Group #19139795 — subgroups

Subgroups — #19139795

Order 5504.

548 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13063317	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₁₇₂	C₂	1	1	yes	maximal, characteristic
3	#14582821	C₂	1	4	yes	maximal, complemented
4	#14582801	C₂	1	1	yes	maximal, characteristic
5	#14582801	C₂	1	1	yes	maximal, characteristic
6	#14582773	C₂	1	4	yes	maximal, complemented
7	#14582769	C₂	1	1	yes	maximal, characteristic, complemented
8	#14582765	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₄₃	C₂²	1	2	yes	—
12	#13063317	—	2	4	no	—
13	C₄×D₁₇₂	—	2	4	no	—
14	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₄₃	C₂²	1	4	yes	—
18	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
19	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
20	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	—
21	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
22	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	—
23	C₄₃⋊#95	C₂²	1	4	yes	—
24	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
25	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
28	#95×C₄₃	C₂²	1	2	yes	—
29	C₂³×D₄₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₄₃	C₂³	1	2	yes	—
36	C₂²×Dic₄₃	C₂³	1	2	yes	—
37	C₂²×Dic₄₃	C₂³	1	2	yes	—
38	C₂×D₁₇₂	—	2	2	no	—
39	C₂×D₁₇₂	—	2	2	no	—
40	C₂×D₁₇₂	Q₈	1	4	yes	—
41	C₂×C₄×D₄₃	—	2	4	no	—
42	C₂×C₄×D₄₃	—	4	4	no	—
43	C₄⋊C₄×C₄₃	—	2	4	no	—
44	C₄²×C₄₃	—	2	2	no	—
45	C₄²×C₄₃	—	2	2	no	—
46	C₄²×C₄₃	D₄	1	4	yes	—
47	(C₂×D₄₃)⋊C₄	—	2	4	no	—
48	(C₂×D₄₃)⋊C₄	—	2	8	no	—
49	(C₂×D₄₃)⋊C₄	—	4	4	no	—
50	C₄⋊Dic₄₃	—	2	2	no	—
51	C₄⋊Dic₄₃	—	2	2	no	—
52	C₄⋊Dic₄₃	—	2	2	no	—
53	C₄⋊Dic₄₃	—	4	4	no	—
54	Dic₄₃⋊C₄	—	2	4	no	—
55	Dic₄₃⋊C₄	—	2	4	no	—
56	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
57	C₂²×D₄₃	—	2	2	no	—
58	C₂²×D₄₃	—	2	4	no	—
59	C₂²×D₄₃	—	2	2	no	—
60	C₂²×D₄₃	—	2	4	no	—
61	C₂²×D₄₃	C₂×Q₈	1	4	yes	—
62	C₂×C₄×C₄₃	—	2	2	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	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₄₃	#41	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	4	no	—
79	C₂×Dic₄₃	—	2	4	no	—
80	C₂×Dic₄₃	—	2	4	no	—
81	D₁₇₂	—	2	8	no	—
82	C₄×D₄₃	—	4	8	no	—
83	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
85	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
86	C₂²×C₄₃	C₂×#41	1	2	yes	—
87	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
88	C₂×D₄₃	—	2	4	no	—
89	C₂×D₄₃	—	2	8	no	—
90	C₂×D₄₃	—	4	2	no	—
91	C₂×D₄₃	—	4	2	no	—
92	C₂×D₄₃	—	4	4	no	—
93	C₄×C₄₃	—	2	2	no	—
94	C₄×C₄₃	—	2	2	no	—
95	C₄×C₄₃	—	2	4	no	—
96	C₄×C₄₃	—	4	2	no	—
97	C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
98	Dic₄₃	—	4	2	no	—
99	Dic₄₃	—	4	2	no	—
100	Dic₄₃	—	4	2	no	—
101	#2566	—	43	1	no	maximal, Sylow
102	C₂×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄₃	#532	1	1	yes	characteristic
104	C₂×C₄₃	#535	1	1	yes	characteristic
105	C₂×C₄₃	#553	1	2	yes	—
106	C₂×C₄₃	#554	1	2	yes	—
107	D₄₃	—	4	4	no	—
