Subgroups — C₅₃×#36717 #27244229

Order 13568.

3394 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#534	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#121×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2847	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2703	C₂	1	8	yes	maximal, complemented
5	C₅₃×#2669	C₂	1	8	yes	maximal, complemented
6	C₅₃×#2617	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2591	C₂	1	2	yes	maximal, complemented
8	C₅₃×#2433	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2403	C₂	1	2	yes	maximal, complemented
10	C₅₃×#2403	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2312	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
15	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
16	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
17	C₅₃×#534	—	2	8	no	—
18	C₂×#127×C₅₃	C₂²	1	4	yes	complemented
19	C₂×#123×C₅₃	C₂²	1	4	yes	complemented
20	C₂×#121×C₅₃	—	2	8	no	—
21	C₂×#121×C₅₃	C₂²	1	2	yes	—
22	C₂×#121×C₅₃	C₂²	1	2	yes	—
23	C₂×#121×C₅₃	C₂²	1	8	yes	complemented
24	C₂×#121×C₅₃	C₂²	1	4	yes	complemented
25	C₂×#120×C₅₃	C₂²	1	4	yes	complemented
26	C₂×#120×C₅₃	C₂²	1	4	yes	complemented
27	C₂×#120×C₅₃	C₂²	1	4	yes	complemented
28	C₂×#120×C₅₃	C₂²	1	4	yes	complemented
29	C₂×#120×C₅₃	C₂²	1	4	yes	complemented
30	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
31	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
32	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
33	C₅₃×#400	C₂²	1	4	yes	complemented
34	C₅₃×#397	C₂²	1	4	yes	complemented
35	C₅₃×#394	C₂²	1	8	yes	complemented
36	C₅₃×#394	C₂²	1	4	yes	complemented
37	C₅₃×#392	C₂²	1	8	yes	complemented
38	C₅₃×#392	C₂²	1	8	yes	complemented
39	C₅₃×#392	C₂²	1	8	yes	complemented
40	C₅₃×#392	C₂²	1	4	yes	complemented
41	C₅₃×#390	C₂²	1	4	yes	complemented
42	C₅₃×#388	C₂²	1	8	yes	complemented
43	C₅₃×#388	C₂²	1	8	yes	complemented
44	C₅₃×#386	C₂²	1	4	yes	complemented
45	C₅₃×#386	C₂²	1	4	yes	complemented
46	C₅₃×#386	C₂²	1	8	yes	complemented
47	C₅₃×#386	C₂²	1	4	yes	complemented
48	C₅₃×#386	C₂²	1	4	yes	complemented
49	C₅₃×#380	C₂²	1	4	yes	complemented
50	C₅₃×#380	C₂²	1	2	yes	complemented
51	C₅₃×#379	C₂²	1	2	yes	—
52	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
53	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
54	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
55	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
56	C₂⁵×C₅₃	C₂³	1	2	yes	—
57	C₂²×D₄×C₅₃	—	2	4	no	—
58	C₂²×D₄×C₅₃	—	2	4	no	—
59	C₂²×D₄×C₅₃	—	2	8	no	—
60	C₂²×D₄×C₅₃	—	2	4	no	—
61	C₂²×D₄×C₅₃	—	4	4	no	—
62	C₂²×D₄×C₅₃	—	4	4	no	—
63	C₂²×D₄×C₅₃	—	4	8	no	—
64	C₂²×D₄×C₅₃	—	4	4	no	—
65	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
66	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
67	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
68	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
