Subgroups — #28316360

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291521	C₂	1	1	yes	maximal, characteristic, complemented
2	#20291520	C₂	1	1	yes	maximal, characteristic, complemented
3	#20290302	C₂	1	2	yes	maximal, complemented
4	#20290300	C₂	1	2	yes	maximal, complemented
5	SD₁₆×#88324	C₂	1	2	yes	maximal, complemented
6	#20290295	C₂	1	2	yes	maximal, complemented
7	#20290263	C₂	1	1	yes	maximal, characteristic, complemented
8	#20290009	C₂	1	1	yes	maximal, characteristic, complemented
9	#20290007	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#13133238	C₂	1	1	yes	maximal, characteristic
12	#16811672	C₂²	1	2	yes	—
13	Q₈×#88324	C₂²	1	2	yes	—
14	#16811651	—	2	2	no	—
15	#16811651	C₂²	1	2	yes	—
16	#16811650	—	2	2	no	—
17	D₄×#88324	C₂²	1	2	yes	—
18	#16811642	—	2	1	no	—
19	#16811642	C₂²	1	1	yes	characteristic
20	#16811642	C₂²	1	1	yes	characteristic
21	#16811640	—	2	1	no	—
22	C₄×D₅₃⋊C₈	—	2	1	no	—
23	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
24	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×#88324	—	2	1	no	—
26	#16811487	C₂²	1	2	yes	—
27	#16811484	C₂²	1	2	yes	—
28	#16811473	C₂²	1	2	yes	—
29	Q₈⋊#88324	C₂²	1	2	yes	—
30	#16811471	C₂²	1	2	yes	—
31	D₄⋊#88324	C₂²	1	2	yes	—
32	C₈×#88324	C₂²	1	2	yes	—
33	C₈×#88324	C₂²	1	2	yes	—
34	C₂×#13133238	C₄	1	1	yes	characteristic
35	C₄×#11719949	C₂²	1	1	yes	characteristic
36	C₄×#11719947	C₂²	1	1	yes	characteristic
37	C₄×#11719925	C₄	1	1	yes	characteristic
38	#16810715	C₂²	1	1	yes	characteristic
39	C₄×#11719913	C₄	1	1	yes	characteristic
40	SD₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
41	#16810710	C₂²	1	1	yes	characteristic
42	C₄×#11719922	C₄	1	1	yes	characteristic
43	#16810703	C₂²	1	1	yes	characteristic
44	#16810658	C₄	1	1	yes	characteristic
45	#16810618	C₄	1	1	yes	characteristic
46	#16810616	C₄	1	1	yes	characteristic
47	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
48	C₄×SD₁₆×C₅₃	C₄	1	1	yes	characteristic
49	C₂×D₅₃⋊C₈	—	2	1	no	—
50	C₂×D₅₃⋊C₈	—	2	2	no	—
51	C₂×D₅₃⋊C₈	D₄	1	2	yes	—
52	C₂²×#88324	—	2	2	no	—
53	#13133284	—	2	2	no	—
54	C₄⋊#88324	—	2	2	no	—
55	C₄⋊#88324	C₂³	1	2	yes	—
56	C₄⋊#88324	C₂³	1	2	yes	—
57	#13133274	—	2	1	no	—
58	#13133274	—	2	1	no	—
59	#13133274	—	2	2	no	—
60	C₂²⋊#88324	—	2	2	no	—
61	C₄×#88324	—	2	1	no	—
62	C₄×#88324	—	2	1	no	—
63	C₄×#88324	—	2	1	no	—
64	C₄×#88324	—	2	2	no	—
65	C₄×#88324	—	2	2	no	—
66	C₄×#88324	D₄	1	1	yes	characteristic
67	C₄×#88324	D₄	1	1	yes	characteristic
68	C₄×#88324	C₂³	1	2	yes	—
69	C₄×#88324	C₂³	1	2	yes	—
70	C₂×#11719949	C₂×C₄	1	1	yes	characteristic
71	C₂×#11719947	C₂×C₄	1	1	yes	characteristic
