Subgroups — C₅₃×#3564 #20272501

Order 6784.

936 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#563	C₂	1	2	yes	maximal, complemented
2	C₅₃×#563	C₂	1	2	yes	maximal, complemented
3	C₅₃×#556	C₂	1	2	yes	maximal, complemented
4	C₅₃×#554	C₂	1	2	yes	maximal, complemented
5	C₅₃×#552	C₂	1	2	yes	maximal, complemented
6	C₅₃×#552	C₂	1	2	yes	maximal, complemented
7	C₅₃×#551	C₂	1	2	yes	maximal, complemented
8	C₅₃×#551	C₂	1	2	yes	maximal, complemented
9	C₅₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#543	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#542	C₂	1	2	yes	maximal, complemented
12	C₅₃×#541	C₂	1	2	yes	maximal, complemented
13	C₅₃×#541	C₂	1	2	yes	maximal, complemented
14	C₅₃×#539	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#539	C₂	1	2	yes	maximal, complemented
16	C₅₃×#538	C₂	1	1	yes	maximal, characteristic
17	C₅₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
18	C₅₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
19	C₂×#126×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
20	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
21	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
22	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
23	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
24	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
25	#126×C₅₃	C₂²	1	2	yes	—
26	#126×C₅₃	C₂²	1	2	yes	—
27	#126×C₅₃	C₂²	1	2	yes	—
28	#126×C₅₃	C₂²	1	2	yes	—
29	#126×C₅₃	C₂²	1	2	yes	—
30	#126×C₅₃	C₂²	1	2	yes	—
31	#126×C₅₃	C₂²	1	2	yes	—
32	#126×C₅₃	C₂²	1	2	yes	—
33	#126×C₅₃	C₂²	1	2	yes	—
34	#126×C₅₃	C₂²	1	2	yes	complemented
35	#126×C₅₃	C₂²	1	2	yes	complemented
36	#126×C₅₃	C₂²	1	2	yes	complemented
37	#126×C₅₃	C₂²	1	2	yes	complemented
38	#125×C₅₃	C₂²	1	1	yes	characteristic, complemented
39	#125×C₅₃	C₂²	1	2	yes	complemented
40	#125×C₅₃	C₂²	1	2	yes	complemented
41	#125×C₅₃	C₂²	1	2	yes	complemented
42	#125×C₅₃	C₂²	1	2	yes	complemented
43	#125×C₅₃	C₂²	1	2	yes	complemented
44	#125×C₅₃	C₂²	1	2	yes	complemented
45	#125×C₅₃	C₂²	1	2	yes	complemented
46	#125×C₅₃	C₂²	1	2	yes	complemented
47	#125×C₅₃	C₂²	1	2	yes	complemented
48	#124×C₅₃	C₂²	1	2	yes	—
49	#124×C₅₃	C₂²	1	2	yes	—
50	#124×C₅₃	C₂²	1	2	yes	—
51	#124×C₅₃	C₂²	1	1	yes	characteristic, complemented
52	#123×C₅₃	C₂²	1	2	yes	—
53	#123×C₅₃	C₂²	1	2	yes	—
54	#123×C₅₃	C₂²	1	2	yes	—
55	#123×C₅₃	C₂²	1	2	yes	—
56	#123×C₅₃	C₂²	1	2	yes	—
57	#123×C₅₃	C₂²	1	2	yes	—
58	#123×C₅₃	C₂²	1	2	yes	—
59	#123×C₅₃	C₂²	1	2	yes	complemented
60	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
61	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
62	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
63	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
64	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
65	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
66	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
67	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
68	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
69	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
70	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
71	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
72	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
73	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
74	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
75	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
76	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
77	#121×C₅₃	C₂²	1	1	yes	characteristic
78	#121×C₅₃	C₂²	1	2	yes	—
79	#120×C₅₃	C₂²	1	2	yes	—
80	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
81	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
82	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
83	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
84	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
85	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
86	C₄×D₄×C₅₃	C₂²	1	2	yes	—
87	C₄×D₄×C₅₃	C₂²	1	2	yes	—
88	C₄×D₄×C₅₃	C₂²	1	2	yes	—
89	C₄×D₄×C₅₃	C₂²	1	2	yes	—
90	C₄×D₄×C₅₃	C₂²	1	2	yes	—
91	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
92	#117×C₅₃	C₂²	1	2	yes	—
93	#117×C₅₃	C₂²	1	2	yes	—
94	#117×C₅₃	C₂²	1	2	yes	—
95	#117×C₅₃	C₂²	1	2	yes	—
96	#117×C₅₃	C₂²	1	2	yes	complemented
97	#117×C₅₃	C₂²	1	2	yes	complemented
98	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
102	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
104	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
