Subgroups — #28319449

Order 13568.

1070 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811634	C₂	1	2	yes	maximal
2	C₂×#16811633	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13133286	C₂	1	1	yes	maximal, characteristic
4	#20291196	C₂	1	2	yes	maximal
5	#20290653	C₂	1	4	yes	maximal, complemented
6	#20290652	C₂	1	2	yes	maximal, complemented
7	#20290651	C₂	1	2	yes	maximal, complemented
8	#405×D₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133296	C₂²	1	2	yes	—
10	C₂²×C₄⋊#6157	C₂²	1	2	yes	—
11	#16811634	—	2	8	no	—
12	#16811633	—	2	4	no	—
13	C₂×#13133286	—	2	4	no	—
14	C₂×#13133286	C₂²	1	2	yes	—
15	C₂×#13133286	C₂²	1	2	yes	—
16	C₂×#13133286	C₂²	1	1	yes	characteristic
17	C₂×#13133286	C₂²	1	1	yes	characteristic
18	C₂×#13133286	C₂²	1	4	yes	—
19	C₂×#13133285	C₂²	1	2	yes	—
20	C₂×#13133285	C₂²	1	4	yes	—
21	#16811574	—	2	4	no	—
22	#16811570	—	2	4	no	—
23	#16811569	C₂²	1	4	yes	—
24	#16811567	C₂²	1	4	yes	—
25	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
27	#117×D₅₃	C₂²	1	1	yes	characteristic
28	#16810581	C₄	1	1	yes	characteristic, complemented
29	#16810580	C₄	1	2	yes	complemented
30	#16810579	C₄	1	1	yes	characteristic, complemented
31	#16810578	C₄	1	1	yes	characteristic, complemented
32	#16810577	C₄	1	2	yes	complemented
33	#97×D₅₃	C₂²	1	2	yes	—
34	C₄×C₈×D₅₃	C₂²	1	2	yes	—
35	C₅₃×#405	C₄	1	1	yes	characteristic, complemented
36	C₂³×#6157	—	2	2	no	—
37	#13133296	—	2	4	no	—
38	#13133296	—	2	4	no	—
39	C₂×C₄⋊#6157	—	2	4	no	—
40	C₂×C₄⋊#6157	—	2	4	no	—
41	C₂×C₄⋊#6157	—	2	2	no	—
42	C₂×C₄⋊#6157	—	2	8	no	—
43	C₂×C₄⋊#6157	C₂³	1	2	yes	—
44	C₂×C₄⋊#6157	C₂³	1	4	yes	—
45	C₂×C₄⋊#6157	C₂³	1	2	yes	—
46	C₂×C₂²⋊#6157	—	2	2	no	—
47	C₂×C₄×#6157	—	2	2	no	—
48	#13133286	—	2	4	no	—
49	#13133286	—	2	4	no	—
50	#13133286	—	2	2	no	—
51	#13133286	—	2	2	no	—
52	#13133286	—	2	8	no	—
53	#13133285	—	2	4	no	—
54	#13133285	—	2	8	no	—
55	(C₂×#6157)⋊C₄	—	2	8	no	—
56	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
57	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
58	C₂×#11719912	C₂×C₄	1	2	yes	—
59	M₁₆×D₅₃	D₄	1	4	yes	—
60	M₁₆×D₅₃	Q₈	1	4	yes	—
61	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
62	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₈×D₅₃	D₄	1	4	yes	—
65	C₂×C₈×D₅₃	Q₈	1	4	yes	—
66	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₈×D₅₃	C₂³	1	2	yes	—
68	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
69	#13133176	C₂×C₄	1	1	yes	characteristic
70	#13133175	Q₈	1	2	yes	—
71	#13133174	C₂×C₄	1	2	yes	—
72	#13133173	Q₈	1	2	yes	—
73	C₄⋊C₄×D₅₃	D₄	1	2	yes	—
74	C₂²⋊C₄×D₅₃	D₄	1	2	yes	—
75	C₄²×D₅₃	C₂³	1	2	yes	—
76	#13133128	C₂×C₄	1	2	yes	complemented
77	#13133127	C₂×C₄	1	2	yes	complemented
78	#13133127	C₂×C₄	1	2	yes	complemented
79	C₄×#88322	C₂×C₄	1	2	yes	complemented
80	C₈×Dic₅₃	C₂×C₄	1	2	yes	complemented
81	C₈×Dic₅₃	C₂×C₄	1	2	yes	complemented
