Subgroups — #28313938

Order 13568.

704 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291524	C₂	1	1	yes	maximal, characteristic
2	#20291524	C₂	1	1	yes	maximal, characteristic
3	#20289938	C₂	1	1	yes	maximal, characteristic, complemented
4	#20289936	C₂	1	1	yes	maximal, characteristic, complemented
5	#20289936	C₂	1	1	yes	maximal, characteristic, complemented
6	#20289934	C₂	1	1	yes	maximal, characteristic, complemented
7	#20289932	C₂	1	1	yes	maximal, characteristic, complemented
8	#20289932	C₂	1	1	yes	maximal, characteristic, complemented
9	#20289930	C₂	1	1	yes	maximal, characteristic, complemented
10	#20289926	C₂	1	1	yes	maximal, characteristic, complemented
11	#20289924	C₂	1	1	yes	maximal, characteristic, complemented
12	#20289920	C₂	1	1	yes	maximal, characteristic, complemented
13	#20289916	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈×#11719940	C₂	1	1	yes	maximal, characteristic, complemented
15	#20283355	C₂	1	1	yes	maximal, characteristic
16	#16811674	C₄	1	2	yes	—
17	#16811674	C₄	1	2	yes	—
18	#16811658	C₄	1	2	yes	—
19	#16811658	C₄	1	2	yes	—
20	#16811654	C₄	1	2	yes	—
21	#16811654	C₄	1	2	yes	—
22	#16811654	C₄	1	2	yes	—
23	#16811654	C₄	1	2	yes	—
24	#16811640	C₂²	1	1	yes	characteristic
25	#16811640	C₂²	1	1	yes	characteristic
26	#16811640	C₂²	1	1	yes	characteristic
27	#16811640	C₂²	1	1	yes	characteristic
28	#16811639	C₂²	1	1	yes	characteristic
29	#16811639	C₂²	1	1	yes	characteristic
30	C₄×#11719940	C₂²	1	1	yes	characteristic
31	C₄×#11719940	C₂²	1	1	yes	characteristic
32	C₄×#11719940	C₂²	1	1	yes	characteristic
33	C₄×#11719940	C₂²	1	1	yes	characteristic
34	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
35	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
36	#88324⋊C₈	C₂²	1	1	yes	characteristic
37	#88324⋊C₈	C₂²	1	1	yes	characteristic
38	#88324⋊C₈	C₂²	1	1	yes	characteristic
39	#88324⋊C₈	C₂²	1	1	yes	characteristic
40	#88324⋊C₈	C₂²	1	1	yes	characteristic
41	#88324⋊C₈	C₂²	1	1	yes	characteristic
42	#88324⋊C₈	C₂²	1	1	yes	characteristic
43	#88324⋊C₈	C₂²	1	1	yes	characteristic
44	#16811464	C₂²	1	1	yes	characteristic
45	#16811464	C₂²	1	1	yes	characteristic
46	#16811464	C₂²	1	1	yes	characteristic
47	#16811464	C₂²	1	1	yes	characteristic
48	C₈×#88324	C₂²	1	1	yes	characteristic
49	C₈×#88324	C₂²	1	1	yes	characteristic
50	C₈×#88324	C₂²	1	1	yes	characteristic
51	C₈×#88324	C₂²	1	1	yes	characteristic
52	#41×Dic₅₃	C₂²	1	1	yes	characteristic
53	#16810681	C₄	1	1	yes	characteristic
54	#16810681	C₄	1	1	yes	characteristic
55	#16810681	C₄	1	1	yes	characteristic
56	#16810681	C₄	1	1	yes	characteristic
57	#16810673	C₂²	1	1	yes	characteristic
58	#16810673	C₂²	1	1	yes	characteristic
59	#16810673	C₂²	1	1	yes	characteristic
60	#16810673	C₂²	1	1	yes	characteristic
61	#16810651	C₄	1	1	yes	characteristic
62	#16810649	C₄	1	1	yes	characteristic
63	#16810590	C₄	1	1	yes	characteristic
64	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
65	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
66	C₅₃×#408	C₄	1	1	yes	characteristic
67	C₂×#11719940	C₂×C₄	1	2	yes	—
68	C₂×#11719940	C₂×C₄	1	2	yes	—
69	C₂×#11719940	C₂×C₄	1	2	yes	—
70	C₂×#11719940	C₂×C₄	1	2	yes	—
71	C₂²×#88324	C₂×C₄	1	2	yes	—
72	C₂²×#88324	C₂×C₄	1	2	yes	—
