Subgroups — #28316260

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291410	C₂	1	2	yes	maximal
2	#20290287	C₂	1	2	yes	maximal, complemented
3	#20290286	C₂	1	2	yes	maximal, complemented
4	#20290284	C₂	1	2	yes	maximal, complemented
5	#20290276	C₂	1	1	yes	maximal, characteristic, complemented
6	#20290275	C₂	1	1	yes	maximal, characteristic, complemented
7	#20290269	C₂	1	1	yes	maximal, characteristic, complemented
8	#20290263	C₂	1	1	yes	maximal, characteristic, complemented
9	#20290262	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16811484	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×#11719947	C₂	1	1	yes	maximal, characteristic
12	#16811648	—	2	2	no	—
13	#16811647	—	2	2	no	—
14	#16811645	C₂²	1	2	yes	—
15	#16811644	C₂²	1	2	yes	—
16	#16811642	C₂²	1	2	yes	—
17	#16811639	C₂²	1	2	yes	—
18	#16811626	C₂²	1	2	yes	—
19	C₂×C₄⋊#88324	C₂²	1	2	yes	—
20	#16811596	—	2	2	no	—
21	#16811587	—	2	2	no	—
22	#16811487	C₂²	1	2	yes	—
23	#16811487	C₂²	1	2	yes	—
24	#16811486	C₂²	1	2	yes	—
25	#16811486	C₂²	1	2	yes	—
26	#16811485	C₂²	1	2	yes	—
27	#16811484	C₂²	1	1	yes	characteristic
28	#16811484	C₂²	1	1	yes	characteristic
29	#16811484	C₂²	1	2	yes	—
30	#16811484	C₂²	1	1	yes	characteristic
31	#16811484	C₂²	1	1	yes	characteristic
32	C₄×#11719947	C₂²	1	1	yes	characteristic
33	C₈×Dic₁₀₆	C₄	1	1	yes	characteristic
34	#16810747	C₄	1	1	yes	characteristic
35	D₄×#88322	C₄	1	1	yes	characteristic
36	C₈×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
37	C₈×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
38	#16810686	C₄	1	1	yes	characteristic
39	#16810686	C₄	1	1	yes	characteristic
40	#16810675	C₂²	1	1	yes	characteristic
41	#16810673	C₂²	1	1	yes	characteristic
42	#16810673	C₂²	1	1	yes	characteristic
43	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
44	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
45	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
46	C₈×D₄×C₅₃	C₄	1	1	yes	characteristic
47	C₂×#11719940	—	2	2	no	—
48	C₂²×#88324	—	2	2	no	—
49	#13133284	—	2	2	no	—
50	C₄⋊#88324	—	2	2	no	—
51	C₄⋊#88324	C₂³	1	2	yes	—
52	C₄⋊#88324	C₂³	1	2	yes	—
53	C₄⋊#88324	C₂³	1	2	yes	—
54	C₄⋊#88324	C₂³	1	2	yes	—
55	#13133274	—	2	2	no	—
56	#13133274	—	2	2	no	—
57	C₂²⋊#88324	—	2	2	no	—
58	C₂²⋊#88324	—	2	2	no	—
59	#13133272	—	2	2	no	—
60	#13133272	—	2	2	no	—
61	C₄×#88324	—	2	2	no	—
62	C₄×#88324	—	2	2	no	—
63	C₂×#11719947	D₄	1	1	yes	characteristic
64	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
65	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
69	#13133182	Q₈	1	1	yes	characteristic
70	D₄×Dic₅₃	Q₈	1	1	yes	characteristic
71	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
73	#13133177	Q₈	1	1	yes	characteristic
74	#13133177	Q₈	1	1	yes	characteristic
75	#13133175	D₄	1	1	yes	characteristic
76	#13133173	D₄	1	1	yes	characteristic
77	#13133173	D₄	1	1	yes	characteristic
78	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
79	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
