Group #27281503 — subgroups

Subgroups — #27281503

Order 13568.

486 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 123 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	#20283800	C₂	1	1	yes	maximal, characteristic
2	#20274711	C₂	1	1	yes	maximal, characteristic
3	#20273387	C₂	1	2	yes	maximal
4	#20272618	C₂	1	2	yes	maximal
5	C₅₃×#2546	C₂	1	1	yes	maximal, characteristic
6	#16810754	—	2	4	no	—
7	#16810680	—	2	2	no	—
8	#16810676	C₂²	1	2	yes	—
9	#16810647	C₂²	1	2	yes	—
10	C₂×#13133136	—	2	1	no	—
11	#16810392	—	2	1	no	—
12	#16810369	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#398	C₄	1	4	yes	—
17	C₅₃×#391	—	2	1	no	—
18	C₅₃×#389	—	2	2	no	—
19	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719917	—	2	2	no	—
21	C₄⋊#88322	—	2	4	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	D₄	1	4	yes	—
25	#13133136	—	4	2	no	—
26	C₂²⋊#88322	—	2	2	no	—
27	#13133128	—	2	4	no	—
28	C₄×#88322	—	2	2	no	—
29	C₂²×Q₈×C₅₃	—	2	1	no	—
30	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₅₃	—	2	2	no	—
32	C₄⋊Q₈×C₅₃	—	2	2	no	—
33	C₄⋊Q₈×C₅₃	—	4	4	no	—
34	C₄×Q₈×C₅₃	—	2	2	no	—
35	C₄×Q₈×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	—	2	1	no	—
43	C₂×C₄²×C₅₃	D₄	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	#95×C₅₃	—	2	4	no	—
46	#95×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×Q₈×C₅₃	—	2	2	no	—
48	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	2	1	no	—
50	C₂×Q₈×C₅₃	—	2	1	no	—
51	C₂×Q₈×C₅₃	—	4	2	no	—
52	C₂×Q₈×C₅₃	—	4	2	no	—
53	C₂×Q₈×C₅₃	—	4	1	no	—
54	C₂×Q₈×C₅₃	—	4	1	no	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₅₃	—	2	4	no	—
63	C₄⋊C₄×C₅₃	—	2	4	no	—
64	C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₄⋊C₄×C₅₃	—	4	8	no	—
67	C₄⋊C₄×C₅₃	—	4	1	no	—
68	C₄⋊C₄×C₅₃	—	4	1	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	4	2	no	—
72	C₄²×C₅₃	C₂×D₄	1	2	yes	—
73	C₄²×C₅₃	#41	1	2	yes	—
74	#11719917	—	4	2	no	—
75	#11719917	—	4	2	no	—
76	C₂×#88322	—	2	4	no	—
77	C₂×#88322	—	4	2	no	—
78	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	Q₈×C₅₃	—	4	2	no	—
80	Q₈×C₅₃	—	4	2	no	—
81	Q₈×C₅₃	—	8	2	no	—
82	Q₈×C₅₃	—	8	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	4	1	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#121	1	2	yes	—
99	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
100	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
102	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
103	#88322	—	4	4	no	—
104	#88322	—	8	2	no	—
105	#9402	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	#386	1	1	yes	characteristic
109	C₂²×C₅₃	#417	1	1	yes	characteristic
110	C₂²×C₅₃	#418	1	1	yes	characteristic
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	2	no	—
113	C₄×C₅₃	—	4	4	no	—
114	C₄×C₅₃	—	4	2	no	—
115	C₄×C₅₃	—	4	2	no	—
116	C₄×C₅₃	—	8	2	no	—
117	C₄×C₅₃	#456	1	2	yes	—
118	C₄×C₅₃	#468	1	2	yes	—
119	C₄×C₅₃	#469	1	2	yes	—
120	#2983	—	53	1	no	—
121	#2546	D₅₃	1	1	yes	characteristic
122	#1801	—	53	1	no	—
123	#1546	—	53	2	no	—
124	#1303	—	53	2	no	—
125	C₂×C₅₃	—	2	1	no	—
126	C₂×C₅₃	—	2	1	no	—
