Subgroups — C₅₃×#3505 #20272442

Order 6784.

1152 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#553	C₂	1	2	yes	maximal, complemented
2	C₅₃×#553	C₂	1	4	yes	maximal, complemented
3	C₅₃×#552	C₂	1	2	yes	maximal, complemented
4	C₅₃×#551	C₂	1	4	yes	maximal, complemented
5	C₅₃×#548	C₂	1	2	yes	maximal, complemented
6	C₅₃×#546	C₂	1	2	yes	maximal, complemented
7	C₅₃×#539	C₂	1	2	yes	maximal, complemented
8	C₅₃×#538	C₂	1	2	yes	maximal, complemented
9	C₅₃×#534	C₂	1	2	yes	maximal, complemented
10	C₅₃×#529	C₂	1	2	yes	maximal, complemented
11	C₅₃×#525	C₂	1	2	yes	maximal, complemented
12	C₂×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#121×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#518	C₂	1	2	yes	maximal, complemented
15	C₂×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×C₅₃	C₂²	1	2	yes	complemented
17	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
18	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×D₄×C₅₃	C₂²	1	2	yes	—
20	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
21	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
23	#126×C₅₃	C₂²	1	4	yes	complemented
24	#126×C₅₃	C₂²	1	4	yes	complemented
25	#126×C₅₃	C₂²	1	4	yes	complemented
26	#125×C₅₃	C₂²	1	2	yes	complemented
27	#125×C₅₃	C₂²	1	4	yes	complemented
28	#124×C₅₃	C₂²	1	2	yes	complemented
29	#124×C₅₃	C₂²	1	4	yes	complemented
30	#123×C₅₃	C₂²	1	4	yes	—
31	#123×C₅₃	C₂²	1	4	yes	complemented
32	#123×C₅₃	C₂²	1	4	yes	complemented
33	#123×C₅₃	C₂²	1	4	yes	complemented
34	#123×C₅₃	C₂²	1	4	yes	complemented
35	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
36	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
37	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
38	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
39	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
40	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
41	#121×C₅₃	C₂²	1	2	yes	—
42	#121×C₅₃	C₂²	1	2	yes	—
43	#121×C₅₃	C₂²	1	2	yes	complemented
44	#121×C₅₃	C₂²	1	2	yes	complemented
45	#121×C₅₃	C₂²	1	2	yes	complemented
46	#121×C₅₃	C₂²	1	4	yes	complemented
47	#121×C₅₃	C₂²	1	4	yes	complemented
48	#121×C₅₃	C₂²	1	4	yes	complemented
49	#120×C₅₃	C₂²	1	4	yes	—
50	#120×C₅₃	C₂²	1	4	yes	complemented
51	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
52	C₄×D₄×C₅₃	C₂²	1	2	yes	—
53	C₄×D₄×C₅₃	C₂²	1	2	yes	—
54	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
55	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
56	C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
57	C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
58	C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
59	C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
60	C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
61	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
62	#117×C₅₃	C₂²	1	4	yes	complemented
63	#117×C₅₃	C₂²	1	2	yes	complemented
64	#117×C₅₃	C₂²	1	4	yes	complemented
65	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
66	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
67	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
68	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
69	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
71	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
72	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
73	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
74	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
75	C₂⁴×C₅₃	C₂³	1	2	yes	—
76	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
