Subgroups — C₄×C₅₃×#536 #27241601

Order 13568.

2962 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#536	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2327	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2326	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2323	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2322	C₂	1	4	yes	maximal, complemented
8	C₅₃×#2317	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2311	C₂	1	1	yes	maximal, characteristic
10	C₅₃×#2310	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₄⋊Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
12	C₄×#124×C₅₃	C₂	1	2	yes	maximal, complemented
13	C₄×C₂²⋊Q₈×C₅₃	C₂	1	4	yes	maximal, complemented
14	C₄×#123×C₅₃	C₂	1	4	yes	maximal, complemented
15	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
17	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#536	C₄	1	16	yes	direct product factor
19	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
20	C₂×#124×C₅₃	C₂²	1	2	yes	—
21	C₂×#123×C₅₃	C₂²	1	4	yes	—
22	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	—
23	C₄×#41×C₅₃	—	2	8	no	—
24	C₂×C₄×Q₈×C₅₃	—	2	8	no	—
25	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
34	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×#117×C₅₃	C₂²	1	2	yes	—
38	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
39	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
40	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
41	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
42	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
43	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
44	C₅₃×#391	C₂²	1	4	yes	—
45	C₅₃×#391	C₂²	1	2	yes	—
46	C₅₃×#391	C₂²	1	2	yes	—
47	C₅₃×#391	C₂²	1	2	yes	—
48	C₅₃×#390	C₂²	1	2	yes	—
49	C₅₃×#389	C₂²	1	4	yes	—
50	C₅₃×#389	C₂²	1	8	yes	—
51	C₅₃×#388	C₂²	1	8	yes	—
52	C₅₃×#388	C₂²	1	8	yes	—
53	C₅₃×#387	C₂²	1	8	yes	—
54	C₅₃×#387	C₂²	1	8	yes	—
55	C₅₃×#386	C₂²	1	4	yes	—
56	C₅₃×#385	C₂²	1	4	yes	—
57	C₅₃×#385	C₂²	1	8	yes	—
58	C₅₃×#384	C₂²	1	2	yes	—
59	C₅₃×#382	C₂²	1	2	yes	—
60	C₅₃×#381	C₂²	1	4	yes	—
61	C₅₃×#380	C₂²	1	4	yes	—
62	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
64	C₄×C₄⋊C₄×C₅₃	C₂²	1	8	yes	—
65	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
66	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
67	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
68	C₅₃×#376	C₂²	1	2	yes	—
69	C₄³×C₅₃	C₂²	1	2	yes	—
70	C₂×#41×C₅₃	—	2	4	no	—
71	C₂×#41×C₅₃	—	2	4	no	—
72	C₂×#41×C₅₃	C₂×C₄	1	8	yes	—
73	C₂²×Q₈×C₅₃	—	2	4	no	—
74	C₂²×Q₈×C₅₃	—	2	4	no	—
75	C₂²×Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
76	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
77	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
78	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
80	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
82	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
83	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
84	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
86	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
87	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
89	C₄⋊Q₈×C₅₃	C₂×C₄	1	16	yes	complemented
90	#124×C₅₃	C₂×C₄	1	16	yes	complemented
91	#123×C₅₃	C₂×C₄	1	32	yes	complemented
92	C₂²⋊Q₈×C₅₃	C₂×C₄	1	32	yes	complemented
93	C₄×Q₈×C₅₃	—	2	8	no	—
94	C₄×Q₈×C₅₃	—	2	8	no	—
95	C₄×Q₈×C₅₃	—	2	8	no	—
96	C₄×Q₈×C₅₃	—	2	8	no	—
97	C₄×Q₈×C₅₃	—	2	4	no	—
98	C₄×Q₈×C₅₃	—	2	4	no	—
99	C₄×D₄×C₅₃	—	2	4	no	—
100	C₄×D₄×C₅₃	—	2	4	no	—
101	C₄×D₄×C₅₃	—	2	8	no	—
102	C₄×D₄×C₅₃	—	2	8	no	—
103	#117×C₅₃	—	2	8	no	—
104	#117×C₅₃	—	2	4	no	—
105	#117×C₅₃	—	2	4	no	—
106	#117×C₅₃	C₂×C₄	1	8	yes	complemented
