Subgroups — C₈×C₄⋊C₄×D₅₃ #27382018

Order 13568.

1438 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₈×D₅₃	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊C₄×D₅₃	C₂	1	1	yes	maximal, characteristic
5	#20274580	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#11719919	C₂	1	2	yes	maximal, complemented
8	C₈×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274576	C₂	1	2	yes	maximal, complemented
10	#336×D₅₃	C₂	1	2	yes	maximal
11	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
13	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×D₅₃	C₄	1	2	yes	—
16	C₂×C₄²×D₅₃	C₂²	1	2	yes	—
17	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
19	C₂×#13133146	C₂²	1	2	yes	—
20	C₄⋊C₈×D₅₃	C₄	1	8	yes	complemented
21	C₂×C₄×#88322	C₂²	1	2	yes	—
22	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
24	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
25	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
27	C₄×C₈×D₅₃	—	2	8	no	—
28	C₄×C₈×D₅₃	C₄	1	8	yes	complemented
29	C₄⋊C₄×Dic₅₃	C₄	1	2	yes	—
30	C₄×#11719919	C₂²	1	2	yes	—
31	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
32	#16810474	C₄	1	2	yes	—
33	#95×D₅₃	C₄	1	2	yes	—
34	#16810378	C₂²	1	2	yes	—
35	#16810377	C₂²	1	2	yes	—
36	#16810377	C₂²	1	2	yes	—
37	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#336	C₂²	1	2	yes	—
42	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×D₅₃	C₂×C₄	1	4	yes	—
44	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
46	C₂²×#88322	C₂³	1	1	yes	characteristic
47	C₂²×#88322	C₂³	1	2	yes	—
48	C₂²×#88322	C₂³	1	1	yes	characteristic
49	C₂×C₈×D₅₃	—	2	4	no	—
50	C₂×C₈×D₅₃	—	2	2	no	—
51	C₂×C₈×D₅₃	—	2	4	no	—
52	C₂×C₈×D₅₃	—	2	2	no	—
53	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
54	C₂×C₈×D₅₃	C₂×C₄	1	8	yes	—
55	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
56	C₂×C₈×D₅₃	D₄	1	4	yes	—
57	C₂×C₈×D₅₃	Q₈	1	4	yes	—
58	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
59	C₂×#11719919	C₂×C₄	1	4	yes	—
60	C₄⋊C₄×D₅₃	C₈	1	16	yes	direct product factor
61	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
62	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
65	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
66	C₄²×D₅₃	—	2	4	no	—
67	C₄²×D₅₃	—	2	4	no	—
68	C₄²×D₅₃	C₂×C₄	1	4	yes	—
69	C₄²×D₅₃	C₂×C₄	1	4	yes	—
70	C₄⋊#88322	C₂×C₄	1	4	yes	complemented
71	#13133146	C₂×C₄	1	8	yes	complemented
72	C₄×#88322	—	2	4	no	—
73	C₄×#88322	C₂×C₄	1	4	yes	complemented
74	C₈×Dic₅₃	—	2	4	no	—
75	C₈×Dic₅₃	—	2	4	no	—
76	C₈×Dic₅₃	C₂×C₄	1	4	yes	complemented
77	C₈×Dic₅₃	C₂×C₄	1	4	yes	complemented
78	#13133124	C₂×C₄	1	2	yes	—
79	#13133124	C₂×C₄	1	4	yes	—
80	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
82	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
84	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
85	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
87	C₄×C₈×C₅₃	—	2	4	no	—
88	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
89	#95×C₅₃	C₂×C₄	1	2	yes	—
90	C₂³×D₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×Dic₅₃	C₂²×C₄	1	4	yes	—
