Subgroups — C₈×D₅₃⋊M₁₆ #28313744

Order 13568.

1046 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊#431	C₂	1	1	yes	maximal, characteristic
2	C₄×D₅₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×#11719936	C₂	1	4	yes	maximal, complemented
4	C₈×#11719940	C₂	1	4	yes	maximal, complemented
5	#20289911	C₂	1	2	yes	maximal, complemented
6	C₈×D₅₃⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×D₅₃⋊M₁₆	C₄	1	2	yes	—
9	#16811642	C₂²	1	2	yes	—
10	#16811641	C₄	1	2	yes	—
11	#16811640	C₂²	1	2	yes	—
12	#16811639	C₄	1	2	yes	—
13	D₅₃⋊(C₄⋊C₈)	C₄	1	2	yes	—
14	D₅₃⋊(C₂²⋊C₈)	C₄	1	2	yes	—
15	C₄×#11719936	C₄	1	2	yes	—
16	C₄×#11719936	C₂²	1	2	yes	—
17	C₄×#11719940	C₄	1	2	yes	—
18	C₄×#11719940	C₂²	1	2	yes	—
19	#16811582	C₄	1	2	yes	—
20	C₄×D₅₃⋊C₈	C₂²	1	2	yes	—
21	C₄×D₅₃⋊C₈	C₂²	1	2	yes	—
22	#88324⋊C₈	C₂²	1	8	yes	complemented
23	C₈×#88324	C₂²	1	8	yes	complemented
24	C₂²×C₈×D₅₃	C₄	1	2	yes	—
25	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×#88322	C₄	1	1	yes	characteristic
27	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
28	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic
29	C₄×C₈×D₅₃	C₄	1	2	yes	—
30	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic
31	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₅₃	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
35	C₂×#11719936	C₂×C₄	1	4	yes	—
36	D₅₃⋊M₁₆	C₈	1	16	yes	direct product factor
37	C₂×#11719940	C₂×C₄	1	4	yes	—
38	C₂×D₅₃⋊C₈	C₂×C₄	1	4	yes	—
39	C₂×D₅₃⋊C₈	C₂×C₄	1	4	yes	—
40	#13133284	C₂×C₄	1	4	yes	—
41	C₄⋊#88324	C₂×C₄	1	4	yes	—
42	#13133274	C₂×C₄	1	4	yes	—
43	C₂²⋊#88324	C₂×C₄	1	4	yes	—
44	#13133272	C₂×C₄	1	4	yes	—
45	#13133272	C₂×C₄	1	4	yes	—
46	C₄×#88324	C₂×C₄	1	4	yes	—
47	C₄×#88324	C₂×C₄	1	4	yes	—
48	C₄×#88324	C₂³	1	4	yes	—
49	C₄×#88324	C₂³	1	4	yes	—
50	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×#88322	C₂×C₄	1	2	yes	—
54	C₂×C₈×D₅₃	—	2	8	no	—
55	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
56	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
57	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
58	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
62	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
67	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄²×D₅₃	C₂³	1	1	yes	characteristic
69	C₄²×D₅₃	C₂³	1	1	yes	characteristic
70	C₄×#88322	C₂×C₄	1	1	yes	characteristic
71	C₄×#88322	C₂×C₄	1	2	yes	—
72	C₄×#88322	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
74	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
75	C₈×Dic₅₃	C₂³	1	2	yes	—
76	C₈×Dic₅₃	C₂³	1	2	yes	—
77	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
78	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
81	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₂³×D₅₃	C₄²	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
87	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×D₅₃	—	2	4	no	—
90	C₂×C₄×D₅₃	—	2	4	no	—
91	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
92	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
93	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
94	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
95	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
