Subgroups — #27366540

Order 13568.

618 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285301	C₂	1	1	yes	maximal, characteristic, complemented
2	#20284503	C₂	1	2	yes	maximal
3	#20274173	C₂	1	2	yes	maximal, complemented
4	#20274166	C₂	1	2	yes	maximal, complemented
5	#20274119	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274114	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274085	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274080	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273370	C₂	1	2	yes	maximal, complemented
11	C₈×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
12	#16811180	C₂²	1	2	yes	—
13	#16811028	C₂²	1	1	yes	characteristic
14	#16811003	C₂²	1	1	yes	characteristic
15	#16810974	C₂²	1	2	yes	—
16	#16810930	—	2	2	no	—
17	#16810923	—	2	2	no	—
18	#16810918	C₂²	1	2	yes	—
19	#16810913	C₂²	1	2	yes	—
20	#16810908	C₂²	1	2	yes	—
21	#16810903	C₂²	1	2	yes	—
22	#16810864	—	2	1	no	—
23	#16810864	—	2	1	no	—
24	C₄×Dic₂₁₂	—	2	1	no	—
25	C₄×Dic₂₁₂	—	2	1	no	—
26	#16810659	—	2	2	no	—
27	C₂×C₄⋊#88322	C₂²	1	2	yes	—
28	#16810619	—	2	2	no	—
29	#16810376	C₂²	1	1	yes	characteristic
30	#16810376	C₂²	1	1	yes	characteristic
31	#16810376	C₂²	1	2	yes	—
32	#16810371	C₂²	1	2	yes	—
33	#16810371	C₂²	1	2	yes	—
34	#16810370	C₂²	1	2	yes	—
35	#16810370	C₂²	1	2	yes	—
36	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
37	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₅₃	—	2	1	no	—
40	C₈×D₄×C₅₃	—	2	1	no	—
41	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
42	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
43	C₅₃×#431	—	2	1	no	—
44	C₅₃×#431	C₂²	1	1	yes	characteristic
45	C₂×C₄×C₈×C₅₃	—	2	1	no	—
46	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₂×Dic₂₁₂	—	2	1	no	—
48	C₂×Dic₂₁₂	—	2	1	no	—
49	C₂²×#88322	—	2	2	no	—
50	#13133217	C₂³	1	2	yes	—
51	#13133217	C₂³	1	2	yes	—
52	#13133209	—	2	2	no	—
53	#13133196	—	2	2	no	—
54	#13133196	—	2	2	no	—
55	C₄×Dic₁₀₆	—	2	2	no	—
56	C₄×Dic₁₀₆	—	2	2	no	—
57	#13133176	—	2	2	no	—
58	#13133151	—	2	1	no	—
59	#13133151	—	2	1	no	—
60	#13133150	—	2	2	no	—
61	#13133150	—	2	2	no	—
62	#13133148	—	2	2	no	—
63	#13133148	—	2	2	no	—
64	C₄⋊#88322	—	2	2	no	—
65	C₄⋊#88322	C₂³	1	2	yes	—
66	C₄⋊#88322	C₂³	1	2	yes	—
67	#13133142	—	2	2	no	—
68	#13133142	—	2	2	no	—
69	#13133141	—	2	2	no	—
70	#13133141	—	2	2	no	—
71	#13133138	—	2	2	no	—
72	#13133138	—	2	2	no	—
73	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
74	C₂²×C₈×C₅₃	—	2	1	no	—
75	C₂²×C₈×C₅₃	—	2	1	no	—
76	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
77	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
78	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
79	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
80	C₄×D₄×C₅₃	—	2	1	no	—
81	C₄×D₄×C₅₃	—	2	1	no	—
82	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
83	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
84	#117×C₅₃	—	2	1	no	—
85	#117×C₅₃	D₄	1	1	yes	characteristic
86	C₂×C₄²×C₅₃	—	2	1	no	—
87	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₅₃	—	2	1	no	—
89	C₄⋊C₈×C₅₃	—	2	1	no	—
90	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₂²⋊C₈×C₅₃	—	2	1	no	—
93	C₂²⋊C₈×C₅₃	—	2	1	no	—
94	C₂²⋊C₈×C₅₃	—	2	1	no	—
95	C₄×C₈×C₅₃	—	2	1	no	—
96	C₄×C₈×C₅₃	—	2	1	no	—
97	C₄×C₈×C₅₃	—	2	1	no	—
98	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
99	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