105	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
106	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
107	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
108	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
109	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
110	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
111	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
112	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
113	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
114	C₂×D₄×C₅₃	C₂³	1	2	yes	—
115	C₂×D₄×C₅₃	C₂³	1	2	yes	—
116	C₂×D₄×C₅₃	C₂³	1	2	yes	—
117	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
119	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
122	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
123	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
124	C₂²×C₄×C₅₃	C₂³	1	2	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	2	yes	—
129	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
130	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
131	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
132	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
133	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
134	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
135	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
136	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
137	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
138	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
139	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
140	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
141	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
142	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
143	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
144	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
145	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
146	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
147	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
148	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
149	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
150	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
151	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
152	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
153	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
154	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
155	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
156	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
157	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
158	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
159	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
160	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
161	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
162	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
163	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
164	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
165	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
166	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
167	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
168	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
169	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
170	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
171	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
172	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
173	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
174	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
175	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
176	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
177	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
178	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
179	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
180	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
181	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
182	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
183	C₄²×C₅₃	C₂³	1	1	yes	characteristic
184	C₄²×C₅₃	C₂³	1	1	yes	characteristic
185	C₄²×C₅₃	C₂³	1	1	yes	characteristic
186	C₄²×C₅₃	C₂³	1	1	yes	characteristic
187	C₄²×C₅₃	C₂³	1	2	yes	—
188	C₄²×C₅₃	C₂³	1	2	yes	—
189	C₄²×C₅₃	C₂³	1	2	yes	—
190	C₄²×C₅₃	C₂³	1	2	yes	—
191	C₄²×C₅₃	C₂³	1	2	yes	—
