Subgroups — C₅₃×#18694 #27226206

Order 13568.

942 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3552	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3217	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3217	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1628	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1627	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1626	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1625	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1556	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1494	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1494	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#563	—	2	2	no	—
13	C₅₃×#557	C₂²	1	1	yes	characteristic, complemented
14	C₅₃×#556	C₂²	1	1	yes	characteristic, complemented
15	C₅₃×#554	—	2	1	no	—
16	C₅₃×#553	—	2	1	no	—
17	D₄×Q₈×C₅₃	—	2	1	no	—
18	D₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
19	C₅₃×#548	—	2	1	no	—
20	C₅₃×#547	C₂²	1	1	yes	characteristic, complemented
21	C₅₃×#544	—	2	1	no	—
22	C₅₃×#542	—	2	1	no	—
23	C₅₃×#541	—	2	1	no	—
24	C₅₃×#539	—	2	2	no	—
25	C₅₃×#536	—	2	1	no	—
26	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#529	C₂²	1	1	yes	characteristic
29	C₅₃×#529	C₂²	1	1	yes	characteristic
30	C₅₃×#496	—	2	1	no	—
31	C₅₃×#496	—	2	1	no	—
32	C₅₃×#490	—	2	1	no	—
33	C₅₃×#490	—	2	1	no	—
34	C₅₃×#486	—	2	2	no	—
35	C₅₃×#486	C₂²	1	2	yes	—
36	C₅₃×#486	C₂²	1	2	yes	—
37	C₅₃×#460	C₂²	1	2	yes	—
38	C₅₃×#460	C₂²	1	2	yes	—
39	C₅₃×#459	C₂²	1	2	yes	—
40	C₅₃×#459	C₂²	1	2	yes	—
41	C₅₃×#450	—	2	1	no	—
42	C₅₃×#450	—	2	1	no	—
43	C₅₃×#447	—	2	1	no	—
44	C₅₃×#447	—	2	1	no	—
45	C₅₃×#432	C₂²	1	1	yes	characteristic
46	C₅₃×#432	C₂²	1	1	yes	characteristic
47	C₂×#102×C₅₃	—	2	2	no	—
48	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
49	C₅₃×#332	C₂²	1	2	yes	complemented
50	C₅₃×#332	C₂²	1	2	yes	complemented
51	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
52	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
53	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
54	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
55	C₂×#41×C₅₃	—	2	1	no	—
56	C₂²×Q₈×C₅₃	—	2	1	no	—
57	C₂²×Q₈×C₅₃	—	2	1	no	—
58	C₂²×D₄×C₅₃	—	2	1	no	—
59	C₂²×D₄×C₅₃	—	2	1	no	—
60	C₂²×D₄×C₅₃	—	2	1	no	—
61	C₂×SD₁₆×C₅₃	—	2	2	no	—
62	C₂×SD₁₆×C₅₃	—	2	2	no	—
63	C₂×D₈×C₅₃	—	2	2	no	—
64	C₂×D₈×C₅₃	—	2	2	no	—
65	C₂²×C₈×C₅₃	—	2	1	no	—
66	C₄⋊Q₈×C₅₃	—	2	2	no	—
67	C₄⋊Q₈×C₅₃	—	2	1	no	—
68	C₄⋊Q₈×C₅₃	—	2	1	no	—
69	C₄⋊Q₈×C₅₃	—	2	1	no	—
70	C₄⋊Q₈×C₅₃	—	2	1	no	—