80	C₄²×D₅₃	C₂³	1	1	yes	characteristic
81	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
82	#13133146	C₂×C₄	1	1	yes	characteristic
83	#13133146	C₂×C₄	1	1	yes	characteristic
84	#13133146	C₂×C₄	1	1	yes	characteristic
85	#13133146	C₂×C₄	1	1	yes	characteristic
86	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
87	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
88	#13133129	C₂×C₄	1	1	yes	characteristic
89	#13133129	C₂×C₄	1	1	yes	characteristic
90	C₄×#88322	C₂×C₄	1	1	yes	characteristic
91	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
92	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
93	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
94	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
95	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
96	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
97	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
98	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
99	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
100	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
101	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
102	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
103	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
104	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
105	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
106	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₂²⋊D₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₂²⋊D₅₃	C₄⋊C₄	1	1	yes	characteristic
111	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
114	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
115	#11719947	—	2	4	no	—
116	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
119	C₂×Dic₁₀₆	C₄⋊C₄	1	1	yes	characteristic
120	#11719940	—	4	4	no	—
121	C₂×#88324	—	2	2	no	—
122	C₂×#88324	—	2	2	no	—
123	C₂×#88324	—	2	2	no	—
124	C₂×#88324	—	2	2	no	—
125	C₂×#88324	—	4	4	no	—
126	C₂×C₈×C₅₃	—	2	2	no	—
127	C₂×C₈×C₅₃	—	2	2	no	—
128	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
133	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
134	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
135	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
137	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
138	#11719921	C₄⋊C₄	1	1	yes	characteristic
139	#11719921	C₄⋊C₄	1	1	yes	characteristic
140	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
141	#11719919	C₄⋊C₄	1	1	yes	characteristic
142	#11719919	C₄⋊C₄	1	1	yes	characteristic
143	#11719919	C₄⋊C₄	1	1	yes	characteristic
144	#11719919	C₄⋊C₄	1	1	yes	characteristic
145	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
148	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
149	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
150	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
151	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
152	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
153	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
