Subgroups — C₂×C₅₃×#3181 #27268199

Order 13568.

1998 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#529	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3181	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#501	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#491	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#489	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#467	C₂	1	2	yes	maximal, complemented
9	C₂×C₅₃×#465	C₂	1	2	yes	maximal
10	C₂×C₅₃×#441	C₂	1	2	yes	maximal, complemented
11	C₂×C₅₃×#440	C₂	1	2	yes	maximal, complemented
12	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#533	—	2	4	no	—
17	C₅₃×#533	C₂²	1	8	yes	complemented
18	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#529	—	2	4	no	—
21	C₅₃×#529	C₂²	1	8	yes	—
22	C₂×#126×C₅₃	—	2	1	no	—
23	C₂×#125×C₅₃	—	2	1	no	—
24	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#124×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#123×C₅₃	—	2	1	no	—
27	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
28	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
29	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#121×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
32	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
33	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×D₄×C₅₃	—	2	1	no	—
35	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×#117×C₅₃	—	2	1	no	—
37	C₂×#117×C₅₃	—	2	1	no	—
38	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#501	C₂²	1	8	yes	complemented
40	C₅₃×#497	C₂²	1	8	yes	complemented
41	C₅₃×#491	C₂²	1	8	yes	complemented
42	C₅₃×#489	C₂²	1	8	yes	complemented
43	C₅₃×#467	C₂²	1	16	yes	complemented
44	C₅₃×#465	C₂²	1	16	yes	—
45	C₅₃×#441	C₂²	1	16	yes	complemented
46	C₅₃×#440	C₂²	1	16	yes	complemented
47	C₂×#107×C₅₃	C₂²	1	2	yes	—
48	C₂×#106×C₅₃	C₂²	1	2	yes	—
49	C₂×#103×C₅₃	C₂²	1	2	yes	—
50	C₂×#103×C₅₃	C₂²	1	2	yes	—
51	C₂×#103×C₅₃	C₂²	1	2	yes	—
52	C₂×#102×C₅₃	C₂²	1	2	yes	—
53	C₂×#97×C₅₃	C₂²	1	2	yes	—
54	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
55	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
56	C₂×#97×C₅₃	C₂²	1	8	yes	complemented
57	C₂²×Q₈×C₅₃	—	2	1	no	—
58	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
60	C₂²×D₄×C₅₃	—	2	1	no	—
61	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₅₃	—	2	1	no	—
63	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
64	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×Q₁₆×C₅₃	—	2	8	no	—
67	C₂×Q₁₆×C₅₃	C₂³	1	8	yes	—
68	C₂×SD₁₆×C₅₃	—	2	8	no	—
69	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	—
70	C₂²×C₈×C₅₃	—	2	8	no	—
71	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
72	C₂²×C₈×C₅₃	C₂³	1	8	yes	—
73	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
74	C₄⋊Q₈×C₅₃	—	2	2	no	—
75	C₄⋊Q₈×C₅₃	—	2	2	no	—
76	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