77	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
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	2	yes	—
82	C₂×D₄×C₅₃	—	2	4	no	—
83	C₂×D₄×C₅₃	—	2	8	no	—
84	C₂×D₄×C₅₃	—	4	2	no	—
85	C₂×D₄×C₅₃	C₂³	1	2	yes	—
86	C₂×D₄×C₅₃	C₂³	1	4	yes	—
87	C₂×D₄×C₅₃	C₂³	1	2	yes	—
88	C₂×D₄×C₅₃	C₂³	1	4	yes	—
89	C₂×D₄×C₅₃	C₂³	1	2	yes	—
90	C₂×D₄×C₅₃	C₂³	1	4	yes	—
91	C₂×D₄×C₅₃	C₂³	1	4	yes	—
92	C₂×D₄×C₅₃	C₂³	1	2	yes	—
93	C₂²×C₄×C₅₃	—	4	2	no	—
94	C₂²×C₄×C₅₃	—	4	2	no	—
95	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
98	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
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	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
108	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
109	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
111	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
112	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
113	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
114	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
115	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
116	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
117	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
118	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
119	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
120	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
121	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
122	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
123	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
124	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
125	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
126	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
127	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
128	C₂²⋊C₄×C₅₃	C₂³	1	2	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	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
135	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
136	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
137	C₄²×C₅₃	C₂³	1	2	yes	—
138	C₄²×C₅₃	C₂³	1	2	yes	—
139	C₄²×C₅₃	C₂³	1	4	yes	—
140	C₄²×C₅₃	C₂³	1	4	yes	—
141	C₂³×C₅₃	—	2	2	no	—
142	C₂³×C₅₃	—	2	4	no	—
143	C₂³×C₅₃	—	4	1	no	—
144	C₂³×C₅₃	—	4	1	no	—
145	C₂³×C₅₃	—	4	1	no	—
146	C₂³×C₅₃	—	4	1	no	—
147	C₂³×C₅₃	—	4	2	no	—
148	C₂³×C₅₃	—	4	2	no	—
149	C₂³×C₅₃	#41	1	4	yes	complemented
150	C₂³×C₅₃	C₂⁴	1	2	yes	—
151	C₂³×C₅₃	C₂⁴	1	4	yes	—
152	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
153	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
154	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
155	Q₈×C₅₃	—	2	4	no	—
156	Q₈×C₅₃	—	4	2	no	—
157	D₄×C₅₃	—	2	4	no	—
158	D₄×C₅₃	—	2	8	no	—
159	D₄×C₅₃	—	2	4	no	—
160	D₄×C₅₃	—	2	8	no	—
161	D₄×C₅₃	—	2	4	no	—
162	D₄×C₅₃	—	4	2	no	—
163	D₄×C₅₃	—	4	4	no	—
164	D₄×C₅₃	—	4	4	no	—
165	C₂×C₄×C₅₃	—	2	4	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	2	4	no	—
171	C₂×C₄×C₅₃	—	4	1	no	—
172	C₂×C₄×C₅₃	—	4	2	no	—
173	C₂×C₄×C₅₃	—	4	1	no	—
174	C₂×C₄×C₅₃	—	4	1	no	—
175	C₂×C₄×C₅₃	—	4	1	no	—
176	C₂×C₄×C₅₃	—	4	1	no	—
177	C₂×C₄×C₅₃	—	4	4	no	—
178	C₂×C₄×C₅₃	—	4	2	no	—
179	C₂×C₄×C₅₃	—	4	2	no	—
180	C₂×C₄×C₅₃	—	4	4	no	—
181	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
183	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
186	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
187	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
188	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
189	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