100	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
102	#41×C₅₃	—	2	2	no	—
103	#41×C₅₃	Q₁₆	1	4	yes	complemented
104	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂×D₄×C₅₃	—	2	1	no	—
106	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	—	2	1	no	—
108	C₂²×C₄×C₅₃	—	2	1	no	—
109	C₂²×C₄×C₅₃	—	2	1	no	—
110	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
113	C₂×Dic₁₀₆	—	2	2	no	—
114	C₂×Dic₁₀₆	—	2	2	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	1	no	—
117	C₂×C₈×C₅₃	—	2	1	no	—
118	C₂×C₈×C₅₃	—	4	1	no	—
119	C₂×C₈×C₅₃	—	4	2	no	—
120	C₂×C₈×C₅₃	—	4	2	no	—
121	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
125	C₄⋊C₄×C₅₃	—	2	1	no	—
126	C₄⋊C₄×C₅₃	—	2	1	no	—
127	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
129	C₂²⋊C₄×C₅₃	—	2	1	no	—
130	C₂²⋊C₄×C₅₃	—	2	1	no	—
131	C₂²⋊C₄×C₅₃	—	2	1	no	—
132	C₄²×C₅₃	—	2	1	no	—
133	C₄²×C₅₃	—	2	1	no	—
134	C₄²×C₅₃	—	2	1	no	—
135	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₅₃	#41	1	1	yes	characteristic
138	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
139	C₂²⋊Dic₅₃	—	4	2	no	—
140	C₄⋊Dic₅₃	—	2	2	no	—
141	C₄⋊Dic₅₃	—	2	2	no	—
142	C₄⋊Dic₅₃	—	2	2	no	—
143	C₄⋊Dic₅₃	—	2	2	no	—
144	#11719919	—	4	2	no	—
145	#11719919	—	4	2	no	—
146	C₄×Dic₅₃	—	4	2	no	—
147	C₂×#88322	—	2	2	no	—
148	C₂×#88322	—	2	2	no	—
149	C₂×#88322	—	4	4	no	—
150	Dic₂₁₂	—	4	2	no	—
151	Dic₂₁₂	—	4	2	no	—
152	C₂³×C₅₃	—	2	1	no	—
153	C₂³×C₅₃	#121	1	1	yes	characteristic
154	Q₈×C₅₃	—	2	1	no	—
155	Q₈×C₅₃	C₂×Q₁₆	1	2	yes	—
156	D₄×C₅₃	—	2	4	no	—
157	D₄×C₅₃	—	2	1	no	—
158	D₄×C₅₃	C₂×Q₁₆	1	2	yes	—
159	C₂×C₄×C₅₃	—	2	1	no	—
160	C₂×C₄×C₅₃	—	2	1	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	4	no	—
163	C₂×C₄×C₅₃	—	2	1	no	—
164	C₂×C₄×C₅₃	—	2	1	no	—
165	C₂×C₄×C₅₃	—	4	1	no	—
166	C₂×C₄×C₅₃	—	4	2	no	—
167	C₂×C₄×C₅₃	—	4	1	no	—
168	C₂×C₄×C₅₃	—	4	1	no	—
169	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
170	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
171	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
173	C₂×C₄×C₅₃	#135	1	2	yes	—
174	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
177	C₂×Dic₅₃	—	4	2	no	—
178	C₂×Dic₅₃	—	4	2	no	—
179	Dic₁₀₆	—	4	2	no	—
180	Dic₁₀₆	—	4	2	no	—
181	Dic₁₀₆	—	8	2	no	—
182	C₈×C₅₃	—	2	1	no	—
183	C₈×C₅₃	—	2	1	no	—
184	C₈×C₅₃	—	2	1	no	—
185	C₈×C₅₃	—	4	1	no	—
186	C₈×C₅₃	—	4	1	no	—
187	C₈×C₅₃	#135	1	2	yes	—
188	#88322	—	4	2	no	—
189	#88322	—	4	2	no	—
190	#19126	—	53	1	no	maximal, Sylow
191	C₂²×C₅₃	—	2	1	no	—
192	C₂²×C₅₃	—	2	1	no	—
193	C₂²×C₅₃	—	2	2	no	—
194	C₂²×C₅₃	—	4	1	no	—
195	C₂²×C₅₃	—	4	1	no	—
196	C₂²×C₅₃	#467	1	2	yes	—
197	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
198	C₄×C₅₃	—	2	1	no	—
199	C₄×C₅₃	—	2	2	no	—
200	C₄×C₅₃	—	2	1	no	—
201	C₄×C₅₃	—	2	1	no	—
202	C₄×C₅₃	—	4	1	no	—
203	C₄×C₅₃	—	4	1	no	—
204	C₄×C₅₃	#467	1	2	yes	—
205	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
206	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
207	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
208	C₄×C₅₃	#575	1	1	yes	characteristic
209	Dic₅₃	—	8	2	no	—
210	Dic₅₃	—	8	2	no	—
211	#3170	—	53	1	no	—
212	#3076	—	53	2	no	—
213	C₈×#41	D₅₃	1	1	yes	characteristic
214	#1684	—	53	2	no	—
215	#1683	—	53	2	no	—
216	#1676	—	53	1	no	—
217	#1675	—	53	1	no	—
218	#1670	—	53	1	no	—
219	#1669	—	53	1	no	—
220	C₂×#335	—	53	1	no	—
221	#1542	—	53	2	no	—
222	C₂×C₅₃	—	2	2	no	—
223	C₂×C₅₃	—	4	2	no	—
