Subgroups — M₁₆×C₄⋊Dic₅₃ #27382083

Order 13568.

1242 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×Dic₅₃	C₂	1	2	yes	maximal
3	C₂×C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#20274587	C₂	1	4	yes	maximal, complemented
5	#20274582	C₂	1	2	yes	maximal, complemented
6	C₈×C₄⋊Dic₅₃	C₂	1	2	yes	maximal, complemented
7	#20274408	C₂	1	2	yes	maximal
8	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11719917	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₅₃	C₄	1	2	yes	—
11	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	—
12	#16810647	C₄	1	8	yes	complemented
13	C₂×C₄⋊#88322	C₂²	1	2	yes	—
14	C₂×C₄⋊#88322	C₂²	1	2	yes	—
15	M₁₆×Dic₅₃	—	2	8	no	—
16	C₂×#13133127	C₂²	1	4	yes	—
17	C₂×C₈×Dic₅₃	C₂²	1	4	yes	—
18	C₄×C₄⋊Dic₅₃	C₄	1	2	yes	—
19	C₄×C₄⋊Dic₅₃	C₄	1	2	yes	—
20	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
23	C₂×#13133124	C₄	1	2	yes	—
24	#16810377	C₂²	1	4	yes	—
25	#16810377	C₂²	1	4	yes	—
26	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₅₃	C₄	1	8	yes	complemented
30	C₂×#97×C₅₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
32	C₂³×Dic₅₃	C₂×C₄	1	4	yes	—
33	C₂×#11719917	—	2	4	no	—
34	C₂×#11719917	—	2	4	no	—
35	C₂×#11719917	C₂×C₄	1	8	yes	—
36	C₂²×#88322	C₂³	1	4	yes	—
37	C₂×C₄⋊Dic₅₃	—	2	8	no	—
38	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄×Dic₅₃	—	2	4	no	—
42	C₂×C₄×Dic₅₃	—	2	4	no	—
43	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
45	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
46	C₄⋊#88322	C₂×C₄	1	8	yes	complemented
47	C₄⋊#88322	C₂×C₄	1	8	yes	complemented
48	#13133127	—	2	8	no	—
49	C₈×Dic₅₃	—	2	8	no	—
50	#13133124	C₂×C₄	1	2	yes	—
51	#13133124	C₂×C₄	1	4	yes	—
52	#13133124	C₂×C₄	1	2	yes	—
53	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×M₁₆×C₅₃	—	2	2	no	—
57	C₂×M₁₆×C₅₃	—	2	2	no	—
58	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
59	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
60	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
61	C₂×M₁₆×C₅₃	Q₈	1	4	yes	—
62	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
63	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
64	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
65	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
66	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	#97×C₅₃	C₂×C₄	1	8	yes	complemented
71	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
72	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	—	2	4	no	—
74	C₂²×C₄×C₅₃	—	4	2	no	—
75	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
76	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
77	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₅₃	#41	1	2	yes	—
87	C₂²×C₄×C₅₃	#41	1	2	yes	—
88	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
89	C₂²×Dic₅₃	—	2	8	no	—
90	C₂²×Dic₅₃	—	2	8	no	—
91	C₂²×Dic₅₃	—	4	4	no	—
92	C₂²×Dic₅₃	C₂²×C₄	1	4	yes	—
93	C₂²×Dic₅₃	C₂²×C₄	1	4	yes	—
94	M₁₆×C₅₃	—	2	8	no	—
95	M₁₆×C₅₃	C₄⋊C₄	1	16	yes	complemented
96	C₂×C₈×C₅₃	—	2	2	no	—
97	C₂×C₈×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
100	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
102	C₄²×C₅₃	C₄²	1	4	yes	—
103	C₄²×C₅₃	C₄²	1	4	yes	—
104	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
105	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
