Group #21720 — subgroups

Subgroups — C₄×#500 #21720

Order 256.

439 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 231 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×#500	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Q₈	C₂	1	2	yes	maximal
3	#1990	C₂	1	2	yes	maximal
4	#1942	C₂	1	4	yes	maximal
5	#1845	C₂	1	1	yes	maximal, characteristic
6	C₄×#107	C₂	1	4	yes	maximal
7	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄×Q₈	—	2	2	no	—
10	#500	C₄	1	8	yes	direct product factor
11	C₂×#107	C₂²	1	4	yes	—
12	C₂×#107	C₂²	1	4	yes	—
13	C₂×C₄×C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂²	1	2	yes	—
17	#391	—	2	2	no	—
18	#389	—	2	2	no	—
19	#389	C₂²	1	4	yes	—
20	#384	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄	—	2	2	no	—
22	C₄×C₄⋊C₄	C₂²	1	4	yes	—
23	C₄³	C₂²	1	1	yes	characteristic
24	#340	C₂²	1	8	yes	—
25	C₂²×Q₈	—	2	2	no	—
26	C₂²×C₈	C₂³	1	2	yes	—
27	C₂²×C₈	C₂³	1	2	yes	—
28	C₄⋊Q₈	—	2	4	no	—
29	C₄⋊Q₈	C₂×C₄	1	8	yes	—
30	C₄×Q₈	—	4	8	no	—
31	C₂×C₄⋊C₄	—	2	2	no	—
32	C₂×C₄⋊C₄	—	2	2	no	—
33	C₂×C₄⋊C₄	—	2	4	no	—
34	C₂×C₄⋊C₄	—	2	2	no	—
35	C₂×C₄⋊C₄	C₂³	1	4	yes	—
36	C₂×C₄⋊C₄	C₂³	1	4	yes	—
37	C₂×C₄²	—	2	4	no	—
38	C₂×C₄²	D₄	1	1	yes	characteristic
39	C₂×C₄²	D₄	1	1	yes	characteristic
40	C₂×C₄²	D₄	1	1	yes	characteristic
41	C₂×C₄²	D₄	1	1	yes	characteristic
42	C₂×C₄²	C₂³	1	1	yes	characteristic
43	C₂×C₄²	C₂³	1	1	yes	characteristic
44	C₂×C₄²	C₂³	1	1	yes	characteristic
45	#107	—	2	8	no	—
46	#107	C₂×C₄	1	16	yes	—
47	C₄×C₈	—	2	4	no	—
48	C₄×C₈	—	2	2	no	—
49	C₄×C₈	C₂×C₄	1	4	yes	—
50	C₄×C₈	Q₈	1	8	yes	complemented
51	#95	—	2	4	no	—
52	#95	—	2	4	no	—
53	C₂×Q₈	—	2	8	no	—
54	C₂×Q₈	—	4	4	no	—
55	C₂×Q₈	—	4	4	no	—
56	C₂²×C₄	—	2	4	no	—
57	C₂²×C₄	—	2	4	no	—
58	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄	C₂⁴	1	1	yes	characteristic
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	C₂²×C₄	1	4	yes	—
68	C₂×C₈	C₂×Q₈	1	4	yes	—
69	C₂×C₈	C₂×Q₈	1	4	yes	—
70	C₂×C₈	#41	1	4	yes	—
71	C₄⋊C₄	—	2	8	no	—
72	C₄⋊C₄	—	2	4	no	—
73	C₄⋊C₄	—	2	8	no	—
74	C₄⋊C₄	—	4	8	no	—
75	C₄⋊C₄	—	4	4	no	—
76	C₄⋊C₄	—	4	4	no	—
77	C₄⋊C₄	C₂²×C₄	1	8	yes	—
78	C₄²	—	2	2	no	—
79	C₄²	—	2	2	no	—
80	C₄²	—	2	2	no	—
81	C₄²	—	4	8	no	—
82	C₄²	D₈	1	4	yes	—
83	C₄²	Q₁₆	1	4	yes	—
84	C₄²	C₂²×C₄	1	2	yes	—
85	C₄²	C₂×Q₈	1	4	yes	—
86	C₄²	#41	1	2	yes	—
87	C₂³	C₂²×D₄	1	1	yes	characteristic
88	Q₈	—	4	16	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	8	no	—
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	4	4	no	—
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	C₄×D₄	1	2	yes	—
102	C₂×C₄	C₄×D₄	1	2	yes	—
103	C₂×C₄	C₄⋊Q₈	1	2	yes	—
104	C₂×C₄	C₄⋊Q₈	1	2	yes	—
105	C₂×C₄	C₂×D₈	1	2	yes	—
106	C₂×C₄	C₂×D₈	1	2	yes	—
107	C₂×C₄	C₂×D₈	1	2	yes	—
108	C₂×C₄	C₂×Q₁₆	1	2	yes	—
109	C₂×C₄	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄	C₂×Q₁₆	1	2	yes	—
111	C₂×C₄	#135	1	2	yes	—
112	C₂×C₄	#135	1	2	yes	—
113	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄	C₂²×Q₈	1	2	yes	—
115	C₂×C₄	C₂×#41	1	2	yes	—
116	C₂×C₄	C₂×#41	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₈	C₄×Q₈	1	8	yes	complemented
119	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
120	C₂²	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₂²	#532	1	1	yes	characteristic
122	C₂²	C₂²×D₈	1	1	yes	characteristic
123	C₂²	C₂²×Q₁₆	1	1	yes	characteristic
124	C₂²	C₂×#135	1	1	yes	characteristic
125	C₂²	C₂×#135	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	—	4	8	no	—
130	C₄	—	8	4	no	—
131	C₄	C₄×D₈	1	4	yes	—
132	C₄	C₄×Q₁₆	1	4	yes	—
133	C₄	#500	1	4	yes	direct product factor
134	C₄	C₂×C₄×Q₈	1	2	yes	—
135	C₄	#533	1	2	yes	—
136	C₂	C₄×C₄⋊Q₈	1	1	yes	characteristic
137	C₂	C₂×C₄×D₈	1	1	yes	characteristic
138	C₂	C₂×C₄×Q₁₆	1	1	yes	characteristic
139	C₂	C₂×#500	1	1	yes	characteristic
140	C₂	#3160	1	1	yes	characteristic
141	C₂	#3169	1	1	yes	characteristic
142	C₂	#3170	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#500

