Group #27281463 — subgroups

Subgroups — #27281463

Order 13568.

506 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 139 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	#20283796	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	Q₈×#88322	—	2	4	no	—
7	#16810676	—	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	C₄×#88322	—	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	2	no	—
34	C₄⋊Q₈×C₅₃	—	4	2	no	—
35	C₄×Q₈×C₅₃	—	2	2	no	—
36	C₄×Q₈×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	—	2	1	no	—
44	C₂×C₄²×C₅₃	D₄	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	#95×C₅₃	—	2	4	no	—
47	#95×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂×Q₈×C₅₃	—	4	2	no	—
51	C₂×Q₈×C₅₃	—	4	1	no	—
52	C₂×Q₈×C₅₃	—	4	1	no	—
53	C₂×Q₈×C₅₃	—	4	2	no	—
54	C₂×Q₈×C₅₃	D₈	1	2	yes	—
55	C₂×Q₈×C₅₃	SD₁₆	1	2	yes	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₅₃	—	2	4	no	—
64	C₄⋊C₄×C₅₃	—	2	4	no	—
65	C₄⋊C₄×C₅₃	—	2	2	no	—
66	C₄⋊C₄×C₅₃	—	2	4	no	—
67	C₄⋊C₄×C₅₃	—	4	8	no	—
68	C₄⋊C₄×C₅₃	—	4	1	no	—
69	C₄⋊C₄×C₅₃	—	4	1	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	—	4	1	no	—
73	C₄²×C₅₃	—	4	1	no	—
74	C₄²×C₅₃	C₂×D₄	1	2	yes	—
75	C₄²×C₅₃	#41	1	2	yes	—
76	#11719917	—	4	2	no	—
77	#11719917	—	4	2	no	—
78	C₂×#88322	—	2	4	no	—
79	C₂×#88322	—	4	2	no	—
80	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	Q₈×C₅₃	—	2	4	no	—
82	Q₈×C₅₃	—	2	4	no	—
83	Q₈×C₅₃	—	8	2	no	—
84	Q₈×C₅₃	—	8	2	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	4	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	4	1	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	—	4	2	no	—
97	C₂×C₄×C₅₃	#102	1	4	yes	—
98	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#121	1	2	yes	—
101	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#135	1	2	yes	—
105	#88322	—	4	4	no	—
106	#88322	—	8	2	no	—
107	#9394	—	53	1	no	maximal, Sylow
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	—	2	1	no	—
110	C₂²×C₅₃	#386	1	1	yes	characteristic
111	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
112	C₂²×C₅₃	#416	1	1	yes	characteristic
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	2	4	no	—
115	C₄×C₅₃	—	2	2	no	—
116	C₄×C₅₃	—	4	4	no	—
117	C₄×C₅₃	—	4	2	no	—
118	C₄×C₅₃	—	8	2	no	—
119	C₄×C₅₃	#456	1	2	yes	—
120	C₄×C₅₃	#465	1	2	yes	—
121	C₄×C₅₃	#466	1	2	yes	—
122	#2982	—	53	1	no	—
123	#2546	D₅₃	1	1	yes	characteristic
124	#1801	—	53	1	no	—
125	#1546	—	53	2	no	—
126	#1303	—	53	2	no	—
127	C₂×C₅₃	—	2	1	no	—
128	C₂×C₅₃	—	2	1	no	—
129	C₂×C₅₃	#1886	1	1	yes	characteristic
130	C₂×C₅₃	#1893	1	1	yes	characteristic
131	C₂×C₅₃	#1915	1	1	yes	characteristic
132	C₂×C₄⋊Q₈	—	2	1	no	—
133	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
134	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
135	C₈×Q₈	—	106	4	no	—
136	#431	—	53	2	no	—
137	#431	—	106	2	no	—
138	#423	—	53	2	no	—