190	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
191	C₂²×C₅₃	—	2	1	no	—
192	C₂²×C₅₃	—	2	1	no	—
193	C₂²×C₅₃	—	2	1	no	—
194	C₂²×C₅₃	—	2	1	no	—
195	C₂²×C₅₃	—	2	1	no	—
196	C₂²×C₅₃	—	2	1	no	—
197	C₂²×C₅₃	—	2	2	no	—
198	C₂²×C₅₃	—	2	4	no	—
199	C₂²×C₅₃	—	2	2	no	—
200	C₂²×C₅₃	—	2	4	no	—
201	C₂²×C₅₃	—	2	4	no	—
202	C₂²×C₅₃	—	2	2	no	—
203	C₂²×C₅₃	—	4	4	no	—
204	C₂²×C₅₃	—	4	4	no	—
205	C₂²×C₅₃	—	8	4	no	—
206	C₂²×C₅₃	C₂×#41	1	2	yes	—
207	C₂²×C₅₃	C₂×#41	1	2	yes	—
208	C₂²×C₅₃	C₂×#41	1	2	yes	—
209	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
210	C₄×C₅₃	—	2	4	no	—
211	C₄×C₅₃	—	2	2	no	—
212	C₄×C₅₃	—	2	1	no	—
213	C₄×C₅₃	—	2	2	no	—
214	C₄×C₅₃	—	2	1	no	—
215	C₄×C₅₃	—	2	1	no	—
216	C₄×C₅₃	—	2	4	no	—
217	C₄×C₅₃	—	2	2	no	—
218	C₄×C₅₃	—	2	4	no	—
219	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
220	#3505	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂×C₅₃	—	2	2	no	—
222	C₂×C₅₃	—	2	2	no	—
223	C₂×C₅₃	—	2	2	no	—
224	C₂×C₅₃	—	4	2	no	—
225	C₂×C₅₃	—	4	4	no	—
226	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
227	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
228	C₂×C₅₃	#585	1	1	yes	characteristic
229	#553	C₂×C₅₃	1	2	yes	complemented
230	#553	C₂×C₅₃	1	4	yes	complemented
231	#552	C₂×C₅₃	1	2	yes	complemented
232	#551	C₂×C₅₃	1	4	yes	complemented
233	#548	C₂×C₅₃	1	2	yes	complemented
234	#546	C₂×C₅₃	1	2	yes	complemented
235	#539	C₂×C₅₃	1	2	yes	complemented
236	#538	C₂×C₅₃	1	2	yes	complemented
237	#534	C₂×C₅₃	1	2	yes	complemented
238	#529	C₂×C₅₃	1	2	yes	complemented
239	#525	C₂×C₅₃	1	2	yes	complemented
240	C₂×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
241	C₂×#121	C₂×C₅₃	1	1	yes	characteristic, complemented
242	#518	C₂×C₅₃	1	2	yes	complemented
243	C₂×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic, complemented
244	C₅₃	#3505	1	1	yes	characteristic, direct product factor, Sylow
245	C₂×#41	C₂²×C₅₃	1	2	yes	complemented
246	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
247	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
248	C₂²×D₄	C₂²×C₅₃	1	2	yes	—
249	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
250	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
251	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
252	#126	C₂²×C₅₃	1	4	yes	complemented
253	#126	C₂²×C₅₃	1	4	yes	complemented
254	#126	C₂²×C₅₃	1	4	yes	complemented
255	#125	C₂²×C₅₃	1	2	yes	complemented
256	#125	C₂²×C₅₃	1	4	yes	complemented
257	#124	C₂²×C₅₃	1	2	yes	complemented
258	#124	C₂²×C₅₃	1	4	yes	complemented
259	#123	C₂²×C₅₃	1	4	yes	—
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	#123	C₂²×C₅₃	1	4	yes	complemented
264	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
265	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
266	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
267	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
268	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
269	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
270	#121	C₂²×C₅₃	1	2	yes	—
271	#121	C₂²×C₅₃	1	2	yes	—
272	#121	C₂²×C₅₃	1	2	yes	complemented
273	#121	C₂²×C₅₃	1	2	yes	complemented
274	#121	C₂²×C₅₃	1	2	yes	complemented
275	#121	C₂²×C₅₃	1	4	yes	complemented
276	#121	C₂²×C₅₃	1	4	yes	complemented
277	#121	C₂²×C₅₃	1	4	yes	complemented
278	#120	C₂²×C₅₃	1	4	yes	—
279	#120	C₂²×C₅₃	1	4	yes	complemented
280	C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
281	C₄×D₄	C₂²×C₅₃	1	2	yes	—
282	C₄×D₄	C₂²×C₅₃	1	2	yes	—
283	C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
