Subgroups — C₅₃×#21332 #27228844

Order 13568.

1202 conjugacy classes of subgroups.

The subgroup lattice has 386 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
2	C₅₃×#2285	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1996	C₂	1	8	yes	maximal
5	C₅₃×#1740	C₂	1	4	yes	maximal
6	C₅₃×#533	—	2	1	no	—
7	C₅₃×#533	—	2	1	no	—
8	C₅₃×#533	—	2	2	no	—
9	C₅₃×#533	—	2	2	no	—
10	C₅₃×#533	—	2	1	no	—
11	C₅₃×#533	—	2	1	no	—
12	C₂×C₄⋊Q₈×C₅₃	C₄	1	2	yes	complemented
13	C₂×#125×C₅₃	C₄	1	2	yes	complemented
14	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	—
15	C₂×C₄×Q₈×C₅₃	C₄	1	4	yes	complemented
16	C₂×#117×C₅₃	C₂²	1	2	yes	—
17	C₂×#117×C₅₃	C₂²	1	2	yes	—
18	C₂×#117×C₅₃	C₂²	1	2	yes	—
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#384	—	2	1	no	—
21	C₅₃×#382	—	2	1	no	—
22	C₅₃×#380	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
26	C₅₃×#376	—	2	1	no	—
27	C₄³×C₅₃	—	2	1	no	—
28	C₅₃×#342	C₂²	1	16	yes	—
29	C₂²×Q₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
31	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₄⋊Q₈×C₅₃	—	2	2	no	—
33	C₄⋊Q₈×C₅₃	—	2	2	no	—
34	C₄⋊Q₈×C₅₃	—	2	2	no	—
35	C₄⋊Q₈×C₅₃	—	2	2	no	—
36	#125×C₅₃	—	2	2	no	—
37	#125×C₅₃	—	2	2	no	—
38	#125×C₅₃	—	2	4	no	—
39	C₂²⋊Q₈×C₅₃	—	2	4	no	—
40	C₂²⋊Q₈×C₅₃	—	2	4	no	—
41	C₂²⋊Q₈×C₅₃	—	2	4	no	—
42	C₂²⋊Q₈×C₅₃	—	2	4	no	—
43	C₄×Q₈×C₅₃	—	2	4	no	—
44	C₄×Q₈×C₅₃	—	2	4	no	—
45	C₄×Q₈×C₅₃	—	2	4	no	—
46	C₄×Q₈×C₅₃	—	2	4	no	—
47	#117×C₅₃	—	2	2	no	—
48	#117×C₅₃	—	2	2	no	—
49	#117×C₅₃	—	2	2	no	—
50	#117×C₅₃	—	2	2	no	—
51	#117×C₅₃	—	2	4	no	—
52	#117×C₅₃	—	2	8	no	—
53	#117×C₅₃	Q₈	1	8	yes	complemented
54	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
57	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
60	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
61	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
62	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
63	C₂×C₄²×C₅₃	—	2	2	no	—
64	C₂×C₄²×C₅₃	—	2	2	no	—
65	C₂×C₄²×C₅₃	—	2	4	no	—
66	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
67	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
68	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	D₄	1	2	yes	—
73	C₂×C₄²×C₅₃	D₄	1	2	yes	—
74	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
76	#95×C₅₃	—	2	4	no	—
77	#95×C₅₃	—	2	4	no	—
78	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	—	2	4	no	—
80	C₂×Q₈×C₅₃	—	2	4	no	—
81	C₂×Q₈×C₅₃	—	4	8	no	—
82	C₂²×C₄×C₅₃	—	2	4	no	—
83	C₂²×C₄×C₅₃	—	2	4	no	—
84	C₂²×C₄×C₅₃	—	4	4	no	—
85	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
87	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
99	C₂²×C₄×C₅₃	#41	1	4	yes	—
100	C₄⋊C₄×C₅₃	—	2	4	no	—
101	C₄⋊C₄×C₅₃	—	2	4	no	—
102	C₄⋊C₄×C₅₃	—	2	8	no	—
103	C₄⋊C₄×C₅₃	—	2	8	no	—
104	C₄⋊C₄×C₅₃	—	2	4	no	—
105	C₄⋊C₄×C₅₃	—	2	4	no	—
106	C₄⋊C₄×C₅₃	—	2	8	no	—
107	C₄⋊C₄×C₅₃	—	4	8	no	—
108	C₄⋊C₄×C₅₃	—	4	4	no	—
109	C₂²⋊C₄×C₅₃	—	2	4	no	—
110	C₂²⋊C₄×C₅₃	—	2	4	no	—
111	C₂²⋊C₄×C₅₃	—	2	4	no	—
112	C₂²⋊C₄×C₅₃	—	2	8	no	—
113	C₂²⋊C₄×C₅₃	C₂×Q₈	1	8	yes	—
114	C₄²×C₅₃	—	2	4	no	—
115	C₄²×C₅₃	—	2	4	no	—
116	C₄²×C₅₃	—	2	8	no	—
117	C₄²×C₅₃	—	4	8	no	—
118	C₄²×C₅₃	—	4	4	no	—
119	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
120	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
121	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