69	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
70	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
71	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
72	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
73	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
74	C₂³×C₄×C₅₃	—	2	4	no	—
75	C₂³×C₄×C₅₃	—	4	4	no	—
76	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
78	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
80	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
82	#127×C₅₃	—	4	8	no	—
83	#123×C₅₃	—	2	16	no	—
84	#121×C₅₃	—	2	32	no	—
85	#121×C₅₃	—	2	8	no	—
86	#121×C₅₃	—	2	8	no	—
87	#121×C₅₃	—	4	8	no	—
88	#120×C₅₃	—	2	16	no	—
89	#120×C₅₃	—	4	8	no	—
90	#120×C₅₃	—	4	8	no	—
91	#120×C₅₃	—	4	8	no	—
92	#120×C₅₃	—	4	8	no	—
93	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
94	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
99	C₂×C₂²⋊C₄×C₅₃	D₄	1	8	yes	complemented
100	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
105	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
108	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
109	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
110	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
111	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
112	#95×C₅₃	C₂³	1	4	yes	—
113	#95×C₅₃	C₂³	1	8	yes	—
114	#95×C₅₃	C₂³	1	4	yes	—
115	#95×C₅₃	C₂³	1	8	yes	—
116	C₂⁴×C₅₃	—	2	2	no	—
117	C₂⁴×C₅₃	—	2	4	no	—
118	C₂⁴×C₅₃	—	2	4	no	—
119	C₂⁴×C₅₃	—	2	2	no	—
120	C₂⁴×C₅₃	—	2	4	no	—
121	C₂⁴×C₅₃	—	2	4	no	—
122	C₂⁴×C₅₃	—	2	4	no	—
123	C₂⁴×C₅₃	—	4	2	no	—
124	C₂⁴×C₅₃	—	4	2	no	—
125	C₂⁴×C₅₃	—	4	2	no	—
126	C₂⁴×C₅₃	—	4	2	no	—
127	C₂⁴×C₅₃	—	4	2	no	—
128	C₂⁴×C₅₃	—	4	4	no	—
129	C₂⁴×C₅₃	—	4	4	no	—
130	C₂⁴×C₅₃	—	4	4	no	—
131	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
132	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
133	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
134	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
135	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
136	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
137	C₂×D₄×C₅₃	—	2	8	no	—
138	C₂×D₄×C₅₃	—	2	8	no	—
139	C₂×D₄×C₅₃	—	2	8	no	—
140	C₂×D₄×C₅₃	—	2	8	no	—
141	C₂×D₄×C₅₃	—	2	8	no	—
142	C₂×D₄×C₅₃	—	2	8	no	—
143	C₂×D₄×C₅₃	—	2	8	no	—
144	C₂×D₄×C₅₃	—	2	8	no	—
145	C₂×D₄×C₅₃	—	2	16	no	—
146	C₂×D₄×C₅₃	—	2	8	no	—
147	C₂×D₄×C₅₃	—	2	4	no	—
148	C₂×D₄×C₅₃	—	2	4	no	—
149	C₂×D₄×C₅₃	—	2	8	no	—
150	C₂×D₄×C₅₃	—	4	4	no	—
151	C₂×D₄×C₅₃	—	4	4	no	—
152	C₂×D₄×C₅₃	—	4	4	no	—
153	C₂×D₄×C₅₃	—	4	4	no	—
154	C₂×D₄×C₅₃	—	4	4	no	—
155	C₂×D₄×C₅₃	—	4	16	no	—
156	C₂×D₄×C₅₃	—	4	8	no	—
157	C₂×D₄×C₅₃	—	4	8	no	—
158	C₂×D₄×C₅₃	—	4	8	no	—
159	C₂×D₄×C₅₃	—	4	4	no	—
160	C₂×D₄×C₅₃	—	4	8	no	—
161	C₂×D₄×C₅₃	—	4	4	no	—
162	C₂×D₄×C₅₃	—	4	8	no	—
163	C₂×D₄×C₅₃	—	4	16	no	—
164	C₂×D₄×C₅₃	—	4	16	no	—
165	C₂×D₄×C₅₃	—	4	32	no	—
