Group #27302605 — subgroups

Subgroups — #27302605

Order 13568.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16811003	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810440	C₂	1	2	yes	maximal
3	#20276436	C₂	1	2	yes	maximal
4	#20274594	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	—	2	1	no	—
7	C₂²×Dic₂₁₂	C₂²	1	2	yes	—
8	C₂×#13133217	—	2	1	no	—
9	#16811003	—	2	2	no	—
10	#16811003	C₄	1	4	yes	complemented
11	C₂×#13133153	C₂²	1	1	yes	characteristic
12	#16810661	C₂²	1	1	yes	characteristic
13	#16810578	—	2	1	no	—
14	#16810440	—	2	4	no	—
15	#16810428	—	2	4	no	—
16	#16810383	—	2	1	no	—
17	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#364	—	2	2	no	—
20	C₂²×Dic₁₀₆	—	2	2	no	—
21	C₂×Dic₂₁₂	—	2	4	no	—
22	C₂×Dic₂₁₂	—	2	2	no	—
23	C₂×Dic₂₁₂	—	2	2	no	—
24	C₂×Dic₂₁₂	—	2	2	no	—
25	C₂×Dic₂₁₂	—	2	2	no	—
26	C₂×Dic₂₁₂	—	2	2	no	—
27	C₂×Dic₂₁₂	—	4	1	no	—
28	C₂×Dic₂₁₂	—	4	1	no	—
29	C₂×Dic₂₁₂	—	4	2	no	—
30	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
31	C₂×#11719917	—	2	1	no	—
32	#13133217	—	2	4	no	—
33	#13133217	—	4	2	no	—
34	#13133176	—	2	1	no	—
35	C₂×C₄⋊Dic₅₃	—	4	2	no	—
36	#13133153	—	2	1	no	—
37	#13133153	D₄	1	2	yes	—
38	#13133151	D₄	1	2	yes	—
39	#13133149	—	2	1	no	—
40	#13133127	—	4	1	no	—
41	C₈×Dic₅₃	—	4	1	no	—
42	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	M₃₂×C₅₃	—	2	2	no	—
46	M₃₂×C₅₃	—	2	2	no	—
47	C₂×C₁₆×C₅₃	—	2	2	no	—
48	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₅₃	D₄	1	2	yes	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₅₃	—	4	2	no	—
53	C₂×Dic₁₀₆	—	2	4	no	—
54	C₂×Dic₁₀₆	—	4	2	no	—
55	C₂×Dic₁₀₆	—	4	2	no	—
56	C₂×Dic₁₀₆	—	4	2	no	—
57	C₂×Dic₁₀₆	—	4	2	no	—
58	C₂×Dic₁₀₆	—	4	2	no	—
59	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
61	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
62	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
68	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
69	C₄²×C₅₃	—	2	1	no	—
70	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
71	C₂²⋊Dic₅₃	—	4	1	no	—
72	C₄⋊Dic₅₃	—	2	2	no	—
73	C₄⋊Dic₅₃	—	4	8	no	—
74	C₄×Dic₅₃	—	4	1	no	—
75	#11719917	—	2	2	no	—
76	#11719917	—	4	1	no	—
77	C₂×#88322	—	4	1	no	—
78	Dic₂₁₂	—	2	4	no	—
79	Dic₂₁₂	—	2	4	no	—
80	Dic₂₁₂	—	4	4	no	—
81	Dic₂₁₂	—	4	4	no	—
82	Dic₂₁₂	—	4	2	no	—
83	Dic₂₁₂	—	4	2	no	—
84	C₁₆×C₅₃	—	4	2	no	—
85	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×Dic₅₃	—	4	4	no	—
95	C₂×Dic₅₃	—	4	2	no	—
96	C₂×Dic₅₃	—	8	2	no	—
97	C₂×Dic₅₃	—	8	2	no	—
98	Dic₁₀₆	—	4	4	no	—
99	Dic₁₀₆	—	4	4	no	—
100	Dic₁₀₆	—	8	2	no	—
101	Dic₁₀₆	—	8	2	no	—
102	C₈×C₅₃	#102	1	2	yes	—
103	C₈×C₅₃	#102	1	2	yes	—
104	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₈×C₅₃	#124	1	1	yes	characteristic
107	C₈×C₅₃	#127	1	1	yes	characteristic
108	#88322	—	4	2	no	—
109	#13262	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	#418	1	1	yes	characteristic
112	C₂²×C₅₃	#459	1	1	yes	characteristic
113	C₂²×C₅₃	#460	1	1	yes	characteristic
114	C₄×C₅₃	—	2	4	no	—
115	C₄×C₅₃	#390	1	1	yes	characteristic
116	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
117	C₄×C₅₃	#459	1	1	yes	characteristic
118	C₄×C₅₃	#460	1	1	yes	characteristic
119	Dic₅₃	—	8	2	no	—
120	Dic₅₃	—	8	4	no	—
121	C₂×#494	—	53	1	no	—
122	C₂×#364	D₅₃	1	1	yes	characteristic
123	C₂×#362	—	53	2	no	—
124	#2141	—	53	2	no	—
125	#1772	—	53	1	no	—
126	C₂×C₅₃	—	2	1	no	—
127	C₂×C₅₃	—	2	1	no	—
128	C₂×C₅₃	#1982	1	1	yes	characteristic
129	C₂×C₅₃	#2214	1	2	yes	—
130	C₂²×Q₁₆	—	53	2	no	—
131	C₂²×Q₁₆	—	106	1	no	—
132	C₂×C₄⋊Q₈	—	106	1	no	—
133	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
