Subgroups — C₂×C₅₃×#3219 #27268301

Order 13568.

2198 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3219	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×#496	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#491	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#490	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#464	C₂	1	2	yes	maximal, complemented
9	C₂×C₅₃×#463	C₂	1	2	yes	maximal, complemented
10	C₂×C₅₃×#452	C₂	1	2	yes	maximal, complemented
11	C₂×C₅₃×#449	C₂	1	2	yes	maximal, complemented
12	C₂×C₅₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₅₃	C₂²	1	2	yes	—
14	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂²×SD₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#532	—	2	4	no	—
19	C₅₃×#532	C₂²	1	8	yes	complemented
20	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#529	—	2	4	no	—
23	C₅₃×#529	C₂²	1	8	yes	complemented
24	C₂×#126×C₅₃	—	2	1	no	—
25	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#124×C₅₃	C₂²	1	2	yes	—
27	C₂×#124×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×#123×C₅₃	—	2	1	no	—
29	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#121×C₅₃	—	2	1	no	—
31	C₂×#121×C₅₃	—	2	1	no	—
32	C₂×#121×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×D₄×C₅₃	—	2	1	no	—
35	C₂×C₄×D₄×C₅₃	—	2	1	no	—
36	C₂×C₄×D₄×C₅₃	—	2	1	no	—
37	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×#117×C₅₃	—	2	1	no	—
39	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#497	C₂²	1	8	yes	complemented
41	C₅₃×#496	C₂²	1	8	yes	complemented
42	C₅₃×#491	C₂²	1	8	yes	complemented
43	C₅₃×#490	C₂²	1	8	yes	complemented
44	C₅₃×#464	C₂²	1	16	yes	complemented
45	C₅₃×#463	C₂²	1	16	yes	complemented
46	C₅₃×#452	C₂²	1	16	yes	complemented
47	C₅₃×#449	C₂²	1	16	yes	complemented
48	C₅₃×#432	C₂²	1	8	yes	complemented
49	C₂×C₄⋊C₈×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×#102×C₅₃	C₂²	1	2	yes	—
54	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
55	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
56	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
57	C₂×#41×C₅₃	—	2	4	no	—
58	C₂×#41×C₅₃	—	4	4	no	—
59	C₂×#41×C₅₃	—	4	4	no	—
60	C₂×#41×C₅₃	C₂³	1	8	yes	—
61	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
63	C₂²×D₄×C₅₃	—	2	1	no	—
64	C₂²×D₄×C₅₃	—	2	2	no	—
65	C₂²×D₄×C₅₃	—	2	1	no	—
66	C₂²×D₄×C₅₃	—	2	1	no	—
67	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
69	C₂³×C₄×C₅₃	—	2	1	no	—
70	C₂³×C₄×C₅₃	—	2	2	no	—
71	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
72	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
73	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×Q₁₆×C₅₃	—	4	4	no	—
75	C₂×Q₁₆×C₅₃	C₂³	1	8	yes	—
76	C₂×SD₁₆×C₅₃	—	4	4	no	—