72	#13133238	C₈	1	8	yes	complemented
73	C₂×#11719925	C₂×C₄	1	1	yes	characteristic
74	C₂×#11719913	C₂×C₄	1	1	yes	characteristic
75	C₂×#11719922	C₂×C₄	1	1	yes	characteristic
76	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
78	Q₈×Dic₅₃	C₂³	1	1	yes	characteristic
79	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
80	#13133182	C₂³	1	1	yes	characteristic
81	C₄×D₂₁₂	—	2	1	no	—
82	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
83	D₄×Dic₅₃	C₂³	1	1	yes	characteristic
84	#13133179	—	2	1	no	—
85	#13133179	C₂³	1	1	yes	characteristic
86	C₄×C₂²⋊D₅₃	—	2	1	no	—
87	#13133177	—	2	1	no	—
88	#13133175	—	2	1	no	—
89	#13133175	C₂×C₄	1	1	yes	characteristic
90	#13133175	C₂×C₄	1	1	yes	characteristic
91	#13133173	—	2	1	no	—
92	C₂×C₄×Dic₅₃	—	2	1	no	—
93	C₄²×D₅₃	—	2	1	no	—
94	C₄²×D₅₃	C₂³	1	1	yes	characteristic
95	#13133150	C₂×C₄	1	1	yes	characteristic
96	#13133150	C₂×C₄	1	1	yes	characteristic
97	#13133149	C₂×C₄	1	1	yes	characteristic
98	#13133142	C₂×C₄	1	1	yes	characteristic
99	#13133141	C₂×C₄	1	1	yes	characteristic
100	#13133140	C₂×C₄	1	1	yes	characteristic
101	#13133139	C₂×C₄	1	1	yes	characteristic
102	#13133138	C₂×C₄	1	1	yes	characteristic
103	#13133137	C₂×C₄	1	1	yes	characteristic
104	C₄×#88322	C₂×C₄	1	1	yes	characteristic
105	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
106	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
107	C₂×SD₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
108	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
109	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
110	#106×C₅₃	C₂×C₄	1	1	yes	characteristic
111	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
112	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
113	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
114	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	—	2	1	no	—
117	C₂×C₂²⋊D₅₃	—	2	1	no	—
118	C₂²×Dic₅₃	—	2	1	no	—
119	C₂²×Dic₅₃	—	2	1	no	—
120	#11719949	—	2	2	no	—
121	#11719949	C₂×C₈	1	4	yes	complemented
122	#11719947	—	2	2	no	—
123	#11719947	C₂×C₈	1	4	yes	complemented
124	C₂×D₂₁₂	—	2	1	no	—
125	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄×D₅₃	—	2	1	no	—
127	C₂×C₄×D₅₃	—	2	1	no	—
128	C₂×C₄×D₅₃	—	2	1	no	—
129	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
132	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
133	C₂×#88324	—	2	2	no	—
134	C₂×#88324	—	2	2	no	—
135	C₂×#88324	—	2	2	no	—
136	C₂×#88324	—	4	4	no	—
137	C₂×#88324	C₂×D₄	1	2	yes	—
138	C₂×#88324	C₂×D₄	1	2	yes	—
139	D₅₃⋊C₈	—	2	4	no	—
140	D₅₃⋊C₈	—	2	4	no	—
141	D₅₃⋊C₈	—	4	4	no	—
142	SD₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