134	#494	—	53	4	no	—
135	#494	—	106	2	no	—
136	C₂×#108	—	53	1	no	—
137	#427	—	53	1	no	—
138	#405	—	106	1	no	—
139	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
140	#364	—	2	2	no	—
141	#362	—	106	4	no	—
142	#358	—	106	4	no	—
143	#339	—	106	1	no	—
144	C₅₃	#13262	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	—	106	2	no	—
146	C₂×Q₁₆	—	53	4	no	—
147	C₂×Q₁₆	—	106	2	no	—
148	C₂×Q₁₆	—	106	2	no	—
149	C₂×Q₁₆	—	106	2	no	—
150	C₂×Q₁₆	—	106	2	no	—
151	C₂×Q₁₆	—	106	2	no	—
152	C₂×Q₁₆	—	106	4	no	—
153	C₂×Q₁₆	—	212	2	no	—
154	C₂×Q₁₆	—	212	1	no	—
155	C₂×Q₁₆	—	212	1	no	—
156	C₂×M₁₆	—	106	1	no	—
157	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
159	C₄⋊Q₈	—	106	4	no	—
160	C₄⋊Q₈	—	212	2	no	—
161	#117	—	106	1	no	—
162	C₂×C₄⋊C₄	—	212	2	no	—
163	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
164	M₃₂	—	2	2	no	—
165	M₃₂	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	#108	—	53	2	no	—
168	#108	—	106	1	no	—
169	#107	—	53	2	no	—
170	#106	—	106	1	no	—
171	#97	—	212	1	no	—
172	C₄×C₈	—	212	1	no	—
173	C₄×C₈	C₄×D₅₃	1	2	yes	—
174	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
175	C₂×Q₈	—	106	4	no	—
176	C₂×Q₈	—	212	2	no	—
177	C₂×Q₈	—	212	2	no	—
178	C₂×Q₈	—	212	2	no	—
179	C₂×Q₈	—	212	2	no	—
180	C₂×Q₈	—	212	2	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	212	2	no	—
183	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
184	Q₁₆	—	106	4	no	—
185	Q₁₆	—	106	4	no	—
186	Q₁₆	—	212	4	no	—
187	Q₁₆	—	212	4	no	—
188	Q₁₆	—	212	2	no	—
189	Q₁₆	—	212	2	no	—
190	M₁₆	—	106	2	no	—
191	M₁₆	—	212	1	no	—
192	C₂×C₈	—	212	1	no	—
193	C₂×C₈	#11719913	1	2	yes	—
194	C₂×C₈	D₄₂₄	1	1	yes	characteristic
195	C₂×C₈	D₄₂₄	1	1	yes	characteristic
196	C₂×C₈	#11719921	1	2	yes	—
197	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
198	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
199	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
200	C₂×C₈	#11719949	1	1	yes	characteristic
201	C₂×C₈	#11719949	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₄⋊C₄	—	106	2	no	—
204	C₄⋊C₄	—	212	8	no	—
205	C₂²⋊C₄	—	212	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	212	1	no	—
208	C₄²	#11719921	1	2	yes	—
209	C₁₆	—	4	2	no	—
210	C₂³	#13133245	1	1	yes	characteristic
211	Q₈	—	424	2	no	—
212	Q₈	—	424	2	no	—
213	Q₈	—	212	4	no	—
214	Q₈	—	212	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	424	2	no	—
218	C₂×C₄	—	424	2	no	—
219	C₂×C₄	—	212	2	no	—
220	C₂×C₄	—	212	4	no	—
221	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
222	C₂×C₄	#13133192	1	1	yes	characteristic
223	C₂×C₄	#13133192	1	1	yes	characteristic
224	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
225	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
226	C₂×C₄	#13133245	1	1	yes	characteristic
227	C₈	—	212	2	no	—
228	C₈	#13133139	1	2	yes	—
229	C₈	#13133139	1	2	yes	—
230	C₈	#13133179	1	1	yes	characteristic
231	C₈	#13133179	1	1	yes	characteristic
232	C₈	#13133206	1	1	yes	characteristic
233	C₈	#13133213	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	#16810629	1	1	yes	characteristic
236	C₂²	#16810819	1	1	yes	characteristic
237	C₂²	#16810824	1	1	yes	characteristic
238	C₄	—	2	4	no	—
239	C₄	—	424	2	no	—
240	C₄	—	424	4	no	—
241	C₄	#16810524	1	1	yes	characteristic
242	C₄	C₂×#13133139	1	1	yes	characteristic
243	C₄	#16810819	1	1	yes	characteristic
244	C₄	#16810824	1	1	yes	characteristic
245	C₂	—	2	1	no	—
246	C₂	—	2	1	no	—
247	C₂	#20275641	1	1	yes	characteristic
248	C₂	#20276797	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811003