77	C₂×SD₁₆×C₅₃	—	4	4	no	—
78	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	—
79	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	—
80	C₂×D₈×C₅₃	—	4	4	no	—
81	C₂×D₈×C₅₃	C₂³	1	8	yes	—
82	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
83	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
84	C₄⋊Q₈×C₅₃	—	2	2	no	—
85	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
86	#127×C₅₃	—	2	2	no	—
87	#127×C₅₃	C₂³	1	4	yes	—
88	#126×C₅₃	—	2	8	no	—
89	#125×C₅₃	—	2	2	no	—
90	#125×C₅₃	C₂³	1	4	yes	—
91	#124×C₅₃	—	2	2	no	—
92	#124×C₅₃	—	2	4	no	—
93	#124×C₅₃	C₂³	1	4	yes	—
94	#124×C₅₃	C₂³	1	8	yes	—
95	#123×C₅₃	—	2	8	no	—
96	C₂²⋊Q₈×C₅₃	—	2	2	no	—
97	C₂²⋊Q₈×C₅₃	C₂³	1	4	yes	—
98	#121×C₅₃	—	2	8	no	—
99	#121×C₅₃	—	2	8	no	—
100	#121×C₅₃	—	2	2	no	—
101	#121×C₅₃	C₂³	1	4	yes	—
102	C₄×Q₈×C₅₃	—	2	2	no	—
103	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
104	C₄×D₄×C₅₃	—	2	8	no	—
105	C₄×D₄×C₅₃	—	2	8	no	—
106	C₄×D₄×C₅₃	—	2	2	no	—
107	C₄×D₄×C₅₃	—	2	8	no	—
108	C₄×D₄×C₅₃	C₂³	1	4	yes	—
109	#117×C₅₃	—	2	8	no	—
110	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
111	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
112	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
113	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
114	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
115	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
116	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
118	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
119	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
120	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
121	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
122	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
123	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
124	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
125	C₂×C₄²×C₅₃	—	2	1	no	—
126	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
127	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
128	C₂×C₄²×C₅₃	D₄	1	4	yes	—
129	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
131	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
132	C₄⋊C₈×C₅₃	C₂³	1	8	yes	—
133	#103×C₅₃	C₂³	1	8	yes	—
134	#103×C₅₃	C₂³	1	8	yes	—
135	#102×C₅₃	C₂³	1	8	yes	—
136	#102×C₅₃	C₂³	1	8	yes	—
137	C₂²⋊C₈×C₅₃	C₂³	1	8	yes	—
138	#97×C₅₃	C₂³	1	4	yes	—
139	#97×C₅₃	C₂³	1	4	yes	—
140	C₂⁴×C₅₃	—	2	1	no	—
141	C₂⁴×C₅₃	—	2	2	no	—
142	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
143	#41×C₅₃	—	4	8	no	—
144	#41×C₅₃	—	4	8	no	—
145	#41×C₅₃	—	4	16	no	—
146	C₂×Q₈×C₅₃	—	2	1	no	—
147	C₂×Q₈×C₅₃	—	2	2	no	—
148	C₂×Q₈×C₅₃	—	2	4	no	—
149	C₂×Q₈×C₅₃	—	2	4	no	—
150	C₂×Q₈×C₅₃	—	2	4	no	—
