Subgroups — C₅₃×#62095 #27269607

Order 13568.

2238 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3499	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×D₈×C₅₃	C₂	1	2	yes	maximal, complemented
4	C₅₃×#3358	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3334	C₂	1	6	yes	maximal, complemented
6	C₅₃×#3294	C₂	1	6	yes	maximal, complemented
7	C₅₃×#3245	C₂	1	3	yes	maximal, complemented
8	C₅₃×#3226	C₂	1	3	yes	maximal, complemented
9	C₅₃×#3100	C₂	1	3	yes	maximal, complemented
10	C₂×C₅₃×#474	C₂	1	3	yes	maximal, complemented
11	C₅₃×#2988	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₅₃	—	2	1	no	—
13	C₂²×#41×C₅₃	—	2	1	no	—
14	C₅₃×#550	C₂²	1	1	yes	characteristic
15	D₄×Q₈×C₅₃	—	2	1	no	—
16	D₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	D₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
18	C₅₃×#548	—	2	1	no	—
19	C₅₃×#547	—	2	3	no	—
20	C₅₃×#547	C₂²	1	3	yes	—
21	C₅₃×#547	C₂²	1	6	yes	complemented
22	C₅₃×#546	—	2	3	no	—
23	D₄²×C₅₃	C₂²	1	3	yes	—
24	C₅₃×#536	—	2	3	no	—
25	C₅₃×#532	C₂²	1	3	yes	—
26	C₅₃×#532	C₂²	1	3	yes	complemented
27	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	complemented
28	C₅₃×#525	—	2	3	no	—
29	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₅₃	C₂²	1	3	yes	—
31	C₅₃×#500	C₂²	1	6	yes	complemented
32	C₅₃×#486	C₂²	1	6	yes	complemented
33	C₅₃×#480	C₂²	1	12	yes	complemented
34	C₅₃×#480	C₂²	1	6	yes	complemented
35	C₅₃×#474	C₂²	1	6	yes	—
36	C₅₃×#474	C₂²	1	3	yes	—
37	C₅₃×#474	C₂²	1	3	yes	complemented
38	C₅₃×#474	C₂²	1	6	yes	complemented
39	C₅₃×#474	C₂²	1	6	yes	complemented
40	C₅₃×#474	C₂²	1	6	yes	complemented
41	C₅₃×#466	C₂²	1	12	yes	complemented
42	C₅₃×#459	C₂²	1	1	yes	characteristic
43	C₅₃×#459	C₂²	1	3	yes	complemented
44	C₅₃×#459	C₂²	1	6	yes	complemented
45	C₅₃×#447	C₂²	1	6	yes	—
46	C₈×Q₈×C₅₃	C₂²	1	2	yes	complemented
47	C₄×D₈×C₅₃	C₂²	1	6	yes	—
48	C₄×D₈×C₅₃	C₂²	1	6	yes	complemented
49	C₈×D₄×C₅₃	C₂²	1	6	yes	complemented
50	C₅₃×#431	C₂²	1	3	yes	complemented
51	C₂×#107×C₅₃	C₂²	1	6	yes	complemented
52	C₂×C₄⋊C₈×C₅₃	C₂²	1	3	yes	complemented
53	C₂×#102×C₅₃	C₂²	1	6	yes	—
54	2₋¹⁺⁴×C₅₃	—	4	8	no	—
55	C₂×#41×C₅₃	—	2	1	no	—
56	C₂×#41×C₅₃	—	2	3	no	—
57	C₂×#41×C₅₃	—	2	6	no	—
58	C₂×#41×C₅₃	—	2	3	no	—
59	C₂×#41×C₅₃	—	2	3	no	—
60	C₂×#41×C₅₃	—	4	8	no	—
61	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
62	C₂²×Q₈×C₅₃	—	2	1	no	—
63	C₂²×Q₈×C₅₃	—	2	2	no	—
64	C₂²×Q₈×C₅₃	—	2	3	no	—
65	C₂²×D₄×C₅₃	—	2	3	no	—