97	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×D₅₃	—	2	8	no	—
100	C₂×C₄×D₅₃	—	2	4	no	—
101	C₂×C₄×D₅₃	—	2	8	no	—
102	C₂×C₄×D₅₃	C₄²	1	4	yes	—
103	C₂×C₄×D₅₃	C₄²	1	4	yes	—
104	C₂×C₄×D₅₃	C₂×C₈	1	16	yes	—
105	C₂×C₄×D₅₃	C₂×C₈	1	8	yes	—
106	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
107	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
108	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
109	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
110	C₂×C₄×D₅₃	#41	1	2	yes	—
111	C₂×C₄×D₅₃	#41	1	2	yes	—
112	C₂×C₈×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	—	2	1	no	—
114	C₂×C₈×C₅₃	—	2	1	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
118	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
120	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
121	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	complemented
122	C₄²×C₅₃	—	2	2	no	—
123	C₄²×C₅₃	—	2	2	no	—
124	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
125	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
126	C₄⋊Dic₅₃	C₂×C₈	1	8	yes	complemented
127	#11719919	C₂×C₈	1	16	yes	complemented
128	C₄×Dic₅₃	—	2	2	no	—
129	C₄×Dic₅₃	—	2	2	no	—
130	C₄×Dic₅₃	—	2	8	no	—
131	C₄×Dic₅₃	C₄²	1	4	yes	—
132	C₄×Dic₅₃	C₄²	1	4	yes	—
133	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
134	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
135	C₂×#88322	—	2	2	no	—
136	C₂×#88322	—	2	1	no	—
137	C₂×#88322	—	2	1	no	—
138	C₂×#88322	—	2	2	no	—
139	C₂×#88322	C₂²×C₄	1	2	yes	—
140	C₂×#88322	C₂²×C₄	1	4	yes	—
141	C₂×#88322	C₂²×C₄	1	2	yes	—
142	C₂×#88322	C₂×D₄	1	2	yes	—
143	C₂×#88322	C₂×Q₈	1	2	yes	—
144	C₈×D₅₃	—	2	8	no	—
145	C₈×D₅₃	—	2	16	no	—
146	C₈×D₅₃	C₄⋊C₄	1	16	yes	direct product factor
147	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
148	C₂²×D₅₃	C₂×C₄²	1	2	yes	—
149	C₂²×D₅₃	C₄×D₄	1	2	yes	—
150	C₂²×D₅₃	C₄×Q₈	1	2	yes	—
151	C₂²×D₅₃	C₂²×C₈	1	4	yes	—
152	C₂²×D₅₃	#131	1	4	yes	—
153	C₂×C₄×C₅₃	—	2	4	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	2	4	no	—
156	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
157	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
158	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
159	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
160	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
166	C₂×Dic₅₃	—	2	4	no	—
167	C₂×Dic₅₃	—	2	2	no	—
168	C₂×Dic₅₃	—	2	4	no	—
169	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
170	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
171	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
172	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
173	C₂×Dic₅₃	C₄×Q₈	1	2	yes	—
174	C₂×Dic₅₃	C₂²×C₈	1	8	yes	—
175	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
176	C₄×D₅₃	—	2	8	no	—
177	C₄×D₅₃	—	2	32	no	—
178	C₄×D₅₃	C₄×C₈	1	16	yes	complemented
179	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
180	C₄×D₅₃	#117	1	4	yes	—
181	C₄×D₅₃	C₄×D₄	1	4	yes	—
182	C₄×D₅₃	C₄×Q₈	1	4	yes	—
183	C₈×C₅₃	—	2	2	no	—
