Subgroups — C₂×#20289774 #28310275

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅₃⋊#108	C₂	1	1	yes	maximal, characteristic
3	#20289774	C₂	1	4	yes	maximal, direct product factor
4	C₂×#108×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×D₅₃⋊M₁₆	—	2	1	no	—
6	C₂×D₅₃⋊C₁₆	—	2	1	no	—
7	C₂×D₅₃⋊C₁₆	—	2	1	no	—
8	C₂×D₅₃⋊C₁₆	—	2	1	no	—
9	C₂×D₅₃⋊C₁₆	C₄	1	2	yes	—
10	C₂×D₅₃⋊C₁₆	C₄	1	2	yes	—
11	C₂×D₅₃⋊C₁₆	C₂²	1	2	yes	—
12	C₂²×#11719910	—	2	1	no	—
13	C₂×#13133288	C₄	1	2	yes	—
14	D₅₃⋊#108	—	2	1	no	—
15	D₅₃⋊#108	—	2	1	no	—
16	D₅₃⋊#108	C₄	1	2	yes	—
17	D₅₃⋊#108	C₂²	1	2	yes	—
18	C₂×#13133287	—	2	1	no	—
19	C₂×M₁₆×D₅₃	—	2	1	no	—
20	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
21	C₂×#13133153	C₄	1	1	yes	characteristic
22	C₂×#13133152	—	2	1	no	—
23	#108×D₅₃	—	2	1	no	—
24	#108×D₅₃	—	2	1	no	—
25	#108×D₅₃	C₄	1	2	yes	—
26	#108×D₅₃	C₂²	1	2	yes	—
27	C₂×#108×C₅₃	C₄	1	1	yes	characteristic
28	C₂×#11719936	—	2	2	no	—
29	D₅₃⋊M₁₆	—	2	2	no	—
30	D₅₃⋊M₁₆	—	2	2	no	—
31	D₅₃⋊M₁₆	—	4	1	no	—
32	D₅₃⋊M₁₆	—	4	1	no	—
33	C₂×#11719940	—	4	1	no	—
34	C₂×D₅₃⋊C₈	—	4	1	no	—
35	D₅₃⋊C₁₆	—	2	2	no	—
36	D₅₃⋊C₁₆	—	2	2	no	—
37	D₅₃⋊C₁₆	—	2	2	no	—
38	D₅₃⋊C₁₆	C₂×C₄	1	4	yes	—
39	C₂×#11719910	—	2	2	no	—
40	C₂×#11719910	—	2	2	no	—
41	C₂×#11719910	—	2	2	no	—
42	#13133288	—	2	2	no	—
43	#13133288	C₂×C₄	1	4	yes	—
44	#13133287	—	2	2	no	—
45	#13133287	—	2	2	no	—
46	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
47	C₂×#11719917	—	2	1	no	—
48	C₂×#11719912	—	4	1	no	—
49	M₁₆×D₅₃	—	2	2	no	—
50	M₁₆×D₅₃	—	2	2	no	—
51	M₁₆×D₅₃	—	4	1	no	—
52	M₁₆×D₅₃	—	4	1	no	—
53	C₂²×#88322	D₄	1	1	yes	characteristic
54	C₂×C₈×D₅₃	—	4	1	no	—
55	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
61	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
62	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
63	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
64	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
65	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
66	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
67	#13133153	—	2	1	no	—
68	#13133153	C₂×C₄	1	2	yes	—
69	#13133152	—	2	2	no	—
70	#13133152	—	2	2	no	—
71	C₂×M₁₆×C₅₃	—	2	1	no	—
72	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	#108×C₅₃	—	2	1	no	—
74	#108×C₅₃	C₂×C₄	1	2	yes	—
75	C₂³×D₅₃	Q₁₆	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×Dic₅₃	SD₁₆	1	1	yes	characteristic
78	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
84	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
85	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
86	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
87	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
