Subgroups — #27391566

Order 13568.

886 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810754	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	#20275656	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275439	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275274	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275274	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274900	C₂	1	1	yes	maximal, characteristic
8	#20274900	C₂	1	1	yes	maximal, characteristic
9	#20274695	C₂	1	1	yes	maximal, characteristic
10	#20274590	C₂	1	1	yes	maximal, characteristic, complemented
11	#20274590	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄⋊C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
13	#20274499	C₂	1	1	yes	maximal, characteristic
14	#20274499	C₂	1	1	yes	maximal, characteristic
15	C₄×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
16	C₂²×#11719917	C₂²	1	1	yes	characteristic
17	C₂²×#11719917	C₂²	1	1	yes	characteristic
18	#16810754	—	2	4	no	—
19	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
27	C₄×#11719917	—	2	4	no	—
28	C₂×#13133128	C₂²	1	1	yes	characteristic
29	C₂×#13133128	C₂²	1	1	yes	characteristic
30	C₂×#13133128	C₂²	1	1	yes	characteristic
31	C₂×#13133128	C₂²	1	1	yes	characteristic
32	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
33	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
34	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
35	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
36	#16810378	C₂²	1	1	yes	characteristic
37	#16810378	C₂²	1	1	yes	characteristic
38	#16810378	C₂²	1	1	yes	characteristic
39	#16810378	C₂²	1	1	yes	characteristic
40	#16810378	C₂²	1	1	yes	characteristic
41	#16810378	C₂²	1	1	yes	characteristic
42	#16810378	C₂²	1	1	yes	characteristic
43	#16810378	C₂²	1	1	yes	characteristic
44	#16810378	C₂²	1	1	yes	characteristic
45	#16810378	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
47	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
48	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
49	C₅₃×#391	C₄	1	1	yes	characteristic
50	C₅₃×#389	C₄	1	1	yes	characteristic
51	C₅₃×#387	C₄	1	1	yes	characteristic
52	C₅₃×#387	C₄	1	1	yes	characteristic
53	C₅₃×#385	C₄	1	1	yes	characteristic
54	C₅₃×#385	C₄	1	1	yes	characteristic
55	C₅₃×#380	C₄	1	1	yes	characteristic
56	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
57	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
58	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
59	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
60	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
61	C₂×#11719917	—	2	2	no	—
62	C₂×#11719917	—	2	2	no	—
63	C₂×#11719917	—	2	2	no	—
64	C₂×#11719917	—	2	2	no	—
65	C₂×#11719917	—	2	2	no	—
66	C₂×#11719917	Q₈	1	4	yes	complemented
67	C₂²×#88322	C₂³	1	1	yes	characteristic
68	C₂²×#88322	C₂³	1	1	yes	characteristic
69	C₂²×#88322	C₂³	1	1	yes	characteristic
70	C₂²×#88322	C₂³	1	1	yes	characteristic
71	C₂²×#88322	C₂³	1	1	yes	characteristic
72	C₂²×#88322	C₂³	1	1	yes	characteristic
73	C₂²×#88322	C₂³	1	1	yes	characteristic
74	C₂²×#88322	C₂³	1	1	yes	characteristic
75	C₄⋊#88322	—	2	2	no	—
76	C₄⋊#88322	—	2	2	no	—
77	C₄⋊#88322	—	2	2	no	—
78	C₄⋊#88322	—	2	2	no	—
79	C₄⋊#88322	—	2	2	no	—
80	C₄⋊#88322	—	2	2	no	—
81	C₂²⋊#88322	—	2	2	no	—
82	C₂²⋊#88322	—	2	2	no	—
83	#13133128	—	2	2	no	—
84	#13133128	—	2	2	no	—