73	#13133284	C₂×C₄	1	2	yes	—
74	#13133284	C₂×C₄	1	2	yes	—
75	#13133284	C₂×C₄	1	2	yes	—
76	#13133284	C₂×C₄	1	2	yes	—
77	C₄⋊#88324	C₂×C₄	1	2	yes	—
78	C₄⋊#88324	C₂×C₄	1	2	yes	—
79	C₂²⋊#88324	C₂×C₄	1	2	yes	—
80	C₂²⋊#88324	C₂×C₄	1	2	yes	—
81	C₂²⋊#88324	C₂×C₄	1	2	yes	—
82	C₂²⋊#88324	C₂×C₄	1	2	yes	—
83	C₂²⋊#88324	C₂×C₄	1	2	yes	—
84	C₂²⋊#88324	C₂×C₄	1	2	yes	—
85	#13133272	C₂×C₄	1	2	yes	—
86	#13133272	C₂×C₄	1	2	yes	—
87	#13133272	C₂×C₄	1	2	yes	—
88	#13133272	C₂×C₄	1	2	yes	—
89	C₄×#88324	C₂×C₄	1	2	yes	—
90	C₄×#88324	C₂×C₄	1	2	yes	—
91	C₄×#88324	C₂³	1	1	yes	characteristic
92	C₄×#88324	C₂³	1	1	yes	characteristic
93	C₄×#88324	C₂³	1	1	yes	characteristic
94	C₄×#88324	C₂³	1	1	yes	characteristic
95	C₄×#88324	C₂³	1	1	yes	characteristic
96	C₄×#88324	C₂³	1	1	yes	characteristic
97	C₄×#88324	C₂³	1	1	yes	characteristic
98	C₄×#88324	C₂³	1	1	yes	characteristic
99	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
100	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
101	Q₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
102	Q₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
103	D₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
104	D₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
105	D₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
106	D₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
107	D₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
108	D₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
109	#13133176	C₂×C₄	1	1	yes	characteristic
110	#13133176	C₂×C₄	1	1	yes	characteristic
111	#13133176	C₂×C₄	1	1	yes	characteristic
112	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
113	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
114	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
115	#13133146	C₂×C₄	1	1	yes	characteristic
116	#13133146	C₂×C₄	1	1	yes	characteristic
117	#13133146	C₂×C₄	1	1	yes	characteristic
118	#13133146	C₂×C₄	1	1	yes	characteristic
119	#13133146	C₂×C₄	1	1	yes	characteristic
120	#13133146	C₂×C₄	1	1	yes	characteristic
121	#13133146	C₂×C₄	1	1	yes	characteristic
122	#13133146	C₂×C₄	1	1	yes	characteristic
123	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
124	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
125	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
126	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
127	#13133127	C₂×C₄	1	1	yes	characteristic
128	#13133127	C₂×C₄	1	1	yes	characteristic
129	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
130	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
131	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
132	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
133	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
134	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
135	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
136	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
137	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
138	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
139	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
140	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
141	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