66	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
67	C₂³×C₄×C₅₃	—	2	3	no	—
68	C₂×D₈×C₅₃	C₂³	1	2	yes	—
69	C₂×D₈×C₅₃	C₂³	1	6	yes	—
70	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
71	C₄⋊Q₈×C₅₃	—	2	3	no	—
72	C₄⋊Q₈×C₅₃	—	2	3	no	—
73	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
74	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
75	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
76	#127×C₅₃	C₂³	1	3	yes	—
77	#127×C₅₃	C₂³	1	1	yes	characteristic
78	#124×C₅₃	—	2	3	no	—
79	#124×C₅₃	—	2	3	no	—
80	#123×C₅₃	—	2	6	no	—
81	#123×C₅₃	—	2	12	no	—
82	C₂²⋊Q₈×C₅₃	—	2	3	no	—
83	C₂²⋊Q₈×C₅₃	—	2	3	no	—
84	C₂²⋊Q₈×C₅₃	—	2	6	no	—
85	C₂²⋊Q₈×C₅₃	—	2	6	no	—
86	#121×C₅₃	—	2	3	no	—
87	#121×C₅₃	—	2	3	no	—
88	#121×C₅₃	—	2	3	no	—
89	#121×C₅₃	—	2	3	no	—
90	#121×C₅₃	C₂³	1	6	yes	—
91	#121×C₅₃	C₂³	1	6	yes	—
92	#120×C₅₃	—	2	6	no	—
93	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
94	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₄×D₄×C₅₃	—	2	3	no	—
96	C₄×D₄×C₅₃	—	2	3	no	—
97	C₄×D₄×C₅₃	—	2	3	no	—
98	C₄×D₄×C₅₃	—	2	3	no	—
99	C₄×D₄×C₅₃	D₄	1	3	yes	—
100	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
101	C₄×D₄×C₅₃	C₂³	1	3	yes	—
102	C₄×D₄×C₅₃	C₂³	1	3	yes	—
103	C₄×D₄×C₅₃	C₂³	1	3	yes	—
104	C₄×D₄×C₅₃	C₂³	1	6	yes	—
105	C₄×D₄×C₅₃	C₂³	1	3	yes	—
106	#117×C₅₃	D₄	1	3	yes	—
107	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
108	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
109	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
110	C₂×C₂²⋊C₄×C₅₃	—	2	3	no	—
111	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
112	#107×C₅₃	C₂³	1	6	yes	—
113	#107×C₅₃	C₂³	1	12	yes	—
114	#107×C₅₃	C₂³	1	6	yes	—
115	C₄⋊C₈×C₅₃	C₂³	1	3	yes	—
116	C₄⋊C₈×C₅₃	C₂³	1	6	yes	—
117	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
118	#102×C₅₃	C₂³	1	6	yes	—
119	#102×C₅₃	C₂³	1	12	yes	—
120	#102×C₅₃	C₂³	1	6	yes	—
121	#102×C₅₃	C₂³	1	6	yes	—
122	#102×C₅₃	C₂³	1	2	yes	—
123	C₂²⋊C₈×C₅₃	C₂³	1	6	yes	—
124	C₄×C₈×C₅₃	C₂³	1	6	yes	—
125	C₂⁴×C₅₃	—	2	3	no	—
126	#41×C₅₃	—	4	4	no	—
127	#41×C₅₃	—	4	12	no	—
128	#41×C₅₃	—	4	24	no	—
129	#41×C₅₃	—	4	12	no	—
130	#41×C₅₃	—	4	12	no	—
131	#41×C₅₃	D₈	1	8	yes	complemented
132	C₂×Q₈×C₅₃	—	2	3	no	—
133	C₂×Q₈×C₅₃	—	2	6	no	—
134	C₂×Q₈×C₅₃	—	2	3	no	—
135	C₂×Q₈×C₅₃	—	4	4	no	—
136	C₂×Q₈×C₅₃	—	4	8	no	—
137	C₂×Q₈×C₅₃	—	4	12	no	—
138	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
139	C₂×D₄×C₅₃	—	2	6	no	—
140	C₂×D₄×C₅₃	—	2	6	no	—
141	C₂×D₄×C₅₃	—	2	6	no	—
142	C₂×D₄×C₅₃	—	2	1	no	—
