Subgroups — C₅₃×#33480 #27240992

Order 13568.

526 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2966	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2169	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2168	C₂	1	2	yes	maximal, complemented
6	D₄×C₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
7	D₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2164	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#2108	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#529	C₄	1	1	yes	characteristic
13	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#446	C₄	1	1	yes	characteristic
18	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#436	C₄	1	1	yes	characteristic
20	C₅₃×#435	C₄	1	2	yes	—
21	C₈×D₄×C₅₃	C₂²	1	2	yes	—
22	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#433	C₄	1	2	yes	—
24	C₅₃×#432	C₄	1	1	yes	characteristic
25	C₅₃×#405	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
31	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₅₃×#346	C₂²	1	2	yes	—
35	C₅₃×#346	C₂²	1	2	yes	—
36	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
37	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
45	#126×C₅₃	C₈	1	2	yes	—
46	#125×C₅₃	C₈	1	1	yes	characteristic
47	#124×C₅₃	C₈	1	1	yes	characteristic
48	#123×C₅₃	C₈	1	2	yes	—
49	C₂²⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
50	#121×C₅₃	C₈	1	1	yes	characteristic
51	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
53	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
54	#117×C₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	M₃₂×C₅₃	—	2	4	no	—
57	C₂×C₁₆×C₅₃	—	2	2	no	—
58	C₂×C₁₆×C₅₃	—	2	4	no	—
59	C₂×C₁₆×C₅₃	—	2	2	no	—
60	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
63	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
64	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
69	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
70	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
71	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
72	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
73	#97×C₅₃	C₂×C₄	1	2	yes	—
74	#97×C₅₃	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₅₃	C₂³	1	2	yes	—
80	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
81	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
82	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
87	M₁₆×C₅₃	—	2	4	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	2	1	no	—
90	C₂×C₈×C₅₃	—	2	1	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
100	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
101	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
102	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
103	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
104	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
105	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
106	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
107	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
108	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
109	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
110	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
112	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
115	C₁₆×C₅₃	—	2	1	no	—
116	C₁₆×C₅₃	—	2	1	no	—
117	C₁₆×C₅₃	—	2	2	no	—
118	C₁₆×C₅₃	—	2	2	no	—
119	C₁₆×C₅₃	#41	1	2	yes	—
120	C₁₆×C₅₃	#41	1	2	yes	—
121	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
122	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
123	Q₈×C₅₃	—	2	2	no	—
124	D₄×C₅₃	—	2	4	no	—
125	D₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	1	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	—	2	2	no	—
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	—	2	2	no	—
145	C₈×C₅₃	#131	1	2	yes	—
146	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
147	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
148	#33480	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	—	2	2	no	—
152	C₂²×C₅₃	—	2	2	no	—
153	C₂²×C₅₃	—	2	2	no	—
154	C₂²×C₅₃	#504	1	2	yes	—
155	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	—	2	2	no	—
161	C₄×C₅₃	—	2	1	no	—
162	C₄×C₅₃	—	2	2	no	—
163	C₄×C₅₃	#504	1	2	yes	—
164	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
165	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
166	#2966	C₂×C₅₃	1	1	yes	characteristic
167	#2183	C₂×C₅₃	1	1	yes	characteristic, complemented
168	Q₈×C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
169	#2169	C₂×C₅₃	1	1	yes	characteristic, complemented
170	#2168	C₂×C₅₃	1	2	yes	complemented
171	D₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
172	D₄×C₁₆	C₂×C₅₃	1	2	yes	complemented
173	#2164	C₂×C₅₃	1	2	yes	complemented
174	#2163	C₂×C₅₃	1	1	yes	characteristic, complemented
175	#2108	C₂×C₅₃	1	2	yes	complemented
176	C₂×C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
177	C₂×C₅₃	—	2	2	no	—
178	C₂×C₅₃	—	4	2	no	—
179	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
180	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
181	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
182	#529	C₄×C₅₃	1	1	yes	characteristic
183	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
184	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
185	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
186	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
187	#446	C₄×C₅₃	1	1	yes	characteristic
188	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic
189	#436	C₄×C₅₃	1	1	yes	characteristic
190	#435	C₄×C₅₃	1	2	yes	—
191	C₈×D₄	C₂²×C₅₃	1	2	yes	—
192	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic
193	#433	C₄×C₅₃	1	2	yes	—
194	#432	C₄×C₅₃	1	1	yes	characteristic
195	#405	C₂²×C₅₃	1	2	yes	—
196	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
197	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
198	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
199	C₄⋊C₁₆	C₂²×C₅₃	1	2	yes	—
