Subgroups — #27290035

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810661	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810578	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810394	C₂	1	2	yes	maximal
4	#20274426	C₂	1	2	yes	maximal
5	C₂²×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	1	yes	characteristic
7	C₂×#13133176	C₂²	1	2	yes	—
8	C₂×#13133176	C₂²	1	1	yes	characteristic
9	#16810661	—	2	2	no	—
10	#16810661	C₄	1	4	yes	complemented
11	C₂×#13133151	—	2	1	no	—
12	C₂×#13133149	—	2	1	no	—
13	#16810578	—	2	1	no	—
14	#16810578	—	2	1	no	—
15	#16810578	C₄	1	2	yes	complemented
16	#16810578	C₄	1	2	yes	complemented
17	C₂×#13133127	—	2	1	no	—
18	C₂×C₈×Dic₅₃	—	2	1	no	—
19	#16810394	—	2	4	no	—
20	#16810384	—	2	4	no	—
21	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
22	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#108×C₅₃	—	2	1	no	—
24	C₂×#108×C₅₃	—	2	1	no	—
25	C₂×#108×C₅₃	—	2	2	no	—
26	C₂×#108×C₅₃	C₄	1	4	yes	complemented
27	C₂×#11719917	—	2	1	no	—
28	C₂×#11719917	—	2	1	no	—
29	C₂×#11719917	—	2	1	no	—
30	C₂×#11719917	C₂×C₄	1	2	yes	—
31	C₂×#11719917	D₄	1	2	yes	—
32	C₂×#11719917	Q₈	1	2	yes	—
33	C₂²×#88322	—	2	1	no	—
34	#13133176	—	2	1	no	—
35	#13133176	—	2	2	no	—
36	#13133176	—	2	2	no	—
37	#13133176	—	2	1	no	—
38	#13133176	—	2	2	no	—
39	#13133176	C₂×C₄	1	4	yes	—
40	#13133176	C₂×C₄	1	2	yes	—
41	#13133176	D₄	1	2	yes	—
42	C₂×C₄⋊Dic₅₃	—	2	2	no	—
43	C₂×C₄⋊Dic₅₃	Q₈	1	2	yes	—
44	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
45	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
46	C₂×C₄×Dic₅₃	—	2	1	no	—
47	C₂×C₄×Dic₅₃	—	2	2	no	—
48	#13133151	—	2	4	no	—
49	#13133149	—	2	4	no	—
50	#13133127	—	2	2	no	—
51	#13133127	—	2	2	no	—
52	C₈×Dic₅₃	—	2	2	no	—
53	C₈×Dic₅₃	—	2	2	no	—
54	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×M₁₆×C₅₃	—	2	2	no	—
56	C₂×M₁₆×C₅₃	—	2	2	no	—
57	C₂×M₁₆×C₅₃	—	2	2	no	—
58	C₂×M₁₆×C₅₃	—	2	2	no	—
59	C₂×M₁₆×C₅₃	—	2	2	no	—
60	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
61	C₂²×C₈×C₅₃	—	2	2	no	—
62	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
65	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
66	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
68	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
69	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
70	C₂²×C₈×C₅₃	D₄	1	2	yes	—
71	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
72	#108×C₅₃	—	2	4	no	—
73	#108×C₅₃	—	2	4	no	—
74	C₂⁴×C₅₃	#41	1	1	yes	characteristic
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	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
89	C₂²×Dic₅₃	—	2	2	no	—
90	C₂²×Dic₅₃	—	4	2	no	—
91	M₁₆×C₅₃	—	2	4	no	—
92	M₁₆×C₅₃	—	2	4	no	—
93	M₁₆×C₅₃	—	4	2	no	—
94	M₁₆×C₅₃	—	4	2	no	—
95	C₂×C₈×C₅₃	—	2	4	no	—
96	C₂×C₈×C₅₃	—	4	2	no	—
97	C₂×C₈×C₅₃	—	4	2	no	—
98	C₂×C₈×C₅₃	C₄²	1	2	yes	—
99	C₂×C₈×C₅₃	C₄²	1	2	yes	—
100	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
105	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
106	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
107	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
108	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
109	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
110	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
111	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
112	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
113	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
114	C₂²⋊Dic₅₃	—	2	2	no	—