192	C₄²×C₅₃	C₂³	1	2	yes	—
193	C₄²×C₅₃	C₂³	1	2	yes	—
194	C₂³×C₅₃	—	2	2	no	—
195	C₂³×C₅₃	—	4	1	no	—
196	C₂³×C₅₃	—	4	1	no	—
197	C₂³×C₅₃	C₂⁴	1	2	yes	—
198	C₂³×C₅₃	C₂⁴	1	2	yes	—
199	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
200	Q₈×C₅₃	—	4	1	no	—
201	Q₈×C₅₃	—	4	2	no	—
202	Q₈×C₅₃	—	4	2	no	—
203	Q₈×C₅₃	—	4	1	no	—
204	Q₈×C₅₃	—	4	2	no	—
205	Q₈×C₅₃	—	4	1	no	—
206	D₄×C₅₃	—	2	4	no	—
207	D₄×C₅₃	—	4	2	no	—
208	D₄×C₅₃	—	4	1	no	—
209	D₄×C₅₃	—	4	2	no	—
210	C₂×C₄×C₅₃	—	2	2	no	—
211	C₂×C₄×C₅₃	—	2	2	no	—
212	C₂×C₄×C₅₃	—	4	2	no	—
213	C₂×C₄×C₅₃	—	4	1	no	—
214	C₂×C₄×C₅₃	—	4	1	no	—
215	C₂×C₄×C₅₃	—	4	1	no	—
216	C₂×C₄×C₅₃	—	4	2	no	—
217	C₂×C₄×C₅₃	—	4	2	no	—
218	C₂×C₄×C₅₃	—	4	2	no	—
219	C₂×C₄×C₅₃	—	4	2	no	—
220	C₂×C₄×C₅₃	—	4	1	no	—
221	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
222	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
223	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
224	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
225	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
226	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
227	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
228	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
229	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
230	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
231	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
232	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
233	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
234	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
235	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
236	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
237	C₂²×C₅₃	—	2	1	no	—
238	C₂²×C₅₃	—	2	1	no	—
239	C₂²×C₅₃	—	2	2	no	—
240	C₂²×C₅₃	—	2	2	no	—
241	C₂²×C₅₃	—	2	2	no	—
242	C₂²×C₅₃	—	2	2	no	—
243	C₂²×C₅₃	—	2	2	no	—
244	C₂²×C₅₃	—	2	2	no	—
245	C₂²×C₅₃	—	8	2	no	—
246	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
247	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
248	C₄×C₅₃	—	2	1	no	—
249	C₄×C₅₃	—	2	1	no	—
250	C₄×C₅₃	—	2	2	no	—
251	C₄×C₅₃	—	2	2	no	—
252	C₄×C₅₃	—	2	1	no	—
253	C₄×C₅₃	—	2	2	no	—
254	C₄×C₅₃	—	2	2	no	—
255	C₄×C₅₃	—	2	2	no	—
256	C₄×C₅₃	—	2	2	no	—
257	C₄×C₅₃	—	2	2	no	—
258	C₄×C₅₃	—	2	1	no	—
259	C₄×C₅₃	—	2	1	no	—
260	C₄×C₅₃	—	2	2	no	—
261	C₄×C₅₃	—	2	2	no	—
262	C₄×C₅₃	—	2	1	no	—
263	C₄×C₅₃	—	2	2	no	—
264	#3564	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
265	C₂×C₅₃	—	2	2	no	—
266	C₂×C₅₃	—	4	2	no	—
267	C₂×C₅₃	—	4	2	no	—
268	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
269	C₂×C₅₃	#585	1	1	yes	characteristic
270	C₂×C₅₃	#585	1	1	yes	characteristic
271	#563	C₂×C₅₃	1	2	yes	complemented
272	#563	C₂×C₅₃	1	2	yes	complemented
273	#556	C₂×C₅₃	1	2	yes	complemented
274	#554	C₂×C₅₃	1	2	yes	complemented
275	#552	C₂×C₅₃	1	2	yes	complemented
276	#552	C₂×C₅₃	1	2	yes	complemented
277	#551	C₂×C₅₃	1	2	yes	complemented
278	#551	C₂×C₅₃	1	2	yes	complemented
279	#544	C₂×C₅₃	1	1	yes	characteristic, complemented
280	#543	C₂×C₅₃	1	1	yes	characteristic, complemented
281	#542	C₂×C₅₃	1	2	yes	complemented
282	#541	C₂×C₅₃	1	2	yes	complemented
283	#541	C₂×C₅₃	1	2	yes	complemented
284	#539	C₂×C₅₃	1	1	yes	characteristic, complemented
285	#539	C₂×C₅₃	1	2	yes	complemented
286	#538	C₂×C₅₃	1	1	yes	characteristic
287	#533	C₂×C₅₃	1	1	yes	characteristic, complemented
288	#529	C₂×C₅₃	1	1	yes	characteristic, complemented
289	C₂×#126	C₂×C₅₃	1	1	yes	characteristic, complemented
290	C₅₃	#3564	1	1	yes	characteristic, direct product factor, Sylow
291	C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
292	C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
293	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
294	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
295	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
296	#126	C₂²×C₅₃	1	2	yes	—
297	#126	C₂²×C₅₃	1	2	yes	—
298	#126	C₂²×C₅₃	1	2	yes	—
299	#126	C₂²×C₅₃	1	2	yes	—
300	#126	C₂²×C₅₃	1	2	yes	—
301	#126	C₂²×C₅₃	1	2	yes	—
302	#126	C₂²×C₅₃	1	2	yes	—
303	#126	C₂²×C₅₃	1	2	yes	—
304	#126	C₂²×C₅₃	1	2	yes	—
305	#126	C₂²×C₅₃	1	2	yes	complemented
306	#126	C₂²×C₅₃	1	2	yes	complemented
307	#126	C₂²×C₅₃	1	2	yes	complemented
308	#126	C₂²×C₅₃	1	2	yes	complemented
309	#125	C₂²×C₅₃	1	1	yes	characteristic, complemented
310	#125	C₂²×C₅₃	1	2	yes	complemented
311	#125	C₂²×C₅₃	1	2	yes	complemented
312	#125	C₂²×C₅₃	1	2	yes	complemented
313	#125	C₂²×C₅₃	1	2	yes	complemented
