Subgroups — #27351282

Order 13568.

1194 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273240	C₂	1	4	yes	maximal, complemented
4	#20273219	C₂	1	4	yes	maximal, complemented
5	#20273214	C₂	1	2	yes	maximal, complemented
6	#20273210	C₂	1	2	yes	maximal, complemented
7	C₅₃×#3444	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
9	C₂²×C₄×Dic₅₃	—	2	1	no	—
10	#16810680	—	2	1	no	—
11	#16810680	C₂²	1	2	yes	—
12	#16810676	—	2	1	no	—
13	#16810676	C₂²	1	2	yes	—
14	#16810647	—	2	1	no	—
15	#16810587	C₂²	1	2	yes	—
16	C₄×#11719917	—	2	1	no	—
17	C₄×C₄⋊Dic₅₃	—	2	1	no	—
18	C₄×C₄⋊Dic₅₃	—	2	1	no	—
19	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
21	C₂×#13133124	C₂²	1	2	yes	—
22	#16810360	C₂²	1	8	yes	—
23	#16810357	C₂²	1	8	yes	—
24	C₅₃×#533	C₄	1	1	yes	characteristic, complemented
25	C₅₃×#533	C₄	1	1	yes	characteristic, complemented
26	C₅₃×#533	C₂²	1	1	yes	characteristic
27	C₅₃×#532	C₄	1	1	yes	characteristic, complemented
28	C₅₃×#532	C₄	1	1	yes	characteristic, complemented
29	C₅₃×#532	C₂²	1	1	yes	characteristic
30	C₅₃×#529	—	2	1	no	—
31	C₅₃×#529	—	2	1	no	—
32	C₅₃×#529	—	2	1	no	—
33	C₅₃×#529	—	2	1	no	—
34	C₅₃×#529	C₄	1	2	yes	complemented
35	C₅₃×#529	C₂²	1	2	yes	—
36	C₅₃×#525	—	2	1	no	—
37	C₅₃×#525	—	2	1	no	—
38	C₅₃×#525	C₂²	1	2	yes	—
39	C₄×#41×C₅₃	—	2	1	no	—
40	C₄×#41×C₅₃	—	2	1	no	—
41	C₄×#41×C₅₃	C₄	1	2	yes	complemented
42	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
43	C₂³×Dic₅₃	—	2	2	no	—
44	C₂×#11719917	—	2	2	no	—
45	C₂×C₄⋊Dic₅₃	—	2	8	no	—
46	C₂×C₄⋊Dic₅₃	—	2	2	no	—
47	C₂×C₄⋊Dic₅₃	—	2	2	no	—
48	C₂×C₄×Dic₅₃	—	2	2	no	—
49	C₂×C₄×Dic₅₃	—	2	2	no	—
50	C₂×C₄×Dic₅₃	—	4	4	no	—
51	C₄⋊#88322	—	2	2	no	—
52	C₄⋊#88322	—	2	2	no	—
53	C₂²⋊#88322	—	2	2	no	—
54	C₂²⋊#88322	C₂³	1	4	yes	—
55	#13133128	—	2	2	no	—
56	C₄×#88322	—	2	2	no	—
57	#13133124	—	2	2	no	—
58	#13133124	C₂³	1	4	yes	—
59	C₂×#41×C₅₃	—	2	1	no	—
60	C₂×#41×C₅₃	—	2	1	no	—
61	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
62	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
64	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
66	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
67	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
68	#127×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
69	#126×C₅₃	—	2	2	no	—
70	#126×C₅₃	—	2	2	no	—
71	#125×C₅₃	—	2	1	no	—
72	#125×C₅₃	—	2	1	no	—
73	#125×C₅₃	C₂×C₄	1	2	yes	complemented
74	#124×C₅₃	—	2	1	no	—
75	#124×C₅₃	—	2	1	no	—
76	#124×C₅₃	C₂×C₄	1	2	yes	complemented
77	#123×C₅₃	—	2	2	no	—
78	#123×C₅₃	—	2	2	no	—
79	#123×C₅₃	—	2	2	no	—
80	C₂²⋊Q₈×C₅₃	—	2	1	no	—
81	C₂²⋊Q₈×C₅₃	—	2	1	no	—
82	C₂²⋊Q₈×C₅₃	—	2	1	no	—
83	C₂²⋊Q₈×C₅₃	—	2	1	no	—
84	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
85	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
86	#121×C₅₃	—	2	1	no	—
87	#121×C₅₃	—	2	1	no	—
88	#121×C₅₃	—	2	1	no	—
89	#121×C₅₃	—	2	1	no	—
90	#121×C₅₃	C₂×C₄	1	2	yes	—
91	#121×C₅₃	C₂³	1	2	yes	—
92	#120×C₅₃	—	2	2	no	—
