Group #19133818 — subgroups

Subgroups — #19133818

Order 5504.

200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	#14583250	C₂	1	1	yes	maximal, characteristic
3	#14583185	C₂	1	1	yes	maximal, characteristic
4	#14583109	C₂	1	1	yes	maximal, characteristic
5	#14582647	C₂	1	1	yes	maximal, characteristic
6	#14582631	C₂	1	1	yes	maximal, characteristic
7	C₄₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×#10621885	—	2	1	no	—
9	#13063325	—	2	1	no	—
10	#13063325	C₂²	1	1	yes	characteristic
11	#13063325	C₂²	1	1	yes	characteristic
12	#13063324	—	2	2	no	—
13	#13063324	—	2	1	no	—
14	Q₈×Dic₄₃	—	2	1	no	—
15	C₄×Dic₈₆	—	2	1	no	—
16	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
17	#13063257	—	2	1	no	—
18	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
19	#13063250	—	2	1	no	—
20	#13063250	—	2	2	no	—
21	#13063248	—	2	1	no	—
22	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₄₃	D₄	1	1	yes	characteristic
26	C₂×Dic₈₆	—	2	1	no	—
27	C₂×Dic₈₆	—	2	2	no	—
28	C₂×Dic₈₆	D₄	1	1	yes	characteristic
29	C₂×Dic₈₆	D₄	1	2	yes	—
30	C₂×C₈×C₄₃	—	2	1	no	—
31	C₂×C₈×C₄₃	—	2	1	no	—
32	C₄⋊C₄×C₄₃	—	2	1	no	—
33	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
34	C₄²×C₄₃	—	2	1	no	—
35	C₄²×C₄₃	C₂³	1	1	yes	characteristic
36	#10621885	—	4	2	no	—
37	C₄⋊Dic₄₃	—	2	2	no	—
38	C₄⋊Dic₄₃	—	2	1	no	—
39	C₄⋊Dic₄₃	—	2	1	no	—
40	C₄⋊Dic₄₃	—	2	1	no	—
41	C₄⋊Dic₄₃	—	2	1	no	—
42	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
43	Dic₄₃⋊C₄	—	2	2	no	—
44	Dic₄₃⋊C₄	—	2	1	no	—
45	C₄×Dic₄₃	—	2	2	no	—
46	C₄×Dic₄₃	—	2	1	no	—
47	C₂×C₄₃⋊C₈	—	2	1	no	—
48	Q₈×C₄₃	—	2	1	no	—
49	Q₈×C₄₃	SD₁₆	1	2	yes	—
50	C₂×C₄×C₄₃	—	2	1	no	—
51	C₂×C₄×C₄₃	—	2	1	no	—
52	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₄₃	—	2	1	no	—
56	C₂×Dic₄₃	—	2	2	no	—
57	C₂×Dic₄₃	—	2	1	no	—
58	C₂×Dic₄₃	—	4	1	no	—
59	Dic₈₆	—	2	1	no	—
60	Dic₈₆	—	4	2	no	—
61	Dic₈₆	—	4	1	no	—
62	Dic₈₆	—	4	1	no	—
63	Dic₈₆	—	8	2	no	—
64	Dic₈₆	Q₁₆	1	2	yes	complemented
65	C₈×C₄₃	—	4	1	no	—
66	C₈×C₄₃	—	4	1	no	—
67	C₄₃⋊C₈	—	4	1	no	—
68	C₂²×C₄₃	#120	1	1	yes	characteristic
69	C₄×C₄₃	—	2	2	no	—
70	C₄×C₄₃	—	2	1	no	—
71	C₄×C₄₃	—	4	1	no	—
72	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₄₃	#137	1	1	yes	characteristic
75	C₄×C₄₃	#137	1	1	yes	characteristic
76	Dic₄₃	—	2	2	no	—
77	Dic₄₃	—	4	2	no	—
78	Dic₄₃	—	4	1	no	—
79	Dic₄₃	—	8	1	no	—
80	#1645	—	43	1	no	maximal, Sylow
81	C₂×C₄₃	#448	1	1	yes	characteristic
82	C₂×C₄₃	#451	1	1	yes	characteristic
83	C₂×C₄₃	#455	1	1	yes	characteristic
84	Q₈²	—	43	1	no	—
85	#487	—	43	1	no	—
86	#475	—	43	1	no	—
87	#462	—	43	1	no	—
88	#333	—	43	1	no	—
89	Q₈⋊C₈	—	43	1	no	—
90	Q₈⋊C₈	D₄₃	1	1	yes	characteristic
91	C₄₃	#1645	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	—	86	1	no	—
93	C₄⋊Q₈	—	43	1	no	—
94	C₄⋊Q₈	—	43	1	no	—
95	C₄⋊Q₈	—	86	1	no	—
96	C₄⋊Q₈	—	86	2	no	—
97	C₄⋊Q₈	—	86	1	no	—
98	C₄×Q₈	—	43	1	no	—
99	C₄×Q₈	—	86	1	no	—
100	C₄×Q₈	—	86	1	no	—
101	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
102	#106	—	86	1	no	—
103	C₄⋊C₈	—	43	1	no	—
104	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
105	#103	—	86	1	no	—
106	#103	—	86	2	no	—
107	#103	—	86	1	no	—
108	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
109	C₂×Q₈	—	43	1	no	—
110	C₂×Q₈	—	43	2	no	—
111	C₂×Q₈	—	86	2	no	—
112	C₂×Q₈	—	86	1	no	—
113	C₂×Q₈	D₁₇₂	1	1	yes	characteristic
114	Q₁₆	—	172	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	86	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	43	1	no	—
120	C₄⋊C₄	—	86	2	no	—
121	C₄⋊C₄	—	86	2	no	—
122	C₄⋊C₄	—	86	1	no	—
123	C₄⋊C₄	—	86	1	no	—
124	C₄⋊C₄	—	86	1	no	—
125	C₄⋊C₄	—	86	1	no	—
126	C₄⋊C₄	—	86	1	no	—
127	C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
128	C₄²	—	2	1	no	—
129	C₄²	—	86	2	no	—
130	C₄²	—	86	1	no	—
131	C₄²	C₂²×D₄₃	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	43	2	no	—
134	Q₈	—	86	1	no	—
135	Q₈	—	344	2	no	—
136	Q₈	—	172	2	no	—
137	Q₈	—	172	1	no	—
138	Q₈	—	172	1	no	—
139	Q₈	#10621873	1	2	yes	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	86	1	no	—
143	C₂×C₄	—	86	1	no	—
144	C₂×C₄	—	86	2	no	—
145	C₂×C₄	—	172	1	no	—
146	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
147	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
148	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	172	1	no	—
152	C₂²	#13063295	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	86	2	no	—
157	C₄	—	344	1	no	—
158	C₄	—	172	1	no	—
159	C₄	—	172	2	no	—
160	C₄	C₂×#10621873	1	1	yes	characteristic
161	C₄	Q₁₆×D₄₃	1	1	yes	characteristic
162	C₄	#13063354	1	1	yes	characteristic
163	C₄	#13063358	1	1	yes	characteristic
164	C₂	#14583040	1	1	yes	characteristic
165	C₂	#14583056	1	1	yes	characteristic
166	C₂	#14583078	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₈₆