107	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
108	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
109	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
112	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
113	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
115	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
118	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
119	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
120	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
121	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
123	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
125	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
126	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
127	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
128	C₂×C₄²×C₅₃	—	2	8	no	—
129	C₂×C₄²×C₅₃	—	2	8	no	—
130	C₂×C₄²×C₅₃	D₄	1	8	yes	—
131	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
132	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
133	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
134	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
135	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
136	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
137	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
138	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
139	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
140	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
141	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
142	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
143	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
144	#95×C₅₃	C₂³	1	4	yes	—
145	#95×C₅₃	C₂³	1	4	yes	—
146	#95×C₅₃	C₂³	1	8	yes	—
147	#95×C₅₃	C₂³	1	8	yes	—
148	#95×C₅₃	C₂³	1	8	yes	—
149	#95×C₅₃	C₂³	1	8	yes	—
150	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
151	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
152	#41×C₅₃	—	2	32	no	—
153	C₂×Q₈×C₅₃	—	2	32	no	—
154	C₂×Q₈×C₅₃	—	2	4	no	—
155	C₂×Q₈×C₅₃	—	2	4	no	—
156	C₂×Q₈×C₅₃	—	2	2	no	—
157	C₂×Q₈×C₅₃	—	2	4	no	—
158	C₂×Q₈×C₅₃	—	2	4	no	—
159	C₂×Q₈×C₅₃	—	2	2	no	—
160	C₂×Q₈×C₅₃	C₂²×C₄	1	8	yes	—
161	C₂×Q₈×C₅₃	C₂²×C₄	1	8	yes	—
162	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
163	C₂×D₄×C₅₃	—	2	4	no	—
164	C₂×D₄×C₅₃	—	2	4	no	—
165	C₂×D₄×C₅₃	—	2	2	no	—
166	C₂×D₄×C₅₃	—	2	2	no	—
167	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
168	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
169	C₂²×C₄×C₅₃	—	2	4	no	—
170	C₂²×C₄×C₅₃	—	2	4	no	—
171	C₂²×C₄×C₅₃	—	2	4	no	—
172	C₂²×C₄×C₅₃	—	2	4	no	—
173	C₂²×C₄×C₅₃	—	2	4	no	—
174	C₂²×C₄×C₅₃	—	2	4	no	—
175	C₂²×C₄×C₅₃	—	2	4	no	—
176	C₂²×C₄×C₅₃	—	2	4	no	—
177	C₂²×C₄×C₅₃	—	2	8	no	—
178	C₂²×C₄×C₅₃	—	4	2	no	—
179	C₂²×C₄×C₅₃	—	4	4	no	—
180	C₂²×C₄×C₅₃	—	4	4	no	—
181	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
182	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
183	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
184	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
185	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
186	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
187	C₂²×C₄×C₅₃	#41	1	4	yes	—
188	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
189	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
190	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
191	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
192	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
193	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
194	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
195	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