85	#13133128	—	2	2	no	—
86	#13133128	D₄	1	4	yes	—
87	C₄×#88322	—	2	2	no	—
88	C₄×#88322	—	2	2	no	—
89	C₄×#88322	—	2	2	no	—
90	C₄×#88322	—	2	2	no	—
91	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
94	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
95	C₂²⋊Q₈×C₅₃	—	2	4	no	—
96	C₄×Q₈×C₅₃	—	2	2	no	—
97	C₄×Q₈×C₅₃	—	2	2	no	—
98	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
107	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
108	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
110	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
111	C₂×C₄²×C₅₃	—	2	2	no	—
112	C₂×C₄²×C₅₃	—	2	2	no	—
113	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
114	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
115	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
121	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
122	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
123	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
124	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
125	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
126	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
127	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
128	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
129	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
130	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
131	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×Q₈×C₅₃	—	2	2	no	—
133	C₂×Q₈×C₅₃	—	2	2	no	—
134	C₂×Q₈×C₅₃	—	4	1	no	—
135	C₂²×C₄×C₅₃	—	2	1	no	—
136	C₂²×C₄×C₅₃	—	2	1	no	—
137	C₂²×C₄×C₅₃	—	2	1	no	—
138	C₂²×C₄×C₅₃	—	2	1	no	—
139	C₂²×C₄×C₅₃	—	2	2	no	—
140	C₂²×C₄×C₅₃	—	2	2	no	—
141	C₂²×C₄×C₅₃	—	2	2	no	—
142	C₂²×C₄×C₅₃	—	2	2	no	—
143	C₂²×C₄×C₅₃	—	4	1	no	—
144	C₂²×C₄×C₅₃	—	4	1	no	—
145	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₅₃	C₂²×C₄	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×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
154	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
159	C₂²×C₄×C₅₃	#41	1	2	yes	—
160	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
161	C₄⋊C₄×C₅₃	—	2	2	no	—
162	C₄⋊C₄×C₅₃	—	2	2	no	—
163	C₄⋊C₄×C₅₃	—	2	2	no	—
164	C₄⋊C₄×C₅₃	—	2	2	no	—
165	C₄⋊C₄×C₅₃	—	2	2	no	—
166	C₄⋊C₄×C₅₃	—	2	2	no	—
167	C₄⋊C₄×C₅₃	—	2	2	no	—
168	C₄⋊C₄×C₅₃	—	4	1	no	—
169	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	complemented
170	C₂²⋊C₄×C₅₃	—	2	2	no	—
171	C₂²⋊C₄×C₅₃	—	2	2	no	—
172	C₂²⋊C₄×C₅₃	—	4	1	no	—
173	C₂²⋊C₄×C₅₃	—	4	1	no	—
174	C₄²×C₅₃	—	2	2	no	—
175	C₄²×C₅₃	—	2	2	no	—
176	C₄²×C₅₃	—	2	1	no	—
177	C₄²×C₅₃	—	2	1	no	—
178	C₄²×C₅₃	—	2	1	no	—
179	C₄²×C₅₃	—	2	1	no	—
180	C₄²×C₅₃	—	2	1	no	—
181	C₄²×C₅₃	—	2	1	no	—
182	C₄²×C₅₃	—	2	1	no	—
183	C₄²×C₅₃	—	2	1	no	—
184	C₄²×C₅₃	—	2	1	no	—
185	C₄²×C₅₃	—	2	1	no	—
186	C₄²×C₅₃	C₂×D₄	1	2	yes	—
187	C₄²×C₅₃	#41	1	2	yes	—
188	#11719917	—	2	8	no	—
189	#11719917	—	2	8	no	—
190	C₂×#88322	—	2	1	no	—
191	C₂×#88322	—	2	1	no	—
192	C₂×#88322	—	2	1	no	—
193	C₂×#88322	—	2	1	no	—
194	C₂×#88322	—	2	1	no	—
195	C₂×#88322	—	2	1	no	—
196	C₂×#88322	—	2	1	no	—
197	C₂×#88322	—	2	1	no	—
198	C₂×#88322	—	2	1	no	—
199	C₂×#88322	—	2	1	no	—
200	C₂×#88322	—	2	1	no	—
201	C₂×#88322	—	2	1	no	—
202	C₂×#88322	—	2	1	no	—
203	C₂×#88322	—	2	1	no	—
204	C₂×#88322	—	2	1	no	—
205	C₂×#88322	—	2	1	no	—
206	C₂×#88322	—	2	1	no	—
207	C₂×#88322	—	2	1	no	—
208	C₂×#88322	C₂×D₄	1	2	yes	—