143	C₂×D₄×C₅₃	—	2	3	no	—
144	C₂×D₄×C₅₃	—	2	3	no	—
145	C₂×D₄×C₅₃	—	2	1	no	—
146	C₂×D₄×C₅₃	—	2	3	no	—
147	C₂×D₄×C₅₃	—	4	3	no	—
148	C₂×D₄×C₅₃	—	4	12	no	—
149	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
150	C₂×D₄×C₅₃	C₂⁴	1	3	yes	—
151	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
152	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
153	C₂²×C₄×C₅₃	—	2	3	no	—
154	C₂²×C₄×C₅₃	—	2	6	no	—
155	C₂²×C₄×C₅₃	—	2	1	no	—
156	C₂²×C₄×C₅₃	—	2	3	no	—
157	C₂²×C₄×C₅₃	—	2	3	no	—
158	C₂²×C₄×C₅₃	—	2	3	no	—
159	C₂²×C₄×C₅₃	—	2	6	no	—
160	C₂²×C₄×C₅₃	—	4	12	no	—
161	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
162	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
163	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
164	D₈×C₅₃	—	2	4	no	—
165	D₈×C₅₃	—	4	6	no	—
166	C₂×C₈×C₅₃	—	4	6	no	—
167	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
168	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
169	C₄⋊C₄×C₅₃	—	2	3	no	—
170	C₄⋊C₄×C₅₃	—	2	6	no	—
171	C₄⋊C₄×C₅₃	—	2	3	no	—
172	C₄⋊C₄×C₅₃	—	2	6	no	—
173	C₄⋊C₄×C₅₃	—	2	3	no	—
174	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
175	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
176	C₄⋊C₄×C₅₃	C₂⁴	1	6	yes	—
177	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
178	C₄⋊C₄×C₅₃	C₂⁴	1	3	yes	—
179	C₂²⋊C₄×C₅₃	—	2	1	no	—
180	C₂²⋊C₄×C₅₃	—	2	3	no	—
181	C₂²⋊C₄×C₅₃	—	2	3	no	—
182	C₂²⋊C₄×C₅₃	—	2	3	no	—
183	C₂²⋊C₄×C₅₃	—	2	6	no	—
184	C₂²⋊C₄×C₅₃	—	2	6	no	—
185	C₂²⋊C₄×C₅₃	—	2	6	no	—
186	C₂²⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
187	C₄²×C₅₃	C₂×D₄	1	3	yes	—
188	C₄²×C₅₃	C₂×D₄	1	3	yes	—
189	C₄²×C₅₃	C₂⁴	1	3	yes	—
190	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
191	C₂³×C₅₃	—	2	1	no	—
192	C₂³×C₅₃	—	2	2	no	—
193	C₂³×C₅₃	—	2	3	no	—
194	C₂³×C₅₃	—	4	6	no	—
195	C₂³×C₅₃	—	4	6	no	—
196	C₂³×C₅₃	—	8	3	no	—
197	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
198	Q₈×C₅₃	—	4	6	no	—
199	Q₈×C₅₃	—	4	12	no	—
200	Q₈×C₅₃	—	4	6	no	—
201	Q₈×C₅₃	C₂×D₈	1	4	yes	—
202	D₄×C₅₃	—	2	3	no	—
203	D₄×C₅₃	—	2	2	no	—
204	D₄×C₅₃	—	2	1	no	—
205	D₄×C₅₃	—	2	3	no	—
206	D₄×C₅₃	—	2	2	no	—
207	D₄×C₅₃	—	4	6	no	—
208	D₄×C₅₃	—	4	2	no	—
209	D₄×C₅₃	—	4	12	no	—
210	D₄×C₅₃	—	4	12	no	—
211	D₄×C₅₃	—	4	6	no	—
212	D₄×C₅₃	—	4	2	no	—
213	D₄×C₅₃	—	8	3	no	—
214	D₄×C₅₃	C₂×D₈	1	12	yes	—
215	D₄×C₅₃	2₋¹⁺⁴	1	2	yes	complemented
216	C₂×C₄×C₅₃	—	2	3	no	—
217	C₂×C₄×C₅₃	—	2	3	no	—
218	C₂×C₄×C₅₃	—	2	1	no	—
219	C₂×C₄×C₅₃	—	2	3	no	—