196	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
197	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
198	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
199	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
200	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
201	C₄⋊C₄×C₅₃	—	2	8	no	—
202	C₄⋊C₄×C₅₃	—	2	8	no	—
203	C₄⋊C₄×C₅₃	—	2	4	no	—
204	C₄⋊C₄×C₅₃	—	2	4	no	—
205	C₄⋊C₄×C₅₃	—	2	4	no	—
206	C₄⋊C₄×C₅₃	—	2	4	no	—
207	C₄⋊C₄×C₅₃	—	2	2	no	—
208	C₄⋊C₄×C₅₃	—	2	2	no	—
209	C₄⋊C₄×C₅₃	—	2	4	no	—
210	C₄⋊C₄×C₅₃	—	4	8	no	—
211	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
212	C₄⋊C₄×C₅₃	C₂²×C₄	1	32	yes	—
213	C₂²⋊C₄×C₅₃	—	2	4	no	—
214	C₂²⋊C₄×C₅₃	—	2	4	no	—
215	C₂²⋊C₄×C₅₃	—	2	4	no	—
216	C₂²⋊C₄×C₅₃	—	2	4	no	—
217	C₂²⋊C₄×C₅₃	—	4	8	no	—
218	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
219	C₂²⋊C₄×C₅₃	C₂²×C₄	1	32	yes	—
220	C₄²×C₅₃	—	2	4	no	—
221	C₄²×C₅₃	—	2	4	no	—
222	C₄²×C₅₃	—	2	4	no	—
223	C₄²×C₅₃	—	2	8	no	—
224	C₄²×C₅₃	—	2	8	no	—
225	C₄²×C₅₃	—	2	4	no	—
226	C₄²×C₅₃	—	2	8	no	—
227	C₄²×C₅₃	—	4	4	no	—
228	C₄²×C₅₃	C₂²×C₄	1	8	yes	—
229	C₄²×C₅₃	C₂×D₄	1	4	yes	—
230	C₄²×C₅₃	C₂×D₄	1	4	yes	—
231	C₄²×C₅₃	C₂×D₄	1	4	yes	—
232	C₄²×C₅₃	C₂×D₄	1	4	yes	—
233	C₂³×C₅₃	—	2	2	no	—
234	C₂³×C₅₃	—	2	2	no	—
235	C₂³×C₅₃	—	2	2	no	—
236	C₂³×C₅₃	—	2	2	no	—
237	C₂³×C₅₃	—	2	4	no	—
238	C₂³×C₅₃	—	2	4	no	—
239	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
240	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
241	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
242	C₂³×C₅₃	C₂×#41	1	2	yes	—
243	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
244	Q₈×C₅₃	—	2	16	no	—
245	Q₈×C₅₃	—	2	16	no	—
246	Q₈×C₅₃	—	2	8	no	—
247	D₄×C₅₃	—	2	16	no	—
248	D₄×C₅₃	—	2	8	no	—
249	C₂×C₄×C₅₃	—	2	16	no	—
250	C₂×C₄×C₅₃	—	2	2	no	—
251	C₂×C₄×C₅₃	—	2	4	no	—
252	C₂×C₄×C₅₃	—	2	2	no	—
253	C₂×C₄×C₅₃	—	2	4	no	—
254	C₂×C₄×C₅₃	—	2	2	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	4	no	—
259	C₂×C₄×C₅₃	—	2	16	no	—
260	C₂×C₄×C₅₃	—	2	8	no	—
261	C₂×C₄×C₅₃	—	2	2	no	—
262	C₂×C₄×C₅₃	—	2	2	no	—
263	C₂×C₄×C₅₃	—	2	4	no	—
264	C₂×C₄×C₅₃	—	2	4	no	—
265	C₂×C₄×C₅₃	—	2	4	no	—
266	C₂×C₄×C₅₃	—	2	4	no	—
267	C₂×C₄×C₅₃	—	2	8	no	—
268	C₂×C₄×C₅₃	—	2	4	no	—
269	C₂×C₄×C₅₃	—	2	4	no	—
270	C₂×C₄×C₅₃	—	4	4	no	—
271	C₂×C₄×C₅₃	—	4	8	no	—
272	C₂×C₄×C₅₃	—	4	8	no	—
273	C₂×C₄×C₅₃	—	4	8	no	—
274	C₂×C₄×C₅₃	C₄×D₄	1	16	yes	—
275	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
276	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
277	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
278	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
279	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
280	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
281	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
282	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
283	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
284	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
285	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
286	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
287	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
288	C₂×C₄×C₅₃	2₋¹⁺⁴	1	4	yes	—
289	C₄×#536	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
290	C₂²×C₅₃	—	2	4	no	—
291	C₂²×C₅₃	—	2	2	no	—
292	C₂²×C₅₃	—	2	2	no	—
293	C₂²×C₅₃	—	2	4	no	—
294	C₂²×C₅₃	—	2	4	no	—
295	C₂²×C₅₃	—	2	8	no	—
296	C₂²×C₅₃	—	4	2	no	—
297	C₂²×C₅₃	—	4	2	no	—
298	C₂²×C₅₃	—	4	4	no	—
299	C₂²×C₅₃	C₂×C₄×D₄	1	4	yes	—
300	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
301	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
302	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
303	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
304	C₂²×C₅₃	#585	1	2	yes	—