166	C₂²×C₄×C₅₃	—	2	2	no	—
167	C₂²×C₄×C₅₃	—	2	4	no	—
168	C₂²×C₄×C₅₃	—	2	2	no	—
169	C₂²×C₄×C₅₃	—	2	4	no	—
170	C₂²×C₄×C₅₃	—	2	8	no	—
171	C₂²×C₄×C₅₃	—	2	8	no	—
172	C₂²×C₄×C₅₃	—	4	4	no	—
173	C₂²×C₄×C₅₃	—	4	2	no	—
174	C₂²×C₄×C₅₃	—	4	2	no	—
175	C₂²×C₄×C₅₃	—	4	2	no	—
176	C₂²×C₄×C₅₃	—	4	4	no	—
177	C₂²×C₄×C₅₃	—	4	16	no	—
178	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
179	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
180	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
181	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
182	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
186	C₄⋊C₄×C₅₃	—	2	8	no	—
187	C₄⋊C₄×C₅₃	—	2	4	no	—
188	C₄⋊C₄×C₅₃	—	2	4	no	—
189	C₄⋊C₄×C₅₃	—	4	4	no	—
190	C₂²⋊C₄×C₅₃	—	2	8	no	—
191	C₂²⋊C₄×C₅₃	—	2	8	no	—
192	C₂²⋊C₄×C₅₃	—	2	8	no	—
193	C₂²⋊C₄×C₅₃	—	2	8	no	—
194	C₂²⋊C₄×C₅₃	—	2	16	no	—
195	C₂²⋊C₄×C₅₃	—	4	8	no	—
196	C₂²⋊C₄×C₅₃	—	4	8	no	—
197	C₂²⋊C₄×C₅₃	—	4	8	no	—
198	C₂²⋊C₄×C₅₃	—	4	4	no	—
199	C₂²⋊C₄×C₅₃	—	4	4	no	—
200	C₂²⋊C₄×C₅₃	C₂×D₄	1	16	yes	—
201	C₄²×C₅₃	—	4	4	no	—
202	C₂³×C₅₃	—	2	2	no	—
203	C₂³×C₅₃	—	2	2	no	—
204	C₂³×C₅₃	—	2	8	no	—
205	C₂³×C₅₃	—	2	2	no	—
206	C₂³×C₅₃	—	2	4	no	—
207	C₂³×C₅₃	—	2	4	no	—
208	C₂³×C₅₃	—	2	4	no	—
209	C₂³×C₅₃	—	2	4	no	—
210	C₂³×C₅₃	—	2	2	no	—
211	C₂³×C₅₃	—	2	4	no	—
212	C₂³×C₅₃	—	2	8	no	—
213	C₂³×C₅₃	—	2	4	no	—
214	C₂³×C₅₃	—	2	2	no	—
215	C₂³×C₅₃	—	2	4	no	—
216	C₂³×C₅₃	—	2	4	no	—
217	C₂³×C₅₃	—	2	4	no	—
218	C₂³×C₅₃	—	2	4	no	—
219	C₂³×C₅₃	—	2	8	no	—
220	C₂³×C₅₃	—	2	8	no	—
221	C₂³×C₅₃	—	2	4	no	—
222	C₂³×C₅₃	—	4	4	no	—
223	C₂³×C₅₃	—	4	4	no	—
224	C₂³×C₅₃	—	4	16	no	—
225	C₂³×C₅₃	—	4	4	no	—
226	C₂³×C₅₃	—	4	4	no	—
227	C₂³×C₅₃	—	4	4	no	—
228	C₂³×C₅₃	—	4	8	no	—
229	C₂³×C₅₃	—	4	4	no	—
230	C₂³×C₅₃	—	4	4	no	—
231	C₂³×C₅₃	—	8	8	no	—
232	C₂³×C₅₃	—	8	8	no	—
233	C₂³×C₅₃	—	8	4	no	—
234	C₂³×C₅₃	—	8	4	no	—
235	C₂³×C₅₃	—	8	16	no	—
236	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
237	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
238	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
239	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
240	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
241	D₄×C₅₃	—	4	16	no	—
242	D₄×C₅₃	—	4	16	no	—
243	D₄×C₅₃	—	4	16	no	—
244	D₄×C₅₃	—	4	16	no	—
245	D₄×C₅₃	—	4	16	no	—
246	D₄×C₅₃	—	8	4	no	—
247	D₄×C₅₃	—	8	4	no	—
248	D₄×C₅₃	—	8	16	no	—
249	D₄×C₅₃	—	8	16	no	—
250	C₂×C₄×C₅₃	—	2	8	no	—
251	C₂×C₄×C₅₃	—	2	2	no	—
252	C₂×C₄×C₅₃	—	2	2	no	—
253	C₂×C₄×C₅₃	—	2	4	no	—
254	C₂×C₄×C₅₃	—	2	4	no	—
255	C₂×C₄×C₅₃	—	2	4	no	—
256	C₂×C₄×C₅₃	—	2	2	no	—
257	C₂×C₄×C₅₃	—	2	4	no	—
258	C₂×C₄×C₅₃	—	2	4	no	—
259	C₂×C₄×C₅₃	—	2	4	no	—
260	C₂×C₄×C₅₃	—	2	8	no	—
261	C₂×C₄×C₅₃	—	2	8	no	—
262	C₂×C₄×C₅₃	—	2	8	no	—
263	C₂×C₄×C₅₃	—	4	4	no	—
264	C₂×C₄×C₅₃	—	4	4	no	—
265	C₂×C₄×C₅₃	—	4	16	no	—
266	C₂×C₄×C₅₃	—	4	8	no	—
267	C₂×C₄×C₅₃	—	4	4	no	—
268	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