93	C₄×Q₈×C₅₃	—	2	1	no	—
94	C₄×Q₈×C₅₃	—	2	1	no	—
95	C₄×Q₈×C₅₃	—	2	1	no	—
96	C₄×Q₈×C₅₃	—	2	1	no	—
97	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	complemented
98	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	complemented
99	C₄×D₄×C₅₃	—	2	1	no	—
100	C₄×D₄×C₅₃	—	2	1	no	—
101	C₄×D₄×C₅₃	—	2	1	no	—
102	C₄×D₄×C₅₃	—	2	1	no	—
103	C₄×D₄×C₅₃	—	2	2	no	—
104	C₄×D₄×C₅₃	—	2	2	no	—
105	C₄×D₄×C₅₃	—	2	2	no	—
106	C₄×D₄×C₅₃	—	2	2	no	—
107	C₄×D₄×C₅₃	—	2	2	no	—
108	C₄×D₄×C₅₃	—	2	2	no	—
109	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	complemented
110	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	complemented
111	#117×C₅₃	—	2	2	no	—
112	#117×C₅₃	—	2	2	no	—
113	#117×C₅₃	—	2	2	no	—
114	#117×C₅₃	—	2	1	no	—
115	#117×C₅₃	—	2	1	no	—
116	#117×C₅₃	C₂×C₄	1	2	yes	—
117	C₂×C₄²×C₅₃	—	2	2	no	—
118	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
119	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
120	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
121	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
122	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
123	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
124	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
125	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
126	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
127	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
128	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
130	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
131	#41×C₅₃	—	4	4	no	—
132	#41×C₅₃	—	4	4	no	—
133	#41×C₅₃	—	4	2	no	—
134	#41×C₅₃	—	4	2	no	—
135	C₂×Q₈×C₅₃	—	2	1	no	—
136	C₂×Q₈×C₅₃	—	2	1	no	—
137	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
138	C₂×D₄×C₅₃	—	2	2	no	—
139	C₂×D₄×C₅₃	—	2	2	no	—
140	C₂×D₄×C₅₃	—	2	2	no	—
141	C₂×D₄×C₅₃	—	2	1	no	—
142	C₂×D₄×C₅₃	—	2	1	no	—
143	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
144	C₂²×C₄×C₅₃	—	2	4	no	—
145	C₂²×C₄×C₅₃	—	2	2	no	—
146	C₂²×C₄×C₅₃	—	2	2	no	—
147	C₂²×C₄×C₅₃	—	2	2	no	—
148	C₂²×C₄×C₅₃	—	2	2	no	—
149	C₂²×C₄×C₅₃	—	2	2	no	—
150	C₂²×C₄×C₅₃	—	4	2	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
169	C₂²×Dic₅₃	—	2	8	no	—
170	C₂²×Dic₅₃	—	2	4	no	—
171	C₂²×Dic₅₃	—	4	4	no	—
172	C₂²×Dic₅₃	—	4	4	no	—
173	C₄⋊C₄×C₅₃	—	2	1	no	—
174	C₄⋊C₄×C₅₃	—	2	1	no	—
175	C₄⋊C₄×C₅₃	—	2	1	no	—
176	C₄⋊C₄×C₅₃	—	2	1	no	—
177	C₄⋊C₄×C₅₃	—	2	2	no	—
178	C₄⋊C₄×C₅₃	—	2	2	no	—
179	C₄⋊C₄×C₅₃	—	2	2	no	—
180	C₄⋊C₄×C₅₃	—	2	2	no	—
181	C₄⋊C₄×C₅₃	—	2	2	no	—
182	C₄⋊C₄×C₅₃	—	2	2	no	—
183	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
184	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
185	C₂²⋊C₄×C₅₃	—	2	2	no	—
186	C₂²⋊C₄×C₅₃	—	2	2	no	—
187	C₂²⋊C₄×C₅₃	—	2	2	no	—
188	C₂²⋊C₄×C₅₃	—	2	2	no	—
189	C₂²⋊C₄×C₅₃	—	2	2	no	—
190	C₂²⋊C₄×C₅₃	—	2	2	no	—
191	C₂²⋊C₄×C₅₃	—	2	2	no	—
192	C₂²⋊C₄×C₅₃	—	2	2	no	—
193	C₂²⋊C₄×C₅₃	—	2	2	no	—
194	C₄²×C₅₃	—	2	2	no	—
195	C₄²×C₅₃	—	2	2	no	—