220	C₂×C₄×C₅₃	—	2	6	no	—
221	C₂×C₄×C₅₃	—	4	3	no	—
222	C₂×C₄×C₅₃	—	4	6	no	—
223	C₂×C₄×C₅₃	—	4	12	no	—
224	C₂×C₄×C₅₃	—	4	2	no	—
225	C₂×C₄×C₅₃	—	4	6	no	—
226	C₂×C₄×C₅₃	—	4	6	no	—
227	C₂×C₄×C₅₃	—	4	6	no	—
228	C₂×C₄×C₅₃	—	4	12	no	—
229	C₂×C₄×C₅₃	C₂×D₈	1	12	yes	—
230	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
232	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
233	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
235	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
236	C₈×C₅₃	—	2	2	no	—
237	C₈×C₅₃	—	2	6	no	—
238	#62095	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
239	C₂²×C₅₃	—	2	3	no	—
240	C₂²×C₅₃	—	2	3	no	—
241	C₂²×C₅₃	—	2	2	no	—
242	C₂²×C₅₃	—	4	1	no	—
243	C₂²×C₅₃	—	4	1	no	—
244	C₂²×C₅₃	—	4	2	no	—
245	C₂²×C₅₃	—	4	3	no	—
246	C₂²×C₅₃	—	4	3	no	—
247	C₂²×C₅₃	—	4	2	no	—
248	C₂²×C₅₃	—	4	2	no	—
249	C₂²×C₅₃	—	4	3	no	—
250	C₂²×C₅₃	—	8	12	no	—
251	C₂²×C₅₃	C₂²×D₈	1	6	yes	—
252	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
253	C₄×C₅₃	—	2	1	no	—
254	C₄×C₅₃	—	2	3	no	—
255	C₄×C₅₃	—	2	3	no	—
256	C₄×C₅₃	—	4	1	no	—
257	C₄×C₅₃	—	4	3	no	—
258	C₄×C₅₃	—	4	6	no	—
259	C₄×C₅₃	C₂²×D₈	1	2	yes	—
260	C₄×C₅₃	C₂²×D₈	1	6	yes	—
261	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
262	C₄×C₅₃	#585	1	1	yes	characteristic
263	#3499	C₂×C₅₃	1	1	yes	characteristic, complemented
264	D₄×#41	C₂×C₅₃	1	1	yes	characteristic, complemented
265	Q₈×D₈	C₂×C₅₃	1	2	yes	complemented
266	#3358	C₂×C₅₃	1	2	yes	complemented
267	#3334	C₂×C₅₃	1	6	yes	complemented
268	#3294	C₂×C₅₃	1	6	yes	complemented
269	#3245	C₂×C₅₃	1	3	yes	complemented
270	#3226	C₂×C₅₃	1	3	yes	complemented
271	#3100	C₂×C₅₃	1	3	yes	complemented
272	C₂×#474	C₂×C₅₃	1	3	yes	complemented
273	#2988	C₂×C₅₃	1	1	yes	characteristic, complemented
274	C₂×C₅₃	—	2	6	no	—
275	C₂×C₅₃	—	4	2	no	—
276	C₂×C₅₃	—	8	2	no	—
277	C₂×C₅₃	—	8	3	no	—
278	C₂×C₅₃	#3489	1	1	yes	characteristic
279	C₂×C₅₃	C₂³×D₈	1	1	yes	characteristic
280	C₂×C₅₃	#3586	1	1	yes	characteristic
281	C₂×2₋¹⁺⁴	—	2	1	no	—
282	C₂²×#41	—	2	1	no	—
283	#550	C₂²×C₅₃	1	1	yes	characteristic
284	D₄×Q₈	—	2	1	no	—
285	D₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
286	D₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
287	#548	—	2	1	no	—
288	#547	—	2	3	no	—
289	#547	C₂²×C₅₃	1	3	yes	—
290	#547	C₂²×C₅₃	1	6	yes	complemented
291	#546	—	2	3	no	—
292	D₄²	C₂²×C₅₃	1	3	yes	—
293	#536	—	2	3	no	—
294	#532	C₂²×C₅₃	1	3	yes	—
295	#532	C₂²×C₅₃	1	3	yes	complemented