127	C₂×C₅₃	#1886	1	1	yes	characteristic
128	C₂×C₅₃	#1895	1	1	yes	characteristic
129	C₂×C₅₃	#1915	1	1	yes	characteristic
130	C₂×C₄⋊Q₈	—	2	1	no	—
131	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
132	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
133	#446	—	106	4	no	—
134	#432	—	106	2	no	—
135	#431	—	53	2	no	—
136	#423	—	53	2	no	—
137	C₂×#101	—	106	1	no	—
138	#398	Dic₅₃	1	4	yes	—
139	#391	—	2	1	no	—
140	#389	—	2	2	no	—
141	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
142	#343	—	106	1	no	—
143	#332	—	106	4	no	—
144	C₅₃	#9402	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	—	2	1	no	—
146	C₂²×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
147	C₂×M₁₆	—	106	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	4	4	no	—
151	C₄×Q₈	—	2	2	no	—
152	C₄×Q₈	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
157	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
158	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂×C₄²	—	2	1	no	—
160	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
161	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
162	C₄⋊C₈	—	53	4	no	—
163	C₄⋊C₈	—	106	4	no	—
164	C₄⋊C₈	—	106	2	no	—
165	C₄⋊C₈	—	106	2	no	—
166	#101	—	212	2	no	—
167	C₂²⋊C₈	—	106	2	no	—
168	#97	—	106	4	no	—
169	C₄×C₈	—	106	2	no	—
170	#95	—	2	4	no	—
171	#95	C₂×Dic₅₃	1	2	yes	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	—	2	1	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	—	4	1	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	4	1	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
183	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
184	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
187	M₁₆	—	212	2	no	—
188	M₁₆	—	212	2	no	—
189	C₂×C₈	—	106	4	no	—
190	C₂×C₈	—	212	2	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	8	no	—
196	C₄⋊C₄	—	4	1	no	—
197	C₄⋊C₄	—	4	1	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	4	2	no	—
201	C₄²	D₄×D₅₃	1	2	yes	—
202	C₄²	#11719947	1	2	yes	—
203	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
204	Q₈	—	4	2	no	—
205	Q₈	—	4	2	no	—
206	Q₈	—	8	2	no	—
207	Q₈	—	8	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
222	C₂×C₄	#13133185	1	1	yes	characteristic
223	C₂×C₄	#13133189	1	2	yes	—
224	C₂×C₄	#13133198	1	1	yes	characteristic
225	C₂×C₄	#13133242	1	1	yes	characteristic
226	C₂×C₄	#13133243	1	2	yes	—
227	C₂×C₄	#13133247	1	1	yes	characteristic
228	C₈	—	424	2	no	—
229	C₈	—	212	4	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	#16810503	1	1	yes	characteristic
233	C₂²	#16810624	1	1	yes	characteristic
234	C₂²	#16810628	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	1	no	—
237	C₄	—	4	4	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	8	2	no	—
241	C₄	#16810805	1	2	yes	—
242	C₄	#16810867	1	2	yes	—
243	C₄	#16810874	1	2	yes	—
244	C₂	—	2	1	no	—
245	C₂	—	2	1	no	—
246	C₂	#20275109	1	1	yes	characteristic
247	C₂	#20275154	1	1	yes	characteristic
248	C₂	#20275268	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20283800

