Subgroups — C₅₃×#3558 #20272495

Order 6784.

1160 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#562	C₂	1	2	yes	maximal, complemented
2	C₅₃×H(3,𝔽₄)	C₂	1	2	yes	maximal, complemented
3	C₅₃×#559	C₂	1	2	yes	maximal, complemented
4	C₅₃×#548	C₂	1	2	yes	maximal, complemented
5	C₅₃×#546	C₂	1	2	yes	maximal, complemented
6	C₅₃×#546	C₂	1	4	yes	maximal, complemented
7	D₄²×C₅₃	C₂	1	2	yes	maximal, complemented
8	C₅₃×#542	C₂	1	2	yes	maximal, complemented
9	C₅₃×#538	C₂	1	2	yes	maximal, complemented
10	C₅₃×#538	C₂	1	4	yes	maximal, complemented
11	C₅₃×#535	C₂	1	2	yes	maximal, complemented
12	C₅₃×#534	C₂	1	2	yes	maximal, complemented
13	C₅₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#124×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×C₅₃	C₂²	1	2	yes	complemented
17	C₂×#41×C₅₃	C₂²	1	2	yes	complemented
18	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×D₄×C₅₃	C₂²	1	4	yes	complemented
21	C₂²×D₄×C₅₃	C₂²	1	2	yes	complemented
22	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
23	#127×C₅₃	C₂²	1	2	yes	—
24	#127×C₅₃	C₂²	1	1	yes	characteristic, complemented
25	#126×C₅₃	C₂²	1	2	yes	complemented
26	#126×C₅₃	C₂²	1	4	yes	complemented
27	#125×C₅₃	C₂²	1	1	yes	characteristic, complemented
28	#124×C₅₃	C₂²	1	2	yes	complemented
29	#124×C₅₃	C₂²	1	2	yes	complemented
30	#124×C₅₃	C₂²	1	2	yes	complemented
31	#124×C₅₃	C₂²	1	2	yes	complemented
32	#124×C₅₃	C₂²	1	1	yes	characteristic, complemented
33	#124×C₅₃	C₂²	1	2	yes	complemented
34	#124×C₅₃	C₂²	1	2	yes	complemented
35	#124×C₅₃	C₂²	1	4	yes	complemented
36	#124×C₅₃	C₂²	1	4	yes	complemented
37	#124×C₅₃	C₂²	1	4	yes	complemented
38	#123×C₅₃	C₂²	1	2	yes	complemented
39	#123×C₅₃	C₂²	1	4	yes	complemented
40	#123×C₅₃	C₂²	1	4	yes	complemented
41	#123×C₅₃	C₂²	1	4	yes	complemented
42	#123×C₅₃	C₂²	1	4	yes	complemented
43	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
44	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
45	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
46	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
47	#121×C₅₃	C₂²	1	1	yes	characteristic
48	#121×C₅₃	C₂²	1	2	yes	—
49	#121×C₅₃	C₂²	1	2	yes	complemented
50	#121×C₅₃	C₂²	1	2	yes	complemented
51	#121×C₅₃	C₂²	1	4	yes	complemented
52	#121×C₅₃	C₂²	1	4	yes	complemented
53	#121×C₅₃	C₂²	1	4	yes	complemented
54	#121×C₅₃	C₂²	1	4	yes	complemented
55	#121×C₅₃	C₂²	1	4	yes	complemented
56	#120×C₅₃	C₂²	1	4	yes	complemented
57	#120×C₅₃	C₂²	1	4	yes	complemented
58	#120×C₅₃	C₂²	1	4	yes	complemented
59	#120×C₅₃	C₂²	1	4	yes	complemented
60	#120×C₅₃	C₂²	1	4	yes	complemented
61	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
62	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
63	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
64	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
65	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
66	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
67	C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
68	C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
69	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
