Subgroups — #27289936

Order 13568.

1086 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810385	C₂	1	4	yes	maximal
4	C₂×C₅₃×#384	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₂²	1	2	yes	—
6	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₅₃	—	2	1	no	—
8	C₂×C₄⋊#88322	—	2	2	no	—
9	C₂×C₄⋊#88322	—	2	2	no	—
10	C₂×C₄⋊#88322	C₄	1	4	yes	complemented
11	C₄×C₄⋊Dic₅₃	—	2	2	no	—
12	C₄×C₄⋊Dic₅₃	C₄	1	4	yes	complemented
13	C₂×#13133124	—	2	1	no	—
14	#16810385	—	2	8	no	—
15	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
17	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
18	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#384	—	2	2	no	—
20	C₅₃×#384	C₄	1	4	yes	complemented
21	C₂³×Dic₅₃	—	2	2	no	—
22	C₂²×#88322	—	2	4	no	—
23	C₂²×#88322	—	2	2	no	—
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	—	2	4	no	—
26	C₂²×#88322	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊Dic₅₃	—	2	2	no	—
28	C₂×C₄⋊Dic₅₃	—	2	2	no	—
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊Dic₅₃	D₄	1	4	yes	—
32	C₂×C₄⋊Dic₅₃	Q₈	1	4	yes	—
33	C₂×C₄×Dic₅₃	—	2	4	no	—
34	C₂×C₄×Dic₅₃	—	4	2	no	—
35	C₂×C₄×Dic₅₃	—	4	2	no	—
36	C₄⋊#88322	—	2	8	no	—
37	#13133124	—	2	4	no	—
38	C₂³×C₄×C₅₃	—	2	2	no	—
39	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
48	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
50	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₅₃	D₄	1	2	yes	—
55	C₂×C₄²×C₅₃	D₄	1	2	yes	—
56	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
57	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	—	2	4	no	—
61	C₂²×C₄×C₅₃	—	4	4	no	—
62	C₂²×C₄×C₅₃	—	4	2	no	—
63	C₂²×C₄×C₅₃	—	4	2	no	—
64	C₂²×C₄×C₅₃	—	4	4	no	—
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅₃	D₈	1	2	yes	—
72	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
73	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
74	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₅₃	#41	1	2	yes	—
84	C₂²×C₄×C₅₃	#41	1	2	yes	—
85	C₂²×Dic₅₃	—	2	4	no	—
86	C₂²×Dic₅₃	—	2	4	no	—
87	C₂²×Dic₅₃	—	2	4	no	—
88	C₂²×Dic₅₃	—	4	2	no	—
89	C₂²×Dic₅₃	—	4	2	no	—
90	C₂²×Dic₅₃	—	4	2	no	—
91	C₂²×Dic₅₃	—	4	2	no	—
92	C₄⋊C₄×C₅₃	—	2	8	no	—
93	C₄⋊C₄×C₅₃	—	4	8	no	—
94	C₄⋊C₄×C₅₃	—	4	8	no	—
95	C₄⋊C₄×C₅₃	—	4	2	no	—
96	C₄⋊C₄×C₅₃	—	4	2	no	—
97	C₄²×C₅₃	C₄²	1	4	yes	—
98	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
99	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
100	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
101	C₄⋊Dic₅₃	—	2	4	no	—
102	C₄⋊Dic₅₃	—	2	4	no	—
103	C₄⋊Dic₅₃	—	2	4	no	—
104	C₄⋊Dic₅₃	C₄⋊C₄	1	8	yes	complemented
105	C₄×Dic₅₃	—	4	8	no	—
106	C₂×#88322	—	2	4	no	—
107	C₂×#88322	—	2	8	no	—
108	C₂×#88322	—	4	4	no	—
109	C₂×#88322	—	4	2	no	—
110	C₂×#88322	—	4	2	no	—
111	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂³×C₅₃	#121	1	2	yes	—
115	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
116	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
117	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
120	C₂³×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
121	C₂³×C₅₃	#137	1	2	yes	—
122	C₂×C₄×C₅₃	—	2	4	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	4	no	—
126	C₂×C₄×C₅₃	—	4	4	no	—
127	C₂×C₄×C₅₃	—	4	4	no	—
128	C₂×C₄×C₅₃	—	4	4	no	—
129	C₂×C₄×C₅₃	—	8	4	no	—
130	C₂×C₄×C₅₃	—	8	4	no	—
131	C₂×C₄×C₅₃	#95	1	4	yes	—
132	C₂×C₄×C₅₃	#95	1	4	yes	—
