Subgroups — C₅₃×#34195 #27241707

Order 13568.

2370 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2356	C₂	1	2	yes	maximal, complemented
3	C₅₃×#2353	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2349	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2347	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2343	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2339	C₂	1	2	yes	maximal
8	C₅₃×#2320	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2318	C₂	1	4	yes	maximal, complemented
10	C₅₃×#2283	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₅₃×#383	C₂	1	2	yes	maximal, complemented
14	C₅₃×#2275	C₂	1	2	yes	maximal, complemented
15	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×#41×C₅₃	C₄	1	16	yes	complemented
17	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
24	C₂×#117×C₅₃	C₂²	1	2	yes	—
25	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
27	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#391	C₂²	1	4	yes	—
29	C₅₃×#389	C₂²	1	4	yes	—
30	C₅₃×#388	C₂²	1	8	yes	—
31	C₅₃×#387	C₂²	1	8	yes	—
32	C₅₃×#387	C₂²	1	8	yes	—
33	C₅₃×#386	C₂²	1	8	yes	—
34	C₅₃×#385	C₂²	1	8	yes	—
35	C₅₃×#383	C₂²	1	1	yes	characteristic
36	C₅₃×#383	C₂²	1	2	yes	—
37	C₅₃×#383	C₂²	1	4	yes	—
38	C₅₃×#383	C₂²	1	4	yes	—
39	C₅₃×#383	C₂²	1	4	yes	—
40	C₅₃×#383	C₂²	1	1	yes	characteristic
41	C₅₃×#382	C₂²	1	4	yes	—
42	C₅₃×#382	C₂²	1	2	yes	—
43	C₅₃×#382	C₂²	1	2	yes	—
44	C₅₃×#381	C₂²	1	2	yes	—
45	C₅₃×#381	C₂²	1	4	yes	—
46	C₅₃×#381	C₂²	1	2	yes	—
47	C₅₃×#380	C₂²	1	2	yes	—
48	C₅₃×#380	C₂²	1	2	yes	—
49	C₅₃×#380	C₂²	1	4	yes	—
50	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
51	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
52	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
53	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
54	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
55	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
56	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
57	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
58	C₅₃×#376	C₂²	1	2	yes	—
59	C₅₃×#376	C₂²	1	2	yes	—
60	C₅₃×#376	C₂²	1	4	yes	—
61	C₂×#95×C₅₃	C₂²	1	4	yes	—
62	C₂×#95×C₅₃	C₂²	1	4	yes	—
63	C₂×#95×C₅₃	C₂²	1	4	yes	—
64	C₂×#41×C₅₃	—	2	8	no	—
65	C₂×#41×C₅₃	C₂×C₄	1	8	yes	—
66	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
68	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
70	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
72	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
73	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
74	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
76	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
77	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
78	C₄×D₄×C₅₃	—	2	16	no	—
79	C₄×D₄×C₅₃	C₂×C₄	1	8	yes	—
80	C₄×D₄×C₅₃	C₂×C₄	1	8	yes	—
81	C₄×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
82	C₄×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
83	#117×C₅₃	C₂×C₄	1	16	yes	complemented
84	#117×C₅₃	C₂×C₄	1	8	yes	complemented
85	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
88	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
93	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
96	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
97	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
98	C₂×C₄²×C₅₃	C₂×C₄	1	16	yes	complemented
99	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	complemented
100	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
101	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
102	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
103	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
106	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
108	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
110	#95×C₅₃	C₂³	1	2	yes	—
111	#95×C₅₃	C₂³	1	2	yes	—
112	#95×C₅₃	C₂³	1	2	yes	—
113	#95×C₅₃	C₂³	1	4	yes	—
114	#95×C₅₃	C₂³	1	4	yes	—
115	#95×C₅₃	C₂³	1	4	yes	—
116	#95×C₅₃	C₂³	1	8	yes	—
117	#95×C₅₃	C₂³	1	8	yes	—
118	#95×C₅₃	C₂³	1	8	yes	—
119	#95×C₅₃	C₂³	1	2	yes	—
120	#95×C₅₃	C₂³	1	2	yes	—
121	#95×C₅₃	C₂³	1	4	yes	—