C₂

yes

maximal, characteristic

C₂×#16810440

C₂

yes

maximal

#20276436

C₂

yes

maximal

#20274594

C₂

yes

maximal, characteristic

C₂×C₅₃×#364

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₂

—

C₂²×Dic₂₁₂

C₂²

yes

—

C₂×#13133217

—

#16811003

—

#16811003

C₄

yes

complemented

C₂×#13133153

C₂²

yes

characteristic

#16810661

C₂²

yes

—

—

—

—

C₂×M₃₂×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#364

—

C₂²×Dic₁₀₆

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

C₂×C₄

yes

—

—

—

—

—

C₂×C₄⋊Dic₅₃

—

#13133153

—

#13133153

D₄

yes

—

#13133151

D₄

yes

—

#13133149

—

#13133127

—

C₈×Dic₅₃

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

characteristic

C₂×C₈×C₅₃

D₈

yes

characteristic

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

C₂×C₈×C₅₃

yes

C₂×C₈×C₅₃

yes

C₄²×C₅₃

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

C₁₆×C₅₃

—

C₂³×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

100

Dic₁₀₆

—

101

Dic₁₀₆

—

102

C₈×C₅₃

#102

yes

—

103

C₈×C₅₃

#102

yes

—

104

C₈×C₅₃

C₄×D₄

yes

characteristic

105

C₈×C₅₃

C₄×D₄

yes

characteristic

106

C₈×C₅₃

#124

yes

characteristic

107

C₈×C₅₃

#127

yes

characteristic

108

#88322

—

109

#13262

—

maximal, Sylow

110

C₂²×C₅₃

—

111

C₂²×C₅₃

#418

yes

characteristic

112

C₂²×C₅₃

#459

yes

characteristic

113

C₂²×C₅₃

#460

yes

characteristic

114

C₄×C₅₃

—

115

C₄×C₅₃

#390

yes

characteristic

116

C₄×C₅₃

C₂×#102

yes

characteristic

117

C₄×C₅₃

#459

yes

characteristic

118

C₄×C₅₃

#460

yes

characteristic

119

Dic₅₃

—

120

Dic₅₃

—

121

C₂×#494

—

122

C₂×#364

D₅₃

yes

characteristic

123

C₂×#362

—

124

#2141

—

125

#1772

—

126

C₂×C₅₃

—

127

C₂×C₅₃

—

128

C₂×C₅₃

#1982

yes

characteristic

129

C₂×C₅₃

#2214

yes

—

130

C₂²×Q₁₆

—

131

C₂²×Q₁₆

—

106

—

132

C₂×C₄⋊Q₈

—

106

—

133

C₂×M₃₂

C₂×D₅₃

yes

—

134

#494

—

135

#494

—

106

—

136

C₂×#108

—

137

#427

—

138

#405

—

106

—

139

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

140

#364

—

141

#362

—

106

—

142

#358

—

106

—

143

#339

—

106

—

144

C₅₃

#13262

yes

characteristic, complemented, Sylow

145

C₂²×Q₈

—

106

—

146

C₂×Q₁₆

—

147

C₂×Q₁₆

—

106

—

148

C₂×Q₁₆

—

106

—

149

C₂×Q₁₆

—

106

—

150

C₂×Q₁₆

—

106

—

151

C₂×Q₁₆

—

106

—

152

C₂×Q₁₆

—

106

—

153

C₂×Q₁₆

—

212

—

154

C₂×Q₁₆

—

212

—

155

C₂×Q₁₆

—

212

—

156

C₂×M₁₆

—

106

—

157

C₂²×C₈

D₂₁₂

yes

characteristic

158

C₂²×C₈

C₂²×D₅₃

yes

characteristic

159

C₄⋊Q₈

—

106

—

160

C₄⋊Q₈

—

212

—

161

#117

—

106

—

162

C₂×C₄⋊C₄

—

212

—

163

C₂×C₄²

D₂₁₂

yes

characteristic

164

M₃₂

—

165

M₃₂

—

166

C₂×C₁₆

—

167

#108

—

168

#108

—

106

—

169

#107

—

170

#106

—

106

—

171

#97

—

212

—

172

C₄×C₈

—

212

—

173

C₄×C₈

C₄×D₅₃

yes

—

174

C₄×C₈

C₂²⋊D₅₃

yes

—

175

C₂×Q₈

—

106

—

176

