Subgroups — #339×Dic₅₃ #27289263

Order 13568.

1214 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
2	#117×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810384	C₂	1	1	yes	maximal, characteristic
5	C₂×#16810383	C₂	1	1	yes	maximal, characteristic
6	C₂×#16810382	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅₃×#339	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11719917	C₂²	1	1	yes	characteristic
9	C₂×#13133176	C₂²	1	1	yes	characteristic
10	C₂²×C₄×Dic₅₃	—	2	2	no	—
11	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
13	M₁₆×Dic₅₃	—	2	2	no	—
14	M₁₆×Dic₅₃	C₄	1	4	yes	complemented
15	C₂×#13133127	—	2	1	no	—
16	C₂×C₈×Dic₅₃	—	2	1	no	—
17	C₄⋊C₄×Dic₅₃	C₄	1	4	yes	complemented
18	C₂²⋊C₄×Dic₅₃	—	2	2	no	—
19	#16810469	—	2	1	no	—
20	C₄²×Dic₅₃	—	2	2	no	—
21	C₄²×Dic₅₃	—	2	1	no	—
22	C₄²×Dic₅₃	C₄	1	4	yes	complemented
23	#16810384	C₄	1	4	yes	complemented
24	#16810383	C₄	1	4	yes	complemented
25	#16810382	C₄	1	4	yes	complemented
26	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#339	C₄	1	4	yes	complemented
30	C₂³×Dic₅₃	—	2	1	no	—
31	C₂³×Dic₅₃	D₄	1	2	yes	—
32	C₂×#11719917	—	2	1	no	—
33	C₂×#11719917	—	2	1	no	—
34	C₂×#11719917	—	2	1	no	—
35	C₂×#11719917	C₂×C₄	1	2	yes	—
36	C₂×#11719917	C₂×C₄	1	2	yes	—
37	C₂×#11719917	C₂×C₄	1	2	yes	—
38	C₂²×#88322	—	2	1	no	—
39	#13133176	—	2	1	no	—
40	#13133176	—	2	1	no	—
41	#13133176	C₂×C₄	1	2	yes	—
42	#13133176	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
45	C₂×C₄×Dic₅₃	—	2	16	no	—
46	C₂×C₄×Dic₅₃	—	2	1	no	—
47	C₂×C₄×Dic₅₃	—	2	1	no	—
48	C₂×C₄×Dic₅₃	—	2	1	no	—
49	C₂×C₄×Dic₅₃	—	2	1	no	—
50	C₂×C₄×Dic₅₃	—	2	8	no	—
51	C₂×C₄×Dic₅₃	—	2	4	no	—
52	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
56	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
57	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
58	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
60	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
61	C₂×C₄×Dic₅₃	Q₈	1	2	yes	—
62	#13133127	—	2	4	no	—
63	C₈×Dic₅₃	—	2	4	no	—
64	#13133124	—	2	4	no	—
65	C₂³×C₄×C₅₃	—	2	1	no	—
66	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₅₃	—	2	1	no	—
68	C₂×M₁₆×C₅₃	—	2	1	no	—
69	C₂×M₁₆×C₅₃	—	2	1	no	—
70	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
71	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
72	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
73	C₂²×C₈×C₅₃	—	2	1	no	—
74	#117×C₅₃	—	2	1	no	—
75	#117×C₅₃	—	2	1	no	—
76	#117×C₅₃	C₂×C₄	1	2	yes	—
77	#117×C₅₃	C₂×C₄	1	2	yes	—
78	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
79	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
80	C₂×C₄²×C₅₃	—	2	1	no	—
81	C₂×C₄²×C₅₃	—	2	1	no	—
82	C₂×C₄²×C₅₃	—	2	1	no	—
83	C₂×C₄²×C₅₃	—	2	1	no	—
84	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
85	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
86	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
87	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	—	2	8	no	—
89	C₂²×C₄×C₅₃	—	2	2	no	—
90	C₂²×C₄×C₅₃	—	2	2	no	—
91	C₂²×C₄×C₅₃	—	2	2	no	—
92	C₂²×C₄×C₅₃	—	2	2	no	—
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
107	C₂²×Dic₅₃	—	2	8	no	—
108	C₂²×Dic₅₃	—	2	2	no	—
109	C₂²×Dic₅₃	—	2	2	no	—
110	C₂²×Dic₅₃	—	2	2	no	—
111	C₂²×Dic₅₃	—	2	8	no	—
112	C₂²×Dic₅₃	—	2	2	no	—
113	C₂²×Dic₅₃	C₂²⋊C₄	1	2	yes	—
