Group #19210 — subgroups

Subgroups — #19210

Order 256.

285 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 121 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	#3249	C₂	1	2	yes	maximal
2	#3160	C₂	1	1	yes	maximal, characteristic
3	#2960	C₂	1	1	yes	maximal, characteristic
4	#1696	C₂	1	2	yes	maximal, complemented
5	#1694	C₂	1	2	yes	maximal, complemented
6	#1672	C₂	1	4	yes	maximal, complemented
7	C₂×#334	C₂	1	1	yes	maximal, characteristic
8	#1550	C₂	1	2	yes	maximal
9	#533	C₂²	1	2	yes	—
10	#533	C₂²	1	1	yes	characteristic
11	#500	C₂²	1	1	yes	characteristic
12	#500	C₂²	1	1	yes	characteristic
13	#499	—	2	1	no	—
14	#499	—	2	1	no	—
15	#498	C₂²	1	2	yes	—
16	#477	—	2	2	no	—
17	#477	C₂²	1	4	yes	—
18	#467	—	2	4	no	—
19	#462	—	2	2	no	—
20	#462	C₂²	1	4	yes	—
21	#446	—	2	1	no	—
22	#446	C₂²	1	2	yes	—
23	C₈×Q₈	—	2	1	no	—
24	C₈×Q₈	C₂²	1	2	yes	—
25	#427	—	2	1	no	—
26	#427	—	2	1	no	—
27	#424	—	2	1	no	—
28	#423	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈	C₂²	1	2	yes	—
30	#405	—	2	1	no	—
31	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
32	#334	C₂²	1	2	yes	—
33	#334	C₂²	1	2	yes	—
34	#333	C₂²	1	4	yes	—
35	#332	C₂²	1	4	yes	—
36	C₂×Q₁₆	—	2	4	no	—
37	C₂×M₁₆	—	2	1	no	—
38	C₂²×C₈	—	2	2	no	—
39	C₂²×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₈	D₄	1	1	yes	characteristic
41	C₄⋊Q₈	D₄	1	1	yes	characteristic
42	C₄⋊Q₈	D₄	1	1	yes	characteristic
43	C₄⋊Q₈	C₂³	1	2	yes	—
44	C₄⋊Q₈	C₂³	1	2	yes	—
45	#125	—	2	2	no	—
46	#125	—	2	1	no	—
47	C₂²⋊Q₈	—	2	1	no	—
48	C₂²⋊Q₈	—	2	2	no	—
49	C₂²⋊Q₈	—	2	2	no	—
50	C₂²⋊Q₈	D₄	1	2	yes	—
51	C₄×Q₈	—	2	1	no	—
52	C₄×Q₈	—	2	2	no	—
53	C₄×Q₈	—	2	2	no	—
54	C₄×Q₈	C₂³	1	2	yes	—
55	#117	—	2	1	no	—
56	#117	—	2	2	no	—
57	C₂×C₄²	C₂³	1	1	yes	characteristic
58	#107	—	2	1	no	—
59	#107	—	2	1	no	—
60	#107	—	2	1	no	—
61	#107	—	2	1	no	—
62	#107	—	2	4	no	—
63	#106	—	2	2	no	—
64	#106	—	2	2	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	—	2	2	no	—
68	C₄⋊C₈	—	2	1	no	—
69	C₄⋊C₈	C₂³	1	2	yes	—
70	C₄⋊C₈	C₂³	1	4	yes	—
71	#103	—	2	4	no	—
72	#103	—	2	4	no	—
73	#97	—	2	2	no	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	D₄	1	1	yes	characteristic
76	C₄×C₈	D₄	1	1	yes	characteristic
77	C₄×C₈	C₂³	1	1	yes	characteristic
78	C₄×C₈	C₂³	1	1	yes	characteristic
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	C₂×D₄	1	2	yes	—
83	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄	#41	1	1	yes	characteristic
86	Q₁₆	—	8	4	no	—
87	M₁₆	—	4	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	4	no	—
91	C₂×C₈	—	8	2	no	—
92	C₂×C₈	Q₁₆	1	4	yes	complemented
93	C₂×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈	C₂×D₄	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	C₂×D₄	1	2	yes	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₂²⋊C₄	—	4	2	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	4	2	no	—
113	C₄²	C₂×D₄	1	1	yes	characteristic
114	C₄²	C₂×D₄	1	1	yes	characteristic
115	C₄²	#41	1	1	yes	characteristic
116	C₄²	C₂⁴	1	1	yes	characteristic
117	C₂³	#121	1	1	yes	characteristic
118	Q₈	—	2	4	no	—
119	Q₈	—	4	2	no	—
120	Q₈	—	4	2	no	—
121	Q₈	—	4	2	no	—
122	Q₈	—	8	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	#121	1	1	yes	characteristic
130	C₂×C₄	#121	1	1	yes	characteristic
131	C₂×C₄	#121	1	1	yes	characteristic
132	C₂×C₄	C₂×Q₁₆	1	2	yes	—
133	C₂×C₄	#137	1	2	yes	—
134	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄	C₂×#41	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	4	2	no	—
140	C₈	—	4	4	no	—
141	C₈	C₂×Q₁₆	1	2	yes	—
142	C₈	C₂×Q₁₆	1	2	yes	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	#462	1	2	yes	—
146	C₂²	C₂×#121	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	4	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	8	2	no	—
153	C₄	—	8	2	no	—
154	C₄	#462	1	2	yes	—
155	C₄	C₂²×Q₁₆	1	1	yes	characteristic
156	C₄	C₂×#137	1	1	yes	characteristic
157	C₄	#577	1	2	yes	—
158	C₂	—	2	2	no	—
159	C₂	C₂×#462	1	1	yes	characteristic
160	C₂	#3060	1	1	yes	characteristic
161	C₂	#3067	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3249