224	C₂×C₅₃	#3035	1	1	yes	characteristic
225	C₂×C₅₃	C₂×#467	1	1	yes	characteristic
226	C₂×C₅₃	#3067	1	1	yes	characteristic
227	#533	—	53	2	no	—
228	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
229	#500	—	53	1	no	—
230	#494	—	53	1	no	—
231	#487	—	53	2	no	—
232	#479	—	106	2	no	—
233	#478	—	106	2	no	—
234	#477	—	53	2	no	—
235	#476	—	53	2	no	—
236	#475	—	53	2	no	—
237	#474	—	53	2	no	—
238	#467	—	106	1	no	—
239	#467	—	106	1	no	—
240	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
241	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
242	C₄×Q₁₆	—	106	1	no	—
243	C₄×Q₁₆	—	106	1	no	—
244	C₈×D₄	—	2	1	no	—
245	C₈×D₄	—	2	1	no	—
246	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
247	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
248	#431	—	2	1	no	—
249	#431	C₂×D₅₃	1	1	yes	characteristic
250	#427	—	106	2	no	—
251	C₂×C₄⋊C₈	—	53	2	no	—
252	#416	—	106	2	no	—
253	C₂×C₄×C₈	—	2	1	no	—
254	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
255	#335	—	53	1	no	—
256	#335	—	53	1	no	—
257	#335	—	53	2	no	—
258	#333	—	53	2	no	—
259	#333	—	53	2	no	—
260	#332	—	53	2	no	—
261	#332	—	53	2	no	—
262	C₅₃	#19126	1	1	yes	characteristic, complemented, Sylow
263	C₂×#41	D₂₁₂	1	1	yes	characteristic
264	C₂×Q₁₆	—	106	1	no	—
265	C₂×Q₁₆	—	106	1	no	—
266	C₂²×C₈	—	2	1	no	—
267	C₂²×C₈	—	2	1	no	—
268	C₂²×C₈	—	106	2	no	—
269	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
270	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
271	C₄⋊Q₈	—	53	2	no	—
272	C₄⋊Q₈	—	53	2	no	—
273	#125	—	106	2	no	—
274	C₂²⋊Q₈	—	106	2	no	—
275	C₂²⋊Q₈	—	106	2	no	—
276	C₄×Q₈	—	106	2	no	—
277	C₄×Q₈	—	106	2	no	—
278	C₄×Q₈	D₂₁₂	1	1	yes	characteristic
279	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
280	C₄×D₄	—	2	1	no	—
281	C₄×D₄	—	2	1	no	—
282	C₄×D₄	D₂₁₂	1	1	yes	characteristic
283	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
284	#117	—	2	1	no	—
285	#117	—	106	2	no	—
286	#117	D₂₁₂	1	1	yes	characteristic
287	C₂×C₄²	—	2	1	no	—
288	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
289	#107	—	106	1	no	—
290	#107	—	106	1	no	—
291	#107	—	106	2	no	—
292	#107	—	106	2	no	—
293	#106	—	106	2	no	—
294	#106	—	106	2	no	—
295	C₄⋊C₈	—	2	1	no	—
296	C₄⋊C₈	—	2	1	no	—
297	C₄⋊C₈	—	53	2	no	—
298	C₄⋊C₈	—	53	2	no	—
299	C₄⋊C₈	—	106	2	no	—
300	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
301	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
302	#103	—	106	2	no	—
303	#103	—	106	2	no	—
304	#103	—	106	2	no	—
305	#103	—	106	2	no	—
306	#102	—	106	2	no	—
307	#102	—	106	2	no	—
308	C₂²⋊C₈	—	2	1	no	—
309	C₂²⋊C₈	—	2	1	no	—
310	C₂²⋊C₈	—	2	1	no	—
311	C₄×C₈	—	2	1	no	—
312	C₄×C₈	—	2	1	no	—
313	C₄×C₈	—	2	1	no	—
314	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
315	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
316	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
317	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
318	#41	—	2	2	no	—
319	#41	Dic₂₁₂	1	4	yes	complemented
320	C₂×Q₈	—	106	2	no	—
321	C₂×Q₈	—	106	2	no	—
322	C₂×Q₈	C₂×D₂₁₂	1	1	yes	characteristic
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	C₂×D₂₁₂	1	1	yes	characteristic
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
329	C₂²×C₄	#11719945	1	1	yes	characteristic
330	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
331	Q₁₆	—	212	2	no	—
332	Q₁₆	—	212	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	4	1	no	—