71	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
72	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
73	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
74	#127×C₅₃	—	2	1	no	—
75	#127×C₅₃	—	2	1	no	—
76	#127×C₅₃	—	2	1	no	—
77	#127×C₅₃	C₂³	1	2	yes	—
78	#126×C₅₃	—	2	2	no	—
79	#126×C₅₃	—	2	2	no	—
80	#126×C₅₃	—	2	2	no	—
81	#126×C₅₃	—	2	1	no	—
82	#126×C₅₃	—	2	1	no	—
83	#125×C₅₃	—	2	2	no	—
84	#125×C₅₃	—	2	2	no	—
85	#125×C₅₃	—	2	2	no	—
86	#125×C₅₃	—	2	1	no	—
87	#125×C₅₃	D₄	1	1	yes	characteristic
88	#125×C₅₃	D₄	1	1	yes	characteristic
89	#124×C₅₃	—	2	2	no	—
90	#124×C₅₃	—	2	1	no	—
91	#124×C₅₃	D₄	1	1	yes	characteristic
92	#124×C₅₃	D₄	1	1	yes	characteristic
93	#123×C₅₃	—	2	2	no	—
94	#123×C₅₃	—	2	2	no	—
95	#123×C₅₃	—	2	2	no	—
96	#123×C₅₃	—	2	1	no	—
97	#123×C₅₃	—	2	1	no	—
98	C₂²⋊Q₈×C₅₃	—	2	2	no	—
99	C₂²⋊Q₈×C₅₃	—	2	2	no	—
100	C₂²⋊Q₈×C₅₃	—	2	2	no	—
101	C₂²⋊Q₈×C₅₃	—	2	2	no	—
102	C₂²⋊Q₈×C₅₃	—	2	2	no	—
103	C₂²⋊Q₈×C₅₃	—	2	2	no	—
104	C₂²⋊Q₈×C₅₃	—	2	1	no	—
105	C₂²⋊Q₈×C₅₃	—	2	1	no	—
106	C₂²⋊Q₈×C₅₃	—	2	1	no	—
107	C₂²⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
108	C₂²⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
109	#121×C₅₃	—	2	2	no	—
110	#121×C₅₃	—	2	1	no	—
111	#121×C₅₃	D₄	1	1	yes	characteristic
112	#121×C₅₃	D₄	1	1	yes	characteristic
113	C₄×Q₈×C₅₃	—	2	1	no	—
114	C₄×Q₈×C₅₃	—	2	1	no	—
115	C₄×Q₈×C₅₃	—	2	1	no	—
116	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
117	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
118	C₄×D₄×C₅₃	—	2	2	no	—
119	C₄×D₄×C₅₃	—	2	2	no	—
120	C₄×D₄×C₅₃	—	2	1	no	—
121	C₄×D₄×C₅₃	—	2	1	no	—
122	C₄×D₄×C₅₃	—	2	1	no	—
123	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
124	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
125	#117×C₅₃	—	2	1	no	—
126	#117×C₅₃	—	2	1	no	—
127	#117×C₅₃	—	2	1	no	—
128	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
129	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
131	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
132	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
133	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
134	#102×C₅₃	—	2	4	no	—
135	#102×C₅₃	—	2	4	no	—
136	#102×C₅₃	—	2	2	no	—
137	#102×C₅₃	—	2	2	no	—
138	#102×C₅₃	—	2	2	no	—
139	#102×C₅₃	—	2	2	no	—
140	C₂²⋊C₈×C₅₃	—	2	1	no	—
141	C₂²⋊C₈×C₅₃	—	2	1	no	—
142	#97×C₅₃	—	2	1	no	—
143	#97×C₅₃	—	2	1	no	—
144	C₄×C₈×C₅₃	—	2	1	no	—
145	C₄×C₈×C₅₃	C₂³	1	2	yes	—
146	C₂⁴×C₅₃	—	4	1	no	—
147	#41×C₅₃	—	8	2	no	—
148	C₂×Q₈×C₅₃	—	2	2	no	—
149	C₂×Q₈×C₅₃	—	2	2	no	—
150	C₂×Q₈×C₅₃	—	2	1	no	—