284	C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
285	C₄×D₄	C₂²×C₅₃	1	4	yes	complemented
286	C₄×D₄	C₂²×C₅₃	1	4	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	4	yes	complemented
290	C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
291	#117	C₂²×C₅₃	1	4	yes	complemented
292	#117	C₂²×C₅₃	1	2	yes	complemented
293	#117	C₂²×C₅₃	1	4	yes	complemented
294	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
295	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
296	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
297	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
298	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
299	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
300	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
301	C₂×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
302	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	complemented
303	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic, complemented
304	C₂⁴	C₂³×C₅₃	1	2	yes	—
305	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
306	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
307	#41	—	4	4	no	—
308	C₂×Q₈	—	4	2	no	—
309	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
310	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
311	C₂×D₄	—	2	4	no	—
312	C₂×D₄	—	2	8	no	—
313	C₂×D₄	—	4	2	no	—
314	C₂×D₄	C₂³×C₅₃	1	2	yes	—
315	C₂×D₄	C₂³×C₅₃	1	4	yes	—
316	C₂×D₄	C₂³×C₅₃	1	2	yes	—
317	C₂×D₄	C₂³×C₅₃	1	4	yes	—
318	C₂×D₄	C₂³×C₅₃	1	2	yes	—
319	C₂×D₄	C₂³×C₅₃	1	4	yes	—
320	C₂×D₄	C₂³×C₅₃	1	4	yes	—
321	C₂×D₄	C₂³×C₅₃	1	2	yes	—
322	C₂²×C₄	—	4	2	no	—
323	C₂²×C₄	—	4	2	no	—
324	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
325	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
326	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
327	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
328	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
329	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
330	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
331	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
332	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
333	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
334	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
335	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
336	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
337	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
338	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
339	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
340	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
341	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
342	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
343	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
344	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
345	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
346	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
347	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
348	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
349	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
350	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
351	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
352	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
353	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
354	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
355	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