151	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
152	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
153	C₂×D₄×C₅₃	—	2	4	no	—
154	C₂×D₄×C₅₃	—	2	8	no	—
155	C₂×D₄×C₅₃	—	2	1	no	—
156	C₂×D₄×C₅₃	—	2	4	no	—
157	C₂×D₄×C₅₃	—	2	4	no	—
158	C₂×D₄×C₅₃	—	2	2	no	—
159	C₂×D₄×C₅₃	—	2	4	no	—
160	C₂×D₄×C₅₃	—	2	4	no	—
161	C₂×D₄×C₅₃	—	2	4	no	—
162	C₂×D₄×C₅₃	—	4	4	no	—
163	C₂×D₄×C₅₃	—	4	4	no	—
164	C₂×D₄×C₅₃	—	4	4	no	—
165	C₂×D₄×C₅₃	—	4	8	no	—
166	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
167	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₅₃	—	2	4	no	—
169	C₂²×C₄×C₅₃	—	2	1	no	—
170	C₂²×C₄×C₅₃	—	2	1	no	—
171	C₂²×C₄×C₅₃	—	2	1	no	—
172	C₂²×C₄×C₅₃	—	2	2	no	—
173	C₂²×C₄×C₅₃	—	2	4	no	—
174	C₂²×C₄×C₅₃	—	2	8	no	—
175	C₂²×C₄×C₅₃	—	4	2	no	—
176	C₂²×C₄×C₅₃	—	4	2	no	—
177	C₂²×C₄×C₅₃	—	4	4	no	—
178	C₂²×C₄×C₅₃	—	4	8	no	—
179	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
180	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
183	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
186	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
187	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
189	Q₁₆×C₅₃	—	4	8	no	—
190	SD₁₆×C₅₃	—	4	8	no	—
191	SD₁₆×C₅₃	—	4	8	no	—
192	D₈×C₅₃	—	4	8	no	—
193	C₂×C₈×C₅₃	—	2	4	no	—
194	C₂×C₈×C₅₃	—	2	4	no	—
195	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
196	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
197	C₄⋊C₄×C₅₃	—	2	4	no	—
198	C₄⋊C₄×C₅₃	—	2	4	no	—
199	C₄⋊C₄×C₅₃	—	2	4	no	—
200	C₄⋊C₄×C₅₃	—	2	4	no	—
201	C₄⋊C₄×C₅₃	—	2	4	no	—
202	C₄⋊C₄×C₅₃	—	2	4	no	—
203	C₄⋊C₄×C₅₃	—	2	1	no	—
204	C₄⋊C₄×C₅₃	—	2	1	no	—
205	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
206	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
207	C₂²⋊C₄×C₅₃	—	2	4	no	—
208	C₂²⋊C₄×C₅₃	—	2	4	no	—
209	C₂²⋊C₄×C₅₃	—	2	4	no	—
210	C₂²⋊C₄×C₅₃	—	2	8	no	—
211	C₂²⋊C₄×C₅₃	—	2	8	no	—
212	C₂²⋊C₄×C₅₃	—	2	4	no	—
213	C₂²⋊C₄×C₅₃	—	2	4	no	—
214	C₄²×C₅₃	—	2	4	no	—
215	C₄²×C₅₃	C₂×D₄	1	4	yes	—
216	C₄²×C₅₃	C₂×D₄	1	4	yes	—
217	C₄²×C₅₃	C₂×D₄	1	2	yes	—
218	C₄²×C₅₃	C₂×D₄	1	2	yes	—
219	C₄²×C₅₃	C₂⁴	1	2	yes	—
220	C₄²×C₅₃	C₂⁴	1	2	yes	—
221	C₂³×C₅₃	—	2	2	no	—
222	C₂³×C₅₃	—	2	2	no	—
223	C₂³×C₅₃	—	2	2	no	—
224	C₂³×C₅₃	—	2	4	no	—
225	C₂³×C₅₃	—	2	2	no	—
226	C₂³×C₅₃	—	4	4	no	—
227	C₂³×C₅₃	—	4	2	no	—
228	C₂³×C₅₃	—	4	2	no	—
229	C₂³×C₅₃	—	4	4	no	—
230	C₂³×C₅₃	—	4	4	no	—
231	C₂³×C₅₃	—	4	2	no	—
232	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
233	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
234	Q₈×C₅₃	—	2	4	no	—
235	Q₈×C₅₃	—	2	4	no	—
236	Q₈×C₅₃	—	2	4	no	—
237	Q₈×C₅₃	—	4	2	no	—
238	Q₈×C₅₃	—	4	4	no	—
239	D₄×C₅₃	—	2	4	no	—
240	D₄×C₅₃	—	2	4	no	—
241	D₄×C₅₃	—	2	4	no	—
242	D₄×C₅₃	—	4	8	no	—
243	D₄×C₅₃	—	4	8	no	—
244	D₄×C₅₃	—	4	8	no	—