151	C₂×Q₈×C₅₃	—	2	1	no	—
152	C₂×Q₈×C₅₃	—	8	2	no	—
153	C₂×Q₈×C₅₃	—	8	2	no	—
154	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
155	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
156	C₂×D₄×C₅₃	—	2	1	no	—
157	C₂×D₄×C₅₃	—	2	1	no	—
158	C₂×D₄×C₅₃	—	2	2	no	—
159	C₂×D₄×C₅₃	—	2	2	no	—
160	C₂×D₄×C₅₃	—	2	2	no	—
161	C₂×D₄×C₅₃	—	2	2	no	—
162	C₂×D₄×C₅₃	—	4	4	no	—
163	C₂×D₄×C₅₃	—	4	1	no	—
164	C₂×D₄×C₅₃	—	4	1	no	—
165	C₂×D₄×C₅₃	—	4	1	no	—
166	C₂×D₄×C₅₃	—	8	2	no	—
167	C₂×D₄×C₅₃	—	8	2	no	—
168	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
169	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₅₃	—	2	2	no	—
171	C₂²×C₄×C₅₃	—	2	2	no	—
172	C₂²×C₄×C₅₃	—	2	2	no	—
173	C₂²×C₄×C₅₃	—	2	1	no	—
174	C₂²×C₄×C₅₃	—	2	1	no	—
175	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
178	SD₁₆×C₅₃	—	8	2	no	—
179	SD₁₆×C₅₃	—	8	2	no	—
180	D₈×C₅₃	—	8	2	no	—
181	D₈×C₅₃	—	8	2	no	—
182	C₂×C₈×C₅₃	—	2	2	no	—
183	C₂×C₈×C₅₃	—	2	2	no	—
184	C₂×C₈×C₅₃	—	2	2	no	—
185	C₂×C₈×C₅₃	—	4	2	no	—
186	C₄⋊C₄×C₅₃	—	2	2	no	—
187	C₄⋊C₄×C₅₃	—	2	2	no	—
188	C₄⋊C₄×C₅₃	—	2	2	no	—
189	C₄⋊C₄×C₅₃	—	2	2	no	—
190	C₄⋊C₄×C₅₃	—	2	2	no	—
191	C₄⋊C₄×C₅₃	—	2	2	no	—
192	C₄⋊C₄×C₅₃	—	2	2	no	—
193	C₄⋊C₄×C₅₃	—	2	2	no	—
194	C₄⋊C₄×C₅₃	—	2	1	no	—
195	C₄⋊C₄×C₅₃	—	2	1	no	—
196	C₄⋊C₄×C₅₃	—	2	1	no	—
197	C₄⋊C₄×C₅₃	—	2	1	no	—
198	C₄⋊C₄×C₅₃	—	2	1	no	—
199	C₄⋊C₄×C₅₃	—	2	1	no	—
200	C₄⋊C₄×C₅₃	—	2	1	no	—
201	C₄⋊C₄×C₅₃	—	2	1	no	—
202	C₄⋊C₄×C₅₃	—	2	1	no	—
203	C₄⋊C₄×C₅₃	—	2	1	no	—
204	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
205	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
206	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
207	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
208	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
209	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
210	C₂²⋊C₄×C₅₃	—	2	2	no	—
211	C₂²⋊C₄×C₅₃	—	2	2	no	—
212	C₂²⋊C₄×C₅₃	—	2	2	no	—
213	C₂²⋊C₄×C₅₃	—	2	2	no	—
214	C₂²⋊C₄×C₅₃	—	2	1	no	—
215	C₂²⋊C₄×C₅₃	—	2	1	no	—
216	C₂²⋊C₄×C₅₃	—	2	1	no	—
217	C₂²⋊C₄×C₅₃	—	2	1	no	—
218	C₂²⋊C₄×C₅₃	—	2	1	no	—
219	C₂²⋊C₄×C₅₃	—	2	1	no	—
220	C₂²⋊C₄×C₅₃	—	2	1	no	—
221	C₂²⋊C₄×C₅₃	—	2	1	no	—
222	C₂²⋊C₄×C₅₃	—	2	1	no	—
223	C₂²⋊C₄×C₅₃	—	2	1	no	—
224	C₄²×C₅₃	—	2	2	no	—
225	C₄²×C₅₃	—	2	2	no	—
226	C₄²×C₅₃	—	2	1	no	—
227	C₄²×C₅₃	—	2	1	no	—
228	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
229	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
230	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
231	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
232	C₂³×C₅₃	—	2	1	no	—