184	C₈×C₅₃	—	2	4	no	—
185	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	complemented
186	#88322	—	2	2	no	—
187	#88322	—	2	4	no	—
188	#88322	C₂×C₄⋊C₄	1	4	yes	complemented
189	C₂×C₈×C₄⋊C₄	—	53	1	no	maximal, Sylow
190	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
191	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
193	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
194	C₂²×C₅₃	C₂×#131	1	2	yes	—
195	C₂×D₅₃	C₄×C₄⋊C₄	1	4	yes	—
196	C₂×D₅₃	C₂×C₄×C₈	1	4	yes	—
197	C₂×D₅₃	#405	1	4	yes	—
198	C₂×D₅₃	C₈×D₄	1	8	yes	—
199	C₂×D₅₃	C₈×Q₈	1	8	yes	—
200	C₄×C₅₃	—	2	2	no	—
201	C₄×C₅₃	—	2	8	no	—
202	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	complemented
203	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
204	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
205	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
206	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
207	Dic₅₃	—	2	2	no	—
208	Dic₅₃	—	2	8	no	—
209	Dic₅₃	C₄×C₄⋊C₄	1	4	yes	—
210	Dic₅₃	C₂×C₄×C₈	1	4	yes	complemented
211	C₂²×C₄⋊C₈	—	53	1	no	—
212	C₂²×C₄×C₈	—	53	1	no	—
213	C₂²×C₄×C₈	—	53	2	no	—
214	C₂×C₄×C₄⋊C₄	—	53	1	no	—
215	C₈×C₄⋊C₄	—	53	1	no	—
216	C₈×C₄⋊C₄	—	53	1	no	—
217	C₈×C₄⋊C₄	—	53	2	no	—
218	C₈×C₄⋊C₄	—	53	1	no	—
219	C₈×C₄⋊C₄	—	53	2	no	—
220	C₈×C₄⋊C₄	D₅₃	1	1	yes	characteristic, direct product factor
221	C₂×#336	—	53	2	no	—
222	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
224	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
225	C₂×C₅₃	C₂×C₈×D₄	1	2	yes	—
226	C₂×C₅₃	C₂×C₈×Q₈	1	2	yes	—
227	D₅₃	C₈×C₄⋊C₄	1	8	yes	direct product factor
228	C₂³×C₈	—	53	1	no	—
229	C₂³×C₈	—	53	2	no	—
230	C₂³×C₈	—	53	1	no	—
231	C₂²×C₄⋊C₄	—	53	2	no	—
232	C₂²×C₄²	—	53	1	no	—
233	C₂²×C₄²	—	53	2	no	—
234	C₂×C₄⋊C₈	—	53	2	no	—
235	C₂×C₄⋊C₈	—	53	1	no	—
236	C₂×C₄⋊C₈	—	53	8	no	—
237	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
238	C₂×C₄×C₈	—	53	1	no	—
239	C₂×C₄×C₈	—	53	1	no	—
240	C₂×C₄×C₈	—	53	1	no	—
241	C₂×C₄×C₈	—	53	8	no	—
242	C₂×C₄×C₈	—	53	2	no	—
243	C₂×C₄×C₈	—	53	2	no	—
244	C₂×C₄×C₈	—	53	2	no	—
245	C₂×C₄×C₈	—	106	8	no	—
246	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
247	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
248	C₄×C₄⋊C₄	—	53	2	no	—
249	C₄×C₄⋊C₄	—	53	1	no	—
250	C₄×C₄⋊C₄	—	53	2	no	—
251	C₄×C₄⋊C₄	—	53	2	no	—
252	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
253	C₂×#95	—	53	2	no	—
254	#336	—	53	2	no	—
255	#336	—	53	2	no	—
256	#336	—	53	2	no	—
257	#336	C₂×D₅₃	1	2	yes	—
258	C₅₃	C₂×C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
259	C₂³×C₄	—	53	1	no	—
260	C₂³×C₄	—	53	1	no	—
261	C₂³×C₄	—	53	4	no	—
262	C₂³×C₄	—	53	1	no	—
263	C₂²×C₈	—	53	1	no	—
264	C₂²×C₈	—	53	4	no	—
265	C₂²×C₈	—	53	4	no	—
266	C₂²×C₈	—	53	4	no	—
267	C₂²×C₈	—	53	1	no	—
268	C₂²×C₈	—	53	2	no	—
269	C₂²×C₈	—	53	4	no	—
270	C₂²×C₈	—	53	8	no	—
271	C₂²×C₈	—	106	4	no	—
272	C₂²×C₈	—	106	2	no	—
273	C₂²×C₈	—	106	4	no	—
274	C₂²×C₈	—	106	2	no	—
275	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
276	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
