Subgroups — C₅₃×#17799 #27225311

Order 13568.

666 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2989	C₂	1	2	yes	maximal
2	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1569	C₂	1	4	yes	maximal, complemented
4	C₅₃×#1568	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1567	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1565	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
10	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
11	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#446	—	2	2	no	—
13	C₅₃×#436	—	2	2	no	—
14	C₅₃×#435	—	2	4	no	—
15	C₅₃×#432	C₂²	1	4	yes	—
16	C₅₃×#431	C₂²	1	2	yes	—
17	C₅₃×#423	C₂²	1	4	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₅₃×#405	C₄	1	1	yes	characteristic
20	C₅₃×#405	C₄	1	2	yes	—
21	C₅₃×#405	C₄	1	2	yes	—
22	C₅₃×#405	C₄	1	1	yes	characteristic
23	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
27	C₅₃×#335	C₂²	1	2	yes	—
28	C₅₃×#335	C₂²	1	2	yes	—
29	C₅₃×#335	C₂²	1	1	yes	characteristic
30	C₅₃×#335	C₂²	1	1	yes	characteristic
31	C₅₃×#335	C₂²	1	4	yes	—
32	C₅₃×#334	C₂²	1	2	yes	—
33	C₅₃×#334	C₂²	1	4	yes	—
34	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
35	#131×C₅₃	—	2	4	no	—
36	#131×C₅₃	—	2	4	no	—
37	C₂×M₁₆×C₅₃	—	2	4	no	—
38	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	—	2	2	no	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
48	C₄×Q₈×C₅₃	Q₈	1	1	yes	characteristic
49	C₄×D₄×C₅₃	D₄	1	2	yes	—
50	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
51	C₄×D₄×C₅₃	Q₈	1	1	yes	characteristic
52	C₄×D₄×C₅₃	Q₈	1	2	yes	—
53	#117×C₅₃	C₂×C₄	1	2	yes	—
54	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
57	C₄⋊C₈×C₅₃	—	2	2	no	—
58	C₄⋊C₈×C₅₃	—	2	4	no	—
59	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
60	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
61	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
62	C₂²⋊C₈×C₅₃	—	2	2	no	—
63	C₂²⋊C₈×C₅₃	—	2	4	no	—
64	#97×C₅₃	—	2	4	no	—
65	#97×C₅₃	C₂×C₄	1	2	yes	—
66	#97×C₅₃	C₂×C₄	1	2	yes	—
67	#97×C₅₃	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₅₃	—	2	2	no	—
69	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	2	yes	—
75	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
76	#41×C₅₃	—	2	2	no	—
77	#41×C₅₃	—	2	2	no	—
78	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
80	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
87	M₁₆×C₅₃	—	2	2	no	—
88	M₁₆×C₅₃	—	2	4	no	—
89	M₁₆×C₅₃	—	2	4	no	—
90	M₁₆×C₅₃	—	8	4	no	—
91	M₁₆×C₅₃	M₁₆	1	4	yes	complemented
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	2	4	no	—
94	C₂×C₈×C₅₃	—	2	4	no	—
95	C₂×C₈×C₅₃	—	2	2	no	—
96	C₂×C₈×C₅₃	—	2	2	no	—
97	C₂×C₈×C₅₃	—	2	4	no	—
98	C₂×C₈×C₅₃	—	8	2	no	—
99	C₂×C₈×C₅₃	M₁₆	1	4	yes	complemented
100	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
107	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
108	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
109	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
111	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
112	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
115	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂×D₄	1	2	yes	—
117	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
118	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
119	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
120	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	Q₈×C₅₃	—	2	2	no	—
123	D₄×C₅₃	—	2	4	no	—
124	D₄×C₅₃	—	2	2	no	—
125	C₂×C₄×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	—	4	1	no	—
132	C₂×C₄×C₅₃	—	4	2	no	—
133	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
140	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
141	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	—	2	1	no	—
146	C₈×C₅₃	—	2	2	no	—
147	C₈×C₅₃	—	4	4	no	—
148	C₈×C₅₃	—	4	2	no	—
149	C₈×C₅₃	—	4	2	no	—
150	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
151	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
152	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
153	#17799	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂²×C₅₃	—	2	1	no	—