305	C₄×C₅₃	—	2	4	no	—
306	C₄×C₅₃	—	2	8	no	—
307	C₄×C₅₃	—	2	8	no	—
308	C₄×C₅₃	—	2	2	no	—
309	C₄×C₅₃	—	2	2	no	—
310	C₄×C₅₃	—	2	2	no	—
311	C₄×C₅₃	—	4	2	no	—
312	C₄×C₅₃	—	4	2	no	—
313	C₄×C₅₃	—	4	4	no	—
314	C₄×C₅₃	—	4	4	no	—
315	C₄×C₅₃	C₂×C₄×D₄	1	4	yes	—
316	C₄×C₅₃	C₂×C₄×D₄	1	4	yes	—
317	C₄×C₅₃	#536	1	4	yes	direct product factor
318	C₂×#536	C₂×C₅₃	1	1	yes	characteristic
319	C₂×C₄×#41	C₂×C₅₃	1	1	yes	characteristic
320	C₂²×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
321	#2327	C₂×C₅₃	1	2	yes	complemented
322	#2326	C₂×C₅₃	1	2	yes	complemented
323	#2323	C₂×C₅₃	1	4	yes	complemented
324	#2322	C₂×C₅₃	1	4	yes	complemented
325	#2317	C₂×C₅₃	1	1	yes	characteristic, complemented
326	#2311	C₂×C₅₃	1	1	yes	characteristic
327	#2310	C₂×C₅₃	1	1	yes	characteristic, complemented
328	C₄×C₄⋊Q₈	C₂×C₅₃	1	2	yes	complemented
329	C₄×#124	C₂×C₅₃	1	2	yes	complemented
330	C₄×C₂²⋊Q₈	C₂×C₅₃	1	4	yes	complemented
331	C₄×#123	C₂×C₅₃	1	4	yes	complemented
332	C₄×#117	C₂×C₅₃	1	1	yes	characteristic, complemented
333	C₂×C₅₃	—	2	4	no	—
334	C₂×C₅₃	—	4	4	no	—
335	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
336	C₂×C₅₃	C₄×2₋¹⁺⁴	1	2	yes	—
337	C₂×C₅₃	C₂×#536	1	1	yes	characteristic
338	C₂×C₅₃	#3449	1	1	yes	characteristic
339	C₂×C₅₃	#3461	1	2	yes	—
340	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
341	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
342	#536	C₄×C₅₃	1	16	yes	direct product factor
343	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
344	C₂×#124	C₂²×C₅₃	1	2	yes	—
345	C₂×#123	C₂²×C₅₃	1	4	yes	—
346	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	—
347	C₄×#41	—	2	8	no	—
348	C₂×C₄×Q₈	—	2	8	no	—
349	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
350	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
351	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
352	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
353	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
354	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
355	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
356	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
357	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
358	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
359	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
360	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
361	C₂×#117	C₂²×C₅₃	1	2	yes	—
362	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
363	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
364	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
365	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
366	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
367	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
368	#391	C₂²×C₅₃	1	2	yes	—
369	#391	C₂²×C₅₃	1	2	yes	—
370	#391	C₂²×C₅₃	1	4	yes	—
371	#391	C₂²×C₅₃	1	2	yes	—
372	#390	C₂²×C₅₃	1	2	yes	—
373	#389	C₂²×C₅₃	1	8	yes	—
374	#389	C₂²×C₅₃	1	4	yes	—
375	#388	C₂²×C₅₃	1	8	yes	—
376	#388	C₂²×C₅₃	1	8	yes	—
377	#387	C₂²×C₅₃	1	8	yes	—
378	#387	C₂²×C₅₃	1	8	yes	—
379	#386	C₂²×C₅₃	1	4	yes	—
380	#385	C₂²×C₅₃	1	4	yes	—
381	#385	C₂²×C₅₃	1	8	yes	—
382	#384	C₂²×C₅₃	1	2	yes	—
383	#382	C₂²×C₅₃	1	2	yes	—
384	#381	C₂²×C₅₃	1	4	yes	—
385	#380	C₂²×C₅₃	1	4	yes	—
386	C₄×C₄⋊C₄	C₂²×C₅₃	1	8	yes	—
387	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
388	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
389	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
390	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
391	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
392	#376	C₂²×C₅₃	1	2	yes	—
393	C₄³	C₂²×C₅₃	1	2	yes	—
394	C₅₃	C₄×#536	1	1	yes	characteristic, direct product factor, Sylow