114	C₂²×Dic₅₃	C₂²⋊C₄	1	4	yes	—
115	C₂²×Dic₅₃	C₂²⋊C₄	1	2	yes	—
116	C₂²×Dic₅₃	C₄⋊C₄	1	4	yes	—
117	M₁₆×C₅₃	—	2	2	no	—
118	M₁₆×C₅₃	—	2	2	no	—
119	M₁₆×C₅₃	—	2	2	no	—
120	M₁₆×C₅₃	C₄²	1	4	yes	complemented
121	C₂×C₈×C₅₃	—	2	2	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	2	no	—
125	C₄⋊C₄×C₅₃	C₄²	1	4	yes	complemented
126	C₂²⋊C₄×C₅₃	—	2	2	no	—
127	C₂²⋊C₄×C₅₃	—	2	2	no	—
128	C₄²×C₅₃	—	2	2	no	—
129	C₄²×C₅₃	—	2	2	no	—
130	C₄²×C₅₃	—	2	2	no	—
131	C₄²×C₅₃	—	2	2	no	—
132	C₄²×C₅₃	—	2	2	no	—
133	C₄²×C₅₃	C₄²	1	4	yes	complemented
134	C₂²⋊Dic₅₃	—	2	2	no	—
135	C₂²⋊Dic₅₃	—	2	2	no	—
136	C₄⋊Dic₅₃	—	2	2	no	—
137	C₄⋊Dic₅₃	C₄²	1	4	yes	complemented
138	C₄×Dic₅₃	—	2	32	no	—
139	C₄×Dic₅₃	—	2	2	no	—
140	C₄×Dic₅₃	—	2	2	no	—
141	C₄×Dic₅₃	—	2	2	no	—
142	C₄×Dic₅₃	—	2	2	no	—
143	C₄×Dic₅₃	—	2	2	no	—
144	C₄×Dic₅₃	—	4	8	no	—
145	C₄×Dic₅₃	C₄²	1	4	yes	—
146	C₄×Dic₅₃	C₄²	1	4	yes	complemented
147	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
148	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
149	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
150	C₄×Dic₅₃	C₄⋊C₄	1	4	yes	—
151	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
152	#11719917	—	2	2	no	—
153	#11719917	—	2	2	no	—
154	#11719917	—	2	2	no	—
155	#11719917	C₄²	1	4	yes	complemented
156	C₂×#88322	—	2	2	no	—
157	C₂×#88322	—	2	2	no	—
158	C₂×#88322	—	2	2	no	—
159	C₂³×C₅₃	—	2	4	no	—
160	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
164	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
165	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
166	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
167	C₂×C₄×C₅₃	—	2	8	no	—
168	C₂×C₄×C₅₃	—	2	8	no	—
169	C₂×C₄×C₅₃	—	2	8	no	—
170	C₂×C₄×C₅₃	—	2	4	no	—
171	C₂×C₄×C₅₃	—	2	4	no	—
172	C₂×C₄×C₅₃	—	4	4	no	—
173	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
184	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
185	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
186	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
187	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
188	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
198	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
199	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
200	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
201	C₂×Dic₅₃	—	2	8	no	—
202	C₂×Dic₅₃	—	2	8	no	—
203	C₂×Dic₅₃	—	2	8	no	—
204	C₂×Dic₅₃	—	2	4	no	—
205	C₂×Dic₅₃	—	2	4	no	—
206	C₂×Dic₅₃	—	2	8	no	—
207	C₂×Dic₅₃	—	2	8	no	—
208	C₂×Dic₅₃	—	2	4	no	—
209	C₂×Dic₅₃	—	4	4	no	—
210	C₂×Dic₅₃	#95	1	4	yes	—
211	C₂×Dic₅₃	#95	1	4	yes	—
212	C₂×Dic₅₃	#104	1	8	yes	—
213	C₈×C₅₃	—	2	4	no	—
214	#88322	—	2	4	no	—
215	C₄×#339	—	53	1	no	maximal, Sylow
216	C₂²×C₅₃	—	2	2	no	—
217	C₂²×C₅₃	—	2	2	no	—
218	C₂²×C₅₃	—	2	2	no	—
219	C₂²×C₅₃	—	2	2	no	—
220	C₂²×C₅₃	—	2	2	no	—
221	C₂²×C₅₃	—	2	2	no	—
222	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
223	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
224	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
225	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
226	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
227	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
228	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
229	C₂²×C₅₃	C₂×#104	1	4	yes	—
230	C₄×C₅₃	—	2	8	no	—