155	C₂²×C₅₃	—	2	1	no	—
156	C₂²×C₅₃	—	2	2	no	—
157	C₂²×C₅₃	—	2	2	no	—
158	C₂²×C₅₃	—	2	2	no	—
159	C₂²×C₅₃	#423	1	2	yes	—
160	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₄×C₅₃	—	2	2	no	—
162	C₄×C₅₃	—	2	1	no	—
163	C₄×C₅₃	—	2	2	no	—
164	C₄×C₅₃	—	2	1	no	—
165	C₄×C₅₃	—	2	1	no	—
166	C₄×C₅₃	—	2	2	no	—
167	C₄×C₅₃	#423	1	2	yes	—
168	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
169	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
170	C₄×C₅₃	#576	1	1	yes	characteristic
171	C₄×C₅₃	#578	1	1	yes	characteristic
172	#2989	C₂×C₅₃	1	2	yes	—
173	C₄×#131	C₂×C₅₃	1	1	yes	characteristic
174	#1569	C₂×C₅₃	1	4	yes	complemented
175	#1568	C₂×C₅₃	1	2	yes	complemented
176	#1567	C₂×C₅₃	1	2	yes	complemented
177	#1565	C₂×C₅₃	1	2	yes	complemented
178	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
179	C₂×#335	C₂×C₅₃	1	1	yes	characteristic, complemented
180	C₂×C₅₃	—	2	2	no	—
181	C₂×C₅₃	—	4	2	no	—
182	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
183	C₂×C₅₃	#2913	1	1	yes	characteristic
184	C₂×C₅₃	#2916	1	1	yes	characteristic
185	C₂×#131	C₄×C₅₃	1	1	yes	characteristic
186	C₂×#131	C₄×C₅₃	1	1	yes	characteristic
187	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
188	#446	—	2	2	no	—
189	#436	—	2	2	no	—
190	#435	—	2	4	no	—
191	#432	C₂²×C₅₃	1	4	yes	—
192	#431	C₂²×C₅₃	1	2	yes	—
193	#423	C₂²×C₅₃	1	4	yes	—
194	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
195	#405	C₄×C₅₃	1	1	yes	characteristic
196	#405	C₄×C₅₃	1	2	yes	—
197	#405	C₄×C₅₃	1	2	yes	—
198	#405	C₄×C₅₃	1	1	yes	characteristic
199	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
200	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
201	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
202	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
203	#335	C₂²×C₅₃	1	2	yes	—
204	#335	C₂²×C₅₃	1	2	yes	—
205	#335	C₂²×C₅₃	1	1	yes	characteristic
206	#335	C₂²×C₅₃	1	1	yes	characteristic
207	#335	C₂²×C₅₃	1	4	yes	—
208	#334	C₂²×C₅₃	1	2	yes	—
209	#334	C₂²×C₅₃	1	4	yes	—
210	C₅₃	#17799	1	1	yes	characteristic, direct product factor, Sylow
211	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
212	#131	—	2	4	no	—
213	#131	—	2	4	no	—
214	C₂×M₁₆	—	2	4	no	—
215	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
216	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
217	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
218	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
219	C₂²×C₈	—	2	2	no	—
220	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
221	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
222	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
223	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
224	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
225	C₄×Q₈	Q₈×C₅₃	1	1	yes	characteristic
226	C₄×D₄	D₄×C₅₃	1	2	yes	—
227	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
228	C₄×D₄	Q₈×C₅₃	1	2	yes	—
229	C₄×D₄	Q₈×C₅₃	1	1	yes	characteristic
230	#117	C₂×C₄×C₅₃	1	2	yes	—
231	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
232	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
233	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
234	C₄⋊C₈	—	2	2	no	—
235	C₄⋊C₈	—	2	4	no	—
236	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
237	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
238	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
239	C₂²⋊C₈	—	2	2	no	—
240	C₂²⋊C₈	—	2	4	no	—
241	#97	—	2	4	no	—
242	#97	C₂×C₄×C₅₃	1	2	yes	—
243	#97	C₂×C₄×C₅₃	1	2	yes	—
244	#97	C₂×C₄×C₅₃	1	2	yes	—
245	C₄×C₈	—	2	2	no	—
246	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
247	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
248	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
249	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
250	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
251	C₄×C₈	C₂³×C₅₃	1	2	yes	—
252	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
253	#41	—	2	2	no	—
254	#41	—	2	2	no	—
255	C₂×Q₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
256	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