245	D₄×C₅₃	—	4	16	no	—
246	D₄×C₅₃	—	4	2	no	—
247	D₄×C₅₃	—	4	4	no	—
248	C₂×C₄×C₅₃	—	2	2	no	—
249	C₂×C₄×C₅₃	—	2	2	no	—
250	C₂×C₄×C₅₃	—	2	2	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	2	no	—
258	C₂×C₄×C₅₃	—	2	2	no	—
259	C₂×C₄×C₅₃	—	4	4	no	—
260	C₂×C₄×C₅₃	—	4	4	no	—
261	C₂×C₄×C₅₃	—	4	8	no	—
262	C₂×C₄×C₅₃	—	4	16	no	—
263	C₂×C₄×C₅₃	—	4	4	no	—
264	C₂×C₄×C₅₃	—	4	2	no	—
265	C₂×C₄×C₅₃	—	4	2	no	—
266	C₂×C₄×C₅₃	—	4	2	no	—
267	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
268	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
269	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
270	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
271	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
274	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
275	C₈×C₅₃	—	2	4	no	—
276	C₈×C₅₃	—	2	4	no	—
277	C₂×#3219	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
278	C₂²×C₅₃	—	2	2	no	—
279	C₂²×C₅₃	—	2	2	no	—
280	C₂²×C₅₃	—	2	2	no	—
281	C₂²×C₅₃	—	4	4	no	—
282	C₂²×C₅₃	—	4	4	no	—
283	C₂²×C₅₃	—	4	2	no	—
284	C₂²×C₅₃	—	4	2	no	—
285	C₂²×C₅₃	—	4	4	no	—
286	C₂²×C₅₃	—	4	4	no	—
287	C₂²×C₅₃	—	4	2	no	—
288	C₂²×C₅₃	—	8	8	no	—
289	C₂²×C₅₃	—	8	4	no	—
290	C₂²×C₅₃	#535	1	2	yes	—
291	C₂²×C₅₃	#575	1	2	yes	—
292	C₂²×C₅₃	#575	1	2	yes	—
293	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
294	C₄×C₅₃	—	2	2	no	—
295	C₄×C₅₃	—	2	2	no	—
296	C₄×C₅₃	—	2	2	no	—
297	C₄×C₅₃	—	2	2	no	—
298	C₄×C₅₃	—	2	2	no	—
299	C₄×C₅₃	—	4	4	no	—
300	C₄×C₅₃	—	4	2	no	—
301	C₄×C₅₃	—	4	2	no	—
302	C₄×C₅₃	—	4	2	no	—
303	C₄×C₅₃	#535	1	2	yes	—
304	C₄×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
305	C₂×#532	C₂×C₅₃	1	1	yes	characteristic, complemented
306	C₂×#529	C₂×C₅₃	1	1	yes	characteristic, complemented
307	#3219	C₂×C₅₃	1	16	yes	direct product factor
308	C₂×#497	C₂×C₅₃	1	1	yes	characteristic, complemented
309	C₂×#496	C₂×C₅₃	1	1	yes	characteristic, complemented
310	C₂×#491	C₂×C₅₃	1	1	yes	characteristic, complemented
311	C₂×#490	C₂×C₅₃	1	1	yes	characteristic, complemented
312	C₂×#464	C₂×C₅₃	1	2	yes	complemented
313	C₂×#463	C₂×C₅₃	1	2	yes	complemented
314	C₂×#452	C₂×C₅₃	1	2	yes	complemented
315	C₂×#449	C₂×C₅₃	1	2	yes	complemented
316	C₂×#432	C₂×C₅₃	1	1	yes	characteristic, complemented
317	C₂×C₅₃	—	4	2	no	—
318	C₂×C₅₃	—	8	4	no	—
319	C₂×C₅₃	—	8	2	no	—
320	C₂×C₅₃	#3219	1	4	yes	direct product factor
321	C₂×C₅₃	C₂×#535	1	1	yes	characteristic
322	C₂×C₅₃	C₂×#575	1	1	yes	characteristic
323	C₂×C₅₃	C₂×#575	1	1	yes	characteristic
324	C₂²×#41	C₂²×C₅₃	1	2	yes	—
325	C₂²×Q₁₆	C₂²×C₅₃	1	2	yes	—
326	C₂²×SD₁₆	C₂²×C₅₃	1	2	yes	—
327	C₂²×SD₁₆	C₂²×C₅₃	1	2	yes	—
328	C₂²×D₈	C₂²×C₅₃	1	2	yes	—
329	#532	—	2	4	no	—
330	#532	C₂²×C₅₃	1	8	yes	complemented