133	C₂×C₄×C₅₃	#102	1	4	yes	—
134	C₂×C₄×C₅₃	#102	1	4	yes	—
135	C₂×C₄×C₅₃	#103	1	4	yes	—
136	C₂×C₄×C₅₃	#103	1	4	yes	—
137	C₂×C₄×C₅₃	#106	1	4	yes	—
138	C₂×C₄×C₅₃	#107	1	4	yes	—
139	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#117	1	2	yes	—
145	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
146	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
147	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
148	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
149	C₂×C₄×C₅₃	#121	1	2	yes	—
150	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
151	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
152	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
153	C₂×C₄×C₅₃	#125	1	2	yes	—
154	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
157	C₂×Dic₅₃	—	2	8	no	—
158	C₂×Dic₅₃	—	4	4	no	—
159	C₂×Dic₅₃	—	4	4	no	—
160	C₂×Dic₅₃	—	4	4	no	—
161	C₂×Dic₅₃	—	4	4	no	—
162	C₂×Dic₅₃	—	4	4	no	—
163	C₂×Dic₅₃	—	4	4	no	—
164	#88322	—	4	4	no	—
165	#10849	—	53	1	no	maximal, Sylow
166	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
167	C₂²×C₅₃	#380	1	2	yes	—
168	C₂²×C₅₃	#389	1	2	yes	—
169	C₂²×C₅₃	#390	1	1	yes	characteristic
170	C₂²×C₅₃	#391	1	1	yes	characteristic
171	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
172	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
173	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
174	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
175	C₂²×C₅₃	#417	1	2	yes	—
176	C₂²×C₅₃	#418	1	1	yes	characteristic
177	C₂²×C₅₃	#419	1	1	yes	characteristic
178	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
179	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
180	C₂²×C₅₃	#428	1	2	yes	—
181	C₂²×C₅₃	#459	1	2	yes	—
182	C₂²×C₅₃	#460	1	2	yes	—
183	C₂²×C₅₃	#461	1	2	yes	—
184	C₂²×C₅₃	#462	1	2	yes	—
185	C₂²×C₅₃	#474	1	2	yes	—
186	C₂²×C₅₃	#475	1	2	yes	—
187	C₂²×C₅₃	#476	1	2	yes	—
188	C₂²×C₅₃	#477	1	2	yes	—
189	C₄×C₅₃	—	2	4	no	—
190	C₄×C₅₃	—	8	4	no	—
191	C₄×C₅₃	#340	1	8	yes	complemented
192	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
193	C₄×C₅₃	#382	1	1	yes	characteristic
194	C₄×C₅₃	#384	1	1	yes	characteristic
195	C₄×C₅₃	#389	1	2	yes	—
196	C₄×C₅₃	#390	1	1	yes	characteristic
197	C₄×C₅₃	#391	1	1	yes	characteristic
198	Dic₅₃	—	4	8	no	—
199	C₂²×C₄⋊C₈	—	53	1	no	—
200	C₂×#384	D₅₃	1	1	yes	characteristic
201	C₂×C₄×C₄⋊C₄	—	53	1	no	—
202	C₂×#340	—	53	4	no	—
203	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
204	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
205	C₂×C₅₃	#1736	1	1	yes	characteristic
206	C₂×C₅₃	#1802	1	2	yes	—
207	C₂×C₅₃	#1803	1	2	yes	—
208	C₂×C₅₃	#1947	1	2	yes	—
209	C₂×C₅₃	#1948	1	2	yes	—
210	C₂×C₅₃	#1981	1	1	yes	characteristic
211	C₂×C₅₃	#1982	1	1	yes	characteristic
212	C₂×C₅₃	#1987	1	1	yes	characteristic
213	C₂×C₅₃	#1988	1	1	yes	characteristic
214	C₂³×C₈	—	53	2	no	—
215	C₂²×C₄⋊C₄	—	2	1	no	—
216	C₂²×C₄⋊C₄	—	2	1	no	—
217	C₂²×C₄⋊C₄	—	53	2	no	—
218	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
219	C₂²×C₄²	—	106	1	no	—
220	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
221	C₂×C₄⋊C₈	—	53	4	no	—
222	C₂×C₄⋊C₈	—	106	2	no	—
223	C₂×C₄⋊C₈	—	106	2	no	—
224	#384	—	2	2	no	—
225	#384	Dic₅₃	1	4	yes	complemented
226	C₄×C₄⋊C₄	—	53	4	no	—
227	C₄×C₄⋊C₄	—	106	2	no	—
228	C₂×#95	—	106	1	no	—
229	#340	—	106	8	no	—
230	C₅₃	#10849	1	1	yes	characteristic, complemented, Sylow
