Subgroups — C₂²×#16810480 #27542349

Order 13568.

8670 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×#11719921	C₂	1	2	yes	maximal
4	C₂×#16810480	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#13133124	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₅₃	C₂²	1	2	yes	—
9	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×D₅₃	—	2	8	no	—
11	C₂³×C₄×D₅₃	—	2	8	no	—
12	C₂³×C₄×D₅₃	C₄	1	16	yes	complemented
13	C₂³×C₄×D₅₃	C₂²	1	12	yes	—
14	C₂²×C₄⋊Dic₅₃	—	2	8	no	—
15	C₂²×C₄⋊Dic₅₃	C₂²	1	12	yes	complemented
16	C₂²×#11719921	—	2	16	no	—
17	C₂²×#11719921	C₂²	1	24	yes	—
18	#16810480	C₂²	1	64	yes	direct product factor
19	C₂×#13133124	C₂²	1	24	yes	complemented
20	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
21	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₅₃	—	2	8	no	—
23	C₂²×C₄⋊C₄×C₅₃	C₂²	1	12	yes	complemented
24	C₂⁴×D₅₃	—	2	4	no	—
25	C₂⁴×D₅₃	—	2	4	no	—
26	C₂⁴×D₅₃	—	2	4	no	—
27	C₂⁴×D₅₃	—	2	4	no	—
28	C₂⁴×D₅₃	C₂×C₄	1	8	yes	—
29	C₂⁴×D₅₃	D₄	1	8	yes	—
30	C₂⁴×D₅₃	Q₈	1	8	yes	—
31	C₂⁴×D₅₃	C₂³	1	6	yes	—
32	C₂³×Dic₅₃	—	2	4	no	—
33	C₂³×Dic₅₃	—	2	4	no	—
34	C₂³×Dic₅₃	—	2	8	no	—
35	C₂³×Dic₅₃	—	2	8	no	—
36	C₂³×Dic₅₃	—	2	8	no	—
37	C₂³×Dic₅₃	C₂×C₄	1	8	yes	—
38	C₂³×Dic₅₃	C₂³	1	12	yes	—
39	C₂³×Dic₅₃	C₂³	1	6	yes	—
40	C₂²×C₄×D₅₃	—	2	128	no	—
41	C₂²×C₄×D₅₃	—	2	48	no	—
42	C₂²×C₄×D₅₃	—	2	48	no	—
43	C₂²×C₄×D₅₃	C₂×C₄	1	96	yes	complemented
44	C₂²×C₄×D₅₃	C₂³	1	16	yes	—
45	C₂×C₄⋊Dic₅₃	—	2	48	no	—
46	C₂×C₄⋊Dic₅₃	C₂³	1	16	yes	complemented
47	C₂×#11719921	—	2	96	no	—
48	C₂×#11719921	C₂³	1	32	yes	—
49	#13133124	C₂³	1	32	yes	complemented
50	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₅₃	—	2	8	no	—
52	C₂³×C₄×C₅₃	—	2	8	no	—
53	C₂³×C₄×C₅₃	—	2	8	no	—
54	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
55	C₂³×C₄×C₅₃	D₄	1	8	yes	—
56	C₂³×C₄×C₅₃	D₄	1	8	yes	—
57	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
58	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
59	C₂×C₄⋊C₄×C₅₃	—	2	48	no	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	complemented
61	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
62	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
63	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
64	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
65	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
66	C₂⁴×C₅₃	#41	1	4	yes	—
67	C₂⁴×C₅₃	#41	1	4	yes	—
