Subgroups — C₅₃×#33676 #27241188

Order 13568.

548 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2237	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2227	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2226	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#2209	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2208	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2189	C₂	1	2	yes	maximal
9	C₅₃×#2176	C₂	1	2	yes	maximal, complemented
10	C₅₃×#2175	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#533	—	2	1	no	—
14	C₄×#41×C₅₃	—	2	1	no	—
15	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
16	C₂×SD₃₂×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#500	C₂²	1	1	yes	characteristic
18	C₅₃×#494	C₂²	1	1	yes	characteristic
19	C₅₃×#487	—	2	1	no	—
20	C₅₃×#467	—	2	1	no	—
21	C₅₃×#467	C₂²	1	2	yes	—
22	C₄×Q₁₆×C₅₃	—	2	1	no	—
23	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
26	C₄×SD₁₆×C₅₃	—	2	1	no	—
27	C₄×SD₁₆×C₅₃	—	2	1	no	—
28	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#427	C₂²	1	1	yes	characteristic
31	C₅₃×#416	—	2	1	no	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#367	C₂²	1	2	yes	—
34	C₅₃×#366	C₂²	1	2	yes	—
35	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₅₃×#358	C₂²	1	2	yes	—
38	C₅₃×#358	C₂²	1	2	yes	—
39	C₅₃×#358	C₂²	1	2	yes	—
40	C₅₃×#357	C₂²	1	2	yes	—
41	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
42	C₅₃×#346	C₂²	1	2	yes	—
43	C₂×#41×C₅₃	—	2	1	no	—
44	#135×C₅₃	—	2	2	no	—
45	#135×C₅₃	—	2	2	no	—
46	C₂×Q₁₆×C₅₃	—	2	1	no	—
47	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
48	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×SD₁₆×C₅₃	—	2	1	no	—
50	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₅₃	—	2	1	no	—
54	C₄⋊Q₈×C₅₃	—	2	1	no	—
55	#125×C₅₃	—	4	1	no	—
56	C₂²⋊Q₈×C₅₃	—	2	2	no	—
57	C₂²⋊Q₈×C₅₃	—	4	1	no	—
58	C₄×Q₈×C₅₃	—	2	1	no	—
59	C₄×Q₈×C₅₃	—	2	1	no	—
60	C₄×Q₈×C₅₃	—	2	2	no	—
61	C₄×Q₈×C₅₃	—	4	1	no	—
62	C₄×D₄×C₅₃	—	2	1	no	—
63	C₄×D₄×C₅₃	—	2	1	no	—
64	C₄×D₄×C₅₃	—	4	1	no	—
65	C₄×D₄×C₅₃	—	4	1	no	—
66	#117×C₅₃	—	2	1	no	—
67	#117×C₅₃	—	4	1	no	—
68	#117×C₅₃	—	4	1	no	—
69	C₂×C₄²×C₅₃	—	4	1	no	—
70	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
71	Q₃₂×C₅₃	—	2	4	no	—
72	SD₃₂×C₅₃	—	2	4	no	—
73	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
74	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
75	#107×C₅₃	—	2	1	no	—
76	#107×C₅₃	C₂³	1	2	yes	—
77	#107×C₅₃	C₂³	1	1	yes	characteristic
78	#107×C₅₃	C₂³	1	1	yes	characteristic
79	#106×C₅₃	—	2	1	no	—
80	#103×C₅₃	—	2	2	no	—
81	#103×C₅₃	—	2	2	no	—
82	#103×C₅₃	—	2	1	no	—
83	#103×C₅₃	—	2	1	no	—
84	#102×C₅₃	—	2	1	no	—
85	#102×C₅₃	—	2	1	no	—
86	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
87	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
88	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
90	#41×C₅₃	—	2	4	no	—
91	#41×C₅₃	—	4	2	no	—
92	C₂×Q₈×C₅₃	—	2	1	no	—
93	C₂×Q₈×C₅₃	—	2	2	no	—
94	C₂×Q₈×C₅₃	—	4	1	no	—
95	C₂×D₄×C₅₃	—	2	1	no	—
96	C₂×D₄×C₅₃	—	4	1	no	—
97	C₂²×C₄×C₅₃	—	4	1	no	—
98	C₂²×C₄×C₅₃	—	4	1	no	—
99	C₂²×C₄×C₅₃	—	4	1	no	—
100	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
103	Q₁₆×C₅₃	—	2	1	no	—
