Subgroups — C₂×#20275490 #27387612

Order 13568.

938 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133152	C₂	1	2	yes	maximal
2	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
4	#20275490	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810387	C₂	1	2	yes	maximal
6	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	2	yes	—
8	C₂³×#88322	C₂²	1	2	yes	—
9	C₂×#13133152	—	2	8	no	—
10	C₂×#13133152	C₂²	1	8	yes	—
11	#16810647	—	2	2	no	—
12	#16810647	C₂²	1	4	yes	—
13	C₂×C₄⋊#88322	—	2	1	no	—
14	C₂×C₄⋊#88322	—	2	1	no	—
15	C₂×C₄⋊#88322	C₄	1	2	yes	—
16	C₂×C₄⋊#88322	C₄	1	4	yes	—
17	C₂×C₄⋊#88322	C₄	1	2	yes	—
18	C₂×C₄⋊#88322	C₂²	1	4	yes	—
19	#16810387	C₂²	1	8	yes	—
20	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×M₁₆×C₅₃	—	2	2	no	—
24	C₄×M₁₆×C₅₃	C₂²	1	4	yes	—
25	C₂×#97×C₅₃	—	2	1	no	—
26	C₂×C₄×C₈×C₅₃	—	2	1	no	—
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719917	—	2	4	no	—
29	C₂×#11719917	—	2	4	no	—
30	C₂×#11719917	C₂³	1	4	yes	—
31	C₂²×#88322	—	2	4	no	—
32	C₂²×#88322	—	2	2	no	—
33	C₂²×#88322	—	2	4	no	—
34	C₂²×#88322	C₂×C₄	1	4	yes	—
35	C₂²×#88322	C₂×C₄	1	4	yes	—
36	C₂²×#88322	C₂³	1	4	yes	—
37	#13133152	—	2	16	no	—
38	C₄⋊#88322	—	2	4	no	—
39	C₄⋊#88322	—	2	4	no	—
40	C₄⋊#88322	C₂×C₄	1	8	yes	—
41	C₄⋊#88322	C₂×C₄	1	8	yes	—
42	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₅₃	—	2	1	no	—
46	C₂×M₁₆×C₅₃	—	2	1	no	—
47	C₂×M₁₆×C₅₃	—	2	2	no	—
48	C₂×M₁₆×C₅₃	—	2	2	no	—
49	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
50	C₂×M₁₆×C₅₃	Q₈	1	4	yes	—
51	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
52	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
53	C₂²×C₈×C₅₃	—	2	1	no	—
54	C₂²×C₈×C₅₃	—	2	1	no	—
55	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	D₄	1	2	yes	—
61	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
62	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
63	#97×C₅₃	—	2	4	no	—
64	C₄×C₈×C₅₃	—	2	4	no	—
65	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	—	2	4	no	—
67	C₂²×C₄×C₅₃	—	4	2	no	—
68	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²×C₄	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	2	yes	—
79	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₅₃	#41	1	2	yes	—
84	C₂²×C₄×C₅₃	#41	1	2	yes	—
85	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
86	M₁₆×C₅₃	—	2	4	no	—
87	M₁₆×C₅₃	—	2	4	no	—
88	M₁₆×C₅₃	—	2	2	no	—
89	M₁₆×C₅₃	—	2	2	no	—
90	M₁₆×C₅₃	C₂×D₄	1	4	yes	—
91	M₁₆×C₅₃	C₂×Q₈	1	4	yes	—
92	C₂×C₈×C₅₃	—	2	4	no	—
93	C₂×C₈×C₅₃	—	2	4	no	—
94	C₂×C₈×C₅₃	—	2	2	no	—
95	C₂×C₈×C₅₃	—	2	2	no	—
96	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
97	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
98	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
99	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
100	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
101	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
102	C₄²×C₅₃	C₂×D₄	1	2	yes	—
103	C₄²×C₅₃	C₂×D₄	1	2	yes	—
104	#11719917	—	2	4	no	—
105	#11719917	—	2	4	no	—
106	#11719917	—	4	4	no	—
107	C₂×#88322	—	2	8	no	—
108	C₂×#88322	—	2	8	no	—
109	C₂×#88322	—	2	4	no	—
110	C₂×#88322	—	2	4	no	—
111	C₂×#88322	—	2	4	no	—
112	C₂×#88322	—	4	4	no	—
113	C₂×#88322	C₂²×C₄	1	8	yes	—
114	C₂³×C₅₃	—	2	2	no	—
115	C₂³×C₅₃	—	2	2	no	—
116	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂³×C₅₃	#125	1	2	yes	—
119	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
120	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	4	no	—
125	C₂×C₄×C₅₃	—	2	4	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	#108	1	4	yes	complemented
129	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
138	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
139	C₂×C₄×C₅₃	#121	1	2	yes	—
140	C₂×C₄×C₅₃	#121	1	2	yes	—
141	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
142	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₅₃	#125	1	2	yes	—
