Group #20248589 — subgroups

Subgroups — #20248589

Order 6016.

548 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 161 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×C₄×D₁₈₈	C₂	1	1	yes	maximal, characteristic
2	#15689045	C₂	1	4	yes	maximal, complemented
3	#15689024	C₂	1	1	yes	maximal, characteristic
4	#15689024	C₂	1	1	yes	maximal, characteristic
5	#15688994	C₂	1	4	yes	maximal, complemented
6	#15688992	C₂	1	1	yes	maximal, characteristic, complemented
7	#15688988	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#125×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
12	C₄×D₁₈₈	—	2	4	no	—
13	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
16	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
17	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
18	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
19	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
21	C₄₇⋊#95	C₂²	1	2	yes	—
22	C₄₇⋊#95	C₂²	1	4	yes	—
23	#125×C₄₇	—	2	4	no	—
24	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
28	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
29	C₂³×D₄₇	C₂³	1	2	yes	—
30	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
32	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂²×Dic₄₇	C₂³	1	2	yes	—
36	C₂²×Dic₄₇	C₂³	1	2	yes	—
37	C₂²×Dic₄₇	C₂³	1	2	yes	—
38	C₂×D₁₈₈	—	2	2	no	—
39	C₂×D₁₈₈	—	2	2	no	—
40	C₂×D₁₈₈	Q₈	1	4	yes	—
41	C₂×C₄×D₄₇	—	2	4	no	—
42	C₂×C₄×D₄₇	—	4	4	no	—
43	C₄⋊C₄×C₄₇	—	2	4	no	—
44	C₄⋊C₄×C₄₇	—	2	4	no	—
45	C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₄²×C₄₇	D₄	1	4	yes	—
48	(C₂×D₄₇)⋊C₄	—	2	4	no	—
49	(C₂×D₄₇)⋊C₄	—	2	8	no	—
50	(C₂×D₄₇)⋊C₄	—	4	4	no	—
51	C₄⋊Dic₄₇	—	2	2	no	—
52	C₄⋊Dic₄₇	—	2	4	no	—
53	Dic₄₇⋊C₄	—	4	4	no	—
54	C₄×Dic₄₇	—	2	2	no	—
55	C₄×Dic₄₇	—	2	2	no	—
56	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
57	C₂²×D₄₇	—	2	2	no	—
58	C₂²×D₄₇	—	2	4	no	—
59	C₂²×D₄₇	—	2	2	no	—
60	C₂²×D₄₇	—	2	4	no	—
61	C₂²×D₄₇	C₂×Q₈	1	4	yes	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	—	2	4	no	—
67	C₂×C₄×C₄₇	—	2	4	no	—
68	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
72	C₂×C₄×C₄₇	#41	1	2	yes	—
73	C₂×Dic₄₇	—	2	2	no	—
74	C₂×Dic₄₇	—	2	2	no	—
75	C₂×Dic₄₇	—	2	2	no	—
76	C₂×Dic₄₇	—	2	2	no	—
77	C₂×Dic₄₇	—	2	2	no	—
78	C₂×Dic₄₇	—	2	4	no	—
79	C₂×Dic₄₇	—	2	2	no	—
80	C₂×Dic₄₇	#41	1	4	yes	—
81	D₁₈₈	—	2	8	no	—
82	C₄×D₄₇	—	4	8	no	—
83	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
85	C₂²×C₄₇	C₂×#41	1	2	yes	—
86	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
87	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
88	C₂×D₄₇	—	2	4	no	—
89	C₂×D₄₇	—	2	8	no	—
90	C₂×D₄₇	—	4	2	no	—
91	C₂×D₄₇	—	4	2	no	—
92	C₂×D₄₇	—	4	4	no	—
93	C₄×C₄₇	—	2	2	no	—
94	C₄×C₄₇	—	2	2	no	—
95	C₄×C₄₇	—	4	2	no	—
96	C₄×C₄₇	—	4	2	no	—
97	C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
98	Dic₄₇	—	2	4	no	—
99	Dic₄₇	—	4	2	no	—
100	Dic₄₇	—	4	2	no	—
101	#2566	—	47	1	no	maximal, Sylow
102	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄₇	#532	1	1	yes	characteristic