88	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
89	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
90	#11719940	—	4	2	no	—
91	#11719940	—	4	2	no	—
92	C₂×#88324	—	4	2	no	—
93	#11719936	—	2	4	no	—
94	#11719936	—	4	2	no	—
95	D₅₃⋊C₈	—	4	2	no	—
96	D₅₃⋊C₈	—	4	2	no	—
97	M₁₆×C₅₃	—	2	2	no	—
98	M₁₆×C₅₃	—	4	1	no	—
99	C₂×C₈×C₅₃	—	4	1	no	—
100	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	#11719917	—	2	2	no	—
107	#11719917	—	4	1	no	—
108	C₂×#88322	—	4	1	no	—
109	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
111	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
112	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
113	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
114	C₂×#88322	C₂×D₄	1	1	yes	characteristic
115	#11719912	—	4	2	no	—
116	#11719912	—	4	2	no	—
117	C₈×D₅₃	—	4	2	no	—
118	C₈×D₅₃	—	4	2	no	—
119	C₈×D₅₃	C₄²	1	1	yes	characteristic
120	C₈×D₅₃	C₄²	1	1	yes	characteristic
121	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
122	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
123	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
124	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
125	C₈×D₅₃	C₄⋊C₄	1	1	yes	characteristic
126	C₈×D₅₃	C₄⋊C₄	1	1	yes	characteristic
127	C₈×D₅₃	C₄⋊C₄	1	1	yes	characteristic
128	C₈×D₅₃	C₄⋊C₄	1	1	yes	characteristic
129	C₈×D₅₃	C₄⋊C₄	1	1	yes	characteristic
130	C₈×D₅₃	C₄⋊C₄	1	1	yes	characteristic
131	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
133	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
134	C₈×D₅₃	C₂×Q₈	1	1	yes	characteristic
135	#11719910	—	2	4	no	—
136	C₂³×C₅₃	#103	1	1	yes	characteristic
137	C₂²×D₅₃	—	2	4	no	—
138	C₂²×D₅₃	#103	1	1	yes	characteristic
139	C₂²×D₅₃	#103	1	1	yes	characteristic
140	C₂²×D₅₃	#103	1	1	yes	characteristic
141	C₂²×D₅₃	#107	1	1	yes	characteristic
142	C₂²×D₅₃	#107	1	1	yes	characteristic
143	C₂²×D₅₃	C₂×Q₁₆	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
146	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
147	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
148	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×Dic₅₃	#102	1	1	yes	characteristic
151	C₂×Dic₅₃	#102	1	1	yes	characteristic
152	C₂×Dic₅₃	#103	1	1	yes	characteristic
153	C₂×Dic₅₃	#106	1	1	yes	characteristic
154	C₂×Dic₅₃	#106	1	1	yes	characteristic
155	C₂×Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
156	C₄×D₅₃	#95	1	1	yes	characteristic
157	C₄×D₅₃	#95	1	1	yes	characteristic
158	C₄×D₅₃	#102	1	1	yes	characteristic
159	C₄×D₅₃	#102	1	1	yes	characteristic
160	C₄×D₅₃	#102	1	1	yes	characteristic
161	C₄×D₅₃	#102	1	1	yes	characteristic
162	C₄×D₅₃	#103	1	1	yes	characteristic