C₂×Q₈

—

212

—

177

C₂×Q₈

—

212

—

178

C₂×Q₈

—

212

—

179

C₂×Q₈

—

212

—

180

C₂×Q₈

—

212

—

181

C₂²×C₄

—

182

C₂²×C₄

—

212

—

183

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

184

Q₁₆

—

106

—

185

Q₁₆

—

106

—

186

Q₁₆

—

212

—

187

Q₁₆

—

212

—

188

Q₁₆

—

212

—

189

Q₁₆

—

212

—

190

M₁₆

—

106

—

191

M₁₆

—

212

—

192

C₂×C₈

—

212

—

193

C₂×C₈

#11719913

yes

—

194

C₂×C₈

D₄₂₄

yes

characteristic

195

C₂×C₈

D₄₂₄

yes

characteristic

196

C₂×C₈

#11719921

yes

—

197

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

198

C₂×C₈

D₄×D₅₃

yes

characteristic

199

C₂×C₈

D₄×D₅₃

yes

characteristic

200

C₂×C₈

#11719949

yes

characteristic

201

C₂×C₈

#11719949

yes

characteristic

202

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

203

C₄⋊C₄

—

106

—

204

C₄⋊C₄

—

212

—

205

C₂²⋊C₄

—

212

—

206

C₄²

—

207

C₄²

—

212

—

208

C₄²

#11719921

yes

—

209

C₁₆

—

210

C₂³

#13133245

yes

characteristic

211

Q₈

—

424

—

212

Q₈

—

424

—

213

Q₈

—

212

—

214

Q₈

—

212

—

215

C₂×C₄

—

216

C₂×C₄

—

217

C₂×C₄

—

424

—

218

C₂×C₄

—

424

—

219

C₂×C₄

—

212

—

220

C₂×C₄

—

212

—

221

C₂×C₄

C₂×#11719921

yes

characteristic

222

C₂×C₄

#13133192

yes

characteristic

223

C₂×C₄

#13133192

yes

characteristic

224

C₂×C₄

C₂×D₄₂₄

yes

characteristic

225

C₂×C₄

C₂×#11719913

yes

characteristic

226

C₂×C₄

#13133245

yes

characteristic

227

C₈

—

212

—

228

C₈

#13133139

yes

—

229

C₈

#13133139

yes

—

230

C₈

#13133179

yes

characteristic

231

C₈

#13133179

yes

characteristic

232

C₈

#13133206

yes

characteristic

233

C₈

#13133213

yes

characteristic

234

C₂²

—

235

C₂²

#16810629

yes

characteristic

236

C₂²

#16810819

yes

characteristic

237

C₂²

#16810824

yes

characteristic

238

C₄

—

239

C₄

—

424

—

240

C₄

—

424

—

241

C₄

#16810524

yes

characteristic

242

C₄

C₂×#13133139

yes

characteristic

243

C₄

#16810819

yes

characteristic

244

C₄

#16810824

yes

characteristic

245

C₂

—

246

C₂

—

247

C₂

#20275641

yes

characteristic

248

C₂

#20276797

yes

—

	subgroup	G/H
center	C₂²	#16810819
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂²×C₅₃	#459
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#13262
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

subgroup

G/H

center

C₂²

#16810819

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#364

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

N	H ≅ G/N	type
10. #16811003	C₄	semidirect
144. C₅₃	#13262	semidirect

H ≅ G/N

type

10. #16811003

C₄

semidirect

144. C₅₃

#13262

semidirect

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

subgroup	index
C₂×#16811003	2
C₂×#16810440	2
#20276436	2
#20274594	2
C₂×C₅₃×#364	2
#13262	53

subgroup

index

C₂×C₅₃×#364

Subgroups — #27302605

Conjugacy classes of subgroups

Complements

Maximal subgroups