Subgroups — #27402540

Order 13568.

858 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810745	C₂	1	2	yes	maximal
2	#20283283	C₂	1	2	yes	maximal, complemented
3	#20275469	C₂	1	2	yes	maximal, complemented
4	#20274992	C₂	1	4	yes	maximal, complemented
5	#20274887	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274735	C₂	1	2	yes	maximal, complemented
7	#20274582	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133224	C₂²	1	4	yes	—
10	#16810745	—	2	8	no	—
11	C₂×#13133153	C₂²	1	2	yes	—
12	C₂×#13133153	C₂²	1	2	yes	—
13	#16810650	—	2	4	no	—
14	#16810650	C₂²	1	4	yes	—
15	#16810647	—	2	2	no	—
16	#16810647	C₂²	1	2	yes	—
17	C₂×C₄⋊#88322	—	2	2	no	—
18	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
19	C₂×#13133144	C₂²	1	4	yes	—
20	C₂×#13133127	C₂²	1	2	yes	—
21	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
22	#16810382	C₂²	1	4	yes	—
23	#16810377	C₂²	1	2	yes	—
24	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
25	C₄×#41×C₅₃	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
27	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
28	C₄×SD₁₆×C₅₃	C₄	1	2	yes	complemented
29	C₄×SD₁₆×C₅₃	C₄	1	2	yes	complemented
30	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic, complemented
31	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic, complemented
32	C₅₃×#427	C₂²	1	1	yes	characteristic
33	C₅₃×#416	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
35	#13133224	—	2	8	no	—
36	#13133224	—	4	4	no	—
37	C₂×#11719917	—	2	4	no	—
38	C₂×#11719917	C₂³	1	4	yes	—
39	C₂²×#88322	—	2	4	no	—
40	C₂²×#88322	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₅₃	—	2	1	no	—
42	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
43	#13133153	—	2	4	no	—
44	#13133153	—	4	2	no	—
45	C₄⋊#88322	—	2	2	no	—
46	C₄⋊#88322	—	2	2	no	—
47	C₄⋊#88322	—	2	4	no	—
48	C₄⋊#88322	—	2	4	no	—
49	C₄⋊#88322	D₄	1	2	yes	—
50	C₄⋊#88322	D₄	1	2	yes	—
51	#13133144	—	2	8	no	—
52	#13133127	—	2	4	no	—
53	#13133124	—	2	1	no	—
54	C₂×#41×C₅₃	C₂³	1	2	yes	—
55	#135×C₅₃	D₄	1	4	yes	—
56	#135×C₅₃	Q₈	1	4	yes	—
57	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
63	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
64	C₄×D₄×C₅₃	—	2	2	no	—
65	C₄×D₄×C₅₃	—	2	2	no	—
66	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
67	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
68	#117×C₅₃	—	2	2	no	—
69	#117×C₅₃	C₂³	1	2	yes	—
70	C₂×C₄²×C₅₃	—	2	2	no	—
71	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
72	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
73	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
74	#106×C₅₃	C₂×C₄	1	2	yes	—
75	#103×C₅₃	C₂×C₄	1	2	yes	—
76	#103×C₅₃	C₂×C₄	1	2	yes	—
77	#102×C₅₃	C₂×C₄	1	2	yes	—
78	#102×C₅₃	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
83	#41×C₅₃	—	2	2	no	—
84	#41×C₅₃	—	2	2	no	—
85	#41×C₅₃	C₂×D₄	1	4	yes	—
86	#41×C₅₃	C₂×Q₈	1	4	yes	—
87	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂×D₄×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅₃	—	2	2	no	—
91	C₂²×C₄×C₅₃	—	2	2	no	—
92	C₂²×C₄×C₅₃	—	2	2	no	—
93	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
96	C₂²×Dic₅₃	—	2	2	no	—
97	Q₁₆×C₅₃	C₄⋊C₄	1	4	yes	complemented
98	SD₁₆×C₅₃	C₄⋊C₄	1	8	yes	complemented
99	D₈×C₅₃	C₄⋊C₄	1	4	yes	complemented