163	C₄×D₅₃	#103	1	1	yes	characteristic
164	C₄×D₅₃	#106	1	1	yes	characteristic
165	C₄×D₅₃	#106	1	1	yes	characteristic
166	C₄×D₅₃	#107	1	1	yes	characteristic
167	C₄×D₅₃	#107	1	1	yes	characteristic
168	C₄×D₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₄×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
171	C₄×D₅₃	C₂×SD₁₆	1	1	yes	characteristic
172	#88324	—	4	4	no	—
173	C₈×C₅₃	—	4	2	no	—
174	C₈×C₅₃	#95	1	1	yes	characteristic
175	C₈×C₅₃	#95	1	1	yes	characteristic
176	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
177	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
178	#88322	—	4	2	no	—
179	#88322	#95	1	1	yes	characteristic
180	#88322	#95	1	1	yes	characteristic
181	#88322	C₂×C₂²⋊C₄	1	1	yes	characteristic
182	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂×#1381	—	53	1	no	maximal, Sylow
184	C₂²×C₅₃	—	2	2	no	—
185	C₂²×C₅₃	#340	1	1	yes	characteristic
186	C₂²×C₅₃	#340	1	1	yes	characteristic
187	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
188	C₂×D₅₃	—	2	2	no	—
189	C₂×D₅₃	—	2	2	no	—
190	C₂×D₅₃	—	2	2	no	—
191	C₂×D₅₃	—	2	2	no	—
192	C₂×D₅₃	—	2	2	no	—
193	C₂×D₅₃	#340	1	1	yes	characteristic
194	C₂×D₅₃	#340	1	1	yes	characteristic
195	C₂×D₅₃	#368	1	4	yes	—
196	C₂×D₅₃	C₂×#103	1	1	yes	characteristic
197	C₂×D₅₃	C₂×#107	1	1	yes	characteristic
198	C₄×C₅₃	#340	1	1	yes	characteristic
199	C₄×C₅₃	#340	1	1	yes	characteristic
200	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
201	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
202	Dic₅₃	#340	1	1	yes	characteristic
203	Dic₅₃	#340	1	1	yes	characteristic
204	Dic₅₃	C₂×#102	1	1	yes	characteristic
205	Dic₅₃	C₂×#106	1	1	yes	characteristic
206	C₂³×C₁₆	—	53	1	no	—
207	C₂²×#108	—	53	1	no	—
208	C₂²×#108	—	53	1	no	—
209	#1381	—	53	4	no	—
210	C₂×C₅₃	—	2	1	no	—
211	C₂×C₅₃	—	2	1	no	—
212	C₂×C₅₃	#1381	1	2	yes	complemented
213	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
214	D₅₃	—	2	1	no	—
215	D₅₃	—	2	1	no	—
216	D₅₃	#1381	1	2	yes	complemented
217	D₅₃	C₂×#368	1	2	yes	—
218	C₂²×M₁₆	—	106	1	no	—
219	C₂²×M₁₆	—	106	1	no	—
220	C₂³×C₈	—	53	1	no	—
221	C₂²×C₁₆	—	53	2	no	—
222	C₂²×C₁₆	—	53	2	no	—
223	C₂²×C₁₆	—	53	2	no	—
224	C₂²×C₁₆	—	106	1	no	—
225	C₂²×C₁₆	—	106	1	no	—
226	C₂²×C₁₆	—	106	1	no	—
227	C₂²×C₁₆	—	106	1	no	—
228	C₂×#108	—	53	2	no	—
229	C₂×#108	—	53	2	no	—
230	C₂×#108	—	53	2	no	—
231	C₂×#108	—	53	1	no	—
232	C₂×#108	—	53	2	no	—
233	C₂×#108	—	53	2	no	—
234	C₂×#108	—	106	1	no	—
235	C₂×#108	—	106	1	no	—
236	C₂×#108	—	106	1	no	—
237	C₂×#108	—	106	1	no	—
238	C₂×#108	—	106	1	no	—
239	C₂×#108	—	106	1	no	—
240	C₂×#108	#6157	1	1	yes	characteristic
241	C₅₃	C₂×#1381	1	1	yes	characteristic, complemented, Sylow
242	C₂³×C₄	—	53	1	no	—