154	C₂×#88322	—	4	1	no	—
155	C₂×#88322	—	4	1	no	—
156	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
157	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
158	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
159	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
160	C₈×D₅₃	—	2	2	no	—
161	C₈×D₅₃	—	4	1	no	—
162	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
165	D₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	1	no	—
167	C₂×C₄×C₅₃	—	2	1	no	—
168	C₂×C₄×C₅₃	—	2	1	no	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
175	C₂²⋊D₅₃	—	2	2	no	—
176	C₂²⋊D₅₃	—	2	2	no	—
177	C₂×Dic₅₃	—	2	2	no	—
178	C₂×Dic₅₃	—	4	1	no	—
179	C₂×Dic₅₃	—	4	1	no	—
180	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
185	C₂×Dic₅₃	#135	1	4	yes	—
186	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
187	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
188	C₄×D₅₃	—	2	1	no	—
189	C₄×D₅₃	—	2	1	no	—
190	C₄×D₅₃	—	2	2	no	—
191	C₄×D₅₃	—	4	1	no	—
192	Dic₁₀₆	—	2	2	no	—
193	#88324	—	4	2	no	—
194	#88324	—	4	2	no	—
195	#88324	—	4	2	no	—
196	#88324	—	4	2	no	—
197	C₈×C₅₃	—	2	1	no	—
198	C₈×C₅₃	—	2	1	no	—
199	C₈×C₅₃	—	2	1	no	—
200	C₈×C₅₃	#131	1	2	yes	—
201	#88322	—	2	1	no	—
202	#88322	—	2	1	no	—
203	#88322	—	2	1	no	—
204	#88322	—	2	1	no	—
205	#17827	—	53	1	no	maximal, Sylow
206	C₂²×C₅₃	—	2	1	no	—
207	C₂²×C₅₃	—	2	1	no	—
208	C₂²×C₅₃	—	2	1	no	—
209	C₂²×C₅₃	—	2	1	no	—
210	C₂²×C₅₃	—	2	1	no	—
211	C₂²×C₅₃	#427	1	2	yes	—
212	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
213	C₂×D₅₃	—	2	1	no	—
214	C₂×D₅₃	—	2	1	no	—
215	C₂×D₅₃	—	2	1	no	—
216	C₄×C₅₃	—	2	1	no	—
217	C₄×C₅₃	—	2	1	no	—
218	C₄×C₅₃	—	2	1	no	—
219	C₄×C₅₃	—	2	1	no	—
220	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
221	C₄×C₅₃	#568	1	1	yes	characteristic
222	Dic₅₃	—	2	1	no	—
223	Dic₅₃	—	2	1	no	—
224	Dic₅₃	—	2	1	no	—
225	Dic₅₃	—	2	1	no	—
226	Dic₅₃	—	2	1	no	—
227	Dic₅₃	#427	1	2	yes	—
228	Dic₅₃	C₂×#135	1	2	yes	—
229	C₈×#41	—	53	1	no	—
230	#2925	—	53	2	no	—
231	#1572	—	53	2	no	—
232	#1572	—	53	2	no	—
233	#1571	—	53	2	no	—
234	#1568	—	53	1	no	—
235	#1568	—	53	1	no	—
236	#1566	—	53	1	no	—
237	#1564	—	53	1	no	—
238	#1564	—	53	1	no	—
239	C₂×#334	—	53	1	no	—
240	C₂×C₅₃	—	2	2	no	—
241	C₂×C₅₃	—	4	1	no	—
242	C₂×C₅₃	#2906	1	1	yes	characteristic
243	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
244	C₂×C₅₃	#2916	1	1	yes	characteristic
245	D₅₃	—	4	1	no	—
246	C₄×#41	—	53	1	no	—
247	C₈×Q₈	—	53	1	no	—
248	C₈×Q₈	—	53	1	no	—
249	C₈×D₄	—	53	1	no	—
250	C₈×D₄	—	53	1	no	—
251	C₈×D₄	—	53	1	no	—
252	C₈×D₄	—	53	1	no	—
253	C₈×D₄	—	53	1	no	—
254	C₈×D₄	#6157	1	1	yes	characteristic
255	#433	—	106	2	no	—
256	#433	—	106	2	no	—
257	#432	—	53	2	no	—
258	#432	—	53	2	no	—
259	#431	—	53	1	no	—
260	#431	—	53	1	no	—
261	#431	—	53	1	no	—
262	#431	—	53	2	no	—
263	#431	—	53	2	no	—
264	#423	—	53	2	no	—
265	C₂×C₄⋊C₈	—	53	2	no	—
266	#408	—	106	2	no	—
267	#405	—	106	2	no	—
268	C₂×C₄×C₈	—	53	1	no	—
269	C₂×C₄×C₈	—	53	1	no	—
270	C₂×C₄×C₈	—	53	1	no	—
271	#335	—	53	2	no	—