233	C₂³×C₅₃	—	2	1	no	—
234	C₂³×C₅₃	—	4	2	no	—
235	C₂³×C₅₃	—	8	2	no	—
236	C₂³×C₅₃	—	8	2	no	—
237	C₂³×C₅₃	—	8	2	no	—
238	C₂³×C₅₃	#120	1	1	yes	characteristic
239	Q₈×C₅₃	—	2	2	no	—
240	Q₈×C₅₃	—	2	2	no	—
241	Q₈×C₅₃	—	4	2	no	—
242	Q₈×C₅₃	—	4	2	no	—
243	Q₈×C₅₃	—	8	2	no	—
244	Q₈×C₅₃	—	8	2	no	—
245	D₄×C₅₃	—	2	2	no	—
246	D₄×C₅₃	—	2	2	no	—
247	D₄×C₅₃	—	4	4	no	—
248	D₄×C₅₃	—	4	2	no	—
249	D₄×C₅₃	—	4	2	no	—
250	D₄×C₅₃	—	4	2	no	—
251	D₄×C₅₃	—	4	2	no	—
252	D₄×C₅₃	—	4	2	no	—
253	D₄×C₅₃	—	4	2	no	—
254	D₄×C₅₃	—	8	2	no	—
255	D₄×C₅₃	—	8	2	no	—
256	D₄×C₅₃	—	8	2	no	—
257	D₄×C₅₃	—	8	2	no	—
258	C₂×C₄×C₅₃	—	2	2	no	—
259	C₂×C₄×C₅₃	—	2	2	no	—
260	C₂×C₄×C₅₃	—	2	2	no	—
261	C₂×C₄×C₅₃	—	2	1	no	—
262	C₂×C₄×C₅₃	—	2	2	no	—
263	C₂×C₄×C₅₃	—	2	2	no	—
264	C₂×C₄×C₅₃	—	2	1	no	—
265	C₂×C₄×C₅₃	—	2	1	no	—
266	C₂×C₄×C₅₃	—	2	1	no	—
267	C₂×C₄×C₅₃	—	2	1	no	—
268	C₂×C₄×C₅₃	—	2	1	no	—
269	C₂×C₄×C₅₃	—	4	2	no	—
270	C₂×C₄×C₅₃	—	4	2	no	—
271	C₂×C₄×C₅₃	—	8	2	no	—
272	C₂×C₄×C₅₃	—	8	2	no	—
273	C₂×C₄×C₅₃	—	8	2	no	—
274	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
275	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
276	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
277	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
278	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
279	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
280	C₈×C₅₃	—	4	2	no	—
281	C₈×C₅₃	—	4	2	no	—
282	C₈×C₅₃	—	4	2	no	—
283	#18694	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
284	C₂²×C₅₃	—	2	1	no	—
285	C₂²×C₅₃	—	2	1	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	—	4	1	no	—
291	C₂²×C₅₃	—	4	1	no	—
292	C₂²×C₅₃	—	8	2	no	—
293	C₂²×C₅₃	—	8	2	no	—
294	C₂²×C₅₃	—	8	2	no	—
295	C₂²×C₅₃	—	16	4	no	—
296	C₂²×C₅₃	#454	1	2	yes	—
297	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
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	—	4	1	no	—
303	C₄×C₅₃	—	4	2	no	—
304	C₄×C₅₃	—	4	1	no	—
305	C₄×C₅₃	—	4	2	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	#576	1	2	yes	—
311	C₄×C₅₃	#577	1	2	yes	—
312	#3552	C₂×C₅₃	1	1	yes	characteristic, complemented
313	#3217	C₂×C₅₃	1	1	yes	characteristic, complemented
314	#3217	C₂×C₅₃	1	1	yes	characteristic, complemented
315	C₂×#486	C₂×C₅₃	1	1	yes	characteristic, complemented
316	#1628	C₂×C₅₃	1	2	yes	complemented
317	#1627	C₂×C₅₃	1	2	yes	complemented
318	#1626	C₂×C₅₃	1	2	yes	complemented
319	#1625	C₂×C₅₃	1	2	yes	complemented
320	#1556	C₂×C₅₃	1	1	yes	characteristic, complemented
321	#1494	C₂×C₅₃	1	1	yes	characteristic, complemented