96	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
97	C₂×C₄×D₅₃	C₂×C₈	1	8	yes	—
98	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
104	#11719940	C₂×C₈	1	16	yes	complemented
105	C₂×#88324	C₂²×C₄	1	8	yes	—
106	C₂×#88324	C₂²×C₄	1	8	yes	—
107	#11719936	C₂×C₈	1	16	yes	complemented
108	D₅₃⋊C₈	—	2	8	no	—
109	D₅₃⋊C₈	C₂×C₈	1	16	yes	complemented
110	C₂×C₈×C₅₃	—	2	4	no	—
111	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
113	C₄²×C₅₃	C₄²	1	1	yes	characteristic
114	C₄²×C₅₃	C₄²	1	1	yes	characteristic
115	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
118	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
119	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
120	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
121	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
125	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
129	C₂×#88322	—	2	4	no	—
130	C₂×#88322	C₂²×C₄	1	2	yes	—
131	C₂×#88322	C₂²×C₄	1	2	yes	—
132	C₈×D₅₃	—	2	4	no	—
133	C₈×D₅₃	—	2	4	no	—
134	C₈×D₅₃	—	2	4	no	—
135	C₈×D₅₃	M₁₆	1	8	yes	complemented
136	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
137	C₂²×D₅₃	—	2	2	no	—
138	C₂²×D₅₃	—	2	2	no	—
139	C₂²×D₅₃	C₄×C₈	1	4	yes	—
140	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
141	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
145	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
151	C₂×Dic₅₃	—	2	2	no	—
152	C₂×Dic₅₃	—	2	2	no	—
153	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
154	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
155	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
156	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
157	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
158	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
159	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
160	C₄×D₅₃	—	2	2	no	—
161	C₄×D₅₃	—	2	2	no	—
162	C₄×D₅₃	—	2	2	no	—
163	C₄×D₅₃	—	2	2	no	—
164	C₄×D₅₃	—	2	2	no	—
165	C₄×D₅₃	—	2	2	no	—
166	C₄×D₅₃	C₄×C₈	1	4	yes	—
167	C₄×D₅₃	C₄×C₈	1	4	yes	—
168	C₄×D₅₃	C₂²×C₈	1	4	yes	—
169	C₄×D₅₃	C₂²×C₈	1	4	yes	—
170	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
171	C₄×D₅₃	#131	1	4	yes	—
172	#88324	—	2	8	no	—
173	#88324	C₂²×C₈	1	16	yes	complemented
174	C₈×C₅₃	—	2	2	no	—
175	C₈×C₅₃	C₂×M₁₆	1	4	yes	complemented
176	#88322	—	2	2	no	—
177	#88322	C₂×M₁₆	1	4	yes	complemented
178	C₂×C₈×M₁₆	—	53	1	no	maximal, Sylow
179	C₂²×C₅₃	—	2	1	no	—
180	C₂²×C₅₃	—	2	1	no	—
181	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
182	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
183	C₂×D₅₃	—	2	1	no	—
184	C₂×D₅₃	—	2	1	no	—
185	C₂×D₅₃	—	2	8	no	—
186	C₂×D₅₃	C₂×C₄×C₈	1	2	yes	—
187	C₂×D₅₃	C₂×C₄×C₈	1	2	yes	—
188	C₂×D₅₃	C₄×M₁₆	1	2	yes	—
189	C₂×D₅₃	#405	1	2	yes	—
190	C₄×C₅₃	—	2	1	no	—
191	C₄×C₅₃	—	2	1	no	—
192	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
193	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
194	C₄×C₅₃	#405	1	2	yes	—
195	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
196	Dic₅₃	—	2	1	no	—
197	Dic₅₃	—	2	1	no	—
198	Dic₅₃	C₂×C₄×C₈	1	2	yes	—
199	Dic₅₃	C₄×M₁₆	1	2	yes	—
200	Dic₅₃	C₂³×C₈	1	2	yes	—
201	Dic₅₃	C₂×#131	1	2	yes	—
202	C₂×#431	—	53	1	no	—
203	C₂×C₄×M₁₆	—	53	1	no	—
204	C₂²×C₄×C₈	—	53	1	no	—