115	C₂²⋊Dic₅₃	—	2	2	no	—
116	C₂²⋊Dic₅₃	—	2	4	no	—
117	C₂²⋊Dic₅₃	—	4	2	no	—
118	C₄⋊Dic₅₃	—	2	2	no	—
119	C₄⋊Dic₅₃	—	2	4	no	—
120	C₄⋊Dic₅₃	—	4	2	no	—
121	C₄⋊Dic₅₃	C₄⋊C₄	1	4	yes	—
122	C₄×Dic₅₃	—	2	2	no	—
123	C₄×Dic₅₃	—	2	2	no	—
124	C₄×Dic₅₃	—	2	4	no	—
125	C₄×Dic₅₃	—	4	2	no	—
126	#11719917	—	2	2	no	—
127	#11719917	—	2	2	no	—
128	#11719917	—	2	2	no	—
129	#11719917	C₄⋊C₄	1	4	yes	—
130	C₂×#88322	—	2	2	no	—
131	C₂×#88322	—	2	2	no	—
132	C₂×#88322	—	2	2	no	—
133	C₂³×C₅₃	—	4	2	no	—
134	C₂³×C₅₃	#117	1	1	yes	characteristic
135	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
136	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
137	C₂³×C₅₃	#121	1	1	yes	characteristic
138	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂³×C₅₃	#124	1	1	yes	characteristic
140	C₂³×C₅₃	#125	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
157	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
158	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
159	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
163	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
164	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
165	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
169	C₂×Dic₅₃	—	2	4	no	—
170	C₂×Dic₅₃	—	4	4	no	—
171	C₂×Dic₅₃	—	4	4	no	—
172	C₂×Dic₅₃	—	8	4	no	—
173	C₈×C₅₃	—	4	4	no	—
174	C₈×C₅₃	#95	1	2	yes	—
175	C₈×C₅₃	#95	1	2	yes	—
176	C₈×C₅₃	#95	1	2	yes	—
177	C₈×C₅₃	#95	1	2	yes	—
178	#88322	—	2	4	no	—
179	#10872	—	53	1	no	maximal, Sylow
180	C₂²×C₅₃	—	2	2	no	—
181	C₂²×C₅₃	—	2	2	no	—
182	C₂²×C₅₃	—	2	2	no	—
183	C₂²×C₅₃	—	4	1	no	—
184	C₂²×C₅₃	—	4	1	no	—
185	C₂²×C₅₃	—	4	1	no	—
186	C₂²×C₅₃	—	4	1	no	—
187	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
188	C₂²×C₅₃	#380	1	1	yes	characteristic
189	C₂²×C₅₃	#382	1	1	yes	characteristic
190	C₂²×C₅₃	#389	1	1	yes	characteristic
191	C₂²×C₅₃	#389	1	1	yes	characteristic
192	C₂²×C₅₃	#390	1	1	yes	characteristic
193	C₂²×C₅₃	#391	1	1	yes	characteristic
194	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
195	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
196	C₄×C₅₃	#380	1	1	yes	characteristic
197	C₄×C₅₃	#384	1	1	yes	characteristic
198	C₄×C₅₃	#389	1	1	yes	characteristic
199	C₄×C₅₃	#389	1	1	yes	characteristic
200	C₄×C₅₃	#390	1	1	yes	characteristic
201	C₄×C₅₃	#391	1	1	yes	characteristic
202	Dic₅₃	—	4	4	no	—
203	Dic₅₃	—	8	4	no	—
204	C₂²×#108	D₅₃	1	1	yes	characteristic
205	C₂×#427	—	53	1	no	—
206	C₂×#405	—	53	1	no	—
207	C₂×#344	—	53	2	no	—
208	#1733	—	53	2	no	—
209	C₂×C₅₃	—	2	1	no	—
210	C₂×C₅₃	—	2	1	no	—
211	C₂×C₅₃	—	2	1	no	—
212	C₂×C₅₃	—	2	1	no	—
213	C₂×C₅₃	—	2	1	no	—
214	C₂×C₅₃	—	2	1	no	—
215	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
216	C₂×C₅₃	#1814	1	2	yes	—
217	C₂²×M₁₆	—	53	1	no	—
218	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
219	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
220	C₂×#117	—	53	1	no	—
221	C₂×#117	—	53	2	no	—
222	C₂×#108	—	2	1	no	—
223	C₂×#108	—	2	1	no	—
224	C₂×#108	—	2	2	no	—
225	C₂×#108	Dic₅₃	1	4	yes	complemented
226	#427	—	53	4	no	—
227	#427	—	106	2	no	—
228	C₂×#107	—	106	1	no	—
229	C₂×#106	—	106	1	no	—
230	#405	—	53	2	no	—
231	#405	—	53	2	no	—
232	#405	—	106	1	no	—
233	#405	—	106	1	no	—
234	C₂×#97	—	106	1	no	—
235	C₂×C₄×C₈	—	106	1	no	—
236	#344	—	106	4	no	—
237	#339	—	106	4	no	—
238	C₅₃	#10872	1	1	yes	characteristic, complemented, Sylow
239	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
240	C₂×M₁₆	—	2	2	no	—
241	C₂×M₁₆	—	2	2	no	—