142	#41×C₅₃	—	4	1	no	—
143	#41×C₅₃	—	4	1	no	—
144	C₂×Q₈×C₅₃	C₄²	1	1	yes	characteristic
145	C₂×D₄×C₅₃	C₄²	1	1	yes	characteristic
146	C₂×D₄×C₅₃	C₄²	1	1	yes	characteristic
147	C₂×D₄×C₅₃	C₄²	1	1	yes	characteristic
148	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
153	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
154	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
156	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
157	#11719940	—	2	4	no	—
158	#11719940	—	2	4	no	—
159	#11719940	—	4	2	no	—
160	#11719940	—	4	2	no	—
161	C₂×#88324	—	2	4	no	—
162	C₂×#88324	—	4	2	no	—
163	C₂×#88324	C₂²×C₄	1	2	yes	—
164	C₂×#88324	C₂²×C₄	1	2	yes	—
165	C₂×#88324	C₂²×C₄	1	2	yes	—
166	C₂×#88324	C₂²×C₄	1	2	yes	—
167	C₂×#88324	C₂²×C₄	1	2	yes	—
168	C₂×#88324	C₂²×C₄	1	2	yes	—
169	C₂×#88324	C₂²×C₄	1	2	yes	—
170	C₂×#88324	C₂²×C₄	1	2	yes	—
171	M₁₆×C₅₃	—	4	1	no	—
172	C₂×C₈×C₅₃	—	4	1	no	—
173	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
174	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
176	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
177	C₂²⋊Dic₅₃	C₄²	1	1	yes	characteristic
178	C₂²⋊Dic₅₃	C₄²	1	1	yes	characteristic
179	C₂²⋊Dic₅₃	C₄²	1	1	yes	characteristic
180	C₂²⋊Dic₅₃	C₄²	1	1	yes	characteristic
181	C₂²⋊Dic₅₃	C₄²	1	1	yes	characteristic
182	C₂²⋊Dic₅₃	C₄²	1	1	yes	characteristic
183	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
184	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
185	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
186	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
187	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
188	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
189	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
190	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
191	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
192	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
193	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
194	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
195	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
196	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
197	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
198	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
199	#11719917	—	4	1	no	—
200	C₂×#88322	—	4	1	no	—
201	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
202	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
203	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
204	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
205	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
206	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
207	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
208	Q₈×C₅₃	—	4	1	no	—
209	D₄×C₅₃	—	4	1	no	—
210	D₄×C₅₃	—	4	1	no	—
211	D₄×C₅₃	—	4	1	no	—
212	C₂×C₄×C₅₃	—	2	1	no	—
213	C₂×C₄×C₅₃	—	2	1	no	—
214	C₂×C₄×C₅₃	—	2	1	no	—
215	C₂×C₄×C₅₃	—	2	1	no	—
216	C₂×C₄×C₅₃	—	2	1	no	—
217	C₂×C₄×C₅₃	—	2	1	no	—
218	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
219	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
220	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
221	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