296	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	complemented
297	#525	—	2	3	no	—
298	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
299	C₂×C₄×D₄	C₂²×C₅₃	1	3	yes	—
300	#500	C₂²×C₅₃	1	6	yes	complemented
301	#486	C₂²×C₅₃	1	6	yes	complemented
302	#480	C₂²×C₅₃	1	6	yes	complemented
303	#480	C₂²×C₅₃	1	12	yes	complemented
304	#474	C₂²×C₅₃	1	3	yes	—
305	#474	C₂²×C₅₃	1	6	yes	—
306	#474	C₂²×C₅₃	1	6	yes	complemented
307	#474	C₂²×C₅₃	1	6	yes	complemented
308	#474	C₂²×C₅₃	1	3	yes	complemented
309	#474	C₂²×C₅₃	1	6	yes	complemented
310	#466	C₂²×C₅₃	1	12	yes	complemented
311	#459	C₂²×C₅₃	1	1	yes	characteristic
312	#459	C₂²×C₅₃	1	3	yes	complemented
313	#459	C₂²×C₅₃	1	6	yes	complemented
314	#447	C₂²×C₅₃	1	6	yes	—
315	C₈×Q₈	C₂²×C₅₃	1	2	yes	complemented
316	C₄×D₈	C₂²×C₅₃	1	6	yes	—
317	C₄×D₈	C₂²×C₅₃	1	6	yes	complemented
318	C₈×D₄	C₂²×C₅₃	1	6	yes	complemented
319	#431	C₂²×C₅₃	1	3	yes	complemented
320	C₂×#107	C₂²×C₅₃	1	6	yes	complemented
321	C₂×C₄⋊C₈	C₂²×C₅₃	1	3	yes	complemented
322	C₂×#102	C₂²×C₅₃	1	6	yes	—
323	C₅₃	#62095	1	1	yes	characteristic, direct product factor, Sylow
324	2₋¹⁺⁴	—	4	8	no	—
325	C₂×#41	—	2	1	no	—
326	C₂×#41	—	2	3	no	—
327	C₂×#41	—	2	6	no	—
328	C₂×#41	—	2	3	no	—
329	C₂×#41	—	2	3	no	—
330	C₂×#41	—	4	8	no	—
331	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
332	C₂²×Q₈	—	2	1	no	—
333	C₂²×Q₈	—	2	2	no	—
334	C₂²×Q₈	—	2	3	no	—
335	C₂²×D₄	—	2	3	no	—
336	C₂²×D₄	C₂³×C₅₃	1	3	yes	—
337	C₂³×C₄	—	2	3	no	—
338	C₂×D₈	C₂³×C₅₃	1	2	yes	—
339	C₂×D₈	C₂³×C₅₃	1	6	yes	—
340	C₂²×C₈	C₂³×C₅₃	1	6	yes	—
341	C₄⋊Q₈	—	2	3	no	—
342	C₄⋊Q₈	—	2	3	no	—
343	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
344	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
345	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
346	#127	C₂³×C₅₃	1	3	yes	—
347	#127	C₂³×C₅₃	1	1	yes	characteristic
348	#124	—	2	3	no	—
349	#124	—	2	3	no	—
350	#123	—	2	6	no	—
351	#123	—	2	12	no	—
352	C₂²⋊Q₈	—	2	3	no	—
353	C₂²⋊Q₈	—	2	3	no	—
354	C₂²⋊Q₈	—	2	6	no	—
355	C₂²⋊Q₈	—	2	6	no	—
356	#121	—	2	3	no	—
357	#121	—	2	3	no	—
358	#121	—	2	3	no	—
359	#121	—	2	3	no	—
360	#121	C₂³×C₅₃	1	6	yes	—
361	#121	C₂³×C₅₃	1	6	yes	—
362	#120	—	2	6	no	—
363	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
364	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
365	C₄×D₄	—	2	3	no	—
366	C₄×D₄	—	2	3	no	—
367	C₄×D₄	—	2	3	no	—
368	C₄×D₄	—	2	3	no	—
369	C₄×D₄	D₄×C₅₃	1	3	yes	—
370	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