70	#117×C₅₃	C₂²	1	2	yes	complemented
71	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
72	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
73	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
74	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
75	C₂⁴×C₅₃	C₂³	1	2	yes	—
76	C₂⁴×C₅₃	C₂³	1	4	yes	—
77	#41×C₅₃	—	4	4	no	—
78	#41×C₅₃	—	4	4	no	—
79	C₂×Q₈×C₅₃	—	4	2	no	—
80	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
81	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
83	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
84	C₂×D₄×C₅₃	—	4	2	no	—
85	C₂×D₄×C₅₃	—	4	4	no	—
86	C₂×D₄×C₅₃	—	4	8	no	—
87	C₂×D₄×C₅₃	C₂³	1	4	yes	—
88	C₂×D₄×C₅₃	C₂³	1	2	yes	—
89	C₂×D₄×C₅₃	C₂³	1	2	yes	—
90	C₂×D₄×C₅₃	C₂³	1	2	yes	—
91	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₅₃	C₂³	1	4	yes	—
93	C₂×D₄×C₅₃	C₂³	1	2	yes	—
94	C₂×D₄×C₅₃	C₂³	1	4	yes	—
95	C₂×D₄×C₅₃	C₂³	1	4	yes	—
96	C₂×D₄×C₅₃	C₂³	1	2	yes	—
97	C₂×D₄×C₅₃	C₂³	1	2	yes	—
98	C₂×D₄×C₅₃	C₂³	1	4	yes	—
99	C₂×D₄×C₅₃	C₂³	1	4	yes	—
100	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
103	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
105	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
106	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
107	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
108	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
109	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
110	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
111	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
112	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
113	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
114	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
115	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
116	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
117	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
118	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
119	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
120	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
121	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
122	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
123	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
124	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
125	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
126	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
127	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
128	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
129	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
130	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
131	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
132	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
133	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
134	C₄²×C₅₃	C₂³	1	1	yes	characteristic
135	C₄²×C₅₃	C₂³	1	1	yes	characteristic
136	C₄²×C₅₃	C₂³	1	1	yes	characteristic
137	C₄²×C₅₃	C₂³	1	1	yes	characteristic
138	C₄²×C₅₃	C₂³	1	2	yes	—
139	C₄²×C₅₃	C₂³	1	4	yes	—
140	C₂³×C₅₃	—	2	4	no	—
141	C₂³×C₅₃	—	4	2	no	—
