Group #21660 — subgroups

Subgroups — #21660

Order 256.

373 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 167 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	#3131	C₂	1	1	yes	maximal, characteristic
2	#2960	C₂	1	1	yes	maximal, characteristic, complemented
3	#2931	C₂	1	1	yes	maximal, characteristic
4	#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	#1962	C₂	1	1	yes	maximal, characteristic
6	#1938	C₂	1	4	yes	maximal, complemented
7	#1837	C₂	1	1	yes	maximal, characteristic, complemented
8	#1765	C₂	1	4	yes	maximal, complemented
9	C₄×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#131	C₂²	1	2	yes	—
11	#533	C₂²	1	1	yes	characteristic
12	#532	C₂²	1	1	yes	characteristic
13	#488	C₄	1	4	yes	complemented
14	#487	C₄	1	2	yes	complemented
15	#486	C₄	1	2	yes	complemented
16	#446	—	2	2	no	—
17	C₈×Q₈	—	2	2	no	—
18	#436	—	2	2	no	—
19	C₈×D₄	—	2	2	no	—
20	#424	C₂²	1	2	yes	—
21	#423	C₂²	1	1	yes	characteristic
22	#423	C₂²	1	1	yes	characteristic
23	C₂×#104	C₂²	1	4	yes	—
24	#416	C₂²	1	4	yes	—
25	#408	C₂²	1	2	yes	—
26	#405	C₂²	1	2	yes	—
27	C₂×#97	C₂²	1	2	yes	—
28	C₂×#97	C₂²	1	2	yes	—
29	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
31	C₄³	C₂²	1	1	yes	characteristic
32	#341	C₂²	1	4	yes	—
33	#339	C₂²	1	4	yes	—
34	C₂×#41	C₂³	1	2	yes	—
35	#131	—	2	8	no	—
36	C₂×M₁₆	—	2	2	no	—
37	C₂×M₁₆	C₂³	1	2	yes	—
38	C₂×M₁₆	C₂³	1	2	yes	—
39	C₂²×C₈	—	2	2	no	—
40	C₂²×C₈	C₂³	1	2	yes	—
41	C₂²×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₈	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
44	C₄⋊Q₈	C₂×C₄	1	2	yes	—
45	#127	C₂×C₄	1	1	yes	characteristic
46	#125	C₂×C₄	1	2	yes	—
47	#124	C₂×C₄	1	2	yes	—
48	C₂²⋊Q₈	—	2	2	no	—
49	#121	—	2	2	no	—
50	C₄×Q₈	—	2	2	no	—
51	C₄×D₄	—	2	2	no	—
52	#117	C₂³	1	2	yes	—
53	C₂×C₄²	—	2	2	no	—
54	C₂×C₄²	C₂³	1	2	yes	—
55	C₂×C₄²	C₂³	1	1	yes	characteristic
56	C₄⋊C₈	—	2	4	no	—
57	C₄⋊C₈	—	2	2	no	—
58	C₄⋊C₈	—	2	2	no	—
59	#104	—	2	8	no	—
60	#103	C₂×C₄	1	8	yes	—
61	#102	C₂×C₄	1	8	yes	—
62	C₂²⋊C₈	—	2	4	no	—
63	#97	—	2	2	no	—
64	#97	—	2	4	no	—
65	#97	—	2	4	no	—
66	C₄×C₈	—	2	2	no	—
67	C₄×C₈	C₂×C₄	1	2	yes	—
68	C₄×C₈	C₂×C₄	1	2	yes	—
69	C₄×C₈	D₄	1	2	yes	—
70	C₄×C₈	D₄	1	2	yes	—
71	#41	—	2	4	no	—
72	#41	—	2	4	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	C₂²×C₄	1	2	yes	—
75	C₂×D₄	—	2	2	no	—
76	C₂×D₄	—	2	2	no	—
77	C₂×D₄	C₂²×C₄	1	2	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	#41	1	2	yes	—
81	C₂²×C₄	C₂⁴	1	1	yes	characteristic
82	M₁₆	—	2	4	no	—
83	M₁₆	—	2	4	no	—
84	M₁₆	—	4	4	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	4	4	no	—
91	C₂×C₈	C₂²×C₄	1	4	yes	—
92	C₂×C₈	C₂×D₄	1	2	yes	—
93	C₂×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈	#41	1	4	yes	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	4	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	C₂²×C₄	1	4	yes	—
100	C₂²⋊C₄	—	2	2	no	—
101	C₂²⋊C₄	—	2	4	no	—
102	C₄²	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	4	4	no	—
105	C₄²	C₂²×C₄	1	1	yes	characteristic
106	C₄²	C₂²×C₄	1	1	yes	characteristic
107	C₄²	C₂×D₄	1	2	yes	—
108	C₄²	#41	1	4	yes	—
109	C₂³	—	2	2	no	—
110	C₂³	C₂×#41	1	1	yes	characteristic
111	Q₈	—	2	4	no	—
112	Q₈	—	4	4	no	—
113	D₄	—	2	4	no	—
114	D₄	—	4	4	no	—
115	D₄	—	4	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	4	4	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	C₄×D₄	1	2	yes	—
124	C₂×C₄	C₄×D₄	1	2	yes	—
125	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄	C₂×#41	1	2	yes	—
128	C₂×C₄	C₂×#41	1	2	yes	—
129	C₈	—	2	4	no	—
130	C₈	—	2	2	no	—
131	C₈	—	4	2	no	—
132	C₈	—	4	2	no	—
133	C₈	#124	1	2	yes	—
134	C₈	#124	1	2	yes	—
135	C₂²	—	2	2	no	—
136	C₂²	—	4	4	no	—
137	C₂²	—	4	2	no	—
138	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂²	#529	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	—	4	4	no	—
143	C₄	—	4	2	no	—
144	C₄	#444	1	4	yes	—
145	C₄	C₄×#41	1	2	yes	—
146	C₄	C₂×#124	1	1	yes	characteristic
147	C₄	#532	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	—	8	2	no	—
150	C₂	C₄×#124	1	1	yes	characteristic
151	C₂	C₂×#444	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3131