77	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
78	#126×C₅₃	—	2	8	no	—
79	#125×C₅₃	—	2	8	no	—
80	#125×C₅₃	—	2	2	no	—
81	#125×C₅₃	C₂³	1	4	yes	—
82	#124×C₅₃	—	2	2	no	—
83	#124×C₅₃	C₂³	1	4	yes	—
84	#123×C₅₃	—	2	8	no	—
85	C₂²⋊Q₈×C₅₃	—	2	8	no	—
86	C₂²⋊Q₈×C₅₃	—	2	2	no	—
87	C₂²⋊Q₈×C₅₃	—	2	4	no	—
88	C₂²⋊Q₈×C₅₃	C₂³	1	4	yes	—
89	C₂²⋊Q₈×C₅₃	C₂³	1	8	yes	—
90	#121×C₅₃	—	2	2	no	—
91	#121×C₅₃	C₂³	1	4	yes	—
92	C₄×Q₈×C₅₃	—	2	8	no	—
93	C₄×Q₈×C₅₃	—	2	2	no	—
94	C₄×Q₈×C₅₃	—	2	4	no	—
95	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
96	C₄×Q₈×C₅₃	C₂³	1	8	yes	—
97	C₄×D₄×C₅₃	—	2	2	no	—
98	C₄×D₄×C₅₃	—	2	8	no	—
99	C₄×D₄×C₅₃	C₂³	1	4	yes	—
100	#117×C₅₃	—	2	8	no	—
101	#117×C₅₃	—	2	8	no	—
102	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
103	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
104	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
105	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
106	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
107	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
108	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
109	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
110	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
111	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
115	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
117	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
118	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
119	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
120	C₂×C₄²×C₅₃	—	2	1	no	—
121	C₂×C₄²×C₅₃	—	2	2	no	—
122	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
123	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
124	C₂×C₄²×C₅₃	D₄	1	4	yes	—
125	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
128	#107×C₅₃	C₂³	1	8	yes	—
129	#106×C₅₃	C₂³	1	8	yes	—
130	#103×C₅₃	C₂³	1	8	yes	—
131	#103×C₅₃	C₂³	1	8	yes	—
132	#103×C₅₃	C₂³	1	8	yes	—
133	#102×C₅₃	C₂³	1	8	yes	—
134	#97×C₅₃	C₂³	1	4	yes	—
135	#97×C₅₃	C₂³	1	4	yes	—
136	#97×C₅₃	C₂³	1	8	yes	—
137	C₂⁴×C₅₃	—	2	1	no	—
138	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
139	C₂×Q₈×C₅₃	—	2	1	no	—
140	C₂×Q₈×C₅₃	—	2	2	no	—
141	C₂×Q₈×C₅₃	—	2	4	no	—
142	C₂×Q₈×C₅₃	—	2	4	no	—
143	C₂×Q₈×C₅₃	—	2	4	no	—
144	C₂×Q₈×C₅₃	—	2	4	no	—
145	C₂×Q₈×C₅₃	—	4	4	no	—
146	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
147	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
148	C₂×D₄×C₅₃	—	2	4	no	—
149	C₂×D₄×C₅₃	—	2	1	no	—
150	C₂×D₄×C₅₃	—	2	4	no	—
151	C₂×D₄×C₅₃	—	4	4	no	—
152	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₅₃	—	2	4	no	—
154	C₂²×C₄×C₅₃	—	2	4	no	—
155	C₂²×C₄×C₅₃	—	2	1	no	—
156	C₂²×C₄×C₅₃	—	2	2	no	—
157	C₂²×C₄×C₅₃	—	2	1	no	—
158	C₂²×C₄×C₅₃	—	2	2	no	—
159	C₂²×C₄×C₅₃	—	2	1	no	—
160	C₂²×C₄×C₅₃	—	2	4	no	—
161	C₂²×C₄×C₅₃	—	4	2	no	—
162	C₂²×C₄×C₅₃	—	4	4	no	—
163	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