100	C₂×C₈×C₅₃	—	2	1	no	—
101	C₂×C₈×C₅₃	—	2	1	no	—
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	1	yes	characteristic
105	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
108	C₄⋊C₄×C₅₃	—	2	2	no	—
109	C₄⋊C₄×C₅₃	—	2	2	no	—
110	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
111	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
112	C₂²⋊C₄×C₅₃	—	2	2	no	—
113	C₂²⋊C₄×C₅₃	—	2	2	no	—
114	C₂²⋊C₄×C₅₃	—	2	2	no	—
115	C₄²×C₅₃	—	2	2	no	—
116	C₄²×C₅₃	—	2	2	no	—
117	C₄²×C₅₃	—	2	2	no	—
118	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₄⋊Dic₅₃	—	4	2	no	—
123	C₄×Dic₅₃	—	2	2	no	—
124	C₄×Dic₅₃	—	2	2	no	—
125	#11719917	—	2	4	no	—
126	#11719917	—	2	4	no	—
127	#11719917	—	4	8	no	—
128	C₂×#88322	—	2	2	no	—
129	C₂×#88322	—	2	2	no	—
130	C₂×#88322	—	2	4	no	—
131	C₂×#88322	—	4	8	no	—
132	C₂×#88322	C₂×D₄	1	4	yes	—
133	C₂³×C₅₃	—	2	2	no	—
134	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
135	Q₈×C₅₃	—	2	2	no	—
136	Q₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
137	D₄×C₅₃	—	2	4	no	—
138	D₄×C₅₃	—	2	4	no	—
139	D₄×C₅₃	—	2	2	no	—
140	D₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	4	no	—
143	C₂×C₄×C₅₃	—	2	4	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	4	1	no	—
148	C₂×C₄×C₅₃	—	4	4	no	—
149	C₂×C₄×C₅₃	—	8	2	no	—
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×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
158	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
160	C₂×Dic₅₃	—	4	2	no	—
161	C₂×Dic₅₃	—	4	2	no	—
162	C₂×Dic₅₃	—	4	2	no	—
163	C₈×C₅₃	—	2	1	no	—
164	C₈×C₅₃	—	2	1	no	—
165	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
166	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
167	#88322	—	2	4	no	—
168	#88322	—	4	4	no	—
169	#22686	—	53	1	no	maximal, Sylow
170	C₂²×C₅₃	—	2	1	no	—
171	C₂²×C₅₃	—	2	1	no	—
172	C₂²×C₅₃	—	2	4	no	—
173	C₂²×C₅₃	—	4	2	no	—
174	C₂²×C₅₃	—	4	2	no	—
175	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂²×C₅₃	#547	1	1	yes	characteristic
177	C₂²×C₅₃	#554	1	1	yes	characteristic
178	C₄×C₅₃	—	2	4	no	—
179	C₄×C₅₃	—	2	1	no	—
180	C₄×C₅₃	—	2	1	no	—
181	C₄×C₅₃	—	2	1	no	—
182	C₄×C₅₃	—	4	2	no	—
183	C₄×C₅₃	—	4	2	no	—
184	C₄×C₅₃	#444	1	2	yes	complemented
185	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
186	C₄×C₅₃	#520	1	1	yes	characteristic
187	C₄×C₅₃	D₄²	1	1	yes	characteristic
188	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
189	Dic₅₃	—	8	2	no	—
190	C₂×#444	—	53	2	no	—
191	C₄×#135	D₅₃	1	1	yes	characteristic
192	#2906	—	53	2	no	—
193	#1946	—	53	2	no	—
194	#1866	—	53	4	no	—
195	#1848	—	53	1	no	—
196	#1806	—	53	2	no	—
197	#1770	—	53	1	no	—
198	C₂×C₅₃	—	2	1	no	—
199	C₂×C₅₃	—	2	1	no	—
200	C₂×C₅₃	—	4	4	no	—
201	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
203	C₂×C₅₃	#2957	1	1	yes	characteristic
204	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
205	C₂×#131	—	53	4	no	—
206	C₄×#41	C₂×D₅₃	1	2	yes	—
207	#444	—	106	8	no	—
208	C₄×Q₁₆	Dic₅₃	1	1	yes	characteristic, complemented
209	C₄×Q₁₆	Dic₅₃	1	1	yes	characteristic, complemented
210	C₄×SD₁₆	Dic₅₃	1	2	yes	complemented
211	C₄×SD₁₆	Dic₅₃	1	2	yes	complemented
212	C₄×D₈	Dic₅₃	1	1	yes	characteristic, complemented
213	C₄×D₈	Dic₅₃	1	1	yes	characteristic, complemented
214	C₂×#108	—	53	2	no	—
215	C₂×#108	—	53	2	no	—