196	C₄²×C₅₃	—	2	2	no	—
197	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
198	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
199	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
200	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
201	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
202	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
203	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
204	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
205	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
206	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
207	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
208	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
209	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
211	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
212	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
213	C₄⋊Dic₅₃	—	2	8	no	—
214	C₄⋊Dic₅₃	—	2	8	no	—
215	C₄×Dic₅₃	—	4	4	no	—
216	C₄×Dic₅₃	—	4	4	no	—
217	#11719917	—	4	4	no	—
218	C₂×#88322	—	2	4	no	—
219	C₂³×C₅₃	—	2	1	no	—
220	C₂³×C₅₃	—	2	1	no	—
221	C₂³×C₅₃	—	2	2	no	—
222	C₂³×C₅₃	—	2	2	no	—
223	C₂³×C₅₃	—	2	2	no	—
224	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
225	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
226	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
227	Q₈×C₅₃	—	4	2	no	—
228	Q₈×C₅₃	—	4	2	no	—
229	Q₈×C₅₃	—	4	2	no	—
230	D₄×C₅₃	—	4	2	no	—
231	D₄×C₅₃	—	4	2	no	—
232	D₄×C₅₃	—	4	2	no	—
233	D₄×C₅₃	—	4	4	no	—
234	D₄×C₅₃	—	4	4	no	—
235	D₄×C₅₃	—	4	2	no	—
236	D₄×C₅₃	—	4	2	no	—
237	C₂×C₄×C₅₃	—	2	1	no	—
238	C₂×C₄×C₅₃	—	2	1	no	—
239	C₂×C₄×C₅₃	—	2	2	no	—
240	C₂×C₄×C₅₃	—	2	2	no	—
241	C₂×C₄×C₅₃	—	2	2	no	—
242	C₂×C₄×C₅₃	—	2	2	no	—
243	C₂×C₄×C₅₃	—	2	2	no	—
244	C₂×C₄×C₅₃	—	2	2	no	—
245	C₂×C₄×C₅₃	—	2	2	no	—
246	C₂×C₄×C₅₃	—	2	2	no	—
247	C₂×C₄×C₅₃	—	2	2	no	—
248	C₂×C₄×C₅₃	—	2	2	no	—
249	C₂×C₄×C₅₃	—	2	2	no	—
250	C₂×C₄×C₅₃	—	2	2	no	—
251	C₂×C₄×C₅₃	—	2	2	no	—
252	C₂×C₄×C₅₃	—	2	2	no	—
253	C₂×C₄×C₅₃	—	4	1	no	—
254	C₂×C₄×C₅₃	—	4	1	no	—
255	C₂×C₄×C₅₃	—	4	1	no	—
256	C₂×C₄×C₅₃	—	4	1	no	—
257	C₂×C₄×C₅₃	—	4	4	no	—
258	C₂×C₄×C₅₃	—	4	4	no	—
259	C₂×C₄×C₅₃	—	4	4	no	—
260	C₂×C₄×C₅₃	#104	1	4	yes	—
261	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
272	C₂×C₄×C₅₃	#137	1	2	yes	—
273	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
274	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
275	C₂×Dic₅₃	—	2	8	no	—
276	C₂×Dic₅₃	—	4	16	no	—
277	C₂×Dic₅₃	—	4	4	no	—
278	C₂×Dic₅₃	—	4	4	no	—
279	#88322	—	4	4	no	—
280	#17546	—	53	1	no	maximal, Sylow
281	C₂²×C₅₃	—	2	4	no	—
282	C₂²×C₅₃	—	2	4	no	—
283	C₂²×C₅₃	—	2	1	no	—
284	C₂²×C₅₃	—	2	1	no	—
285	C₂²×C₅₃	—	2	1	no	—
286	C₂²×C₅₃	—	2	1	no	—
287	C₂²×C₅₃	—	2	1	no	—
288	C₂²×C₅₃	—	2	1	no	—
289	C₂²×C₅₃	—	4	2	no	—
290	C₂²×C₅₃	—	4	2	no	—
291	C₂²×C₅₃	—	4	2	no	—
292	C₂²×C₅₃	#410	1	2	yes	—
293	C₂²×C₅₃	C₂×#104	1	2	yes	—
294	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