272	#335	—	53	2	no	—
273	#335	—	53	2	no	—
274	#335	—	53	2	no	—
275	#334	—	53	1	no	—
276	#334	—	53	1	no	—
277	#334	—	53	2	no	—
278	#334	—	53	1	no	—
279	#334	—	53	1	no	—
280	#334	—	53	2	no	—
281	C₅₃	#17827	1	1	yes	characteristic, complemented, Sylow
282	C₂×#41	—	53	1	no	—
283	C₂×M₁₆	—	106	2	no	—
284	C₂²×C₈	—	53	1	no	—
285	C₂²×C₈	—	53	1	no	—
286	C₂²×C₈	—	53	1	no	—
287	C₂²×C₈	—	53	1	no	—
288	C₂²×C₈	—	106	2	no	—
289	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
290	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
291	C₄×Q₈	—	53	1	no	—
292	C₄×Q₈	—	53	1	no	—
293	C₄×D₄	—	53	1	no	—
294	C₄×D₄	—	53	1	no	—
295	C₄×D₄	—	53	1	no	—
296	C₄×D₄	—	53	1	no	—
297	C₄×D₄	—	53	1	no	—
298	C₄×D₄	C₂×#6157	1	1	yes	characteristic
299	#117	—	53	1	no	—
300	#117	—	53	1	no	—
301	#117	—	53	1	no	—
302	C₂×C₄²	—	53	1	no	—
303	C₂×C₄²	—	53	1	no	—
304	C₂×C₄²	—	53	1	no	—
305	C₄⋊C₈	—	53	1	no	—
306	C₄⋊C₈	—	53	1	no	—
307	C₄⋊C₈	—	53	1	no	—
308	C₄⋊C₈	—	53	1	no	—
309	C₄⋊C₈	—	53	2	no	—
310	C₄⋊C₈	—	53	2	no	—
311	C₄⋊C₈	—	53	2	no	—
312	C₄⋊C₈	—	53	2	no	—
313	C₄⋊C₈	—	53	1	no	—
314	C₄⋊C₈	—	106	2	no	—
315	C₄⋊C₈	—	106	2	no	—
316	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
317	C₂²⋊C₈	—	53	1	no	—
318	C₂²⋊C₈	—	53	1	no	—
319	C₂²⋊C₈	—	53	1	no	—
320	C₂²⋊C₈	—	53	1	no	—
321	C₂²⋊C₈	—	106	2	no	—
322	C₂²⋊C₈	—	106	2	no	—
323	C₂²⋊C₈	—	106	2	no	—
324	C₂²⋊C₈	—	106	2	no	—
325	C₂²⋊C₈	C₂×#6157	1	1	yes	characteristic
326	C₂²⋊C₈	C₂×#6157	1	1	yes	characteristic
327	#97	—	106	2	no	—
328	#97	—	106	2	no	—
329	C₄×C₈	—	53	1	no	—
330	C₄×C₈	—	53	1	no	—
331	C₄×C₈	—	53	1	no	—
332	C₄×C₈	—	53	1	no	—
333	C₄×C₈	—	53	1	no	—
334	C₄×C₈	—	53	1	no	—
335	C₄×C₈	—	53	1	no	—
336	C₄×C₈	—	53	1	no	—
337	C₄×C₈	—	53	1	no	—
338	C₄×C₈	—	106	2	no	—
339	C₄×C₈	—	106	2	no	—
340	C₄×C₈	C₂×#6157	1	1	yes	characteristic
341	#41	—	106	4	no	—
342	C₂×Q₈	—	53	1	no	—
343	C₂×D₄	—	53	1	no	—
344	C₂×D₄	—	53	1	no	—
345	C₂×D₄	C₄⋊#6157	1	1	yes	characteristic
346	C₂²×C₄	—	53	1	no	—
347	C₂²×C₄	—	53	1	no	—
348	C₂²×C₄	—	53	1	no	—
349	C₂²×C₄	—	53	1	no	—
350	C₂²×C₄	—	53	1	no	—
351	C₂²×C₄	—	53	1	no	—
352	C₂²×C₄	—	53	1	no	—
353	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
354	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
355	M₁₆	—	212	4	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	53	1	no	—
359	C₂×C₈	—	53	1	no	—
360	C₂×C₈	—	53	1	no	—
361	C₂×C₈	—	53	1	no	—
362	C₂×C₈	—	106	2	no	—
363	C₂×C₈	—	106	2	no	—
364	C₂×C₈	—	106	2	no	—
365	C₂×C₈	—	106	2	no	—
366	C₂×C₈	—	106	2	no	—
367	C₂×C₈	—	212	1	no	—
368	C₂×C₈	—	212	1	no	—
369	C₂×C₈	—	212	1	no	—
370	C₂×C₈	—	212	4	no	—
371	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
372	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
373	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
374	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
375	C₄⋊C₄	—	53	1	no	—
376	C₄⋊C₄	—	53	1	no	—
377	C₄⋊C₄	—	53	1	no	—
378	C₄⋊C₄	—	53	1	no	—
379	C₄⋊C₄	—	53	1	no	—
380	C₄⋊C₄	C₄⋊#6157	1	1	yes	characteristic
381	C₂²⋊C₄	—	53	1	no	—
382	C₂²⋊C₄	—	53	1	no	—
