Subgroups — #27290017

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810661	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810394	C₂	1	2	yes	maximal
4	C₂×#16810379	C₂	1	2	yes	maximal
5	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂×#13133176	C₂²	1	2	yes	—
8	C₂×#13133153	—	2	1	no	—
9	C₂×#13133153	—	2	1	no	—
10	C₂×#13133153	—	2	2	no	—
11	C₂×#13133153	C₄	1	4	yes	complemented
12	#16810661	—	2	2	no	—
13	#16810661	C₄	1	4	yes	complemented
14	C₂×#13133151	—	2	1	no	—
15	C₂×#13133149	—	2	1	no	—
16	#16810394	—	2	4	no	—
17	#16810379	—	2	4	no	—
18	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#405	—	2	2	no	—
22	C₅₃×#405	—	2	2	no	—
23	C₂×#97×C₅₃	C₄	1	2	yes	complemented
24	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	complemented
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	—	2	2	no	—
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719917	—	2	2	no	—
29	C₂×#11719917	—	2	2	no	—
30	C₂×#11719917	C₂×C₄	1	4	yes	—
31	C₂²×#88322	—	2	2	no	—
32	#13133176	—	2	2	no	—
33	#13133176	—	2	2	no	—
34	#13133176	—	2	2	no	—
35	#13133176	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊Dic₅₃	—	2	2	no	—
37	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
38	C₂×C₄×Dic₅₃	—	2	2	no	—
39	#13133153	—	2	4	no	—
40	#13133153	—	2	4	no	—
41	#13133151	—	2	4	no	—
42	#13133149	—	2	4	no	—
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₅₃	—	2	1	no	—
45	C₂×M₁₆×C₅₃	—	2	1	no	—
46	C₂×M₁₆×C₅₃	—	2	1	no	—
47	C₂×M₁₆×C₅₃	—	2	1	no	—
48	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
49	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	D₄	1	2	yes	—
58	C₂²×C₈×C₅₃	D₄	1	2	yes	—
59	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
61	#117×C₅₃	—	2	1	no	—
62	#117×C₅₃	—	2	1	no	—
63	#117×C₅₃	D₄	1	2	yes	—
64	#117×C₅₃	Q₈	1	2	yes	—
65	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
67	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
68	#97×C₅₃	—	2	4	no	—
69	C₄×C₈×C₅₃	—	2	4	no	—
70	C₂⁴×C₅₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	—	2	2	no	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×D₄	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×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
86	C₂²×Dic₅₃	—	4	2	no	—
87	M₁₆×C₅₃	—	2	2	no	—
88	M₁₆×C₅₃	—	2	2	no	—
89	M₁₆×C₅₃	—	2	2	no	—
90	M₁₆×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	C₄²	1	2	yes	—
94	C₂×C₈×C₅₃	C₄²	1	2	yes	—
95	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
97	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
98	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
99	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
100	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
101	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
102	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
103	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
104	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
105	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
106	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₄⋊C₄×C₅₃	—	2	2	no	—
111	C₄⋊C₄×C₅₃	—	2	2	no	—
112	C₂²⋊C₄×C₅₃	—	2	2	no	—
113	C₂²⋊C₄×C₅₃	—	2	2	no	—
114	C₄²×C₅₃	—	2	2	no	—
115	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
116	C₂²⋊Dic₅₃	—	2	4	no	—
117	C₂²⋊Dic₅₃	—	4	2	no	—
118	C₄⋊Dic₅₃	—	2	4	no	—
119	C₄⋊Dic₅₃	—	4	2	no	—
120	C₄×Dic₅₃	—	2	4	no	—
121	C₄×Dic₅₃	—	4	2	no	—
122	#11719917	—	2	4	no	—
123	#11719917	—	2	4	no	—
124	#11719917	—	4	2	no	—