142	C₂³×C₅₃	—	4	4	no	—
143	C₂³×C₅₃	—	4	4	no	—
144	C₂³×C₅₃	—	4	2	no	—
145	C₂³×C₅₃	—	4	4	no	—
146	C₂³×C₅₃	C₂⁴	1	4	yes	—
147	C₂³×C₅₃	C₂⁴	1	2	yes	—
148	C₂³×C₅₃	C₂⁴	1	4	yes	—
149	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
150	Q₈×C₅₃	—	2	4	no	—
151	Q₈×C₅₃	—	4	2	no	—
152	Q₈×C₅₃	—	4	1	no	—
153	Q₈×C₅₃	—	4	2	no	—
154	D₄×C₅₃	—	2	4	no	—
155	D₄×C₅₃	—	2	8	no	—
156	D₄×C₅₃	—	2	4	no	—
157	D₄×C₅₃	—	2	4	no	—
158	D₄×C₅₃	—	4	4	no	—
159	D₄×C₅₃	—	4	2	no	—
160	D₄×C₅₃	—	4	4	no	—
161	D₄×C₅₃	—	4	4	no	—
162	D₄×C₅₃	—	4	2	no	—
163	D₄×C₅₃	—	4	1	no	—
164	D₄×C₅₃	—	4	4	no	—
165	D₄×C₅₃	—	4	2	no	—
166	D₄×C₅₃	—	4	4	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	4	no	—
170	C₂×C₄×C₅₃	—	4	1	no	—
171	C₂×C₄×C₅₃	—	4	4	no	—
172	C₂×C₄×C₅₃	—	4	2	no	—
173	C₂×C₄×C₅₃	—	4	4	no	—
174	C₂×C₄×C₅₃	—	4	4	no	—
175	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
177	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
178	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
179	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
180	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
185	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
186	C₂²×C₅₃	—	2	1	no	—
187	C₂²×C₅₃	—	2	1	no	—
188	C₂²×C₅₃	—	2	2	no	—
189	C₂²×C₅₃	—	2	4	no	—
190	C₂²×C₅₃	—	2	4	no	—
191	C₂²×C₅₃	—	2	4	no	—
192	C₂²×C₅₃	—	2	2	no	—
193	C₂²×C₅₃	—	2	4	no	—
194	C₂²×C₅₃	—	2	4	no	—
195	C₂²×C₅₃	—	2	2	no	—
196	C₂²×C₅₃	—	2	4	no	—
197	C₂²×C₅₃	—	8	4	no	—
198	C₂²×C₅₃	—	8	8	no	—
199	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
200	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
201	C₄×C₅₃	—	2	1	no	—
202	C₄×C₅₃	—	2	2	no	—
203	C₄×C₅₃	—	2	1	no	—
204	C₄×C₅₃	—	2	1	no	—
205	C₄×C₅₃	—	2	2	no	—
206	C₄×C₅₃	—	2	4	no	—
207	C₄×C₅₃	—	2	4	no	—
208	C₄×C₅₃	—	2	1	no	—
209	C₄×C₅₃	—	2	1	no	—
210	C₄×C₅₃	—	2	2	no	—
211	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
212	#3558	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂×C₅₃	—	2	2	no	—
214	C₂×C₅₃	—	4	4	no	—
215	C₂×C₅₃	—	4	2	no	—
216	C₂×C₅₃	—	4	4	no	—
217	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
218	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
219	C₂×C₅₃	#585	1	1	yes	characteristic
220	#562	C₂×C₅₃	1	2	yes	complemented
221	H(3,𝔽₄)	C₂×C₅₃	1	2	yes	complemented
222	#559	C₂×C₅₃	1	2	yes	complemented
223	#548	C₂×C₅₃	1	2	yes	complemented
224	#546	C₂×C₅₃	1	2	yes	complemented
225	#546	C₂×C₅₃	1	4	yes	complemented
226	D₄²	C₂×C₅₃	1	2	yes	complemented
227	#542	C₂×C₅₃	1	2	yes	complemented
228	#538	C₂×C₅₃	1	2	yes	complemented
229	#538	C₂×C₅₃	1	4	yes	complemented
230	#535	C₂×C₅₃	1	2	yes	complemented
231	#534	C₂×C₅₃	1	2	yes	complemented
232	#532	C₂×C₅₃	1	1	yes	characteristic, complemented
233	#529	C₂×C₅₃	1	1	yes	characteristic, complemented
234	C₂×#124	C₂×C₅₃	1	1	yes	characteristic, complemented
235	C₅₃	#3558	1	1	yes	characteristic, direct product factor, Sylow
236	C₂×#41	C₂²×C₅₃	1	2	yes	complemented
237	C₂×#41	C₂²×C₅₃	1	2	yes	complemented
238	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
239	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
240	C₂²×D₄	C₂²×C₅₃	1	4	yes	complemented