165	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
171	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
173	Q₁₆×C₅₃	—	2	16	no	—
174	SD₁₆×C₅₃	—	2	16	no	—
175	C₂×C₈×C₅₃	—	2	4	no	—
176	C₂×C₈×C₅₃	—	2	16	no	—
177	C₂×C₈×C₅₃	#41	1	8	yes	—
178	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
179	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
180	C₄⋊C₄×C₅₃	—	2	4	no	—
181	C₄⋊C₄×C₅₃	—	2	4	no	—
182	C₄⋊C₄×C₅₃	—	2	4	no	—
183	C₄⋊C₄×C₅₃	—	2	4	no	—
184	C₄⋊C₄×C₅₃	—	2	4	no	—
185	C₄⋊C₄×C₅₃	—	2	4	no	—
186	C₄⋊C₄×C₅₃	—	2	8	no	—
187	C₄⋊C₄×C₅₃	—	2	8	no	—
188	C₄⋊C₄×C₅₃	—	2	4	no	—
189	C₄⋊C₄×C₅₃	—	2	1	no	—
190	C₄⋊C₄×C₅₃	—	2	1	no	—
191	C₄⋊C₄×C₅₃	—	2	2	no	—
192	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
193	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
194	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
195	C₂²⋊C₄×C₅₃	—	2	8	no	—
196	C₂²⋊C₄×C₅₃	—	2	4	no	—
197	C₂²⋊C₄×C₅₃	—	2	4	no	—
198	C₂²⋊C₄×C₅₃	—	2	4	no	—
199	C₂²⋊C₄×C₅₃	—	2	4	no	—
200	C₂²⋊C₄×C₅₃	—	2	4	no	—
201	C₄²×C₅₃	—	2	4	no	—
202	C₄²×C₅₃	—	2	8	no	—
203	C₄²×C₅₃	C₂×D₄	1	4	yes	—
204	C₄²×C₅₃	C₂×D₄	1	2	yes	—
205	C₄²×C₅₃	C₂×D₄	1	4	yes	—
206	C₄²×C₅₃	C₂×D₄	1	2	yes	—
207	C₄²×C₅₃	C₂⁴	1	2	yes	—
208	C₄²×C₅₃	C₂⁴	1	2	yes	—
209	C₂³×C₅₃	—	2	2	no	—
210	C₂³×C₅₃	—	2	2	no	—
211	C₂³×C₅₃	—	2	2	no	—
212	C₂³×C₅₃	—	4	2	no	—
213	C₂³×C₅₃	—	4	2	no	—
214	C₂³×C₅₃	—	4	2	no	—
215	C₂³×C₅₃	#137	1	4	yes	—
216	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
217	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
218	Q₈×C₅₃	—	2	4	no	—
219	Q₈×C₅₃	—	2	4	no	—
220	Q₈×C₅₃	—	2	4	no	—
221	Q₈×C₅₃	—	4	8	no	—
222	Q₈×C₅₃	—	4	2	no	—
223	Q₈×C₅₃	—	4	4	no	—
224	D₄×C₅₃	—	2	4	no	—
225	D₄×C₅₃	—	4	8	no	—
226	D₄×C₅₃	—	4	2	no	—
227	C₂×C₄×C₅₃	—	2	2	no	—
228	C₂×C₄×C₅₃	—	2	8	no	—
229	C₂×C₄×C₅₃	—	2	2	no	—
230	C₂×C₄×C₅₃	—	2	2	no	—
231	C₂×C₄×C₅₃	—	2	2	no	—
232	C₂×C₄×C₅₃	—	2	2	no	—
233	C₂×C₄×C₅₃	—	2	4	no	—
234	C₂×C₄×C₅₃	—	2	4	no	—
235	C₂×C₄×C₅₃	—	2	2	no	—
236	C₂×C₄×C₅₃	—	2	2	no	—
237	C₂×C₄×C₅₃	—	2	4	no	—
238	C₂×C₄×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	—	2	4	no	—
240	C₂×C₄×C₅₃	—	4	4	no	—
241	C₂×C₄×C₅₃	—	4	8	no	—
242	C₂×C₄×C₅₃	—	4	2	no	—
243	C₂×C₄×C₅₃	—	4	4	no	—
244	C₂×C₄×C₅₃	—	4	2	no	—
245	C₂×C₄×C₅₃	—	4	2	no	—
246	C₂×C₄×C₅₃	—	4	4	no	—
247	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
248	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
249	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
250	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
251	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
254	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
255	C₈×C₅₃	—	2	4	no	—
256	C₈×C₅₃	C₂×#41	1	8	yes	—
257	C₂×#3181	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
258	C₂²×C₅₃	—	2	8	no	—
259	C₂²×C₅₃	—	2	2	no	—
260	C₂²×C₅₃	—	2	2	no	—
261	C₂²×C₅₃	—	4	2	no	—
262	C₂²×C₅₃	—	4	2	no	—
263	C₂²×C₅₃	—	4	2	no	—
264	C₂²×C₅₃	—	8	4	no	—
265	C₂²×C₅₃	#532	1	2	yes	—