383	C₂²⋊C₄	—	53	1	no	—
384	C₂²⋊C₄	—	53	1	no	—
385	C₂²⋊C₄	C₄⋊#6157	1	1	yes	characteristic
386	C₂²⋊C₄	C₄⋊#6157	1	1	yes	characteristic
387	C₄²	—	53	1	no	—
388	C₄²	—	53	1	no	—
389	C₄²	—	53	1	no	—
390	C₄²	—	53	1	no	—
391	C₄²	—	53	1	no	—
392	C₄²	—	53	1	no	—
393	C₄²	—	53	1	no	—
394	C₄²	—	53	1	no	—
395	C₄²	—	53	1	no	—
396	C₄²	C₂²×#6157	1	1	yes	characteristic
397	C₂³	—	53	1	no	—
398	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
399	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
400	Q₈	—	106	2	no	—
401	D₄	—	2	2	no	—
402	D₄	—	106	2	no	—
403	D₄	—	106	2	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	53	4	no	—
409	C₂×C₄	—	53	1	no	—
410	C₂×C₄	—	53	1	no	—
411	C₂×C₄	—	53	1	no	—
412	C₂×C₄	—	53	1	no	—
413	C₂×C₄	—	53	1	no	—
414	C₂×C₄	—	53	1	no	—
415	C₂×C₄	—	53	1	no	—
416	C₂×C₄	—	106	2	no	—
417	C₂×C₄	—	106	2	no	—
418	C₂×C₄	—	106	1	no	—
419	C₂×C₄	—	106	1	no	—
420	C₂×C₄	—	212	1	no	—
421	C₂×C₄	—	212	1	no	—
422	C₂×C₄	—	212	1	no	—
423	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
424	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
425	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
426	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
427	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
428	C₈	—	2	1	no	—
429	C₈	—	2	1	no	—
430	C₈	—	2	1	no	—
431	C₈	—	106	1	no	—
432	C₈	—	106	1	no	—
433	C₈	—	106	1	no	—
434	C₈	—	106	1	no	—
435	C₈	—	212	2	no	—
436	C₈	—	212	2	no	—
437	C₈	—	212	2	no	—
438	C₈	—	212	2	no	—
439	C₈	#13133304	1	2	yes	—
440	C₂²	—	2	1	no	—
441	C₂²	—	2	1	no	—
442	C₂²	—	2	1	no	—
443	C₂²	—	2	1	no	—
444	C₂²	—	2	1	no	—
445	C₂²	—	106	1	no	—
446	C₂²	—	106	1	no	—
447	C₂²	—	106	1	no	—
448	C₂²	#16811633	1	2	yes	—
449	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
450	C₄	—	2	1	no	—
451	C₄	—	2	1	no	—
452	C₄	—	2	1	no	—
453	C₄	—	2	1	no	—
454	C₄	—	53	2	no	—
455	C₄	—	53	2	no	—
456	C₄	—	106	1	no	—
457	C₄	—	106	1	no	—
458	C₄	—	106	1	no	—
459	C₄	—	106	1	no	—
460	C₄	—	106	1	no	—
461	C₄	C₂×#13133304	1	1	yes	characteristic
462	C₄	#16811702	1	1	yes	characteristic
463	C₂	—	2	2	no	—
464	C₂	—	4	1	no	—
465	C₂	—	212	1	no	—
466	C₂	#20291360	1	1	yes	characteristic
467	C₂	C₂×#16811633	1	1	yes	characteristic
468	C₂	#20291376	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₈×D₄×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#11719947	D₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₈×D₄	#6157
hypocenter	C₅₃	#17827
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #20290287	C₂	semidirect
3. #20290286	C₂	semidirect
4. #20290284	C₂	semidirect
5. #20290276	C₂	semidirect
6. #20290275	C₂	semidirect
7. #20290269	C₂	semidirect
8. #20290263	C₂	semidirect
9. #20290262	C₂	semidirect
10. C₂×#16811484	C₂	semidirect
281. C₅₃	#17827	semidirect

Maximal subgroups

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

subgroup	index
#20291410	2
#20290287	2
#20290286	2
#20290284	2
#20290276	2
#20290275	2
#20290269	2
#20290263	2
#20290262	2
C₂×#16811484	2
C₈×#11719947	2
#17827	53