257	C₂×D₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
258	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
259	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
262	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
263	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
264	M₁₆	—	2	2	no	—
265	M₁₆	—	2	4	no	—
266	M₁₆	—	2	4	no	—
267	M₁₆	—	8	4	no	—
268	M₁₆	M₁₆×C₅₃	1	4	yes	complemented
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	4	no	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	8	2	no	—
276	C₂×C₈	M₁₆×C₅₃	1	4	yes	complemented
277	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
278	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
279	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
280	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
281	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
282	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
283	C₄⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
284	C₄⋊C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
285	C₄⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
286	C₄⋊C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
287	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
288	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
289	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
290	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
291	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
292	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
293	C₄²	C₂×D₄×C₅₃	1	2	yes	—
294	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
295	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
296	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
297	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
298	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
299	Q₈	—	2	2	no	—
300	D₄	—	2	4	no	—
301	D₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
311	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
312	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
314	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
315	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
316	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
317	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
318	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
319	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
320	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	2	2	no	—
324	C₈	—	4	4	no	—
325	C₈	—	4	2	no	—
326	C₈	—	4	2	no	—
327	C₈	C₂×M₁₆×C₅₃	1	2	yes	—
328	C₈	C₂×M₁₆×C₅₃	1	4	yes	—
329	C₈	C₂×M₁₆×C₅₃	1	2	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	C₅₃×#423	1	2	yes	—
336	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
337	C₄	—	2	2	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	2	no	—
343	C₄	C₅₃×#423	1	2	yes	—
344	C₄	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
345	C₄	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
346	C₄	C₅₃×#576	1	1	yes	characteristic
347	C₄	C₅₃×#578	1	1	yes	characteristic
348	C₂	—	2	2	no	—
349	C₂	—	4	2	no	—
350	C₂	C₂×C₅₃×#423	1	1	yes	characteristic
351	C₂	C₅₃×#2913	1	1	yes	characteristic
352	C₂	C₅₃×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₅₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1569	C₂	semidirect
4. C₅₃×#1568	C₂	semidirect
5. C₅₃×#1567	C₂	semidirect
6. C₅₃×#1565	C₂	semidirect
7. C₅₃×#1564	C₂	semidirect
8. C₂×C₅₃×#335	C₂	semidirect
91. M₁₆×C₅₃	M₁₆	semidirect
99. C₂×C₈×C₅₃	M₁₆	semidirect
153. #17799	C₅₃	direct (N × H)
174. #1569	C₂×C₅₃	semidirect
175. #1568	C₂×C₅₃	semidirect
176. #1567	C₂×C₅₃	semidirect
177. #1565	C₂×C₅₃	semidirect
178. #1564	C₂×C₅₃	semidirect
179. C₂×#335	C₂×C₅₃	semidirect
210. C₅₃	#17799	direct (N × H)
268. M₁₆	M₁₆×C₅₃	semidirect
276. C₂×C₈	M₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2989	2
C₄×#131×C₅₃	2
C₅₃×#1569	2
C₅₃×#1568	2
C₅₃×#1567	2
C₅₃×#1565	2
C₅₃×#1564	2
C₂×C₅₃×#335	2
#17799	53

Subgroups — C53×#17799 #27225311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17799 #27225311