216	#427	C₂×D₅₃	1	1	yes	characteristic
217	#424	—	53	4	no	—
218	#424	—	106	4	no	—
219	#423	—	53	2	no	—
220	#423	—	106	2	no	—
221	C₂×C₄⋊C₈	—	53	1	no	—
222	C₂×C₄⋊C₈	—	106	2	no	—
223	C₂×#104	—	53	4	no	—
224	#416	C₂×D₅₃	1	2	yes	—
225	C₂×#97	—	53	2	no	—
226	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
227	C₄×C₄⋊C₄	—	53	1	no	—
228	#339	—	53	4	no	—
229	#336	—	53	2	no	—
230	C₅₃	#22686	1	1	yes	characteristic, complemented, Sylow
231	C₂×#41	C₂²×D₅₃	1	2	yes	—
232	#135	Dic₁₀₆	1	4	yes	—
233	#135	D₂₁₂	1	4	yes	—
234	C₂×Q₁₆	C₂×Dic₅₃	1	1	yes	characteristic
235	C₂×SD₁₆	C₂×Dic₅₃	1	2	yes	—
236	C₂×D₈	C₂×Dic₅₃	1	1	yes	characteristic
237	#131	—	106	8	no	—
238	#131	—	212	4	no	—
239	C₂×M₁₆	—	53	4	no	—
240	C₂×M₁₆	—	106	4	no	—
241	C₂²×C₈	—	53	2	no	—
242	C₂²×C₈	—	106	4	no	—
243	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
244	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
245	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
246	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
247	C₄×D₄	—	2	2	no	—
248	C₄×D₄	—	2	2	no	—
249	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
250	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
251	#117	—	2	2	no	—
252	#117	C₂²×D₅₃	1	2	yes	—
253	C₂×C₄⋊C₄	—	106	1	no	—
254	C₂×C₄²	—	2	2	no	—
255	C₂×C₄²	—	53	2	no	—
256	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
257	#108	—	106	4	no	—
258	#108	—	212	2	no	—
259	#107	C₂×Dic₅₃	1	1	yes	characteristic
260	#107	C₂×Dic₅₃	1	1	yes	characteristic
261	#106	C₂×Dic₅₃	1	2	yes	—
262	C₄⋊C₈	—	53	2	no	—
263	C₄⋊C₈	—	53	2	no	—
264	C₄⋊C₈	—	106	4	no	—
265	C₄⋊C₈	—	106	4	no	—
266	C₄⋊C₈	—	106	2	no	—
267	C₄⋊C₈	—	106	2	no	—
268	#104	—	106	8	no	—
269	#103	C₂×Dic₅₃	1	2	yes	—
270	#103	C₂×Dic₅₃	1	2	yes	—
271	#102	C₂×Dic₅₃	1	2	yes	—
272	#102	C₂×Dic₅₃	1	2	yes	—
273	#97	—	106	4	no	—
274	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
275	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
276	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
277	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
278	#95	—	106	1	no	—
279	#41	—	2	2	no	—
280	#41	—	2	2	no	—
281	#41	C₂×Dic₁₀₆	1	4	yes	—
282	#41	C₂×D₂₁₂	1	4	yes	—
283	C₂×Q₈	C₂²×Dic₅₃	1	2	yes	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	106	2	no	—
290	C₂²×C₄	#11719947	1	1	yes	characteristic
291	C₂²×C₄	#11719947	1	1	yes	characteristic
292	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
293	Q₁₆	C₄⋊Dic₅₃	1	4	yes	complemented
294	SD₁₆	C₄⋊Dic₅₃	1	8	yes	complemented
295	D₈	C₄⋊Dic₅₃	1	4	yes	complemented
296	M₁₆	—	106	4	no	—
297	M₁₆	—	106	4	no	—
298	M₁₆	—	212	8	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	53	4	no	—
302	C₂×C₈	—	106	2	no	—
303	C₂×C₈	—	106	2	no	—
304	C₂×C₈	—	106	4	no	—
305	C₂×C₈	—	212	8	no	—
306	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
307	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
308	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
309	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
310	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
311	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	212	2	no	—
315	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
316	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
317	C₂²⋊C₄	—	2	2	no	—
