Subgroups — #27353247

Order 13568.

640 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283347	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810679	C₂	1	1	yes	maximal, characteristic
3	#431×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20273492	C₂	1	4	yes	maximal, complemented
5	#20273491	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273489	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273486	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273475	C₂	1	4	yes	maximal, complemented
9	C₅₃×#1570	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
11	#16810679	—	2	1	no	—
12	#16810679	—	2	1	no	—
13	#16810679	—	2	1	no	—
14	#16810679	C₂²	1	2	yes	—
15	#16810679	C₂²	1	2	yes	—
16	#16810679	C₂²	1	2	yes	—
17	#16810676	C₂²	1	1	yes	characteristic
18	#16810675	—	2	1	no	—
19	#16810675	C₄	1	2	yes	—
20	#16810675	C₂²	1	2	yes	—
21	#16810674	C₂²	1	2	yes	—
22	#16810673	C₄	1	2	yes	—
23	#16810647	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×D₅₃	C₄	1	2	yes	—
26	#16810586	—	2	1	no	—
27	C₂×#13133129	—	2	1	no	—
28	C₂²⋊C₈×D₅₃	C₄	1	2	yes	—
29	M₁₆×Dic₅₃	—	2	1	no	—
30	C₂×#13133127	—	2	1	no	—
31	C₄×C₈×D₅₃	C₂²	1	2	yes	—
32	#16810375	C₂²	1	4	yes	complemented
33	#16810375	C₂²	1	4	yes	complemented
34	#16810374	C₂²	1	4	yes	complemented
35	C₅₃×#431	C₂²	1	1	yes	characteristic
36	C₅₃×#423	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#334	C₂²	1	4	yes	complemented
39	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
41	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
42	C₂×#11719917	—	2	1	no	—
43	C₂²×#88322	—	2	1	no	—
44	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
45	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
48	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄²×D₅₃	D₄	1	1	yes	characteristic
51	C₄²×D₅₃	D₄	1	1	yes	characteristic
52	C₄²×D₅₃	Q₈	1	1	yes	characteristic
53	C₄²×D₅₃	Q₈	1	1	yes	characteristic
54	C₄²×D₅₃	C₂³	1	1	yes	characteristic
55	C₄²×D₅₃	C₂³	1	1	yes	characteristic
56	C₄⋊#88322	—	2	1	no	—
57	C₄⋊#88322	—	2	1	no	—
58	C₄⋊#88322	C₂×C₄	1	2	yes	—
59	C₄⋊#88322	C₂³	1	2	yes	—
60	C₄⋊#88322	C₂³	1	2	yes	—
61	#13133146	C₂×C₄	1	4	yes	—
62	C₂²⋊#88322	C₂×C₄	1	2	yes	—
63	#13133129	—	2	2	no	—
64	#13133129	—	2	2	no	—
65	#13133129	—	2	2	no	—
66	#13133129	—	2	2	no	—
67	#13133129	C₂×C₄	1	4	yes	—
68	#13133127	—	2	2	no	—
69	#13133127	—	2	2	no	—
70	#13133127	—	2	2	no	—
71	C₄×#88322	C₂³	1	2	yes	—
72	C₈×Dic₅₃	—	2	2	no	—
73	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
74	C₂×M₁₆×C₅₃	—	2	1	no	—
75	C₂²×C₈×C₅₃	—	2	1	no	—
76	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₅₃	—	2	1	no	—
78	C₄⋊C₈×C₅₃	—	2	1	no	—
79	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
80	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
81	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
82	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₅₃	C₂³	1	2	yes	—
84	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
88	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×D₅₃	—	2	2	no	—
92	C₂×C₄×D₅₃	—	2	2	no	—
93	C₂×C₄×D₅₃	—	4	2	no	—
94	C₂×C₄×D₅₃	—	4	2	no	—
