Subgroups — #27394520

Order 13568.

734 conjugacy classes of subgroups.

The subgroup lattice has 382 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	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#20275490	C₂	1	4	yes	maximal, complemented
4	#20275488	C₂	1	4	yes	maximal, complemented
5	#20274751	C₂	1	4	yes	maximal
6	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88322	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×#13133152	C₂²	1	8	yes	—
13	#16810647	C₂²	1	2	yes	—
14	C₂×C₄⋊#88322	C₄	1	2	yes	—
15	C₂×C₄⋊#88322	C₄	1	2	yes	—
16	C₂×C₄⋊#88322	C₄	1	2	yes	—
17	C₂×C₄⋊#88322	C₄	1	2	yes	—
18	C₂×C₄⋊#88322	C₂²	1	2	yes	—
19	C₂×C₄⋊#88322	C₂²	1	2	yes	—
20	#16810587	C₂²	1	2	yes	—
21	C₄×#11719917	C₂²	1	2	yes	—
22	#16810387	C₂²	1	8	yes	—
23	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#432	—	2	1	no	—
25	C₅₃×#432	—	2	1	no	—
26	C₅₃×#431	—	2	1	no	—
27	C₅₃×#431	—	2	1	no	—
28	C₅₃×#423	C₂²	1	2	yes	—
29	C₅₃×#407	C₂²	1	2	yes	—
30	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
31	C₂×#11719917	C₂³	1	4	yes	—
32	C₂²×#88322	—	4	4	no	—
33	C₂²×#88322	C₂×C₄	1	4	yes	—
34	C₂²×#88322	C₂×C₄	1	4	yes	—
35	C₂²×#88322	C₂³	1	4	yes	—
36	#13133152	—	4	8	no	—
37	C₄⋊#88322	—	2	2	no	—
38	C₄⋊#88322	—	2	2	no	—
39	C₄⋊#88322	C₂×C₄	1	4	yes	—
40	C₄⋊#88322	C₂×C₄	1	4	yes	—
41	C₄⋊#88322	C₂×C₄	1	4	yes	—
42	C₄⋊#88322	C₂×C₄	1	4	yes	—
43	C₂²⋊#88322	—	2	2	no	—
44	C₂²⋊#88322	—	2	2	no	—
45	#13133128	—	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	C₂×M₁₆×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	—	2	2	no	—
64	C₄⋊C₈×C₅₃	—	2	2	no	—
65	C₂²⋊C₈×C₅₃	—	2	2	no	—
66	C₂²⋊C₈×C₅₃	—	2	2	no	—
67	#97×C₅₃	—	2	2	no	—
68	C₄×C₈×C₅₃	—	2	2	no	—
69	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	—	2	4	no	—
71	C₂²×C₄×C₅₃	—	2	2	no	—
72	C₂²×C₄×C₅₃	—	2	2	no	—
73	C₂²×C₄×C₅₃	—	4	2	no	—
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	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×D₄	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	C₂×Q₈	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
92	M₁₆×C₅₃	—	2	4	no	—
93	M₁₆×C₅₃	—	2	4	no	—
94	C₂×C₈×C₅₃	—	2	4	no	—
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	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
111	#11719917	—	2	4	no	—
112	#11719917	—	2	4	no	—
113	C₂×#88322	—	2	8	no	—
114	C₂×#88322	—	2	4	no	—
115	C₂×#88322	—	4	4	no	—
116	C₂×#88322	—	4	4	no	—
117	C₂×#88322	C₂²×C₄	1	8	yes	—
118	C₂³×C₅₃	—	2	2	no	—
119	C₂³×C₅₃	—	2	2	no	—
120	C₂³×C₅₃	—	2	2	no	—
121	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
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	1	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	1	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	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	4	1	no	—
139	C₂×C₄×C₅₃	—	4	1	no	—
140	C₂×C₄×C₅₃	—	4	1	no	—
141	C₂×C₄×C₅₃	—	4	1	no	—
142	C₂×C₄×C₅₃	#108	1	4	yes	complemented
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	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
156	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
157	C₈×C₅₃	—	4	4	no	—
158	#88322	—	2	8	no	—
159	#88322	—	4	4	no	—
160	#21969	—	53	1	no	maximal, Sylow
161	C₂²×C₅₃	—	2	8	no	—
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	—	2	1	no	—
164	C₂²×C₅₃	—	2	1	no	—
165	C₂²×C₅₃	—	2	1	no	—
166	C₂²×C₅₃	—	2	1	no	—
167	C₂²×C₅₃	—	2	1	no	—
168	C₂²×C₅₃	C₂×#108	1	2	yes	—
169	C₂²×C₅₃	#430	1	2	yes	—