295	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
296	C₂²×C₅₃	C₂×#137	1	1	yes	characteristic
297	C₄×C₅₃	—	2	1	no	—
298	C₄×C₅₃	—	2	1	no	—
299	C₄×C₅₃	—	2	1	no	—
300	C₄×C₅₃	—	2	1	no	—
301	C₄×C₅₃	—	2	1	no	—
302	C₄×C₅₃	—	2	1	no	—
303	C₄×C₅₃	—	2	1	no	—
304	C₄×C₅₃	—	2	1	no	—
305	C₄×C₅₃	—	4	2	no	—
306	C₄×C₅₃	—	4	2	no	—
307	C₄×C₅₃	—	4	2	no	—
308	C₄×C₅₃	#417	1	2	yes	—
309	C₄×C₅₃	#417	1	2	yes	—
310	C₄×C₅₃	C₂×#104	1	2	yes	—
311	C₄×C₅₃	C₂×#104	1	2	yes	—
312	Dic₅₃	—	4	8	no	—
313	#3444	D₅₃	1	1	yes	characteristic
314	#2920	—	53	1	no	—
315	C₂×C₄×C₄⋊C₄	—	53	1	no	—
316	#1509	—	53	4	no	—
317	#1504	—	53	4	no	—
318	#1503	—	53	2	no	—
319	#1502	—	53	2	no	—
320	C₂×C₅₃	—	2	1	no	—
321	C₂×C₅₃	—	2	1	no	—
322	C₂×C₅₃	—	2	2	no	—
323	C₂×C₅₃	—	2	2	no	—
324	C₂×C₅₃	—	8	2	no	—
325	C₂×C₅₃	C₂×#410	1	1	yes	characteristic
326	C₂×C₅₃	C₂×#417	1	1	yes	characteristic
327	C₂×C₅₃	C₂²×#104	1	1	yes	characteristic
328	#533	Dic₅₃	1	1	yes	characteristic, complemented
329	#533	Dic₅₃	1	1	yes	characteristic, complemented
330	#533	C₂×D₅₃	1	1	yes	characteristic
331	#532	Dic₅₃	1	1	yes	characteristic, complemented
332	#532	Dic₅₃	1	1	yes	characteristic, complemented
333	#532	C₂×D₅₃	1	1	yes	characteristic
334	#529	—	2	1	no	—
335	#529	—	2	1	no	—
336	#529	—	2	1	no	—
337	#529	—	2	1	no	—
338	#529	Dic₅₃	1	2	yes	complemented
339	#529	C₂×D₅₃	1	2	yes	—
340	#525	—	2	1	no	—
341	#525	—	2	1	no	—
342	#525	C₂×D₅₃	1	2	yes	—
343	C₄×#41	—	2	1	no	—
344	C₄×#41	—	2	1	no	—
345	C₄×#41	Dic₅₃	1	2	yes	complemented
346	C₂²×C₄⋊C₄	—	106	1	no	—
347	C₂²×C₄²	—	106	1	no	—
348	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
349	#432	—	53	2	no	—
350	#432	—	106	1	no	—
351	#431	—	53	2	no	—
352	#431	—	106	1	no	—
353	#423	—	106	1	no	—
354	#407	—	53	2	no	—
355	C₄×M₁₆	—	106	1	no	—
356	C₄×C₄⋊C₄	—	53	2	no	—
357	C₄×C₄⋊C₄	—	53	2	no	—
358	C₄×C₄⋊C₄	—	106	1	no	—
359	C₄×C₄⋊C₄	—	106	1	no	—
360	C₂×#95	—	53	2	no	—
361	#328	—	53	8	no	—
362	#327	—	53	8	no	—
363	C₅₃	#17546	1	1	yes	characteristic, complemented, Sylow
364	C₂×#41	—	2	1	no	—
365	C₂×#41	—	2	1	no	—
366	C₂×#41	C₂×Dic₅₃	1	2	yes	—
367	C₂³×C₄	—	106	2	no	—
368	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
369	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
370	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
371	C₂×M₁₆	—	106	2	no	—
372	C₄⋊Q₈	C₂×Dic₅₃	1	1	yes	characteristic, complemented
373	C₄⋊Q₈	C₂×Dic₅₃	1	1	yes	characteristic, complemented
374	C₄⋊Q₈	C₂×Dic₅₃	1	1	yes	characteristic, complemented
375	#127	C₂×Dic₅₃	1	1	yes	characteristic, complemented
376	#126	—	2	2	no	—
377	#126	—	2	2	no	—
378	#125	—	2	1	no	—
379	#125	—	2	1	no	—
380	#125	C₂×Dic₅₃	1	2	yes	complemented
381	#124	—	2	1	no	—
382	#124	—	2	1	no	—
383	#124	C₂×Dic₅₃	1	2	yes	complemented
384	#123	—	2	2	no	—
385	#123	—	2	2	no	—
386	#123	—	2	2	no	—
387	C₂²⋊Q₈	—	2	1	no	—
388	C₂²⋊Q₈	—	2	1	no	—
389	C₂²⋊Q₈	—	2	1	no	—
390	C₂²⋊Q₈	—	2	1	no	—
391	C₂²⋊Q₈	C₂×Dic₅₃	1	2	yes	—