104	C₂×C₄₇	#535	1	1	yes	characteristic
105	C₂×C₄₇	#553	1	2	yes	—
106	C₂×C₄₇	#554	1	2	yes	—
107	D₄₇	—	4	4	no	—
108	C₂×#125	D₄₇	1	1	yes	characteristic, complemented
109	C₂×C₄×D₄	—	47	1	no	—
110	#396	—	47	4	no	—
111	#390	—	47	1	no	—
112	#390	—	47	1	no	—
113	#385	—	47	4	no	—
114	#384	—	47	1	no	—
115	#382	—	47	1	no	—
116	C₄×C₄⋊C₄	—	47	1	no	—
117	C₄₇	#2566	1	1	yes	characteristic, complemented, Sylow
118	C₂²×D₄	—	47	1	no	—
119	C₂³×C₄	—	47	2	no	—
120	#125	—	2	4	no	—
121	C₄×D₄	—	94	4	no	—
122	C₂×C₄⋊C₄	—	47	1	no	—
123	C₂×C₄⋊C₄	—	47	2	no	—
124	C₂×C₄⋊C₄	—	47	4	no	—
125	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
127	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
128	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
129	C₂×C₂²⋊C₄	—	47	2	no	—
130	C₂×C₂²⋊C₄	—	47	4	no	—
131	C₂×C₂²⋊C₄	—	47	4	no	—
132	C₂×C₄²	—	47	1	no	—
133	C₂×C₄²	—	47	1	no	—
134	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
135	#95	—	47	2	no	—
136	#95	—	47	4	no	—
137	C₂⁴	—	47	2	no	—
138	C₂×D₄	—	47	4	no	—
139	C₂×D₄	—	94	2	no	—
140	C₂×D₄	—	94	2	no	—
141	C₂²×C₄	—	47	2	no	—
142	C₂²×C₄	—	47	2	no	—
143	C₂²×C₄	—	47	2	no	—
144	C₂²×C₄	—	94	4	no	—
145	C₂²×C₄	—	188	4	no	—
146	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
147	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
148	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	94	4	no	—
156	C₄⋊C₄	—	94	2	no	—
157	C₄⋊C₄	—	188	4	no	—
158	C₂²⋊C₄	—	94	8	no	—
159	C₂²⋊C₄	—	94	4	no	—
160	C₂²⋊C₄	—	188	4	no	—
161	C₄²	—	94	2	no	—
162	C₄²	—	94	2	no	—
163	C₄²	C₂²⋊D₄₇	1	4	yes	—
164	C₂³	—	47	4	no	—
165	C₂³	—	94	2	no	—
166	C₂³	—	94	4	no	—
167	C₂³	—	94	2	no	—
168	C₂³	—	94	4	no	—
169	C₂³	C₂³×D₄₇	1	1	yes	characteristic
170	D₄	—	94	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	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	47	4	no	—
178	C₂×C₄	—	94	2	no	—
179	C₂×C₄	—	94	2	no	—
180	C₂×C₄	—	94	2	no	—
181	C₂×C₄	—	94	2	no	—
182	C₂×C₄	—	94	2	no	—
183	C₂×C₄	—	94	2	no	—
184	C₂×C₄	—	94	4	no	—
185	C₂×C₄	—	188	8	no	—
186	C₂×C₄	Q₈×D₄₇	1	2	yes	—
187	C₂×C₄	#10625354	1	2	yes	—
188	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
189	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
190	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
191	C₂²	—	94	8	no	—
192	C₂²	—	94	4	no	—
193	C₂²	—	188	4	no	—
194	C₂²	—	188	2	no	—
195	C₂²	—	188	2	no	—
196	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
197	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
198	C₂²	#41×D₄₇	1	2	yes	—
199	C₂²	C₂×#10625354	1	1	yes	characteristic
200	C₂²	#13094070	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	2	no	—
205	C₄	—	94	4	no	—
206	C₄	—	188	2	no	—
207	C₄	—	188	2	no	—
208	C₄	#13094000	1	4	yes	—
209	C₂	—	188	4	no	—
210	C₂	C₂×#13094000	1	1	yes	characteristic
211	C₂	#15689669	1	1	yes	characteristic
212	C₂	#15689687	1	1	yes	characteristic
213	C₂	#15689826	1	2	yes	—
214	C₂	#15689829	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₁₈₈