269	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
270	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
271	#36717	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
272	C₂²×C₅₃	—	2	2	no	—
273	C₂²×C₅₃	—	2	2	no	—
274	C₂²×C₅₃	—	2	8	no	—
275	C₂²×C₅₃	—	2	4	no	—
276	C₂²×C₅₃	—	2	4	no	—
277	C₂²×C₅₃	—	2	4	no	—
278	C₂²×C₅₃	—	2	4	no	—
279	C₂²×C₅₃	—	2	8	no	—
280	C₂²×C₅₃	—	4	16	no	—
281	C₂²×C₅₃	—	4	2	no	—
282	C₂²×C₅₃	—	4	2	no	—
283	C₂²×C₅₃	—	4	4	no	—
284	C₂²×C₅₃	—	4	8	no	—
285	C₂²×C₅₃	—	4	4	no	—
286	C₂²×C₅₃	—	4	2	no	—
287	C₂²×C₅₃	—	4	2	no	—
288	C₂²×C₅₃	—	4	2	no	—
289	C₂²×C₅₃	—	4	4	no	—
290	C₂²×C₅₃	—	4	4	no	—
291	C₂²×C₅₃	—	4	8	no	—
292	C₂²×C₅₃	—	4	8	no	—
293	C₂²×C₅₃	—	4	8	no	—
294	C₂²×C₅₃	—	8	8	no	—
295	C₂²×C₅₃	—	8	8	no	—
296	C₂²×C₅₃	—	8	16	no	—
297	C₂²×C₅₃	#535	1	4	yes	—
298	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
299	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
300	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
301	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
302	C₄×C₅₃	—	2	4	no	—
303	C₄×C₅₃	—	2	4	no	—
304	C₄×C₅₃	—	2	4	no	—
305	C₄×C₅₃	—	4	4	no	—
306	C₄×C₅₃	—	4	2	no	—
307	C₄×C₅₃	—	4	8	no	—
308	C₂×#534	C₂×C₅₃	1	1	yes	characteristic, complemented
309	C₂²×#121	C₂×C₅₃	1	1	yes	characteristic, complemented
310	#2847	C₂×C₅₃	1	2	yes	complemented
311	#2703	C₂×C₅₃	1	8	yes	complemented
312	#2669	C₂×C₅₃	1	8	yes	complemented
313	#2617	C₂×C₅₃	1	2	yes	complemented
314	#2591	C₂×C₅₃	1	2	yes	complemented
315	#2433	C₂×C₅₃	1	2	yes	complemented
316	#2403	C₂×C₅₃	1	2	yes	complemented
317	#2403	C₂×C₅₃	1	2	yes	complemented
318	#2312	C₂×C₅₃	1	1	yes	characteristic, complemented
319	C₂×C₅₃	—	2	4	no	—
320	C₂×C₅₃	—	4	4	no	—
321	C₂×C₅₃	—	4	4	no	—
322	C₂×C₅₃	—	8	4	no	—
323	C₂×C₅₃	—	8	2	no	—
324	C₂×C₅₃	C₂×#535	1	1	yes	characteristic
325	C₂×C₅₃	#3484	1	1	yes	characteristic
326	C₂×C₅₃	#3484	1	1	yes	characteristic
327	C₂×C₅₃	#3537	1	4	yes	—
328	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
329	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
330	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
331	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
332	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
333	#534	—	2	8	no	—
334	C₂×#127	C₂²×C₅₃	1	4	yes	complemented
335	C₂×#123	C₂²×C₅₃	1	4	yes	complemented
336	C₂×#121	—	2	8	no	—
337	C₂×#121	C₂²×C₅₃	1	2	yes	—
338	C₂×#121	C₂²×C₅₃	1	2	yes	—
339	C₂×#121	C₂²×C₅₃	1	8	yes	complemented
340	C₂×#121	C₂²×C₅₃	1	4	yes	complemented
341	C₂×#120	C₂²×C₅₃	1	4	yes	complemented
342	C₂×#120	C₂²×C₅₃	1	4	yes	complemented
343	C₂×#120	C₂²×C₅₃	1	4	yes	complemented
344	C₂×#120	C₂²×C₅₃	1	4	yes	complemented
345	C₂×#120	C₂²×C₅₃	1	4	yes	complemented
346	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
347	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
348	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