222	C₂×Dic₅₃	—	2	2	no	—
223	C₂×Dic₅₃	—	2	2	no	—
224	C₂×Dic₅₃	—	2	2	no	—
225	C₂×Dic₅₃	—	2	2	no	—
226	C₂×Dic₅₃	—	2	2	no	—
227	C₂×Dic₅₃	—	2	2	no	—
228	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
229	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
230	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
231	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
232	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
233	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
234	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
235	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
236	#88324	—	2	2	no	—
237	#88324	—	2	2	no	—
238	#88324	—	2	2	no	—
239	#88324	—	2	2	no	—
240	#88324	—	2	2	no	—
241	#88324	—	2	2	no	—
242	#88324	—	2	2	no	—
243	#88324	#131	1	4	yes	—
244	C₈×C₅₃	—	2	1	no	—
245	C₈×C₅₃	—	2	1	no	—
246	C₈×C₅₃	—	2	1	no	—
247	C₈×C₅₃	—	2	1	no	—
248	#88322	—	2	1	no	—
249	#88322	—	2	1	no	—
250	#88322	—	2	1	no	—
251	#88322	—	2	1	no	—
252	#17340	—	53	1	no	maximal, Sylow
253	C₂²×C₅₃	—	2	1	no	—
254	C₂²×C₅₃	—	2	1	no	—
255	C₂²×C₅₃	—	2	1	no	—
256	C₂²×C₅₃	—	2	1	no	—
257	C₂²×C₅₃	—	2	1	no	—
258	C₂²×C₅₃	—	2	1	no	—
259	C₂²×C₅₃	—	2	1	no	—
260	C₂²×C₅₃	—	2	1	no	—
261	C₂²×C₅₃	—	2	1	no	—
262	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
263	C₄×C₅₃	—	2	1	no	—
264	C₄×C₅₃	—	2	1	no	—
265	C₄×C₅₃	—	2	1	no	—
266	C₄×C₅₃	#568	1	1	yes	characteristic
267	C₄×C₅₃	#568	1	1	yes	characteristic
268	Dic₅₃	—	2	1	no	—
269	Dic₅₃	—	2	1	no	—
270	Dic₅₃	—	2	1	no	—
271	Dic₅₃	—	2	1	no	—
272	Dic₅₃	—	2	1	no	—
273	Dic₅₃	—	2	1	no	—
274	Dic₅₃	C₂×#131	1	2	yes	—
275	Dic₅₃	#568	1	2	yes	—
276	#2965	—	53	1	no	—
277	#2965	—	53	1	no	—
278	#2922	—	53	1	no	—
279	#1455	—	53	1	no	—
280	#1454	—	53	1	no	—
281	#1454	—	53	1	no	—
282	#1453	—	53	1	no	—
283	#1453	—	53	1	no	—
284	#1453	—	53	1	no	—
285	#1452	—	53	1	no	—
286	#1451	—	53	1	no	—
287	#1451	—	53	1	no	—
288	#1450	—	53	1	no	—
289	C₈×M₁₆	—	53	1	no	—
290	C₈×M₁₆	—	53	1	no	—
291	C₂×C₅₃	—	4	1	no	—
292	C₂×C₅₃	—	4	1	no	—
293	C₂×C₅₃	—	4	1	no	—
294	C₂×C₅₃	#2873	1	1	yes	characteristic
295	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
296	C₂×C₅₃	#2875	1	1	yes	characteristic
297	C₄×#41	—	53	1	no	—
298	#446	—	53	2	no	—
299	#446	—	53	2	no	—
300	#436	—	53	2	no	—
301	#436	—	53	2	no	—
302	#435	—	53	2	no	—
303	#435	—	53	2	no	—
304	#435	—	53	2	no	—
305	#435	—	53	2	no	—
306	#433	—	53	1	no	—
307	#433	—	53	1	no	—
308	#433	—	53	1	no	—
309	#433	—	53	1	no	—
310	#431	—	53	1	no	—
311	#431	—	53	1	no	—
312	#431	—	53	1	no	—
313	#431	—	53	1	no	—
314	#431	—	53	1	no	—
315	#431	—	53	1	no	—
316	#431	—	53	1	no	—
317	#431	—	53	1	no	—
318	#431	—	53	1	no	—
319	#431	—	53	1	no	—
320	#424	—	53	1	no	—
321	#424	—	53	1	no	—
322	#408	—	53	1	no	—
323	#408	#6157	1	1	yes	characteristic
324	C₄×M₁₆	—	53	1	no	—
325	C₄×M₁₆	—	53	1	no	—
326	C₄×M₁₆	—	53	1	no	—
327	C₄×M₁₆	—	53	1	no	—
328	C₄×M₁₆	—	53	1	no	—
329	C₂×C₄×C₈	—	53	1	no	—
330	C₂×C₄×C₈	—	53	1	no	—
331	C₂×C₄×C₈	—	53	1	no	—
332	#322	—	53	1	no	—
333	#322	—	53	1	no	—
334	#322	—	53	1	no	—
335	#322	—	53	1	no	—
336	#322	—	53	1	no	—
337	#322	—	53	1	no	—
338	#322	—	53	1	no	—
339	#322	—	53	1	no	—
340	#322	—	53	1	no	—
341	#322	—	53	1	no	—
342	#322	—	53	1	no	—