C₂

yes

maximal, characteristic

#15689045

C₂

yes

maximal, complemented

#15689024

C₂

yes

maximal, characteristic

#15689024

C₂

yes

maximal, characteristic

#15688994

C₂

yes

maximal, complemented

#15688992

C₂

yes

maximal, characteristic, complemented

#15688988

C₂

yes

maximal, characteristic, complemented

C₄×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

characteristic

C₂²×C₄×D₄₇

C₂²

yes

—

C₄×D₁₈₈

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

#125×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

Q₈

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#41

yes

—

D₁₈₈

—

C₄×D₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

Dic₄₇

—

Dic₄₇

—

100

Dic₄₇

—

101

#2566

—

maximal, Sylow

102

C₂×C₄₇

C₂×C₂²⋊Q₈

yes

characteristic

103

C₂×C₄₇

#532

yes

characteristic

104

C₂×C₄₇

#535

yes

characteristic

105

C₂×C₄₇

#553

yes

—

106

C₂×C₄₇

#554

yes

—

107

D₄₇

—

108

C₂×#125

D₄₇

yes

characteristic, complemented

109

C₂×C₄×D₄

—

110

#396

—

111

#390

—

112

#390

—

113

#385

—

114

#384

—

115

#382

—

116

C₄×C₄⋊C₄

—

117

C₄₇

#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

C₂×D₄₇

yes

characteristic

126

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

127

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

128

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

129

C₂×C₂²⋊C₄

—

130

C₂×C₂²⋊C₄

—

131

C₂×C₂²⋊C₄

—

132

C₂×C₄²

—

133

C₂×C₄²

—

134

C₂×C₄²

C₂×D₄₇

yes

characteristic

135

#95

—

136

#95

—

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

—

188

—

146

C₂²×C₄

C₂²×D₄₇

yes

—

147

C₂²×C₄

C₂²×D₄₇

yes

—

148

C₂²×C₄

C₂²×D₄₇

yes

characteristic

149

C₂²×C₄

C₂²×D₄₇

yes

characteristic

150

C₂²×C₄

C₂²×D₄₇

yes

characteristic

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

188

—

158

C₂²⋊C₄

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

188

—

161

C₄²

—

162

C₄²

—

163

C₄²

C₂²⋊D₄₇

yes

—

164

C₂³

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

C₂³×D₄₇

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

—

185

C₂×C₄

—

188

—

186

C₂×C₄

Q₈×D₄₇

yes

—

187

C₂×C₄

#10625354

yes

—

188

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

189

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

190

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

188

—

194

C₂²

—

188

—

195

C₂²

—

188

—

196

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

197

C₂²

C₂×Q₈×D₄₇

yes

characteristic

198

C₂²

#41×D₄₇

yes

—

199

C₂²

C₂×#10625354

yes

characteristic

200

C₂²

#13094070

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

188

—

207

C₄

—

188

—

208

C₄

#13094000

yes

—

209

C₂

—

188

—

210

C₂

C₂×#13094000

yes

characteristic

211

C₂

#15689669

yes

characteristic

212

C₂

#15689687

yes

characteristic

213

C₂

#15689826

yes

—

214

C₂

#15689829

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×#125×C₄₇	C₂
hypercenter	C₂×#125	D₄₇
hypocenter	C₄₇	#2566
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×#125×C₄₇

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
2. #15689045	C₂	semidirect
5. #15688994	C₂	semidirect
6. #15688992	C₂	semidirect
7. #15688988	C₂	semidirect
8. C₄×C₄⋊Dic₄₇	C₂	semidirect
9. C₂×#125×C₄₇	C₂	semidirect
108. C₂×#125	D₄₇	semidirect
117. C₄₇	#2566	semidirect

H ≅ G/N

type

2. #15689045

C₂

semidirect

5. #15688994

C₂

semidirect

6. #15688992

C₂

semidirect

7. #15688988

C₂

semidirect

8. C₄×C₄⋊Dic₄₇

C₂

semidirect

9. C₂×#125×C₄₇

C₂

semidirect

108. C₂×#125

D₄₇

semidirect

117. C₄₇

#2566

semidirect

Subgroups — #20248589

Conjugacy classes of subgroups

Complements