205	C₈×M₁₆	—	53	4	no	—
206	C₈×M₁₆	—	53	4	no	—
207	C₂×#322	—	53	2	no	—
208	C₂×C₈²	—	53	2	no	—
209	C₂×C₅₃	—	2	2	no	—
210	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
211	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
212	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
213	D₅₃	—	2	2	no	—
214	D₅₃	C₈×M₁₆	1	4	yes	complemented
215	C₂²×M₁₆	—	53	2	no	—
216	C₂³×C₈	—	53	2	no	—
217	C₂²×C₄²	—	53	1	no	—
218	#431	—	53	2	no	—
219	#431	—	53	2	no	—
220	#431	—	53	2	no	—
221	#431	—	53	2	no	—
222	C₂×C₄⋊C₈	—	53	2	no	—
223	C₂×C₂²⋊C₈	—	53	2	no	—
224	C₄×M₁₆	—	53	2	no	—
225	C₄×M₁₆	—	53	2	no	—
226	C₄×M₁₆	—	53	2	no	—
227	C₄×M₁₆	—	53	2	no	—
228	C₂×#97	—	53	2	no	—
229	C₂×C₄×C₈	—	53	2	no	—
230	C₂×C₄×C₈	—	53	2	no	—
231	C₂×C₄×C₈	—	53	2	no	—
232	C₂×C₄×C₈	—	53	1	no	—
233	C₂×C₄×C₈	—	53	2	no	—
234	C₂×C₄×C₈	—	53	1	no	—
235	C₂×C₄×C₈	—	53	1	no	—
236	C₂×C₄×C₈	—	53	2	no	—
237	C₂×C₄×C₈	—	53	1	no	—
238	C₂×C₄×C₈	—	53	1	no	—
239	C₂×C₄×C₈	#6157	1	1	yes	characteristic
240	#322	—	53	8	no	—
241	C₈²	—	53	8	no	—
242	C₅₃	C₂×C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
243	C₂³×C₄	—	53	1	no	—
244	C₂³×C₄	—	53	1	no	—
245	C₂³×C₄	—	53	1	no	—
246	C₂×M₁₆	—	53	4	no	—
247	C₂×M₁₆	—	53	4	no	—
248	C₂×M₁₆	—	53	16	no	—
249	C₂²×C₈	—	53	4	no	—
250	C₂²×C₈	—	53	2	no	—
251	C₂²×C₈	—	53	4	no	—
252	C₂²×C₈	—	53	2	no	—
253	C₂²×C₈	—	53	2	no	—
254	C₂²×C₈	—	53	2	no	—
255	C₂²×C₈	—	53	2	no	—
256	C₂²×C₈	—	106	8	no	—
257	C₂²×C₈	C₂×#6157	1	2	yes	—
258	C₂×C₄²	—	53	1	no	—
259	C₂×C₄²	—	53	1	no	—
260	C₂×C₄²	—	53	1	no	—
261	C₂×C₄²	—	53	1	no	—
262	C₂×C₄²	—	53	1	no	—
263	C₂×C₄²	—	53	1	no	—
264	C₂×C₄²	—	53	1	no	—
265	C₂×C₄²	—	53	1	no	—
266	C₂×C₄²	—	53	1	no	—
267	C₂×C₄²	—	53	1	no	—
268	C₂×C₄²	—	53	1	no	—
269	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
270	C₄⋊C₈	—	53	4	no	—
271	C₄⋊C₈	—	53	4	no	—
272	C₂²⋊C₈	—	53	4	no	—
273	C₂²⋊C₈	—	53	4	no	—
274	#97	—	53	4	no	—
275	#97	—	53	4	no	—
276	C₄×C₈	—	53	4	no	—
277	C₄×C₈	—	53	4	no	—
278	C₄×C₈	—	53	4	no	—
279	C₄×C₈	—	53	4	no	—
280	C₄×C₈	—	53	2	no	—
281	C₄×C₈	—	53	2	no	—
282	C₄×C₈	—	53	2	no	—
283	C₄×C₈	—	53	2	no	—
284	C₄×C₈	—	53	2	no	—
285	C₄×C₈	—	53	1	no	—
286	C₄×C₈	—	53	1	no	—
287	C₄×C₈	C₂×#6157	1	2	yes	—
288	C₄×C₈	C₂×#6157	1	1	yes	characteristic
289	C₄×C₈	C₂×#6157	1	1	yes	characteristic
290	C₂⁴	—	53	1	no	—
291	C₂²×C₄	—	53	1	no	—
292	C₂²×C₄	—	53	8	no	—
293	C₂²×C₄	—	53	1	no	—
294	C₂²×C₄	—	53	1	no	—
295	C₂²×C₄	—	53	1	no	—
296	C₂²×C₄	—	53	1	no	—
297	C₂²×C₄	—	53	1	no	—
298	C₂²×C₄	—	53	1	no	—
299	C₂²×C₄	—	53	1	no	—
300	C₂²×C₄	—	53	1	no	—
301	C₂²×C₄	—	53	1	no	—
302	C₂²×C₄	—	53	1	no	—
303	C₂²×C₄	—	53	1	no	—
304	C₂²×C₄	—	53	1	no	—
305	C₂²×C₄	—	53	1	no	—
306	C₂²×C₄	—	53	1	no	—
307	C₂²×C₄	—	106	4	no	—
308	C₂²×C₄	—	106	4	no	—
309	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
310	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
311	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
312	M₁₆	—	53	16	no	—
313	M₁₆	—	53	16	no	—
314	C₂×C₈	—	2	4	no	—
315	C₂×C₈	—	53	8	no	—
316	C₂×C₈	—	53	16	no	—
317	C₂×C₈	—	53	2	no	—
318	C₂×C₈	—	53	8	no	—
319	C₂×C₈	—	53	8	no	—
320	C₂×C₈	—	53	2	no	—
321	C₂×C₈	—	106	4	no	—
322	C₂×C₈	—	106	4	no	—
323	C₂×C₈	—	106	4	no	—
324	C₂×C₈	—	106	8	no	—
325	C₂×C₈	—	106	4	no	—
326	C₂×C₈	C₂²×#6157	1	2	yes	—