343	#322	—	53	1	no	—
344	C₈²	—	53	1	no	—
345	C₈²	—	53	1	no	—
346	C₈²	—	53	1	no	—
347	C₈²	—	53	1	no	—
348	C₅₃	#17340	1	1	yes	characteristic, complemented, Sylow
349	C₂×#41	C₂×#6157	1	1	yes	characteristic
350	C₂×M₁₆	—	53	2	no	—
351	C₂×M₁₆	—	53	2	no	—
352	C₂×M₁₆	—	53	2	no	—
353	C₂×M₁₆	—	53	2	no	—
354	C₂×M₁₆	—	53	1	no	—
355	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
356	C₂²×C₈	—	53	2	no	—
357	C₂²×C₈	—	53	2	no	—
358	C₂²×C₈	—	53	1	no	—
359	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
360	C₄×Q₈	—	53	1	no	—
361	C₄×Q₈	—	53	1	no	—
362	C₄×D₄	—	53	1	no	—
363	C₄×D₄	—	53	1	no	—
364	C₄×D₄	—	53	1	no	—
365	C₄×D₄	—	53	1	no	—
366	C₄×D₄	—	53	1	no	—
367	C₄×D₄	—	53	1	no	—
368	#117	—	53	1	no	—
369	#117	—	53	1	no	—
370	#117	—	53	1	no	—
371	C₂×C₄²	—	53	1	no	—
372	C₂×C₄²	—	53	1	no	—
373	C₂×C₄²	—	53	1	no	—
374	C₄⋊C₈	—	53	2	no	—
375	C₄⋊C₈	—	53	2	no	—
376	C₄⋊C₈	—	53	2	no	—
377	C₄⋊C₈	—	53	2	no	—
378	C₄⋊C₈	—	53	2	no	—
379	C₄⋊C₈	—	53	2	no	—
380	C₄⋊C₈	—	53	1	no	—
381	C₄⋊C₈	—	53	1	no	—
382	C₄⋊C₈	—	53	1	no	—
383	C₄⋊C₈	—	53	1	no	—
384	C₄⋊C₈	—	53	1	no	—
385	C₄⋊C₈	—	53	1	no	—
386	C₄⋊C₈	—	53	1	no	—
387	C₄⋊C₈	—	53	1	no	—
388	C₂²⋊C₈	—	53	2	no	—
389	C₂²⋊C₈	—	53	2	no	—
390	C₂²⋊C₈	—	53	2	no	—
391	C₂²⋊C₈	—	53	2	no	—
392	C₂²⋊C₈	—	53	2	no	—
393	C₂²⋊C₈	—	53	2	no	—
394	C₂²⋊C₈	—	53	1	no	—
395	C₂²⋊C₈	—	53	1	no	—
396	C₂²⋊C₈	—	53	1	no	—
397	C₂²⋊C₈	—	53	1	no	—
398	C₂²⋊C₈	C₂×#6157	1	1	yes	characteristic
399	C₂²⋊C₈	C₂×#6157	1	1	yes	characteristic
400	C₂²⋊C₈	C₂×#6157	1	1	yes	characteristic
401	C₂²⋊C₈	C₂×#6157	1	1	yes	characteristic
402	#97	—	53	2	no	—
403	#97	—	53	2	no	—
404	#97	—	53	2	no	—
405	#97	—	53	2	no	—
406	#97	—	53	1	no	—
407	#97	—	53	1	no	—
408	C₄×C₈	—	53	1	no	—
409	C₄×C₈	—	53	1	no	—
410	C₄×C₈	—	53	1	no	—
411	C₄×C₈	—	53	1	no	—
412	C₄×C₈	—	53	1	no	—
413	C₄×C₈	—	53	1	no	—
414	C₄×C₈	—	53	1	no	—
415	C₄×C₈	—	53	1	no	—
416	C₄×C₈	—	53	2	no	—
417	C₄×C₈	—	53	2	no	—
418	C₄×C₈	—	53	1	no	—
419	C₄×C₈	—	53	1	no	—
420	C₄×C₈	—	53	1	no	—
421	C₄×C₈	—	53	1	no	—
422	C₄×C₈	—	53	1	no	—
423	C₄×C₈	—	53	1	no	—
424	#41	—	4	1	no	—
425	#41	—	4	1	no	—
426	C₂×Q₈	C₄×#6157	1	1	yes	characteristic
427	C₂×D₄	C₄×#6157	1	1	yes	characteristic
428	C₂×D₄	C₄×#6157	1	1	yes	characteristic
429	C₂×D₄	C₄×#6157	1	1	yes	characteristic
430	C₂²×C₄	—	53	1	no	—
431	C₂²×C₄	—	53	1	no	—
432	C₂²×C₄	—	53	1	no	—
433	C₂²×C₄	—	53	1	no	—
434	C₂²×C₄	—	53	1	no	—
435	C₂²×C₄	—	53	1	no	—
436	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
437	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
438	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
439	M₁₆	—	4	1	no	—
440	M₁₆	—	106	4	no	—
441	M₁₆	—	106	4	no	—
442	M₁₆	—	212	1	no	—
443	M₁₆	—	212	2	no	—
444	M₁₆	—	212	2	no	—
445	C₂×C₈	—	4	1	no	—
446	C₂×C₈	—	53	2	no	—
447	C₂×C₈	—	53	2	no	—
448	C₂×C₈	—	53	2	no	—
449	C₂×C₈	—	53	2	no	—
450	C₂×C₈	—	53	2	no	—
451	C₂×C₈	—	53	2	no	—
452	C₂×C₈	—	53	2	no	—
453	C₂×C₈	—	53	2	no	—
454	C₂×C₈	—	53	1	no	—
455	C₂×C₈	—	53	1	no	—
456	C₂×C₈	—	53	1	no	—
457	C₂×C₈	—	53	1	no	—
458	C₂×C₈	—	106	4	no	—
459	C₂×C₈	—	212	1	no	—
460	C₂×C₈	—	212	2	no	—
461	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
462	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
463	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