266	C₂²×C₅₃	C₂×#137	1	4	yes	—
267	C₂²×C₅₃	C₂×#137	1	2	yes	—
268	C₂²×C₅₃	#575	1	2	yes	—
269	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
270	C₄×C₅₃	—	2	2	no	—
271	C₄×C₅₃	—	2	2	no	—
272	C₄×C₅₃	—	2	2	no	—
273	C₄×C₅₃	—	2	2	no	—
274	C₄×C₅₃	—	2	2	no	—
275	C₄×C₅₃	—	4	2	no	—
276	C₄×C₅₃	—	4	4	no	—
277	C₄×C₅₃	—	4	2	no	—
278	C₄×C₅₃	—	4	2	no	—
279	C₄×C₅₃	—	4	4	no	—
280	C₄×C₅₃	#532	1	2	yes	—
281	C₄×C₅₃	C₂²×#41	1	2	yes	—
282	C₂×#533	C₂×C₅₃	1	1	yes	characteristic, complemented
283	C₂×#529	C₂×C₅₃	1	1	yes	characteristic
284	#3181	C₂×C₅₃	1	16	yes	direct product factor
285	C₂×#501	C₂×C₅₃	1	1	yes	characteristic, complemented
286	C₂×#497	C₂×C₅₃	1	1	yes	characteristic, complemented
287	C₂×#491	C₂×C₅₃	1	1	yes	characteristic, complemented
288	C₂×#489	C₂×C₅₃	1	1	yes	characteristic, complemented
289	C₂×#467	C₂×C₅₃	1	2	yes	complemented
290	C₂×#465	C₂×C₅₃	1	2	yes	—
291	C₂×#441	C₂×C₅₃	1	2	yes	complemented
292	C₂×#440	C₂×C₅₃	1	2	yes	complemented
293	C₂²×#97	C₂×C₅₃	1	1	yes	characteristic, complemented
294	C₂×C₅₃	—	2	4	no	—
295	C₂×C₅₃	—	8	2	no	—
296	C₂×C₅₃	#3181	1	4	yes	direct product factor
297	C₂×C₅₃	C₂×#532	1	1	yes	characteristic
298	C₂×C₅₃	C₂²×#137	1	1	yes	characteristic
299	C₂×C₅₃	C₂×#575	1	1	yes	characteristic
300	C₂²×Q₁₆	C₂²×C₅₃	1	2	yes	—
301	C₂²×SD₁₆	C₂²×C₅₃	1	2	yes	—
302	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
303	#533	—	2	4	no	—
304	#533	C₂²×C₅₃	1	8	yes	complemented
305	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
306	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
307	#529	—	2	4	no	—
308	#529	C₂²×C₅₃	1	8	yes	—
309	C₂×#126	—	2	1	no	—
310	C₂×#125	—	2	1	no	—
311	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic
312	C₂×#124	C₂²×C₅₃	1	1	yes	characteristic
313	C₂×#123	—	2	1	no	—
314	C₂×C₂²⋊Q₈	—	2	1	no	—
315	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
316	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
317	C₂×#121	C₂²×C₅₃	1	1	yes	characteristic
318	C₂×C₄×Q₈	—	2	1	no	—
319	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
320	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
321	C₂×C₄×D₄	—	2	1	no	—
322	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
323	C₂×#117	—	2	1	no	—
324	C₂×#117	—	2	1	no	—
325	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
326	#501	C₂²×C₅₃	1	8	yes	complemented
327	#497	C₂²×C₅₃	1	8	yes	complemented
328	#491	C₂²×C₅₃	1	8	yes	complemented
329	#489	C₂²×C₅₃	1	8	yes	complemented
330	#467	C₂²×C₅₃	1	16	yes	complemented
331	#465	C₂²×C₅₃	1	16	yes	—
332	#441	C₂²×C₅₃	1	16	yes	complemented
333	#440	C₂²×C₅₃	1	16	yes	complemented
334	C₂×#107	C₂²×C₅₃	1	2	yes	—
335	C₂×#106	C₂²×C₅₃	1	2	yes	—
336	C₂×#103	C₂²×C₅₃	1	2	yes	—
337	C₂×#103	C₂²×C₅₃	1	2	yes	—
338	C₂×#103	C₂²×C₅₃	1	2	yes	—
339	C₂×#102	C₂²×C₅₃	1	2	yes	—
340	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
341	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