243	C₂×M₁₆	—	2	1	no	—
244	C₂×M₁₆	—	106	2	no	—
245	C₂×M₁₆	—	106	2	no	—
246	C₂×M₁₆	—	106	2	no	—
247	C₂×M₁₆	—	106	2	no	—
248	C₂×M₁₆	—	106	2	no	—
249	C₂×M₁₆	—	106	1	no	—
250	C₂×M₁₆	—	212	1	no	—
251	C₂×M₁₆	—	212	1	no	—
252	C₂×M₁₆	—	212	1	no	—
253	C₂×M₁₆	—	212	1	no	—
254	C₂×M₁₆	—	212	1	no	—
255	C₂×M₁₆	—	212	1	no	—
256	C₂²×C₈	—	53	1	no	—
257	C₂²×C₈	—	53	1	no	—
258	C₂²×C₈	—	53	1	no	—
259	C₂²×C₈	—	53	1	no	—
260	C₂²×C₈	—	53	1	no	—
261	C₂²×C₈	—	53	1	no	—
262	C₂²×C₈	—	53	1	no	—
263	C₂²×C₈	—	53	1	no	—
264	C₂²×C₈	—	53	1	no	—
265	C₂²×C₈	—	53	1	no	—
266	C₂²×C₈	—	53	1	no	—
267	C₂²×C₈	—	53	1	no	—
268	C₂²×C₈	—	53	1	no	—
269	C₂²×C₈	—	212	1	no	—
270	C₂²×C₈	—	212	1	no	—
271	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
272	C₂×C₁₆	—	53	4	no	—
273	C₂×C₁₆	—	106	2	no	—
274	C₂×C₁₆	—	106	2	no	—
275	C₂×C₁₆	—	106	2	no	—
276	C₂×C₁₆	—	106	2	no	—
277	C₂×C₁₆	—	106	2	no	—
278	C₂×C₁₆	—	106	2	no	—
279	#108	—	2	1	no	—
280	#108	—	53	4	no	—
281	#108	—	53	2	no	—
282	#108	—	106	2	no	—
283	#108	—	106	2	no	—
284	#108	—	106	2	no	—
285	#108	—	106	2	no	—
286	#108	—	106	2	no	—
287	#108	—	106	1	no	—
288	#108	C₂×#6157	1	2	yes	—
289	C₂⁴	—	53	1	no	—
290	C₂²×C₄	—	53	1	no	—
291	C₂²×C₄	—	53	1	no	—
292	C₂²×C₄	—	53	1	no	—
293	C₂²×C₄	—	53	1	no	—
294	C₂²×C₄	—	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	C₂²⋊#6157	1	1	yes	characteristic
304	M₁₆	—	2	2	no	—
305	M₁₆	—	4	1	no	—
306	M₁₆	—	106	4	no	—
307	M₁₆	—	106	2	no	—
308	M₁₆	—	212	2	no	—
309	M₁₆	—	212	2	no	—
310	M₁₆	—	212	2	no	—
311	M₁₆	—	212	2	no	—
312	M₁₆	—	212	2	no	—
313	M₁₆	—	212	1	no	—
314	C₂×C₈	—	4	1	no	—
315	C₂×C₈	—	53	1	no	—
316	C₂×C₈	—	53	1	no	—
317	C₂×C₈	—	53	1	no	—
318	C₂×C₈	—	53	1	no	—
319	C₂×C₈	—	53	1	no	—
320	C₂×C₈	—	53	1	no	—
321	C₂×C₈	—	53	1	no	—
322	C₂×C₈	—	53	1	no	—
323	C₂×C₈	—	53	1	no	—
324	C₂×C₈	—	53	1	no	—
325	C₂×C₈	—	53	1	no	—
326	C₂×C₈	—	53	1	no	—
327	C₂×C₈	—	53	1	no	—
328	C₂×C₈	—	53	1	no	—
329	C₂×C₈	—	53	1	no	—
330	C₂×C₈	—	53	1	no	—
331	C₂×C₈	—	53	1	no	—
332	C₂×C₈	—	53	1	no	—
333	C₂×C₈	—	53	1	no	—
334	C₂×C₈	—	53	1	no	—
335	C₂×C₈	—	53	1	no	—
336	C₂×C₈	—	53	1	no	—
337	C₂×C₈	—	212	2	no	—
338	C₂×C₈	—	212	2	no	—
339	C₂×C₈	—	212	2	no	—
340	C₂×C₈	—	212	2	no	—
341	C₂×C₈	—	212	2	no	—
342	C₂×C₈	—	212	1	no	—
343	C₂×C₈	C₄×#6157	1	1	yes	characteristic
344	C₂×C₈	C₄×#6157	1	1	yes	characteristic
345	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
346	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
347	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
348	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
349	C₁₆	—	106	4	no	—
350	C₂³	—	53	1	no	—
351	C₂³	—	53	1	no	—
352	C₂³	—	53	1	no	—
353	C₂³	—	53	1	no	—
354	C₂³	—	53	1	no	—
355	C₂³	—	53	1	no	—