241	C₂²×D₄	C₂²×C₅₃	1	2	yes	complemented
242	C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
243	#127	C₂²×C₅₃	1	2	yes	—
244	#127	C₂²×C₅₃	1	1	yes	characteristic, complemented
245	#126	C₂²×C₅₃	1	2	yes	complemented
246	#126	C₂²×C₅₃	1	4	yes	complemented
247	#125	C₂²×C₅₃	1	1	yes	characteristic, complemented
248	#124	C₂²×C₅₃	1	2	yes	complemented
249	#124	C₂²×C₅₃	1	2	yes	complemented
250	#124	C₂²×C₅₃	1	2	yes	complemented
251	#124	C₂²×C₅₃	1	2	yes	complemented
252	#124	C₂²×C₅₃	1	1	yes	characteristic, complemented
253	#124	C₂²×C₅₃	1	2	yes	complemented
254	#124	C₂²×C₅₃	1	2	yes	complemented
255	#124	C₂²×C₅₃	1	4	yes	complemented
256	#124	C₂²×C₅₃	1	4	yes	complemented
257	#124	C₂²×C₅₃	1	4	yes	complemented
258	#123	C₂²×C₅₃	1	2	yes	complemented
259	#123	C₂²×C₅₃	1	4	yes	complemented
260	#123	C₂²×C₅₃	1	4	yes	complemented
261	#123	C₂²×C₅₃	1	4	yes	complemented
262	#123	C₂²×C₅₃	1	4	yes	complemented
263	C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
264	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
265	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
266	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
267	#121	C₂²×C₅₃	1	1	yes	characteristic
268	#121	C₂²×C₅₃	1	2	yes	—
269	#121	C₂²×C₅₃	1	2	yes	complemented
270	#121	C₂²×C₅₃	1	2	yes	complemented
271	#121	C₂²×C₅₃	1	4	yes	complemented
272	#121	C₂²×C₅₃	1	4	yes	complemented
273	#121	C₂²×C₅₃	1	4	yes	complemented
274	#121	C₂²×C₅₃	1	4	yes	complemented
275	#121	C₂²×C₅₃	1	4	yes	complemented
276	#120	C₂²×C₅₃	1	4	yes	complemented
277	#120	C₂²×C₅₃	1	4	yes	complemented
278	#120	C₂²×C₅₃	1	4	yes	complemented
279	#120	C₂²×C₅₃	1	4	yes	complemented
280	#120	C₂²×C₅₃	1	4	yes	complemented
281	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
282	C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
283	C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
284	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
285	C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
286	C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
287	C₄×D₄	C₂²×C₅₃	1	4	yes	complemented
288	C₄×D₄	C₂²×C₅₃	1	4	yes	complemented
289	C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
290	#117	C₂²×C₅₃	1	2	yes	complemented
291	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	complemented
292	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	complemented
293	C₂×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
294	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic, complemented
295	C₂⁴	C₂³×C₅₃	1	2	yes	—
296	C₂⁴	C₂³×C₅₃	1	4	yes	—
297	#41	—	4	4	no	—
298	#41	—	4	4	no	—
299	C₂×Q₈	—	4	2	no	—
300	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
301	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
302	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
303	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
304	C₂×D₄	—	4	2	no	—
305	C₂×D₄	—	4	4	no	—
306	C₂×D₄	—	4	8	no	—
307	C₂×D₄	C₂³×C₅₃	1	4	yes	—
308	C₂×D₄	C₂³×C₅₃	1	2	yes	—
309	C₂×D₄	C₂³×C₅₃	1	2	yes	—
310	C₂×D₄	C₂³×C₅₃	1	2	yes	—
311	C₂×D₄	C₂³×C₅₃	1	1	yes	characteristic
312	C₂×D₄	C₂³×C₅₃	1	4	yes	—