108	C₂×#125	—	43	1	no	—
109	C₂×C₄×D₄	—	43	1	no	—
110	#396	—	43	4	no	—
111	#390	—	43	1	no	—
112	#390	—	43	1	no	—
113	#385	—	43	4	no	—
114	#384	—	43	1	no	—
115	#382	—	43	1	no	—
116	C₄×C₄⋊C₄	D₄₃	1	1	yes	characteristic, complemented
117	C₄₃	#2566	1	1	yes	characteristic, complemented, Sylow
118	C₂²×D₄	—	43	1	no	—
119	C₂³×C₄	—	43	2	no	—
120	#125	—	86	4	no	—
121	C₄×D₄	—	86	4	no	—
122	C₂×C₄⋊C₄	—	43	1	no	—
123	C₂×C₄⋊C₄	—	43	2	no	—
124	C₂×C₄⋊C₄	—	43	2	no	—
125	C₂×C₄⋊C₄	—	43	1	no	—
126	C₂×C₄⋊C₄	—	43	1	no	—
127	C₂×C₄⋊C₄	—	43	4	no	—
128	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
129	C₂×C₂²⋊C₄	—	43	2	no	—
130	C₂×C₂²⋊C₄	—	43	4	no	—
131	C₂×C₂²⋊C₄	—	43	4	no	—
132	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
133	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
134	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
135	#95	—	43	4	no	—
136	#95	C₂×D₄₃	1	2	yes	—
137	C₂⁴	—	43	2	no	—
138	C₂×D₄	—	43	4	no	—
139	C₂×D₄	—	86	2	no	—
140	C₂×D₄	—	86	2	no	—
141	C₂²×C₄	—	43	2	no	—
142	C₂²×C₄	—	43	2	no	—
143	C₂²×C₄	—	43	2	no	—
144	C₂²×C₄	—	86	4	no	—
145	C₂²×C₄	—	172	4	no	—
146	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
147	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
150	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	86	2	no	—
153	C₄⋊C₄	—	86	4	no	—
154	C₄⋊C₄	—	86	4	no	—
155	C₄⋊C₄	—	86	2	no	—
156	C₄⋊C₄	—	86	2	no	—
157	C₄⋊C₄	—	172	4	no	—
158	C₂²⋊C₄	—	86	4	no	—
159	C₂²⋊C₄	—	86	8	no	—
160	C₂²⋊C₄	—	172	4	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	D₁₇₂	1	4	yes	—
164	C₂³	—	43	4	no	—
165	C₂³	—	86	2	no	—
166	C₂³	—	86	2	no	—
167	C₂³	—	86	4	no	—
168	C₂³	—	86	4	no	—
169	C₂³	C₂³×D₄₃	1	1	yes	characteristic
170	D₄	—	86	8	no	—
171	C₂×C₄	—	2	2	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	—	86	2	no	—
179	C₂×C₄	—	86	2	no	—
180	C₂×C₄	—	86	2	no	—
181	C₂×C₄	—	86	4	no	—
182	C₂×C₄	—	86	4	no	—
183	C₂×C₄	—	86	4	no	—
184	C₂×C₄	—	172	8	no	—
185	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
186	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
187	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
188	C₂×C₄	#10621898	1	4	yes	—
189	C₂×C₄	#10621900	1	2	yes	—
190	C₂×C₄	Q₈×D₄₃	1	2	yes	—
191	C₂²	—	86	8	no	—
192	C₂²	—	86	4	no	—
193	C₂²	—	172	4	no	—
194	C₂²	—	172	2	no	—
195	C₂²	—	172	2	no	—
196	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
197	C₂²	C₂×Q₈×D₄₃	1	1	yes	characteristic
198	C₂²	C₂×#10621900	1	1	yes	characteristic
199	C₂²	C₂×#10621898	1	2	yes	—
200	C₂²	#13063371	1	2	yes	—
201	C₄	—	2	4	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	2	no	—
204	C₄	—	4	2	no	—
205	C₄	—	172	2	no	—
206	C₄	—	172	2	no	—
207	C₄	—	172	2	no	—
208	C₄	#13063305	1	4	yes	—
209	C₂	—	172	4	no	—
210	C₂	C₂×#13063305	1	1	yes	characteristic
211	C₂	#14583449	1	1	yes	characteristic
212	C₂	#14583468	1	1	yes	characteristic
213	C₂	#14583595	1	2	yes	—
214	C₂	#14583609	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063317