314	#125	C₂²×C₅₃	1	2	yes	complemented
315	#125	C₂²×C₅₃	1	2	yes	complemented
316	#125	C₂²×C₅₃	1	2	yes	complemented
317	#125	C₂²×C₅₃	1	2	yes	complemented
318	#125	C₂²×C₅₃	1	2	yes	complemented
319	#124	C₂²×C₅₃	1	2	yes	—
320	#124	C₂²×C₅₃	1	2	yes	—
321	#124	C₂²×C₅₃	1	2	yes	—
322	#124	C₂²×C₅₃	1	1	yes	characteristic, complemented
323	#123	C₂²×C₅₃	1	2	yes	—
324	#123	C₂²×C₅₃	1	2	yes	—
325	#123	C₂²×C₅₃	1	2	yes	—
326	#123	C₂²×C₅₃	1	2	yes	—
327	#123	C₂²×C₅₃	1	2	yes	—
328	#123	C₂²×C₅₃	1	2	yes	—
329	#123	C₂²×C₅₃	1	2	yes	—
330	#123	C₂²×C₅₃	1	2	yes	complemented
331	C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
332	C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
333	C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
334	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
335	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
336	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
337	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
338	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
339	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
340	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
341	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
342	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
343	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
344	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
345	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
346	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
347	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
348	#121	C₂²×C₅₃	1	1	yes	characteristic
349	#121	C₂²×C₅₃	1	2	yes	—
350	#120	C₂²×C₅₃	1	2	yes	—
351	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
352	C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
353	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
354	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
355	C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
356	C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
357	C₄×D₄	C₂²×C₅₃	1	2	yes	—
358	C₄×D₄	C₂²×C₅₃	1	2	yes	—
359	C₄×D₄	C₂²×C₅₃	1	2	yes	—
360	C₄×D₄	C₂²×C₅₃	1	2	yes	—
361	C₄×D₄	C₂²×C₅₃	1	2	yes	—
362	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
363	#117	C₂²×C₅₃	1	2	yes	—
364	#117	C₂²×C₅₃	1	2	yes	—
365	#117	C₂²×C₅₃	1	2	yes	—
366	#117	C₂²×C₅₃	1	2	yes	—
367	#117	C₂²×C₅₃	1	2	yes	complemented
368	#117	C₂²×C₅₃	1	2	yes	complemented
369	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
370	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
372	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
373	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
374	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
375	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
376	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
377	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
378	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
379	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
380	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
381	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
382	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
383	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
384	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
385	C₂×D₄	C₂³×C₅₃	1	2	yes	—
386	C₂×D₄	C₂³×C₅₃	1	2	yes	—
387	C₂×D₄	C₂³×C₅₃	1	2	yes	—
388	C₂×D₄	C₂³×C₅₃	1	1	yes	characteristic
389	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
390	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
391	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
392	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
393	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
394	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
395	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
396	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
397	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
398	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
399	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
400	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
401	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
402	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
403	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
404	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
405	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