356	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
357	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
358	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
359	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
360	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
361	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
362	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
363	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
364	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
365	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
366	C₄²	C₂³×C₅₃	1	2	yes	—
367	C₄²	C₂³×C₅₃	1	2	yes	—
368	C₄²	C₂³×C₅₃	1	4	yes	—
369	C₄²	C₂³×C₅₃	1	4	yes	—
370	C₂³	—	2	2	no	—
371	C₂³	—	2	4	no	—
372	C₂³	—	4	1	no	—
373	C₂³	—	4	1	no	—
374	C₂³	—	4	1	no	—
375	C₂³	—	4	1	no	—
376	C₂³	—	4	2	no	—
377	C₂³	—	4	2	no	—
378	C₂³	#41×C₅₃	1	4	yes	complemented
379	C₂³	C₂⁴×C₅₃	1	2	yes	—
380	C₂³	C₂⁴×C₅₃	1	4	yes	—
381	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
382	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
383	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
384	Q₈	—	2	4	no	—
385	Q₈	—	4	2	no	—
386	D₄	—	2	4	no	—
387	D₄	—	2	8	no	—
388	D₄	—	2	4	no	—
389	D₄	—	2	8	no	—
390	D₄	—	2	4	no	—
391	D₄	—	4	2	no	—
392	D₄	—	4	4	no	—
393	D₄	—	4	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	4	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	4	no	—
410	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
411	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
412	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
413	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
414	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
415	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
416	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
417	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
418	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
419	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
420	C₂²	—	2	1	no	—
421	C₂²	—	2	1	no	—
422	C₂²	—	2	1	no	—
423	C₂²	—	2	1	no	—
424	C₂²	—	2	1	no	—
425	C₂²	—	2	1	no	—
426	C₂²	—	2	2	no	—
427	C₂²	—	2	4	no	—
428	C₂²	—	2	2	no	—
429	C₂²	—	2	4	no	—
430	C₂²	—	2	4	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	4	4	no	—
433	C₂²	—	4	4	no	—
434	C₂²	—	8	4	no	—
435	C₂²	C₂×#41×C₅₃	1	2	yes	—
436	C₂²	C₂×#41×C₅₃	1	2	yes	—
437	C₂²	C₂×#41×C₅₃	1	2	yes	—
438	C₂²	C₂⁵×C₅₃	1	1	yes	characteristic
439	C₄	—	2	4	no	—
440	C₄	—	2	2	no	—
441	C₄	—	2	1	no	—
442	C₄	—	2	2	no	—
443	C₄	—	2	1	no	—
444	C₄	—	2	1	no	—
445	C₄	—	2	4	no	—
446	C₄	—	2	2	no	—
447	C₄	—	2	4	no	—
448	C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
449	C₂	—	2	2	no	—
450	C₂	—	2	2	no	—
451	C₂	—	2	2	no	—
452	C₂	—	4	2	no	—
453	C₂	—	4	4	no	—
454	C₂	C₂²×#41×C₅₃	1	1	yes	characteristic
455	C₂	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
456	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×#553	C₂	semidirect
2. C₅₃×#553	C₂	semidirect
3. C₅₃×#552	C₂	semidirect
4. C₅₃×#551	C₂	semidirect
5. C₅₃×#548	C₂	semidirect
6. C₅₃×#546	C₂	semidirect
7. C₅₃×#539	C₂	semidirect
8. C₅₃×#538	C₂	semidirect
9. C₅₃×#534	C₂	semidirect
10. C₅₃×#529	C₂	semidirect
11. C₅₃×#525	C₂	semidirect
12. C₂×C₂²⋊Q₈×C₅₃	C₂	semidirect
13. C₂×#121×C₅₃	C₂	semidirect
14. C₅₃×#518	C₂	semidirect
15. C₂×C₄×D₄×C₅₃	C₂	semidirect
16. C₂×#41×C₅₃	C₂²	semidirect
17. C₂²×Q₈×C₅₃	C₂²	semidirect
20. C₂²×D₄×C₅₃	C₂²	semidirect
22. C₂³×C₄×C₅₃	C₂²	semidirect
23. #126×C₅₃	C₂²	semidirect
24. #126×C₅₃	C₂²	semidirect
25. #126×C₅₃	C₂²	semidirect