125	#11719917	—	4	2	no	—
126	C₂×#88322	—	2	4	no	—
127	C₂×#88322	—	4	2	no	—
128	C₂×#88322	—	4	2	no	—
129	C₂³×C₅₃	—	4	2	no	—
130	C₂³×C₅₃	#117	1	1	yes	characteristic
131	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
132	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
133	C₂³×C₅₃	#121	1	1	yes	characteristic
134	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂³×C₅₃	#124	1	1	yes	characteristic
136	C₂³×C₅₃	#125	1	1	yes	characteristic
137	C₂×C₄×C₅₃	—	2	4	no	—
138	C₂×C₄×C₅₃	—	4	4	no	—
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×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×C₄⋊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	#117	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
155	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
156	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
157	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
161	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
162	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
163	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
167	C₂×Dic₅₃	—	4	4	no	—
168	C₂×Dic₅₃	—	8	4	no	—
169	C₈×C₅₃	—	2	4	no	—
170	C₈×C₅₃	#95	1	2	yes	—
171	C₈×C₅₃	#95	1	2	yes	—
172	C₈×C₅₃	#95	1	2	yes	—
173	C₈×C₅₃	#95	1	2	yes	—
174	#88322	—	4	4	no	—
175	#10868	—	53	1	no	maximal, Sylow
176	C₂²×C₅₃	—	2	2	no	—
177	C₂²×C₅₃	—	2	2	no	—
178	C₂²×C₅₃	—	2	2	no	—
179	C₂²×C₅₃	—	4	1	no	—
180	C₂²×C₅₃	—	4	1	no	—
181	C₂²×C₅₃	—	4	1	no	—
182	C₂²×C₅₃	—	4	1	no	—
183	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
184	C₂²×C₅₃	#380	1	1	yes	characteristic
185	C₂²×C₅₃	#382	1	1	yes	characteristic
186	C₂²×C₅₃	#389	1	1	yes	characteristic
187	C₂²×C₅₃	#389	1	1	yes	characteristic
188	C₂²×C₅₃	#390	1	1	yes	characteristic
189	C₂²×C₅₃	#391	1	1	yes	characteristic
190	C₄×C₅₃	—	4	4	no	—
191	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
192	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
193	C₄×C₅₃	#380	1	1	yes	characteristic
194	C₄×C₅₃	#384	1	1	yes	characteristic
195	C₄×C₅₃	#389	1	1	yes	characteristic
196	C₄×C₅₃	#389	1	1	yes	characteristic
197	C₄×C₅₃	#390	1	1	yes	characteristic
198	C₄×C₅₃	#391	1	1	yes	characteristic
199	Dic₅₃	—	8	4	no	—
200	C₂²×#108	—	53	1	no	—
201	C₂×#427	—	53	1	no	—
202	C₂×#405	D₅₃	1	1	yes	characteristic
203	C₂×#344	—	53	2	no	—
204	C₂×H(3,ℤ/4)	—	53	2	no	—
205	C₂×C₅₃	—	2	1	no	—
206	C₂×C₅₃	—	2	1	no	—
207	C₂×C₅₃	—	2	1	no	—
208	C₂×C₅₃	—	2	1	no	—
209	C₂×C₅₃	—	2	1	no	—
210	C₂×C₅₃	—	2	1	no	—
211	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
212	C₂×C₅₃	#1833	1	2	yes	—
213	C₂²×M₁₆	—	53	2	no	—
214	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
215	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
216	C₂×#117	—	53	2	no	—
217	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
218	C₂×#108	—	53	4	no	—
219	C₂×#108	—	106	1	no	—
220	C₂×#108	—	106	1	no	—
221	C₂×#108	—	106	2	no	—
222	#427	—	53	4	no	—
223	#427	—	106	2	no	—
224	C₂×#107	—	106	1	no	—
225	C₂×#106	—	106	1	no	—
226	#405	—	2	2	no	—
227	#405	—	2	2	no	—
228	C₂×#97	Dic₅₃	1	2	yes	complemented
229	C₂×C₄×C₈	Dic₅₃	1	2	yes	complemented
230	#344	—	106	4	no	—
231	H(3,ℤ/4)	—	106	4	no	—
232	C₅₃	#10868	1	1	yes	characteristic, complemented, Sylow
233	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
234	C₂×M₁₆	—	2	1	no	—
235	C₂×M₁₆	—	2	1	no	—
236	C₂×M₁₆	—	2	1	no	—
237	C₂×M₁₆	—	2	1	no	—
238	C₂×M₁₆	—	53	4	no	—
239	C₂×M₁₆	—	106	2	no	—
240	C₂×M₁₆	—	106	2	no	—