342	C₂×#97	C₂²×C₅₃	1	2	yes	—
343	C₂×#97	C₂²×C₅₃	1	8	yes	complemented
344	C₅₃	C₂×#3181	1	1	yes	characteristic, direct product factor, Sylow
345	C₂²×Q₈	—	2	1	no	—
346	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
347	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
348	C₂²×D₄	—	2	1	no	—
349	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
350	C₂³×C₄	—	2	1	no	—
351	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
352	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
353	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
354	C₂×Q₁₆	—	2	8	no	—
355	C₂×Q₁₆	C₂³×C₅₃	1	8	yes	—
356	C₂×SD₁₆	—	2	8	no	—
357	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	—
358	C₂²×C₈	—	2	8	no	—
359	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
360	C₂²×C₈	C₂³×C₅₃	1	8	yes	—
361	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
362	C₄⋊Q₈	—	2	2	no	—
363	C₄⋊Q₈	—	2	2	no	—
364	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
365	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
366	#126	—	2	8	no	—
367	#125	—	2	8	no	—
368	#125	—	2	2	no	—
369	#125	C₂³×C₅₃	1	4	yes	—
370	#124	—	2	2	no	—
371	#124	C₂³×C₅₃	1	4	yes	—
372	#123	—	2	8	no	—
373	C₂²⋊Q₈	—	2	8	no	—
374	C₂²⋊Q₈	—	2	2	no	—
375	C₂²⋊Q₈	—	2	4	no	—
376	C₂²⋊Q₈	C₂³×C₅₃	1	4	yes	—
377	C₂²⋊Q₈	C₂³×C₅₃	1	8	yes	—
378	#121	—	2	2	no	—
379	#121	C₂³×C₅₃	1	4	yes	—
380	C₄×Q₈	—	2	8	no	—
381	C₄×Q₈	—	2	2	no	—
382	C₄×Q₈	—	2	4	no	—
383	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
384	C₄×Q₈	C₂³×C₅₃	1	8	yes	—
385	C₄×D₄	—	2	8	no	—
386	C₄×D₄	—	2	2	no	—
387	C₄×D₄	C₂³×C₅₃	1	4	yes	—
388	#117	—	2	8	no	—
389	#117	—	2	8	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	C₂×C₄⋊C₄	—	2	1	no	—
392	C₂×C₄⋊C₄	—	2	1	no	—
393	C₂×C₄⋊C₄	—	2	1	no	—
394	C₂×C₄⋊C₄	—	2	1	no	—
395	C₂×C₄⋊C₄	—	2	1	no	—
396	C₂×C₄⋊C₄	—	2	2	no	—
397	C₂×C₄⋊C₄	—	2	2	no	—
398	C₂×C₄⋊C₄	—	2	1	no	—
399	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
400	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
401	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
402	C₂×C₂²⋊C₄	—	2	2	no	—
403	C₂×C₂²⋊C₄	—	2	1	no	—
404	C₂×C₂²⋊C₄	—	2	1	no	—
405	C₂×C₂²⋊C₄	—	2	1	no	—
406	C₂×C₂²⋊C₄	—	2	1	no	—
407	C₂×C₂²⋊C₄	—	2	1	no	—
408	C₂×C₄²	—	2	1	no	—
409	C₂×C₄²	—	2	2	no	—
410	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
411	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
412	C₂×C₄²	D₄×C₅₃	1	4	yes	—
413	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
414	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
415	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
416	#107	C₂³×C₅₃	1	8	yes	—
417	#106	C₂³×C₅₃	1	8	yes	—
418	#103	C₂³×C₅₃	1	8	yes	—
419	#103	C₂³×C₅₃	1	8	yes	—
420	#103	C₂³×C₅₃	1	8	yes	—
421	#102	C₂³×C₅₃	1	8	yes	—
422	#97	C₂³×C₅₃	1	8	yes	—
423	#97	C₂³×C₅₃	1	4	yes	—
424	#97	C₂³×C₅₃	1	4	yes	—
425	C₂⁴	—	2	1	no	—
426	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