313	C₂×D₄	C₂³×C₅₃	1	2	yes	—
314	C₂×D₄	C₂³×C₅₃	1	4	yes	—
315	C₂×D₄	C₂³×C₅₃	1	4	yes	—
316	C₂×D₄	C₂³×C₅₃	1	2	yes	—
317	C₂×D₄	C₂³×C₅₃	1	2	yes	—
318	C₂×D₄	C₂³×C₅₃	1	4	yes	—
319	C₂×D₄	C₂³×C₅₃	1	4	yes	—
320	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
321	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
322	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
323	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
324	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
325	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
326	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
327	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
328	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
329	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
330	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
331	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
332	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
333	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
334	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
335	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
336	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
337	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
338	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
339	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
340	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
341	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
342	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
343	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
344	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
345	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
346	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
347	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
348	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
349	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
350	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
351	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
352	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
353	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
354	C₄²	C₂³×C₅₃	1	1	yes	characteristic
355	C₄²	C₂³×C₅₃	1	1	yes	characteristic
356	C₄²	C₂³×C₅₃	1	1	yes	characteristic
357	C₄²	C₂³×C₅₃	1	1	yes	characteristic
358	C₄²	C₂³×C₅₃	1	2	yes	—
359	C₄²	C₂³×C₅₃	1	4	yes	—
360	C₂³	—	2	4	no	—
361	C₂³	—	4	2	no	—
362	C₂³	—	4	4	no	—
363	C₂³	—	4	4	no	—
364	C₂³	—	4	2	no	—
365	C₂³	—	4	4	no	—
366	C₂³	C₂⁴×C₅₃	1	2	yes	—
367	C₂³	C₂⁴×C₅₃	1	4	yes	—
368	C₂³	C₂⁴×C₅₃	1	4	yes	—
369	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
370	Q₈	—	2	4	no	—
371	Q₈	—	4	2	no	—
372	Q₈	—	4	1	no	—
373	Q₈	—	4	2	no	—
374	D₄	—	2	4	no	—
375	D₄	—	2	8	no	—
376	D₄	—	2	4	no	—
377	D₄	—	2	4	no	—
378	D₄	—	4	4	no	—
379	D₄	—	4	2	no	—
380	D₄	—	4	4	no	—
381	D₄	—	4	4	no	—
382	D₄	—	4	2	no	—
383	D₄	—	4	1	no	—
384	D₄	—	4	4	no	—
385	D₄	—	4	2	no	—
386	D₄	—	4	4	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	4	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	4	no	—