68	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
69	C₂³×D₅₃	—	2	32	no	—
70	C₂³×D₅₃	—	2	32	no	—
71	C₂³×D₅₃	—	2	32	no	—
72	C₂³×D₅₃	—	2	32	no	—
73	C₂³×D₅₃	—	2	32	no	—
74	C₂³×D₅₃	—	2	32	no	—
75	C₂³×D₅₃	—	2	12	no	—
76	C₂³×D₅₃	—	2	12	no	—
77	C₂³×D₅₃	—	2	12	no	—
78	C₂³×D₅₃	—	2	12	no	—
79	C₂³×D₅₃	C₄⋊C₄	1	64	yes	complemented
80	C₂³×D₅₃	C₂²×C₄	1	24	yes	—
81	C₂³×D₅₃	C₂×D₄	1	24	yes	—
82	C₂³×D₅₃	C₂×Q₈	1	24	yes	—
83	C₂³×D₅₃	C₂⁴	1	4	yes	—
84	C₂²×C₄×C₅₃	—	2	24	no	—
85	C₂²×C₄×C₅₃	—	2	24	no	—
86	C₂²×C₄×C₅₃	—	2	24	no	—
87	C₂²×C₄×C₅₃	—	4	32	no	—
88	C₂²×C₄×C₅₃	C₂²⋊C₄	1	64	yes	complemented
89	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
90	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
91	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
92	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
93	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
94	C₂²×Dic₅₃	—	2	32	no	—
95	C₂²×Dic₅₃	—	2	12	no	—
96	C₂²×Dic₅₃	—	2	12	no	—
97	C₂²×Dic₅₃	—	2	24	no	—
98	C₂²×Dic₅₃	—	2	24	no	—
99	C₂²×Dic₅₃	—	2	24	no	—
100	C₂²×Dic₅₃	—	4	32	no	—
101	C₂²×Dic₅₃	C₂²×C₄	1	24	yes	—
102	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
103	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
104	C₂×C₄×D₅₃	—	2	384	no	—
105	C₂×C₄×D₅₃	—	2	32	no	—
106	C₂×C₄×D₅₃	—	2	32	no	—
107	C₂×C₄×D₅₃	C₂²×C₄	1	64	yes	complemented
108	C₄⋊C₄×C₅₃	—	2	32	no	—
109	#11719921	—	2	64	no	—
110	C₄⋊Dic₅₃	—	2	32	no	—
111	C₂³×C₅₃	C₂×C₂²⋊C₄	1	16	yes	—
112	C₂³×C₅₃	C₂×C₄⋊C₄	1	16	yes	—
113	C₂³×C₅₃	#117	1	16	yes	—
114	C₂³×C₅₃	#121	1	16	yes	—
115	C₂³×C₅₃	#121	1	16	yes	—
116	C₂³×C₅₃	C₂²⋊Q₈	1	16	yes	—
117	C₂³×C₅₃	C₂²⋊Q₈	1	16	yes	—
118	C₂³×C₅₃	C₂³×C₄	1	6	yes	—
119	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
120	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
121	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
122	C₂³×C₅₃	C₂²×Q₈	1	6	yes	—
123	C₂³×C₅₃	C₂×#41	1	6	yes	—
124	C₂³×C₅₃	C₂×#41	1	6	yes	—
125	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
126	C₂²×D₅₃	—	2	48	no	—
127	C₂²×D₅₃	—	2	48	no	—
128	C₂²×D₅₃	—	2	48	no	—
129	C₂²×D₅₃	—	2	48	no	—
130	C₂²×D₅₃	—	2	48	no	—
131	C₂²×D₅₃	—	2	48	no	—
132	C₂²×D₅₃	—	2	4	no	—
133	C₂²×D₅₃	—	2	4	no	—
134	C₂²×D₅₃	—	2	4	no	—
135	C₂²×D₅₃	—	2	4	no	—
136	C₂²×D₅₃	—	2	256	no	—
137	C₂²×D₅₃	C₂×C₄⋊C₄	1	96	yes	complemented
138	C₂²×D₅₃	C₂³×C₄	1	8	yes	—
139	C₂²×D₅₃	C₂²×D₄	1	8	yes	—