406	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
407	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
408	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
409	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
410	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
411	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
412	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
413	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
414	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
415	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
416	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
417	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
418	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
419	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
420	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
421	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
422	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
423	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
424	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
425	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
426	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
427	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
428	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
429	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
430	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
431	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
432	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
433	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
434	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
435	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
436	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
437	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
438	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
439	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
440	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
441	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
442	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
443	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
444	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
445	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
446	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
447	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
448	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
449	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
450	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
451	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
452	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
453	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
454	C₄²	C₂³×C₅₃	1	1	yes	characteristic
455	C₄²	C₂³×C₅₃	1	1	yes	characteristic
456	C₄²	C₂³×C₅₃	1	1	yes	characteristic
457	C₄²	C₂³×C₅₃	1	1	yes	characteristic
458	C₄²	C₂³×C₅₃	1	2	yes	—
459	C₄²	C₂³×C₅₃	1	2	yes	—
460	C₄²	C₂³×C₅₃	1	2	yes	—
461	C₄²	C₂³×C₅₃	1	2	yes	—
462	C₄²	C₂³×C₅₃	1	2	yes	—
463	C₄²	C₂³×C₅₃	1	2	yes	—
464	C₄²	C₂³×C₅₃	1	2	yes	—
465	C₂³	—	2	2	no	—
466	C₂³	—	4	1	no	—
467	C₂³	—	4	1	no	—
468	C₂³	C₂⁴×C₅₃	1	2	yes	—
469	C₂³	C₂⁴×C₅₃	1	2	yes	—
470	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
471	Q₈	—	4	1	no	—
472	Q₈	—	4	2	no	—
473	Q₈	—	4	2	no	—
474	Q₈	—	4	1	no	—
475	Q₈	—	4	2	no	—
476	Q₈	—	4	1	no	—
477	D₄	—	2	4	no	—
478	D₄	—	4	2	no	—
479	D₄	—	4	1	no	—
480	D₄	—	4	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	1	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
493	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
494	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
495	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
496	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
497	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
498	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
499	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
500	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
501	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
502	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
503	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
504	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
505	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
506	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