106	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
107	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
108	C₄⋊Dic₅₃	—	2	4	no	—
109	C₄⋊Dic₅₃	—	2	8	no	—
110	C₄⋊Dic₅₃	M₁₆	1	8	yes	direct product factor
111	C₄×Dic₅₃	—	2	8	no	—
112	C₄×Dic₅₃	—	2	2	no	—
113	C₄×Dic₅₃	—	2	2	no	—
114	C₄×Dic₅₃	—	2	2	no	—
115	C₄×Dic₅₃	—	2	2	no	—
116	#11719917	—	2	16	no	—
117	C₂×#88322	—	2	4	no	—
118	C₂×#88322	—	2	4	no	—
119	C₂×#88322	C₂²×C₄	1	8	yes	—
120	C₂³×C₅₃	—	2	2	no	—
121	C₂³×C₅₃	—	2	2	no	—
122	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
123	C₂³×C₅₃	C₄×D₄	1	2	yes	—
124	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
125	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
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	8	no	—
130	C₂×C₄×C₅₃	—	4	4	no	—
131	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
134	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
135	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
136	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
137	C₂×C₄×C₅₃	#117	1	4	yes	—
138	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
139	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
141	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
142	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
143	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
149	C₂×Dic₅₃	—	2	4	no	—
150	C₂×Dic₅₃	—	2	4	no	—
151	C₂×Dic₅₃	—	2	4	no	—
152	C₂×Dic₅₃	—	2	4	no	—
153	C₂×Dic₅₃	—	2	4	no	—
154	C₂×Dic₅₃	—	4	16	no	—
155	C₂×Dic₅₃	C₂×M₁₆	1	8	yes	—
156	C₈×C₅₃	—	2	4	no	—
157	C₈×C₅₃	C₂×C₄⋊C₄	1	8	yes	complemented
158	#88322	—	2	8	no	—
159	C₄⋊C₄×M₁₆	—	53	1	no	maximal, Sylow
160	C₂²×C₅₃	—	2	2	no	—
161	C₂²×C₅₃	—	2	2	no	—
162	C₂²×C₅₃	—	2	4	no	—
163	C₂²×C₅₃	—	2	4	no	—
164	C₂²×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
165	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
166	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₂²×C₅₃	C₂²×M₁₆	1	2	yes	—
169	C₂²×C₅₃	#568	1	2	yes	—
170	C₄×C₅₃	—	2	2	no	—
171	C₄×C₅₃	—	2	2	no	—
172	C₄×C₅₃	—	2	2	no	—
173	C₄×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
174	C₄×C₅₃	C₄×M₁₆	1	4	yes	complemented
175	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
176	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
177	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
178	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
179	Dic₅₃	—	2	8	no	—
180	Dic₅₃	—	4	4	no	—
181	C₂×#423	—	53	1	no	—
182	C₂×C₄×M₁₆	—	53	2	no	—
183	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
184	C₂×C₄×C₄⋊C₄	—	53	1	no	—
185	#1771	—	53	4	no	—
186	#1770	—	53	2	no	—
187	C₈×C₄⋊C₄	—	53	2	no	—
188	#1728	—	53	2	no	—
189	C₂×C₅₃	—	2	4	no	—
190	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
192	C₂×C₅₃	#2873	1	1	yes	characteristic
193	C₂×C₅₃	D₄×M₁₆	1	2	yes	—
194	C₂×C₅₃	Q₈×M₁₆	1	2	yes	—
195	C₂²×M₁₆	—	53	2	no	—
196	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
197	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
198	C₂²×C₄⋊C₄	—	53	2	no	—
199	C₂²×C₄²	—	53	2	no	—
200	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
201	#423	—	53	8	no	—
202	C₂×C₄⋊C₈	—	53	2	no	—
203	C₂×C₄⋊C₈	—	53	2	no	—
204	C₄×M₁₆	—	106	8	no	—
205	C₄×M₁₆	Dic₅₃	1	8	yes	complemented
206	C₂×#97	—	53	4	no	—
207	C₂×#97	C₂×D₅₃	1	2	yes	—
208	C₂×C₄×C₈	—	53	4	no	—