C₂

yes

maximal, characteristic

#20274711

C₂

yes

maximal, characteristic

#20273387

C₂

yes

maximal

#20272618

C₂

yes

maximal

C₅₃×#2546

C₂

yes

maximal, characteristic

#16810754

—

#16810680

—

#16810676

C₂²

yes

—

#16810647

C₂²

yes

—

C₂×#13133136

—

#16810392

—

#16810369

—

C₂×C₄⋊Q₈×C₅₃

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₅₃×#398

C₄

yes

—

C₅₃×#391

—

C₅₃×#389

—

C₄×C₄⋊C₄×C₅₃

C₂²

yes

—

—

—

—

D₄

yes

—

—

—

—

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

D₄

yes

characteristic

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

#95×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

—

—

—

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

#123

yes

characteristic

100

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

characteristic

101

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

102

C₂×C₄×C₅₃

#137

yes

characteristic

103

#88322

—

104

#88322

—

105

#9402

—

maximal, Sylow

106

C₂²×C₅₃

—

107

C₂²×C₅₃

—

108

C₂²×C₅₃

#386

yes

characteristic

109

C₂²×C₅₃

#417

yes

characteristic

110

C₂²×C₅₃

#418

yes

characteristic

111

C₄×C₅₃

—

112

C₄×C₅₃

—

113

C₄×C₅₃

—

114

C₄×C₅₃

—

115

C₄×C₅₃

—

116

C₄×C₅₃

—

117

C₄×C₅₃

#456

yes

—

118

C₄×C₅₃

#468

yes

—

119

C₄×C₅₃

#469

yes

—

120

#2983

—

121

#2546

D₅₃

yes

characteristic

122

#1801

—

123

#1546

—

124

#1303

—

125

C₂×C₅₃

—

126

C₂×C₅₃

—

127

C₂×C₅₃

#1886

yes

characteristic

128

C₂×C₅₃

#1895

yes

characteristic

129

C₂×C₅₃

#1915

yes

characteristic

130

C₂×C₄⋊Q₈

—

131

C₂×C₄⋊Q₈

C₂×D₅₃

yes

characteristic

132

C₂×C₄×Q₈

C₂×D₅₃

yes

characteristic

133

#446

—

106

—

134

#432

—

106

—

135

#431

—

136

#423

—

137

C₂×#101

—

106

—

138

#398

Dic₅₃

yes

—

139

#391

—

140

#389

—

141

C₄×C₄⋊C₄

C₂×D₅₃

yes

characteristic

142

#343

—

106

—

143

#332

—

106

—

144

C₅₃

#9402

yes

characteristic, complemented, Sylow

145

C₂²×Q₈

—

146

C₂²×Q₈

C₂²⋊D₅₃

yes

characteristic

147

C₂×M₁₆

—

106

—

148

C₄⋊Q₈

—

149

C₄⋊Q₈

—

150

C₄⋊Q₈

—

151

C₄×Q₈

—

152

C₄×Q₈

—

153

C₂×C₄⋊C₄

—

154

C₂×C₄⋊C₄

—

155

C₂×C₄⋊C₄

—

156

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

—

157

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

—

158

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

159

C₂×C₄²

—

160

C₂×C₄²

C₂²⋊D₅₃

yes

—

161

C₂×C₄²

C₂²×D₅₃

yes

characteristic

162

C₄⋊C₈

—

163

C₄⋊C₈

—

106

—

164

C₄⋊C₈

—

106

—

165

C₄⋊C₈

—

106

—

166

#101

—

212

—

167

C₂²⋊C₈

—

106

—

168

#97

—

106

—

169

C₄×C₈

—

106

—

170

#95

—

171

#95

C₂×Dic₅₃

yes

—

172

C₂×Q₈

—

173

C₂×Q₈

—

174

C₂×Q₈

—

175

C₂×Q₈

—

176

C₂×Q₈

—

177

C₂×Q₈

—

178

C₂×Q₈

—

179

C₂×Q₈

—

180

C₂²×C₄

—

181

C₂²×C₄

—

182

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

183

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

184

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

185

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

186

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

187

M₁₆

—

212

—

188

M₁₆

—

212

—

189

C₂×C₈

—

106

—

190

C₂×C₈

—

212

—

191

C₄⋊C₄

—

192

C₄⋊C₄

—

193

C₄⋊C₄

—

194

C₄⋊C₄

—

195

C₄⋊C₄

—

196

C₄⋊C₄

—

197

C₄⋊C₄

—

198

C₄²

—

199

C₄²

—

200

C₄²

—

201

C₄²

D₄×D₅₃

yes

—

202

C₄²

#11719947

yes

—

203

C₂³

C₂×C₂²⋊Dic₅₃

yes

characteristic

204

Q₈

—

205

Q₈

—

206

Q₈

—

207

Q₈

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

—

216

C₂×C₄

—

217

C₂×C₄

—

218

C₂×C₄

—

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

D₄×Dic₅₃

yes

—

222

C₂×C₄

#13133185

yes

characteristic

223

C₂×C₄

#13133189

yes

—

224

C₂×C₄

#13133198

yes

characteristic

225

C₂×C₄

#13133242

yes

characteristic

226

C₂×C₄

#13133243

yes

—

227

C₂×C₄

#13133247

yes

characteristic

228

C₈

—

424

—

229

C₈

—

212

—

230

C₂²

—

231

C₂²

—

232

C₂²

#16810503

yes

characteristic

233

C₂²

#16810624

yes

characteristic

234

C₂²

#16810628

yes

characteristic

235

C₄

—

236

C₄

—

237

C₄

—

238

C₄

—

239

C₄

—

240

C₄

—

241

C₄

#16810805

yes

—

242

C₄

#16810867

yes

—

243

C₄

#16810874

yes

—

244

C₂

—

245

C₂

—

246

C₂

#20275109

yes

characteristic

247

C₂

#20275154

yes

characteristic

248

C₂

#20275268

yes

characteristic

	subgroup	G/H
center	C₂²	#16810503
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2546	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2546	D₅₃
hypocenter	C₅₃	#9402
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2546

subgroup

G/H

center

C₂²

#16810503

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2546

C₂

socle

C₂²×C₅₃

#386

prime radical

C₂³×C₅₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2546

N	H ≅ G/N	type
144. C₅₃	#9402	semidirect

H ≅ G/N

type

144. C₅₃

#9402

semidirect

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

subgroup	index
#20283800	2
#20274711	2
#20273387	2
#20272618	2
C₅₃×#2546	2
#9402	53

subgroup

index

C₅₃×#2546

Subgroups — #27281503

Conjugacy classes of subgroups

Complements

Maximal subgroups