427	C₂×Q₈	—	2	1	no	—
428	C₂×Q₈	—	2	2	no	—
429	C₂×Q₈	—	2	4	no	—
430	C₂×Q₈	—	2	4	no	—
431	C₂×Q₈	—	2	4	no	—
432	C₂×Q₈	—	2	4	no	—
433	C₂×Q₈	—	4	4	no	—
434	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
435	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
436	C₂×D₄	—	2	4	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	—	2	4	no	—
439	C₂×D₄	—	4	4	no	—
440	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
441	C₂²×C₄	—	2	4	no	—
442	C₂²×C₄	—	2	4	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	2	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	4	no	—
449	C₂²×C₄	—	4	2	no	—
450	C₂²×C₄	—	4	4	no	—
451	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
455	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
456	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
457	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
458	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
459	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
460	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
461	Q₁₆	—	2	16	no	—
462	SD₁₆	—	2	16	no	—
463	C₂×C₈	—	2	4	no	—
464	C₂×C₈	—	2	16	no	—
465	C₂×C₈	#41×C₅₃	1	8	yes	—
466	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
467	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
468	C₄⋊C₄	—	2	4	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	4	no	—
471	C₄⋊C₄	—	2	4	no	—
472	C₄⋊C₄	—	2	4	no	—
473	C₄⋊C₄	—	2	4	no	—
474	C₄⋊C₄	—	2	8	no	—
475	C₄⋊C₄	—	2	8	no	—
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	2	no	—
480	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
481	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
482	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
483	C₂²⋊C₄	—	2	8	no	—
484	C₂²⋊C₄	—	2	4	no	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	2	4	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	—	2	4	no	—
489	C₄²	—	2	4	no	—
490	C₄²	—	2	8	no	—
491	C₄²	C₂×D₄×C₅₃	1	4	yes	—
492	C₄²	C₂×D₄×C₅₃	1	2	yes	—
493	C₄²	C₂×D₄×C₅₃	1	4	yes	—
494	C₄²	C₂×D₄×C₅₃	1	2	yes	—
495	C₄²	C₂⁴×C₅₃	1	2	yes	—
496	C₄²	C₂⁴×C₅₃	1	2	yes	—
497	C₂³	—	2	2	no	—
498	C₂³	—	2	2	no	—
499	C₂³	—	2	2	no	—
500	C₂³	—	4	2	no	—
501	C₂³	—	4	2	no	—
502	C₂³	—	4	2	no	—
503	C₂³	#137×C₅₃	1	4	yes	—
504	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
505	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
506	Q₈	—	2	4	no	—
507	Q₈	—	2	4	no	—
508	Q₈	—	2	4	no	—
509	Q₈	—	4	8	no	—
510	Q₈	—	4	2	no	—
511	Q₈	—	4	4	no	—
512	D₄	—	2	4	no	—
513	D₄	—	4	8	no	—
514	D₄	—	4	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	8	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	4	4	no	—
529	C₂×C₄	—	4	8	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	—	4	4	no	—
532	C₂×C₄	—	4	2	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	4	4	no	—
535	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
536	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