209	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
210	C₄×C₄⋊C₄	—	53	2	no	—
211	C₄×C₄⋊C₄	—	53	2	no	—
212	C₄×C₄⋊C₄	—	53	1	no	—
213	C₄×C₄⋊C₄	—	53	1	no	—
214	C₄×C₄⋊C₄	—	53	2	no	—
215	C₂×#95	—	53	2	no	—
216	#336	—	53	4	no	—
217	#336	—	53	4	no	—
218	C₅₃	C₄⋊C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
219	C₂³×C₄	—	53	4	no	—
220	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
221	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
222	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
223	C₂×M₁₆	—	2	2	no	—
224	C₂×M₁₆	—	2	2	no	—
225	C₂×M₁₆	—	53	8	no	—
226	C₂×M₁₆	—	106	4	no	—
227	C₂×M₁₆	—	106	4	no	—
228	C₂×M₁₆	Dic₁₀₆	1	4	yes	—
229	C₂×M₁₆	D₂₁₂	1	4	yes	—
230	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
231	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
232	C₂²×C₈	—	53	4	no	—
233	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
234	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
235	C₂×C₄⋊C₄	—	53	2	no	—
236	C₂×C₄⋊C₄	—	53	4	no	—
237	C₂×C₄⋊C₄	—	53	2	no	—
238	C₂×C₄⋊C₄	—	106	8	no	—
239	C₂×C₄²	—	53	4	no	—
240	C₂×C₄²	—	53	2	no	—
241	C₂×C₄²	—	53	2	no	—
242	C₂×C₄²	—	106	4	no	—
243	C₂×C₄²	—	106	4	no	—
244	C₂×C₄²	C₄×D₅₃	1	4	yes	—
245	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
246	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
247	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
248	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
249	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
250	C₄⋊C₈	—	53	8	no	—
251	C₄⋊C₈	—	53	8	no	—
252	#97	—	106	8	no	—
253	#97	C₂×Dic₅₃	1	8	yes	complemented
254	C₄×C₈	—	106	8	no	—
255	C₄×C₈	C₂×Dic₅₃	1	8	yes	complemented
256	#95	—	53	2	no	—
257	#95	—	53	4	no	—
258	#95	—	53	2	no	—
259	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	53	4	no	—
263	C₂²×C₄	—	53	4	no	—
264	C₂²×C₄	—	106	8	no	—
265	C₂²×C₄	—	106	8	no	—
266	C₂²×C₄	—	212	4	no	—
267	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
268	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
269	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
270	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
271	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
276	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
277	C₂²×C₄	#11719945	1	2	yes	—
278	C₂²×C₄	#11719945	1	2	yes	—
279	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
280	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
281	M₁₆	—	2	8	no	—
282	M₁₆	—	106	16	no	—
283	M₁₆	C₄⋊Dic₅₃	1	16	yes	direct product factor
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	53	8	no	—
287	C₂×C₈	—	106	4	no	—
288	C₂×C₈	—	106	4	no	—
289	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
290	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
291	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
292	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
293	C₄⋊C₄	—	53	8	no	—
294	C₄⋊C₄	—	106	8	no	—
295	C₄⋊C₄	—	106	4	no	—
296	C₄²	—	106	8	no	—
297	C₄²	—	106	2	no	—
298	C₄²	—	106	2	no	—
299	C₄²	—	106	2	no	—
300	C₄²	—	106	2	no	—
301	C₄²	C₄×Dic₅₃	1	4	yes	—
302	C₄²	C₄×Dic₅₃	1	4	yes	—
303	C₄²	C₂×C₄×D₅₃	1	2	yes	—
304	C₄²	C₂×C₄×D₅₃	1	2	yes	—
305	C₄²	C₂²×Dic₅₃	1	2	yes	—
306	C₄²	C₂²×Dic₅₃	1	2	yes	—
307	C₂³	—	2	2	no	—
308	C₂³	—	2	2	no	—