318	C₂²⋊C₄	—	2	2	no	—
319	C₂²⋊C₄	—	2	2	no	—
320	C₄²	—	2	2	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	106	2	no	—
324	C₄²	—	106	2	no	—
325	C₄²	D₄×D₅₃	1	1	yes	characteristic
326	C₄²	D₄×D₅₃	1	1	yes	characteristic
327	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
328	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
329	C₂³	—	2	2	no	—
330	C₂³	C₂×#11719947	1	1	yes	characteristic
331	Q₈	—	2	2	no	—
332	Q₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
333	D₄	—	2	4	no	—
334	D₄	—	2	4	no	—
335	D₄	—	2	2	no	—
336	D₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	4	no	—
345	C₂×C₄	—	8	2	no	—
346	C₂×C₄	—	212	2	no	—
347	C₂×C₄	—	212	2	no	—
348	C₂×C₄	—	212	2	no	—
349	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
350	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
351	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
352	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
353	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
354	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
356	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
357	C₂×C₄	#13133263	1	1	yes	characteristic
358	C₂×C₄	#13133264	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	106	4	no	—
362	C₈	—	212	4	no	—
363	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
364	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	4	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	4	2	no	—
370	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
371	C₂²	#16811277	1	1	yes	characteristic
372	C₂²	#16811330	1	1	yes	characteristic
373	C₄	—	2	4	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	2	no	—
379	C₄	—	424	2	no	—
380	C₄	#16810745	1	2	yes	complemented
381	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
382	C₄	#16811106	1	1	yes	characteristic
383	C₄	D₄×D₂₁₂	1	1	yes	characteristic
384	C₄	D₄×Dic₁₀₆	1	1	yes	characteristic
385	C₂	—	2	1	no	—
386	C₂	—	2	1	no	—
387	C₂	—	4	4	no	—
388	C₂	D₄×C₄⋊Dic₅₃	1	1	yes	characteristic
389	C₂	C₂×#16810745	1	1	yes	characteristic
390	C₂	#20283632	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#135×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#135×C₅₃	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#135	D₅₃
hypocenter	C₅₃	#22686
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. #20283283	C₂	semidirect
3. #20275469	C₂	semidirect
4. #20274992	C₂	semidirect
5. #20274887	C₂	semidirect
6. #20274735	C₂	semidirect
7. #20274582	C₂	semidirect
26. C₄×Q₁₆×C₅₃	C₄	semidirect
27. C₄×Q₁₆×C₅₃	C₄	semidirect
28. C₄×SD₁₆×C₅₃	C₄	semidirect
29. C₄×SD₁₆×C₅₃	C₄	semidirect
30. C₄×D₈×C₅₃	C₄	semidirect
31. C₄×D₈×C₅₃	C₄	semidirect
97. Q₁₆×C₅₃	C₄⋊C₄	semidirect
98. SD₁₆×C₅₃	C₄⋊C₄	semidirect
99. D₈×C₅₃	C₄⋊C₄	semidirect
184. C₄×C₅₃	#444	semidirect
208. C₄×Q₁₆	Dic₅₃	semidirect
209. C₄×Q₁₆	Dic₅₃	semidirect
210. C₄×SD₁₆	Dic₅₃	semidirect
211. C₄×SD₁₆	Dic₅₃	semidirect
212. C₄×D₈	Dic₅₃	semidirect
213. C₄×D₈	Dic₅₃	semidirect
230. C₅₃	#22686	semidirect
293. Q₁₆	C₄⋊Dic₅₃	semidirect
294. SD₁₆	C₄⋊Dic₅₃	semidirect
295. D₈	C₄⋊Dic₅₃	semidirect
380. C₄	#16810745	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810745	2
#20283283	2
#20275469	2
#20274992	2
#20274887	2
#20274735	2
#20274582	2
C₄×#135×C₅₃	2
#22686	53