26. #125×C₅₃	C₂²	semidirect
27. #125×C₅₃	C₂²	semidirect
28. #124×C₅₃	C₂²	semidirect
29. #124×C₅₃	C₂²	semidirect
31. #123×C₅₃	C₂²	semidirect
32. #123×C₅₃	C₂²	semidirect
33. #123×C₅₃	C₂²	semidirect
34. #123×C₅₃	C₂²	semidirect
35. C₂²⋊Q₈×C₅₃	C₂²	semidirect
36. C₂²⋊Q₈×C₅₃	C₂²	semidirect
37. C₂²⋊Q₈×C₅₃	C₂²	semidirect
38. C₂²⋊Q₈×C₅₃	C₂²	semidirect
39. C₂²⋊Q₈×C₅₃	C₂²	semidirect
40. C₂²⋊Q₈×C₅₃	C₂²	semidirect
43. #121×C₅₃	C₂²	semidirect
44. #121×C₅₃	C₂²	semidirect
45. #121×C₅₃	C₂²	semidirect
46. #121×C₅₃	C₂²	semidirect
47. #121×C₅₃	C₂²	semidirect
48. #121×C₅₃	C₂²	semidirect
50. #120×C₅₃	C₂²	semidirect
51. C₄×Q₈×C₅₃	C₂²	semidirect
54. C₄×D₄×C₅₃	C₂²	semidirect
55. C₄×D₄×C₅₃	C₂²	semidirect
56. C₄×D₄×C₅₃	C₂²	semidirect
57. C₄×D₄×C₅₃	C₂²	semidirect
58. C₄×D₄×C₅₃	C₂²	semidirect
59. C₄×D₄×C₅₃	C₂²	semidirect
60. C₄×D₄×C₅₃	C₂²	semidirect
61. C₄×D₄×C₅₃	C₂²	semidirect
62. #117×C₅₃	C₂²	semidirect
63. #117×C₅₃	C₂²	semidirect
64. #117×C₅₃	C₂²	semidirect
65. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
66. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
67. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
68. C₂×C₄⋊C₄×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
149. C₂³×C₅₃	#41	semidirect
220. #3505	C₅₃	direct (N × H)
229. #553	C₂×C₅₃	semidirect
230. #553	C₂×C₅₃	semidirect
231. #552	C₂×C₅₃	semidirect
232. #551	C₂×C₅₃	semidirect
233. #548	C₂×C₅₃	semidirect
234. #546	C₂×C₅₃	semidirect
235. #539	C₂×C₅₃	semidirect
236. #538	C₂×C₅₃	semidirect
237. #534	C₂×C₅₃	semidirect
238. #529	C₂×C₅₃	semidirect
239. #525	C₂×C₅₃	semidirect
240. C₂×C₂²⋊Q₈	C₂×C₅₃	semidirect
241. C₂×#121	C₂×C₅₃	semidirect
242. #518	C₂×C₅₃	semidirect
243. C₂×C₄×D₄	C₂×C₅₃	semidirect
244. C₅₃	#3505	direct (N × H)
245. C₂×#41	C₂²×C₅₃	semidirect
246. C₂²×Q₈	C₂²×C₅₃	semidirect
249. C₂²×D₄	C₂²×C₅₃	semidirect
251. C₂³×C₄	C₂²×C₅₃	semidirect
252. #126	C₂²×C₅₃	semidirect
253. #126	C₂²×C₅₃	semidirect
254. #126	C₂²×C₅₃	semidirect
255. #125	C₂²×C₅₃	semidirect
256. #125	C₂²×C₅₃	semidirect
257. #124	C₂²×C₅₃	semidirect
258. #124	C₂²×C₅₃	semidirect
260. #123	C₂²×C₅₃	semidirect
261. #123	C₂²×C₅₃	semidirect
262. #123	C₂²×C₅₃	semidirect
263. #123	C₂²×C₅₃	semidirect
264. C₂²⋊Q₈	C₂²×C₅₃	semidirect
265. C₂²⋊Q₈	C₂²×C₅₃	semidirect
266. C₂²⋊Q₈	C₂²×C₅₃	semidirect
267. C₂²⋊Q₈	C₂²×C₅₃	semidirect
268. C₂²⋊Q₈	C₂²×C₅₃	semidirect
269. C₂²⋊Q₈	C₂²×C₅₃	semidirect
272. #121	C₂²×C₅₃	semidirect
273. #121	C₂²×C₅₃	semidirect
274. #121	C₂²×C₅₃	semidirect
275. #121	C₂²×C₅₃	semidirect
276. #121	C₂²×C₅₃	semidirect
277. #121	C₂²×C₅₃	semidirect
279. #120	C₂²×C₅₃	semidirect
280. 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. C₄×D₄	C₂²×C₅₃	semidirect
291. #117	C₂²×C₅₃	semidirect
292. #117	C₂²×C₅₃	semidirect
293. #117	C₂²×C₅₃	semidirect
294. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
295. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
296. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
297. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
300. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
301. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
302. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
303. C₂×C₄²	C₂²×C₅₃	semidirect
378. C₂³	#41×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#553	2
C₅₃×#552	2
C₅₃×#551	2
C₅₃×#548	2
C₅₃×#546	2
C₅₃×#539	2
C₅₃×#538	2
C₅₃×#534	2
C₅₃×#529	2
C₅₃×#525	2
C₂×C₂²⋊Q₈×C₅₃	2
C₂×#121×C₅₃	2
C₅₃×#518	2
C₂×C₄×D₄×C₅₃	2
#3505	53

Subgroups — C53×#3505 #20272442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3505 #20272442