242	C₂×M₁₆	—	2	2	no	—
243	C₂×M₁₆	—	2	2	no	—
244	C₂×M₁₆	—	2	2	no	—
245	C₂×M₁₆	—	53	2	no	—
246	C₂×M₁₆	—	53	2	no	—
247	C₂×M₁₆	—	53	2	no	—
248	C₂×M₁₆	—	106	1	no	—
249	C₂×M₁₆	—	106	1	no	—
250	C₂×M₁₆	—	106	1	no	—
251	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
252	C₂²×C₈	—	2	2	no	—
253	C₂²×C₈	—	106	1	no	—
254	C₂²×C₈	Dic₁₀₆	1	2	yes	—
255	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
256	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
257	C₂²×C₈	C₄×D₅₃	1	2	yes	—
258	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
259	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
260	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
261	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
262	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
263	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
264	#117	—	53	2	no	—
265	#117	—	53	2	no	—
266	#117	—	53	4	no	—
267	#117	—	106	1	no	—
268	#117	—	106	1	no	—
269	#117	—	106	2	no	—
270	#117	—	106	2	no	—
271	#117	—	106	2	no	—
272	C₂×C₄⋊C₄	—	53	2	no	—
273	C₂×C₄⋊C₄	—	106	2	no	—
274	C₂×C₂²⋊C₄	—	106	1	no	—
275	C₂×C₂²⋊C₄	—	106	2	no	—
276	C₂×C₄²	—	106	1	no	—
277	C₂×C₄²	—	106	2	no	—
278	#108	—	2	4	no	—
279	#108	—	2	4	no	—
280	#107	—	106	4	no	—
281	#106	—	106	4	no	—
282	#97	—	106	2	no	—
283	#97	—	106	2	no	—
284	C₄×C₈	—	106	2	no	—
285	C₄×C₈	—	106	2	no	—
286	C₂⁴	#11719949	1	1	yes	characteristic
287	C₂²×C₄	—	106	2	no	—
288	C₂²×C₄	—	212	2	no	—
289	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
291	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
293	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
294	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
295	C₂²×C₄	#11719947	1	1	yes	characteristic
296	C₂²×C₄	#11719947	1	1	yes	characteristic
297	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
298	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
299	C₂²×C₄	#11719949	1	1	yes	characteristic
300	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
301	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
302	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
303	M₁₆	—	2	4	no	—
304	M₁₆	—	2	4	no	—
305	M₁₆	—	4	2	no	—
306	M₁₆	—	4	2	no	—
307	M₁₆	—	53	4	no	—
308	M₁₆	—	106	2	no	—
309	M₁₆	—	106	2	no	—
310	M₁₆	—	106	2	no	—
311	C₂×C₈	—	2	4	no	—
312	C₂×C₈	—	4	2	no	—
313	C₂×C₈	—	4	2	no	—
314	C₂×C₈	—	106	2	no	—
315	C₂×C₈	—	106	2	no	—
316	C₂×C₈	—	106	2	no	—
317	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
318	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
319	C₂×C₈	#11719919	1	2	yes	—
320	C₂×C₈	#11719919	1	2	yes	—
321	C₂×C₈	#11719919	1	2	yes	—
322	C₂×C₈	#11719919	1	2	yes	—
323	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
324	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
325	C₂×C₈	#11719921	1	1	yes	characteristic
326	C₂×C₈	#11719921	1	1	yes	characteristic
327	C₂×C₈	#11719921	1	1	yes	characteristic
328	C₂×C₈	#11719921	1	1	yes	characteristic
329	C₂×C₈	#11719921	1	2	yes	—
330	C₂×C₈	#11719921	1	2	yes	—
331	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
332	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
333	C₄⋊C₄	—	53	4	no	—
334	C₄⋊C₄	—	106	2	no	—
335	C₄⋊C₄	—	106	4	no	—
336	C₄⋊C₄	—	212	2	no	—
337	C₂²⋊C₄	—	106	2	no	—
338	C₂²⋊C₄	—	106	2	no	—
339	C₂²⋊C₄	—	106	4	no	—
340	C₂²⋊C₄	—	212	2	no	—
341	C₄²	—	106	2	no	—
342	C₄²	—	106	2	no	—
343	C₄²	—	106	4	no	—
344	C₄²	—	212	2	no	—
345	C₂³	—	4	2	no	—
346	C₂³	#13133175	1	1	yes	characteristic