349	#400	C₂²×C₅₃	1	4	yes	complemented
350	#397	C₂²×C₅₃	1	4	yes	complemented
351	#394	C₂²×C₅₃	1	4	yes	complemented
352	#394	C₂²×C₅₃	1	8	yes	complemented
353	#392	C₂²×C₅₃	1	4	yes	complemented
354	#392	C₂²×C₅₃	1	8	yes	complemented
355	#392	C₂²×C₅₃	1	8	yes	complemented
356	#392	C₂²×C₅₃	1	8	yes	complemented
357	#390	C₂²×C₅₃	1	4	yes	complemented
358	#388	C₂²×C₅₃	1	8	yes	complemented
359	#388	C₂²×C₅₃	1	8	yes	complemented
360	#386	C₂²×C₅₃	1	4	yes	complemented
361	#386	C₂²×C₅₃	1	4	yes	complemented
362	#386	C₂²×C₅₃	1	8	yes	complemented
363	#386	C₂²×C₅₃	1	4	yes	complemented
364	#386	C₂²×C₅₃	1	4	yes	complemented
365	#380	C₂²×C₅₃	1	2	yes	complemented
366	#380	C₂²×C₅₃	1	4	yes	complemented
367	#379	C₂²×C₅₃	1	2	yes	—
368	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	complemented
369	C₅₃	#36717	1	1	yes	characteristic, direct product factor, Sylow
370	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
371	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
372	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
373	C₂⁵	C₂³×C₅₃	1	2	yes	—
374	C₂²×D₄	—	2	4	no	—
375	C₂²×D₄	—	2	4	no	—
376	C₂²×D₄	—	2	8	no	—
377	C₂²×D₄	—	2	4	no	—
378	C₂²×D₄	—	4	4	no	—
379	C₂²×D₄	—	4	4	no	—
380	C₂²×D₄	—	4	8	no	—
381	C₂²×D₄	—	4	4	no	—
382	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
383	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
384	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
385	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
386	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
387	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
388	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
389	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
390	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
391	C₂³×C₄	—	2	4	no	—
392	C₂³×C₄	—	4	4	no	—
393	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
394	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
395	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
396	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
397	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
398	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
399	#127	—	4	8	no	—
400	#123	—	2	16	no	—
401	#121	—	2	8	no	—
402	#121	—	2	8	no	—
403	#121	—	2	32	no	—
404	#121	—	4	8	no	—
405	#120	—	2	16	no	—
406	#120	—	4	8	no	—
407	#120	—	4	8	no	—
408	#120	—	4	8	no	—
409	#120	—	4	8	no	—
410	C₂×C₄⋊C₄	—	2	4	no	—
411	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
412	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
413	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
414	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
415	C₂×C₂²⋊C₄	—	2	4	no	—
416	C₂×C₂²⋊C₄	D₄×C₅₃	1	8	yes	complemented
417	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
425	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