95	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
106	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
107	M₁₆×C₅₃	—	4	2	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	—	4	2	no	—
111	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
112	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
115	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
116	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
117	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
118	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
119	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
120	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
121	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
122	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
126	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
127	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
128	#11719917	—	4	2	no	—
129	C₂×#88322	—	2	2	no	—
130	C₂×#88322	—	2	2	no	—
131	C₂×#88322	—	4	2	no	—
132	C₂×#88322	C₂²×C₄	1	4	yes	—
133	C₈×D₅₃	—	4	4	no	—
134	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×D₅₃	—	2	2	no	—
136	C₂²×D₅₃	—	2	2	no	—
137	C₂²×D₅₃	—	4	1	no	—
138	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	4	1	no	—
142	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#135	1	2	yes	—
146	C₂×C₄×C₅₃	#135	1	2	yes	—
147	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
150	C₂×Dic₅₃	—	2	2	no	—
151	C₂×Dic₅₃	—	4	1	no	—
152	C₂×Dic₅₃	—	4	1	no	—
153	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₄×D₅₃	—	2	2	no	—
160	C₄×D₅₃	—	2	2	no	—
161	C₄×D₅₃	—	2	2	no	—
162	C₄×D₅₃	—	2	2	no	—
163	C₄×D₅₃	—	2	1	no	—
164	C₄×D₅₃	—	2	1	no	—
165	C₄×D₅₃	—	2	1	no	—
166	C₄×D₅₃	—	2	1	no	—
167	C₄×D₅₃	—	2	2	no	—
168	C₄×D₅₃	—	2	2	no	—
169	C₄×D₅₃	—	4	1	no	—
170	C₄×D₅₃	—	4	1	no	—
171	C₄×D₅₃	—	4	1	no	—
172	C₄×D₅₃	—	4	1	no	—
173	C₈×C₅₃	—	2	4	no	—
174	C₈×C₅₃	—	4	2	no	—
175	C₈×C₅₃	—	4	2	no	—
176	#88322	—	2	4	no	—
177	#88322	—	4	2	no	—
178	#88322	—	4	2	no	—
179	#17816	—	53	1	no	maximal, Sylow
180	C₂²×C₅₃	—	2	1	no	—
181	C₂²×C₅₃	—	2	1	no	—
182	C₂²×C₅₃	#427	1	2	yes	—
183	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
184	C₂×D₅₃	—	2	1	no	—
185	C₂×D₅₃	—	2	1	no	—
186	C₂×D₅₃	—	2	1	no	—
187	C₂×D₅₃	—	2	1	no	—
188	C₂×D₅₃	—	2	1	no	—
189	C₂×D₅₃	—	4	4	no	—
190	C₂×D₅₃	#430	1	2	yes	—
191	C₄×C₅₃	—	2	1	no	—
192	C₄×C₅₃	—	2	1	no	—
193	C₄×C₅₃	—	2	1	no	—
194	C₄×C₅₃	#427	1	2	yes	—
195	C₄×C₅₃	#568	1	2	yes	—
196	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
197	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
198	Dic₅₃	—	2	1	no	—
199	Dic₅₃	—	2	1	no	—
200	Dic₅₃	—	2	1	no	—
201	Dic₅₃	—	2	1	no	—
202	Dic₅₃	—	2	1	no	—
203	Dic₅₃	—	2	1	no	—
204	#2920	—	53	1	no	—
205	C₂×#432	—	53	1	no	—
206	C₂×#431	—	53	1	no	—
207	#1572	—	53	4	no	—
208	#1571	—	53	1	no	—
209	#1571	—	53	1	no	—
210	#1570	—	53	1	no	—
211	#1570	D₅₃	1	1	yes	characteristic, complemented
212	#1568	—	53	4	no	—
213	C₂×C₅₃	—	2	2	no	—
214	C₂×C₅₃	#2905	1	1	yes	characteristic
215	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
216	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
217	D₅₃	—	2	2	no	—