395	C₂×#41	—	2	4	no	—
396	C₂×#41	—	2	4	no	—
397	C₂×#41	C₂×C₄×C₅₃	1	8	yes	—
398	C₂²×Q₈	—	2	4	no	—
399	C₂²×Q₈	—	2	4	no	—
400	C₂²×Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
401	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
402	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
403	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
404	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
405	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
406	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
407	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
408	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
409	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
410	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
411	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
412	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
413	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
414	C₄⋊Q₈	C₂×C₄×C₅₃	1	16	yes	complemented
415	#124	C₂×C₄×C₅₃	1	16	yes	complemented
416	#123	C₂×C₄×C₅₃	1	32	yes	complemented
417	C₂²⋊Q₈	C₂×C₄×C₅₃	1	32	yes	complemented
418	C₄×Q₈	—	2	8	no	—
419	C₄×Q₈	—	2	8	no	—
420	C₄×Q₈	—	2	8	no	—
421	C₄×Q₈	—	2	8	no	—
422	C₄×Q₈	—	2	4	no	—
423	C₄×Q₈	—	2	4	no	—
424	C₄×D₄	—	2	8	no	—
425	C₄×D₄	—	2	8	no	—
426	C₄×D₄	—	2	4	no	—
427	C₄×D₄	—	2	4	no	—
428	#117	—	2	4	no	—
429	#117	—	2	4	no	—
430	#117	—	2	8	no	—
431	#117	C₂×C₄×C₅₃	1	8	yes	complemented
432	C₂×C₄⋊C₄	—	2	8	no	—
433	C₂×C₄⋊C₄	—	2	4	no	—
434	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
435	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
436	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
437	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
438	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
439	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
440	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
441	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
442	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
443	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
444	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
445	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
446	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
447	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
448	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
449	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
450	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
451	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
452	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
453	C₂×C₄²	—	2	8	no	—
454	C₂×C₄²	—	2	8	no	—
455	C₂×C₄²	D₄×C₅₃	1	8	yes	—
456	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
457	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
458	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
459	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
460	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
461	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
462	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
463	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
464	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
465	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
466	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
467	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
468	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
469	#95	C₂³×C₅₃	1	4	yes	—
470	#95	C₂³×C₅₃	1	4	yes	—
471	#95	C₂³×C₅₃	1	8	yes	—
472	#95	C₂³×C₅₃	1	8	yes	—
473	#95	C₂³×C₅₃	1	8	yes	—
474	#95	C₂³×C₅₃	1	8	yes	—
475	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