C₂

yes

maximal, characteristic

#2960

C₂

yes

maximal, characteristic, complemented

#2931

C₂

yes

maximal, characteristic

#1994

C₂

yes

maximal, characteristic, complemented

#1962

C₂

yes

maximal, characteristic

#1938

C₂

yes

maximal, complemented

#1837

C₂

yes

maximal, characteristic, complemented

#1765

C₂

yes

maximal, complemented

C₄×#97

C₂

yes

maximal, characteristic, complemented

C₂×#131

C₂²

yes

—

#533

C₂²

yes

characteristic

#532

C₂²

yes

characteristic

#488

C₄

yes

complemented

#487

C₄

yes

complemented

#486

C₄

yes

complemented

#446

—

C₈×Q₈

—

#436

—

C₈×D₄

—

#424

C₂²

yes

—

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

C₂×#104

C₂²

yes

—

#416

C₂²

yes

—

#408

C₂²

yes

—

#405

C₂²

yes

—

C₂×#97

C₂²

yes

—

C₂×#97

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄³

C₂²

yes

characteristic

#341

C₂²

yes

—

#339

C₂²

yes

—

C₂×#41

C₂³

yes

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

—

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

C₂×C₄

yes

characteristic

C₄⋊Q₈

C₂×C₄

yes

—

#127

C₂×C₄

yes

characteristic

#125

C₂×C₄

yes

—

#124

C₂×C₄

yes

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

C₂³

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#104

—

#103

C₂×C₄

yes

—

#102

C₂×C₄

yes

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

C₂²×C₄

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×C₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#41

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×C₄

yes

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂²×C₄

yes

characteristic

106

C₄²

C₂²×C₄

yes

characteristic

107

C₄²

C₂×D₄

yes

—

108

C₄²

#41

yes

—

109

C₂³

—

110

C₂³

C₂×#41

yes

characteristic

111

Q₈

—

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₄×D₄

yes

—

124

C₂×C₄

C₄×D₄

yes

—

125

C₂×C₄

C₂³×C₄

yes

characteristic

126

C₂×C₄

C₂²×D₄

yes

characteristic

127

C₂×C₄

C₂×#41

yes

—

128

C₂×C₄

C₂×#41

yes

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

#124

yes

—

134

C₈

#124

yes

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂×C₄×D₄

yes

characteristic

139

C₂²

#529

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#444

yes

—

145

C₄

C₄×#41

yes

—

146

C₄

C₂×#124

yes

characteristic

147

C₄

#532

yes

characteristic

148

C₂

—

149

C₂

—

150

C₂

C₄×#124

yes

characteristic

151

C₂

C₂×#444

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

C₂²×D₄

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₂²×C₄

C₂⁴

socle

C₂²

C₂×C₄×D₄

C₂²

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2960	C₂	semidirect
4. #1994	C₂	semidirect
6. #1938	C₂	semidirect
7. #1837	C₂	semidirect
8. #1765	C₂	semidirect
9. C₄×#97	C₂	semidirect
13. #488	C₄	semidirect
14. #487	C₄	semidirect
15. #486	C₄	semidirect

H ≅ G/N

type

2. #2960

C₂

semidirect

4. #1994

C₂

semidirect

6. #1938

C₂

semidirect

7. #1837

C₂

semidirect

8. #1765

C₂

semidirect

9. C₄×#97

C₂

semidirect

13. #488

C₄

semidirect

14. #487

C₄

semidirect

15. #486

C₄

semidirect

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

subgroup	index
#3131	2
#2960	2
#2931	2
#1994	2
#1962	2
#1938	2
#1837	2
#1765	2
C₄×#97	2

subgroup

index

C₄×#97

Subgroups — #21660

Conjugacy classes of subgroups

Complements

Maximal subgroups