537	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
538	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
539	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
540	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
541	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
542	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
543	C₈	—	2	4	no	—
544	C₈	C₂×#41×C₅₃	1	8	yes	—
545	C₂²	—	2	2	no	—
546	C₂²	—	2	8	no	—
547	C₂²	—	2	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	4	2	no	—
550	C₂²	—	4	2	no	—
551	C₂²	—	8	4	no	—
552	C₂²	C₅₃×#532	1	2	yes	—
553	C₂²	C₂×#137×C₅₃	1	4	yes	—
554	C₂²	C₂×#137×C₅₃	1	2	yes	—
555	C₂²	C₅₃×#575	1	2	yes	—
556	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
557	C₄	—	2	2	no	—
558	C₄	—	2	2	no	—
559	C₄	—	2	2	no	—
560	C₄	—	2	2	no	—
561	C₄	—	2	2	no	—
562	C₄	—	4	2	no	—
563	C₄	—	4	2	no	—
564	C₄	—	4	4	no	—
565	C₄	—	4	2	no	—
566	C₄	—	4	4	no	—
567	C₄	C₅₃×#532	1	2	yes	—
568	C₄	C₂²×#41×C₅₃	1	2	yes	—
569	C₂	—	2	4	no	—
570	C₂	—	8	2	no	—
571	C₂	C₅₃×#3181	1	4	yes	direct product factor
572	C₂	C₂×C₅₃×#532	1	1	yes	characteristic
573	C₂	C₂²×#137×C₅₃	1	1	yes	characteristic
574	C₂	C₂×C₅₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₂×#121×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#533	C₂	semidirect
3. C₅₃×#3181	C₂	direct (N × H)
4. C₂×C₅₃×#501	C₂	semidirect
5. C₂×C₅₃×#497	C₂	semidirect
6. C₂×C₅₃×#491	C₂	semidirect
7. C₂×C₅₃×#489	C₂	semidirect
8. C₂×C₅₃×#467	C₂	semidirect
10. C₂×C₅₃×#441	C₂	semidirect
11. C₂×C₅₃×#440	C₂	semidirect
12. C₂²×#97×C₅₃	C₂	semidirect
17. C₅₃×#533	C₂²	semidirect
39. C₅₃×#501	C₂²	semidirect
40. C₅₃×#497	C₂²	semidirect
41. C₅₃×#491	C₂²	semidirect
42. C₅₃×#489	C₂²	semidirect
43. C₅₃×#467	C₂²	semidirect
45. C₅₃×#441	C₂²	semidirect
46. C₅₃×#440	C₂²	semidirect
56. C₂×#97×C₅₃	C₂²	semidirect
257. C₂×#3181	C₅₃	direct (N × H)
282. C₂×#533	C₂×C₅₃	semidirect
284. #3181	C₂×C₅₃	direct (N × H)
285. C₂×#501	C₂×C₅₃	semidirect
286. C₂×#497	C₂×C₅₃	semidirect
287. C₂×#491	C₂×C₅₃	semidirect
288. C₂×#489	C₂×C₅₃	semidirect
289. C₂×#467	C₂×C₅₃	semidirect
291. C₂×#441	C₂×C₅₃	semidirect
292. C₂×#440	C₂×C₅₃	semidirect
293. C₂²×#97	C₂×C₅₃	semidirect
296. C₂×C₅₃	#3181	direct (N × H)
304. #533	C₂²×C₅₃	semidirect
326. #501	C₂²×C₅₃	semidirect
327. #497	C₂²×C₅₃	semidirect
328. #491	C₂²×C₅₃	semidirect
329. #489	C₂²×C₅₃	semidirect
330. #467	C₂²×C₅₃	semidirect
332. #441	C₂²×C₅₃	semidirect
333. #440	C₂²×C₅₃	semidirect
343. C₂×#97	C₂²×C₅₃	semidirect
344. C₅₃	C₂×#3181	direct (N × H)
571. C₂	C₅₃×#3181	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#533	2
C₂×C₅₃×#529	2
C₅₃×#3181	2
C₂×C₅₃×#501	2
C₂×C₅₃×#497	2
C₂×C₅₃×#491	2
C₂×C₅₃×#489	2
C₂×C₅₃×#467	2
C₂×C₅₃×#465	2
C₂×C₅₃×#441	2
C₂×C₅₃×#440	2
C₂²×#97×C₅₃	2
C₂×#3181	53

Subgroups — C2×C53×#3181 #27268199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3181 #27268199