476	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
477	#41	—	2	32	no	—
478	C₂×Q₈	—	2	32	no	—
479	C₂×Q₈	—	2	4	no	—
480	C₂×Q₈	—	2	4	no	—
481	C₂×Q₈	—	2	2	no	—
482	C₂×Q₈	—	2	4	no	—
483	C₂×Q₈	—	2	4	no	—
484	C₂×Q₈	—	2	2	no	—
485	C₂×Q₈	C₂²×C₄×C₅₃	1	8	yes	—
486	C₂×Q₈	C₂²×C₄×C₅₃	1	8	yes	—
487	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
488	C₂×D₄	—	2	4	no	—
489	C₂×D₄	—	2	4	no	—
490	C₂×D₄	—	2	2	no	—
491	C₂×D₄	—	2	2	no	—
492	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
493	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
494	C₂²×C₄	—	2	4	no	—
495	C₂²×C₄	—	2	4	no	—
496	C₂²×C₄	—	2	4	no	—
497	C₂²×C₄	—	2	4	no	—
498	C₂²×C₄	—	2	4	no	—
499	C₂²×C₄	—	2	4	no	—
500	C₂²×C₄	—	2	4	no	—
501	C₂²×C₄	—	2	4	no	—
502	C₂²×C₄	—	2	8	no	—
503	C₂²×C₄	—	4	2	no	—
504	C₂²×C₄	—	4	4	no	—
505	C₂²×C₄	—	4	4	no	—
506	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
507	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
508	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
509	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
510	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
511	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
512	C₂²×C₄	#41×C₅₃	1	4	yes	—
513	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
514	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
515	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
516	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
517	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
518	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
519	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
520	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
521	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
522	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
523	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
524	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
525	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
526	C₄⋊C₄	—	2	8	no	—
527	C₄⋊C₄	—	2	8	no	—
528	C₄⋊C₄	—	2	4	no	—
529	C₄⋊C₄	—	2	4	no	—
530	C₄⋊C₄	—	2	4	no	—
531	C₄⋊C₄	—	2	4	no	—
532	C₄⋊C₄	—	2	2	no	—
533	C₄⋊C₄	—	2	2	no	—
534	C₄⋊C₄	—	2	4	no	—
535	C₄⋊C₄	—	4	8	no	—
536	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
537	C₄⋊C₄	C₂²×C₄×C₅₃	1	32	yes	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₂²⋊C₄	—	2	4	no	—
540	C₂²⋊C₄	—	2	4	no	—
541	C₂²⋊C₄	—	2	4	no	—
542	C₂²⋊C₄	—	4	8	no	—
543	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
544	C₂²⋊C₄	C₂²×C₄×C₅₃	1	32	yes	—
545	C₄²	—	2	4	no	—
546	C₄²	—	2	4	no	—
547	C₄²	—	2	4	no	—
548	C₄²	—	2	8	no	—
549	C₄²	—	2	8	no	—
550	C₄²	—	2	4	no	—
551	C₄²	—	2	8	no	—
552	C₄²	—	4	4	no	—
553	C₄²	C₂²×C₄×C₅₃	1	8	yes	—
554	C₄²	C₂×D₄×C₅₃	1	4	yes	—
555	C₄²	C₂×D₄×C₅₃	1	4	yes	—
556	C₄²	C₂×D₄×C₅₃	1	4	yes	—
557	C₄²	C₂×D₄×C₅₃	1	4	yes	—
558	C₂³	—	2	2	no	—
559	C₂³	—	2	2	no	—
560	C₂³	—	2	2	no	—
561	C₂³	—	2	2	no	—
562	C₂³	—	2	4	no	—
563	C₂³	—	2	4	no	—
564	C₂³	C₂³×C₄×C₅₃	1	4	yes	—
565	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
566	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
567	C₂³	C₂×#41×C₅₃	1	2	yes	—
568	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
569	Q₈	—	2	16	no	—
570	Q₈	—	2	16	no	—
571	Q₈	—	2	8	no	—
572	D₄	—	2	16	no	—
573	D₄	—	2	8	no	—
574	C₂×C₄	—	2	16	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	4	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	16	no	—
585	C₂×C₄	—	2	8	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	4	no	—
589	C₂×C₄	—	2	4	no	—
590	C₂×C₄	—	2	4	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	4	no	—
593	C₂×C₄	—	2	8	no	—
594	C₂×C₄	—	2	4	no	—
595	C₂×C₄	—	4	4	no	—
596	C₂×C₄	—	4	8	no	—
597	C₂×C₄	—	4	8	no	—
598	C₂×C₄	—	4	8	no	—