371	C₄×D₄	C₂³×C₅₃	1	3	yes	—
372	C₄×D₄	C₂³×C₅₃	1	3	yes	—
373	C₄×D₄	C₂³×C₅₃	1	3	yes	—
374	C₄×D₄	C₂³×C₅₃	1	6	yes	—
375	C₄×D₄	C₂³×C₅₃	1	3	yes	—
376	#117	D₄×C₅₃	1	3	yes	—
377	C₂×C₄⋊C₄	—	2	3	no	—
378	C₂×C₄⋊C₄	C₂³×C₅₃	1	3	yes	—
379	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
380	C₂×C₂²⋊C₄	—	2	3	no	—
381	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
382	#107	C₂³×C₅₃	1	6	yes	—
383	#107	C₂³×C₅₃	1	12	yes	—
384	#107	C₂³×C₅₃	1	6	yes	—
385	C₄⋊C₈	C₂³×C₅₃	1	3	yes	—
386	C₄⋊C₈	C₂³×C₅₃	1	6	yes	—
387	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
388	#102	C₂³×C₅₃	1	6	yes	—
389	#102	C₂³×C₅₃	1	12	yes	—
390	#102	C₂³×C₅₃	1	6	yes	—
391	#102	C₂³×C₅₃	1	2	yes	—
392	#102	C₂³×C₅₃	1	6	yes	—
393	C₂²⋊C₈	C₂³×C₅₃	1	6	yes	—
394	C₄×C₈	C₂³×C₅₃	1	6	yes	—
395	C₂⁴	—	2	3	no	—
396	#41	—	4	4	no	—
397	#41	—	4	12	no	—
398	#41	—	4	24	no	—
399	#41	—	4	12	no	—
400	#41	—	4	12	no	—
401	#41	D₈×C₅₃	1	8	yes	complemented
402	C₂×Q₈	—	2	3	no	—
403	C₂×Q₈	—	2	6	no	—
404	C₂×Q₈	—	2	3	no	—
405	C₂×Q₈	—	4	4	no	—
406	C₂×Q₈	—	4	8	no	—
407	C₂×Q₈	—	4	12	no	—
408	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
409	C₂×D₄	—	2	6	no	—
410	C₂×D₄	—	2	6	no	—
411	C₂×D₄	—	2	6	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	2	3	no	—
414	C₂×D₄	—	2	3	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	3	no	—
417	C₂×D₄	—	4	3	no	—
418	C₂×D₄	—	4	12	no	—
419	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
420	C₂×D₄	C₂⁴×C₅₃	1	3	yes	—
421	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
422	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
423	C₂²×C₄	—	2	3	no	—
424	C₂²×C₄	—	2	6	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	3	no	—
427	C₂²×C₄	—	2	3	no	—
428	C₂²×C₄	—	2	3	no	—
429	C₂²×C₄	—	2	6	no	—
430	C₂²×C₄	—	4	12	no	—
431	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
432	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
433	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
434	D₈	—	2	4	no	—
435	D₈	—	4	6	no	—
436	C₂×C₈	—	4	6	no	—
437	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
438	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
439	C₄⋊C₄	—	2	3	no	—
440	C₄⋊C₄	—	2	6	no	—
441	C₄⋊C₄	—	2	3	no	—
442	C₄⋊C₄	—	2	6	no	—
443	C₄⋊C₄	—	2	3	no	—
444	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
445	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
446	C₄⋊C₄	C₂⁴×C₅₃	1	6	yes	—
447	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
448	C₄⋊C₄	C₂⁴×C₅₃	1	3	yes	—