231	C₄×C₅₃	—	4	4	no	—
232	C₄×C₅₃	C₄³	1	1	yes	characteristic
233	C₄×C₅₃	#376	1	1	yes	characteristic
234	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
235	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
236	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
237	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
238	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
239	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
240	Dic₅₃	—	2	8	no	—
241	Dic₅₃	—	2	4	no	—
242	Dic₅₃	—	4	4	no	—
243	Dic₅₃	#339	1	8	yes	direct product factor
244	C₂×C₄×M₁₆	—	53	1	no	—
245	C₄×#117	—	53	1	no	—
246	C₂×C₄³	—	53	1	no	—
247	C₂×#339	—	53	1	no	—
248	C₂×#339	—	53	1	no	—
249	C₂×#339	—	53	1	no	—
250	C₂×#339	D₅₃	1	1	yes	characteristic
251	C₂×C₅₃	—	2	1	no	—
252	C₂×C₅₃	—	2	1	no	—
253	C₂×C₅₃	—	2	1	no	—
254	C₂×C₅₃	—	2	1	no	—
255	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
256	C₂×C₅₃	C₂×#339	1	2	yes	—
257	C₂×C₅₃	C₄×#104	1	2	yes	—
258	C₂×C₅₃	C₄×#104	1	2	yes	—
259	C₂²×M₁₆	—	53	1	no	—
260	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
261	C₂×#117	—	53	1	no	—
262	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
263	C₂²×C₄²	—	53	1	no	—
264	C₂²×C₄²	—	53	1	no	—
265	C₂²×C₄²	—	106	2	no	—
266	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
267	C₄×M₁₆	—	53	4	no	—
268	C₄×M₁₆	—	106	2	no	—
269	C₂×#97	—	106	1	no	—
270	C₂×C₄×C₈	—	106	1	no	—
271	C₄×C₄⋊C₄	—	53	4	no	—
272	C₄×C₂²⋊C₄	—	106	2	no	—
273	#376	—	106	1	no	—
274	C₄³	—	53	4	no	—
275	C₄³	—	106	2	no	—
276	C₄³	—	106	1	no	—
277	#339	—	53	4	no	—
278	#339	—	53	4	no	—
279	#339	—	53	4	no	—
280	#339	Dic₅₃	1	4	yes	direct product factor
281	C₅₃	C₄×#339	1	1	yes	characteristic, complemented, Sylow
282	C₂³×C₄	—	2	1	no	—
283	C₂³×C₄	—	53	2	no	—
284	C₂³×C₄	—	106	1	no	—
285	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
286	C₂×M₁₆	—	2	1	no	—
287	C₂×M₁₆	—	2	1	no	—
288	C₂×M₁₆	—	2	1	no	—
289	C₂×M₁₆	—	53	2	no	—
290	C₂×M₁₆	—	53	2	no	—
291	C₂×M₁₆	—	53	2	no	—
292	C₂×M₁₆	—	106	1	no	—
293	C₂×M₁₆	—	106	1	no	—
294	C₂×M₁₆	—	106	1	no	—
295	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
296	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
297	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
298	C₂²×C₈	—	2	1	no	—
299	C₂²×C₈	—	106	1	no	—
300	#117	—	2	1	no	—
301	#117	—	2	1	no	—
302	#117	—	53	2	no	—
303	#117	—	53	2	no	—
304	#117	—	106	1	no	—
305	#117	—	106	1	no	—
306	#117	C₄×D₅₃	1	2	yes	—
307	#117	C₂×Dic₅₃	1	2	yes	—
308	C₂×C₄⋊C₄	—	53	2	no	—
309	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
310	C₂×C₂²⋊C₄	—	2	1	no	—
311	C₂×C₂²⋊C₄	—	106	1	no	—
312	C₂×C₄²	—	2	1	no	—
313	C₂×C₄²	—	2	1	no	—
314	C₂×C₄²	—	2	1	no	—
315	C₂×C₄²	—	2	1	no	—
316	C₂×C₄²	—	53	2	no	—
317	C₂×C₄²	—	53	2	no	—
318	C₂×C₄²	—	53	2	no	—
319	C₂×C₄²	—	53	2	no	—
320	C₂×C₄²	—	53	1	no	—
321	C₂×C₄²	—	53	1	no	—
322	C₂×C₄²	—	53	2	no	—
323	C₂×C₄²	—	53	2	no	—
324	C₂×C₄²	—	53	2	no	—
325	C₂×C₄²	—	53	2	no	—
326	C₂×C₄²	—	106	8	no	—
327	C₂×C₄²	—	106	16	no	—
328	C₂×C₄²	—	106	1	no	—
329	C₂×C₄²	—	106	1	no	—
330	C₂×C₄²	—	106	1	no	—
331	C₂×C₄²	—	106	1	no	—
332	C₂×C₄²	—	106	4	no	—
333	C₂×C₄²	C₄×D₅₃	1	2	yes	—
334	C₂×C₄²	C₄×D₅₃	1	2	yes	—
335	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
336	#97	—	106	4	no	—
337	C₄×C₈	—	106	4	no	—
338	#95	—	106	4	no	—
339	C₂⁴	D₄×D₅₃	1	1	yes	characteristic