C₂

yes

maximal, characteristic

C₄×C₄⋊Q₈

C₂

yes

maximal

#1990

C₂

yes

maximal

#1942

C₂

yes

maximal

#1845

C₂

yes

maximal, characteristic

C₄×#107

C₂

yes

maximal

C₄²×C₈

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

C₂×C₄×Q₈

—

#500

C₄

yes

direct product factor

C₂×#107

C₂²

yes

—

C₂×#107

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

#391

—

#389

—

#389

C₂²

yes

—

#384

C₂²

yes

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

C₂²

yes

—

C₄³

C₂²

yes

characteristic

#340

C₂²

yes

—

C₂²×Q₈

—

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₄

yes

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄²

—

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

#107

—

#107

C₂×C₄

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

Q₈

yes

complemented

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂⁴

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×C₄

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

D₈

yes

—

C₄²

Q₁₆

yes

—

C₄²

C₂²×C₄

yes

—

C₄²

C₂×Q₈

yes

—

C₄²

#41

yes

—

C₂³

C₂²×D₄

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄×D₄

yes

—

102

C₂×C₄

C₄×D₄

yes

—

103

C₂×C₄

C₄⋊Q₈

yes

—

104

C₂×C₄

C₄⋊Q₈

yes

—

105

C₂×C₄

C₂×D₈

yes

—

106

C₂×C₄

C₂×D₈

yes

—

107

C₂×C₄

C₂×D₈

yes

—

108

C₂×C₄

C₂×Q₁₆

yes

—

109

C₂×C₄

C₂×Q₁₆

yes

—

110

C₂×C₄

C₂×Q₁₆

yes

—

111

C₂×C₄

#135

yes

—

112

C₂×C₄

#135

yes

—

113

C₂×C₄

C₂³×C₄

yes

characteristic

114

C₂×C₄

C₂²×Q₈

yes

—

115

C₂×C₄

C₂×#41

yes

—

116

C₂×C₄

C₂×#41

yes

characteristic

117

C₈

—

118

C₈

C₄×Q₈

yes

complemented

119

C₂²

C₂×C₄×D₄

yes

characteristic

120

C₂²

C₂×C₄⋊Q₈

yes

characteristic

121

C₂²

#532

yes

characteristic

122

C₂²

C₂²×D₈

yes

characteristic

123

C₂²

C₂²×Q₁₆

yes

characteristic

124

C₂²

C₂×#135

yes

characteristic

125

C₂²

C₂×#135

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

C₄×D₈

yes

—

132

C₄

C₄×Q₁₆

yes

—

133

C₄

#500

yes

direct product factor

134

C₄

C₂×C₄×Q₈

yes

—

135

C₄

#533

yes

—

136

C₂

C₄×C₄⋊Q₈

yes

characteristic

137

C₂

C₂×C₄×D₈

yes

characteristic

138

C₂

C₂×C₄×Q₁₆

yes

characteristic

139

C₂

C₂×#500

yes

characteristic

140

C₂

#3160

yes

characteristic

141

C₂

#3169

yes

characteristic

142

C₂

#3170

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle, prime radical	C₂³	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₂²×C₄

C₂⁴

socle, prime radical

C₂³

C₂²×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
10. #500	C₄	direct (N × H)
50. C₄×C₈	Q₈	semidirect
118. C₈	C₄×Q₈	semidirect
133. C₄	#500	direct (N × H)

H ≅ G/N

type

10. #500

C₄

direct (N × H)

50. C₄×C₈

Q₈

semidirect

118. C₈

C₄×Q₈

semidirect

133. C₄

#500

direct (N × H)

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

subgroup	index
C₂×#500	2
C₄×C₄⋊Q₈	2
#1990	2
#1942	2
#1845	2
C₄×#107	2
C₄²×C₈	2

subgroup

index

C₄×C₄⋊Q₈

C₄²×C₈

Subgroups — C₄×#500 #21720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#500 #21720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#500 #21720