356	C₂³	—	106	4	no	—
357	C₂³	#13133280	1	1	yes	characteristic
358	C₂×C₄	—	53	1	no	—
359	C₂×C₄	—	53	1	no	—
360	C₂×C₄	—	53	1	no	—
361	C₂×C₄	—	53	1	no	—
362	C₂×C₄	—	53	1	no	—
363	C₂×C₄	—	53	1	no	—
364	C₂×C₄	—	53	1	no	—
365	C₂×C₄	—	53	1	no	—
366	C₂×C₄	—	53	1	no	—
367	C₂×C₄	—	53	1	no	—
368	C₂×C₄	—	53	1	no	—
369	C₂×C₄	—	53	1	no	—
370	C₂×C₄	—	53	1	no	—
371	C₂×C₄	—	53	1	no	—
372	C₂×C₄	—	53	1	no	—
373	C₂×C₄	—	53	1	no	—
374	C₂×C₄	—	53	1	no	—
375	C₂×C₄	—	53	1	no	—
376	C₂×C₄	—	53	1	no	—
377	C₂×C₄	—	53	1	no	—
378	C₂×C₄	—	53	1	no	—
379	C₂×C₄	—	53	1	no	—
380	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
381	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
382	C₂×C₄	#13133279	1	1	yes	characteristic
383	C₂×C₄	#13133279	1	1	yes	characteristic
384	C₂×C₄	#13133280	1	1	yes	characteristic
385	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
386	C₈	—	4	2	no	—
387	C₈	—	53	1	no	—
388	C₈	—	53	1	no	—
389	C₈	—	53	1	no	—
390	C₈	—	53	1	no	—
391	C₈	—	212	4	no	—
392	C₈	—	212	2	no	—
393	C₈	(C₂×#6157)⋊C₄	1	1	yes	characteristic
394	C₈	(C₂×#6157)⋊C₄	1	1	yes	characteristic
395	C₈	C₂×C₄×#6157	1	1	yes	characteristic
396	C₈	C₂×C₄⋊#6157	1	1	yes	characteristic
397	C₂²	—	2	2	no	—
398	C₂²	—	53	1	no	—
399	C₂²	—	53	1	no	—
400	C₂²	—	53	4	no	—
401	C₂²	—	53	1	no	—
402	C₂²	—	53	1	no	—
403	C₂²	—	106	2	no	—
404	C₂²	—	106	2	no	—
405	C₂²	—	106	2	no	—
406	C₂²	—	106	2	no	—
407	C₂²	—	106	2	no	—
408	C₂²	#16811496	1	1	yes	characteristic
409	C₂²	#16811496	1	1	yes	characteristic
410	C₂²	C₂×#13133280	1	1	yes	characteristic
411	C₄	—	53	1	no	—
412	C₄	—	53	1	no	—
413	C₄	—	53	1	no	—
414	C₄	—	53	1	no	—
415	C₄	#16811496	1	1	yes	characteristic
416	C₄	#16811496	1	1	yes	characteristic
417	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
418	C₄	C₂×#13133279	1	1	yes	characteristic
419	C₂	—	2	1	no	—
420	C₂	—	2	1	no	—
421	C₂	—	53	2	no	—
422	C₂	—	53	2	no	—
423	C₂	—	106	1	no	—
424	C₂	—	106	1	no	—
425	C₂	#20289774	1	2	yes	direct product factor
426	C₂	C₂×#16811496	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133279
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂×#108×C₅₃	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂³×D₅₃	Q₁₆
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₂×#108	#6157
hypocenter	C₅₃	C₂×#1381
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
3. #20289774	C₂	direct (N × H)
212. C₂×C₅₃	#1381	semidirect
216. D₅₃	#1381	semidirect
241. C₅₃	C₂×#1381	semidirect
425. C₂	#20289774	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₅₃⋊C₁₆	2
C₂×D₅₃⋊#108	2
#20289774	2
C₂×#108×D₅₃	2
C₂×#1381	53

Subgroups — C2×#20289774 #28310275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289774 #28310275