241	C₂×M₁₆	—	106	2	no	—
242	C₂×M₁₆	—	106	2	no	—
243	C₂×M₁₆	—	106	2	no	—
244	C₂×M₁₆	Dic₁₀₆	1	2	yes	—
245	C₂×M₁₆	D₂₁₂	1	2	yes	—
246	C₂²×C₈	—	106	2	no	—
247	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
248	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
249	C₂²×C₈	C₄×D₅₃	1	2	yes	—
250	C₂²×C₈	C₄×D₅₃	1	2	yes	—
251	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
252	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
253	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
254	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
255	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
256	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
257	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
258	#117	—	2	1	no	—
259	#117	—	2	1	no	—
260	#117	—	53	4	no	—
261	#117	—	106	2	no	—
262	#117	—	106	2	no	—
263	#117	—	106	2	no	—
264	#117	Dic₁₀₆	1	2	yes	—
265	#117	D₂₁₂	1	2	yes	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	106	2	no	—
268	C₂×C₂²⋊C₄	—	2	1	no	—
269	C₂×C₂²⋊C₄	—	106	2	no	—
270	C₂×C₄²	—	106	2	no	—
271	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
272	#108	—	106	4	no	—
273	#108	—	106	4	no	—
274	#107	—	106	4	no	—
275	#106	—	106	4	no	—
276	#97	—	2	4	no	—
277	C₄×C₈	—	2	4	no	—
278	C₂⁴	#11719945	1	1	yes	characteristic
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	212	2	no	—
281	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
282	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
283	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
284	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
287	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
289	C₂²×C₄	#11719945	1	1	yes	characteristic
290	C₂²×C₄	#11719945	1	1	yes	characteristic
291	C₂²×C₄	#11719945	1	1	yes	characteristic
292	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
293	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
294	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
295	M₁₆	—	2	2	no	—
296	M₁₆	—	2	2	no	—
297	M₁₆	—	2	2	no	—
298	M₁₆	—	2	2	no	—
299	M₁₆	—	106	4	no	—
300	M₁₆	—	106	4	no	—
301	M₁₆	—	212	2	no	—
302	M₁₆	—	212	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	106	4	no	—
306	C₂×C₈	—	212	2	no	—
307	C₂×C₈	—	212	2	no	—
308	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
309	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
310	C₂×C₈	#11719919	1	2	yes	—
311	C₂×C₈	#11719919	1	2	yes	—
312	C₂×C₈	#11719919	1	2	yes	—
313	C₂×C₈	#11719919	1	2	yes	—
314	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
315	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
316	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
317	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
318	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
319	C₂×C₈	#11719921	1	2	yes	—
320	C₂×C₈	#11719921	1	2	yes	—
321	C₂×C₈	#11719921	1	2	yes	—
322	C₂×C₈	#11719921	1	2	yes	—
323	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
324	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	106	4	no	—
328	C₄⋊C₄	—	212	2	no	—
329	C₂²⋊C₄	—	2	2	no	—
330	C₂²⋊C₄	—	2	2	no	—
331	C₂²⋊C₄	—	106	4	no	—
332	C₂²⋊C₄	—	212	2	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	106	4	no	—
335	C₄²	—	212	2	no	—
336	C₄²	C₄⋊Dic₅₃	1	4	yes	—
337	C₂³	—	4	2	no	—
338	C₂³	#13133176	1	1	yes	characteristic
339	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
340	C₂³	C₄×Dic₁₀₆	1	1	yes	characteristic
341	C₂³	#13133191	1	1	yes	characteristic