104	Q₁₆×C₅₃	—	2	2	no	—
105	Q₁₆×C₅₃	—	2	2	no	—
106	Q₁₆×C₅₃	—	4	2	no	—
107	Q₁₆×C₅₃	#41	1	2	yes	—
108	SD₁₆×C₅₃	—	2	4	no	—
109	D₈×C₅₃	—	2	1	no	—
110	D₈×C₅₃	#41	1	2	yes	—
111	C₂×C₈×C₅₃	—	2	1	no	—
112	C₂×C₈×C₅₃	—	2	1	no	—
113	C₂×C₈×C₅₃	—	4	1	no	—
114	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
118	C₄⋊C₄×C₅₃	—	2	1	no	—
119	C₄⋊C₄×C₅₃	—	2	1	no	—
120	C₄⋊C₄×C₅₃	—	2	2	no	—
121	C₄⋊C₄×C₅₃	—	4	1	no	—
122	C₄⋊C₄×C₅₃	—	4	1	no	—
123	C₄⋊C₄×C₅₃	—	4	1	no	—
124	C₄⋊C₄×C₅₃	—	4	1	no	—
125	C₄⋊C₄×C₅₃	—	4	2	no	—
126	C₄⋊C₄×C₅₃	—	4	2	no	—
127	C₄⋊C₄×C₅₃	—	4	1	no	—
128	C₂²⋊C₄×C₅₃	—	4	1	no	—
129	C₂²⋊C₄×C₅₃	—	4	2	no	—
130	C₂²⋊C₄×C₅₃	—	4	1	no	—
131	C₂²⋊C₄×C₅₃	—	4	1	no	—
132	C₄²×C₅₃	—	4	1	no	—
133	C₄²×C₅₃	—	4	1	no	—
134	C₄²×C₅₃	—	4	1	no	—
135	C₄²×C₅₃	—	4	2	no	—
136	C₄²×C₅₃	D₈	1	1	yes	characteristic
137	C₄²×C₅₃	D₈	1	1	yes	characteristic
138	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₁₆×C₅₃	—	2	2	no	—
141	C₁₆×C₅₃	—	2	2	no	—
142	C₂³×C₅₃	—	4	1	no	—
143	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
144	Q₈×C₅₃	—	2	2	no	—
145	Q₈×C₅₃	—	4	1	no	—
146	Q₈×C₅₃	—	4	2	no	—
147	Q₈×C₅₃	—	4	2	no	—
148	Q₈×C₅₃	—	8	2	no	—
149	D₄×C₅₃	—	2	2	no	—
150	D₄×C₅₃	—	4	4	no	—
151	D₄×C₅₃	—	4	1	no	—
152	C₂×C₄×C₅₃	—	2	1	no	—
153	C₂×C₄×C₅₃	—	2	1	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	4	1	no	—
156	C₂×C₄×C₅₃	—	4	4	no	—
157	C₂×C₄×C₅₃	—	4	1	no	—
158	C₂×C₄×C₅₃	—	4	1	no	—
159	C₂×C₄×C₅₃	—	4	1	no	—
160	C₂×C₄×C₅₃	—	4	2	no	—
161	C₂×C₄×C₅₃	—	4	2	no	—
162	C₂×C₄×C₅₃	—	8	2	no	—
163	C₂×C₄×C₅₃	—	8	2	no	—
164	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
170	C₈×C₅₃	—	2	1	no	—
171	C₈×C₅₃	—	2	1	no	—
172	C₈×C₅₃	—	2	1	no	—
173	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
174	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
175	#33676	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂²×C₅₃	—	2	1	no	—
177	C₂²×C₅₃	—	2	1	no	—
178	C₂²×C₅₃	—	2	1	no	—
179	C₂²×C₅₃	—	4	2	no	—
180	C₂²×C₅₃	—	8	1	no	—
181	C₂²×C₅₃	—	8	1	no	—
182	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
183	C₄×C₅₃	—	2	1	no	—
184	C₄×C₅₃	—	2	1	no	—
185	C₄×C₅₃	—	2	1	no	—
186	C₄×C₅₃	—	2	1	no	—
187	C₄×C₅₃	—	4	2	no	—
188	C₄×C₅₃	—	8	2	no	—
189	C₄×C₅₃	—	8	1	no	—
190	C₄×C₅₃	—	8	1	no	—
191	C₄×C₅₃	—	8	2	no	—
192	C₄×C₅₃	#510	1	2	yes	—
193	C₄×C₅₃	#525	1	1	yes	characteristic
194	C₄×C₅₃	#575	1	1	yes	characteristic
195	#3170	C₂×C₅₃	1	1	yes	characteristic, complemented
196	C₄×#135	C₂×C₅₃	1	1	yes	characteristic
197	#2237	C₂×C₅₃	1	2	yes	complemented
198	#2227	C₂×C₅₃	1	1	yes	characteristic, complemented
199	#2226	C₂×C₅₃	1	1	yes	characteristic, complemented
200	#2209	C₂×C₅₃	1	1	yes	characteristic, complemented
201	#2208	C₂×C₅₃	1	1	yes	characteristic, complemented
202	#2189	C₂×C₅₃	1	2	yes	—
203	#2176	C₂×C₅₃	1	2	yes	complemented
204	#2175	C₂×C₅₃	1	2	yes	complemented
205	#2163	C₂×C₅₃	1	1	yes	characteristic, complemented
206	C₂×C₅₃	—	4	1	no	—
207	C₂×C₅₃	—	8	2	no	—
208	C₂×C₅₃	#3100	1	1	yes	characteristic
209	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
210	C₂×C₅₃	#3414	1	1	yes	characteristic
211	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
212	#533	—	2	1	no	—
213	C₄×#41	—	2	1	no	—
214	C₂×Q₃₂	C₂²×C₅₃	1	2	yes	—
215	C₂×SD₃₂	C₂²×C₅₃	1	2	yes	—