331	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
332	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
333	#529	—	2	4	no	—
334	#529	C₂²×C₅₃	1	8	yes	complemented
335	C₂×#126	—	2	1	no	—
336	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic
337	C₂×#124	C₂²×C₅₃	1	1	yes	characteristic
338	C₂×#124	C₂²×C₅₃	1	2	yes	—
339	C₂×#123	—	2	1	no	—
340	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
341	C₂×#121	—	2	1	no	—
342	C₂×#121	—	2	1	no	—
343	C₂×#121	C₂²×C₅₃	1	1	yes	characteristic
344	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
345	C₂×C₄×D₄	—	2	1	no	—
346	C₂×C₄×D₄	—	2	1	no	—
347	C₂×C₄×D₄	—	2	1	no	—
348	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
349	C₂×#117	—	2	1	no	—
350	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
351	#497	C₂²×C₅₃	1	8	yes	complemented
352	#496	C₂²×C₅₃	1	8	yes	complemented
353	#491	C₂²×C₅₃	1	8	yes	complemented
354	#490	C₂²×C₅₃	1	8	yes	complemented
355	#464	C₂²×C₅₃	1	16	yes	complemented
356	#463	C₂²×C₅₃	1	16	yes	complemented
357	#452	C₂²×C₅₃	1	16	yes	complemented
358	#449	C₂²×C₅₃	1	16	yes	complemented
359	#432	C₂²×C₅₃	1	8	yes	complemented
360	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
361	C₂×#103	C₂²×C₅₃	1	2	yes	—
362	C₂×#103	C₂²×C₅₃	1	2	yes	—
363	C₂×#102	C₂²×C₅₃	1	2	yes	—
364	C₂×#102	C₂²×C₅₃	1	2	yes	—
365	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
366	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
367	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
368	C₅₃	C₂×#3219	1	1	yes	characteristic, direct product factor, Sylow
369	C₂×#41	—	2	4	no	—
370	C₂×#41	—	4	4	no	—
371	C₂×#41	—	4	4	no	—
372	C₂×#41	C₂³×C₅₃	1	8	yes	—
373	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
374	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
375	C₂²×D₄	—	2	1	no	—
376	C₂²×D₄	—	2	2	no	—
377	C₂²×D₄	—	2	1	no	—
378	C₂²×D₄	—	2	1	no	—
379	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
380	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
381	C₂³×C₄	—	2	1	no	—
382	C₂³×C₄	—	2	2	no	—
383	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
384	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
385	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
386	C₂×Q₁₆	—	4	4	no	—
387	C₂×Q₁₆	C₂³×C₅₃	1	8	yes	—
388	C₂×SD₁₆	—	4	4	no	—
389	C₂×SD₁₆	—	4	4	no	—
390	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	—
391	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	—
392	C₂×D₈	—	4	4	no	—
393	C₂×D₈	C₂³×C₅₃	1	8	yes	—
394	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
395	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
396	C₄⋊Q₈	—	2	2	no	—
397	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
398	#127	—	2	2	no	—
399	#127	C₂³×C₅₃	1	4	yes	—