170	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
171	C₂²×C₅₃	#544	1	2	yes	—
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	—	2	1	no	—
180	C₄×C₅₃	C₂×#108	1	2	yes	—
181	C₄×C₅₃	C₂×#108	1	2	yes	—
182	C₄×C₅₃	#518	1	1	yes	characteristic
183	C₄×C₅₃	#519	1	1	yes	characteristic
184	C₄×C₅₃	#537	1	2	yes	—
185	#2920	—	53	1	no	—
186	#2920	D₅₃	1	1	yes	characteristic
187	C₂²×C₄⋊C₈	—	53	1	no	—
188	#1950	—	53	4	no	—
189	#1950	—	53	4	no	—
190	#1810	—	53	4	no	—
191	C₂×C₅₃	—	2	2	no	—
192	C₂×C₅₃	—	2	2	no	—
193	C₂×C₅₃	—	2	2	no	—
194	C₂×C₅₃	#2316	1	1	yes	characteristic
195	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
196	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
197	C₂³×C₈	—	53	2	no	—
198	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
199	#432	—	2	1	no	—
200	#432	—	2	1	no	—
201	#432	—	106	1	no	—
202	#432	—	106	1	no	—
203	#431	—	2	1	no	—
204	#431	—	2	1	no	—
205	#431	—	106	1	no	—
206	#431	—	106	1	no	—
207	C₂×#108	—	53	8	no	—
208	#423	—	53	2	no	—
209	#423	C₂×D₅₃	1	2	yes	—
210	C₂×C₄⋊C₈	—	53	2	no	—
211	C₂×C₄⋊C₈	—	53	2	no	—
212	C₂×C₄⋊C₈	—	53	2	no	—
213	C₂×C₄⋊C₈	—	53	2	no	—
214	C₂×C₄⋊C₈	—	53	2	no	—
215	C₂×C₄⋊C₈	—	53	2	no	—
216	#407	—	53	2	no	—
217	#407	C₂×D₅₃	1	2	yes	—
218	C₄×M₁₆	—	53	2	no	—
219	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
220	#341	—	53	8	no	—
221	C₅₃	#21969	1	1	yes	characteristic, complemented, Sylow
222	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
223	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
224	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
225	C₂×M₁₆	—	53	4	no	—
226	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
227	C₂²×C₈	—	53	4	no	—
228	C₂²×C₈	—	53	4	no	—
229	C₂²×C₈	—	53	4	no	—
230	C₂²×C₈	—	212	4	no	—
231	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
232	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
233	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
234	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
235	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
236	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
237	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
238	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
239	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
240	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
243	#108	—	212	8	no	—
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	53	4	no	—
247	C₄⋊C₈	—	53	4	no	—
248	C₄⋊C₈	—	53	4	no	—
249	C₄⋊C₈	—	53	4	no	—
250	C₄⋊C₈	—	106	2	no	—
251	C₄⋊C₈	—	106	2	no	—
252	C₂²⋊C₈	—	2	2	no	—
253	C₂²⋊C₈	—	2	2	no	—
254	C₂²⋊C₈	—	106	2	no	—
255	C₂²⋊C₈	—	106	2	no	—
256	#97	—	2	2	no	—
257	#97	—	106	2	no	—
258	C₄×C₈	—	2	2	no	—
259	C₄×C₈	—	106	2	no	—
260	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
261	C₂²×C₄	—	2	4	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	#11719919	1	1	yes	characteristic
266	C₂²×C₄	#11719919	1	1	yes	characteristic
267	C₂²×C₄	#11719919	1	1	yes	characteristic
268	C₂²×C₄	#11719919	1	1	yes	characteristic
269	C₂²×C₄	#11719919	1	1	yes	characteristic
270	C₂²×C₄	#11719919	1	1	yes	characteristic
271	C₂²×C₄	#11719919	1	1	yes	characteristic
272	C₂²×C₄	#11719919	1	1	yes	characteristic
273	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
274	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
275	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
276	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
277	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