144	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
145	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
151	C₈×C₅₃	—	2	4	no	—
152	C₈×C₅₃	—	2	4	no	—
153	#88322	—	2	8	no	—
154	#88322	—	4	4	no	—
155	C₂×#1950	—	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	4	no	—
161	C₂²×C₅₃	—	2	4	no	—
162	C₂²×C₅₃	#389	1	2	yes	—
163	C₂²×C₅₃	#389	1	2	yes	—
164	C₂²×C₅₃	C₂×#108	1	2	yes	—
165	C₂²×C₅₃	#430	1	2	yes	—
166	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
167	C₂²×C₅₃	C₂×#125	1	1	yes	characteristic
168	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
169	C₄×C₅₃	—	2	2	no	—
170	C₄×C₅₃	—	2	2	no	—
171	C₄×C₅₃	—	2	2	no	—
172	C₄×C₅₃	#389	1	2	yes	—
173	C₄×C₅₃	#389	1	2	yes	—
174	C₄×C₅₃	C₂×#108	1	4	yes	complemented
175	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
176	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
177	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
178	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
179	C₂²×#108	—	53	2	no	—
180	C₂×#423	—	53	1	no	—
181	C₂²×C₄⋊C₈	—	53	1	no	—
182	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
183	#1950	—	53	8	no	—
184	C₂×#341	—	53	2	no	—
185	C₂×C₅₃	—	2	1	no	—
186	C₂×C₅₃	—	2	1	no	—
187	C₂×C₅₃	—	2	2	no	—
188	C₂×C₅₃	#1950	1	4	yes	complemented
189	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
190	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
191	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
192	C₂²×M₁₆	—	53	2	no	—
193	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
194	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
195	C₂³×C₈	—	53	2	no	—
196	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
197	C₂×#108	—	53	8	no	—
198	C₂×#108	—	106	8	no	—
199	#423	—	53	4	no	—
200	#423	—	106	2	no	—
201	C₂×C₄⋊C₈	—	53	2	no	—
202	C₂×C₄⋊C₈	—	53	4	no	—
203	C₂×C₄⋊C₈	—	53	2	no	—
204	C₂×C₄⋊C₈	—	53	4	no	—
205	C₂×C₄⋊C₈	—	106	1	no	—
206	C₂×C₄⋊C₈	—	106	1	no	—
207	C₄×M₁₆	—	2	2	no	—
208	C₄×M₁₆	C₂×D₅₃	1	4	yes	—
209	C₂×#97	—	2	1	no	—
210	C₂×C₄×C₈	—	2	1	no	—
211	#341	—	53	8	no	—
212	C₅₃	C₂×#1950	1	1	yes	characteristic, complemented, Sylow
213	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
214	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
215	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
216	C₂×M₁₆	—	2	2	no	—
217	C₂×M₁₆	—	2	2	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	—	2	1	no	—
220	C₂×M₁₆	—	53	4	no	—
221	C₂×M₁₆	—	106	2	no	—
222	C₂×M₁₆	—	106	4	no	—
223	C₂×M₁₆	—	106	4	no	—
224	C₂×M₁₆	Dic₁₀₆	1	4	yes	—
225	C₂×M₁₆	D₂₁₂	1	4	yes	—
226	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
227	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
228	C₂²×C₈	—	2	1	no	—
229	C₂²×C₈	—	2	1	no	—
230	C₂²×C₈	—	53	4	no	—
231	C₂²×C₈	—	53	4	no	—
232	C₂²×C₈	—	53	4	no	—
233	C₂²×C₈	—	106	4	no	—
234	C₂²×C₈	—	106	4	no	—
235	C₂²×C₈	—	106	2	no	—
236	C₂×C₄²	Dic₁₀₆	1	2	yes	—
237	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
238	C₂×C₄²	C₄×D₅₃	1	2	yes	—
239	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
240	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
241	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
242	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
243	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
244	#108	—	106	16	no	—
245	C₄⋊C₈	—	53	8	no	—
246	C₄⋊C₈	—	53	8	no	—
247	C₄⋊C₈	—	106	4	no	—
248	C₄⋊C₈	—	106	4	no	—
249	#97	—	2	4	no	—
250	C₄×C₈	—	2	4	no	—
251	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	#11719919	1	1	yes	characteristic
255	C₂²×C₄	#11719919	1	1	yes	characteristic
256	C₂²×C₄	#11719919	1	1	yes	characteristic
257	C₂²×C₄	#11719919	1	2	yes	—
258	C₂²×C₄	#11719919	1	2	yes	—
259	C₂²×C₄	#11719919	1	1	yes	characteristic
260	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
261	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
262	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
263	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