507	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
508	C₂²	—	2	1	no	—
509	C₂²	—	2	1	no	—
510	C₂²	—	2	2	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	2	2	no	—
513	C₂²	—	2	2	no	—
514	C₂²	—	2	2	no	—
515	C₂²	—	2	2	no	—
516	C₂²	—	8	2	no	—
517	C₂²	2₋¹⁺⁴×C₅₃	1	2	yes	—
518	C₂²	C₂⁵×C₅₃	1	1	yes	characteristic
519	C₄	—	2	1	no	—
520	C₄	—	2	1	no	—
521	C₄	—	2	2	no	—
522	C₄	—	2	2	no	—
523	C₄	—	2	1	no	—
524	C₄	—	2	2	no	—
525	C₄	—	2	2	no	—
526	C₄	—	2	2	no	—
527	C₄	—	2	2	no	—
528	C₄	—	2	2	no	—
529	C₄	—	2	1	no	—
530	C₄	—	2	1	no	—
531	C₄	—	2	2	no	—
532	C₄	—	2	2	no	—
533	C₄	—	2	1	no	—
534	C₄	—	2	2	no	—
535	C₂	—	2	2	no	—
536	C₂	—	4	2	no	—
537	C₂	—	4	2	no	—
538	C₂	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
539	C₂	C₅₃×#585	1	1	yes	characteristic
540	C₂	C₅₃×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₅₃
prime radical	C₅₃×#538	C₂
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×#563	C₂	semidirect
2. C₅₃×#563	C₂	semidirect
3. C₅₃×#556	C₂	semidirect
4. C₅₃×#554	C₂	semidirect
5. C₅₃×#552	C₂	semidirect
6. C₅₃×#552	C₂	semidirect
7. C₅₃×#551	C₂	semidirect
8. C₅₃×#551	C₂	semidirect
9. C₅₃×#544	C₂	semidirect
10. C₅₃×#543	C₂	semidirect
11. C₅₃×#542	C₂	semidirect
12. C₅₃×#541	C₂	semidirect
13. C₅₃×#541	C₂	semidirect
14. C₅₃×#539	C₂	semidirect
15. C₅₃×#539	C₂	semidirect
17. C₅₃×#533	C₂	semidirect
18. C₅₃×#529	C₂	semidirect
19. C₂×#126×C₅₃	C₂	semidirect
20. C₄⋊Q₈×C₅₃	C₂²	semidirect
21. C₄⋊Q₈×C₅₃	C₂²	semidirect
22. C₄⋊Q₈×C₅₃	C₂²	semidirect
23. C₄⋊Q₈×C₅₃	C₂²	semidirect
24. C₄⋊Q₈×C₅₃	C₂²	semidirect
34. #126×C₅₃	C₂²	semidirect
35. #126×C₅₃	C₂²	semidirect
36. #126×C₅₃	C₂²	semidirect
37. #126×C₅₃	C₂²	semidirect
38. #125×C₅₃	C₂²	semidirect
39. #125×C₅₃	C₂²	semidirect
40. #125×C₅₃	C₂²	semidirect
41. #125×C₅₃	C₂²	semidirect
42. #125×C₅₃	C₂²	semidirect
43. #125×C₅₃	C₂²	semidirect
44. #125×C₅₃	C₂²	semidirect
45. #125×C₅₃	C₂²	semidirect
46. #125×C₅₃	C₂²	semidirect
47. #125×C₅₃	C₂²	semidirect
51. #124×C₅₃	C₂²	semidirect
59. #123×C₅₃	C₂²	semidirect
72. C₂²⋊Q₈×C₅₃	C₂²	semidirect
73. C₂²⋊Q₈×C₅₃	C₂²	semidirect
74. C₂²⋊Q₈×C₅₃	C₂²	semidirect
75. C₂²⋊Q₈×C₅₃	C₂²	semidirect
76. C₂²⋊Q₈×C₅₃	C₂²	semidirect
80. C₄×Q₈×C₅₃	C₂²	semidirect
81. C₄×Q₈×C₅₃	C₂²	semidirect
82. C₄×Q₈×C₅₃	C₂²	semidirect
83. C₄×Q₈×C₅₃	C₂²	semidirect
84. C₄×Q₈×C₅₃	C₂²	semidirect
85. C₄×Q₈×C₅₃	C₂²	semidirect
91. C₄×D₄×C₅₃	C₂²	semidirect
96. #117×C₅₃	C₂²	semidirect
97. #117×C₅₃	C₂²	semidirect
101. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
264. #3564	C₅₃	direct (N × H)
271. #563	C₂×C₅₃	semidirect
272. #563	C₂×C₅₃	semidirect
273. #556	C₂×C₅₃	semidirect
274. #554	C₂×C₅₃	semidirect
275. #552	C₂×C₅₃	semidirect
276. #552	C₂×C₅₃	semidirect
277. #551	C₂×C₅₃	semidirect
278. #551	C₂×C₅₃	semidirect
279. #544	C₂×C₅₃	semidirect
280. #543	C₂×C₅₃	semidirect
281. #542	C₂×C₅₃	semidirect
282. #541	C₂×C₅₃	semidirect
283. #541	C₂×C₅₃	semidirect
284. #539	C₂×C₅₃	semidirect
285. #539	C₂×C₅₃	semidirect
287. #533	C₂×C₅₃	semidirect
288. #529	C₂×C₅₃	semidirect
289. C₂×#126	C₂×C₅₃	semidirect
290. C₅₃	#3564	direct (N × H)
291. C₄⋊Q₈	C₂²×C₅₃	semidirect
292. C₄⋊Q₈	C₂²×C₅₃	semidirect
293. C₄⋊Q₈	C₂²×C₅₃	semidirect
294. C₄⋊Q₈	C₂²×C₅₃	semidirect
295. C₄⋊Q₈	C₂²×C₅₃	semidirect
305. #126	C₂²×C₅₃	semidirect
306. #126	C₂²×C₅₃	semidirect
307. #126	C₂²×C₅₃	semidirect
308. #126	C₂²×C₅₃	semidirect
309. #125	C₂²×C₅₃	semidirect
310. #125	C₂²×C₅₃	semidirect
311. #125	C₂²×C₅₃	semidirect
312. #125	C₂²×C₅₃	semidirect
313. #125	C₂²×C₅₃	semidirect
314. #125	C₂²×C₅₃	semidirect
315. #125	C₂²×C₅₃	semidirect
316. #125	C₂²×C₅₃	semidirect
317. #125	C₂²×C₅₃	semidirect
318. #125	C₂²×C₅₃	semidirect
322. #124	C₂²×C₅₃	semidirect
330. #123	C₂²×C₅₃	semidirect
343. C₂²⋊Q₈	C₂²×C₅₃	semidirect
344. C₂²⋊Q₈	C₂²×C₅₃	semidirect
345. C₂²⋊Q₈	C₂²×C₅₃	semidirect
346. C₂²⋊Q₈	C₂²×C₅₃	semidirect
347. C₂²⋊Q₈	C₂²×C₅₃	semidirect
351. C₄×Q₈	C₂²×C₅₃	semidirect
352. C₄×Q₈	C₂²×C₅₃	semidirect
353. C₄×Q₈	C₂²×C₅₃	semidirect
354. C₄×Q₈	C₂²×C₅₃	semidirect
355. C₄×Q₈	C₂²×C₅₃	semidirect
356. C₄×Q₈	C₂²×C₅₃	semidirect
362. C₄×D₄	C₂²×C₅₃	semidirect
367. #117	C₂²×C₅₃	semidirect
368. #117	C₂²×C₅₃	semidirect
372. C₂×C₄⋊C₄	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×#563	2
C₅₃×#556	2
C₅₃×#554	2
C₅₃×#552	2
C₅₃×#551	2
C₅₃×#544	2
C₅₃×#543	2
C₅₃×#542	2
C₅₃×#541	2
C₅₃×#539	2
C₅₃×#538	2
C₅₃×#533	2
C₅₃×#529	2
C₂×#126×C₅₃	2
#3564	53

Subgroups — C53×#3564 #20272501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3564 #20272501