231	C₂³×C₄	—	2	2	no	—
232	C₂³×C₄	—	106	2	no	—
233	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
234	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
235	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
236	C₂²×C₈	—	53	4	no	—
237	C₂²×C₈	—	106	2	no	—
238	C₂²×C₈	—	106	4	no	—
239	C₂²×C₈	—	106	4	no	—
240	C₂²×C₈	—	106	2	no	—
241	C₂×C₄⋊C₄	—	2	4	no	—
242	C₂×C₄⋊C₄	—	2	4	no	—
243	C₂×C₄⋊C₄	—	2	2	no	—
244	C₂×C₄⋊C₄	—	2	4	no	—
245	C₂×C₄⋊C₄	—	2	4	no	—
246	C₂×C₄⋊C₄	—	4	4	no	—
247	C₂×C₄⋊C₄	—	4	2	no	—
248	C₂×C₄⋊C₄	—	4	2	no	—
249	C₂×C₄⋊C₄	—	53	4	no	—
250	C₂×C₄⋊C₄	—	53	4	no	—
251	C₂×C₄⋊C₄	—	53	4	no	—
252	C₂×C₄⋊C₄	—	106	2	no	—
253	C₂×C₄⋊C₄	—	106	2	no	—
254	C₂×C₄⋊C₄	—	106	2	no	—
255	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
256	C₂×C₄²	—	106	4	no	—
257	C₂×C₄²	—	212	2	no	—
258	C₂×C₄²	—	212	2	no	—
259	C₂×C₄²	Dic₁₀₆	1	2	yes	—
260	C₂×C₄²	C₄×D₅₃	1	2	yes	—
261	C₂×C₄²	C₄×D₅₃	1	2	yes	—
262	C₂×C₄²	D₂₁₂	1	2	yes	—
263	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
264	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
265	C₄⋊C₈	—	106	8	no	—
266	#95	—	106	4	no	—
267	C₂⁴	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	4	no	—
271	C₂²×C₄	—	4	4	no	—
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	—	4	4	no	—
275	C₂²×C₄	—	106	4	no	—
276	C₂²×C₄	—	106	4	no	—
277	C₂²×C₄	—	106	4	no	—
278	C₂²×C₄	—	212	2	no	—
279	C₂²×C₄	—	212	2	no	—
280	C₂²×C₄	—	212	2	no	—
281	C₂²×C₄	—	212	2	no	—
282	C₂²×C₄	#11719919	1	2	yes	—
283	C₂²×C₄	#11719919	1	2	yes	—
284	C₂²×C₄	#11719921	1	2	yes	—
285	C₂²×C₄	#11719921	1	2	yes	—
286	C₂²×C₄	#11719922	1	2	yes	—
287	C₂²×C₄	#11719923	1	2	yes	—
288	C₂²×C₄	#11719924	1	2	yes	—
289	C₂²×C₄	#11719925	1	2	yes	—
290	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
291	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
292	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
293	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
294	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
295	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
296	C₂²×C₄	D₄×D₅₃	1	2	yes	—
297	C₂²×C₄	#11719947	1	2	yes	—
298	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
299	C₂²×C₄	#11719949	1	2	yes	—
300	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
301	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
302	C₂×C₈	—	106	4	no	—
303	C₂×C₈	—	106	8	no	—
304	C₂×C₈	—	212	4	no	—
305	C₂×C₈	—	212	2	no	—
306	C₂×C₈	—	212	2	no	—
307	C₄⋊C₄	—	2	8	no	—
308	C₄⋊C₄	—	4	8	no	—
309	C₄⋊C₄	—	4	8	no	—
310	C₄⋊C₄	—	4	2	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₄⋊C₄	—	53	8	no	—
313	C₄⋊C₄	—	106	4	no	—
314	C₄⋊C₄	—	106	4	no	—
315	C₄⋊C₄	—	106	4	no	—
316	C₄²	—	212	8	no	—
317	C₄²	C₄×Dic₅₃	1	4	yes	—
318	C₄²	#11719919	1	4	yes	—
319	C₄²	C₄⋊Dic₅₃	1	4	yes	—
320	C₄²	#11719921	1	4	yes	—
321	C₂³	C₂×#11719921	1	1	yes	characteristic
322	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
323	C₂³	C₂×#11719919	1	1	yes	characteristic
324	C₂³	#13133188	1	2	yes	—
325	C₂³	#13133193	1	2	yes	—
326	C₂³	C₂×#11719922	1	1	yes	characteristic
327	C₂³	C₂×#11719924	1	1	yes	characteristic
328	C₂³	C₂×#11719923	1	1	yes	characteristic
329	C₂³	C₂×#11719925	1	1	yes	characteristic
330	C₂³	#13133243	1	2	yes	—
331	C₂³	#13133248	1	2	yes	—
332	C₂×C₄	—	2	4	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	4	no	—
336	C₂×C₄	—	4	4	no	—
337	C₂×C₄	—	4	4	no	—