C₂

yes

maximal, characteristic

#14583250

C₂

yes

maximal, characteristic

#14583185

C₂

yes

maximal, characteristic

#14583109

C₂

yes

maximal, characteristic

#14582647

C₂

yes

maximal, characteristic

#14582631

C₂

yes

maximal, characteristic

C₄₃×Q₈⋊C₈

C₂

yes

maximal, characteristic

C₂×#10621885

—

#13063325

—

#13063325

C₂²

yes

characteristic

#13063325

C₂²

yes

characteristic

#13063324

—

#13063324

—

Q₈×Dic₄₃

—

C₄×Dic₈₆

—

C₄×Dic₈₆

C₂²

yes

characteristic

#13063257

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

—

—

C₄×Q₈×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

D₄

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

D₄

yes

characteristic

C₂×Dic₈₆

D₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

—

C₄²×C₄₃

C₂³

yes

characteristic

#10621885

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

D₄

yes

characteristic

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

Q₈×C₄₃

—

Q₈×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

Q₁₆

yes

complemented

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

#120

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₄×C₄₃

C₂×Q₁₆

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

Q₈²

—

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

D₄₃

yes

characteristic

C₄₃

#1645

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

100

C₄×Q₈

—

101

C₄×Q₈

C₂×D₄₃

yes

characteristic

102

#106

—

103

C₄⋊C₈

—

104

C₄⋊C₈

C₂×D₄₃

yes

characteristic

105

#103

—

106

#103

—

107

#103

—

108

C₄×C₈

C₂×D₄₃

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

D₁₇₂

yes

characteristic

114

Q₁₆

—

172

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

D₁₇₂

yes

characteristic

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²×D₄₃

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

344

—

136

Q₈

—

172

—

137

Q₈

—

172

—

138

Q₈

—

172

—

139

Q₈

#10621873

yes

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

172

—

146

C₂×C₄

C₂×D₁₇₂

yes

characteristic

147

C₂×C₄

D₄×D₄₃

yes

characteristic

148

C₂×C₄

D₄×D₄₃

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

172

—

152

C₂²

#13063295

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

344

—

158

C₄

—

172

—

159

C₄

—

172

—

160

C₄

C₂×#10621873

yes

characteristic

161

C₄

Q₁₆×D₄₃

yes

characteristic

162

C₄

#13063354

yes

characteristic

163

C₄

#13063358

yes

characteristic

164

C₂

#14583040

yes

characteristic

165

C₂

#14583056

yes

characteristic

166

C₂

#14583078

yes

characteristic

	subgroup	G/H
center	C₂²	#13063295
commutator	C₄²×C₄₃	C₂³
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×Q₈⋊C₈	C₂
socle	C₂²×C₄₃	#120
hypercenter	Q₈⋊C₈	D₄₃
hypocenter	C₄₃	#1645
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#13063295

commutator

C₄²×C₄₃

C₂³

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×Q₈⋊C₈

C₂

socle

C₂²×C₄₃

#120

hypercenter

Q₈⋊C₈

D₄₃

hypocenter

C₄₃

#1645

LCS

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
64. Dic₈₆	Q₁₆	semidirect
91. C₄₃	#1645	semidirect

H ≅ G/N

type

64. Dic₈₆

Q₁₆

semidirect

91. C₄₃

#1645

semidirect

Subgroups — #19133818

Conjugacy classes of subgroups

Complements