143	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₄⋊C₄×C₅₃	—	2	1	no	—
146	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₂²⋊C₄×C₅₃	—	2	1	no	—
149	C₄²×C₅₃	—	2	1	no	—
150	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₂²⋊Dic₅₃	—	2	1	no	—
152	#11719925	C₂×C₈	1	4	yes	complemented
153	#11719922	C₂×C₈	1	4	yes	complemented
154	#11719921	—	2	1	no	—
155	#11719921	—	2	1	no	—
156	#11719921	—	2	1	no	—
157	#11719921	—	2	1	no	—
158	C₄⋊Dic₅₃	—	2	1	no	—
159	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
160	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
161	#11719919	—	2	1	no	—
162	#11719919	—	2	1	no	—
163	C₄×Dic₅₃	—	2	1	no	—
164	C₄×Dic₅₃	—	2	1	no	—
165	C₄×Dic₅₃	—	2	1	no	—
166	C₄×Dic₅₃	—	2	1	no	—
167	C₄×Dic₅₃	—	2	1	no	—
168	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
169	C₄×Dic₅₃	#41	1	1	yes	characteristic
170	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
171	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
172	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
173	#11719913	C₂×C₈	1	4	yes	complemented
174	C₈×D₅₃	—	2	2	no	—
175	C₈×D₅₃	C₂×C₈	1	4	yes	complemented
176	C₂³×C₅₃	—	2	1	no	—
177	C₂²×D₅₃	—	2	1	no	—
178	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
179	Q₈×C₅₃	—	2	1	no	—
180	Q₈×C₅₃	C₂²×C₈	1	2	yes	—
181	D₄×C₅₃	—	2	1	no	—
182	D₄×C₅₃	C₂²×C₈	1	2	yes	—
183	C₂×C₄×C₅₃	—	2	1	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	1	no	—
186	C₂×C₄×C₅₃	—	4	2	no	—
187	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
190	C₂²⋊D₅₃	—	2	4	no	—
191	C₂×Dic₅₃	—	2	1	no	—
192	C₂×Dic₅₃	—	2	4	no	—
193	C₂×Dic₅₃	—	2	1	no	—
194	C₂×Dic₅₃	—	2	1	no	—
195	C₂×Dic₅₃	—	4	2	no	—
196	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
197	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
198	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
199	D₂₁₂	—	2	4	no	—
200	D₂₁₂	—	2	1	no	—
201	D₂₁₂	C₂²×C₈	1	2	yes	—
202	C₄×D₅₃	—	2	2	no	—
203	C₄×D₅₃	—	2	2	no	—
204	C₄×D₅₃	—	2	4	no	—
205	C₄×D₅₃	—	4	2	no	—
206	C₄×D₅₃	—	4	2	no	—
207	C₄×D₅₃	C₂²×C₈	1	2	yes	—
208	C₄×D₅₃	#131	1	2	yes	—
209	Dic₁₀₆	—	2	1	no	—
210	Dic₁₀₆	C₂²×C₈	1	2	yes	—
211	#88324	—	2	4	no	—
212	#88324	—	2	2	no	—
213	#88324	—	4	2	no	—
214	#88324	—	4	2	no	—
215	#88324	#135	1	4	yes	complemented
216	C₈×C₅₃	—	2	1	no	—
217	C₈×C₅₃	C₂²×C₈	1	2	yes	—
218	#88322	—	2	1	no	—
219	#88322	C₂²×C₈	1	2	yes	—
220	C₈×#135	—	53	1	no	maximal, Sylow
221	C₂²×C₅₃	—	2	2	no	—
222	C₂²×C₅₃	—	4	1	no	—
223	C₂²×C₅₃	—	4	1	no	—
224	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
225	C₂×D₅₃	—	2	1	no	—
226	C₂×D₅₃	—	2	1	no	—
227	C₂×D₅₃	—	2	2	no	—
228	C₂×D₅₃	—	4	1	no	—