209	C₂×#88322	C₂×D₄	1	2	yes	—
210	C₂×#88322	C₂×Q₈	1	2	yes	—
211	C₂×#88322	C₂×Q₈	1	2	yes	—
212	C₂×#88322	#41	1	2	yes	—
213	C₂×#88322	#41	1	2	yes	—
214	C₂³×C₅₃	—	2	1	no	—
215	C₂³×C₅₃	—	2	1	no	—
216	C₂³×C₅₃	—	2	1	no	—
217	C₂³×C₅₃	—	2	1	no	—
218	C₂³×C₅₃	—	2	1	no	—
219	C₂³×C₅₃	—	2	1	no	—
220	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
221	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
222	Q₈×C₅₃	—	4	4	no	—
223	C₂×C₄×C₅₃	—	2	1	no	—
224	C₂×C₄×C₅₃	—	2	1	no	—
225	C₂×C₄×C₅₃	—	2	1	no	—
226	C₂×C₄×C₅₃	—	2	1	no	—
227	C₂×C₄×C₅₃	—	2	1	no	—
228	C₂×C₄×C₅₃	—	2	1	no	—
229	C₂×C₄×C₅₃	—	2	1	no	—
230	C₂×C₄×C₅₃	—	2	1	no	—
231	C₂×C₄×C₅₃	—	2	1	no	—
232	C₂×C₄×C₅₃	—	2	1	no	—
233	C₂×C₄×C₅₃	—	2	1	no	—
234	C₂×C₄×C₅₃	—	2	1	no	—
235	C₂×C₄×C₅₃	—	2	1	no	—
236	C₂×C₄×C₅₃	—	2	1	no	—
237	C₂×C₄×C₅₃	—	2	2	no	—
238	C₂×C₄×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	—	2	2	no	—
240	C₂×C₄×C₅₃	—	2	2	no	—
241	C₂×C₄×C₅₃	—	2	1	no	—
242	C₂×C₄×C₅₃	—	2	1	no	—
243	C₂×C₄×C₅₃	—	2	1	no	—
244	C₂×C₄×C₅₃	—	2	1	no	—
245	C₂×C₄×C₅₃	—	2	1	no	—
246	C₂×C₄×C₅₃	—	2	1	no	—
247	C₂×C₄×C₅₃	—	2	1	no	—
248	C₂×C₄×C₅₃	—	2	1	no	—
249	C₂×C₄×C₅₃	—	2	1	no	—
250	C₂×C₄×C₅₃	—	2	1	no	—
251	C₂×C₄×C₅₃	—	2	1	no	—
252	C₂×C₄×C₅₃	—	2	1	no	—
253	C₂×C₄×C₅₃	—	2	1	no	—
254	C₂×C₄×C₅₃	—	2	1	no	—
255	C₂×C₄×C₅₃	—	2	1	no	—
256	C₂×C₄×C₅₃	—	2	1	no	—
257	C₂×C₄×C₅₃	—	2	1	no	—
258	C₂×C₄×C₅₃	—	2	1	no	—
259	C₂×C₄×C₅₃	—	4	4	no	—
260	C₂×C₄×C₅₃	—	4	4	no	—
261	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
262	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
263	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
264	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
265	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
266	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
267	C₂×C₄×C₅₃	#131	1	2	yes	—
268	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
269	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
270	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
271	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
272	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
273	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
274	#88322	—	2	2	no	—
275	#88322	—	2	2	no	—
276	#88322	—	2	2	no	—
277	#88322	—	4	1	no	—
278	#88322	—	4	1	no	—
279	#88322	—	4	1	no	—
280	#88322	—	4	1	no	—
281	#88322	C₂²⋊Q₈	1	4	yes	complemented
282	#21600	—	53	1	no	maximal, Sylow
283	C₂²×C₅₃	—	2	2	no	—
284	C₂²×C₅₃	—	2	2	no	—
285	C₂²×C₅₃	—	2	2	no	—
286	C₂²×C₅₃	—	4	1	no	—
287	C₂²×C₅₃	—	4	1	no	—
288	C₂²×C₅₃	—	4	1	no	—
289	C₂²×C₅₃	—	4	1	no	—
290	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
291	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
292	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
293	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
294	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
295	C₂²×C₅₃	#568	1	1	yes	characteristic
296	C₂²×C₅₃	#568	1	1	yes	characteristic
297	C₄×C₅₃	—	2	2	no	—
298	C₄×C₅₃	—	2	2	no	—
299	C₄×C₅₃	—	2	2	no	—
300	C₄×C₅₃	—	2	2	no	—
301	C₄×C₅₃	—	2	2	no	—
302	C₄×C₅₃	—	2	2	no	—
303	C₄×C₅₃	—	2	2	no	—
304	C₄×C₅₃	—	4	1	no	—
305	C₄×C₅₃	—	4	1	no	—
306	C₄×C₅₃	—	4	1	no	—
307	C₄×C₅₃	—	4	1	no	—
308	C₄×C₅₃	—	4	1	no	—
309	C₄×C₅₃	—	4	1	no	—
310	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
311	C₄×C₅₃	#525	1	1	yes	characteristic
312	C₄×C₅₃	#529	1	1	yes	characteristic