82	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
84	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
85	#97×C₅₃	C₂×C₄	1	2	yes	complemented
86	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	complemented
87	C₂³×D₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×#6157	—	2	2	no	—
89	C₂²×#6157	—	2	4	no	—
90	C₂²×#6157	—	2	2	no	—
91	C₂²×#6157	—	4	4	no	—
92	C₂²×#6157	—	8	4	no	—
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×D₅₃	C₂×D₄	1	4	yes	—
97	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
98	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
99	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
101	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
102	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
103	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
104	C₂²⋊#6157	—	4	4	no	—
105	C₄⋊#6157	—	2	2	no	—
106	C₄⋊#6157	—	2	4	no	—
107	C₄⋊#6157	—	2	2	no	—
108	C₄⋊#6157	—	2	2	no	—
109	C₄⋊#6157	—	2	4	no	—
110	C₄⋊#6157	—	2	2	no	—
111	C₄⋊#6157	—	4	4	no	—
112	C₄⋊#6157	—	8	8	no	—
113	C₄×#6157	—	4	4	no	—
114	M₁₆×C₅₃	C₄⋊C₄	1	4	yes	complemented
115	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	complemented
116	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
120	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
121	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
122	C₂²⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
123	#11719921	C₄⋊C₄	1	4	yes	—
124	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
125	#11719919	C₄⋊C₄	1	4	yes	—
126	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
127	C₄×Dic₅₃	C₂×Q₈	1	4	yes	—
128	#11719917	C₄⋊C₄	1	4	yes	complemented
129	C₂×#88322	C₄⋊C₄	1	4	yes	complemented
130	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
131	C₂×#88322	C₂²×C₄	1	2	yes	—
132	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
133	#11719912	C₄⋊C₄	1	8	yes	complemented
134	C₈×D₅₃	C₄⋊C₄	1	4	yes	complemented
135	C₈×D₅₃	C₄⋊C₄	1	4	yes	complemented
136	C₈×D₅₃	C₂×D₄	1	2	yes	—
137	C₈×D₅₃	C₂×D₄	1	2	yes	—
138	C₈×D₅₃	C₂×D₄	1	4	yes	—
139	C₈×D₅₃	C₂×Q₈	1	2	yes	—
140	C₈×D₅₃	C₂×Q₈	1	2	yes	—
141	C₈×D₅₃	C₂×Q₈	1	4	yes	—
142	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×D₅₃	—	2	2	no	—
144	C₂²×D₅₃	—	2	2	no	—
145	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×D₅₃	#127	1	2	yes	—
147	C₂²×D₅₃	C₄⋊Q₈	1	2	yes	—
148	C₂²×D₅₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×#6157	—	4	2	no	—
150	C₂×#6157	—	4	2	no	—
151	C₂×#6157	—	4	4	no	—
152	C₂×#6157	—	4	4	no	—
153	C₂×#6157	—	4	8	no	—
154	C₂×#6157	—	4	4	no	—
155	C₂×#6157	—	8	8	no	—
156	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
157	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
158	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
159	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
161	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