229	C₂×D₅₃	—	4	1	no	—
230	C₂×D₅₃	C₈×D₄	1	2	yes	—
231	C₄×C₅₃	—	2	2	no	—
232	C₄×C₅₃	—	2	1	no	—
233	C₄×C₅₃	—	4	1	no	—
234	C₄×C₅₃	—	4	1	no	—
235	C₄×C₅₃	#443	1	2	yes	—
236	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
237	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
238	Dic₅₃	—	2	2	no	—
239	Dic₅₃	—	2	1	no	—
240	Dic₅₃	—	4	1	no	—
241	Dic₅₃	—	4	1	no	—
242	Dic₅₃	C₈×D₄	1	2	yes	—
243	Dic₅₃	C₂×#135	1	2	yes	—
244	C₈×#41	—	53	1	no	—
245	C₈×#41	—	53	1	no	—
246	C₄×#135	—	53	1	no	—
247	C₈×Q₁₆	—	53	2	no	—
248	C₈×SD₁₆	—	53	2	no	—
249	C₈×SD₁₆	—	53	2	no	—
250	C₈×D₈	—	53	2	no	—
251	#1564	—	53	1	no	—
252	#1476	—	53	1	no	—
253	#1476	—	53	1	no	—
254	C₂×C₈²	—	53	1	no	—
255	C₂×C₅₃	—	4	2	no	—
256	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
257	C₂×C₅₃	C₄×#135	1	1	yes	characteristic
258	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
259	D₅₃	—	2	2	no	—
260	D₅₃	—	4	2	no	—
261	C₂×#135	—	53	1	no	—
262	C₄×#41	—	53	1	no	—
263	C₄×#41	—	53	1	no	—
264	C₈×Q₈	—	53	2	no	—
265	C₈×Q₈	—	53	2	no	—
266	C₄×Q₁₆	—	53	1	no	—
267	C₄×Q₁₆	—	53	1	no	—
268	C₄×SD₁₆	—	53	1	no	—
269	C₄×SD₁₆	—	53	1	no	—
270	C₄×SD₁₆	—	53	1	no	—
271	C₄×SD₁₆	#6157	1	1	yes	characteristic
272	C₄×D₈	—	53	1	no	—
273	C₄×D₈	—	53	1	no	—
274	C₈×D₄	—	53	2	no	—
275	C₈×D₄	—	53	2	no	—
276	C₈×D₄	—	106	2	no	—
277	C₈×D₄	—	106	2	no	—
278	#431	—	53	1	no	—
279	#431	—	53	1	no	—
280	#431	—	106	1	no	—
281	#431	—	106	1	no	—
282	#427	—	53	1	no	—
283	#416	—	53	1	no	—
284	#416	—	53	1	no	—
285	C₂×C₄×C₈	—	53	1	no	—
286	C₂×C₄×C₈	—	53	1	no	—
287	C₂×C₄×C₈	—	53	1	no	—
288	C₂×C₄×C₈	—	106	1	no	—
289	C₂×C₄×C₈	—	106	1	no	—
290	#335	—	53	2	no	—
291	#334	—	53	2	no	—
292	Q₈⋊C₈	—	53	2	no	—
293	Q₈⋊C₈	—	53	2	no	—
294	D₄⋊C₈	—	53	2	no	—
295	D₄⋊C₈	—	53	2	no	—
296	C₈²	—	53	2	no	—
297	C₈²	—	53	2	no	—
298	C₅₃	C₈×#135	1	1	yes	characteristic, complemented, Sylow
299	C₂×#41	—	53	1	no	—
300	C₂×#41	—	53	1	no	—
301	#135	—	53	8	no	—
302	C₂×Q₁₆	—	53	1	no	—
303	C₂×SD₁₆	—	53	1	no	—
304	C₂×SD₁₆	C₂×#6157	1	1	yes	characteristic
305	C₂×D₈	—	53	1	no	—
306	C₂²×C₈	—	53	1	no	—
307	C₂²×C₈	—	53	2	no	—
308	C₂²×C₈	—	53	1	no	—
309	C₂²×C₈	—	106	2	no	—
310	C₂²×C₈	—	106	2	no	—
311	C₂²×C₈	—	106	1	no	—
312	C₄×Q₈	—	53	1	no	—
313	C₄×Q₈	—	53	1	no	—
314	C₄×Q₈	—	53	1	no	—
315	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
316	C₄×D₄	—	53	1	no	—
317	C₄×D₄	—	53	1	no	—
318	C₄×D₄	—	53	1	no	—
319	C₄×D₄	—	106	1	no	—
320	C₄×D₄	—	106	1	no	—
321	C₄×D₄	—	106	1	no	—
322	C₄×D₄	—	106	1	no	—
323	C₄×D₄	C₂×#6157	1	1	yes	characteristic
324	#117	—	53	1	no	—
325	#117	—	53	1	no	—
326	#117	—	106	1	no	—
327	#117	—	106	1	no	—
328	C₂×C₄²	—	53	1	no	—