200	C₄⋊C₁₆	C₂²×C₅₃	1	2	yes	—
201	C₂²⋊C₁₆	C₂²×C₅₃	1	2	yes	—
202	C₂²⋊C₁₆	C₂²×C₅₃	1	2	yes	—
203	C₂²⋊C₁₆	C₂²×C₅₃	1	2	yes	—
204	#346	C₂²×C₅₃	1	2	yes	—
205	#346	C₂²×C₅₃	1	2	yes	—
206	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
207	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
208	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
209	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
210	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
211	C₅₃	#33480	1	1	yes	characteristic, direct product factor, Sylow
212	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
213	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
214	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
215	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
216	#126	C₈×C₅₃	1	2	yes	—
217	#125	C₈×C₅₃	1	1	yes	characteristic
218	#124	C₈×C₅₃	1	1	yes	characteristic
219	#123	C₈×C₅₃	1	2	yes	—
220	C₂²⋊Q₈	C₈×C₅₃	1	1	yes	characteristic
221	#121	C₈×C₅₃	1	1	yes	characteristic
222	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
223	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
224	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
225	#117	C₂×C₄×C₅₃	1	2	yes	—
226	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
227	M₃₂	—	2	4	no	—
228	C₂×C₁₆	—	2	2	no	—
229	C₂×C₁₆	—	2	4	no	—
230	C₂×C₁₆	—	2	2	no	—
231	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
232	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
233	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
234	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
235	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
236	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
237	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
238	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
239	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
240	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
241	C₂²⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
242	C₂²⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
243	C₂²⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
244	#97	C₂×C₄×C₅₃	1	2	yes	—
245	#97	C₂×C₄×C₅₃	1	2	yes	—
246	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
247	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
250	C₄×C₈	C₂³×C₅₃	1	2	yes	—
251	C₂×Q₈	C₂×C₈×C₅₃	1	1	yes	characteristic
252	C₂×D₄	C₂×C₈×C₅₃	1	2	yes	—
253	C₂×D₄	C₂×C₈×C₅₃	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
256	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
257	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
258	M₁₆	—	2	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
265	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
266	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
267	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
268	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
269	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
270	C₄⋊C₄	C₂×C₈×C₅₃	1	2	yes	—
271	C₄⋊C₄	C₂×C₈×C₅₃	1	2	yes	—
272	C₄⋊C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
273	C₄⋊C₄	C₂×C₈×C₅₃	1	2	yes	—
274	C₄⋊C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
275	C₄⋊C₄	C₂×C₈×C₅₃	1	2	yes	—
276	C₂²⋊C₄	C₂×C₈×C₅₃	1	2	yes	—
277	C₂²⋊C₄	C₂×C₈×C₅₃	1	2	yes	—
278	C₂²⋊C₄	C₂×C₈×C₅₃	1	2	yes	—
279	C₂²⋊C₄	C₂×C₈×C₅₃	1	2	yes	—
280	C₂²⋊C₄	C₂×C₈×C₅₃	1	2	yes	—
281	C₄²	C₂×C₈×C₅₃	1	1	yes	characteristic
282	C₄²	C₂×C₈×C₅₃	1	1	yes	characteristic
283	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
284	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
285	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	—	2	2	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	#41×C₅₃	1	2	yes	—
291	C₁₆	#41×C₅₃	1	2	yes	—
292	C₂³	C₂²×C₈×C₅₃	1	2	yes	—
293	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
294	Q₈	—	2	2	no	—
295	D₄	—	2	4	no	—
296	D₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
307	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
308	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
310	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
311	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	2	no	—
314	C₈	—	2	1	no	—
315	C₈	—	2	2	no	—
316	C₈	#131×C₅₃	1	2	yes	—
317	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
318	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	C₅₃×#504	1	2	yes	—
325	C₂²	C₂³×C₈×C₅₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	C₅₃×#504	1	2	yes	—
334	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
335	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	4	2	no	—
338	C₂	C₈×#41×C₅₃	1	1	yes	characteristic
339	C₂	C₂×C₅₃×#504	1	1	yes	characteristic
340	C₂	C₂×C₅₃×#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆×C₅₃	C₂³
commutator	C₂²	C₂³×C₈×C₅₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂³×C₈
prime radical	#121×C₅₃	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₅₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₅₃×#2183	C₂	semidirect
3. Q₈×C₁₆×C₅₃	C₂	semidirect
4. C₅₃×#2169	C₂	semidirect
5. C₅₃×#2168	C₂	semidirect
6. D₄×C₁₆×C₅₃	C₂	semidirect
7. D₄×C₁₆×C₅₃	C₂	semidirect
8. C₅₃×#2164	C₂	semidirect
9. C₅₃×#2163	C₂	semidirect
10. C₅₃×#2108	C₂	semidirect
11. C₂×C₄×C₁₆×C₅₃	C₂	semidirect
148. #33480	C₅₃	direct (N × H)
167. #2183	C₂×C₅₃	semidirect
168. Q₈×C₁₆	C₂×C₅₃	semidirect
169. #2169	C₂×C₅₃	semidirect
170. #2168	C₂×C₅₃	semidirect
171. D₄×C₁₆	C₂×C₅₃	semidirect
172. D₄×C₁₆	C₂×C₅₃	semidirect
173. #2164	C₂×C₅₃	semidirect
174. #2163	C₂×C₅₃	semidirect
175. #2108	C₂×C₅₃	semidirect
176. C₂×C₄×C₁₆	C₂×C₅₃	semidirect
211. C₅₃	#33480	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2966	2
C₅₃×#2183	2
Q₈×C₁₆×C₅₃	2
C₅₃×#2169	2
C₅₃×#2168	2
D₄×C₁₆×C₅₃	2
C₅₃×#2164	2
C₅₃×#2163	2
C₅₃×#2108	2
C₂×C₄×C₁₆×C₅₃	2
#33480	53

Subgroups — C53×#33480 #27240992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33480 #27240992