327	C₂×C₈	C₂²×#6157	1	2	yes	—
328	C₄²	—	53	1	no	—
329	C₄²	—	53	1	no	—
330	C₄²	—	53	1	no	—
331	C₄²	—	53	1	no	—
332	C₄²	—	53	1	no	—
333	C₄²	—	53	1	no	—
334	C₄²	—	53	1	no	—
335	C₄²	—	53	1	no	—
336	C₄²	—	53	1	no	—
337	C₄²	—	53	1	no	—
338	C₄²	—	53	1	no	—
339	C₄²	—	53	1	no	—
340	C₄²	C₄×#6157	1	1	yes	characteristic
341	C₄²	C₄×#6157	1	1	yes	characteristic
342	C₄²	C₂²×#6157	1	1	yes	characteristic
343	C₄²	C₂²×#6157	1	1	yes	characteristic
344	C₂³	—	53	1	no	—
345	C₂³	—	53	4	no	—
346	C₂³	—	53	1	no	—
347	C₂³	—	106	2	no	—
348	C₂³	—	106	2	no	—
349	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	53	4	no	—
353	C₂×C₄	—	53	1	no	—
354	C₂×C₄	—	53	1	no	—
355	C₂×C₄	—	53	1	no	—
356	C₂×C₄	—	53	1	no	—
357	C₂×C₄	—	53	4	no	—
358	C₂×C₄	—	53	4	no	—
359	C₂×C₄	—	53	4	no	—
360	C₂×C₄	—	53	4	no	—
361	C₂×C₄	—	53	4	no	—
362	C₂×C₄	—	53	4	no	—
363	C₂×C₄	—	53	1	no	—
364	C₂×C₄	—	53	1	no	—
365	C₂×C₄	—	106	2	no	—
366	C₂×C₄	—	106	2	no	—
367	C₂×C₄	—	106	2	no	—
368	C₂×C₄	—	106	2	no	—
369	C₂×C₄	—	106	2	no	—
370	C₂×C₄	—	106	2	no	—
371	C₂×C₄	—	106	2	no	—
372	C₂×C₄	—	106	2	no	—
373	C₂×C₄	C₈×#6157	1	4	yes	—
374	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
375	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
376	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
377	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
378	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
379	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
380	C₈	—	2	2	no	—
381	C₈	—	53	16	no	—
382	C₈	—	53	4	no	—
383	C₈	—	106	8	no	—
384	C₈	—	106	2	no	—
385	C₈	D₅₃⋊M₁₆	1	4	yes	direct product factor
386	C₂²	—	2	1	no	—
387	C₂²	—	2	1	no	—
388	C₂²	—	53	2	no	—
389	C₂²	—	53	2	no	—
390	C₂²	—	53	2	no	—
391	C₂²	—	53	2	no	—
392	C₂²	—	106	8	no	—
393	C₂²	—	106	1	no	—
394	C₂²	—	106	1	no	—
395	C₂²	C₂×C₈×#6157	1	2	yes	—
396	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	53	2	no	—
400	C₄	—	53	2	no	—
401	C₄	—	53	2	no	—
402	C₄	—	53	2	no	—
403	C₄	—	106	1	no	—
404	C₄	—	106	1	no	—
405	C₄	C₂×C₈×#6157	1	2	yes	—
406	C₄	C₂×C₈×#6157	1	2	yes	—
407	C₄	#16811586	1	2	yes	—
408	C₄	C₂×D₅₃⋊M₁₆	1	2	yes	—
409	C₂	—	2	2	no	—
410	C₂	—	53	4	no	—
411	C₂	—	106	2	no	—
412	C₂	C₂²×C₈×#6157	1	1	yes	characteristic
413	C₂	C₄×D₅₃⋊M₁₆	1	1	yes	characteristic
414	C₂	C₂×#16811586	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₄²
prime radical	C₂³×D₅₃	C₄²
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	C₂×C₈×M₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₈×#11719936	C₂	semidirect
4. C₈×#11719940	C₂	semidirect
5. #20289911	C₂	semidirect
6. C₈×D₅₃⋊C₈	C₂	semidirect
22. #88324⋊C₈	C₂²	semidirect
23. C₈×#88324	C₂²	semidirect
36. D₅₃⋊M₁₆	C₈	direct (N × H)
104. #11719940	C₂×C₈	semidirect
107. #11719936	C₂×C₈	semidirect
109. D₅₃⋊C₈	C₂×C₈	semidirect
135. C₈×D₅₃	M₁₆	semidirect
173. #88324	C₂²×C₈	semidirect
175. C₈×C₅₃	C₂×M₁₆	semidirect
177. #88322	C₂×M₁₆	semidirect
214. D₅₃	C₈×M₁₆	semidirect
242. C₅₃	C₂×C₈×M₁₆	semidirect
385. C₈	D₅₃⋊M₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
D₅₃⋊#431	2
C₄×D₅₃⋊M₁₆	2
C₈×#11719936	2
C₈×#11719940	2
#20289911	2
C₈×D₅₃⋊C₈	2
C₂×C₄×C₈×D₅₃	2
C₂×C₈×M₁₆	53

Subgroups — C8×D53⋊M16 #28313744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×D₅₃⋊M₁₆ #28313744