122	#95×C₅₃	C₂³	1	4	yes	—
123	#95×C₅₃	C₂³	1	8	yes	—
124	#95×C₅₃	C₂³	1	2	yes	—
125	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
126	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
127	#41×C₅₃	—	2	32	no	—
128	C₂×Q₈×C₅₃	—	2	4	no	—
129	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
130	C₂×D₄×C₅₃	—	2	8	no	—
131	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
132	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
133	C₂×D₄×C₅₃	#41	1	8	yes	—
134	C₂²×C₄×C₅₃	—	2	8	no	—
135	C₂²×C₄×C₅₃	—	2	4	no	—
136	C₂²×C₄×C₅₃	—	2	16	no	—
137	C₂²×C₄×C₅₃	—	4	4	no	—
138	C₂²×C₄×C₅₃	—	4	2	no	—
139	C₂²×C₄×C₅₃	—	4	4	no	—
140	C₂²×C₄×C₅₃	—	4	2	no	—
141	C₂²×C₄×C₅₃	—	4	8	no	—
142	C₂²×C₄×C₅₃	—	4	8	no	—
143	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
144	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
145	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
146	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
147	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
148	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
149	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
151	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
153	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
154	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
155	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
158	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
161	C₄⋊C₄×C₅₃	—	2	8	no	—
162	C₄⋊C₄×C₅₃	—	4	4	no	—
163	C₄⋊C₄×C₅₃	—	4	8	no	—
164	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
165	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
166	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
167	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
168	C₂²⋊C₄×C₅₃	—	2	16	no	—
169	C₂²⋊C₄×C₅₃	—	4	4	no	—
170	C₂²⋊C₄×C₅₃	—	4	4	no	—
171	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
172	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
173	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
174	C₄²×C₅₃	—	2	8	no	—
175	C₄²×C₅₃	—	4	8	no	—
176	C₄²×C₅₃	—	4	4	no	—
177	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
178	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
179	C₄²×C₅₃	C₂²×C₄	1	8	yes	—
180	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
181	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
182	C₄²×C₅₃	C₂²×C₄	1	8	yes	—
183	C₄²×C₅₃	C₂²×C₄	1	8	yes	—
184	C₂³×C₅₃	—	2	4	no	—
185	C₂³×C₅₃	—	2	4	no	—
186	C₂³×C₅₃	—	2	2	no	—
187	C₂³×C₅₃	—	2	2	no	—
188	C₂³×C₅₃	—	2	2	no	—
189	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
190	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
191	C₂³×C₅₃	C₂×#41	1	8	yes	—
192	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
193	Q₈×C₅₃	—	2	8	no	—
194	D₄×C₅₃	—	2	16	no	—
195	D₄×C₅₃	#117	1	16	yes	—
196	C₂×C₄×C₅₃	—	2	16	no	—
197	C₂×C₄×C₅₃	—	2	2	no	—
198	C₂×C₄×C₅₃	—	2	4	no	—
199	C₂×C₄×C₅₃	—	2	2	no	—
200	C₂×C₄×C₅₃	—	2	2	no	—
201	C₂×C₄×C₅₃	—	2	4	no	—
202	C₂×C₄×C₅₃	—	2	2	no	—
203	C₂×C₄×C₅₃	—	2	2	no	—
204	C₂×C₄×C₅₃	—	2	4	no	—
205	C₂×C₄×C₅₃	—	2	8	no	—
206	C₂×C₄×C₅₃	—	2	4	no	—
207	C₂×C₄×C₅₃	—	2	8	no	—
208	C₂×C₄×C₅₃	—	2	4	no	—
209	C₂×C₄×C₅₃	—	2	4	no	—
210	C₂×C₄×C₅₃	—	2	4	no	—
211	C₂×C₄×C₅₃	—	2	8	no	—
212	C₂×C₄×C₅₃	—	2	8	no	—
213	C₂×C₄×C₅₃	—	2	4	no	—
214	C₂×C₄×C₅₃	—	2	4	no	—
215	C₂×C₄×C₅₃	—	4	8	no	—
216	C₂×C₄×C₅₃	—	4	8	no	—
217	C₂×C₄×C₅₃	—	4	4	no	—
218	C₂×C₄×C₅₃	—	4	4	no	—
219	C₂×C₄×C₅₃	—	8	16	no	—
220	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
221	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
222	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
223	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
224	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
225	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
226	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
227	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
228	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
229	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
230	#34195	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
231	C₂²×C₅₃	—	2	4	no	—