313	C₄×C₅₃	#533	1	1	yes	characteristic
314	C₂×#446	—	53	1	no	—
315	C₂×C₄×M₁₆	—	53	1	no	—
316	C₄×C₂²⋊Q₈	D₅₃	1	1	yes	characteristic
317	#1985	—	53	1	no	—
318	#1940	—	53	1	no	—
319	#1916	—	53	1	no	—
320	#1916	—	53	1	no	—
321	#1851	—	53	1	no	—
322	#1851	—	53	1	no	—
323	#1797	—	53	1	no	—
324	#1771	—	53	1	no	—
325	#1771	—	53	1	no	—
326	C₈×C₄⋊C₄	—	53	1	no	—
327	#1753	—	53	1	no	—
328	#1753	—	53	1	no	—
329	C₂×C₅₃	—	4	2	no	—
330	C₂×C₅₃	C₄×C₂²⋊Q₈	1	1	yes	characteristic
331	C₂×C₅₃	#2933	1	1	yes	characteristic
332	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
333	C₂×C₅₃	#2962	1	1	yes	characteristic
334	C₂×C₅₃	Q₈×M₁₆	1	1	yes	characteristic
335	C₂×C₅₃	#2965	1	1	yes	characteristic
336	C₂×C₅₃	#2967	1	1	yes	characteristic
337	C₂²×M₁₆	—	53	1	no	—
338	C₂²×M₁₆	—	53	1	no	—
339	C₂×C₂²⋊Q₈	Dic₅₃	1	1	yes	characteristic
340	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
341	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
342	#446	—	106	4	no	—
343	C₂×C₄⋊C₈	—	53	1	no	—
344	C₂×C₄⋊C₈	—	53	1	no	—
345	C₂×C₄⋊C₈	—	53	1	no	—
346	C₂×C₄⋊C₈	—	53	1	no	—
347	C₂×C₄⋊C₈	—	53	1	no	—
348	C₂×C₄⋊C₈	—	53	1	no	—
349	C₂×C₂²⋊C₈	—	53	1	no	—
350	C₂×C₂²⋊C₈	—	53	1	no	—
351	C₄×M₁₆	—	106	4	no	—
352	C₂×#97	—	53	1	no	—
353	C₂×#97	—	53	1	no	—
354	C₂×#97	—	53	1	no	—
355	C₂×#97	—	53	1	no	—
356	C₂×C₄×C₈	—	53	1	no	—
357	C₂×C₄×C₈	—	53	1	no	—
358	C₂×C₄×C₈	—	53	1	no	—
359	C₂×C₄×C₈	—	53	1	no	—
360	#391	Dic₅₃	1	1	yes	characteristic
361	#389	Dic₅₃	1	1	yes	characteristic
362	#387	Dic₅₃	1	1	yes	characteristic
363	#387	Dic₅₃	1	1	yes	characteristic
364	#385	Dic₅₃	1	1	yes	characteristic
365	#385	Dic₅₃	1	1	yes	characteristic
366	#380	Dic₅₃	1	1	yes	characteristic
367	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
368	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
369	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
370	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
371	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
372	#336	—	53	1	no	—
373	#336	—	53	1	no	—
374	#336	—	53	1	no	—
375	#336	—	53	1	no	—
376	#336	—	53	1	no	—
377	#336	—	53	1	no	—
378	#336	—	53	1	no	—
379	#336	—	53	1	no	—
380	#336	—	53	1	no	—
381	#336	—	53	1	no	—
382	C₅₃	#21600	1	1	yes	characteristic, complemented, Sylow
383	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
384	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
385	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
386	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
387	C₂×M₁₆	—	53	4	no	—
388	C₂×M₁₆	—	106	2	no	—
389	C₂×M₁₆	—	106	2	no	—
390	C₂×M₁₆	—	106	2	no	—
391	C₂×M₁₆	—	106	2	no	—
392	C₂×M₁₆	—	106	2	no	—
393	C₂²×C₈	—	53	1	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	—	53	1	no	—
399	C₂²×C₈	—	53	1	no	—
400	C₂²×C₈	—	53	1	no	—
401	C₂²⋊Q₈	—	2	4	no	—
402	C₄×Q₈	—	2	2	no	—
403	C₄×Q₈	—	2	2	no	—
404	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
405	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
406	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
408	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
410	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
411	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
412	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
413	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
414	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
