Group #19209 — subgroups

Subgroups — #19209

Order 256.

291 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 123 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	#3230	C₂	1	2	yes	maximal
2	C₂×#500	C₂	1	1	yes	maximal, characteristic
3	#2960	C₂	1	1	yes	maximal, characteristic
4	#1696	C₂	1	4	yes	maximal, complemented
5	#1676	C₂	1	4	yes	maximal, complemented
6	#1564	C₂	1	1	yes	maximal, characteristic
7	#1548	C₂	1	2	yes	maximal
8	#533	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
10	#500	—	2	2	no	—
11	#500	—	2	1	no	—
12	#500	—	2	1	no	—
13	#500	C₂²	1	2	yes	—
14	#500	C₂²	1	2	yes	—
15	#487	—	2	2	no	—
16	#484	—	2	4	no	—
17	#477	C₂²	1	8	yes	—
18	#446	—	2	1	no	—
19	#446	C₂²	1	2	yes	—
20	C₈×Q₈	—	2	1	no	—
21	C₈×Q₈	C₂²	1	2	yes	—
22	#431	C₂²	1	2	yes	—
23	C₂×#107	—	2	1	no	—
24	C₂×#107	—	2	1	no	—
25	#424	—	2	1	no	—
26	#423	C₂²	1	1	yes	characteristic
27	#405	—	2	1	no	—
28	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
29	#335	C₂²	1	2	yes	—
30	#334	C₂²	1	2	yes	—
31	#333	C₂²	1	8	yes	—
32	C₂²×Q₈	—	2	2	no	—
33	C₂×M₁₆	—	2	1	no	—
34	C₂²×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₈	D₄	1	1	yes	characteristic
36	C₄⋊Q₈	—	2	2	no	—
37	C₄⋊Q₈	—	2	2	no	—
38	C₄⋊Q₈	—	2	2	no	—
39	C₄⋊Q₈	D₄	1	1	yes	characteristic
40	C₄⋊Q₈	D₄	1	1	yes	characteristic
41	C₄⋊Q₈	C₂³	1	4	yes	—
42	#125	—	2	1	no	—
43	C₂²⋊Q₈	—	2	1	no	—
44	C₂²⋊Q₈	D₄	1	2	yes	—
45	C₄×Q₈	—	2	1	no	—
46	C₄×Q₈	C₂³	1	2	yes	—
47	#117	—	2	1	no	—
48	C₂×C₄⋊C₄	—	2	2	no	—
49	C₂×C₄⋊C₄	—	2	2	no	—
50	C₂×C₄²	C₂³	1	1	yes	characteristic
51	#107	—	2	2	no	—
52	#107	—	2	2	no	—
53	#107	—	2	2	no	—
54	#107	—	2	2	no	—
55	#107	—	2	8	no	—
56	C₄⋊C₈	—	2	2	no	—
57	C₄⋊C₈	—	2	2	no	—
58	C₄⋊C₈	—	2	1	no	—
59	C₄⋊C₈	C₂³	1	2	yes	—
60	C₄⋊C₈	C₂³	1	4	yes	—
61	#103	—	2	8	no	—
62	C₂²⋊C₈	—	2	2	no	—
63	#97	—	2	2	no	—
64	C₄×C₈	—	2	2	no	—
65	C₄×C₈	—	2	2	no	—
66	C₄×C₈	D₄	1	1	yes	characteristic
67	C₄×C₈	D₄	1	1	yes	characteristic
68	C₄×C₈	C₂³	1	1	yes	characteristic
69	C₄×C₈	C₂³	1	1	yes	characteristic
70	C₂×Q₈	—	2	1	no	—
71	C₂×Q₈	—	2	4	no	—
72	C₂×Q₈	—	4	2	no	—
73	C₂×Q₈	—	8	4	no	—
74	C₂×Q₈	C₂×D₄	1	2	yes	—
75	C₂²×C₄	—	4	2	no	—
76	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄	#41	1	1	yes	characteristic
79	M₁₆	—	4	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	Q₁₆	1	4	yes	—
84	C₂×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈	C₂×D₄	1	1	yes	characteristic
88	C₄⋊C₄	—	2	1	no	—
89	C₄⋊C₄	—	2	1	no	—
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	2	4	no	—
95	C₄⋊C₄	—	4	2	no	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄⋊C₄	C₂×D₄	1	2	yes	—
98	C₂²⋊C₄	—	2	2	no	—
99	C₄²	—	2	2	no	—
100	C₄²	C₂×D₄	1	1	yes	characteristic
101	C₄²	C₂×D₄	1	1	yes	characteristic
102	C₄²	#41	1	1	yes	characteristic
103	C₄²	C₂⁴	1	1	yes	characteristic
104	C₂³	#121	1	1	yes	characteristic
105	Q₈	—	2	4	no	—
106	Q₈	—	4	2	no	—
107	Q₈	—	8	4	no	—
108	Q₈	—	8	4	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	4	4	no	—
114	C₂×C₄	—	8	4	no	—
115	C₂×C₄	#121	1	1	yes	characteristic
116	C₂×C₄	#121	1	1	yes	characteristic
117	C₂×C₄	#121	1	1	yes	characteristic
118	C₂×C₄	C₂×Q₁₆	1	2	yes	—
119	C₂×C₄	#137	1	2	yes	—
120	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄	C₂×#41	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	1	no	—
125	C₈	—	4	2	no	—
126	C₈	—	4	4	no	—
127	C₈	C₂×Q₁₆	1	2	yes	—
128	C₈	C₂×Q₁₆	1	2	yes	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#462	1	2	yes	—
132	C₂²	C₂×#121	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	2	no	—
136	C₄	—	4	2	no	—
137	C₄	—	8	4	no	—
138	C₄	#462	1	2	yes	—
139	C₄	#472	1	2	yes	—
140	C₄	C₂²×Q₁₆	1	1	yes	characteristic
141	C₄	C₂×#137	1	1	yes	characteristic
142	C₄	#578	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	C₂×#462	1	1	yes	characteristic
145	C₂	#3059	1	1	yes	characteristic
146	C₂	C₂×#472	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3230

