Group #19139581 — subgroups

Subgroups — #19139581

Order 5504.

540 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 185 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×#13063323	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063323	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
4	#14582832	C₂	1	4	yes	maximal
5	#14582806	C₂	1	1	yes	maximal, characteristic
6	#14582805	C₂	1	1	yes	maximal, characteristic
7	#14582794	C₂	1	4	yes	maximal
8	C₄×Dic₄₃⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
11	#13063323	—	2	4	no	—
12	#13063323	—	2	4	no	—
13	Q₈×Dic₄₃	—	2	4	no	—
14	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
16	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
17	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	—
18	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
19	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
20	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
21	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
23	C₄₃⋊#95	C₂²	1	4	yes	—
24	C₄₃⋊#95	C₂²	1	4	yes	—
25	C₄₃⋊#95	C₂²	1	2	yes	—
26	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄⋊Q₈×C₄₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₄₃	C₂²	1	4	yes	—
30	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×Q₈×C₄₃	—	2	2	no	—
32	C₂×Q₈×C₄₃	—	2	2	no	—
33	C₂×Q₈×C₄₃	—	2	2	no	—
34	C₂×Q₈×C₄₃	—	2	2	no	—
35	C₂×Q₈×C₄₃	—	2	2	no	—
36	C₂×Q₈×C₄₃	D₄	1	4	yes	—
37	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	C₂²×Dic₄₃	C₂³	1	2	yes	—
42	C₂²×Dic₄₃	C₂³	1	2	yes	—
43	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₄₃	C₂³	1	2	yes	—
45	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
46	C₂×Dic₈₆	—	2	2	no	—
47	C₂×Dic₈₆	—	2	2	no	—
48	C₂×Dic₈₆	—	2	2	no	—
49	C₄⋊C₄×C₄₃	—	2	8	no	—
50	C₄²×C₄₃	—	2	2	no	—
51	C₄⋊Dic₄₃	—	2	2	no	—
52	C₄⋊Dic₄₃	—	2	4	no	—
53	Dic₄₃⋊C₄	—	2	8	no	—
54	Dic₄₃⋊C₄	—	2	4	no	—
55	Dic₄₃⋊C₄	—	2	4	no	—
56	Dic₄₃⋊C₄	Q₈	1	8	yes	complemented
57	C₄×Dic₄₃	—	2	2	no	—
58	C₄×Dic₄₃	—	2	4	no	—
59	C₄×Dic₄₃	D₄	1	4	yes	—
60	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
61	Q₈×C₄₃	—	2	8	no	—
62	Q₈×C₄₃	—	4	4	no	—
63	C₂×C₄×C₄₃	—	2	2	no	—
64	C₂×C₄×C₄₃	—	2	2	no	—
65	C₂×C₄×C₄₃	—	2	2	no	—
66	C₂×C₄×C₄₃	—	2	2	no	—
67	C₂×C₄×C₄₃	—	2	2	no	—
68	C₂×C₄×C₄₃	—	2	2	no	—
69	C₂×C₄×C₄₃	—	2	2	no	—
70	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₄₃	C₂×D₄	1	4	yes	—
72	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₄₃	C₂×Q₈	1	4	yes	—
74	C₂×C₄×C₄₃	#41	1	2	yes	—
75	C₂×Dic₄₃	—	2	2	no	—
76	C₂×Dic₄₃	—	2	2	no	—
77	C₂×Dic₄₃	—	2	2	no	—
78	C₂×Dic₄₃	—	2	2	no	—
79	C₂×Dic₄₃	—	2	2	no	—
80	C₂×Dic₄₃	—	2	2	no	—
81	C₂×Dic₄₃	—	2	2	no	—
82	C₂×Dic₄₃	—	2	4	no	—
83	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
84	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
85	C₂×Dic₄₃	C₂×Q₈	1	4	yes	—
86	C₂×Dic₄₃	C₂×Q₈	1	4	yes	—
87	Dic₈₆	—	4	4	no	—
88	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
89	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
90	C₂²×C₄₃	C₂²×Q₈	1	2	yes	—
91	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
92	C₂²×C₄₃	2₋¹⁺⁴	1	2	yes	—
93	C₄×C₄₃	—	2	4	no	—
94	C₄×C₄₃	—	2	4	no	—
95	C₄×C₄₃	—	4	2	no	—
96	C₄×C₄₃	#121	1	4	yes	—
97	Dic₄₃	—	2	2	no	—