122	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
123	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
124	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
125	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
126	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
127	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
128	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
129	C₂³×C₅₃	—	4	2	no	—
130	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
131	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
132	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
133	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
134	C₂³×C₅₃	C₂×#41	1	2	yes	—
135	Q₈×C₅₃	—	4	8	no	—
136	Q₈×C₅₃	—	4	8	no	—
137	C₂×C₄×C₅₃	—	2	4	no	—
138	C₂×C₄×C₅₃	—	2	4	no	—
139	C₂×C₄×C₅₃	—	2	4	no	—
140	C₂×C₄×C₅₃	—	2	8	no	—
141	C₂×C₄×C₅₃	—	2	8	no	—
142	C₂×C₄×C₅₃	—	4	2	no	—
143	C₂×C₄×C₅₃	—	4	2	no	—
144	C₂×C₄×C₅₃	—	4	2	no	—
145	C₂×C₄×C₅₃	—	4	8	no	—
146	C₂×C₄×C₅₃	—	4	8	no	—
147	C₂×C₄×C₅₃	—	4	8	no	—
148	C₂×C₄×C₅₃	—	4	4	no	—
149	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
150	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
151	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
156	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
158	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
159	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
160	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
161	C₂×C₄×C₅₃	#124	1	2	yes	—
162	C₂×C₄×C₅₃	#124	1	2	yes	—
163	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
164	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
165	#21332	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂²×C₅₃	—	2	2	no	—
167	C₂²×C₅₃	—	2	2	no	—
168	C₂²×C₅₃	—	2	2	no	—
169	C₂²×C₅₃	—	4	2	no	—
170	C₂²×C₅₃	—	4	2	no	—
171	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
172	C₂²×C₅₃	C₂×C₄×Q₈	1	2	yes	—
173	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	2	yes	—
174	C₂²×C₅₃	C₂×#124	1	1	yes	characteristic
175	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
176	C₄×C₅₃	—	2	2	no	—
177	C₄×C₅₃	—	2	2	no	—
178	C₄×C₅₃	—	2	2	no	—
179	C₄×C₅₃	—	4	8	no	—
180	C₄×C₅₃	—	4	8	no	—
181	C₄×C₅₃	—	8	4	no	—
182	C₄×C₅₃	#391	1	2	yes	—
183	C₄×C₅₃	#391	1	2	yes	—
184	C₄×C₅₃	#391	1	2	yes	—
185	C₄×C₅₃	#391	1	2	yes	—
186	C₄×C₅₃	#410	1	4	yes	—
187	C₂×#533	C₂×C₅₃	1	1	yes	characteristic
188	#2285	C₂×C₅₃	1	1	yes	characteristic
189	C₄×#117	C₂×C₅₃	1	1	yes	characteristic
190	#1996	C₂×C₅₃	1	8	yes	—
191	#1740	C₂×C₅₃	1	4	yes	—
192	C₂×C₅₃	—	2	2	no	—
193	C₂×C₅₃	—	2	1	no	—
194	C₂×C₅₃	—	2	2	no	—
195	C₂×C₅₃	—	2	1	no	—
196	C₂×C₅₃	C₂×#391	1	1	yes	characteristic
197	C₂×C₅₃	C₂×#410	1	2	yes	—
198	#533	—	2	1	no	—
199	#533	—	2	1	no	—
200	#533	—	2	2	no	—
201	#533	—	2	2	no	—
202	#533	—	2	1	no	—
203	#533	—	2	1	no	—
204	C₂×C₄⋊Q₈	C₄×C₅₃	1	2	yes	complemented
205	C₂×#125	C₄×C₅₃	1	2	yes	complemented
206	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	—
207	C₂×C₄×Q₈	C₄×C₅₃	1	4	yes	complemented
208	C₂×#117	C₂²×C₅₃	1	2	yes	—
209	C₂×#117	C₂²×C₅₃	1	2	yes	—
210	C₂×#117	C₂²×C₅₃	1	2	yes	—
211	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
212	#384	—	2	1	no	—
213	#382	—	2	1	no	—
214	#380	C₂²×C₅₃	1	4	yes	—
215	C₄×C₄⋊C₄	—	2	2	no	—
216	C₄×C₄⋊C₄	—	2	2	no	—
217	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