449	C₂²⋊C₄	—	2	1	no	—
450	C₂²⋊C₄	—	2	3	no	—
451	C₂²⋊C₄	—	2	3	no	—
452	C₂²⋊C₄	—	2	3	no	—
453	C₂²⋊C₄	—	2	6	no	—
454	C₂²⋊C₄	—	2	6	no	—
455	C₂²⋊C₄	—	2	6	no	—
456	C₂²⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
457	C₄²	C₂×D₄×C₅₃	1	3	yes	—
458	C₄²	C₂×D₄×C₅₃	1	3	yes	—
459	C₄²	C₂⁴×C₅₃	1	3	yes	—
460	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
461	C₂³	—	2	1	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	2	3	no	—
464	C₂³	—	4	6	no	—
465	C₂³	—	4	6	no	—
466	C₂³	—	8	3	no	—
467	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
468	Q₈	—	4	6	no	—
469	Q₈	—	4	12	no	—
470	Q₈	—	4	6	no	—
471	Q₈	C₂×D₈×C₅₃	1	4	yes	—
472	D₄	—	2	2	no	—
473	D₄	—	2	3	no	—
474	D₄	—	2	1	no	—
475	D₄	—	2	2	no	—
476	D₄	—	2	3	no	—
477	D₄	—	4	6	no	—
478	D₄	—	4	2	no	—
479	D₄	—	4	12	no	—
480	D₄	—	4	12	no	—
481	D₄	—	4	6	no	—
482	D₄	—	4	2	no	—
483	D₄	—	8	3	no	—
484	D₄	C₂×D₈×C₅₃	1	12	yes	—
485	D₄	2₋¹⁺⁴×C₅₃	1	2	yes	complemented
486	C₂×C₄	—	2	3	no	—
487	C₂×C₄	—	2	3	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	3	no	—
490	C₂×C₄	—	2	6	no	—
491	C₂×C₄	—	4	3	no	—
492	C₂×C₄	—	4	6	no	—
493	C₂×C₄	—	4	12	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	4	6	no	—
496	C₂×C₄	—	4	6	no	—
497	C₂×C₄	—	4	6	no	—
498	C₂×C₄	—	4	12	no	—
499	C₂×C₄	C₂×D₈×C₅₃	1	12	yes	—
500	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
501	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
502	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
503	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
504	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
505	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
506	C₈	—	2	2	no	—
507	C₈	—	2	6	no	—
508	C₂²	—	2	3	no	—
509	C₂²	—	2	3	no	—
510	C₂²	—	2	2	no	—
511	C₂²	—	4	1	no	—
512	C₂²	—	4	1	no	—
513	C₂²	—	4	3	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	4	3	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	3	no	—
519	C₂²	—	8	12	no	—
520	C₂²	C₂²×D₈×C₅₃	1	6	yes	—
521	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
522	C₄	—	2	3	no	—
523	C₄	—	2	1	no	—
524	C₄	—	2	3	no	—
525	C₄	—	4	1	no	—
526	C₄	—	4	3	no	—
527	C₄	—	4	6	no	—
528	C₄	C₂²×D₈×C₅₃	1	2	yes	—
529	C₄	C₂²×D₈×C₅₃	1	6	yes	—
530	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
531	C₄	C₅₃×#585	1	1	yes	characteristic
532	C₂	—	2	6	no	—
533	C₂	—	4	2	no	—
534	C₂	—	8	3	no	—
535	C₂	—	8	2	no	—
536	C₂	C₅₃×#3489	1	1	yes	characteristic
537	C₂	C₂³×D₈×C₅₃	1	1	yes	characteristic