C₂

yes

maximal

C₂×#500

C₂

yes

maximal, characteristic

#2960

C₂

yes

maximal, characteristic

#1696

C₂

yes

maximal, complemented

#1676

C₂

yes

maximal, complemented

#1564

C₂

yes

maximal, characteristic

#1548

C₂

yes

maximal

#533

C₂²

yes

characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

—

—

—

C₂²

yes

—

#500

C₂²

yes

—

#487

—

#484

—

#477

C₂²

yes

—

#446

—

#446

C₂²

yes

—

C₈×Q₈

—

C₈×Q₈

C₂²

yes

—

#431

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

#405

—

C₂×C₄×C₈

C₂²

yes

characteristic

#335

C₂²

yes

—

#334

C₂²

yes

—

#333

C₂²

yes

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₄

yes

characteristic

C₄⋊Q₈

D₄

yes

characteristic

C₄⋊Q₈

C₂³

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₄

yes

—

C₄×Q₈

—

C₄×Q₈

C₂³

yes

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂³

yes

—

C₄⋊C₈

C₂³

yes

—

#103

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

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×Q₈

C₂×D₄

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Q₁₆

yes

—

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄

yes

—

C₂²⋊C₄

—

C₄²

—

100

C₄²

C₂×D₄

yes

characteristic

101

C₄²

C₂×D₄

yes

characteristic

102

C₄²

#41

yes

characteristic

103

C₄²

C₂⁴

yes

characteristic

104

C₂³

#121

yes

characteristic

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

#121

yes

characteristic

116

C₂×C₄

#121

yes

characteristic

117

C₂×C₄

#121

yes

characteristic

118

C₂×C₄

C₂×Q₁₆

yes

—

119

C₂×C₄

#137

yes

—

120

C₂×C₄

C₂²×D₄

yes

characteristic

121

C₂×C₄

C₂²×D₄

yes

characteristic

122

C₂×C₄

C₂×#41

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

C₂×Q₁₆

yes

—

128

C₈

C₂×Q₁₆

yes

—

129

C₂²

—

130

C₂²

—

131

C₂²

#462

yes

—

132

C₂²

C₂×#121

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#462

yes

—

139

C₄

#472

yes

—

140

C₄

C₂²×Q₁₆

yes

characteristic

141

C₄

C₂×#137

yes

characteristic

142

C₄

#578

yes

—

143

C₂

—

144

C₂

C₂×#462

yes

characteristic

145

C₂

#3059

yes

characteristic

146

C₂

C₂×#472

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. #1676	C₂	semidirect

H ≅ G/N

type

4. #1696

C₂

semidirect

5. #1676

C₂

semidirect

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

subgroup	index
#3230	2
C₂×#500	2
#2960	2
#1696	2
#1676	2
#1564	2
#1548	2

subgroup

index

Subgroups — #19209

Conjugacy classes of subgroups

Complements

Maximal subgroups