162	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×Dic₅₃	C₄⋊Q₈	1	2	yes	—
164	C₂×Dic₅₃	C₄⋊Q₈	1	2	yes	—
165	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
166	C₄×D₅₃	—	2	4	no	—
167	C₄×D₅₃	—	2	4	no	—
168	C₄×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
169	C₄×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
170	C₄×D₅₃	#127	1	2	yes	—
171	C₄×D₅₃	C₄⋊Q₈	1	2	yes	—
172	C₄×D₅₃	C₄⋊Q₈	1	2	yes	—
173	C₄×D₅₃	C₄⋊Q₈	1	2	yes	—
174	C₄×D₅₃	C₂²×D₄	1	1	yes	characteristic
175	C₄×D₅₃	C₂²×D₄	1	1	yes	characteristic
176	C₄×D₅₃	C₂²×Q₈	1	1	yes	characteristic
177	C₄×D₅₃	C₂²×Q₈	1	1	yes	characteristic
178	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
179	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
180	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
181	#88322	C₂×C₄⋊C₄	1	2	yes	—
182	#88322	C₂×C₄⋊C₄	1	2	yes	—
183	#88322	C₂×C₄⋊C₄	1	4	yes	—
184	#20803	—	53	1	no	maximal, Sylow
185	C₂²×C₅₃	—	2	2	no	—
186	C₂²×C₅₃	#384	1	2	yes	—
187	C₂²×C₅₃	C₂²×C₄⋊C₄	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	4	no	—
192	C₂×D₅₃	#384	1	2	yes	—
193	C₂×D₅₃	C₂×#127	1	1	yes	characteristic
194	C₂×D₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
195	C₂×D₅₃	#576	1	2	yes	—
196	C₂×D₅₃	#578	1	2	yes	—
197	#6157	—	8	2	no	—
198	#6157	—	8	4	no	—
199	#6157	—	8	2	no	—
200	C₄×C₅₃	—	2	4	no	—
201	C₄×C₅₃	#384	1	2	yes	—
202	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
203	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
204	Dic₅₃	—	2	4	no	—
205	Dic₅₃	#384	1	2	yes	—
206	Dic₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
207	Dic₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
208	C₂×#428	—	53	2	no	—
209	C₂×#427	—	53	1	no	—
210	C₂²×#107	—	53	1	no	—
211	C₂×#405	—	53	1	no	—
212	#2317	—	53	2	no	—
213	#1847	—	53	4	no	—
214	#1846	—	53	2	no	—
215	#1845	—	53	2	no	—
216	C₂×C₅₃	—	2	2	no	—
217	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
218	C₂×C₅₃	#2913	1	2	yes	—
219	D₅₃	—	2	2	no	—
220	D₅₃	#3155	1	1	yes	characteristic
221	D₅₃	#3156	1	1	yes	characteristic
222	D₅₃	#3164	1	2	yes	—
223	C₂²×M₁₆	—	53	1	no	—
224	C₂³×C₈	—	53	1	no	—
225	C₂×#117	—	53	1	no	—
226	C₂×#117	—	53	2	no	—
227	C₂²×C₄⋊C₄	—	53	2	no	—
228	#428	—	106	8	no	—
229	#427	—	106	4	no	—
230	C₂×#107	—	53	1	no	—
231	C₂×#107	—	53	2	no	—
232	C₂×#107	—	53	4	no	—
233	C₂×#107	—	53	1	no	—
234	C₂×#107	—	53	2	no	—
235	C₂×#107	—	106	4	no	—
236	C₂×#106	—	53	4	no	—
237	C₂×#106	—	53	2	no	—
238	#405	—	53	1	no	—
239	#405	—	53	1	no	—
240	#405	—	53	1	no	—
241	#405	—	53	2	no	—
242	#405	—	53	2	no	—
243	#405	#6157	1	1	yes	characteristic, complemented
244	C₂×#97	—	53	2	no	—
245	C₂×C₄×C₈	—	53	2	no	—
246	#389	—	106	4	no	—
247	#385	—	106	4	no	—
248	#384	—	53	4	no	—
249	#382	—	53	4	no	—
250	C₅₃	#20803	1	1	yes	characteristic, complemented, Sylow
251	C₂³×C₄	—	53	1	no	—
252	C₂³×C₄	—	106	2	no	—