218	#376	—	2	1	no	—
219	C₄³	—	2	1	no	—
220	#342	C₂²×C₅₃	1	16	yes	—
221	C₅₃	#21332	1	1	yes	characteristic, direct product factor, Sylow
222	C₂²×Q₈	C₂×C₄×C₅₃	1	4	yes	—
223	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
224	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
225	C₄⋊Q₈	—	2	2	no	—
226	C₄⋊Q₈	—	2	2	no	—
227	C₄⋊Q₈	—	2	2	no	—
228	C₄⋊Q₈	—	2	2	no	—
229	#125	—	2	2	no	—
230	#125	—	2	2	no	—
231	#125	—	2	4	no	—
232	C₂²⋊Q₈	—	2	4	no	—
233	C₂²⋊Q₈	—	2	4	no	—
234	C₂²⋊Q₈	—	2	4	no	—
235	C₂²⋊Q₈	—	2	4	no	—
236	C₄×Q₈	—	2	4	no	—
237	C₄×Q₈	—	2	4	no	—
238	C₄×Q₈	—	2	4	no	—
239	C₄×Q₈	—	2	4	no	—
240	#117	—	2	2	no	—
241	#117	—	2	2	no	—
242	#117	—	2	2	no	—
243	#117	—	2	2	no	—
244	#117	—	2	4	no	—
245	#117	—	2	8	no	—
246	#117	Q₈×C₅₃	1	8	yes	complemented
247	C₂×C₄⋊C₄	—	2	2	no	—
248	C₂×C₄⋊C₄	—	2	2	no	—
249	C₂×C₄⋊C₄	—	2	4	no	—
250	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
251	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
252	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
253	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
254	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
255	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
256	C₂×C₄²	—	2	2	no	—
257	C₂×C₄²	—	2	2	no	—
258	C₂×C₄²	—	2	4	no	—
259	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
260	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
261	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
262	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
263	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
264	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
265	C₂×C₄²	D₄×C₅₃	1	2	yes	—
266	C₂×C₄²	D₄×C₅₃	1	2	yes	—
267	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
268	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
269	#95	—	2	4	no	—
270	#95	—	2	4	no	—
271	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
272	C₂×Q₈	—	2	4	no	—
273	C₂×Q₈	—	2	4	no	—
274	C₂×Q₈	—	4	8	no	—
275	C₂²×C₄	—	2	4	no	—
276	C₂²×C₄	—	2	4	no	—
277	C₂²×C₄	—	4	4	no	—
278	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
279	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
280	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
281	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
282	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
284	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
285	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
286	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
287	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
289	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
290	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
291	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
292	C₂²×C₄	#41×C₅₃	1	4	yes	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	—	2	4	no	—
295	C₄⋊C₄	—	2	8	no	—
296	C₄⋊C₄	—	2	8	no	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	4	no	—
299	C₄⋊C₄	—	2	8	no	—
300	C₄⋊C₄	—	4	8	no	—
301	C₄⋊C₄	—	4	4	no	—
302	C₂²⋊C₄	—	2	4	no	—
303	C₂²⋊C₄	—	2	4	no	—
304	C₂²⋊C₄	—	2	4	no	—
305	C₂²⋊C₄	—	2	8	no	—
306	C₂²⋊C₄	C₂×Q₈×C₅₃	1	8	yes	—
307	C₄²	—	2	4	no	—
308	C₄²	—	2	4	no	—
309	C₄²	—	2	8	no	—
310	C₄²	—	4	8	no	—
311	C₄²	—	4	4	no	—
312	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
313	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