400	#126	—	2	8	no	—
401	#125	—	2	2	no	—
402	#125	C₂³×C₅₃	1	4	yes	—
403	#124	—	2	2	no	—
404	#124	—	2	4	no	—
405	#124	C₂³×C₅₃	1	4	yes	—
406	#124	C₂³×C₅₃	1	8	yes	—
407	#123	—	2	8	no	—
408	C₂²⋊Q₈	—	2	2	no	—
409	C₂²⋊Q₈	C₂³×C₅₃	1	4	yes	—
410	#121	—	2	8	no	—
411	#121	—	2	8	no	—
412	#121	—	2	2	no	—
413	#121	C₂³×C₅₃	1	4	yes	—
414	C₄×Q₈	—	2	2	no	—
415	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
416	C₄×D₄	—	2	8	no	—
417	C₄×D₄	—	2	8	no	—
418	C₄×D₄	—	2	8	no	—
419	C₄×D₄	—	2	2	no	—
420	C₄×D₄	C₂³×C₅₃	1	4	yes	—
421	#117	—	2	8	no	—
422	C₂×C₄⋊C₄	—	2	1	no	—
423	C₂×C₄⋊C₄	—	2	1	no	—
424	C₂×C₄⋊C₄	—	2	1	no	—
425	C₂×C₄⋊C₄	—	2	1	no	—
426	C₂×C₄⋊C₄	—	2	1	no	—
427	C₂×C₄⋊C₄	—	2	1	no	—
428	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
429	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
430	C₂×C₂²⋊C₄	—	2	1	no	—
431	C₂×C₂²⋊C₄	—	2	1	no	—
432	C₂×C₂²⋊C₄	—	2	1	no	—
433	C₂×C₂²⋊C₄	—	2	2	no	—
434	C₂×C₂²⋊C₄	—	2	2	no	—
435	C₂×C₂²⋊C₄	—	2	1	no	—
436	C₂×C₂²⋊C₄	—	2	1	no	—
437	C₂×C₄²	—	2	1	no	—
438	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
439	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
440	C₂×C₄²	D₄×C₅₃	1	4	yes	—
441	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
442	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
443	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
444	C₄⋊C₈	C₂³×C₅₃	1	8	yes	—
445	#103	C₂³×C₅₃	1	8	yes	—
446	#103	C₂³×C₅₃	1	8	yes	—
447	#102	C₂³×C₅₃	1	8	yes	—
448	#102	C₂³×C₅₃	1	8	yes	—
449	C₂²⋊C₈	C₂³×C₅₃	1	8	yes	—
450	#97	C₂³×C₅₃	1	4	yes	—
451	#97	C₂³×C₅₃	1	4	yes	—
452	C₂⁴	—	2	1	no	—
453	C₂⁴	—	2	2	no	—
454	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
455	#41	—	4	8	no	—
456	#41	—	4	8	no	—
457	#41	—	4	16	no	—
458	C₂×Q₈	—	2	1	no	—
459	C₂×Q₈	—	2	2	no	—
460	C₂×Q₈	—	2	4	no	—
461	C₂×Q₈	—	2	4	no	—
462	C₂×Q₈	—	2	4	no	—
463	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
464	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
465	C₂×D₄	—	2	4	no	—
466	C₂×D₄	—	2	8	no	—
467	C₂×D₄	—	2	1	no	—
468	C₂×D₄	—	2	4	no	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	—	2	2	no	—
471	C₂×D₄	—	2	4	no	—
472	C₂×D₄	—	2	4	no	—
473	C₂×D₄	—	2	4	no	—
474	C₂×D₄	—	4	4	no	—
475	C₂×D₄	—	4	4	no	—
476	C₂×D₄	—	4	4	no	—
477	C₂×D₄	—	4	8	no	—
478	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
479	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
480	C₂²×C₄	—	2	4	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	2	no	—
485	C₂²×C₄	—	2	4	no	—
486	C₂²×C₄	—	2	8	no	—
487	C₂²×C₄	—	4	2	no	—
488	C₂²×C₄	—	4	2	no	—
489	C₂²×C₄	—	4	4	no	—
490	C₂²×C₄	—	4	8	no	—
491	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
492	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