337	C₂×C₈	—	4	2	no	—
338	C₂×C₈	—	4	2	no	—
339	C₂×C₈	—	106	2	no	—
340	C₂×C₈	—	106	2	no	—
341	C₂×C₈	—	212	4	no	—
342	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
343	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
344	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
345	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	106	2	no	—
349	C₄⋊C₄	—	106	2	no	—
350	C₄⋊C₄	—	106	2	no	—
351	C₄⋊C₄	—	106	2	no	—
352	C₄⋊C₄	—	212	2	no	—
353	C₄⋊C₄	—	212	2	no	—
354	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
355	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
356	C₂²⋊C₄	—	2	1	no	—
357	C₂²⋊C₄	—	2	1	no	—
358	C₂²⋊C₄	—	2	1	no	—
359	C₂²⋊C₄	—	212	2	no	—
360	C₄²	—	2	1	no	—
361	C₄²	—	2	1	no	—
362	C₄²	—	2	1	no	—
363	C₄²	—	212	2	no	—
364	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
365	C₄²	#11719945	1	1	yes	characteristic
366	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
367	C₄²	C₂³×D₅₃	1	1	yes	characteristic
368	C₂³	—	2	1	no	—
369	C₂³	#13133191	1	1	yes	characteristic
370	Q₈	—	2	1	no	—
371	Q₈	—	424	2	no	—
372	Q₈	—	212	2	no	—
373	Q₈	—	212	2	no	—
374	Q₈	C₂×Dic₂₁₂	1	2	yes	—
375	D₄	—	2	4	no	—
376	D₄	—	2	1	no	—
377	D₄	C₂×Dic₂₁₂	1	2	yes	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	212	2	no	—
389	C₂×C₄	—	212	2	no	—
390	C₂×C₄	#13133191	1	1	yes	characteristic
391	C₂×C₄	#13133191	1	1	yes	characteristic
392	C₂×C₄	#13133191	1	1	yes	characteristic
393	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
394	C₂×C₄	#13133239	1	2	yes	—
395	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
396	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
397	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
398	C₈	—	2	1	no	—
399	C₈	—	2	1	no	—
400	C₈	—	2	1	no	—
401	C₈	—	4	1	no	—
402	C₈	—	4	1	no	—
403	C₈	—	212	2	no	—
404	C₈	—	212	2	no	—
405	C₈	#13133237	1	2	yes	—
406	C₂²	—	2	1	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	4	1	no	—
410	C₂²	—	4	1	no	—
411	C₂²	#16810864	1	2	yes	—
412	C₂²	C₂×#13133191	1	1	yes	characteristic
413	C₄	—	2	1	no	—
414	C₄	—	2	1	no	—
415	C₄	—	2	1	no	—
416	C₄	—	2	2	no	—
417	C₄	—	4	1	no	—
418	C₄	—	4	1	no	—
419	C₄	—	424	2	no	—
420	C₄	—	424	2	no	—
421	C₄	#16810864	1	2	yes	—
422	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
423	C₄	C₂×#13133237	1	1	yes	characteristic
424	C₄	C₂×#13133239	1	1	yes	characteristic
425	C₄	#16811421	1	1	yes	characteristic
426	C₂	—	2	2	no	—
427	C₂	—	4	2	no	—
428	C₂	#20284170	1	1	yes	characteristic
429	C₂	C₂×#16810864	1	1	yes	characteristic
430	C₂	#20284435	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133191
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×#41×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂×#41×C₅₃	D₄
hypercenter	C₈×#41	D₅₃
hypocenter	C₅₃	#19126
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. #20285301	C₂	semidirect
3. #20274173	C₂	semidirect
4. #20274166	C₂	semidirect
5. #20274119	C₂	semidirect
6. #20274114	C₂	semidirect
7. #20274085	C₂	semidirect
8. #20274080	C₂	semidirect
9. C₂×#16810376	C₂	semidirect
10. #20273370	C₂	semidirect
103. #41×C₅₃	Q₁₆	semidirect
262. C₅₃	#19126	semidirect
319. #41	Dic₂₁₂	semidirect

Maximal subgroups

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

subgroup	index
#20285301	2
#20284503	2
#20274173	2
#20274166	2
#20274119	2
#20274114	2
#20274085	2
#20274080	2
C₂×#16810376	2
#20273370	2
C₈×#41×C₅₃	2
#19126	53