218	D₅₃	—	4	1	no	—
219	C₂²×C₄²	—	53	1	no	—
220	#432	—	53	2	no	—
221	#432	—	53	2	no	—
222	#432	—	53	2	no	—
223	#432	—	106	1	no	—
224	#432	—	106	1	no	—
225	#432	—	106	1	no	—
226	#431	—	53	1	no	—
227	#431	—	53	2	no	—
228	#431	—	53	2	no	—
229	#431	—	53	2	no	—
230	#431	—	53	2	no	—
231	#431	—	106	1	no	—
232	#431	C₂×D₅₃	1	1	yes	characteristic
233	#423	—	53	1	no	—
234	#423	C₂×D₅₃	1	1	yes	characteristic
235	C₂×C₄⋊C₈	—	53	2	no	—
236	C₂×C₄⋊C₈	—	53	1	no	—
237	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
238	#407	—	106	1	no	—
239	C₂×C₂²⋊C₈	—	53	2	no	—
240	C₂×C₂²⋊C₈	—	106	1	no	—
241	C₄×M₁₆	—	106	1	no	—
242	C₂×#97	—	106	1	no	—
243	C₂×C₄×C₈	—	53	2	no	—
244	#335	—	53	4	no	—
245	#335	—	53	4	no	—
246	#334	—	53	4	no	—
247	#334	C₂×D₅₃	1	4	yes	complemented
248	C₅₃	#17816	1	1	yes	characteristic, complemented, Sylow
249	C₂³×C₄	—	53	1	no	—
250	C₂³×C₄	—	53	1	no	—
251	C₂³×C₄	—	53	1	no	—
252	C₂×M₁₆	—	2	1	no	—
253	C₂×M₁₆	—	106	1	no	—
254	C₂²×C₈	—	2	1	no	—
255	C₂²×C₈	—	53	4	no	—
256	C₂²×C₈	—	106	1	no	—
257	C₂×C₄²	—	53	1	no	—
258	C₂×C₄²	—	53	1	no	—
259	C₂×C₄²	—	53	1	no	—
260	C₂×C₄²	—	53	1	no	—
261	C₂×C₄²	—	53	1	no	—
262	C₂×C₄²	—	53	1	no	—
263	C₂×C₄²	—	53	1	no	—
264	C₂×C₄²	—	53	1	no	—
265	C₂×C₄²	—	53	1	no	—
266	C₂×C₄²	—	53	1	no	—
267	C₂×C₄²	—	53	1	no	—
268	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	2	1	no	—
271	C₄⋊C₈	—	53	4	no	—
272	C₄⋊C₈	—	53	2	no	—
273	C₄⋊C₈	—	53	2	no	—
274	C₄⋊C₈	—	53	2	no	—
275	C₄⋊C₈	—	106	1	no	—
276	C₄⋊C₈	—	106	1	no	—
277	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
278	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
279	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
280	C₂²⋊C₈	—	53	2	no	—
281	C₂²⋊C₈	—	53	4	no	—
282	C₂²⋊C₈	—	106	2	no	—
283	C₂²⋊C₈	—	106	2	no	—
284	C₂²⋊C₈	—	106	2	no	—
285	C₂²⋊C₈	—	106	2	no	—
286	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
287	#97	—	106	2	no	—
288	#97	—	106	2	no	—
289	#97	—	106	2	no	—
290	C₄×C₈	—	53	4	no	—
291	C₄×C₈	—	53	2	no	—
292	C₄×C₈	—	106	2	no	—
293	C₄×C₈	C₂²×D₅₃	1	2	yes	—
294	C₂⁴	—	53	1	no	—
295	C₂²×C₄	—	53	1	no	—
296	C₂²×C₄	—	53	1	no	—
297	C₂²×C₄	—	53	1	no	—
298	C₂²×C₄	—	53	1	no	—
299	C₂²×C₄	—	53	1	no	—
300	C₂²×C₄	—	53	1	no	—
301	C₂²×C₄	—	53	1	no	—
302	C₂²×C₄	—	53	1	no	—
303	C₂²×C₄	—	53	1	no	—
304	C₂²×C₄	—	53	1	no	—
305	C₂²×C₄	—	53	1	no	—
306	C₂²×C₄	—	53	1	no	—
307	C₂²×C₄	—	53	1	no	—
308	C₂²×C₄	—	53	1	no	—
309	C₂²×C₄	—	53	1	no	—
310	C₂²×C₄	—	106	2	no	—
311	C₂²×C₄	—	106	2	no	—
312	C₂²×C₄	—	212	2	no	—
313	C₂²×C₄	—	212	2	no	—
314	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
315	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
316	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
317	M₁₆	—	4	2	no	—
318	M₁₆	—	212	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	4	2	no	—
322	C₂×C₈	—	53	4	no	—
323	C₂×C₈	—	106	2	no	—
324	C₂×C₈	—	106	2	no	—
325	C₂×C₈	—	212	4	no	—
326	C₂×C₈	—	212	2	no	—
327	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
328	C₄²	—	53	1	no	—
329	C₄²	—	53	1	no	—
330	C₄²	—	53	1	no	—
331	C₄²	—	53	1	no	—
332	C₄²	—	53	1	no	—
333	C₄²	—	53	1	no	—
334	C₄²	—	53	1	no	—
335	C₄²	—	53	1	no	—