140	C₂²×D₅₃	C₂²×Q₈	1	8	yes	—
141	C₂×C₄×C₅₃	—	2	8	no	—
142	C₂×C₄×C₅₃	—	2	8	no	—
143	C₂×C₄×C₅₃	—	2	8	no	—
144	C₂×C₄×C₅₃	—	4	48	no	—
145	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	96	yes	complemented
146	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
147	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
148	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
149	C₂×Dic₅₃	—	2	48	no	—
150	C₂×Dic₅₃	—	2	4	no	—
151	C₂×Dic₅₃	—	2	4	no	—
152	C₂×Dic₅₃	—	2	8	no	—
153	C₂×Dic₅₃	—	2	8	no	—
154	C₂×Dic₅₃	—	2	8	no	—
155	C₂×Dic₅₃	—	4	48	no	—
156	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
157	C₄×D₅₃	—	2	128	no	—
158	C₂²×#380	—	53	1	no	maximal, Sylow
159	C₂²×C₅₃	#380	1	64	yes	complemented
160	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	12	yes	—
161	C₂²×C₅₃	C₂²×C₄⋊C₄	1	12	yes	—
162	C₂²×C₅₃	C₂×#117	1	12	yes	—
163	C₂²×C₅₃	C₂×#121	1	12	yes	—
164	C₂²×C₅₃	C₂×#121	1	12	yes	—
165	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	12	yes	—
166	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	12	yes	—
167	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
168	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
169	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
170	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
171	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
172	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
173	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
174	C₂×D₅₃	—	2	8	no	—
175	C₂×D₅₃	—	2	8	no	—
176	C₂×D₅₃	—	2	8	no	—
177	C₂×D₅₃	—	2	8	no	—
178	C₂×D₅₃	—	2	8	no	—
179	C₂×D₅₃	—	2	8	no	—
180	C₂×D₅₃	—	2	192	no	—
181	C₂×D₅₃	C₂²×C₄⋊C₄	1	16	yes	complemented
182	C₄×C₅₃	—	4	8	no	—
183	C₄×C₅₃	C₂²×C₂²⋊C₄	1	16	yes	complemented
184	Dic₅₃	—	2	8	no	—
185	Dic₅₃	—	4	8	no	—
186	C₂⁵×C₄	—	53	1	no	—
187	C₂³×C₄⋊C₄	—	53	1	no	—
188	C₂³×C₄⋊C₄	D₅₃	1	1	yes	characteristic, complemented
189	C₂³×C₂²⋊C₄	—	53	2	no	—
190	C₂×#380	—	53	24	no	—
191	C₂²×#95	—	53	2	no	—
192	C₂×C₅₃	C₂×#380	1	24	yes	complemented
193	C₂×C₅₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
195	C₂×C₅₃	C₂²×#117	1	1	yes	characteristic
196	C₂×C₅₃	C₂²×#121	1	1	yes	characteristic
197	C₂×C₅₃	C₂²×#121	1	1	yes	characteristic
198	C₂×C₅₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
199	C₂×C₅₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
200	D₅₃	—	2	16	no	—