277	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
278	C₂×C₄⋊C₄	—	53	4	no	—
279	C₂×C₄⋊C₄	—	53	2	no	—
280	C₂×C₄⋊C₄	—	53	16	no	—
281	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
282	C₂×C₄²	—	53	1	no	—
283	C₂×C₄²	—	53	1	no	—
284	C₂×C₄²	—	53	1	no	—
285	C₂×C₄²	—	53	4	no	—
286	C₂×C₄²	—	53	4	no	—
287	C₂×C₄²	—	53	4	no	—
288	C₂×C₄²	—	53	2	no	—
289	C₂×C₄²	—	106	4	no	—
290	C₂×C₄²	—	106	4	no	—
291	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
292	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
293	C₄⋊C₈	—	53	8	no	—
294	C₄⋊C₈	—	53	4	no	—
295	C₄⋊C₈	C₄×D₅₃	1	4	yes	complemented
296	C₄×C₈	—	2	4	no	—
297	C₄×C₈	—	53	4	no	—
298	C₄×C₈	—	53	4	no	—
299	C₄×C₈	—	53	4	no	—
300	C₄×C₈	—	106	4	no	—
301	C₄×C₈	—	106	4	no	—
302	C₄×C₈	—	106	4	no	—
303	C₄×C₈	C₄×D₅₃	1	4	yes	complemented
304	#95	—	53	4	no	—
305	#95	—	53	2	no	—
306	#95	C₄×D₅₃	1	2	yes	—
307	C₂⁴	—	53	1	no	—
308	C₂²×C₄	—	53	1	no	—
309	C₂²×C₄	—	53	1	no	—
310	C₂²×C₄	—	53	4	no	—
311	C₂²×C₄	—	53	8	no	—
312	C₂²×C₄	—	53	2	no	—
313	C₂²×C₄	—	53	2	no	—
314	C₂²×C₄	—	53	2	no	—
315	C₂²×C₄	—	53	2	no	—
316	C₂²×C₄	—	53	2	no	—
317	C₂²×C₄	—	53	2	no	—
318	C₂²×C₄	—	53	1	no	—
319	C₂²×C₄	—	53	4	no	—
320	C₂²×C₄	—	53	4	no	—
321	C₂²×C₄	—	53	16	no	—
322	C₂²×C₄	—	106	8	no	—
323	C₂²×C₄	—	106	8	no	—
324	C₂²×C₄	—	106	4	no	—
325	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
326	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
327	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
328	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	53	2	no	—
334	C₂×C₈	—	53	2	no	—
335	C₂×C₈	—	53	2	no	—
336	C₂×C₈	—	53	16	no	—
337	C₂×C₈	—	53	2	no	—
338	C₂×C₈	—	53	4	no	—
339	C₂×C₈	—	106	8	no	—
340	C₂×C₈	—	106	16	no	—
341	C₂×C₈	—	106	2	no	—
342	C₂×C₈	—	106	1	no	—
343	C₂×C₈	—	106	1	no	—
344	C₂×C₈	—	106	2	no	—
345	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
346	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
347	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
348	C₂×C₈	D₄×D₅₃	1	2	yes	—
349	C₂×C₈	Q₈×D₅₃	1	2	yes	—
350	C₄⋊C₄	—	53	16	no	—
351	C₄⋊C₄	—	53	8	no	—
352	C₄⋊C₄	C₈×D₅₃	1	8	yes	direct product factor
353	C₄²	—	2	2	no	—
354	C₄²	—	2	2	no	—
355	C₄²	—	53	4	no	—
356	C₄²	—	53	4	no	—
357	C₄²	—	53	2	no	—
358	C₄²	—	53	2	no	—
359	C₄²	—	106	8	no	—
360	C₄²	—	106	2	no	—
361	C₄²	—	106	2	no	—
362	C₄²	C₂×C₄×D₅₃	1	2	yes	—
363	C₄²	C₂×C₄×D₅₃	1	2	yes	—
364	C₂³	—	53	2	no	—
365	C₂³	—	53	2	no	—
366	C₂³	—	53	2	no	—
367	C₂³	—	53	4	no	—
368	C₂³	—	53	4	no	—
369	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	53	2	no	—
374	C₂×C₄	—	53	2	no	—
375	C₂×C₄	—	53	2	no	—
376	C₂×C₄	—	53	2	no	—
377	C₂×C₄	—	53	4	no	—
378	C₂×C₄	—	53	4	no	—
379	C₂×C₄	—	53	4	no	—
380	C₂×C₄	—	53	16	no	—
381	C₂×C₄	—	53	4	no	—
382	C₂×C₄	—	53	4	no	—
383	C₂×C₄	—	53	2	no	—
384	C₂×C₄	—	53	8	no	—
385	C₂×C₄	—	106	8	no	—
386	C₂×C₄	—	106	32	no	—
387	C₂×C₄	—	106	4	no	—
388	C₂×C₄	—	106	2	no	—
389	C₂×C₄	—	106	4	no	—
390	C₂×C₄	C₄²×D₅₃	1	2	yes	—
391	C₂×C₄	C₄²×D₅₃	1	2	yes	—
392	C₂×C₄	C₂×C₈×D₅₃	1	8	yes	—