253	C₂×M₁₆	—	53	1	no	—
254	C₂×M₁₆	—	53	4	no	—
255	C₂×M₁₆	—	53	4	no	—
256	C₂×M₁₆	—	53	2	no	—
257	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
258	C₂²×C₈	—	53	1	no	—
259	C₂²×C₈	—	53	4	no	—
260	C₂²×C₈	—	53	4	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	2	no	—
266	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
267	#117	—	53	1	no	—
268	#117	—	53	2	no	—
269	#117	—	53	2	no	—
270	#117	—	53	2	no	—
271	#117	—	106	4	no	—
272	#117	—	106	4	no	—
273	#117	C₂×#6157	1	1	yes	characteristic
274	C₂×C₄⋊C₄	—	53	2	no	—
275	C₂×C₄⋊C₄	—	53	4	no	—
276	C₂×C₄⋊C₄	—	53	2	no	—
277	C₂×C₄⋊C₄	—	53	2	no	—
278	C₂×C₄⋊C₄	—	106	4	no	—
279	C₂×C₄⋊C₄	—	106	4	no	—
280	C₂×C₄⋊C₄	—	106	8	no	—
281	C₂×C₄⋊C₄	—	106	2	no	—
282	C₂×C₂²⋊C₄	—	53	2	no	—
283	C₂×C₂²⋊C₄	—	106	2	no	—
284	C₂×C₄²	—	53	2	no	—
285	C₂×C₄²	—	106	2	no	—
286	#107	—	106	4	no	—
287	#107	—	106	4	no	—
288	#107	—	106	2	no	—
289	#107	—	106	2	no	—
290	#107	—	106	8	no	—
291	#106	—	106	4	no	—
292	#106	—	106	8	no	—
293	#97	—	53	2	no	—
294	#97	—	53	2	no	—
295	#97	—	53	2	no	—
296	#97	C₂×#6157	1	2	yes	complemented
297	C₄×C₈	—	53	2	no	—
298	C₄×C₈	—	53	2	no	—
299	C₄×C₈	—	53	2	no	—
300	C₄×C₈	C₂×#6157	1	2	yes	complemented
301	#95	—	106	8	no	—
302	C₂⁴	—	53	1	no	—
303	C₂²×C₄	—	53	1	no	—
304	C₂²×C₄	—	53	1	no	—
305	C₂²×C₄	—	53	2	no	—
306	C₂²×C₄	—	53	2	no	—
307	C₂²×C₄	—	53	1	no	—
308	C₂²×C₄	—	53	2	no	—
309	C₂²×C₄	—	53	2	no	—
310	C₂²×C₄	—	53	1	no	—
311	C₂²×C₄	—	53	1	no	—
312	C₂²×C₄	—	53	4	no	—
313	C₂²×C₄	—	106	2	no	—
314	C₂²×C₄	—	106	4	no	—
315	C₂²×C₄	—	106	2	no	—
316	C₂²×C₄	—	424	4	no	—
317	C₂²×C₄	—	212	4	no	—
318	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
319	M₁₆	—	53	4	no	—
320	M₁₆	—	53	8	no	—
321	M₁₆	C₄⋊#6157	1	4	yes	complemented
322	C₂×C₈	—	53	4	no	—
323	C₂×C₈	—	53	1	no	—
324	C₂×C₈	—	53	2	no	—
325	C₂×C₈	—	53	1	no	—
326	C₂×C₈	—	53	2	no	—
327	C₂×C₈	—	53	4	no	—
328	C₂×C₈	—	53	2	no	—
329	C₂×C₈	—	53	4	no	—
330	C₂×C₈	—	53	4	no	—
331	C₂×C₈	—	53	2	no	—
332	C₂×C₈	—	53	2	no	—
333	C₂×C₈	—	53	4	no	—
334	C₂×C₈	C₄⋊#6157	1	4	yes	complemented
335	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
336	C₂×C₈	C₂²×#6157	1	2	yes	—
337	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
338	C₄⋊C₄	—	53	4	no	—
339	C₄⋊C₄	—	53	2	no	—
340	C₄⋊C₄	—	106	2	no	—
341	C₄⋊C₄	—	106	4	no	—
342	C₄⋊C₄	—	106	2	no	—
343	C₄⋊C₄	—	106	2	no	—
344	C₄⋊C₄	—	106	4	no	—
345	C₄⋊C₄	—	106	2	no	—
346	C₄⋊C₄	—	424	8	no	—
347	C₄⋊C₄	—	212	4	no	—
348	C₄⋊C₄	C₄⋊#6157	1	2	yes	—
349	C₂²⋊C₄	—	53	2	no	—
350	C₂²⋊C₄	—	53	4	no	—
351	C₂²⋊C₄	—	212	4	no	—
352	C₂²⋊C₄	C₄⋊#6157	1	2	yes	—
353	C₄²	—	53	4	no	—
354	C₄²	—	53	2	no	—
355	C₄²	—	212	4	no	—
356	C₄²	C₂²×#6157	1	2	yes	—
357	C₂³	—	53	1	no	—
358	C₂³	—	53	2	no	—
359	C₂³	—	53	2	no	—
360	C₂³	—	53	1	no	—
361	C₂³	—	106	2	no	—
362	C₂³	—	106	2	no	—
363	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
364	C₂×C₄	—	53	2	no	—
365	C₂×C₄	—	53	2	no	—