232	C₂²×C₅₃	—	2	4	no	—
233	C₂²×C₅₃	—	2	16	no	—
234	C₂²×C₅₃	—	2	2	no	—
235	C₂²×C₅₃	—	2	2	no	—
236	C₂²×C₅₃	—	2	2	no	—
237	C₂²×C₅₃	—	4	4	no	—
238	C₂²×C₅₃	C₂×#117	1	8	yes	—
239	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
240	C₂²×C₅₃	C₂²×#41	1	2	yes	—
241	C₂²×C₅₃	#585	1	2	yes	—
242	C₂²×C₅₃	#585	1	2	yes	—
243	C₄×C₅₃	—	2	2	no	—
244	C₄×C₅₃	—	2	4	no	—
245	C₄×C₅₃	—	2	2	no	—
246	C₄×C₅₃	—	2	2	no	—
247	C₄×C₅₃	—	2	4	no	—
248	C₄×C₅₃	—	2	2	no	—
249	C₄×C₅₃	—	2	4	no	—
250	C₄×C₅₃	—	2	2	no	—
251	C₄×C₅₃	—	4	4	no	—
252	C₄×C₅₃	—	4	8	no	—
253	C₄×C₅₃	—	4	4	no	—
254	C₄×C₅₃	C₂×#117	1	4	yes	—
255	C₂×C₄×#41	C₂×C₅₃	1	1	yes	characteristic
256	#2356	C₂×C₅₃	1	2	yes	complemented
257	#2353	C₂×C₅₃	1	4	yes	complemented
258	#2349	C₂×C₅₃	1	2	yes	complemented
259	#2347	C₂×C₅₃	1	2	yes	complemented
260	#2343	C₂×C₅₃	1	4	yes	complemented
261	#2339	C₂×C₅₃	1	2	yes	—
262	#2320	C₂×C₅₃	1	2	yes	complemented
263	#2318	C₂×C₅₃	1	4	yes	complemented
264	#2283	C₂×C₅₃	1	1	yes	characteristic, complemented
265	#2283	C₂×C₅₃	1	1	yes	characteristic, complemented
266	#2283	C₂×C₅₃	1	2	yes	complemented
267	C₂×#383	C₂×C₅₃	1	2	yes	complemented
268	#2275	C₂×C₅₃	1	2	yes	complemented
269	C₂×C₅₃	—	2	8	no	—
270	C₂×C₅₃	—	4	2	no	—
271	C₂×C₅₃	C₂²×#117	1	1	yes	characteristic
272	C₂×C₅₃	#3428	1	1	yes	characteristic
273	C₂×C₅₃	#3428	1	1	yes	characteristic
274	C₂×C₅₃	#3521	1	4	yes	—
275	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
276	C₄×#41	C₄×C₅₃	1	16	yes	complemented
277	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
278	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
279	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
280	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
281	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
282	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
283	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
284	C₂×#117	C₂²×C₅₃	1	2	yes	—
285	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
286	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
287	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
288	#391	C₂²×C₅₃	1	4	yes	—
289	#389	C₂²×C₅₃	1	4	yes	—
290	#388	C₂²×C₅₃	1	8	yes	—
291	#387	C₂²×C₅₃	1	8	yes	—
292	#387	C₂²×C₅₃	1	8	yes	—
293	#386	C₂²×C₅₃	1	8	yes	—
294	#385	C₂²×C₅₃	1	8	yes	—
295	#383	C₂²×C₅₃	1	4	yes	—
296	#383	C₂²×C₅₃	1	4	yes	—
297	#383	C₂²×C₅₃	1	4	yes	—
298	#383	C₂²×C₅₃	1	1	yes	characteristic
299	#383	C₂²×C₅₃	1	2	yes	—
300	#383	C₂²×C₅₃	1	1	yes	characteristic
301	#382	C₂²×C₅₃	1	4	yes	—
302	#382	C₂²×C₅₃	1	2	yes	—
303	#382	C₂²×C₅₃	1	2	yes	—
304	#381	C₂²×C₅₃	1	2	yes	—
305	#381	C₂²×C₅₃	1	4	yes	—
306	#381	C₂²×C₅₃	1	2	yes	—
307	#380	C₂²×C₅₃	1	2	yes	—
308	#380	C₂²×C₅₃	1	2	yes	—
309	#380	C₂²×C₅₃	1	4	yes	—
310	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
311	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
312	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
313	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
314	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
315	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
316	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
317	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
318	#376	C₂²×C₅₃	1	2	yes	—
319	#376	C₂²×C₅₃	1	2	yes	—
320	#376	C₂²×C₅₃	1	4	yes	—
321	C₂×#95	C₂²×C₅₃	1	4	yes	—
322	C₂×#95	C₂²×C₅₃	1	4	yes	—
323	C₂×#95	C₂²×C₅₃	1	4	yes	—
324	C₅₃	#34195	1	1	yes	characteristic, direct product factor, Sylow
325	C₂×#41	—	2	8	no	—
326	C₂×#41	C₂×C₄×C₅₃	1	8	yes	—
327	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
328	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
329	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
330	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
331	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
332	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
333	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
334	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
335	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
336	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