201	C₂⁶	—	53	1	no	—
202	C₂⁴×C₄	—	53	1	no	—
203	C₂⁴×C₄	—	53	16	no	—
204	C₂⁴×C₄	—	53	12	no	—
205	C₂⁴×C₄	—	53	2	no	—
206	C₂⁴×C₄	—	106	8	no	—
207	C₂⁴×C₄	—	106	8	no	—
208	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
209	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
210	C₂²×C₄⋊C₄	—	2	8	no	—
211	C₂²×C₄⋊C₄	—	53	12	no	—
212	C₂²×C₄⋊C₄	—	106	8	no	—
213	C₂²×C₄⋊C₄	C₂×D₅₃	1	12	yes	complemented
214	C₂²×C₂²⋊C₄	—	53	24	no	—
215	C₂²×C₂²⋊C₄	—	106	16	no	—
216	#380	—	53	64	no	—
217	C₂×#95	—	53	24	no	—
218	C₅₃	C₂²×#380	1	1	yes	characteristic, complemented, Sylow
219	C₂⁵	—	53	8	no	—
220	C₂⁵	—	53	8	no	—
221	C₂⁵	—	53	8	no	—
222	C₂⁵	—	53	6	no	—
223	C₂⁵	—	106	4	no	—
224	C₂⁵	—	106	4	no	—
225	C₂⁵	—	106	4	no	—
226	C₂⁵	—	106	4	no	—
227	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
228	C₂³×C₄	—	2	8	no	—
229	C₂³×C₄	—	2	8	no	—
230	C₂³×C₄	—	2	8	no	—
231	C₂³×C₄	—	53	8	no	—
232	C₂³×C₄	—	53	6	no	—
233	C₂³×C₄	—	53	96	no	—
234	C₂³×C₄	—	53	16	no	—
235	C₂³×C₄	—	53	12	no	—
236	C₂³×C₄	—	106	4	no	—
237	C₂³×C₄	—	106	4	no	—
238	C₂³×C₄	—	106	48	no	—
239	C₂³×C₄	—	106	48	no	—
240	C₂³×C₄	—	106	128	no	—
241	C₂³×C₄	—	106	8	no	—
242	C₂³×C₄	—	106	8	no	—
243	C₂³×C₄	—	106	8	no	—
244	C₂³×C₄	C₄×D₅₃	1	8	yes	—
245	C₂³×C₄	D₂₁₂	1	8	yes	—
246	C₂³×C₄	C₂²⋊D₅₃	1	8	yes	—
247	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
248	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
249	C₂×C₄⋊C₄	—	2	48	no	—
250	C₂×C₄⋊C₄	—	53	16	no	—
251	C₂×C₄⋊C₄	—	106	48	no	—
252	C₂×C₄⋊C₄	C₂²×D₅₃	1	16	yes	complemented
253	C₂×C₂²⋊C₄	—	53	32	no	—
254	C₂×C₂²⋊C₄	—	106	96	no	—
255	#95	—	53	32	no	—
256	C₂⁴	—	53	64	no	—
257	C₂⁴	—	53	24	no	—
258	C₂⁴	—	53	24	no	—
259	C₂⁴	—	53	24	no	—
260	C₂⁴	—	53	4	no	—
261	C₂⁴	—	106	32	no	—
262	C₂⁴	—	106	32	no	—
263	C₂⁴	—	106	32	no	—
264	C₂⁴	—	106	32	no	—
265	C₂⁴	—	106	32	no	—
266	C₂⁴	—	106	32	no	—
267	C₂⁴	—	106	12	no	—
268	C₂⁴	—	106	12	no	—
269	C₂⁴	—	106	12	no	—
270	C₂⁴	—	106	12	no	—
271	C₂⁴	C₂×C₄×D₅₃	1	4	yes	—
272	C₂⁴	C₂×D₂₁₂	1	4	yes	—
273	C₂⁴	D₄×D₅₃	1	4	yes	—
274	C₂⁴	#11719947	1	4	yes	—
275	C₂⁴	Q₈×D₅₃	1	4	yes	—
276	C₂⁴	#11719949	1	4	yes	—
277	C₂⁴	C₂×C₂²⋊D₅₃	1	4	yes	—
278	C₂⁴	C₂³×D₅₃	1	3	yes	—
279	C₂²×C₄	—	2	24	no	—
280	C₂²×C₄	—	2	24	no	—
281	C₂²×C₄	—	2	24	no	—
282	C₂²×C₄	—	4	32	no	—
283	C₂²×C₄	—	53	24	no	—
284	C₂²×C₄	—	53	4	no	—
285	C₂²×C₄	—	53	64	no	—
286	C₂²×C₄	—	53	8	no	—
287	C₂²×C₄	—	106	32	no	—
288	C₂²×C₄	—	106	12	no	—
