Subgroups — #27389030

Order 13568.

780 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	#20275491	C₂	1	4	yes	maximal, complemented
4	#20274922	C₂	1	4	yes	maximal
5	C₄×#13133153	C₂	1	4	yes	maximal, complemented
6	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	2	yes	—
8	#16810680	—	2	1	no	—
9	#16810680	—	2	1	no	—
10	#16810676	—	2	1	no	—
11	#16810676	—	2	1	no	—
12	C₂×#13133153	C₂²	1	8	yes	—
13	#16810647	C₂²	1	2	yes	—
14	#16810587	C₂²	1	2	yes	—
15	C₄×#11719917	—	2	1	no	—
16	C₄×#11719917	—	2	1	no	—
17	C₄×#11719917	C₂²	1	2	yes	—
18	C₄×#11719917	C₂²	1	2	yes	—
19	C₄×#11719917	C₂²	1	2	yes	—
20	C₂×#13133128	—	2	1	no	—
21	C₂×C₄×#88322	—	2	1	no	—
22	#16810388	C₂²	1	8	yes	—
23	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#431	C₄	1	2	yes	—
25	C₅₃×#431	C₄	1	2	yes	—
26	C₅₃×#431	C₄	1	2	yes	—
27	C₅₃×#431	C₄	1	2	yes	—
28	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
31	C₂×#11719917	—	2	8	no	—
32	C₂×#11719917	—	2	2	no	—
33	C₂×#11719917	C₂³	1	4	yes	—
34	C₂×#11719917	C₂³	1	4	yes	—
35	C₂²×#88322	—	2	2	no	—
36	#13133153	—	2	16	no	—
37	C₄⋊#88322	—	2	2	no	—
38	C₄⋊#88322	—	2	2	no	—
39	C₂²⋊#88322	—	2	2	no	—
40	C₂²⋊#88322	—	2	2	no	—
41	#13133128	—	2	2	no	—
42	#13133128	—	2	2	no	—
43	#13133128	—	2	2	no	—
44	C₄×#88322	—	2	2	no	—
45	C₄×#88322	—	2	2	no	—
46	C₄×#88322	—	2	2	no	—
47	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
49	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
64	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
65	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
66	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
67	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
68	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
69	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	—	2	4	no	—
71	C₂²×C₄×C₅₃	—	2	4	no	—
72	C₂²×C₄×C₅₃	—	4	2	no	—
73	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
91	C₂×C₈×C₅₃	—	2	8	no	—
92	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
93	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
94	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
109	#11719917	—	2	8	no	—
110	#11719917	—	2	4	no	—
111	#11719917	—	2	8	no	—
112	#11719917	#41	1	8	yes	—
113	C₂×#88322	—	2	8	no	—
114	C₂×#88322	—	2	4	no	—
115	C₂×#88322	—	2	4	no	—
116	C₂³×C₅₃	—	2	2	no	—
117	C₂³×C₅₃	—	2	2	no	—
118	C₂³×C₅₃	—	4	1	no	—
119	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	4	1	no	—
136	C₂×C₄×C₅₃	—	4	1	no	—
137	C₂×C₄×C₅₃	—	4	1	no	—
138	C₂×C₄×C₅₃	#108	1	4	yes	complemented
139	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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	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	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
152	C₈×C₅₃	—	2	8	no	—
153	#88322	—	2	8	no	—
154	#88322	—	4	4	no	—
155	#21370	—	53	1	no	maximal, Sylow
156	C₂²×C₅₃	—	2	1	no	—
157	C₂²×C₅₃	—	2	1	no	—
158	C₂²×C₅₃	—	2	1	no	—
159	C₂²×C₅₃	—	2	1	no	—
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	—	4	4	no	—
164	C₂²×C₅₃	C₂×#108	1	2	yes	—
165	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₂²×C₅₃	#533	1	2	yes	—
167	C₄×C₅₃	—	2	1	no	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	C₂×#108	1	2	yes	—
175	C₄×C₅₃	C₂×#108	1	2	yes	—
176	C₄×C₅₃	#430	1	2	yes	—
177	C₄×C₅₃	C₄×#41	1	2	yes	—
178	C₄×C₅₃	#525	1	2	yes	—
179	#2920	—	53	1	no	—
180	C₂×#431	D₅₃	1	1	yes	characteristic
181	C₂×C₄×M₁₆	—	53	1	no	—
182	#1950	—	53	4	no	—
183	#1855	—	53	4	no	—
184	C₄×#108	—	53	4	no	—
185	C₂×C₅₃	—	2	2	no	—
186	C₂×C₅₃	—	2	2	no	—
187	C₂×C₅₃	—	4	1	no	—
188	C₂×C₅₃	#2296	1	1	yes	characteristic
189	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
190	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
191	C₂²×M₁₆	—	53	2	no	—
192	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
193	#432	—	106	1	no	—
194	#432	—	106	1	no	—
195	#431	—	106	1	no	—
196	#431	—	106	1	no	—
197	#431	Dic₅₃	1	2	yes	—
198	#431	Dic₅₃	1	2	yes	—
199	#431	Dic₅₃	1	2	yes	—
200	#431	Dic₅₃	1	2	yes	—
201	C₂×#108	—	53	8	no	—
202	#423	—	53	2	no	—
203	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
204	#407	—	53	2	no	—
205	C₂×C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
206	C₄×M₁₆	—	53	2	no	—
207	C₄×M₁₆	—	53	2	no	—
208	C₄×M₁₆	—	53	2	no	—
209	C₄×M₁₆	—	106	1	no	—
210	C₄×M₁₆	—	106	1	no	—
211	C₂×#97	—	106	1	no	—
212	C₂×C₄×C₈	—	106	1	no	—
213	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
214	#341	—	53	8	no	—
215	C₅₃	#21370	1	1	yes	characteristic, complemented, Sylow
216	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
217	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
218	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
219	C₂×M₁₆	—	53	4	no	—
220	C₂×M₁₆	—	53	4	no	—
221	C₂×M₁₆	—	106	8	no	—
222	C₂×M₁₆	—	106	2	no	—
223	C₂²×C₈	—	106	2	no	—
224	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
225	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
226	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
227	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
228	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