338	C₂×C₄	—	4	4	no	—
339	C₂×C₄	—	8	4	no	—
340	C₂×C₄	—	8	4	no	—
341	C₂×C₄	—	106	8	no	—
342	C₂×C₄	—	212	4	no	—
343	C₂×C₄	—	212	4	no	—
344	C₂×C₄	—	212	4	no	—
345	C₂×C₄	—	212	4	no	—
346	C₂×C₄	—	212	4	no	—
347	C₂×C₄	—	212	4	no	—
348	C₂×C₄	#13133124	1	4	yes	—
349	C₂×C₄	#13133124	1	4	yes	—
350	C₂×C₄	#13133137	1	4	yes	—
351	C₂×C₄	#13133138	1	4	yes	—
352	C₂×C₄	#13133140	1	4	yes	—
353	C₂×C₄	#13133141	1	4	yes	—
354	C₂×C₄	#13133148	1	4	yes	—
355	C₂×C₄	#13133150	1	4	yes	—
356	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
357	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
358	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
359	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
360	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
361	C₂×C₄	#13133175	1	2	yes	—
362	C₂×C₄	#13133179	1	2	yes	—
363	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
364	C₂×C₄	#13133182	1	2	yes	—
365	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
366	C₂×C₄	#13133192	1	2	yes	—
367	C₂×C₄	#13133197	1	2	yes	—
368	C₂×C₄	#13133204	1	1	yes	characteristic
369	C₂×C₄	#13133206	1	1	yes	characteristic
370	C₂×C₄	#13133208	1	2	yes	—
371	C₂×C₄	#13133213	1	1	yes	characteristic
372	C₂×C₄	#13133215	1	1	yes	characteristic
373	C₂×C₄	#13133216	1	2	yes	—
374	C₈	—	212	4	no	—
375	C₂²	C₂×#13133124	1	1	yes	characteristic
376	C₂²	#16810480	1	2	yes	—
377	C₂²	#16810519	1	2	yes	—
378	C₂²	#16810526	1	1	yes	characteristic
379	C₂²	#16810530	1	1	yes	characteristic
380	C₂²	C₂×#13133137	1	1	yes	characteristic
381	C₂²	C₂×#13133138	1	1	yes	characteristic
382	C₂²	C₂×#13133140	1	1	yes	characteristic
383	C₂²	C₂×#13133141	1	1	yes	characteristic
384	C₂²	#16810622	1	2	yes	—
385	C₂²	#16810628	1	1	yes	characteristic
386	C₂²	#16810632	1	1	yes	characteristic
387	C₂²	C₂×#13133148	1	1	yes	characteristic
388	C₂²	C₂×#13133150	1	1	yes	characteristic
389	C₂²	#16810663	1	2	yes	—
390	C₂²	#16810815	1	2	yes	—
391	C₂²	#16810820	1	2	yes	—
392	C₂²	#16810825	1	2	yes	—
393	C₂²	#16810830	1	2	yes	—
394	C₂²	#16810900	1	2	yes	—
395	C₂²	#16810905	1	2	yes	—
396	C₂²	#16810910	1	2	yes	—
397	C₂²	#16810915	1	2	yes	—
398	C₄	—	2	4	no	—
399	C₄	—	8	4	no	—
400	C₄	—	212	8	no	—
401	C₄	#16810385	1	8	yes	complemented
402	C₄	C₄⋊C₄×Dic₅₃	1	2	yes	—
403	C₄	#16810488	1	1	yes	characteristic
404	C₄	#16810492	1	1	yes	characteristic
405	C₄	#16810522	1	2	yes	—
406	C₄	#16810524	1	1	yes	characteristic
407	C₄	#16810528	1	1	yes	characteristic
408	C₂	#20272944	1	1	yes	characteristic
409	C₂	C₂×#16810385	1	1	yes	characteristic
410	C₂	#20274434	1	1	yes	characteristic
411	C₂	#20274712	1	2	yes	—
412	C₂	#20274717	1	2	yes	—
413	C₂	#20275471	1	2	yes	—
414	C₂	#20275476	1	2	yes	—
415	C₂	#20275639	1	1	yes	characteristic
416	C₂	#20275643	1	1	yes	characteristic
417	C₂	#20275664	1	1	yes	characteristic
418	C₂	#20275668	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#384	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#384	D₅₃
hypocenter	C₅₃	#10849
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
10. C₂×C₄⋊#88322	C₄	semidirect
12. C₄×C₄⋊Dic₅₃	C₄	semidirect
20. C₅₃×#384	C₄	semidirect
104. C₄⋊Dic₅₃	C₄⋊C₄	semidirect
191. C₄×C₅₃	#340	semidirect
225. #384	Dic₅₃	semidirect
230. C₅₃	#10849	semidirect
401. C₄	#16810385	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#88322	2
C₂×C₄×C₄⋊Dic₅₃	2
C₂×#16810385	2
C₂×C₅₃×#384	2
#10849	53