289	C₂²×C₄	—	106	12	no	—
290	C₂²×C₄	—	106	384	no	—
291	C₂²×C₄	—	106	32	no	—
292	C₂²×C₄	—	106	32	no	—
293	C₂²×C₄	—	106	24	no	—
294	C₂²×C₄	—	106	24	no	—
295	C₂²×C₄	—	106	24	no	—
296	C₂²×C₄	—	212	32	no	—
297	C₂²×C₄	#11719921	1	64	yes	complemented
298	C₂²×C₄	C₂×C₄×D₅₃	1	24	yes	—
299	C₂²×C₄	C₂×D₂₁₂	1	24	yes	—
300	C₂²×C₄	C₂×C₂²⋊D₅₃	1	24	yes	—
301	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
302	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
303	C₄⋊C₄	—	2	32	no	—
304	C₄⋊C₄	—	106	32	no	—
305	C₂²⋊C₄	—	106	64	no	—
306	C₂³	—	53	96	no	—
307	C₂³	—	53	8	no	—
308	C₂³	—	53	8	no	—
309	C₂³	—	53	8	no	—
310	C₂³	—	106	48	no	—
311	C₂³	—	106	48	no	—
312	C₂³	—	106	48	no	—
313	C₂³	—	106	48	no	—
314	C₂³	—	106	48	no	—
315	C₂³	—	106	48	no	—
316	C₂³	—	106	4	no	—
317	C₂³	—	106	4	no	—
318	C₂³	—	106	4	no	—
319	C₂³	—	106	4	no	—
320	C₂³	—	106	256	no	—
321	C₂³	C₂×#11719921	1	16	yes	—
322	C₂³	C₄⋊C₄×D₅₃	1	16	yes	—
323	C₂³	#13133175	1	16	yes	—
324	C₂³	#13133188	1	16	yes	—
325	C₂³	#13133192	1	16	yes	—
326	C₂³	#13133193	1	16	yes	—
327	C₂³	#13133197	1	16	yes	—
328	C₂³	C₂²×C₄×D₅₃	1	6	yes	—
329	C₂³	C₂×D₄×D₅₃	1	6	yes	—
330	C₂³	C₂²×C₂²⋊D₅₃	1	6	yes	—
331	C₂³	C₂²×D₂₁₂	1	6	yes	—
332	C₂³	C₂×Q₈×D₅₃	1	6	yes	—
333	C₂³	C₂×#11719947	1	6	yes	—
334	C₂³	C₂×#11719949	1	6	yes	—
335	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
336	C₂×C₄	—	2	8	no	—
337	C₂×C₄	—	2	8	no	—
338	C₂×C₄	—	2	8	no	—
339	C₂×C₄	—	4	48	no	—
340	C₂×C₄	—	53	8	no	—
341	C₂×C₄	—	106	48	no	—
342	C₂×C₄	—	106	4	no	—
343	C₂×C₄	—	106	4	no	—
344	C₂×C₄	—	106	128	no	—
345	C₂×C₄	—	106	8	no	—
346	C₂×C₄	—	106	8	no	—
347	C₂×C₄	—	106	8	no	—
348	C₂×C₄	—	212	48	no	—
349	C₂×C₄	C₂×#11719921	1	96	yes	complemented
350	C₂×C₄	C₂²×C₄×D₅₃	1	8	yes	—
351	C₂×C₄	C₂²×C₂²⋊D₅₃	1	8	yes	—
352	C₂×C₄	C₂²×D₂₁₂	1	8	yes	—
353	C₂²	—	53	16	no	—
354	C₂²	—	106	192	no	—
355	C₂²	—	106	8	no	—
356	C₂²	—	106	8	no	—
357	C₂²	—	106	8	no	—
358	C₂²	—	106	8	no	—
359	C₂²	—	106	8	no	—
360	C₂²	—	106	8	no	—
361	C₂²	#16810480	1	64	yes	direct product factor
362	C₂²	C₂²×#11719921	1	12	yes	—
363	C₂²	C₂×C₄⋊C₄×D₅₃	1	12	yes	—
364	C₂²	C₂×#13133175	1	12	yes	—
365	C₂²	C₂×#13133188	1	12	yes	—
366	C₂²	C₂×#13133192	1	12	yes	—
367	C₂²	C₂×#13133193	1	12	yes	—
368	C₂²	C₂×#13133197	1	12	yes	—
369	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
370	C₂²	C₂²×D₄×D₅₃	1	1	yes	characteristic