493	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
494	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
495	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
496	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
497	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
498	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
499	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
500	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
501	Q₁₆	—	4	8	no	—
502	SD₁₆	—	4	8	no	—
503	SD₁₆	—	4	8	no	—
504	D₈	—	4	8	no	—
505	C₂×C₈	—	2	4	no	—
506	C₂×C₈	—	2	4	no	—
507	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
508	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
509	C₄⋊C₄	—	2	4	no	—
510	C₄⋊C₄	—	2	4	no	—
511	C₄⋊C₄	—	2	4	no	—
512	C₄⋊C₄	—	2	4	no	—
513	C₄⋊C₄	—	2	4	no	—
514	C₄⋊C₄	—	2	4	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
518	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
519	C₂²⋊C₄	—	2	4	no	—
520	C₂²⋊C₄	—	2	4	no	—
521	C₂²⋊C₄	—	2	4	no	—
522	C₂²⋊C₄	—	2	8	no	—
523	C₂²⋊C₄	—	2	8	no	—
524	C₂²⋊C₄	—	2	4	no	—
525	C₂²⋊C₄	—	2	4	no	—
526	C₄²	—	2	4	no	—
527	C₄²	C₂×D₄×C₅₃	1	4	yes	—
528	C₄²	C₂×D₄×C₅₃	1	4	yes	—
529	C₄²	C₂×D₄×C₅₃	1	2	yes	—
530	C₄²	C₂×D₄×C₅₃	1	2	yes	—
531	C₄²	C₂⁴×C₅₃	1	2	yes	—
532	C₄²	C₂⁴×C₅₃	1	2	yes	—
533	C₂³	—	2	2	no	—
534	C₂³	—	2	2	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	2	no	—
537	C₂³	—	2	4	no	—
538	C₂³	—	4	4	no	—
539	C₂³	—	4	2	no	—
540	C₂³	—	4	2	no	—
541	C₂³	—	4	4	no	—
542	C₂³	—	4	4	no	—
543	C₂³	—	4	2	no	—
544	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
545	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
546	Q₈	—	2	4	no	—
547	Q₈	—	2	4	no	—
548	Q₈	—	2	4	no	—
549	Q₈	—	4	2	no	—
550	Q₈	—	4	4	no	—
551	D₄	—	2	4	no	—
552	D₄	—	2	4	no	—
553	D₄	—	2	4	no	—
554	D₄	—	4	8	no	—
555	D₄	—	4	8	no	—
556	D₄	—	4	8	no	—
557	D₄	—	4	16	no	—
558	D₄	—	4	2	no	—
559	D₄	—	4	4	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	4	4	no	—
572	C₂×C₄	—	4	4	no	—
573	C₂×C₄	—	4	8	no	—
574	C₂×C₄	—	4	16	no	—
575	C₂×C₄	—	4	4	no	—
576	C₂×C₄	—	4	2	no	—
577	C₂×C₄	—	4	2	no	—
578	C₂×C₄	—	4	2	no	—
579	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
580	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
581	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
582	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
583	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
584	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
585	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
586	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
587	C₈	—	2	4	no	—
588	C₈	—	2	4	no	—
589	C₂²	—	2	2	no	—
590	C₂²	—	2	2	no	—
591	C₂²	—	2	2	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	4	2	no	—