394	C₂×C₄	—	4	4	no	—
395	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
396	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
397	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
398	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
399	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
400	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
401	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
402	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
403	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
404	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
405	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
406	C₂²	—	2	1	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	2	4	no	—
410	C₂²	—	2	4	no	—
411	C₂²	—	2	4	no	—
412	C₂²	—	2	2	no	—
413	C₂²	—	2	4	no	—
414	C₂²	—	2	4	no	—
415	C₂²	—	2	2	no	—
416	C₂²	—	2	4	no	—
417	C₂²	—	8	4	no	—
418	C₂²	—	8	8	no	—
419	C₂²	H(5,𝔽₂)×C₅₃	1	2	yes	—
420	C₂²	C₂⁵×C₅₃	1	1	yes	characteristic
421	C₄	—	2	1	no	—
422	C₄	—	2	2	no	—
423	C₄	—	2	1	no	—
424	C₄	—	2	1	no	—
425	C₄	—	2	2	no	—
426	C₄	—	2	4	no	—
427	C₄	—	2	4	no	—
428	C₄	—	2	1	no	—
429	C₄	—	2	1	no	—
430	C₄	—	2	2	no	—
431	C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
432	C₂	—	2	2	no	—
433	C₂	—	4	2	no	—
434	C₂	—	4	4	no	—
435	C₂	—	4	4	no	—
436	C₂	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
437	C₂	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
438	C₂	C₅₃×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₅₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#562	C₂	semidirect
2. C₅₃×H(3,𝔽₄)	C₂	semidirect
3. C₅₃×#559	C₂	semidirect
4. C₅₃×#548	C₂	semidirect
5. C₅₃×#546	C₂	semidirect
6. C₅₃×#546	C₂	semidirect
7. D₄²×C₅₃	C₂	semidirect
8. C₅₃×#542	C₂	semidirect
9. C₅₃×#538	C₂	semidirect
10. C₅₃×#538	C₂	semidirect
11. C₅₃×#535	C₂	semidirect
12. C₅₃×#534	C₂	semidirect
13. C₅₃×#532	C₂	semidirect
14. C₅₃×#529	C₂	semidirect
15. C₂×#124×C₅₃	C₂	semidirect
16. C₂×#41×C₅₃	C₂²	semidirect
17. C₂×#41×C₅₃	C₂²	semidirect
18. C₂²×Q₈×C₅₃	C₂²	semidirect
20. C₂²×D₄×C₅₃	C₂²	semidirect
21. C₂²×D₄×C₅₃	C₂²	semidirect
22. C₄⋊Q₈×C₅₃	C₂²	semidirect
24. #127×C₅₃	C₂²	semidirect
25. #126×C₅₃	C₂²	semidirect
26. #126×C₅₃	C₂²	semidirect
27. #125×C₅₃	C₂²	semidirect
28. #124×C₅₃	C₂²	semidirect
29. #124×C₅₃	C₂²	semidirect
30. #124×C₅₃	C₂²	semidirect
31. #124×C₅₃	C₂²	semidirect
32. #124×C₅₃	C₂²	semidirect
33. #124×C₅₃	C₂²	semidirect
34. #124×C₅₃	C₂²	semidirect
35. #124×C₅₃	C₂²	semidirect
36. #124×C₅₃	C₂²	semidirect
37. #124×C₅₃	C₂²	semidirect
38. #123×C₅₃	C₂²	semidirect
39. #123×C₅₃	C₂²	semidirect
40. #123×C₅₃	C₂²	semidirect
41. #123×C₅₃	C₂²	semidirect
42. #123×C₅₃	C₂²	semidirect
43. C₂²⋊Q₈×C₅₃	C₂²	semidirect
44. C₂²⋊Q₈×C₅₃	C₂²	semidirect
45. C₂²⋊Q₈×C₅₃	C₂²	semidirect
46. C₂²⋊Q₈×C₅₃	C₂²	semidirect
49. #121×C₅₃	C₂²	semidirect
50. #121×C₅₃	C₂²	semidirect
51. #121×C₅₃	C₂²	semidirect
52. #121×C₅₃	C₂²	semidirect
53. #121×C₅₃	C₂²	semidirect
54. #121×C₅₃	C₂²	semidirect
55. #121×C₅₃	C₂²	semidirect
56. #120×C₅₃	C₂²	semidirect
57. #120×C₅₃	C₂²	semidirect
58. #120×C₅₃	C₂²	semidirect