393	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
394	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
395	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
396	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
397	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
398	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
399	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
400	C₈	—	2	2	no	—
401	C₈	—	2	4	no	—
402	C₈	—	53	4	no	—
403	C₈	—	106	2	no	—
404	C₈	—	106	4	no	—
405	C₈	C₄⋊C₄×D₅₃	1	4	yes	direct product factor
406	C₂²	—	53	4	no	—
407	C₂²	—	53	4	no	—
408	C₂²	—	53	8	no	—
409	C₂²	—	53	4	no	—
410	C₂²	—	53	8	no	—
411	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
412	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
413	C₂²	C₄×Q₈×D₅₃	1	1	yes	characteristic
414	C₂²	C₂²×C₈×D₅₃	1	2	yes	—
415	C₂²	#131×D₅₃	1	2	yes	—
416	C₄	—	2	2	no	—
417	C₄	—	2	8	no	—
418	C₄	—	53	4	no	—
419	C₄	—	53	4	no	—
420	C₄	—	106	2	no	—
421	C₄	—	106	8	no	—
422	C₄	C₄×C₈×D₅₃	1	4	yes	complemented
423	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
424	C₄	#117×D₅₃	1	1	yes	characteristic
425	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
426	C₄	C₄×Q₈×D₅₃	1	1	yes	characteristic
427	C₂	—	53	8	no	—
428	C₂	C₄×C₄⋊C₄×D₅₃	1	1	yes	characteristic
429	C₂	C₂×C₄×C₈×D₅₃	1	1	yes	characteristic
430	C₂	#405×D₅₃	1	1	yes	characteristic
431	C₂	C₈×D₄×D₅₃	1	2	yes	—
432	C₂	C₈×Q₈×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₈×C₄⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂³×D₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂×C₈×C₄⋊C₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
5. #20274580	C₂	semidirect
6. C₄⋊C₄×#88322	C₂	semidirect
7. C₈×#11719919	C₂	semidirect
8. C₈×C₄⋊Dic₅₃	C₂	semidirect
9. #20274576	C₂	semidirect
11. C₈×C₄⋊C₄×C₅₃	C₂	semidirect
20. C₄⋊C₈×D₅₃	C₄	semidirect
28. C₄×C₈×D₅₃	C₄	semidirect
60. C₄⋊C₄×D₅₃	C₈	direct (N × H)
70. C₄⋊#88322	C₂×C₄	semidirect
71. #13133146	C₂×C₄	semidirect
73. C₄×#88322	C₂×C₄	semidirect
76. C₈×Dic₅₃	C₂×C₄	semidirect
77. C₈×Dic₅₃	C₂×C₄	semidirect
86. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
88. C₄×C₈×C₅₃	C₂×C₄	semidirect
121. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
126. C₄⋊Dic₅₃	C₂×C₈	semidirect
127. #11719919	C₂×C₈	semidirect
146. C₈×D₅₃	C₄⋊C₄	direct (N × H)
178. C₄×D₅₃	C₄×C₈	semidirect
185. C₈×C₅₃	C₂×C₄⋊C₄	semidirect
188. #88322	C₂×C₄⋊C₄	semidirect
202. C₄×C₅₃	C₂×C₄×C₈	semidirect
210. Dic₅₃	C₂×C₄×C₈	semidirect
220. C₈×C₄⋊C₄	D₅₃	direct (N × H)
227. D₅₃	C₈×C₄⋊C₄	direct (N × H)
258. C₅₃	C₂×C₈×C₄⋊C₄	semidirect
295. C₄⋊C₈	C₄×D₅₃	semidirect
303. C₄×C₈	C₄×D₅₃	semidirect
352. C₄⋊C₄	C₈×D₅₃	direct (N × H)
405. C₈	C₄⋊C₄×D₅₃	direct (N × H)
422. C₄	C₄×C₈×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₈×D₅₃	2
C₂×C₄×C₈×D₅₃	2
C₄×C₄⋊C₄×D₅₃	2
#20274580	2
C₄⋊C₄×#88322	2
C₈×#11719919	2
C₈×C₄⋊Dic₅₃	2
#20274576	2
#336×D₅₃	2
C₈×C₄⋊C₄×C₅₃	2
C₂×C₈×C₄⋊C₄	53

Subgroups — C8×C4⋊C4×D53 #27382018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₄⋊C₄×D₅₃ #27382018