594	C₂²	—	4	4	no	—
595	C₂²	—	4	4	no	—
596	C₂²	—	4	2	no	—
597	C₂²	—	4	4	no	—
598	C₂²	—	4	2	no	—
599	C₂²	—	8	8	no	—
600	C₂²	—	8	4	no	—
601	C₂²	C₅₃×#535	1	2	yes	—
602	C₂²	C₅₃×#575	1	2	yes	—
603	C₂²	C₅₃×#575	1	2	yes	—
604	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
605	C₄	—	2	2	no	—
606	C₄	—	2	2	no	—
607	C₄	—	2	2	no	—
608	C₄	—	2	2	no	—
609	C₄	—	2	2	no	—
610	C₄	—	4	2	no	—
611	C₄	—	4	2	no	—
612	C₄	—	4	4	no	—
613	C₄	—	4	2	no	—
614	C₄	C₅₃×#535	1	2	yes	—
615	C₄	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
616	C₂	—	4	2	no	—
617	C₂	—	8	2	no	—
618	C₂	—	8	4	no	—
619	C₂	C₅₃×#3219	1	4	yes	direct product factor
620	C₂	C₂×C₅₃×#535	1	1	yes	characteristic
621	C₂	C₂×C₅₃×#575	1	1	yes	characteristic
622	C₂	C₂×C₅₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
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×#532	C₂	semidirect
2. C₂×C₅₃×#529	C₂	semidirect
3. C₅₃×#3219	C₂	direct (N × H)
4. C₂×C₅₃×#497	C₂	semidirect
5. C₂×C₅₃×#496	C₂	semidirect
6. C₂×C₅₃×#491	C₂	semidirect
7. C₂×C₅₃×#490	C₂	semidirect
8. C₂×C₅₃×#464	C₂	semidirect
9. C₂×C₅₃×#463	C₂	semidirect
10. C₂×C₅₃×#452	C₂	semidirect
11. C₂×C₅₃×#449	C₂	semidirect
12. C₂×C₅₃×#432	C₂	semidirect
19. C₅₃×#532	C₂²	semidirect
23. C₅₃×#529	C₂²	semidirect
40. C₅₃×#497	C₂²	semidirect
41. C₅₃×#496	C₂²	semidirect
42. C₅₃×#491	C₂²	semidirect
43. C₅₃×#490	C₂²	semidirect
44. C₅₃×#464	C₂²	semidirect
45. C₅₃×#463	C₂²	semidirect
46. C₅₃×#452	C₂²	semidirect
47. C₅₃×#449	C₂²	semidirect
48. C₅₃×#432	C₂²	semidirect
277. C₂×#3219	C₅₃	direct (N × H)
305. C₂×#532	C₂×C₅₃	semidirect
306. C₂×#529	C₂×C₅₃	semidirect
307. #3219	C₂×C₅₃	direct (N × H)
308. C₂×#497	C₂×C₅₃	semidirect
309. C₂×#496	C₂×C₅₃	semidirect
310. C₂×#491	C₂×C₅₃	semidirect
311. C₂×#490	C₂×C₅₃	semidirect
312. C₂×#464	C₂×C₅₃	semidirect
313. C₂×#463	C₂×C₅₃	semidirect
314. C₂×#452	C₂×C₅₃	semidirect
315. C₂×#449	C₂×C₅₃	semidirect
316. C₂×#432	C₂×C₅₃	semidirect
320. C₂×C₅₃	#3219	direct (N × H)
330. #532	C₂²×C₅₃	semidirect
334. #529	C₂²×C₅₃	semidirect
351. #497	C₂²×C₅₃	semidirect
352. #496	C₂²×C₅₃	semidirect
353. #491	C₂²×C₅₃	semidirect
354. #490	C₂²×C₅₃	semidirect
355. #464	C₂²×C₅₃	semidirect
356. #463	C₂²×C₅₃	semidirect
357. #452	C₂²×C₅₃	semidirect
358. #449	C₂²×C₅₃	semidirect
359. #432	C₂²×C₅₃	semidirect
368. C₅₃	C₂×#3219	direct (N × H)
619. C₂	C₅₃×#3219	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×#532	2
C₂×C₅₃×#529	2
C₅₃×#3219	2
C₂×C₅₃×#497	2
C₂×C₅₃×#496	2
C₂×C₅₃×#491	2
C₂×C₅₃×#490	2
C₂×C₅₃×#464	2
C₂×C₅₃×#463	2
C₂×C₅₃×#452	2
C₂×C₅₃×#449	2
C₂×C₅₃×#432	2
C₂×#3219	53

Subgroups — C2×C53×#3219 #27268301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3219 #27268301