216	#500	C₂²×C₅₃	1	1	yes	characteristic
217	#494	C₂²×C₅₃	1	1	yes	characteristic
218	#487	—	2	1	no	—
219	#467	—	2	1	no	—
220	#467	C₂²×C₅₃	1	2	yes	—
221	C₄×Q₁₆	—	2	1	no	—
222	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
223	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
224	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
225	C₄×SD₁₆	—	2	1	no	—
226	C₄×SD₁₆	—	2	1	no	—
227	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
228	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
229	#427	C₂²×C₅₃	1	1	yes	characteristic
230	#416	—	2	1	no	—
231	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
232	#367	C₂²×C₅₃	1	2	yes	—
233	#366	C₂²×C₅₃	1	2	yes	—
234	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
235	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
236	#358	C₂²×C₅₃	1	2	yes	—
237	#358	C₂²×C₅₃	1	2	yes	—
238	#358	C₂²×C₅₃	1	2	yes	—
239	#357	C₂²×C₅₃	1	2	yes	—
240	C₂²⋊C₁₆	C₂²×C₅₃	1	2	yes	—
241	#346	C₂²×C₅₃	1	2	yes	—
242	C₅₃	#33676	1	1	yes	characteristic, direct product factor, Sylow
243	C₂×#41	—	2	1	no	—
244	#135	—	2	2	no	—
245	#135	—	2	2	no	—
246	C₂×Q₁₆	—	2	1	no	—
247	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
248	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
249	C₂×SD₁₆	—	2	1	no	—
250	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
251	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
252	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
253	C₄⋊Q₈	—	2	1	no	—
254	C₄⋊Q₈	—	2	1	no	—
255	#125	—	4	1	no	—
256	C₂²⋊Q₈	—	2	2	no	—
257	C₂²⋊Q₈	—	4	1	no	—
258	C₄×Q₈	—	2	1	no	—
259	C₄×Q₈	—	2	1	no	—
260	C₄×Q₈	—	2	2	no	—
261	C₄×Q₈	—	4	1	no	—
262	C₄×D₄	—	2	1	no	—
263	C₄×D₄	—	2	1	no	—
264	C₄×D₄	—	4	1	no	—
265	C₄×D₄	—	4	1	no	—
266	#117	—	2	1	no	—
267	#117	—	4	1	no	—
268	#117	—	4	1	no	—
269	C₂×C₄²	—	4	1	no	—
270	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
271	Q₃₂	—	2	4	no	—
272	SD₃₂	—	2	4	no	—
273	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
274	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
275	#107	—	2	1	no	—
276	#107	C₂³×C₅₃	1	2	yes	—
277	#107	C₂³×C₅₃	1	1	yes	characteristic
278	#107	C₂³×C₅₃	1	1	yes	characteristic
279	#106	—	2	1	no	—
280	#103	—	2	1	no	—
281	#103	—	2	1	no	—
282	#103	—	2	2	no	—
283	#103	—	2	2	no	—
284	#102	—	2	1	no	—
285	#102	—	2	1	no	—
286	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
287	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
288	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
289	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
290	#41	—	2	4	no	—
291	#41	—	4	2	no	—
292	C₂×Q₈	—	2	1	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	4	1	no	—
295	C₂×D₄	—	2	1	no	—
296	C₂×D₄	—	4	1	no	—
297	C₂²×C₄	—	4	1	no	—
298	C₂²×C₄	—	4	1	no	—
299	C₂²×C₄	—	4	1	no	—
300	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
301	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
302	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
303	Q₁₆	—	2	1	no	—
304	Q₁₆	—	2	2	no	—
305	Q₁₆	—	2	2	no	—
306	Q₁₆	—	4	2	no	—
307	Q₁₆	#41×C₅₃	1	2	yes	—
308	SD₁₆	—	2	4	no	—
309	D₈	—	2	1	no	—
310	D₈	#41×C₅₃	1	2	yes	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	4	1	no	—