322	#1494	C₂×C₅₃	1	1	yes	characteristic, complemented
323	C₂×C₅₃	—	2	2	no	—
324	C₂×C₅₃	—	8	1	no	—
325	C₂×C₅₃	—	8	1	no	—
326	C₂×C₅₃	—	16	2	no	—
327	C₂×C₅₃	#3008	1	1	yes	characteristic
328	C₂×C₅₃	#3010	1	1	yes	characteristic
329	C₂×C₅₃	C₂×#454	1	1	yes	characteristic
330	#563	—	2	2	no	—
331	#557	C₂²×C₅₃	1	1	yes	characteristic, complemented
332	#556	C₂²×C₅₃	1	1	yes	characteristic, complemented
333	#554	—	2	1	no	—
334	#553	—	2	1	no	—
335	D₄×Q₈	—	2	1	no	—
336	D₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
337	#548	—	2	1	no	—
338	#547	C₂²×C₅₃	1	1	yes	characteristic, complemented
339	#544	—	2	1	no	—
340	#542	—	2	1	no	—
341	#541	—	2	1	no	—
342	#539	—	2	2	no	—
343	#536	—	2	1	no	—
344	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
345	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
346	#529	C₂²×C₅₃	1	1	yes	characteristic
347	#529	C₂²×C₅₃	1	1	yes	characteristic
348	#496	—	2	1	no	—
349	#496	—	2	1	no	—
350	#490	—	2	1	no	—
351	#490	—	2	1	no	—
352	#486	—	2	2	no	—
353	#486	C₂²×C₅₃	1	2	yes	—
354	#486	C₂²×C₅₃	1	2	yes	—
355	#460	C₂²×C₅₃	1	2	yes	—
356	#460	C₂²×C₅₃	1	2	yes	—
357	#459	C₂²×C₅₃	1	2	yes	—
358	#459	C₂²×C₅₃	1	2	yes	—
359	#450	—	2	1	no	—
360	#450	—	2	1	no	—
361	#447	—	2	1	no	—
362	#447	—	2	1	no	—
363	#432	C₂²×C₅₃	1	1	yes	characteristic
364	#432	C₂²×C₅₃	1	1	yes	characteristic
365	C₂×#102	—	2	2	no	—
366	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
367	#332	C₂²×C₅₃	1	2	yes	complemented
368	#332	C₂²×C₅₃	1	2	yes	complemented
369	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	complemented
370	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	complemented
371	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
372	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
373	C₅₃	#18694	1	1	yes	characteristic, direct product factor, Sylow
374	C₂×#41	—	2	1	no	—
375	C₂²×Q₈	—	2	1	no	—
376	C₂²×Q₈	—	2	1	no	—
377	C₂²×D₄	—	2	1	no	—
378	C₂²×D₄	—	2	1	no	—
379	C₂²×D₄	—	2	1	no	—
380	C₂×SD₁₆	—	2	2	no	—
381	C₂×SD₁₆	—	2	2	no	—
382	C₂×D₈	—	2	2	no	—
383	C₂×D₈	—	2	2	no	—
384	C₂²×C₈	—	2	1	no	—
385	C₄⋊Q₈	—	2	2	no	—
386	C₄⋊Q₈	—	2	1	no	—
387	C₄⋊Q₈	—	2	1	no	—
388	C₄⋊Q₈	—	2	1	no	—
389	C₄⋊Q₈	—	2	1	no	—
390	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
391	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
392	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
393	#127	—	2	1	no	—
394	#127	—	2	1	no	—
395	#127	—	2	1	no	—
396	#127	C₂³×C₅₃	1	2	yes	—
397	#126	—	2	2	no	—
398	#126	—	2	2	no	—
399	#126	—	2	2	no	—
400	#126	—	2	1	no	—