309	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
310	C₂³	C₄×D₂₁₂	1	2	yes	—
311	C₂³	C₄×Dic₁₀₆	1	2	yes	—
312	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	8	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	53	8	no	—
319	C₂×C₄	—	106	4	no	—
320	C₂×C₄	—	106	4	no	—
321	C₂×C₄	—	106	4	no	—
322	C₂×C₄	—	106	4	no	—
323	C₂×C₄	—	106	4	no	—
324	C₂×C₄	—	212	16	no	—
325	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
326	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
327	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
328	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
329	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
330	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
331	C₂×C₄	#13133176	1	4	yes	—
332	C₂×C₄	C₄×D₂₁₂	1	4	yes	—
333	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
334	C₂×C₄	C₄×Dic₁₀₆	1	4	yes	—
335	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
336	C₂×C₄	M₁₆×D₅₃	1	4	yes	—
337	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
338	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
339	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
340	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
341	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
342	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
343	C₈	—	2	4	no	—
344	C₈	—	106	8	no	—
345	C₈	C₂×C₄⋊Dic₅₃	1	8	yes	complemented
346	C₂²	—	2	4	no	—
347	C₂²	—	2	4	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	2	no	—
350	C₂²	C₄×C₄⋊Dic₅₃	1	4	yes	—
351	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
352	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
353	C₂²	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
354	C₂²	C₂×M₁₆×D₅₃	1	2	yes	—
355	C₂²	#16811389	1	2	yes	—
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	2	2	no	—
359	C₄	—	106	8	no	—
360	C₄	—	212	4	no	—
361	C₄	C₄×C₄⋊Dic₅₃	1	4	yes	—
362	C₄	M₁₆×Dic₅₃	1	4	yes	complemented
363	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
364	C₄	C₂×#13133176	1	1	yes	characteristic
365	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
366	C₄	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
367	C₂	—	2	4	no	—
368	C₂	C₂×C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
369	C₂	C₂×M₁₆×Dic₅₃	1	1	yes	characteristic
370	C₂	#20283133	1	1	yes	characteristic
371	C₂	M₁₆×D₂₁₂	1	2	yes	—
372	C₂	M₁₆×Dic₁₀₆	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. #20274587	C₂	semidirect
5. #20274582	C₂	semidirect
6. C₈×C₄⋊Dic₅₃	C₂	semidirect
12. #16810647	C₄	semidirect
29. C₄×M₁₆×C₅₃	C₄	semidirect
46. C₄⋊#88322	C₂×C₄	semidirect
47. C₄⋊#88322	C₂×C₄	semidirect
70. #97×C₅₃	C₂×C₄	semidirect
71. C₄×C₈×C₅₃	C₂×C₄	semidirect
95. M₁₆×C₅₃	C₄⋊C₄	semidirect
110. C₄⋊Dic₅₃	M₁₆	direct (N × H)
157. C₈×C₅₃	C₂×C₄⋊C₄	semidirect
174. C₄×C₅₃	C₄×M₁₆	semidirect
205. C₄×M₁₆	Dic₅₃	semidirect
218. C₅₃	C₄⋊C₄×M₁₆	semidirect
253. #97	C₂×Dic₅₃	semidirect
255. C₄×C₈	C₂×Dic₅₃	semidirect
283. M₁₆	C₄⋊Dic₅₃	direct (N × H)
345. C₈	C₂×C₄⋊Dic₅₃	semidirect
362. C₄	M₁₆×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810647	2
C₂×M₁₆×Dic₅₃	2
C₂×C₄×C₄⋊Dic₅₃	2
#20274587	2
#20274582	2
C₈×C₄⋊Dic₅₃	2
#20274408	2
C₂×C₄×M₁₆×C₅₃	2
C₄⋊C₄×M₁₆	53

Subgroups — M16×C4⋊Dic53 #27382083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×C₄⋊Dic₅₃ #27382083