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₁₇₂

C₂

yes

maximal, characteristic

#14582821

C₂

yes

maximal, complemented

#14582801

C₂

yes

maximal, characteristic

#14582801

C₂

yes

maximal, characteristic

#14582773

C₂

yes

maximal, complemented

#14582769

C₂

yes

maximal, characteristic, complemented

#14582765

C₂

yes

maximal, characteristic, complemented

C₄×C₄⋊C₄×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₇₂

C₂²

yes

characteristic

C₂²×C₄×D₄₃

C₂²

yes

—

#13063317

—

C₄×D₁₇₂

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#95×C₄₃

C₂²

yes

—

C₂³×D₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

Q₈

yes

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

D₄

yes

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₂³×C₄₃

C₂⁴

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

C₂×Q₈

yes

—

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×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

C₄×D₄₃

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂²×Q₈

yes

characteristic

C₂²×C₄₃

C₂×#41

yes

characteristic

C₂²×C₄₃

C₂×#41

yes

—

C₂²×C₄₃

H(5,𝔽₂)

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊Q₈

yes

—

Dic₄₃

—

Dic₄₃

—

100

Dic₄₃

—

101

#2566

—

maximal, Sylow

102

C₂×C₄₃

C₂×C₂²⋊Q₈

yes

characteristic

103

C₂×C₄₃

#532

yes

characteristic

104

C₂×C₄₃

#535

yes

characteristic

105

C₂×C₄₃

#553

yes

—

106

C₂×C₄₃

#554

yes

—

107

D₄₃

—

108

C₂×#125

—

109

C₂×C₄×D₄

—

110

#396

—

111

#390

—

112

#390

—

113

#385

—

114

#384

—

115

#382

—

116

C₄×C₄⋊C₄

D₄₃

yes

characteristic, complemented

117

C₄₃

#2566

yes

characteristic, complemented, Sylow

118

C₂²×D₄

—

119

C₂³×C₄

—

120

#125

—

121

C₄×D₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

129

C₂×C₂²⋊C₄

—

130

C₂×C₂²⋊C₄

—

131

C₂×C₂²⋊C₄

—

132

C₂×C₄²

C₂×D₄₃

yes

characteristic

133

C₂×C₄²

C₂×D₄₃

yes

characteristic

134

C₂×C₄²

C₂×D₄₃

yes

characteristic

135

#95

—

136

#95

C₂×D₄₃

yes

—

137

C₂⁴

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

172

—

146

C₂²×C₄

C₂²×D₄₃

yes

—

147

C₂²×C₄

C₂²×D₄₃

yes

characteristic

148

C₂²×C₄

C₂²×D₄₃

yes

characteristic

149

C₂²×C₄

C₂²×D₄₃

yes

—

150

C₂²×C₄

C₂²×D₄₃

yes

characteristic

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

172

—

158

C₂²⋊C₄

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

172

—

161

C₄²

—

162

C₄²

—

163

C₄²

D₁₇₂

yes

—

164

C₂³

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

C₂³×D₄₃

yes

characteristic

170

D₄

—

171

C₂×C₄

—

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

—

172

—

185

C₂×C₄

C₂×D₁₇₂

yes

—

186

C₂×C₄

C₂×D₁₇₂

yes

—

187

C₂×C₄

C₂×D₁₇₂

yes

—

188

C₂×C₄

#10621898

yes

—

189

C₂×C₄

#10621900

yes

—

190

C₂×C₄

Q₈×D₄₃

yes

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

172

—

194

C₂²

—

172

—

195

C₂²

—

172

—

196

C₂²

C₂²×D₁₇₂

yes

characteristic

197

C₂²

C₂×Q₈×D₄₃

yes

characteristic

198

C₂²

C₂×#10621900

yes

characteristic

199

C₂²

C₂×#10621898

yes

—

200

C₂²

#13063371

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

172

—

206

C₄

—

172

—

207

C₄

—

172

—

208

C₄

#13063305

yes

—

209

C₂

—

172

—

210

C₂

C₂×#13063305

yes

characteristic

211

C₂

#14583449

yes

characteristic

212

C₂

#14583468

yes

characteristic

213

C₂

#14583595

yes

—

214

C₂

#14583609

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₄×C₄⋊C₄×C₄₃	C₂
hypercenter	C₄×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#2566
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator, socle

C₂³×C₄₃

C₂⁴

Fitting

C₄×C₄⋊C₄×C₄₃

C₂

hypercenter

C₄×C₄⋊C₄

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
1. C₂×#13063317	C₂	semidirect
3. #14582821	C₂	semidirect
6. #14582773	C₂	semidirect
7. #14582769	C₂	semidirect
8. #14582765	C₂	semidirect
9. C₄×C₄⋊C₄×C₄₃	C₂	semidirect
116. C₄×C₄⋊C₄	D₄₃	semidirect
117. C₄₃	#2566	semidirect

H ≅ G/N

type

1. C₂×#13063317

C₂

semidirect

3. #14582821

C₂

semidirect

6. #14582773

C₂

semidirect

7. #14582769

C₂

semidirect

8. #14582765

C₂

semidirect

9. C₄×C₄⋊C₄×C₄₃

C₂

semidirect

116. C₄×C₄⋊C₄

D₄₃

semidirect

117. C₄₃

#2566

semidirect

Subgroups — #19139795

Conjugacy classes of subgroups

Complements