347	C₂³	#13133179	1	1	yes	characteristic
348	C₂³	Q₈×Dic₅₃	1	1	yes	characteristic
349	C₂³	#13133192	1	1	yes	characteristic
350	C₂³	#13133193	1	1	yes	characteristic
351	C₂³	#13133206	1	1	yes	characteristic
352	C₂³	#13133208	1	1	yes	characteristic
353	C₂×C₄	—	106	4	no	—
354	C₂×C₄	—	424	4	no	—
355	C₂×C₄	—	212	4	no	—
356	C₂×C₄	—	212	4	no	—
357	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
358	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
359	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
360	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
361	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
362	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
363	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
364	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
365	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
366	C₂×C₄	#13133175	1	1	yes	characteristic
367	C₂×C₄	#13133179	1	1	yes	characteristic
368	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
369	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
370	C₂×C₄	#13133182	1	1	yes	characteristic
371	C₂×C₄	#13133182	1	1	yes	characteristic
372	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
373	C₂×C₄	#13133188	1	1	yes	characteristic
374	C₂×C₄	#13133188	1	1	yes	characteristic
375	C₂×C₄	#13133192	1	1	yes	characteristic
376	C₂×C₄	#13133193	1	1	yes	characteristic
377	C₂×C₄	#13133197	1	1	yes	characteristic
378	C₂×C₄	#13133197	1	1	yes	characteristic
379	C₂×C₄	#13133204	1	1	yes	characteristic
380	C₂×C₄	#13133208	1	1	yes	characteristic
381	C₂×C₄	#13133213	1	1	yes	characteristic
382	C₂×C₄	#13133215	1	1	yes	characteristic
383	C₂×C₄	#13133216	1	1	yes	characteristic
384	C₂×C₄	#13133216	1	1	yes	characteristic
385	C₈	—	4	4	no	—
386	C₈	—	106	4	no	—
387	C₈	#13133124	1	2	yes	—
388	C₈	#13133124	1	2	yes	—
389	C₈	#13133124	1	2	yes	—
390	C₈	#13133124	1	2	yes	—
391	C₂²	—	2	2	no	—
392	C₂²	—	2	2	no	—
393	C₂²	—	2	2	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	4	1	no	—
397	C₂²	—	4	1	no	—
398	C₂²	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
399	C₂²	#16810480	1	1	yes	characteristic
400	C₂²	#16810488	1	1	yes	characteristic
401	C₂²	#16810519	1	1	yes	characteristic
402	C₂²	#16810522	1	1	yes	characteristic
403	C₂²	#16810524	1	1	yes	characteristic
404	C₂²	#16810530	1	1	yes	characteristic
405	C₄	—	424	4	no	—
406	C₄	—	212	4	no	—
407	C₄	C₂×#13133124	1	1	yes	characteristic
408	C₄	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
409	C₄	#16810480	1	1	yes	characteristic
410	C₄	#16810492	1	1	yes	characteristic
411	C₄	#16810519	1	1	yes	characteristic
412	C₄	#16810522	1	1	yes	characteristic
413	C₄	#16810526	1	1	yes	characteristic
414	C₄	#16810528	1	1	yes	characteristic
415	C₂	—	2	1	no	—
416	C₂	—	2	1	no	—
417	C₂	—	2	1	no	—
418	C₂	—	2	1	no	—
419	C₂	—	2	1	no	—
420	C₂	—	2	1	no	—
421	C₂	#20272944	1	1	yes	characteristic
422	C₂	#20274770	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133188
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂²×#108×C₅₃	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×#108	D₅₃
hypocenter	C₅₃	#10872
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
10. #16810661	C₄	semidirect
15. #16810578	C₄	semidirect
16. #16810578	C₄	semidirect
26. C₂×#108×C₅₃	C₄	semidirect
225. C₂×#108	Dic₅₃	semidirect
238. C₅₃	#10872	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810661	2
C₂×#16810578	2
C₂×#16810394	2
#20274426	2
C₂²×#108×C₅₃	2
#10872	53