98	Dic₄₃	—	2	4	no	—
99	Dic₄₃	—	4	2	no	—
100	Dic₄₃	—	4	2	no	—
101	Dic₄₃	C₄⋊Q₈	1	4	yes	complemented
102	#2546	—	43	1	no	maximal, Sylow
103	C₂×C₄₃	C₂×#121	1	1	yes	characteristic
104	C₂×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₄₃	#536	1	1	yes	characteristic
106	C₂×C₄₃	D₄×Q₈	1	2	yes	—
107	C₂×C₄₃	#557	1	2	yes	—
108	C₂×C₄⋊Q₈	—	43	1	no	—
109	C₂×C₄⋊Q₈	—	43	1	no	—
110	C₂×C₄⋊Q₈	D₄₃	1	1	yes	characteristic
111	C₂×C₄×Q₈	—	43	1	no	—
112	#398	—	43	4	no	—
113	#391	—	43	1	no	—
114	#391	—	43	1	no	—
115	#389	—	43	4	no	—
116	C₄×C₄⋊C₄	—	43	1	no	—
117	C₄₃	#2546	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₈	—	43	1	no	—
119	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
120	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
121	C₄⋊Q₈	—	2	4	no	—
122	C₄⋊Q₈	—	86	4	no	—
123	C₄⋊Q₈	—	86	4	no	—
124	C₄×Q₈	—	86	4	no	—
125	C₂×C₄⋊C₄	—	43	2	no	—
126	C₂×C₄⋊C₄	—	43	1	no	—
127	C₂×C₄⋊C₄	—	43	2	no	—
128	C₂×C₄⋊C₄	—	43	2	no	—
129	C₂×C₄⋊C₄	—	43	4	no	—
130	C₂×C₄⋊C₄	—	43	2	no	—
131	C₂×C₄⋊C₄	C₂×D₄₃	1	4	yes	—
132	C₂×C₄²	—	43	1	no	—
133	C₂×C₄²	—	43	2	no	—
134	C₂×C₄²	—	43	1	no	—
135	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
136	#95	—	43	2	no	—
137	#95	—	43	4	no	—
138	#95	—	43	4	no	—
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	—	86	2	no	—
145	C₂×Q₈	—	86	2	no	—
146	C₂×Q₈	—	86	2	no	—
147	C₂×Q₈	C₂²⋊D₄₃	1	4	yes	—
148	C₂²×C₄	—	43	1	no	—
149	C₂²×C₄	—	43	2	no	—
150	C₂²×C₄	—	43	1	no	—
151	C₂²×C₄	—	43	2	no	—
152	C₂²×C₄	—	43	2	no	—
153	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
154	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
155	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
156	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
157	C₄⋊C₄	—	2	8	no	—
158	C₄⋊C₄	—	43	8	no	—
159	C₄⋊C₄	—	86	4	no	—
160	C₄⋊C₄	—	86	4	no	—
161	C₄⋊C₄	—	86	4	no	—
162	C₄⋊C₄	—	86	2	no	—
163	C₄⋊C₄	—	86	8	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	43	4	no	—
166	C₄²	—	86	4	no	—
167	C₄²	—	86	2	no	—
168	C₂³	C₂³×D₄₃	1	1	yes	characteristic
169	Q₈	—	2	8	no	—
170	Q₈	—	4	4	no	—
171	Q₈	—	172	4	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	43	2	no	—
180	C₂×C₄	—	43	4	no	—
181	C₂×C₄	—	43	2	no	—
182	C₂×C₄	—	43	4	no	—
183	C₂×C₄	—	86	2	no	—
184	C₂×C₄	—	86	2	no	—
185	C₂×C₄	—	86	2	no	—
186	C₂×C₄	—	86	2	no	—
187	C₂×C₄	—	86	2	no	—
188	C₂×C₄	—	86	2	no	—
189	C₂×C₄	—	86	2	no	—
190	C₂×C₄	—	86	4	no	—
191	C₂×C₄	D₄×D₄₃	1	2	yes	—
192	C₂×C₄	#10621900	1	2	yes	—
193	C₂×C₄	Q₈×D₄₃	1	4	yes	—
194	C₂×C₄	C₂×C₂²⋊D₄₃	1	4	yes	—
195	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
196	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
197	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
198	C₂²	C₂×Q₈×D₄₃	1	2	yes	—
199	C₂²	C₂×#10621900	1	1	yes	characteristic
200	C₂²	#13063373	1	2	yes	—
201	C₄	—	2	4	no	—
202	C₄	—	2	4	no	—
203	C₄	—	4	2	no	—
204	C₄	—	43	4	no	—
205	C₄	—	86	4	no	—
206	C₄	—	86	2	no	—
207	C₄	—	172	2	no	—
208	C₄	—	172	2	no	—
209	C₄	#13063297	1	4	yes	—
210	C₂	C₂×#13063297	1	1	yes	characteristic
211	C₂	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
212	C₂	#14583474	1	1	yes	characteristic
213	C₂	Q₈×C₂²⋊D₄₃	1	2	yes	—
214	C₂	#14583624	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063323