C₂

yes

maximal

#3160

C₂

yes

maximal, characteristic

#2960

C₂

yes

maximal, characteristic

#1696

C₂

yes

maximal, complemented

#1694

C₂

yes

maximal, complemented

#1672

C₂

yes

maximal, complemented

C₂×#334

C₂

yes

maximal, characteristic

#1550

C₂

yes

maximal

#533

C₂²

yes

—

#533

C₂²

yes

characteristic

#500

C₂²

yes

characteristic

#500

C₂²

yes

—

—

C₂²

yes

—

#477

—

#477

C₂²

yes

—

#467

—

#462

—

#462

C₂²

yes

—

#446

—

#446

C₂²

yes

—

C₈×Q₈

—

C₈×Q₈

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

#405

—

C₂×C₄×C₈

C₂²

yes

characteristic

#334

C₂²

yes

—

#334

C₂²

yes

—

#333

C₂²

yes

—

#332

C₂²

yes

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

D₄

yes

characteristic

C₄⋊Q₈

D₄

yes

characteristic

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₄

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂³

yes

—

#117

—

#117

—

C₂×C₄²

C₂³

yes

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂³

yes

—

C₄⋊C₈

C₂³

yes

—

#103

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

Q₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Q₁₆

yes

complemented

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

C₂×D₄

yes

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₄²

C₂×D₄

yes

characteristic

114

C₄²

C₂×D₄

yes

characteristic

115

C₄²

#41

yes

characteristic

116

C₄²

C₂⁴

yes

characteristic

117

C₂³

#121

yes

characteristic

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#121

yes

characteristic

130

C₂×C₄

#121

yes

characteristic

131

C₂×C₄

#121

yes

characteristic

132

C₂×C₄

C₂×Q₁₆

yes

—

133

C₂×C₄

#137

yes

—

134

C₂×C₄

C₂²×D₄

yes

characteristic

135

C₂×C₄

C₂²×D₄

yes

characteristic

136

C₂×C₄

C₂×#41

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

C₂×Q₁₆

yes

—

142

C₈

C₂×Q₁₆

yes

—

143

C₂²

—

144

C₂²

—

145

C₂²

#462

yes

—

146

C₂²

C₂×#121

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#462

yes

—

155

C₄

C₂²×Q₁₆

yes

characteristic

156

C₄

C₂×#137

yes

characteristic

157

C₄

#577

yes

—

158

C₂

—

159

C₂

C₂×#462

yes

characteristic

160

C₂

#3060

yes

characteristic

161

C₂

#3067

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂³	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#121

commutator, Frattini, cosocle

C₄²

C₂⁴

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
4. #1696	C₂	semidirect
5. #1694	C₂	semidirect
6. #1672	C₂	semidirect
92. C₂×C₈	Q₁₆	semidirect

H ≅ G/N

type

4. #1696

C₂

semidirect

5. #1694

C₂

semidirect

6. #1672

C₂

semidirect

92. C₂×C₈

Q₁₆

semidirect

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

subgroup	index
#3249	2
#3160	2
#2960	2
#1696	2
#1694	2
#1672	2
C₂×#334	2
#1550	2

subgroup

index

Subgroups — #19210

Conjugacy classes of subgroups

Complements

Maximal subgroups