426	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
427	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
428	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
429	#95	C₂³×C₅₃	1	4	yes	—
430	#95	C₂³×C₅₃	1	8	yes	—
431	#95	C₂³×C₅₃	1	4	yes	—
432	#95	C₂³×C₅₃	1	8	yes	—
433	C₂⁴	—	2	2	no	—
434	C₂⁴	—	2	4	no	—
435	C₂⁴	—	2	4	no	—
436	C₂⁴	—	2	2	no	—
437	C₂⁴	—	2	4	no	—
438	C₂⁴	—	2	4	no	—
439	C₂⁴	—	2	4	no	—
440	C₂⁴	—	4	2	no	—
441	C₂⁴	—	4	2	no	—
442	C₂⁴	—	4	2	no	—
443	C₂⁴	—	4	2	no	—
444	C₂⁴	—	4	2	no	—
445	C₂⁴	—	4	4	no	—
446	C₂⁴	—	4	4	no	—
447	C₂⁴	—	4	4	no	—
448	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
449	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
450	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
451	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
452	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
453	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
454	C₂×D₄	—	2	8	no	—
455	C₂×D₄	—	2	8	no	—
456	C₂×D₄	—	2	8	no	—
457	C₂×D₄	—	2	8	no	—
458	C₂×D₄	—	2	8	no	—
459	C₂×D₄	—	2	8	no	—
460	C₂×D₄	—	2	8	no	—
461	C₂×D₄	—	2	8	no	—
462	C₂×D₄	—	2	16	no	—
463	C₂×D₄	—	2	8	no	—
464	C₂×D₄	—	2	4	no	—
465	C₂×D₄	—	2	4	no	—
466	C₂×D₄	—	2	8	no	—
467	C₂×D₄	—	4	4	no	—
468	C₂×D₄	—	4	4	no	—
469	C₂×D₄	—	4	4	no	—
470	C₂×D₄	—	4	4	no	—
471	C₂×D₄	—	4	4	no	—
472	C₂×D₄	—	4	16	no	—
473	C₂×D₄	—	4	8	no	—
474	C₂×D₄	—	4	8	no	—
475	C₂×D₄	—	4	8	no	—
476	C₂×D₄	—	4	4	no	—
477	C₂×D₄	—	4	8	no	—
478	C₂×D₄	—	4	4	no	—
479	C₂×D₄	—	4	8	no	—
480	C₂×D₄	—	4	16	no	—
481	C₂×D₄	—	4	16	no	—
482	C₂×D₄	—	4	32	no	—
483	C₂²×C₄	—	2	2	no	—
484	C₂²×C₄	—	2	4	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	4	no	—
487	C₂²×C₄	—	2	8	no	—
488	C₂²×C₄	—	2	8	no	—
489	C₂²×C₄	—	4	2	no	—
490	C₂²×C₄	—	4	4	no	—
491	C₂²×C₄	—	4	2	no	—
492	C₂²×C₄	—	4	2	no	—
493	C₂²×C₄	—	4	4	no	—
494	C₂²×C₄	—	4	16	no	—
495	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
496	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
497	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
498	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
499	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
500	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
501	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
502	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
503	C₄⋊C₄	—	2	8	no	—
504	C₄⋊C₄	—	2	4	no	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	4	4	no	—
507	C₂²⋊C₄	—	2	8	no	—
508	C₂²⋊C₄	—	2	8	no	—
509	C₂²⋊C₄	—	2	8	no	—
510	C₂²⋊C₄	—	2	8	no	—
511	C₂²⋊C₄	—	2	16	no	—
512	C₂²⋊C₄	—	4	4	no	—
513	C₂²⋊C₄	—	4	8	no	—
514	C₂²⋊C₄	—	4	8	no	—
515	C₂²⋊C₄	—	4	8	no	—
516	C₂²⋊C₄	—	4	4	no	—
517	C₂²⋊C₄	C₂×D₄×C₅₃	1	16	yes	—
518	C₄²	—	4	4	no	—
519	C₂³	—	2	2	no	—
520	C₂³	—	2	2	no	—
521	C₂³	—	2	8	no	—
522	C₂³	—	2	2	no	—
523	C₂³	—	2	4	no	—
524	C₂³	—	2	4	no	—
525	C₂³	—	2	4	no	—
526	C₂³	—	2	4	no	—
527	C₂³	—	2	4	no	—
528	C₂³	—	2	2	no	—