337	C₄×Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
338	C₄×Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
339	C₄×D₄	—	2	16	no	—
340	C₄×D₄	C₂×C₄×C₅₃	1	8	yes	—
341	C₄×D₄	C₂×C₄×C₅₃	1	8	yes	—
342	C₄×D₄	C₂×C₄×C₅₃	1	16	yes	complemented
343	C₄×D₄	C₂×C₄×C₅₃	1	16	yes	complemented
344	#117	C₂×C₄×C₅₃	1	16	yes	complemented
345	#117	C₂×C₄×C₅₃	1	8	yes	complemented
346	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
347	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
348	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
349	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
350	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
351	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
352	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
353	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
359	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	complemented
360	C₂×C₄²	C₂×C₄×C₅₃	1	16	yes	complemented
361	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
362	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
363	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
364	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
365	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
366	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
367	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
368	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
369	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
370	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
371	#95	C₂³×C₅₃	1	2	yes	—
372	#95	C₂³×C₅₃	1	2	yes	—
373	#95	C₂³×C₅₃	1	2	yes	—
374	#95	C₂³×C₅₃	1	4	yes	—
375	#95	C₂³×C₅₃	1	4	yes	—
376	#95	C₂³×C₅₃	1	4	yes	—
377	#95	C₂³×C₅₃	1	8	yes	—
378	#95	C₂³×C₅₃	1	8	yes	—
379	#95	C₂³×C₅₃	1	8	yes	—
380	#95	C₂³×C₅₃	1	2	yes	—
381	#95	C₂³×C₅₃	1	2	yes	—
382	#95	C₂³×C₅₃	1	4	yes	—
383	#95	C₂³×C₅₃	1	4	yes	—
384	#95	C₂³×C₅₃	1	8	yes	—
385	#95	C₂³×C₅₃	1	2	yes	—
386	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
387	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
388	#41	—	2	32	no	—
389	C₂×Q₈	—	2	4	no	—
390	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
391	C₂×D₄	—	2	8	no	—
392	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
393	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
394	C₂×D₄	#41×C₅₃	1	8	yes	—
395	C₂²×C₄	—	2	8	no	—
396	C₂²×C₄	—	2	4	no	—
397	C₂²×C₄	—	2	16	no	—
398	C₂²×C₄	—	4	4	no	—
399	C₂²×C₄	—	4	2	no	—
400	C₂²×C₄	—	4	4	no	—
401	C₂²×C₄	—	4	2	no	—
402	C₂²×C₄	—	4	8	no	—
403	C₂²×C₄	—	4	8	no	—
404	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
405	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
406	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
407	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
408	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
409	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
410	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
411	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
412	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
413	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
414	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
415	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
416	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
417	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
418	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
419	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
420	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
421	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
422	C₄⋊C₄	—	2	8	no	—
423	C₄⋊C₄	—	4	4	no	—
424	C₄⋊C₄	—	4	8	no	—
425	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
426	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
427	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
428	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
429	C₂²⋊C₄	—	2	16	no	—
430	C₂²⋊C₄	—	4	4	no	—
431	C₂²⋊C₄	—	4	4	no	—
432	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
433	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