C₂

yes

maximal, characteristic

C₂×#13063323

C₂

yes

maximal, characteristic

C₂×Q₈×Dic₄₃

C₂

yes

maximal, characteristic

#14582832

C₂

yes

maximal

#14582806

C₂

yes

maximal, characteristic

#14582805

C₂

yes

maximal, characteristic

#14582794

C₂

yes

maximal

C₄×Dic₄₃⋊C₄

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

characteristic

#13063323

—

#13063323

—

Q₈×Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

characteristic

C₂×C₄×Dic₄₃

C₂²

yes

characteristic

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₃

C₂²

yes

characteristic

C₂²×Q₈×C₄₃

C₂²

yes

characteristic

C₄⋊Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

Q₈

yes

complemented

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

D₄

yes

—

C₂³×C₄₃

C₂⁴

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₄

yes

—

C₂×Dic₄₃

C₂×D₄

yes

—

C₂×Dic₄₃

C₂×Q₈

yes

—

C₂×Dic₄₃

C₂×Q₈

yes

—

Dic₈₆

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂²×Q₈

yes

—

C₂²×C₄₃

C₂×#41

yes

characteristic

C₂²×C₄₃

2₋¹⁺⁴

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#121

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

100

Dic₄₃

—

101

Dic₄₃

C₄⋊Q₈

yes

complemented

102

#2546

—

maximal, Sylow

103

C₂×C₄₃

C₂×#121

yes

characteristic

104

C₂×C₄₃

C₂×C₄⋊Q₈

yes

characteristic

105

C₂×C₄₃

#536

yes

characteristic

106

C₂×C₄₃

D₄×Q₈

yes

—

107

C₂×C₄₃

#557

yes

—

108

C₂×C₄⋊Q₈

—

109

C₂×C₄⋊Q₈

—

110

C₂×C₄⋊Q₈

D₄₃

yes

characteristic

111

C₂×C₄×Q₈

—

112

#398

—

113

#391

—

114

#391

—

115

#389

—

116

C₄×C₄⋊C₄

—

117

C₄₃

#2546

yes

characteristic, complemented, Sylow

118

C₂²×Q₈

—

119

C₂²×Q₈

C₂×D₄₃

yes

characteristic

120

C₂²×Q₈

C₂×D₄₃

yes

characteristic

121

C₄⋊Q₈

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

C₄×Q₈

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

132

C₂×C₄²

—

133

C₂×C₄²

—

134

C₂×C₄²

—

135

C₂×C₄²

C₂×D₄₃

yes

characteristic

136

#95

—

137

#95

—

138

#95

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

C₂²⋊D₄₃

yes

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

—

153

C₂²×C₄

C₂²×D₄₃

yes

—

154

C₂²×C₄

C₂²×D₄₃

yes

—

155

C₂²×C₄

C₂²×D₄₃

yes

—

156

C₂²×C₄

C₂²×D₄₃

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₂³

C₂³×D₄₃

yes

characteristic

169

Q₈

—

170

Q₈

—

171

Q₈

—

172

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

D₄×D₄₃

yes

—

192

C₂×C₄

#10621900

yes

—

193

C₂×C₄

Q₈×D₄₃

yes

—

194

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

195

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

196

C₂²

C₂×D₄×D₄₃

yes

characteristic

197

C₂²

C₂²×C₂²⋊D₄₃

yes

characteristic

198

C₂²

C₂×Q₈×D₄₃

yes

—

199

C₂²

C₂×#10621900

yes

characteristic

200

C₂²

#13063373

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

172

—

208

C₄

—

172

—

209

C₄

#13063297

yes

—

210

C₂

C₂×#13063297

yes

characteristic

211

C₂

C₄⋊Q₈×D₄₃

yes

characteristic

212

C₂

#14583474

yes

characteristic

213

C₂

Q₈×C₂²⋊D₄₃

yes

—

214

C₂

#14583624

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₄₃	C₂
hypercenter	C₂×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	#2546
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator, socle

C₂³×C₄₃

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₄₃

C₂

hypercenter

C₂×C₄⋊Q₈

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
56. Dic₄₃⋊C₄	Q₈	semidirect
101. Dic₄₃	C₄⋊Q₈	semidirect
117. C₄₃	#2546	semidirect

H ≅ G/N

type

56. Dic₄₃⋊C₄

Q₈

semidirect

101. Dic₄₃

C₄⋊Q₈

semidirect

117. C₄₃

#2546

semidirect

Subgroups — #19139581

Conjugacy classes of subgroups

Complements