366	C₂×C₄	—	53	1	no	—
367	C₂×C₄	—	53	1	no	—
368	C₂×C₄	—	53	2	no	—
369	C₂×C₄	—	53	2	no	—
370	C₂×C₄	—	53	2	no	—
371	C₂×C₄	—	53	2	no	—
372	C₂×C₄	—	53	2	no	—
373	C₂×C₄	—	53	2	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	4	no	—
379	C₂×C₄	—	106	4	no	—
380	C₂×C₄	—	106	4	no	—
381	C₂×C₄	—	424	8	no	—
382	C₂×C₄	—	212	2	no	—
383	C₂×C₄	—	212	2	no	—
384	C₂×C₄	—	212	4	no	—
385	C₂×C₄	—	212	4	no	—
386	C₂×C₄	—	212	8	no	—
387	C₂×C₄	—	212	4	no	—
388	C₂×C₄	C₂×C₄⋊#6157	1	4	yes	—
389	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
390	C₂×C₄	C₂×C₄⋊#6157	1	2	yes	—
391	C₂×C₄	C₂×C₄⋊#6157	1	2	yes	—
392	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
393	C₈	—	53	2	no	—
394	C₈	—	53	2	no	—
395	C₈	—	53	4	no	—
396	C₈	C₂×C₄⋊#6157	1	2	yes	—
397	C₈	C₂×C₄⋊#6157	1	2	yes	—
398	C₈	C₂×C₄⋊#6157	1	4	yes	—
399	C₂²	—	2	2	no	—
400	C₂²	—	53	2	no	—
401	C₂²	—	53	2	no	—
402	C₂²	—	53	1	no	—
403	C₂²	—	53	1	no	—
404	C₂²	—	53	2	no	—
405	C₂²	—	106	2	no	—
406	C₂²	—	106	2	no	—
407	C₂²	—	106	4	no	—
408	C₂²	—	106	2	no	—
409	C₂²	#16811569	1	2	yes	—
410	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
411	C₄	—	2	4	no	—
412	C₄	—	53	1	no	—
413	C₄	—	53	1	no	—
414	C₄	—	53	2	no	—
415	C₄	—	106	4	no	—
416	C₄	—	424	4	no	—
417	C₄	—	424	2	no	—
418	C₄	—	424	2	no	—
419	C₄	#16811569	1	2	yes	—
420	C₄	C₂²×C₄⋊#6157	1	1	yes	characteristic
421	C₄	C₂²×C₄⋊#6157	1	1	yes	characteristic
422	C₂	—	2	2	no	—
423	C₂	—	53	1	no	—
424	C₂	—	53	1	no	—
425	C₂	—	53	2	no	—
426	C₂	—	106	2	no	—
427	C₂	C₂×#16811569	1	1	yes	characteristic
428	C₂	#20291370	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#405	C₄
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×D₄
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	#405	#6157
hypocenter	C₅₃	#20803
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
5. #20290653	C₂	semidirect
6. #20290652	C₂	semidirect
7. #20290651	C₂	semidirect
28. #16810581	C₄	semidirect
29. #16810580	C₄	semidirect
30. #16810579	C₄	semidirect
31. #16810578	C₄	semidirect
32. #16810577	C₄	semidirect
35. C₅₃×#405	C₄	semidirect
76. #13133128	C₂×C₄	semidirect
77. #13133127	C₂×C₄	semidirect
78. #13133127	C₂×C₄	semidirect
79. C₄×#88322	C₂×C₄	semidirect
80. C₈×Dic₅₃	C₂×C₄	semidirect
81. C₈×Dic₅₃	C₂×C₄	semidirect
85. #97×C₅₃	C₂×C₄	semidirect
86. C₄×C₈×C₅₃	C₂×C₄	semidirect
114. M₁₆×C₅₃	C₄⋊C₄	semidirect
115. C₂×C₈×C₅₃	C₄⋊C₄	semidirect
128. #11719917	C₄⋊C₄	semidirect
129. C₂×#88322	C₄⋊C₄	semidirect
133. #11719912	C₄⋊C₄	semidirect
134. C₈×D₅₃	C₄⋊C₄	semidirect
135. C₈×D₅₃	C₄⋊C₄	semidirect
243. #405	#6157	semidirect
250. C₅₃	#20803	semidirect
296. #97	C₂×#6157	semidirect
300. C₄×C₈	C₂×#6157	semidirect
321. M₁₆	C₄⋊#6157	semidirect
334. C₂×C₈	C₄⋊#6157	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811634	2
C₂×#16811633	2
C₂²×#13133286	2
#20291196	2
#20290653	2
#20290652	2
#20290651	2
#405×D₅₃	2
#20803	53