329	C₂×C₄²	—	106	1	no	—
330	C₂×C₄²	—	106	1	no	—
331	#107	—	53	1	no	—
332	#107	—	53	1	no	—
333	#106	—	53	1	no	—
334	#106	C₂×#6157	1	1	yes	characteristic
335	C₄⋊C₈	—	53	2	no	—
336	C₄⋊C₈	—	53	2	no	—
337	C₄⋊C₈	—	106	2	no	—
338	C₄⋊C₈	—	106	2	no	—
339	#103	—	53	1	no	—
340	#103	—	53	1	no	—
341	#103	—	53	1	no	—
342	#103	C₂×#6157	1	1	yes	characteristic
343	#102	—	53	1	no	—
344	#102	—	53	1	no	—
345	#102	—	53	1	no	—
346	#102	C₂×#6157	1	1	yes	characteristic
347	C₂²⋊C₈	—	106	2	no	—
348	C₂²⋊C₈	—	106	2	no	—
349	C₂²⋊C₈	—	106	1	no	—
350	C₂²⋊C₈	—	106	1	no	—
351	C₄×C₈	—	53	2	no	—
352	C₄×C₈	—	53	1	no	—
353	C₄×C₈	—	53	1	no	—
354	C₄×C₈	—	53	1	no	—
355	C₄×C₈	—	53	1	no	—
356	C₄×C₈	—	53	2	no	—
357	C₄×C₈	—	53	1	no	—
358	C₄×C₈	—	106	1	no	—
359	C₄×C₈	—	106	1	no	—
360	C₄×C₈	—	106	1	no	—
361	C₄×C₈	—	106	2	no	—
362	C₄×C₈	—	106	2	no	—
363	C₄×C₈	C₂×#6157	1	1	yes	characteristic
364	#41	—	53	4	no	—
365	#41	—	53	4	no	—
366	#41	—	106	2	no	—
367	#41	—	106	2	no	—
368	C₂×Q₈	—	53	1	no	—
369	C₂×Q₈	C₂²×#6157	1	1	yes	characteristic
370	C₂×D₄	—	53	1	no	—
371	C₂×D₄	—	106	1	no	—
372	C₂×D₄	—	106	1	no	—
373	C₂×D₄	C₂²×#6157	1	1	yes	characteristic
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	53	1	no	—
376	C₂²×C₄	—	53	1	no	—
377	C₂²×C₄	—	53	1	no	—
378	C₂²×C₄	—	106	1	no	—
379	C₂²×C₄	—	106	1	no	—
380	C₂²×C₄	—	106	1	no	—
381	C₂²×C₄	—	106	1	no	—
382	C₂²×C₄	—	106	1	no	—
383	Q₁₆	—	53	4	no	—
384	SD₁₆	—	53	4	no	—
385	SD₁₆	D₅₃⋊C₈	1	4	yes	complemented
386	D₈	—	53	4	no	—
387	C₂×C₈	—	53	2	no	—
388	C₂×C₈	—	53	1	no	—
389	C₂×C₈	—	53	4	no	—
390	C₂×C₈	—	53	2	no	—
391	C₂×C₈	—	53	1	no	—
392	C₂×C₈	—	106	2	no	—
393	C₂×C₈	—	106	4	no	—
394	C₂×C₈	—	106	2	no	—
395	C₂×C₈	—	106	4	no	—
396	C₂×C₈	—	106	2	no	—
397	C₂×C₈	—	106	2	no	—
398	C₂×C₈	—	212	4	no	—
399	C₂×C₈	—	212	4	no	—
400	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
401	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	53	1	no	—
404	C₄⋊C₄	—	53	1	no	—
405	C₄⋊C₄	—	106	1	no	—
406	C₄⋊C₄	—	106	1	no	—
407	C₄⋊C₄	—	106	1	no	—
408	C₄⋊C₄	C₂²×#6157	1	1	yes	characteristic
409	C₄⋊C₄	C₂²×#6157	1	1	yes	characteristic
410	C₂²⋊C₄	—	2	1	no	—
411	C₂²⋊C₄	—	106	1	no	—
412	C₂²⋊C₄	—	106	1	no	—
413	C₂²⋊C₄	—	106	1	no	—
414	C₂²⋊C₄	—	106	1	no	—
415	C₂²⋊C₄	—	106	1	no	—
416	C₄²	—	2	1	no	—
417	C₄²	—	53	1	no	—
418	C₄²	—	53	1	no	—
419	C₄²	—	53	1	no	—
420	C₄²	—	106	1	no	—
421	C₄²	—	106	1	no	—
422	C₄²	—	106	1	no	—
423	C₄²	—	106	1	no	—
424	C₄²	—	106	1	no	—
425	C₄²	C₂²×#6157	1	1	yes	characteristic
426	C₂³	—	2	1	no	—
427	C₂³	—	53	1	no	—
428	C₂³	—	106	1	no	—
429	Q₈	—	2	1	no	—
430	Q₈	—	53	2	no	—
431	Q₈	—	106	1	no	—
432	Q₈	C₂×D₅₃⋊C₈	1	2	yes	—
433	D₄	—	2	1	no	—