401	#126	—	2	1	no	—
402	#125	—	2	2	no	—
403	#125	—	2	2	no	—
404	#125	—	2	2	no	—
405	#125	—	2	1	no	—
406	#125	D₄×C₅₃	1	1	yes	characteristic
407	#125	D₄×C₅₃	1	1	yes	characteristic
408	#124	—	2	2	no	—
409	#124	—	2	1	no	—
410	#124	D₄×C₅₃	1	1	yes	characteristic
411	#124	D₄×C₅₃	1	1	yes	characteristic
412	#123	—	2	2	no	—
413	#123	—	2	2	no	—
414	#123	—	2	1	no	—
415	#123	—	2	1	no	—
416	#123	—	2	2	no	—
417	C₂²⋊Q₈	—	2	2	no	—
418	C₂²⋊Q₈	—	2	2	no	—
419	C₂²⋊Q₈	—	2	2	no	—
420	C₂²⋊Q₈	—	2	2	no	—
421	C₂²⋊Q₈	—	2	2	no	—
422	C₂²⋊Q₈	—	2	1	no	—
423	C₂²⋊Q₈	—	2	2	no	—
424	C₂²⋊Q₈	—	2	1	no	—
425	C₂²⋊Q₈	—	2	1	no	—
426	C₂²⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
427	C₂²⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
428	#121	—	2	2	no	—
429	#121	—	2	1	no	—
430	#121	D₄×C₅₃	1	1	yes	characteristic
431	#121	D₄×C₅₃	1	1	yes	characteristic
432	C₄×Q₈	—	2	1	no	—
433	C₄×Q₈	—	2	1	no	—
434	C₄×Q₈	—	2	1	no	—
435	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
436	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
437	C₄×D₄	—	2	2	no	—
438	C₄×D₄	—	2	2	no	—
439	C₄×D₄	—	2	1	no	—
440	C₄×D₄	—	2	1	no	—
441	C₄×D₄	—	2	1	no	—
442	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
443	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
444	#117	—	2	1	no	—
445	#117	—	2	1	no	—
446	#117	—	2	1	no	—
447	C₂×C₄⋊C₄	—	2	1	no	—
448	C₂×C₄⋊C₄	—	2	1	no	—
449	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
450	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
451	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
452	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
453	#102	—	2	4	no	—
454	#102	—	2	4	no	—
455	#102	—	2	2	no	—
456	#102	—	2	2	no	—
457	#102	—	2	2	no	—
458	#102	—	2	2	no	—
459	C₂²⋊C₈	—	2	1	no	—
460	C₂²⋊C₈	—	2	1	no	—
461	#97	—	2	1	no	—
462	#97	—	2	1	no	—
463	C₄×C₈	—	2	1	no	—
464	C₄×C₈	C₂³×C₅₃	1	2	yes	—
465	C₂⁴	—	4	1	no	—
466	#41	—	8	2	no	—
467	C₂×Q₈	—	2	2	no	—
468	C₂×Q₈	—	2	2	no	—
469	C₂×Q₈	—	2	1	no	—
470	C₂×Q₈	—	2	1	no	—
471	C₂×Q₈	—	8	2	no	—
472	C₂×Q₈	—	8	2	no	—
473	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
474	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
475	C₂×D₄	—	2	1	no	—
476	C₂×D₄	—	2	1	no	—
477	C₂×D₄	—	2	2	no	—
478	C₂×D₄	—	2	2	no	—
479	C₂×D₄	—	2	2	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	4	4	no	—
482	C₂×D₄	—	4	1	no	—
483	C₂×D₄	—	4	1	no	—
484	C₂×D₄	—	4	1	no	—
485	C₂×D₄	—	8	2	no	—
486	C₂×D₄	—	8	2	no	—
487	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
488	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	2	no	—