538	C₂	C₅₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3499	C₂	semidirect
2. D₄×#41×C₅₃	C₂	semidirect
3. Q₈×D₈×C₅₃	C₂	semidirect
4. C₅₃×#3358	C₂	semidirect
5. C₅₃×#3334	C₂	semidirect
6. C₅₃×#3294	C₂	semidirect
7. C₅₃×#3245	C₂	semidirect
8. C₅₃×#3226	C₂	semidirect
9. C₅₃×#3100	C₂	semidirect
10. C₂×C₅₃×#474	C₂	semidirect
11. C₅₃×#2988	C₂	semidirect
17. D₄×Q₈×C₅₃	C₂²	semidirect
21. C₅₃×#547	C₂²	semidirect
26. C₅₃×#532	C₂²	semidirect
27. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
31. C₅₃×#500	C₂²	semidirect
32. C₅₃×#486	C₂²	semidirect
33. C₅₃×#480	C₂²	semidirect
34. C₅₃×#480	C₂²	semidirect
37. C₅₃×#474	C₂²	semidirect
38. C₅₃×#474	C₂²	semidirect
39. C₅₃×#474	C₂²	semidirect
40. C₅₃×#474	C₂²	semidirect
41. C₅₃×#466	C₂²	semidirect
43. C₅₃×#459	C₂²	semidirect
44. C₅₃×#459	C₂²	semidirect
46. C₈×Q₈×C₅₃	C₂²	semidirect
48. C₄×D₈×C₅₃	C₂²	semidirect
49. C₈×D₄×C₅₃	C₂²	semidirect
50. C₅₃×#431	C₂²	semidirect
51. C₂×#107×C₅₃	C₂²	semidirect
52. C₂×C₄⋊C₈×C₅₃	C₂²	semidirect
131. #41×C₅₃	D₈	semidirect
215. D₄×C₅₃	2₋¹⁺⁴	semidirect
238. #62095	C₅₃	direct (N × H)
263. #3499	C₂×C₅₃	semidirect
264. D₄×#41	C₂×C₅₃	semidirect
265. Q₈×D₈	C₂×C₅₃	semidirect
266. #3358	C₂×C₅₃	semidirect
267. #3334	C₂×C₅₃	semidirect
268. #3294	C₂×C₅₃	semidirect
269. #3245	C₂×C₅₃	semidirect
270. #3226	C₂×C₅₃	semidirect
271. #3100	C₂×C₅₃	semidirect
272. C₂×#474	C₂×C₅₃	semidirect
273. #2988	C₂×C₅₃	semidirect
286. D₄×Q₈	C₂²×C₅₃	semidirect
290. #547	C₂²×C₅₃	semidirect
295. #532	C₂²×C₅₃	semidirect
296. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
300. #500	C₂²×C₅₃	semidirect
301. #486	C₂²×C₅₃	semidirect
302. #480	C₂²×C₅₃	semidirect
303. #480	C₂²×C₅₃	semidirect
306. #474	C₂²×C₅₃	semidirect
307. #474	C₂²×C₅₃	semidirect
308. #474	C₂²×C₅₃	semidirect
309. #474	C₂²×C₅₃	semidirect
310. #466	C₂²×C₅₃	semidirect
312. #459	C₂²×C₅₃	semidirect
313. #459	C₂²×C₅₃	semidirect
315. C₈×Q₈	C₂²×C₅₃	semidirect
317. C₄×D₈	C₂²×C₅₃	semidirect
318. C₈×D₄	C₂²×C₅₃	semidirect
319. #431	C₂²×C₅₃	semidirect
320. C₂×#107	C₂²×C₅₃	semidirect
321. C₂×C₄⋊C₈	C₂²×C₅₃	semidirect
323. C₅₃	#62095	direct (N × H)
401. #41	D₈×C₅₃	semidirect
485. D₄	2₋¹⁺⁴×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3499	2
D₄×#41×C₅₃	2
Q₈×D₈×C₅₃	2
C₅₃×#3358	2
C₅₃×#3334	2
C₅₃×#3294	2
C₅₃×#3245	2
C₅₃×#3226	2
C₅₃×#3100	2
C₂×C₅₃×#474	2
C₅₃×#2988	2
#62095	53

Subgroups — C53×#62095 #27269607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#62095 #27269607