434	D₄	—	53	2	no	—
435	D₄	—	106	4	no	—
436	D₄	—	106	4	no	—
437	D₄	—	106	1	no	—
438	D₄	C₂×D₅₃⋊C₈	1	2	yes	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	53	1	no	—
444	C₂×C₄	—	53	1	no	—
445	C₂×C₄	—	53	2	no	—
446	C₂×C₄	—	53	2	no	—
447	C₂×C₄	—	53	1	no	—
448	C₂×C₄	—	106	2	no	—
449	C₂×C₄	—	106	4	no	—
450	C₂×C₄	—	106	4	no	—
451	C₂×C₄	—	106	1	no	—
452	C₂×C₄	—	106	2	no	—
453	C₂×C₄	—	106	1	no	—
454	C₂×C₄	—	106	1	no	—
455	C₂×C₄	—	212	2	no	—
456	C₂×C₄	—	212	2	no	—
457	C₂×C₄	—	212	2	no	—
458	C₂×C₄	D₄×#6157	1	1	yes	characteristic
459	C₂×C₄	D₄×#6157	1	1	yes	characteristic
460	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
461	C₈	—	2	1	no	—
462	C₈	—	53	4	no	—
463	C₈	—	53	2	no	—
464	C₈	—	106	2	no	—
465	C₈	—	106	4	no	—
466	C₈	—	106	1	no	—
467	C₈	—	212	2	no	—
468	C₈	—	212	2	no	—
469	C₈	C₂×D₅₃⋊C₈	1	2	yes	—
470	C₂²	—	2	2	no	—
471	C₂²	—	4	1	no	—
472	C₂²	—	4	1	no	—
473	C₂²	—	53	2	no	—
474	C₂²	—	106	1	no	—
475	C₂²	—	106	1	no	—
476	C₂²	—	106	2	no	—
477	C₂²	—	212	1	no	—
478	C₂²	—	212	1	no	—
479	C₂²	C₂×D₄×#6157	1	1	yes	characteristic
480	C₄	—	2	2	no	—
481	C₄	—	2	1	no	—
482	C₄	—	4	1	no	—
483	C₄	—	4	1	no	—
484	C₄	—	53	2	no	—
485	C₄	—	53	2	no	—
486	C₄	—	106	2	no	—
487	C₄	—	106	1	no	—
488	C₄	—	212	1	no	—
489	C₄	—	212	1	no	—
490	C₄	#16811666	1	2	yes	—
491	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
492	C₄	D₅₃⋊#131	1	1	yes	characteristic
493	C₂	—	4	2	no	—
494	C₂	—	106	2	no	—
495	C₂	—	212	2	no	—
496	C₂	D₄×D₅₃⋊C₈	1	1	yes	characteristic
497	C₂	#135×#6157	1	1	yes	characteristic
498	C₂	C₂×#16811666	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₄×SD₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133238	C₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₄×SD₁₆	#6157
hypocenter	C₅₃	C₈×#135
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #20291521	C₂	semidirect
2. #20291520	C₂	semidirect
3. #20290302	C₂	semidirect
4. #20290300	C₂	semidirect
5. SD₁₆×#88324	C₂	semidirect
6. #20290295	C₂	semidirect
7. #20290263	C₂	semidirect
8. #20290009	C₂	semidirect
9. #20290007	C₂	semidirect
10. C₈×D₅₃⋊C₈	C₂	semidirect
72. #13133238	C₈	semidirect
121. #11719949	C₂×C₈	semidirect
123. #11719947	C₂×C₈	semidirect
142. SD₁₆×C₅₃	C₂×C₈	semidirect
152. #11719925	C₂×C₈	semidirect
153. #11719922	C₂×C₈	semidirect
173. #11719913	C₂×C₈	semidirect
175. C₈×D₅₃	C₂×C₈	semidirect
215. #88324	#135	semidirect
298. C₅₃	C₈×#135	semidirect
385. SD₁₆	D₅₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20291521	2
#20291520	2
#20290302	2
#20290300	2
SD₁₆×#88324	2
#20290295	2
#20290263	2
#20290009	2
#20290007	2
C₈×D₅₃⋊C₈	2
C₄×#13133238	2
C₈×#135	53