491	C₂²×C₄	—	2	2	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
495	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
496	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
497	SD₁₆	—	8	2	no	—
498	SD₁₆	—	8	2	no	—
499	D₈	—	8	2	no	—
500	D₈	—	8	2	no	—
501	C₂×C₈	—	2	2	no	—
502	C₂×C₈	—	2	2	no	—
503	C₂×C₈	—	2	2	no	—
504	C₂×C₈	—	4	2	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	2	2	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	2	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	1	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	1	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	1	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
524	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
525	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
526	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
527	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
528	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
529	C₂²⋊C₄	—	2	2	no	—
530	C₂²⋊C₄	—	2	2	no	—
531	C₂²⋊C₄	—	2	2	no	—
532	C₂²⋊C₄	—	2	2	no	—
533	C₂²⋊C₄	—	2	1	no	—
534	C₂²⋊C₄	—	2	1	no	—
535	C₂²⋊C₄	—	2	1	no	—
536	C₂²⋊C₄	—	2	1	no	—
537	C₂²⋊C₄	—	2	1	no	—
538	C₂²⋊C₄	—	2	1	no	—
539	C₂²⋊C₄	—	2	1	no	—
540	C₂²⋊C₄	—	2	1	no	—
541	C₂²⋊C₄	—	2	1	no	—
542	C₂²⋊C₄	—	2	1	no	—
543	C₄²	—	2	2	no	—
544	C₄²	—	2	2	no	—
545	C₄²	—	2	1	no	—
546	C₄²	—	2	1	no	—
547	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
548	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
549	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
550	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
551	C₂³	—	2	1	no	—
552	C₂³	—	2	1	no	—
553	C₂³	—	4	2	no	—
554	C₂³	—	8	2	no	—
555	C₂³	—	8	2	no	—
556	C₂³	—	8	2	no	—
557	C₂³	#120×C₅₃	1	1	yes	characteristic
558	Q₈	—	2	2	no	—
559	Q₈	—	2	2	no	—
560	Q₈	—	4	2	no	—
561	Q₈	—	4	2	no	—
562	Q₈	—	8	2	no	—
563	Q₈	—	8	2	no	—
564	D₄	—	2	2	no	—
565	D₄	—	2	2	no	—
566	D₄	—	4	4	no	—
567	D₄	—	4	2	no	—
568	D₄	—	4	2	no	—
569	D₄	—	4	2	no	—
570	D₄	—	4	2	no	—
571	D₄	—	4	2	no	—
572	D₄	—	4	2	no	—
573	D₄	—	8	2	no	—
574	D₄	—	8	2	no	—
575	D₄	—	8	2	no	—
576	D₄	—	8	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	4	2	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	8	2	no	—
591	C₂×C₄	—	8	2	no	—
592	C₂×C₄	—	8	2	no	—
593	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
594	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
595	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
596	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