336	C₄²	—	53	1	no	—
337	C₄²	—	53	1	no	—
338	C₄²	—	53	1	no	—
339	C₄²	—	53	1	no	—
340	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
341	C₄²	D₄×D₅₃	1	1	yes	characteristic
342	C₄²	Q₈×D₅₃	1	1	yes	characteristic
343	C₄²	C₂³×D₅₃	1	1	yes	characteristic
344	C₂³	—	53	1	no	—
345	C₂³	—	53	1	no	—
346	C₂³	—	106	2	no	—
347	C₂³	—	106	2	no	—
348	C₂³	—	212	1	no	—
349	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	4	1	no	—
352	C₂×C₄	—	53	1	no	—
353	C₂×C₄	—	53	1	no	—
354	C₂×C₄	—	53	1	no	—
355	C₂×C₄	—	53	1	no	—
356	C₂×C₄	—	53	1	no	—
357	C₂×C₄	—	53	1	no	—
358	C₂×C₄	—	106	2	no	—
359	C₂×C₄	—	106	2	no	—
360	C₂×C₄	—	106	2	no	—
361	C₂×C₄	—	106	2	no	—
362	C₂×C₄	—	106	2	no	—
363	C₂×C₄	—	106	1	no	—
364	C₂×C₄	—	106	1	no	—
365	C₂×C₄	—	106	1	no	—
366	C₂×C₄	—	106	1	no	—
367	C₂×C₄	—	106	2	no	—
368	C₂×C₄	—	106	2	no	—
369	C₂×C₄	—	212	1	no	—
370	C₂×C₄	—	212	1	no	—
371	C₂×C₄	—	212	1	no	—
372	C₂×C₄	—	212	1	no	—
373	C₂×C₄	—	212	1	no	—
374	C₂×C₄	—	212	1	no	—
375	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
376	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
377	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
378	C₂×C₄	#13133238	1	2	yes	—
379	C₂×C₄	#13133238	1	2	yes	—
380	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
381	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
382	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
383	C₈	—	2	4	no	—
384	C₈	—	4	2	no	—
385	C₈	—	4	2	no	—
386	C₈	—	106	4	no	—
387	C₈	—	212	2	no	—
388	C₈	—	212	2	no	—
389	C₂²	—	2	1	no	—
390	C₂²	—	2	1	no	—
391	C₂²	—	53	2	no	—
392	C₂²	—	106	1	no	—
393	C₂²	—	106	1	no	—
394	C₂²	—	106	1	no	—
395	C₂²	—	106	1	no	—
396	C₂²	—	106	1	no	—
397	C₂²	—	212	4	no	—
398	C₂²	#16810658	1	2	yes	—
399	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
400	C₄	—	2	1	no	—
401	C₄	—	2	1	no	—
402	C₄	—	2	1	no	—
403	C₄	—	106	1	no	—
404	C₄	—	106	1	no	—
405	C₄	—	106	1	no	—
406	C₄	—	106	1	no	—
407	C₄	—	106	1	no	—
408	C₄	—	106	1	no	—
409	C₄	#16810658	1	2	yes	—
410	C₄	#16811388	1	2	yes	—
411	C₄	C₂×#13133238	1	1	yes	characteristic
412	C₄	C₂×#13133238	1	1	yes	characteristic
413	C₂	—	2	2	no	—
414	C₂	—	106	2	no	—
415	C₂	—	212	1	no	—
416	C₂	#20283276	1	1	yes	characteristic
417	C₂	C₂×#16810658	1	1	yes	characteristic
418	C₂	#430×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1570	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1570	D₅₃
hypocenter	C₅₃	#17816
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1570

Complements

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

N	H ≅ G/N	type
4. #20273492	C₂	semidirect
5. #20273491	C₂	semidirect
6. #20273489	C₂	semidirect
7. #20273486	C₂	semidirect
8. #20273475	C₂	semidirect
9. C₅₃×#1570	C₂	semidirect
32. #16810375	C₂²	semidirect
33. #16810375	C₂²	semidirect
34. #16810374	C₂²	semidirect
38. C₅₃×#334	C₂²	semidirect
211. #1570	D₅₃	semidirect
247. #334	C₂×D₅₃	semidirect
248. C₅₃	#17816	semidirect

Maximal subgroups

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

subgroup	index
#20283347	2
C₂×#16810679	2
#431×D₅₃	2
#20273492	2
#20273491	2
#20273489	2
#20273486	2
#20273475	2
C₅₃×#1570	2
#17816	53