340	C₂²×C₄	—	2	8	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	53	2	no	—
346	C₂²×C₄	—	53	2	no	—
347	C₂²×C₄	—	53	4	no	—
348	C₂²×C₄	—	53	4	no	—
349	C₂²×C₄	—	106	8	no	—
350	C₂²×C₄	—	106	8	no	—
351	C₂²×C₄	—	106	2	no	—
352	C₂²×C₄	—	106	2	no	—
353	C₂²×C₄	—	106	2	no	—
354	C₂²×C₄	—	106	2	no	—
355	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
356	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
357	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
358	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
359	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
360	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
361	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
362	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
363	C₂²×C₄	#11719947	1	1	yes	characteristic
364	C₂²×C₄	#11719947	1	1	yes	characteristic
365	C₂²×C₄	#11719947	1	1	yes	characteristic
366	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
367	C₂²×C₄	#11719949	1	1	yes	characteristic
368	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
369	M₁₆	—	2	2	no	—
370	M₁₆	—	2	2	no	—
371	M₁₆	—	2	2	no	—
372	M₁₆	—	53	4	no	—
373	M₁₆	—	106	2	no	—
374	M₁₆	—	106	2	no	—
375	M₁₆	—	106	2	no	—
376	M₁₆	C₄×Dic₅₃	1	4	yes	complemented
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	106	2	no	—
381	C₂×C₈	—	106	2	no	—
382	C₂×C₈	—	106	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	53	4	no	—
385	C₄⋊C₄	—	106	2	no	—
386	C₄⋊C₄	C₄×Dic₅₃	1	4	yes	complemented
387	C₂²⋊C₄	—	2	2	no	—
388	C₂²⋊C₄	—	2	2	no	—
389	C₂²⋊C₄	—	106	2	no	—
390	C₂²⋊C₄	—	106	2	no	—
391	C₄²	—	2	2	no	—
392	C₄²	—	2	2	no	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	2	no	—
395	C₄²	—	2	2	no	—
396	C₄²	—	53	4	no	—
397	C₄²	—	53	2	no	—
398	C₄²	—	53	2	no	—
399	C₄²	—	53	2	no	—
400	C₄²	—	53	4	no	—
401	C₄²	—	53	2	no	—
402	C₄²	—	53	4	no	—
403	C₄²	—	106	32	no	—
404	C₄²	—	106	2	no	—
405	C₄²	—	106	2	no	—
406	C₄²	—	106	2	no	—
407	C₄²	—	106	2	no	—
408	C₄²	—	106	2	no	—
409	C₄²	—	212	8	no	—
410	C₄²	C₄×Dic₅₃	1	4	yes	complemented
411	C₂³	—	2	4	no	—
412	C₂³	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
413	C₂³	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
414	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
415	C₂³	#13133177	1	1	yes	characteristic
416	C₂³	#13133177	1	1	yes	characteristic
417	C₂³	#13133179	1	1	yes	characteristic
418	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
419	C₂×C₄	—	2	8	no	—
420	C₂×C₄	—	2	8	no	—
421	C₂×C₄	—	2	8	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	4	4	no	—
425	C₂×C₄	—	53	8	no	—
426	C₂×C₄	—	53	4	no	—
427	C₂×C₄	—	53	4	no	—
428	C₂×C₄	—	106	8	no	—
429	C₂×C₄	—	106	8	no	—
430	C₂×C₄	—	106	8	no	—
431	C₂×C₄	—	106	8	no	—
432	C₂×C₄	—	106	8	no	—
433	C₂×C₄	—	106	4	no	—
434	C₂×C₄	—	106	4	no	—
435	C₂×C₄	—	106	4	no	—
436	C₂×C₄	—	212	4	no	—
437	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
438	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
439	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
440	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
441	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
442	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
443	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
444	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