597	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
598	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
599	C₈	—	4	2	no	—
600	C₈	—	4	2	no	—
601	C₈	—	4	2	no	—
602	C₂²	—	2	1	no	—
603	C₂²	—	2	1	no	—
604	C₂²	—	4	1	no	—
605	C₂²	—	4	1	no	—
606	C₂²	—	4	1	no	—
607	C₂²	—	4	1	no	—
608	C₂²	—	4	1	no	—
609	C₂²	—	4	1	no	—
610	C₂²	—	8	2	no	—
611	C₂²	—	8	2	no	—
612	C₂²	—	8	2	no	—
613	C₂²	—	16	4	no	—
614	C₂²	C₅₃×#454	1	2	yes	—
615	C₂²	C₂×#120×C₅₃	1	1	yes	characteristic
616	C₄	—	2	1	no	—
617	C₄	—	2	1	no	—
618	C₄	—	2	1	no	—
619	C₄	—	2	1	no	—
620	C₄	—	4	1	no	—
621	C₄	—	4	2	no	—
622	C₄	—	4	1	no	—
623	C₄	—	4	2	no	—
624	C₄	—	4	1	no	—
625	C₄	—	4	1	no	—
626	C₄	—	4	1	no	—
627	C₄	—	4	1	no	—
628	C₄	C₅₃×#576	1	2	yes	—
629	C₄	C₅₃×#577	1	2	yes	—
630	C₂	—	2	2	no	—
631	C₂	—	8	1	no	—
632	C₂	—	8	1	no	—
633	C₂	—	16	2	no	—
634	C₂	C₅₃×#3008	1	1	yes	characteristic
635	C₂	C₅₃×#3010	1	1	yes	characteristic
636	C₂	C₂×C₅₃×#454	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3552	C₂	semidirect
2. C₅₃×#3217	C₂	semidirect
3. C₅₃×#3217	C₂	semidirect
4. C₂×C₅₃×#486	C₂	semidirect
5. C₅₃×#1628	C₂	semidirect
6. C₅₃×#1627	C₂	semidirect
7. C₅₃×#1626	C₂	semidirect
8. C₅₃×#1625	C₂	semidirect
9. C₅₃×#1556	C₂	semidirect
10. C₅₃×#1494	C₂	semidirect
11. C₅₃×#1494	C₂	semidirect
13. C₅₃×#557	C₂²	semidirect
14. C₅₃×#556	C₂²	semidirect
18. D₄×Q₈×C₅₃	C₂²	semidirect
20. C₅₃×#547	C₂²	semidirect
49. C₅₃×#332	C₂²	semidirect
50. C₅₃×#332	C₂²	semidirect
51. C₅₃×Q₈⋊C₈	C₂²	semidirect
52. C₅₃×Q₈⋊C₈	C₂²	semidirect
53. C₅₃×D₄⋊C₈	C₂²	semidirect
54. C₅₃×D₄⋊C₈	C₂²	semidirect
283. #18694	C₅₃	direct (N × H)
312. #3552	C₂×C₅₃	semidirect
313. #3217	C₂×C₅₃	semidirect
314. #3217	C₂×C₅₃	semidirect
315. C₂×#486	C₂×C₅₃	semidirect
316. #1628	C₂×C₅₃	semidirect
317. #1627	C₂×C₅₃	semidirect
318. #1626	C₂×C₅₃	semidirect
319. #1625	C₂×C₅₃	semidirect
320. #1556	C₂×C₅₃	semidirect
321. #1494	C₂×C₅₃	semidirect
322. #1494	C₂×C₅₃	semidirect
331. #557	C₂²×C₅₃	semidirect
332. #556	C₂²×C₅₃	semidirect
336. D₄×Q₈	C₂²×C₅₃	semidirect
338. #547	C₂²×C₅₃	semidirect
367. #332	C₂²×C₅₃	semidirect
368. #332	C₂²×C₅₃	semidirect
369. Q₈⋊C₈	C₂²×C₅₃	semidirect
370. Q₈⋊C₈	C₂²×C₅₃	semidirect
371. D₄⋊C₈	C₂²×C₅₃	semidirect
372. D₄⋊C₈	C₂²×C₅₃	semidirect
373. C₅₃	#18694	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3552	2
C₅₃×#3217	2
C₂×C₅₃×#486	2
C₅₃×#1628	2
C₅₃×#1627	2
C₅₃×#1626	2
C₅₃×#1625	2
C₅₃×#1556	2
C₅₃×#1494	2
#18694	53

Subgroups — C53×#18694 #27226206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18694 #27226206