529	C₂³	—	2	8	no	—
530	C₂³	—	2	2	no	—
531	C₂³	—	2	4	no	—
532	C₂³	—	2	8	no	—
533	C₂³	—	2	4	no	—
534	C₂³	—	2	8	no	—
535	C₂³	—	2	4	no	—
536	C₂³	—	2	4	no	—
537	C₂³	—	2	4	no	—
538	C₂³	—	2	4	no	—
539	C₂³	—	4	4	no	—
540	C₂³	—	4	4	no	—
541	C₂³	—	4	16	no	—
542	C₂³	—	4	4	no	—
543	C₂³	—	4	4	no	—
544	C₂³	—	4	4	no	—
545	C₂³	—	4	8	no	—
546	C₂³	—	4	4	no	—
547	C₂³	—	4	4	no	—
548	C₂³	—	8	8	no	—
549	C₂³	—	8	8	no	—
550	C₂³	—	8	4	no	—
551	C₂³	—	8	4	no	—
552	C₂³	—	8	16	no	—
553	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
554	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
555	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
556	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
557	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
558	D₄	—	4	16	no	—
559	D₄	—	4	16	no	—
560	D₄	—	4	16	no	—
561	D₄	—	4	16	no	—
562	D₄	—	4	16	no	—
563	D₄	—	8	4	no	—
564	D₄	—	8	4	no	—
565	D₄	—	8	16	no	—
566	D₄	—	8	16	no	—
567	C₂×C₄	—	2	8	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	2	8	no	—
575	C₂×C₄	—	2	8	no	—
576	C₂×C₄	—	2	8	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	4	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	4	4	no	—
581	C₂×C₄	—	4	4	no	—
582	C₂×C₄	—	4	16	no	—
583	C₂×C₄	—	4	4	no	—
584	C₂×C₄	—	4	8	no	—
585	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
586	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
587	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
588	C₂²	—	2	2	no	—
589	C₂²	—	2	2	no	—
590	C₂²	—	2	8	no	—
591	C₂²	—	2	4	no	—
592	C₂²	—	2	4	no	—
593	C₂²	—	2	4	no	—
594	C₂²	—	2	8	no	—
595	C₂²	—	2	4	no	—
596	C₂²	—	4	16	no	—
597	C₂²	—	4	2	no	—
598	C₂²	—	4	4	no	—
599	C₂²	—	4	8	no	—
600	C₂²	—	4	2	no	—
601	C₂²	—	4	2	no	—
602	C₂²	—	4	2	no	—
603	C₂²	—	4	4	no	—
604	C₂²	—	4	4	no	—
605	C₂²	—	4	8	no	—
606	C₂²	—	4	8	no	—
607	C₂²	—	4	8	no	—
608	C₂²	—	4	4	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	8	8	no	—
611	C₂²	—	8	8	no	—
612	C₂²	—	8	16	no	—
613	C₂²	C₅₃×#535	1	4	yes	—
614	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
615	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
616	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
617	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
618	C₄	—	2	4	no	—
619	C₄	—	2	4	no	—
620	C₄	—	2	4	no	—
621	C₄	—	4	2	no	—
622	C₄	—	4	8	no	—
623	C₄	—	4	4	no	—
624	C₂	—	2	4	no	—
625	C₂	—	4	4	no	—
626	C₂	—	4	4	no	—
627	C₂	—	8	2	no	—
628	C₂	—	8	4	no	—
629	C₂	C₂×C₅₃×#535	1	1	yes	characteristic
630	C₂	C₅₃×#3484	1	1	yes	characteristic
631	C₂	C₅₃×#3484	1	1	yes	characteristic
632	C₂	C₅₃×#3537	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#534	C₂	semidirect
2. C₂²×#121×C₅₃	C₂	semidirect
3. C₅₃×#2847	C₂	semidirect
4. C₅₃×#2703	C₂	semidirect
5. C₅₃×#2669	C₂	semidirect
6. C₅₃×#2617	C₂	semidirect
7. C₅₃×#2591	C₂	semidirect
8. C₅₃×#2433	C₂	semidirect
9. C₅₃×#2403	C₂	semidirect
10. C₅₃×#2403	C₂	semidirect