445	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
446	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
447	C₂×C₄	#13133173	1	1	yes	characteristic
448	C₂×C₄	#13133173	1	1	yes	characteristic
449	C₂×C₄	#13133173	1	1	yes	characteristic
450	C₂×C₄	#13133175	1	1	yes	characteristic
451	C₂×C₄	#13133175	1	1	yes	characteristic
452	C₂×C₄	#13133175	1	1	yes	characteristic
453	C₂×C₄	#13133177	1	1	yes	characteristic
454	C₂×C₄	#13133177	1	1	yes	characteristic
455	C₂×C₄	#13133177	1	1	yes	characteristic
456	C₂×C₄	#13133177	1	1	yes	characteristic
457	C₂×C₄	#13133179	1	1	yes	characteristic
458	C₂×C₄	#13133179	1	1	yes	characteristic
459	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
460	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
461	C₂×C₄	#13133182	1	1	yes	characteristic
462	C₂×C₄	#13133182	1	1	yes	characteristic
463	C₂×C₄	#13133182	1	1	yes	characteristic
464	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
465	C₈	—	2	4	no	—
466	C₈	—	106	4	no	—
467	C₂²	—	2	2	no	—
468	C₂²	—	2	2	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	2	2	no	—
473	C₂²	#95×D₅₃	1	1	yes	characteristic
474	C₂²	#16810470	1	1	yes	characteristic
475	C₂²	#16810470	1	1	yes	characteristic
476	C₂²	C₂²⋊C₄×Dic₅₃	1	1	yes	characteristic
477	C₂²	C₂²⋊C₄×Dic₅₃	1	1	yes	characteristic
478	C₂²	#16810474	1	1	yes	characteristic
479	C₂²	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
480	C₂²	#104×D₅₃	1	4	yes	—
481	C₄	—	2	8	no	—
482	C₄	—	4	4	no	—
483	C₄	—	53	8	no	—
484	C₄	—	106	8	no	—
485	C₄	—	106	4	no	—
486	C₄	—	212	4	no	—
487	C₄	C₄²×Dic₅₃	1	1	yes	characteristic
488	C₄	#16810468	1	1	yes	characteristic
489	C₄	#16810470	1	1	yes	characteristic
490	C₄	C₂²⋊C₄×Dic₅₃	1	1	yes	characteristic
491	C₄	#16810474	1	1	yes	characteristic
492	C₄	#16810474	1	1	yes	characteristic
493	C₄	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
494	C₄	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
495	C₂	—	2	1	no	—
496	C₂	—	2	1	no	—
497	C₂	—	2	1	no	—
498	C₂	—	2	1	no	—
499	C₂	#95×Dic₅₃	1	1	yes	characteristic
500	C₂	#339×D₅₃	1	2	yes	—
501	C₂	#20274519	1	2	yes	—
502	C₂	#104×Dic₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₄³
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#339	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#339	D₅₃
hypocenter	C₅₃	C₄×#339
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#339

Complements

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

N	H ≅ G/N	type
14. M₁₆×Dic₅₃	C₄	semidirect
17. C₄⋊C₄×Dic₅₃	C₄	semidirect
22. C₄²×Dic₅₃	C₄	semidirect
23. #16810384	C₄	semidirect
24. #16810383	C₄	semidirect
25. #16810382	C₄	semidirect
29. C₅₃×#339	C₄	semidirect
120. M₁₆×C₅₃	C₄²	semidirect
125. C₄⋊C₄×C₅₃	C₄²	semidirect
133. C₄²×C₅₃	C₄²	semidirect
137. C₄⋊Dic₅₃	C₄²	semidirect
146. C₄×Dic₅₃	C₄²	semidirect
155. #11719917	C₄²	semidirect
243. Dic₅₃	#339	direct (N × H)
280. #339	Dic₅₃	direct (N × H)
281. C₅₃	C₄×#339	semidirect
376. M₁₆	C₄×Dic₅₃	semidirect
386. C₄⋊C₄	C₄×Dic₅₃	semidirect
410. C₄²	C₄×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×Dic₅₃	2
#117×Dic₅₃	2
C₂×C₄²×Dic₅₃	2
C₂×#16810384	2
C₂×#16810383	2
C₂×#16810382	2
C₂×C₅₃×#339	2
C₄×#339	53

Subgroups — #339×Dic53 #27289263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #339×Dic₅₃ #27289263