392	C₂²⋊Q₈	C₂²×D₅₃	1	2	yes	—
393	#121	—	2	1	no	—
394	#121	—	2	1	no	—
395	#121	—	2	1	no	—
396	#121	—	2	1	no	—
397	#121	C₂×Dic₅₃	1	2	yes	—
398	#121	C₂²×D₅₃	1	2	yes	—
399	#120	—	2	2	no	—
400	C₄×Q₈	—	2	1	no	—
401	C₄×Q₈	—	2	1	no	—
402	C₄×Q₈	—	2	1	no	—
403	C₄×Q₈	—	2	1	no	—
404	C₄×Q₈	C₂×Dic₅₃	1	2	yes	complemented
405	C₄×Q₈	C₂×Dic₅₃	1	2	yes	complemented
406	C₄×D₄	—	2	2	no	—
407	C₄×D₄	—	2	2	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	2	2	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	—	2	1	no	—
413	C₄×D₄	—	2	1	no	—
414	C₄×D₄	—	2	1	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	C₂×Dic₅₃	1	2	yes	complemented
417	C₄×D₄	C₂×Dic₅₃	1	2	yes	complemented
418	#117	—	2	1	no	—
419	#117	—	2	1	no	—
420	#117	—	2	2	no	—
421	#117	—	2	2	no	—
422	#117	—	2	2	no	—
423	#117	C₂×Dic₅₃	1	2	yes	—
424	C₂×C₄⋊C₄	—	106	8	no	—
425	C₂×C₄⋊C₄	—	106	2	no	—
426	C₂×C₄⋊C₄	—	106	2	no	—
427	C₂×C₄²	—	2	2	no	—
428	C₂×C₄²	—	106	2	no	—
429	C₂×C₄²	—	106	2	no	—
430	C₂×C₄²	—	212	4	no	—
431	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
432	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
433	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
434	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
435	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
436	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
437	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
438	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
439	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
440	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
441	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
442	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
443	C₄⋊C₈	—	106	2	no	—
444	C₄⋊C₈	—	106	2	no	—
445	C₂²⋊C₈	—	53	4	no	—
446	C₂²⋊C₈	—	106	2	no	—
447	#97	—	106	2	no	—
448	C₄×C₈	—	106	2	no	—
449	#95	—	53	4	no	—
450	#95	—	106	2	no	—
451	C₂⁴	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
452	#41	—	4	4	no	—
453	#41	—	4	4	no	—
454	#41	—	4	2	no	—
455	#41	—	4	2	no	—
456	C₂×Q₈	—	2	1	no	—
457	C₂×Q₈	—	2	1	no	—
458	C₂×Q₈	C₂²×Dic₅₃	1	2	yes	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	2	no	—
461	C₂×D₄	—	2	2	no	—
462	C₂×D₄	—	2	1	no	—
463	C₂×D₄	—	2	1	no	—
464	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	4	2	no	—
472	C₂²×C₄	—	106	8	no	—
473	C₂²×C₄	—	106	4	no	—
474	C₂²×C₄	—	212	4	no	—
475	C₂²×C₄	—	212	4	no	—
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	C₂²⋊Dic₅₃	1	1	yes	characteristic
483	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
484	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
485	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
486	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
487	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
488	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
489	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
490	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