314	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
315	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
316	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
317	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
318	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
319	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
320	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
321	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
322	C₂³	—	4	2	no	—
323	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
324	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
325	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
326	C₂³	C₂²×Q₈×C₅₃	1	2	yes	—
327	C₂³	C₂×#41×C₅₃	1	2	yes	—
328	Q₈	—	4	8	no	—
329	Q₈	—	4	8	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	4	no	—
333	C₂×C₄	—	2	8	no	—
334	C₂×C₄	—	2	8	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	8	no	—
339	C₂×C₄	—	4	8	no	—
340	C₂×C₄	—	4	8	no	—
341	C₂×C₄	—	4	4	no	—
342	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
343	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
344	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
345	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
346	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
347	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
348	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
349	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
350	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
351	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
352	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
353	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
354	C₂×C₄	#124×C₅₃	1	2	yes	—
355	C₂×C₄	#124×C₅₃	1	2	yes	—
356	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
357	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
358	C₂²	—	2	2	no	—
359	C₂²	—	2	2	no	—
360	C₂²	—	2	2	no	—
361	C₂²	—	4	2	no	—
362	C₂²	—	4	2	no	—
363	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
364	C₂²	C₂×C₄×Q₈×C₅₃	1	2	yes	—
365	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	2	yes	—
366	C₂²	C₂×#124×C₅₃	1	1	yes	characteristic
367	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
368	C₄	—	2	2	no	—
369	C₄	—	2	2	no	—
370	C₄	—	2	2	no	—
371	C₄	—	4	8	no	—
372	C₄	—	4	8	no	—
373	C₄	—	8	4	no	—
374	C₄	C₅₃×#391	1	2	yes	—
375	C₄	C₅₃×#391	1	2	yes	—
376	C₄	C₅₃×#391	1	2	yes	—
377	C₄	C₅₃×#391	1	2	yes	—
378	C₄	C₅₃×#410	1	4	yes	—
379	C₂	—	2	2	no	—
380	C₂	—	2	1	no	—
381	C₂	—	2	2	no	—
382	C₂	—	2	1	no	—
383	C₂	C₂×C₅₃×#391	1	1	yes	characteristic
384	C₂	C₂×C₅₃×#410	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂⁴	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂⁴
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂×C₄⋊Q₈×C₅₃	C₄	semidirect
13. C₂×#125×C₅₃	C₄	semidirect
15. C₂×C₄×Q₈×C₅₃	C₄	semidirect
53. #117×C₅₃	Q₈	semidirect
165. #21332	C₅₃	direct (N × H)
204. C₂×C₄⋊Q₈	C₄×C₅₃	semidirect
205. C₂×#125	C₄×C₅₃	semidirect
207. C₂×C₄×Q₈	C₄×C₅₃	semidirect
221. C₅₃	#21332	direct (N × H)
246. #117	Q₈×C₅₃	semidirect

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
C₅₃×#2285	2
C₄×#117×C₅₃	2
C₅₃×#1996	2
C₅₃×#1740	2
#21332	53

Subgroups — C53×#21332 #27228844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21332 #27228844