59. #120×C₅₃	C₂²	semidirect
60. #120×C₅₃	C₂²	semidirect
61. C₄×Q₈×C₅₃	C₂²	semidirect
62. C₄×Q₈×C₅₃	C₂²	semidirect
63. C₄×D₄×C₅₃	C₂²	semidirect
64. C₄×D₄×C₅₃	C₂²	semidirect
65. C₄×D₄×C₅₃	C₂²	semidirect
66. C₄×D₄×C₅₃	C₂²	semidirect
67. C₄×D₄×C₅₃	C₂²	semidirect
68. C₄×D₄×C₅₃	C₂²	semidirect
69. C₄×D₄×C₅₃	C₂²	semidirect
70. #117×C₅₃	C₂²	semidirect
71. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
72. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
73. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
74. C₂×C₄²×C₅₃	C₂²	semidirect
212. #3558	C₅₃	direct (N × H)
220. #562	C₂×C₅₃	semidirect
221. H(3,𝔽₄)	C₂×C₅₃	semidirect
222. #559	C₂×C₅₃	semidirect
223. #548	C₂×C₅₃	semidirect
224. #546	C₂×C₅₃	semidirect
225. #546	C₂×C₅₃	semidirect
226. D₄²	C₂×C₅₃	semidirect
227. #542	C₂×C₅₃	semidirect
228. #538	C₂×C₅₃	semidirect
229. #538	C₂×C₅₃	semidirect
230. #535	C₂×C₅₃	semidirect
231. #534	C₂×C₅₃	semidirect
232. #532	C₂×C₅₃	semidirect
233. #529	C₂×C₅₃	semidirect
234. C₂×#124	C₂×C₅₃	semidirect
235. C₅₃	#3558	direct (N × H)
236. C₂×#41	C₂²×C₅₃	semidirect
237. C₂×#41	C₂²×C₅₃	semidirect
238. C₂²×Q₈	C₂²×C₅₃	semidirect
240. C₂²×D₄	C₂²×C₅₃	semidirect
241. C₂²×D₄	C₂²×C₅₃	semidirect
242. C₄⋊Q₈	C₂²×C₅₃	semidirect
244. #127	C₂²×C₅₃	semidirect
245. #126	C₂²×C₅₃	semidirect
246. #126	C₂²×C₅₃	semidirect
247. #125	C₂²×C₅₃	semidirect
248. #124	C₂²×C₅₃	semidirect
249. #124	C₂²×C₅₃	semidirect
250. #124	C₂²×C₅₃	semidirect
251. #124	C₂²×C₅₃	semidirect
252. #124	C₂²×C₅₃	semidirect
253. #124	C₂²×C₅₃	semidirect
254. #124	C₂²×C₅₃	semidirect
255. #124	C₂²×C₅₃	semidirect
256. #124	C₂²×C₅₃	semidirect
257. #124	C₂²×C₅₃	semidirect
258. #123	C₂²×C₅₃	semidirect
259. #123	C₂²×C₅₃	semidirect
260. #123	C₂²×C₅₃	semidirect
261. #123	C₂²×C₅₃	semidirect
262. #123	C₂²×C₅₃	semidirect
263. C₂²⋊Q₈	C₂²×C₅₃	semidirect
264. C₂²⋊Q₈	C₂²×C₅₃	semidirect
265. C₂²⋊Q₈	C₂²×C₅₃	semidirect
266. C₂²⋊Q₈	C₂²×C₅₃	semidirect
269. #121	C₂²×C₅₃	semidirect
270. #121	C₂²×C₅₃	semidirect
271. #121	C₂²×C₅₃	semidirect
272. #121	C₂²×C₅₃	semidirect
273. #121	C₂²×C₅₃	semidirect
274. #121	C₂²×C₅₃	semidirect
275. #121	C₂²×C₅₃	semidirect
276. #120	C₂²×C₅₃	semidirect
277. #120	C₂²×C₅₃	semidirect
278. #120	C₂²×C₅₃	semidirect
279. #120	C₂²×C₅₃	semidirect
280. #120	C₂²×C₅₃	semidirect
281. C₄×Q₈	C₂²×C₅₃	semidirect
282. C₄×Q₈	C₂²×C₅₃	semidirect
283. C₄×D₄	C₂²×C₅₃	semidirect
284. C₄×D₄	C₂²×C₅₃	semidirect
285. C₄×D₄	C₂²×C₅₃	semidirect
286. C₄×D₄	C₂²×C₅₃	semidirect
287. C₄×D₄	C₂²×C₅₃	semidirect
288. C₄×D₄	C₂²×C₅₃	semidirect
289. C₄×D₄	C₂²×C₅₃	semidirect
290. #117	C₂²×C₅₃	semidirect
291. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
292. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
293. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
294. C₂×C₄²	C₂²×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#562	2
C₅₃×H(3,𝔽₄)	2
C₅₃×#559	2
C₅₃×#548	2
C₅₃×#546	2
D₄²×C₅₃	2
C₅₃×#542	2
C₅₃×#538	2
C₅₃×#535	2
C₅₃×#534	2
C₅₃×#532	2
C₅₃×#529	2
C₂×#124×C₅₃	2
#3558	53

Subgroups — C53×#3558 #20272495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3558 #20272495