229	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
230	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
231	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
232	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
233	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
234	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
236	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
237	#108	—	106	16	no	—
238	C₄⋊C₈	—	106	2	no	—
239	C₄⋊C₈	—	106	2	no	—
240	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
241	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
242	C₂²⋊C₈	—	106	2	no	—
243	C₂²⋊C₈	—	106	2	no	—
244	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
245	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
246	#97	—	106	2	no	—
247	#97	—	106	2	no	—
248	#97	—	106	2	no	—
249	C₄×C₈	—	106	2	no	—
250	C₄×C₈	—	106	2	no	—
251	C₄×C₈	—	106	2	no	—
252	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
253	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
254	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
259	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
260	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
261	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
262	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
263	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
264	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
265	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
266	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
267	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
268	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
269	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
270	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
273	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
274	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
275	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
276	M₁₆	—	53	8	no	—
277	M₁₆	—	106	8	no	—
278	M₁₆	—	106	4	no	—
279	M₁₆	—	106	8	no	—
280	C₂×C₈	—	2	8	no	—
281	C₂×C₈	—	106	8	no	—
282	C₂×C₈	—	106	4	no	—
283	C₂×C₈	—	106	4	no	—
284	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
285	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
286	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
287	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
288	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
289	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
290	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
291	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
292	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
293	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
294	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
295	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
296	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
297	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
298	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
299	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
300	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
301	C₂³	—	2	2	no	—
302	C₂³	—	2	2	no	—
303	C₂³	—	4	1	no	—
304	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
305	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
306	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	#13133153	1	4	yes	complemented
324	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
325	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
326	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
327	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
328	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
329	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
330	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
331	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
332	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
333	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
334	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
335	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
336	C₂×C₄	#41×D₅₃	1	4	yes	—
337	C₈	—	2	8	no	—
338	C₈	—	106	8	no	—
339	C₈	—	212	4	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	4	no	—
348	C₂²	C₂×#13133153	1	2	yes	—
349	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
350	C₂²	#16811176	1	2	yes	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	C₂×#13133153	1	2	yes	—
359	C₄	C₂×#13133153	1	2	yes	—
360	C₄	#16810672	1	2	yes	—
361	C₄	#41×Dic₅₃	1	2	yes	—
362	C₄	#16811135	1	2	yes	—
363	C₂	—	2	2	no	—
364	C₂	—	2	2	no	—
365	C₂	—	4	1	no	—
366	C₂	#20277199	1	1	yes	characteristic
367	C₂	C₂²×#13133153	1	1	yes	characteristic
368	C₂	C₂×#16810672	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#21370
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20275491	C₂	semidirect
5. C₄×#13133153	C₂	semidirect
138. C₂×C₄×C₅₃	#108	semidirect
215. C₅₃	#21370	semidirect
323. C₂×C₄	#13133153	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
C₂×C₄×#11719917	2
#20275491	2
#20274922	2
C₄×#13133153	2
C₂×C₅₃×#431	2
#21370	53