139	C₂×#101	—	106	1	no	—
140	#398	Dic₅₃	1	4	yes	—
141	#391	—	2	1	no	—
142	#389	—	2	2	no	—
143	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
144	#343	—	106	1	no	—
145	#332	—	106	4	no	—
146	C₅₃	#9394	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	—	2	1	no	—
148	C₂²×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
149	C₂×M₁₆	—	106	2	no	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	4	2	no	—
153	C₄⋊Q₈	—	4	2	no	—
154	C₄×Q₈	—	2	2	no	—
155	C₄×Q₈	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	—	2	4	no	—
159	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
160	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
161	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
164	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
165	C₄⋊C₈	—	53	4	no	—
166	C₄⋊C₈	—	106	4	no	—
167	C₄⋊C₈	—	106	2	no	—
168	C₄⋊C₈	—	106	2	no	—
169	#101	—	212	2	no	—
170	C₂²⋊C₈	—	106	2	no	—
171	C₄×C₈	—	106	4	no	—
172	C₄×C₈	—	106	2	no	—
173	#95	—	2	4	no	—
174	#95	C₂×Dic₅₃	1	2	yes	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	1	no	—
179	C₂×Q₈	—	4	1	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×Q₈	#11719922	1	2	yes	—
182	C₂×Q₈	#11719923	1	2	yes	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
186	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
187	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
190	M₁₆	—	212	2	no	—
191	M₁₆	—	212	2	no	—
192	C₂×C₈	—	106	4	no	—
193	C₂×C₈	—	212	2	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	8	no	—
199	C₄⋊C₄	—	4	1	no	—
200	C₄⋊C₄	—	4	1	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	4	1	no	—
204	C₄²	—	4	1	no	—
205	C₄²	D₄×D₅₃	1	2	yes	—
206	C₄²	#11719947	1	2	yes	—
207	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
208	Q₈	—	2	4	no	—
209	Q₈	—	2	4	no	—
210	Q₈	—	8	2	no	—
211	Q₈	—	8	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	#13133138	1	4	yes	—
225	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
226	C₂×C₄	#13133185	1	1	yes	characteristic
227	C₂×C₄	#13133189	1	2	yes	—
228	C₂×C₄	#13133198	1	1	yes	characteristic
229	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
230	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
231	C₂×C₄	#13133237	1	2	yes	—
232	C₈	—	424	2	no	—
233	C₈	—	212	4	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	#16810503	1	1	yes	characteristic
237	C₂²	C₂×#13133138	1	1	yes	characteristic
238	C₂²	#16810619	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	4	no	—
242	C₄	—	4	4	no	—
243	C₄	—	4	2	no	—
244	C₄	—	8	2	no	—
245	C₄	#16810805	1	2	yes	—
246	C₄	#16810850	1	2	yes	—
247	C₄	#16810857	1	2	yes	—
248	C₂	—	2	1	no	—
249	C₂	—	2	1	no	—
250	C₂	#20275109	1	1	yes	characteristic
251	C₂	#20275144	1	1	yes	characteristic
252	C₂	#20275268	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20283796