599	C₂×C₄	C₄×D₄×C₅₃	1	16	yes	—
600	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
601	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
602	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
603	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
604	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
605	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
606	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
607	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
608	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
609	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
610	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
611	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
612	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
613	C₂×C₄	2₋¹⁺⁴×C₅₃	1	4	yes	—
614	C₂²	—	2	2	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	2	4	no	—
617	C₂²	—	2	4	no	—
618	C₂²	—	2	4	no	—
619	C₂²	—	2	8	no	—
620	C₂²	—	4	2	no	—
621	C₂²	—	4	2	no	—
622	C₂²	—	4	4	no	—
623	C₂²	C₂×C₄×D₄×C₅₃	1	4	yes	—
624	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
625	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
626	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
627	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	2	yes	—
628	C₂²	C₅₃×#585	1	2	yes	—
629	C₄	—	2	2	no	—
630	C₄	—	2	2	no	—
631	C₄	—	2	2	no	—
632	C₄	—	2	4	no	—
633	C₄	—	2	8	no	—
634	C₄	—	2	8	no	—
635	C₄	—	4	2	no	—
636	C₄	—	4	2	no	—
637	C₄	—	4	4	no	—
638	C₄	—	4	4	no	—
639	C₄	C₂×C₄×D₄×C₅₃	1	4	yes	—
640	C₄	C₂×C₄×D₄×C₅₃	1	4	yes	—
641	C₄	C₅₃×#536	1	4	yes	direct product factor
642	C₂	—	2	4	no	—
643	C₂	—	4	4	no	—
644	C₂	C₂²×C₄×D₄×C₅₃	1	1	yes	characteristic
645	C₂	C₄×2₋¹⁺⁴×C₅₃	1	2	yes	—
646	C₂	C₂×C₅₃×#536	1	1	yes	characteristic
647	C₂	C₅₃×#3449	1	1	yes	characteristic
648	C₂	C₅₃×#3461	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
socle	C₂³×C₅₃	C₂⁵
prime radical	C₂²×#41×C₅₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄×Q₈×C₅₃	C₂	semidirect
4. C₅₃×#2327	C₂	semidirect
5. C₅₃×#2326	C₂	semidirect
6. C₅₃×#2323	C₂	semidirect
7. C₅₃×#2322	C₂	semidirect
8. C₅₃×#2317	C₂	semidirect
10. C₅₃×#2310	C₂	semidirect
11. C₄×C₄⋊Q₈×C₅₃	C₂	semidirect
12. C₄×#124×C₅₃	C₂	semidirect
13. C₄×C₂²⋊Q₈×C₅₃	C₂	semidirect
14. C₄×#123×C₅₃	C₂	semidirect
15. C₄×#117×C₅₃	C₂	semidirect
18. C₅₃×#536	C₄	direct (N × H)
75. C₂²×Q₈×C₅₃	C₂×C₄	semidirect
89. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
90. #124×C₅₃	C₂×C₄	semidirect
91. #123×C₅₃	C₂×C₄	semidirect
92. C₂²⋊Q₈×C₅₃	C₂×C₄	semidirect
106. #117×C₅₃	C₂×C₄	semidirect
289. C₄×#536	C₅₃	direct (N × H)
317. C₄×C₅₃	#536	direct (N × H)
320. C₂²×C₄×Q₈	C₂×C₅₃	semidirect
321. #2327	C₂×C₅₃	semidirect
322. #2326	C₂×C₅₃	semidirect
323. #2323	C₂×C₅₃	semidirect
324. #2322	C₂×C₅₃	semidirect
325. #2317	C₂×C₅₃	semidirect
327. #2310	C₂×C₅₃	semidirect
328. C₄×C₄⋊Q₈	C₂×C₅₃	semidirect
329. C₄×#124	C₂×C₅₃	semidirect
330. C₄×C₂²⋊Q₈	C₂×C₅₃	semidirect
331. C₄×#123	C₂×C₅₃	semidirect
332. C₄×#117	C₂×C₅₃	semidirect
342. #536	C₄×C₅₃	direct (N × H)
394. C₅₃	C₄×#536	direct (N × H)
400. C₂²×Q₈	C₂×C₄×C₅₃	semidirect
414. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
415. #124	C₂×C₄×C₅₃	semidirect
416. #123	C₂×C₄×C₅₃	semidirect
417. C₂²⋊Q₈	C₂×C₄×C₅₃	semidirect
431. #117	C₂×C₄×C₅₃	semidirect
641. C₄	C₅₃×#536	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×#536	2
C₂×C₄×#41×C₅₃	2
C₂²×C₄×Q₈×C₅₃	2
C₅₃×#2327	2
C₅₃×#2326	2
C₅₃×#2323	2
C₅₃×#2322	2
C₅₃×#2317	2
C₅₃×#2311	2
C₅₃×#2310	2
C₄×C₄⋊Q₈×C₅₃	2
C₄×#124×C₅₃	2
C₄×C₂²⋊Q₈×C₅₃	2
C₄×#123×C₅₃	2
C₄×#117×C₅₃	2
C₄×#536	53

Subgroups — C4×C53×#536 #27241601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×#536 #27241601