11. C₅₃×#2312	C₂	semidirect
15. C₂³×D₄×C₅₃	C₂²	semidirect
16. C₂⁴×C₄×C₅₃	C₂²	semidirect
18. C₂×#127×C₅₃	C₂²	semidirect
19. C₂×#123×C₅₃	C₂²	semidirect
23. C₂×#121×C₅₃	C₂²	semidirect
24. C₂×#121×C₅₃	C₂²	semidirect
25. C₂×#120×C₅₃	C₂²	semidirect
26. C₂×#120×C₅₃	C₂²	semidirect
27. C₂×#120×C₅₃	C₂²	semidirect
28. C₂×#120×C₅₃	C₂²	semidirect
29. C₂×#120×C₅₃	C₂²	semidirect
30. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
32. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
33. C₅₃×#400	C₂²	semidirect
34. C₅₃×#397	C₂²	semidirect
35. C₅₃×#394	C₂²	semidirect
36. C₅₃×#394	C₂²	semidirect
37. C₅₃×#392	C₂²	semidirect
38. C₅₃×#392	C₂²	semidirect
39. C₅₃×#392	C₂²	semidirect
40. C₅₃×#392	C₂²	semidirect
41. C₅₃×#390	C₂²	semidirect
42. C₅₃×#388	C₂²	semidirect
43. C₅₃×#388	C₂²	semidirect
44. C₅₃×#386	C₂²	semidirect
45. C₅₃×#386	C₂²	semidirect
46. C₅₃×#386	C₂²	semidirect
47. C₅₃×#386	C₂²	semidirect
48. C₅₃×#386	C₂²	semidirect
49. C₅₃×#380	C₂²	semidirect
50. C₅₃×#380	C₂²	semidirect
52. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
99. C₂×C₂²⋊C₄×C₅₃	D₄	semidirect
271. #36717	C₅₃	direct (N × H)
308. C₂×#534	C₂×C₅₃	semidirect
309. C₂²×#121	C₂×C₅₃	semidirect
310. #2847	C₂×C₅₃	semidirect
311. #2703	C₂×C₅₃	semidirect
312. #2669	C₂×C₅₃	semidirect
313. #2617	C₂×C₅₃	semidirect
314. #2591	C₂×C₅₃	semidirect
315. #2433	C₂×C₅₃	semidirect
316. #2403	C₂×C₅₃	semidirect
317. #2403	C₂×C₅₃	semidirect
318. #2312	C₂×C₅₃	semidirect
331. C₂³×D₄	C₂²×C₅₃	semidirect
332. C₂⁴×C₄	C₂²×C₅₃	semidirect
334. C₂×#127	C₂²×C₅₃	semidirect
335. C₂×#123	C₂²×C₅₃	semidirect
339. C₂×#121	C₂²×C₅₃	semidirect
340. C₂×#121	C₂²×C₅₃	semidirect
341. C₂×#120	C₂²×C₅₃	semidirect
342. C₂×#120	C₂²×C₅₃	semidirect
343. C₂×#120	C₂²×C₅₃	semidirect
344. C₂×#120	C₂²×C₅₃	semidirect
345. C₂×#120	C₂²×C₅₃	semidirect
346. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
348. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
349. #400	C₂²×C₅₃	semidirect
350. #397	C₂²×C₅₃	semidirect
351. #394	C₂²×C₅₃	semidirect
352. #394	C₂²×C₅₃	semidirect
353. #392	C₂²×C₅₃	semidirect
354. #392	C₂²×C₅₃	semidirect
355. #392	C₂²×C₅₃	semidirect
356. #392	C₂²×C₅₃	semidirect
357. #390	C₂²×C₅₃	semidirect
358. #388	C₂²×C₅₃	semidirect
359. #388	C₂²×C₅₃	semidirect
360. #386	C₂²×C₅₃	semidirect
361. #386	C₂²×C₅₃	semidirect
362. #386	C₂²×C₅₃	semidirect
363. #386	C₂²×C₅₃	semidirect
364. #386	C₂²×C₅₃	semidirect
365. #380	C₂²×C₅₃	semidirect
366. #380	C₂²×C₅₃	semidirect
368. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
369. C₅₃	#36717	direct (N × H)
416. C₂×C₂²⋊C₄	D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#534	2
C₂²×#121×C₅₃	2
C₅₃×#2847	2
C₅₃×#2703	2
C₅₃×#2669	2
C₅₃×#2617	2
C₅₃×#2591	2
C₅₃×#2433	2
C₅₃×#2403	2
C₅₃×#2312	2
#36717	53

Subgroups — C53×#36717 #27244229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36717 #27244229