342	C₂³	#13133196	1	1	yes	characteristic
343	C₂³	#13133205	1	1	yes	characteristic
344	C₂³	#13133209	1	1	yes	characteristic
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	4	4	no	—
347	C₂×C₄	—	424	4	no	—
348	C₂×C₄	—	212	4	no	—
349	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
350	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
351	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
352	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
353	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
354	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
355	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
356	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
357	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
358	C₂×C₄	#13133176	1	1	yes	characteristic
359	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
360	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
361	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
362	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
363	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
364	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
365	C₂×C₄	#13133191	1	1	yes	characteristic
366	C₂×C₄	#13133191	1	1	yes	characteristic
367	C₂×C₄	#13133191	1	1	yes	characteristic
368	C₂×C₄	#13133196	1	1	yes	characteristic
369	C₂×C₄	#13133196	1	1	yes	characteristic
370	C₂×C₄	#13133196	1	1	yes	characteristic
371	C₂×C₄	#13133205	1	1	yes	characteristic
372	C₂×C₄	#13133209	1	1	yes	characteristic
373	C₂×C₄	#13133214	1	1	yes	characteristic
374	C₂×C₄	#13133217	1	1	yes	characteristic
375	C₂×C₄	#13133217	1	1	yes	characteristic
376	C₂×C₄	#13133217	1	1	yes	characteristic
377	C₈	—	2	4	no	—
378	C₈	—	212	4	no	—
379	C₈	#13133124	1	2	yes	—
380	C₈	#13133124	1	2	yes	—
381	C₈	#13133124	1	2	yes	—
382	C₈	#13133124	1	2	yes	—
383	C₂²	—	2	2	no	—
384	C₂²	—	2	2	no	—
385	C₂²	—	2	2	no	—
386	C₂²	—	4	1	no	—
387	C₂²	—	4	1	no	—
388	C₂²	—	4	1	no	—
389	C₂²	—	4	1	no	—
390	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
391	C₂²	#16810483	1	1	yes	characteristic
392	C₂²	#16810489	1	1	yes	characteristic
393	C₂²	#16810521	1	1	yes	characteristic
394	C₂²	#16810521	1	1	yes	characteristic
395	C₂²	#16810525	1	1	yes	characteristic
396	C₂²	#16810529	1	1	yes	characteristic
397	C₄	—	4	4	no	—
398	C₄	—	424	4	no	—
399	C₄	C₂×#13133124	1	1	yes	characteristic
400	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
401	C₄	#16810483	1	1	yes	characteristic
402	C₄	#16810493	1	1	yes	characteristic
403	C₄	#16810521	1	1	yes	characteristic
404	C₄	#16810521	1	1	yes	characteristic
405	C₄	#16810525	1	1	yes	characteristic
406	C₄	#16810529	1	1	yes	characteristic
407	C₂	—	2	1	no	—
408	C₂	—	2	1	no	—
409	C₂	—	2	1	no	—
410	C₂	—	2	1	no	—
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	#20272945	1	1	yes	characteristic
414	C₂	#20274839	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133176
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	#382
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#10868
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
11. C₂×#13133153	C₄	semidirect
13. #16810661	C₄	semidirect
23. C₂×#97×C₅₃	C₄	semidirect
24. C₂×C₄×C₈×C₅₃	C₄	semidirect
228. C₂×#97	Dic₅₃	semidirect
229. C₂×C₄×C₈	Dic₅₃	semidirect
232. C₅₃	#10868	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133153	2
C₂×#16810661	2
C₂×#16810394	2
C₂×#16810379	2
C₂×C₅₃×#405	2
#10868	53