434	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
435	C₄²	—	2	8	no	—
436	C₄²	—	4	8	no	—
437	C₄²	—	4	4	no	—
438	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
439	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
440	C₄²	C₂²×C₄×C₅₃	1	8	yes	—
441	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
442	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
443	C₄²	C₂²×C₄×C₅₃	1	8	yes	—
444	C₄²	C₂²×C₄×C₅₃	1	8	yes	—
445	C₂³	—	2	4	no	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	2	no	—
450	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
451	C₂³	C₂³×C₄×C₅₃	1	4	yes	—
452	C₂³	C₂×#41×C₅₃	1	8	yes	—
453	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
454	Q₈	—	2	8	no	—
455	D₄	—	2	16	no	—
456	D₄	#117×C₅₃	1	16	yes	—
457	C₂×C₄	—	2	16	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	8	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	8	no	—
472	C₂×C₄	—	2	8	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	4	8	no	—
477	C₂×C₄	—	4	8	no	—
478	C₂×C₄	—	4	4	no	—
479	C₂×C₄	—	4	4	no	—
480	C₂×C₄	—	8	16	no	—
481	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
482	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
483	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
484	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
485	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
486	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
487	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
488	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
489	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
490	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
491	C₂²	—	2	16	no	—
492	C₂²	—	2	4	no	—
493	C₂²	—	2	4	no	—
494	C₂²	—	2	2	no	—
495	C₂²	—	2	2	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	4	4	no	—
498	C₂²	C₂×#117×C₅₃	1	8	yes	—
499	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
500	C₂²	C₂²×#41×C₅₃	1	2	yes	—
501	C₂²	C₅₃×#585	1	2	yes	—
502	C₂²	C₅₃×#585	1	2	yes	—
503	C₄	—	2	2	no	—
504	C₄	—	2	4	no	—
505	C₄	—	2	2	no	—
506	C₄	—	2	2	no	—
507	C₄	—	2	4	no	—
508	C₄	—	2	2	no	—
509	C₄	—	2	4	no	—
510	C₄	—	2	2	no	—
511	C₄	—	4	4	no	—
512	C₄	—	4	8	no	—
513	C₄	—	4	4	no	—
514	C₄	C₂×#117×C₅₃	1	4	yes	—
515	C₂	—	2	8	no	—
516	C₂	—	4	2	no	—
517	C₂	C₂²×#117×C₅₃	1	1	yes	characteristic
518	C₂	C₅₃×#3428	1	1	yes	characteristic
519	C₂	C₅₃×#3428	1	1	yes	characteristic
520	C₂	C₅₃×#3521	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2356	C₂	semidirect
3. C₅₃×#2353	C₂	semidirect
4. C₅₃×#2349	C₂	semidirect
5. C₅₃×#2347	C₂	semidirect
6. C₅₃×#2343	C₂	semidirect
8. C₅₃×#2320	C₂	semidirect
9. C₅₃×#2318	C₂	semidirect
10. C₅₃×#2283	C₂	semidirect
11. C₅₃×#2283	C₂	semidirect
12. C₅₃×#2283	C₂	semidirect
13. C₂×C₅₃×#383	C₂	semidirect
14. C₅₃×#2275	C₂	semidirect
16. C₄×#41×C₅₃	C₄	semidirect
76. C₄×Q₈×C₅₃	C₂×C₄	semidirect
77. C₄×Q₈×C₅₃	C₂×C₄	semidirect
81. C₄×D₄×C₅₃	C₂×C₄	semidirect
82. C₄×D₄×C₅₃	C₂×C₄	semidirect
83. #117×C₅₃	C₂×C₄	semidirect
84. #117×C₅₃	C₂×C₄	semidirect
98. C₂×C₄²×C₅₃	C₂×C₄	semidirect
99. C₂×C₄²×C₅₃	C₂×C₄	semidirect
230. #34195	C₅₃	direct (N × H)
256. #2356	C₂×C₅₃	semidirect
257. #2353	C₂×C₅₃	semidirect
258. #2349	C₂×C₅₃	semidirect
259. #2347	C₂×C₅₃	semidirect
260. #2343	C₂×C₅₃	semidirect
262. #2320	C₂×C₅₃	semidirect
263. #2318	C₂×C₅₃	semidirect
264. #2283	C₂×C₅₃	semidirect
265. #2283	C₂×C₅₃	semidirect
266. #2283	C₂×C₅₃	semidirect
267. C₂×#383	C₂×C₅₃	semidirect
268. #2275	C₂×C₅₃	semidirect
276. C₄×#41	C₄×C₅₃	semidirect
324. C₅₃	#34195	direct (N × H)
337. C₄×Q₈	C₂×C₄×C₅₃	semidirect
338. C₄×Q₈	C₂×C₄×C₅₃	semidirect
342. C₄×D₄	C₂×C₄×C₅₃	semidirect
343. C₄×D₄	C₂×C₄×C₅₃	semidirect
344. #117	C₂×C₄×C₅₃	semidirect
345. #117	C₂×C₄×C₅₃	semidirect
359. C₂×C₄²	C₂×C₄×C₅₃	semidirect
360. C₂×C₄²	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₅₃	2
C₅₃×#2356	2
C₅₃×#2353	2
C₅₃×#2349	2
C₅₃×#2347	2
C₅₃×#2343	2
C₅₃×#2339	2
C₅₃×#2320	2
C₅₃×#2318	2
C₅₃×#2283	2
C₂×C₅₃×#383	2
C₅₃×#2275	2
#34195	53

Subgroups — C53×#34195 #27241707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34195 #27241707