264	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
267	C₂²×C₄	#11719945	1	2	yes	—
268	C₂²×C₄	#11719945	1	2	yes	—
269	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
271	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
272	M₁₆	—	2	4	no	—
273	M₁₆	—	2	4	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	2	2	no	—
276	M₁₆	—	106	4	no	—
277	M₁₆	—	106	4	no	—
278	M₁₆	—	212	4	no	—
279	M₁₆	C₂×Dic₁₀₆	1	4	yes	—
280	M₁₆	C₂×D₂₁₂	1	4	yes	—
281	C₂×C₈	—	2	4	no	—
282	C₂×C₈	—	2	4	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	53	8	no	—
286	C₂×C₈	—	106	8	no	—
287	C₂×C₈	—	106	8	no	—
288	C₂×C₈	—	106	4	no	—
289	C₂×C₈	—	106	4	no	—
290	C₂×C₈	—	106	4	no	—
291	C₂×C₈	—	212	4	no	—
292	C₄²	#11719919	1	2	yes	—
293	C₄²	#11719919	1	2	yes	—
294	C₄²	#11719919	1	2	yes	—
295	C₄²	#11719919	1	2	yes	—
296	C₄²	C₂×C₄×D₅₃	1	2	yes	—
297	C₄²	C₂×C₄×D₅₃	1	2	yes	—
298	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
299	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
300	C₂³	—	2	2	no	—
301	C₂³	—	2	2	no	—
302	C₂³	C₂×#11719919	1	1	yes	characteristic
303	C₂³	C₂×#11719919	1	1	yes	characteristic
304	C₂³	#13133209	1	2	yes	—
305	C₂³	#13133217	1	2	yes	—
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	4	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	#13133152	1	4	yes	complemented
315	C₂×C₄	C₂×#11719919	1	2	yes	—
316	C₂×C₄	C₂×#11719919	1	2	yes	—
317	C₂×C₄	C₂×#11719919	1	2	yes	—
318	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
319	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
320	C₂×C₄	C₂×#11719919	1	2	yes	—
321	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
322	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
323	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
324	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
325	C₂×C₄	#13133191	1	2	yes	—
326	C₂×C₄	#13133191	1	2	yes	—
327	C₂×C₄	#13133196	1	2	yes	—
328	C₂×C₄	#13133196	1	2	yes	—
329	C₂×C₄	#13133209	1	2	yes	—
330	C₂×C₄	#13133217	1	2	yes	—
331	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
332	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
334	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
335	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
336	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
337	C₈	—	2	4	no	—
338	C₈	—	2	4	no	—
339	C₈	—	106	8	no	—
340	C₈	—	212	4	no	—
341	C₂²	—	2	4	no	—
342	C₂²	—	2	4	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	#16810521	1	2	yes	—
348	C₂²	#16810521	1	2	yes	—
349	C₂²	C₂×#13133152	1	2	yes	—
350	C₂²	#16810671	1	2	yes	—
351	C₂²	C₂²×#11719919	1	1	yes	characteristic
352	C₂²	C₂×#13133209	1	1	yes	characteristic
353	C₂²	C₂×#13133217	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	2	no	—
357	C₄	#16810521	1	2	yes	—
358	C₄	#16810521	1	2	yes	—
359	C₄	C₂×#13133152	1	4	yes	complemented
360	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
361	C₄	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
362	C₄	C₂×#13133191	1	1	yes	characteristic
363	C₄	C₂×#13133196	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	2	1	no	—
366	C₂	—	2	1	no	—
367	C₂	#20275490	1	4	yes	direct product factor
368	C₂	C₂×#16810521	1	1	yes	characteristic
369	C₂	C₂²×#13133152	1	1	yes	characteristic
370	C₂	C₂×#16810671	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	C₂×#1950
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #20275490	C₂	direct (N × H)
128. C₂×C₄×C₅₃	#108	semidirect
174. C₄×C₅₃	C₂×#108	semidirect
188. C₂×C₅₃	#1950	semidirect
212. C₅₃	C₂×#1950	semidirect
314. C₂×C₄	#13133152	semidirect
359. C₄	C₂×#13133152	semidirect
367. C₂	#20275490	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133152	2
C₂×#16810647	2
C₂²×C₄⋊#88322	2
#20275490	2
C₂×#16810387	2
C₂×C₄×M₁₆×C₅₃	2
C₂×#1950	53

Subgroups — C2×#20275490 #27387612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20275490 #27387612