280	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
281	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
282	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
283	M₁₆	—	2	4	no	—
284	M₁₆	—	2	4	no	—
285	M₁₆	—	106	4	no	—
286	M₁₆	—	106	4	no	—
287	C₂×C₈	—	2	4	no	—
288	C₂×C₈	—	53	8	no	—
289	C₂×C₈	—	106	8	no	—
290	C₂×C₈	—	106	4	no	—
291	C₂×C₈	—	212	4	no	—
292	C₂×C₈	—	212	4	no	—
293	C₄²	#11719919	1	1	yes	characteristic
294	C₄²	#11719919	1	1	yes	characteristic
295	C₄²	#11719919	1	1	yes	characteristic
296	C₄²	#11719919	1	1	yes	characteristic
297	C₄²	#11719919	1	1	yes	characteristic
298	C₄²	#11719919	1	1	yes	characteristic
299	C₄²	#11719919	1	1	yes	characteristic
300	C₄²	#11719919	1	1	yes	characteristic
301	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
302	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
303	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
304	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
305	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
306	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
307	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
308	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
309	C₂³	—	2	2	no	—
310	C₂³	—	2	2	no	—
311	C₂³	—	2	2	no	—
312	C₂³	C₂×#11719919	1	1	yes	characteristic
313	C₂³	C₂×#11719919	1	1	yes	characteristic
314	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
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	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	#13133152	1	4	yes	complemented
334	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
335	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
336	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
337	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
338	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
339	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
340	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
341	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
342	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
343	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
344	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
345	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
346	C₂×C₄	#13133263	1	2	yes	—
347	C₂×C₄	#13133264	1	2	yes	—
348	C₈	—	4	4	no	—
349	C₈	—	106	8	no	—
350	C₈	—	212	4	no	—
351	C₂²	—	2	8	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	C₂×#13133152	1	2	yes	—
359	C₂²	#16810671	1	2	yes	—
360	C₂²	C₂²×#11719919	1	1	yes	characteristic
361	C₂²	#16811251	1	2	yes	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	C₂×#13133152	1	2	yes	—
371	C₄	C₂×#13133152	1	2	yes	—
372	C₄	#16811097	1	1	yes	characteristic
373	C₄	#16811100	1	1	yes	characteristic
374	C₄	#16811201	1	2	yes	—
375	C₂	—	2	2	no	—
376	C₂	—	2	2	no	—
377	C₂	—	2	2	no	—
378	C₂	#20277331	1	1	yes	characteristic
379	C₂	C₂²×#13133152	1	1	yes	characteristic
380	C₂	C₂×#16810671	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2920	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#2920	D₅₃
hypocenter	C₅₃	#21969
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
3. #20275490	C₂	semidirect
4. #20275488	C₂	semidirect
142. C₂×C₄×C₅₃	#108	semidirect
221. C₅₃	#21969	semidirect
333. C₂×C₄	#13133152	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
C₂²×C₄⋊#88322	2
#20275490	2
#20275488	2
#20274751	2
C₅₃×#2920	2
#21969	53