C₂

yes

maximal, characteristic

#20274711

C₂

yes

maximal, characteristic

#20273387

C₂

yes

maximal

#20272618

C₂

yes

maximal

C₅₃×#2546

C₂

yes

maximal, characteristic

Q₈×#88322

—

#16810676

—

#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

—

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

D₈

yes

—

C₂×Q₈×C₅₃

SD₁₆

yes

—

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

—

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

#102

yes

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#120

yes

characteristic

100

C₂×C₄×C₅₃

#121

yes

—

101

C₂×C₄×C₅₃

#123

yes

characteristic

102

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

103

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

104

C₂×C₄×C₅₃

#135

yes

—

105

#88322

—

106

#88322

—

107

#9394

—

maximal, Sylow

108

C₂²×C₅₃

—

109

C₂²×C₅₃

—

110

C₂²×C₅₃

#386

yes

characteristic

111

C₂²×C₅₃

C₂×#102

yes

characteristic

112

C₂²×C₅₃

#416

yes

characteristic

113

C₄×C₅₃

—

114

C₄×C₅₃

—

115

C₄×C₅₃

—

116

C₄×C₅₃

—

117

C₄×C₅₃

—

118

C₄×C₅₃

—

119

C₄×C₅₃

#456

yes

—

120

C₄×C₅₃

#465

yes

—

121

C₄×C₅₃

#466

yes

—

122

#2982

—

123

#2546

D₅₃

yes

characteristic

124

#1801

—

125

#1546

—

126

#1303

—

127

C₂×C₅₃

—

128

C₂×C₅₃

—

129

C₂×C₅₃

#1886

yes

characteristic

130

C₂×C₅₃

#1893

yes

characteristic

131

C₂×C₅₃

#1915

yes

characteristic

132

C₂×C₄⋊Q₈

—

133

C₂×C₄⋊Q₈

C₂×D₅₃

yes

characteristic

134

C₂×C₄×Q₈

C₂×D₅₃

yes

characteristic

135

C₈×Q₈

—

106

—

136

#431

—

137

#431

—

106

—

138

#423

—

139

C₂×#101

—

106

—

140

#398

Dic₅₃

yes

—

141

#391

—

142

#389

—

143

C₄×C₄⋊C₄

C₂×D₅₃

yes

characteristic

144

#343

—

106

—

145

#332

—

106

—

146

C₅₃

#9394

yes

characteristic, complemented, Sylow

147

C₂²×Q₈

—

148

C₂²×Q₈

C₂²⋊D₅₃

yes

characteristic

149

C₂×M₁₆

—

106

—

150

C₄⋊Q₈

—

151

C₄⋊Q₈

—

152

C₄⋊Q₈

—

153

C₄⋊Q₈

—

154

C₄×Q₈

—

155

C₄×Q₈

—

156

C₂×C₄⋊C₄

—

157

C₂×C₄⋊C₄

—

158

C₂×C₄⋊C₄

—

159

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

—

160

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

—

161

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

162

C₂×C₄²

—

163

C₂×C₄²

C₂²⋊D₅₃

yes

—

164

C₂×C₄²

C₂²×D₅₃

yes

characteristic

165

C₄⋊C₈

—

166

C₄⋊C₈

—

106

—

167

C₄⋊C₈

—

106

—

168

C₄⋊C₈

—

106

—

169

#101

—

212

—

170

C₂²⋊C₈

—

106

—

171

C₄×C₈

—

106

—

172

C₄×C₈

—

106

—

173

#95

—

174

#95

C₂×Dic₅₃

yes

—

175

C₂×Q₈

—

176

C₂×Q₈

—

177

C₂×Q₈

—

178

C₂×Q₈

—

179

C₂×Q₈

—

180

C₂×Q₈

—

181

C₂×Q₈

#11719922

yes

—

182

C₂×Q₈

#11719923

yes

—

183

C₂²×C₄

—

184

C₂²×C₄

—

185

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

186

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

187

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

188

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

189

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

190

M₁₆

—

212

—

191

M₁₆

—

212

—

192

C₂×C₈

—

106

—

193

C₂×C₈

—

212

—

194

C₄⋊C₄

—

195

C₄⋊C₄

—

196

C₄⋊C₄

—

197

C₄⋊C₄

—

198

C₄⋊C₄

—

199

C₄⋊C₄

—

200

C₄⋊C₄

—

201

C₄²

—

202

C₄²

—

203

C₄²

—

204

C₄²

—

205

C₄²

D₄×D₅₃

yes

—

206

C₄²

#11719947

yes

—

207

C₂³

C₂×C₂²⋊Dic₅₃

yes

characteristic

208

Q₈

—

209

Q₈

—

210

Q₈

—

211

Q₈

—

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

—

222

C₂×C₄

—

223

C₂×C₄

—

224

C₂×C₄

#13133138

yes

—

225

C₂×C₄

D₄×Dic₅₃

yes

—

226

C₂×C₄

#13133185

yes

characteristic

227

C₂×C₄

#13133189

yes

—

228

C₂×C₄

#13133198

yes

characteristic

229

C₂×C₄

C₂×#11719922

yes

characteristic

230

C₂×C₄

C₂×#11719923

yes

characteristic

231

C₂×C₄

#13133237

yes

—

232

C₈

—

424

—

233

C₈

—

212

—

234

C₂²

—

235

C₂²

—

236

C₂²

#16810503

yes

characteristic

237

C₂²

C₂×#13133138

yes

characteristic

238

C₂²

#16810619

yes

characteristic

239

C₄

—

240

C₄

—

241

C₄

—

242

C₄

—

243

C₄

—

244

C₄

—

245

C₄

#16810805

yes

—

246

C₄

#16810850

yes

—

247

C₄

#16810857

yes

—

248

C₂

—

249

C₂

—

250

C₂

#20275109

yes

characteristic

251

C₂

#20275144

yes

characteristic

252

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	#9394
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
146. C₅₃	#9394	semidirect

H ≅ G/N

type

146. C₅₃

#9394

semidirect

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

subgroup	index
#20283796	2
#20274711	2
#20273387	2
#20272618	2
C₅₃×#2546	2
#9394	53

subgroup

index

C₅₃×#2546

Subgroups — #27281463

Conjugacy classes of subgroups

Complements

Maximal subgroups