314	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
315	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
316	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
317	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	2	1	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	4	1	no	—
322	C₄⋊C₄	—	4	1	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	4	1	no	—
325	C₄⋊C₄	—	4	2	no	—
326	C₄⋊C₄	—	4	2	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₂²⋊C₄	—	4	1	no	—
329	C₂²⋊C₄	—	4	2	no	—
330	C₂²⋊C₄	—	4	1	no	—
331	C₂²⋊C₄	—	4	1	no	—
332	C₄²	—	4	1	no	—
333	C₄²	—	4	1	no	—
334	C₄²	—	4	1	no	—
335	C₄²	—	4	2	no	—
336	C₄²	D₈×C₅₃	1	1	yes	characteristic
337	C₄²	D₈×C₅₃	1	1	yes	characteristic
338	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
339	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
340	C₁₆	—	2	2	no	—
341	C₁₆	—	2	2	no	—
342	C₂³	—	4	1	no	—
343	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
344	Q₈	—	2	2	no	—
345	Q₈	—	4	1	no	—
346	Q₈	—	4	2	no	—
347	Q₈	—	4	2	no	—
348	Q₈	—	8	2	no	—
349	D₄	—	2	2	no	—
350	D₄	—	4	4	no	—
351	D₄	—	4	1	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	4	1	no	—
356	C₂×C₄	—	4	4	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	8	2	no	—
363	C₂×C₄	—	8	2	no	—
364	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
365	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
366	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
367	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
368	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
369	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
374	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
375	C₂²	—	2	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	8	1	no	—
380	C₂²	—	8	1	no	—
381	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	4	2	no	—
387	C₄	—	8	2	no	—
388	C₄	—	8	1	no	—
389	C₄	—	8	1	no	—
390	C₄	—	8	2	no	—
391	C₄	C₅₃×#510	1	2	yes	—
392	C₄	C₅₃×#525	1	1	yes	characteristic
393	C₄	C₅₃×#575	1	1	yes	characteristic
394	C₂	—	4	1	no	—
395	C₂	—	8	2	no	—
396	C₂	C₅₃×#3100	1	1	yes	characteristic
397	C₂	C₂×C₅₃×#510	1	1	yes	characteristic
398	C₂	C₅₃×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
prime radical	C₂×#135×C₅₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3170	C₂	semidirect
3. C₅₃×#2237	C₂	semidirect
4. C₅₃×#2227	C₂	semidirect
5. C₅₃×#2226	C₂	semidirect
6. C₅₃×#2209	C₂	semidirect
7. C₅₃×#2208	C₂	semidirect
9. C₅₃×#2176	C₂	semidirect
10. C₅₃×#2175	C₂	semidirect
11. C₅₃×#2163	C₂	semidirect
175. #33676	C₅₃	direct (N × H)
195. #3170	C₂×C₅₃	semidirect
197. #2237	C₂×C₅₃	semidirect
198. #2227	C₂×C₅₃	semidirect
199. #2226	C₂×C₅₃	semidirect
200. #2209	C₂×C₅₃	semidirect
201. #2208	C₂×C₅₃	semidirect
203. #2176	C₂×C₅₃	semidirect
204. #2175	C₂×C₅₃	semidirect
205. #2163	C₂×C₅₃	semidirect
242. C₅₃	#33676	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×#3170	2
C₄×#135×C₅₃	2
C₅₃×#2237	2
C₅₃×#2227	2
C₅₃×#2226	2
C₅₃×#2209	2
C₅₃×#2208	2
C₅₃×#2189	2
C₅₃×#2176	2
C₅₃×#2175	2
C₅₃×#2163	2
#33676	53

Subgroups — C53×#33676 #27241188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33676 #27241188