491	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
492	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
493	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
494	M₁₆	—	212	4	no	—
495	C₂×C₈	—	106	4	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	—	2	2	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	106	8	no	—
507	C₄⋊C₄	—	106	8	no	—
508	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
509	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	2	no	—
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	—	2	2	no	—
516	C₂²⋊C₄	—	2	2	no	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	2	2	no	—
519	C₄²	—	2	2	no	—
520	C₄²	—	2	2	no	—
521	C₄²	—	2	2	no	—
522	C₄²	—	212	4	no	—
523	C₄²	—	212	4	no	—
524	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
525	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
526	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
527	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
528	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
529	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
530	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
531	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
532	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
533	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
534	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
535	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
536	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
537	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
538	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
539	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
540	C₂³	—	2	2	no	—
541	C₂³	—	2	2	no	—
542	C₂³	—	2	1	no	—
543	C₂³	—	2	1	no	—
544	C₂³	—	2	2	no	—
545	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
546	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
547	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
548	Q₈	—	4	2	no	—
549	Q₈	—	4	2	no	—
550	Q₈	—	4	2	no	—
551	D₄	—	4	2	no	—
552	D₄	—	4	4	no	—
553	D₄	—	4	4	no	—
554	D₄	—	4	2	no	—
555	D₄	—	4	2	no	—
556	D₄	—	4	2	no	—
557	D₄	—	4	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	4	no	—
579	C₂×C₄	—	4	4	no	—
580	C₂×C₄	—	4	4	no	—
581	C₂×C₄	—	106	8	no	—
582	C₂×C₄	—	212	16	no	—
583	C₂×C₄	—	212	4	no	—
584	C₂×C₄	—	212	4	no	—
585	C₂×C₄	#13133144	1	4	yes	—
586	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
587	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
588	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
589	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
590	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
591	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
592	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
593	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
594	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