464	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
465	C₄⋊C₄	—	53	1	no	—
466	C₄⋊C₄	—	53	1	no	—
467	C₄⋊C₄	—	53	1	no	—
468	C₄⋊C₄	—	53	1	no	—
469	C₄⋊C₄	—	53	1	no	—
470	C₄⋊C₄	—	53	1	no	—
471	C₂²⋊C₄	—	53	1	no	—
472	C₂²⋊C₄	—	53	1	no	—
473	C₂²⋊C₄	—	53	1	no	—
474	C₂²⋊C₄	—	53	1	no	—
475	C₂²⋊C₄	—	53	1	no	—
476	C₂²⋊C₄	—	53	1	no	—
477	C₄²	—	53	1	no	—
478	C₄²	—	53	1	no	—
479	C₄²	—	53	1	no	—
480	C₄²	—	53	1	no	—
481	C₄²	—	53	1	no	—
482	C₄²	—	53	1	no	—
483	C₄²	—	53	1	no	—
484	C₄²	—	53	1	no	—
485	C₄²	—	53	1	no	—
486	C₄²	—	53	1	no	—
487	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
488	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
489	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
490	Q₈	—	4	1	no	—
491	D₄	—	4	1	no	—
492	D₄	—	4	1	no	—
493	D₄	—	4	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	53	1	no	—
501	C₂×C₄	—	53	1	no	—
502	C₂×C₄	—	53	1	no	—
503	C₂×C₄	—	53	1	no	—
504	C₂×C₄	—	53	1	no	—
505	C₂×C₄	—	53	1	no	—
506	C₂×C₄	—	53	1	no	—
507	C₂×C₄	—	53	1	no	—
508	C₂×C₄	—	106	2	no	—
509	C₂×C₄	—	106	2	no	—
510	C₂×C₄	—	106	2	no	—
511	C₂×C₄	—	106	2	no	—
512	C₂×C₄	—	106	2	no	—
513	C₂×C₄	—	106	2	no	—
514	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
515	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
516	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
517	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
518	C₈	—	2	1	no	—
519	C₈	—	2	1	no	—
520	C₈	—	2	1	no	—
521	C₈	—	2	1	no	—
522	C₈	—	53	4	no	—
523	C₈	—	106	2	no	—
524	C₈	—	106	2	no	—
525	C₈	—	106	2	no	—
526	C₈	—	106	2	no	—
527	C₈	—	106	2	no	—
528	C₈	—	106	2	no	—
529	C₈	—	106	2	no	—
530	C₈	—	106	1	no	—
531	C₈	—	106	1	no	—
532	C₈	—	106	1	no	—
533	C₈	—	106	1	no	—
534	C₂²	—	2	1	no	—
535	C₂²	—	2	1	no	—
536	C₂²	—	2	1	no	—
537	C₂²	—	2	1	no	—
538	C₂²	—	2	1	no	—
539	C₂²	—	2	1	no	—
540	C₂²	—	2	1	no	—
541	C₂²	—	2	1	no	—
542	C₂²	—	2	1	no	—
543	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
544	C₄	—	2	1	no	—
545	C₄	—	2	1	no	—
546	C₄	—	2	1	no	—
547	C₄	—	53	2	no	—
548	C₄	—	53	2	no	—
549	C₄	—	106	1	no	—
550	C₄	—	106	1	no	—
551	C₄	—	106	1	no	—
552	C₄	—	106	1	no	—
553	C₄	—	106	1	no	—
554	C₄	—	106	1	no	—
555	C₄	#16811701	1	1	yes	characteristic
556	C₄	#16811701	1	1	yes	characteristic
557	C₂	—	4	1	no	—
558	C₂	—	4	1	no	—
559	C₂	—	4	1	no	—
560	C₂	#20291279	1	1	yes	characteristic
561	C₂	#131×#6157	1	1	yes	characteristic
562	C₂	#20291285	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#408	C₄
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	#408	#6157
hypocenter	C₅₃	#17340
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
3. #20289938	C₂	semidirect
4. #20289936	C₂	semidirect
5. #20289936	C₂	semidirect
6. #20289934	C₂	semidirect
7. #20289932	C₂	semidirect
8. #20289932	C₂	semidirect
9. #20289930	C₂	semidirect
10. #20289926	C₂	semidirect
11. #20289924	C₂	semidirect
12. #20289920	C₂	semidirect
13. #20289916	C₂	semidirect
14. C₈×#11719940	C₂	semidirect
348. C₅₃	#17340	semidirect

Maximal subgroups

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

subgroup	index
#20291524	2
#20289938	2
#20289936	2
#20289934	2
#20289932	2
#20289930	2
#20289926	2
#20289924	2
#20289920	2
#20289916	2
C₈×#11719940	2
#20283355	2
#17340	53