415	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
416	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
417	C₂×C₄²	—	2	2	no	—
418	C₂×C₄²	—	2	2	no	—
419	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
420	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
421	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
422	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
423	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
424	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
425	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
426	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
427	C₄⋊C₈	—	106	2	no	—
428	C₄⋊C₈	—	106	2	no	—
429	C₄⋊C₈	—	106	2	no	—
430	C₄⋊C₈	—	106	2	no	—
431	C₄⋊C₈	—	106	2	no	—
432	C₄⋊C₈	—	106	2	no	—
433	C₂²⋊C₈	—	106	2	no	—
434	C₂²⋊C₈	—	106	2	no	—
435	#97	—	53	4	no	—
436	#97	—	106	2	no	—
437	#97	—	106	2	no	—
438	#97	—	106	2	no	—
439	C₄×C₈	—	106	2	no	—
440	C₄×C₈	—	106	2	no	—
441	C₄×C₈	—	106	2	no	—
442	C₄×C₈	—	106	2	no	—
443	#95	C₂×Dic₅₃	1	1	yes	characteristic
444	#95	C₂×Dic₅₃	1	1	yes	characteristic
445	#95	C₂×Dic₅₃	1	1	yes	characteristic
446	#95	C₂×Dic₅₃	1	1	yes	characteristic
447	#95	C₂×Dic₅₃	1	1	yes	characteristic
448	#95	C₂×Dic₅₃	1	1	yes	characteristic
449	#95	C₂×Dic₅₃	1	1	yes	characteristic
450	#95	C₂×Dic₅₃	1	1	yes	characteristic
451	#95	C₂×Dic₅₃	1	1	yes	characteristic
452	#95	C₂×Dic₅₃	1	1	yes	characteristic
453	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
454	C₂×Q₈	—	2	2	no	—
455	C₂×Q₈	—	2	2	no	—
456	C₂×Q₈	—	4	1	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	4	1	no	—
466	C₂²×C₄	—	4	1	no	—
467	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
468	C₂²×C₄	#11719949	1	2	yes	—
469	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
470	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
471	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
472	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
473	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
474	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
475	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
476	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
477	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
478	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
479	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
480	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
481	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
482	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
483	M₁₆	—	106	8	no	—
484	M₁₆	—	106	8	no	—
485	C₂×C₈	—	53	2	no	—
486	C₂×C₈	—	53	2	no	—
487	C₂×C₈	—	53	2	no	—
488	C₂×C₈	—	53	2	no	—
489	C₂×C₈	—	53	2	no	—
490	C₂×C₈	—	53	2	no	—
491	C₂×C₈	—	106	1	no	—
492	C₂×C₈	—	106	1	no	—
493	C₂×C₈	—	106	1	no	—
494	C₂×C₈	—	106	1	no	—
495	C₂×C₈	—	106	1	no	—
496	C₂×C₈	—	106	1	no	—
497	C₂×C₈	—	106	1	no	—
498	C₂×C₈	—	106	1	no	—
499	C₂×C₈	—	106	1	no	—
500	C₂×C₈	—	106	1	no	—
501	C₂×C₈	—	106	1	no	—
502	C₂×C₈	—	106	1	no	—
503	C₂×C₈	—	106	1	no	—
504	C₂×C₈	—	106	1	no	—
505	C₂×C₈	—	106	1	no	—
506	C₂×C₈	—	106	1	no	—
507	C₂×C₈	—	106	1	no	—
508	C₂×C₈	—	106	1	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	2	no	—
514	C₄⋊C₄	—	2	2	no	—
515	C₄⋊C₄	—	2	2	no	—
516	C₄⋊C₄	—	4	1	no	—
517	C₄⋊C₄	#11719917	1	4	yes	complemented
518	C₂²⋊C₄	—	2	2	no	—
519	C₂²⋊C₄	—	2	2	no	—
520	C₂²⋊C₄	—	4	1	no	—
521	C₂²⋊C₄	—	4	1	no	—