371	C₂²	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
372	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
373	C₂²	C₂²×Q₈×D₅₃	1	1	yes	characteristic
374	C₂²	C₂²×#11719947	1	1	yes	characteristic
375	C₂²	C₂²×#11719949	1	1	yes	characteristic
376	C₄	—	4	8	no	—
377	C₄	—	106	8	no	—
378	C₄	—	212	8	no	—
379	C₄	C₂²×#11719921	1	16	yes	complemented
380	C₂	—	106	16	no	—
381	C₂	C₂×#16810480	1	24	yes	direct product factor
382	C₂	C₂³×#11719921	1	1	yes	characteristic
383	C₂	C₂²×C₄⋊C₄×D₅₃	1	1	yes	characteristic
384	C₂	C₂²×#13133175	1	1	yes	characteristic
385	C₂	C₂²×#13133188	1	1	yes	characteristic
386	C₂	C₂²×#13133192	1	1	yes	characteristic
387	C₂	C₂²×#13133193	1	1	yes	characteristic
388	C₂	C₂²×#13133197	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂³×C₄⋊C₄×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂³
prime radical	C₂⁵×D₅₃	C₂²
cosocle	C₂³×C₅₃	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂²×#380
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄⋊Dic₅₃	C₂	semidirect
4. C₂×#16810480	C₂	direct (N × H)
5. C₂²×#13133124	C₂	semidirect
6. C₂³×C₄⋊C₄×C₅₃	C₂	semidirect
12. C₂³×C₄×D₅₃	C₄	semidirect
15. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
18. #16810480	C₂²	direct (N × H)
19. C₂×#13133124	C₂²	semidirect
23. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
43. C₂²×C₄×D₅₃	C₂×C₄	semidirect
46. C₂×C₄⋊Dic₅₃	C₂³	semidirect
49. #13133124	C₂³	semidirect
60. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
79. C₂³×D₅₃	C₄⋊C₄	semidirect
88. C₂²×C₄×C₅₃	C₂²⋊C₄	semidirect
107. C₂×C₄×D₅₃	C₂²×C₄	semidirect
137. C₂²×D₅₃	C₂×C₄⋊C₄	semidirect
145. C₂×C₄×C₅₃	C₂×C₂²⋊C₄	semidirect
159. C₂²×C₅₃	#380	semidirect
181. C₂×D₅₃	C₂²×C₄⋊C₄	semidirect
183. C₄×C₅₃	C₂²×C₂²⋊C₄	semidirect
188. C₂³×C₄⋊C₄	D₅₃	semidirect
192. C₂×C₅₃	C₂×#380	semidirect
213. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
218. C₅₃	C₂²×#380	semidirect
252. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
297. C₂²×C₄	#11719921	semidirect
349. C₂×C₄	C₂×#11719921	semidirect
361. C₂²	#16810480	direct (N × H)
379. C₄	C₂²×#11719921	semidirect
381. C₂	C₂×#16810480	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄×D₅₃	2
C₂³×C₄⋊Dic₅₃	2
C₂³×#11719921	2
C₂×#16810480	2
C₂²×#13133124	2
C₂³×C₄⋊C₄×C₅₃	2
C₂²×#380	53

Subgroups — C22×#16810480 #27542349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810480 #27542349