595	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
596	C₂×C₄	#13133242	1	2	yes	—
597	C₂×C₄	#13133247	1	2	yes	—
598	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
599	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
600	C₈	—	212	4	no	—
601	C₂²	—	2	4	no	—
602	C₂²	—	2	4	no	—
603	C₂²	—	2	1	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	2	1	no	—
606	C₂²	—	2	1	no	—
607	C₂²	—	2	1	no	—
608	C₂²	—	2	1	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	4	2	no	—
611	C₂²	—	4	2	no	—
612	C₂²	#16810597	1	2	yes	—
613	C₂²	C₂×#13133144	1	2	yes	—
614	C₂²	C₂²×C₂²⋊Dic₅₃	1	1	yes	characteristic
615	C₂²	C₂×#13133242	1	1	yes	characteristic
616	C₂²	C₂×#13133247	1	1	yes	characteristic
617	C₄	—	2	1	no	—
618	C₄	—	2	1	no	—
619	C₄	—	2	1	no	—
620	C₄	—	2	1	no	—
621	C₄	—	2	1	no	—
622	C₄	—	2	1	no	—
623	C₄	—	2	1	no	—
624	C₄	—	2	1	no	—
625	C₄	—	4	2	no	—
626	C₄	—	4	2	no	—
627	C₄	—	4	2	no	—
628	C₄	—	212	8	no	—
629	C₄	#16810624	1	2	yes	—
630	C₄	#16810624	1	2	yes	—
631	C₄	C₂×#13133144	1	2	yes	—
632	C₄	C₂×#13133144	1	2	yes	—
633	C₂	—	2	1	no	—
634	C₂	—	2	2	no	—
635	C₂	—	2	2	no	—
636	C₂	—	2	1	no	—
637	C₂	—	8	2	no	—
638	C₂	C₂×#16810597	1	1	yes	characteristic
639	C₂	C₂×#16810624	1	1	yes	characteristic
640	C₂	C₂²×#13133144	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3444	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#3444	D₅₃
hypocenter	C₅₃	#17546
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #3444

Complements

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

N	H ≅ G/N	type
1. #20283348	C₂	semidirect
2. C₂×C₄×C₄⋊Dic₅₃	C₂	semidirect
3. #20273240	C₂	semidirect
4. #20273219	C₂	semidirect
5. #20273214	C₂	semidirect
6. #20273210	C₂	semidirect
24. C₅₃×#533	C₄	semidirect
25. C₅₃×#533	C₄	semidirect
27. C₅₃×#532	C₄	semidirect
28. C₅₃×#532	C₄	semidirect
34. C₅₃×#529	C₄	semidirect
41. C₄×#41×C₅₃	C₄	semidirect
65. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
66. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
67. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
68. #127×C₅₃	C₂×C₄	semidirect
73. #125×C₅₃	C₂×C₄	semidirect
76. #124×C₅₃	C₂×C₄	semidirect
97. C₄×Q₈×C₅₃	C₂×C₄	semidirect
98. C₄×Q₈×C₅₃	C₂×C₄	semidirect
109. C₄×D₄×C₅₃	C₂×C₄	semidirect
110. C₄×D₄×C₅₃	C₂×C₄	semidirect
328. #533	Dic₅₃	semidirect
329. #533	Dic₅₃	semidirect
331. #532	Dic₅₃	semidirect
332. #532	Dic₅₃	semidirect
338. #529	Dic₅₃	semidirect
345. C₄×#41	Dic₅₃	semidirect
363. C₅₃	#17546	semidirect
372. C₄⋊Q₈	C₂×Dic₅₃	semidirect
373. C₄⋊Q₈	C₂×Dic₅₃	semidirect
374. C₄⋊Q₈	C₂×Dic₅₃	semidirect
375. #127	C₂×Dic₅₃	semidirect
380. #125	C₂×Dic₅₃	semidirect
383. #124	C₂×Dic₅₃	semidirect
404. C₄×Q₈	C₂×Dic₅₃	semidirect
405. C₄×Q₈	C₂×Dic₅₃	semidirect
416. C₄×D₄	C₂×Dic₅₃	semidirect
417. C₄×D₄	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
C₂×C₄×C₄⋊Dic₅₃	2
#20273240	2
#20273219	2
#20273214	2
#20273210	2
C₅₃×#3444	2
#17546	53