522	C₄²	—	2	2	no	—
523	C₄²	—	2	2	no	—
524	C₄²	—	2	1	no	—
525	C₄²	—	2	1	no	—
526	C₄²	—	2	1	no	—
527	C₄²	—	2	1	no	—
528	C₄²	—	2	1	no	—
529	C₄²	—	2	1	no	—
530	C₄²	—	2	1	no	—
531	C₄²	—	2	1	no	—
532	C₄²	—	2	1	no	—
533	C₄²	—	2	1	no	—
534	C₄²	D₄×D₅₃	1	2	yes	—
535	C₄²	#11719947	1	2	yes	—
536	C₂³	—	2	1	no	—
537	C₂³	—	2	1	no	—
538	C₂³	—	2	1	no	—
539	C₂³	—	2	1	no	—
540	C₂³	—	2	1	no	—
541	C₂³	—	2	1	no	—
542	C₂³	Q₈×Dic₅₃	1	2	yes	—
543	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
544	Q₈	—	4	4	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	4	4	no	—
582	C₂×C₄	—	4	4	no	—
583	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
584	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
585	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
586	C₂×C₄	C₂×#11719917	1	2	yes	—
587	C₂×C₄	C₂×#11719917	1	2	yes	—
588	C₂×C₄	C₂×#11719917	1	2	yes	—
589	C₂×C₄	#13133224	1	2	yes	—
590	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
591	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
592	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
593	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
594	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
595	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
596	C₈	—	53	4	no	—
597	C₈	—	106	2	no	—
598	C₈	—	106	2	no	—
599	C₈	—	106	2	no	—
600	C₈	—	212	1	no	—
601	C₈	—	212	1	no	—
602	C₈	—	212	1	no	—
603	C₈	—	212	1	no	—
604	C₂²	—	2	2	no	—
605	C₂²	—	2	2	no	—
606	C₂²	—	2	2	no	—
607	C₂²	—	4	1	no	—
608	C₂²	—	4	1	no	—
609	C₂²	—	4	1	no	—
610	C₂²	—	4	1	no	—
611	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
612	C₂²	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
613	C₂²	#41×Dic₅₃	1	1	yes	characteristic
614	C₂²	C₂²×#11719917	1	1	yes	characteristic
615	C₂²	C₂×#13133224	1	1	yes	characteristic
616	C₂²	#16811390	1	1	yes	characteristic
617	C₂²	#16811390	1	1	yes	characteristic
618	C₄	—	2	2	no	—
619	C₄	—	2	2	no	—
620	C₄	—	2	2	no	—
621	C₄	—	2	2	no	—
622	C₄	—	2	2	no	—
623	C₄	—	2	2	no	—
624	C₄	—	2	2	no	—
625	C₄	—	4	1	no	—
626	C₄	—	4	1	no	—
627	C₄	—	4	1	no	—
628	C₄	—	4	1	no	—
629	C₄	—	4	1	no	—
630	C₄	—	4	1	no	—
631	C₄	C₂²⋊Q₈×D₅₃	1	1	yes	characteristic
632	C₄	#16811136	1	1	yes	characteristic
633	C₄	#16811159	1	1	yes	characteristic
634	C₄	#16811175	1	1	yes	characteristic
635	C₂	—	4	2	no	—
636	C₂	C₂²⋊Q₈×Dic₅₃	1	1	yes	characteristic
637	C₂	#20283442	1	1	yes	characteristic
638	C₂	D₄×#11719917	1	1	yes	characteristic
639	C₂	#20283664	1	1	yes	characteristic
640	C₂	Q₈×#11719917	1	1	yes	characteristic
641	C₂	#20283682	1	1	yes	characteristic
642	C₂	#20283702	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₄×C₂²⋊Q₈×C₅₃	C₂
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×C₂²⋊Q₈	D₅₃
hypocenter	C₅₃	#21600
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16810754	C₂	semidirect
3. #20275656	C₂	semidirect
4. #20275439	C₂	semidirect
5. #20275274	C₂	semidirect
6. #20275274	C₂	semidirect
10. #20274590	C₂	semidirect
11. #20274590	C₂	semidirect
12. C₄⋊C₄×#88322	C₂	semidirect
66. C₂×#11719917	Q₈	semidirect
169. C₄⋊C₄×C₅₃	M₁₆	semidirect
281. #88322	C₂²⋊Q₈	semidirect
382. C₅₃	#21600	semidirect
517. C₄⋊C₄	#11719917	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810754	2
C₂×C₄×#11719917	2
#20275656	2
#20275439	2
#20275274	2
#20274900	2
#20274695	2
#20274590	2
C₄⋊C₄×#88322	2
#20274499	2
C₄×C₂²⋊Q₈×C₅₃	2
#21600	53