	2	no	—
149	C₄	#895	1	2	yes	—
150	C₄	#937	1	1	yes	characteristic
151	C₄	#947	1	1	yes	characteristic
152	C₃	#2050	1	1	yes	characteristic, complemented, Sylow
153	C₂	—	2	1	no	—
154	C₂	—	2	2	no	—
155	C₂	—	24	2	no	—
156	C₂	#4991	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#5748

C₂

yes

maximal, characteristic

C₃×#423

C₂

yes

maximal, characteristic, complemented

#5132

C₂

yes

maximal, complemented

C₂×#805

C₂

yes

maximal, complemented

C₃⋊#338

C₂

yes

maximal, characteristic, complemented

#2050

—

maximal, Sylow

C₂×#235

C₂²

yes

—

C₂×C₃×M₁₆

C₂²

yes

characteristic

C₂×C₃×M₁₆

C₂²

yes

characteristic

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×#208

C₂²

yes

—

#909

—

#908

—

#904

—

C₃×C₄⋊C₈

—

—

—

—

—

—

D₃

yes

characteristic, complemented

C₂×#104

—

#413

—

#338

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂×#67

—

—

—

—

—

C₂×D₁₂

D₄

yes

—

C₂×C₄×D₃

—

C₂×Dic₆

D₄

yes

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₄

yes

—

C₃×C₄²

—

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

#208

—

#208

—

C₂×C₃⋊C₈

—

Dic₁₂

—

D₂₄

—

#204

—

C₂×#41

—

#137

—

AGL(1,ℤ/8)

—

#135

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃

yes

characteristic

C₂×M₁₆

C₂×D₃

yes

characteristic

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

—

—

—

—

C₂³×C₃

#41

yes

characteristic

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

—

#67

—

#67

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

C₄×D₃

—

C₄×D₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

characteristic

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

#67

yes

—

102

C₄²

—

103

C₄²

D₁₂

yes

characteristic

104

C₄²

D₁₂

yes

characteristic

105

C₂²×C₃

—

106

C₂²×C₃

#121

yes

characteristic

107

C₂²×C₃

#123

yes

—

108

C₂×D₃

—

109

C₂×D₃

—

110

C₃×C₄

—

111

C₃×C₄

—

112

C₃×C₄

#120

yes

characteristic

113

C₃×C₄

#121

yes

—

114

C₃×C₄

#124

yes

characteristic

115

Dic₃

—

116

C₂³

—

117

C₂³

#235

yes

characteristic

118

Q₈

—

119

Q₈

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×D₁₂

yes

characteristic

131

C₂×C₄

D₃×D₄

yes

—

132

C₂×C₄

#239

yes

—

133

C₂×C₄

C₂×#67

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₂×C₃

—

138

C₂×C₃

—

139

C₂×C₃

#394

yes

characteristic

140

D₃

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#894

yes

—

145

C₂²

#930

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#895

yes

—

150

C₄

#937

yes

characteristic

151

C₄

#947

yes

characteristic

152

C₃

#2050

yes

characteristic, complemented, Sylow

153

C₂

—

154

C₂

—

155

C₂

—

156

C₂

#4991

yes

characteristic

	subgroup	G/H
center	C₄	#947
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#423	C₂
socle	C₂×C₃	#394
prime radical	#5748	C₂
hypercenter	#423	D₃
hypocenter	C₃	#2050
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

subgroup

G/H

center

C₄

#947

commutator, cosocle

C₂²×C₃×C₄

C₂³

Frattini

C₂²×C₄

C₂²×D₃

C₂

C₂×C₃

C₂

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

N	H ≅ G/N	type
2. C₃×#423	C₂	semidirect
3. #5132	C₂	semidirect
4. C₂×#805	C₂	semidirect
5. C₃⋊#338	C₂	semidirect
21. #423	D₃	semidirect
152. C₃	#2050	semidirect

H ≅ G/N

type

2. C₃×#423

C₂

semidirect

3. #5132

C₂

semidirect

4. C₂×#805

C₂

semidirect

5. C₃⋊#338

C₂

semidirect

21. #423

D₃

semidirect

152. C₃

#2050

semidirect

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

subgroup	index
#5748	2
C₃×#423	2
#5132	2
C₂×#805	2
C₃⋊#338	2
#2050	3

subgroup

index

Subgroups — #70531

Conjugacy classes of subgroups

Complements

Maximal subgroups