Subgroups — Q₈⋊C₈×Dic₅₃ #27274691

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊#88322	C₂	1	1	yes	maximal, characteristic
5	#20272555	C₂	1	1	yes	maximal, characteristic
6	#20272554	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₅₃	C₄	1	2	yes	—
9	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
10	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	—	2	2	no	—
12	C₂×C₈×Dic₅₃	—	2	1	no	—
13	C₄⋊C₄×Dic₅₃	—	2	2	no	—
14	C₄⋊C₄×Dic₅₃	C₄	1	2	yes	—
15	C₄×C₄⋊Dic₅₃	—	2	1	no	—
16	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
17	C₄²×Dic₅₃	—	2	1	no	—
18	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
19	#16810377	—	2	1	no	—
20	Q₈⋊#88322	C₄	1	4	yes	complemented
21	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×Q₈⋊C₈	C₄	1	4	yes	complemented
25	C₂²×#88322	—	2	1	no	—
26	C₂²×#88322	—	2	1	no	—
27	Q₈×Dic₅₃	—	2	4	no	—
28	Q₈×Dic₅₃	C₈	1	8	yes	complemented
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₅₃	—	2	1	no	—
32	C₂×C₄×Dic₅₃	—	2	2	no	—
33	C₂×C₄×Dic₅₃	—	2	2	no	—
34	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
36	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
38	C₄⋊#88322	—	2	1	no	—
39	C₄⋊#88322	C₂×C₄	1	2	yes	—
40	C₄×#88322	—	2	1	no	—
41	C₄×#88322	C₂×C₄	1	2	yes	—
42	C₈×Dic₅₃	—	2	8	no	—
43	C₈×Dic₅₃	—	4	2	no	—
44	#13133124	—	2	1	no	—
45	#13133124	—	2	2	no	—
46	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	—	2	1	no	—
48	C₂²×C₈×C₅₃	—	2	1	no	—
49	C₄×Q₈×C₅₃	—	2	1	no	—
50	C₄×Q₈×C₅₃	—	2	1	no	—
51	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	—	2	1	no	—
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₅₃	—	2	1	no	—
58	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₅₃	—	2	1	no	—
60	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₂×Q₈×C₅₃	—	2	1	no	—
62	C₂×Q₈×C₅₃	—	2	2	no	—
63	C₂×Q₈×C₅₃	C₄²	1	2	yes	—
64	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
65	C₂²×C₄×C₅₃	—	2	1	no	—
66	C₂²×C₄×C₅₃	—	2	1	no	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₅₃	—	2	2	no	—
71	C₂²×Dic₅₃	C₂²⋊C₄	1	2	yes	—
72	C₂²×Dic₅₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	—	4	1	no	—
78	C₄⋊C₄×C₅₃	—	2	1	no	—
79	C₄⋊C₄×C₅₃	—	2	2	no	—
80	C₄⋊C₄×C₅₃	—	2	2	no	—
81	C₄⋊C₄×C₅₃	C₄²	1	2	yes	—
82	C₄²×C₅₃	—	2	2	no	—
83	C₄²×C₅₃	—	2	2	no	—
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₅₃	#41	1	1	yes	characteristic
87	C₄²×C₅₃	#41	1	1	yes	characteristic
88	C₄⋊Dic₅₃	—	2	2	no	—
89	C₄⋊Dic₅₃	—	2	8	no	—
90	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
91	C₄×Dic₅₃	—	2	1	no	—
92	C₄×Dic₅₃	—	2	1	no	—
93	C₄×Dic₅₃	—	2	8	no	—
94	C₄×Dic₅₃	—	2	4	no	—
95	C₄×Dic₅₃	—	4	1	no	—
96	C₄×Dic₅₃	—	4	1	no	—
97	C₄×Dic₅₃	—	4	4	no	—
98	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
99	C₄×Dic₅₃	M₁₆	1	4	yes	—
100	C₄×Dic₅₃	SD₁₆	1	2	yes	—
101	C₄×Dic₅₃	Q₁₆	1	2	yes	—
102	C₂×#88322	—	2	4	no	—
103	C₂×#88322	—	2	2	no	—
104	C₂×#88322	—	2	2	no	—
105	C₂×#88322	—	4	1	no	—
106	C₂×#88322	—	4	1	no	—
107	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	Q₈×C₅₃	—	2	2	no	—
109	Q₈×C₅₃	—	2	4	no	—
110	Q₈×C₅₃	C₄×C₈	1	4	yes	complemented
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	4	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	4	2	no	—
116	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
124	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
127	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
128	C₂×Dic₅₃	—	2	2	no	—
129	C₂×Dic₅₃	—	2	2	no	—
130	C₂×Dic₅₃	—	2	4	no	—
131	C₂×Dic₅₃	—	4	2	no	—
132	C₂×Dic₅₃	—	4	2	no	—
133	C₂×Dic₅₃	C₂²⋊C₈	1	4	yes	—
134	C₂×Dic₅₃	C₂²⋊C₈	1	4	yes	—
135	C₂×Dic₅₃	#103	1	4	yes	—
136	C₂×Dic₅₃	#104	1	4	yes	—
137	C₈×C₅₃	—	2	4	no	—
138	C₈×C₅₃	—	4	2	no	—
139	#88322	—	2	4	no	—
140	#88322	—	4	2	no	—
141	C₄×Q₈⋊C₈	—	53	1	no	maximal, Sylow
142	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
144	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
145	C₂²×C₅₃	#416	1	1	yes	characteristic
146	C₂²×C₅₃	C₂×#104	1	2	yes	—
147	C₄×C₅₃	—	2	4	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	4	2	no	—
151	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
152	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
153	C₄×C₅₃	#431	1	2	yes	—
154	C₄×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
155	C₄×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
156	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
157	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
158	Dic₅₃	—	2	4	no	—
159	Dic₅₃	—	4	4	no	—
160	Dic₅₃	Q₈⋊C₈	1	8	yes	direct product factor
161	C₄²×Q₈	—	53	1	no	—
162	C₄×C₄⋊C₈	—	53	1	no	—
163	C₄²×C₈	—	53	1	no	—
164	C₂×Q₈⋊C₈	—	53	1	no	—
165	C₂×Q₈⋊C₈	D₅₃	1	1	yes	characteristic
166	#1279	—	53	1	no	—
167	#1279	—	53	1	no	—
168	C₂×C₅₃	C₂×Q₈⋊C₈	1	2	yes	—
169	C₂×C₅₃	#1476	1	2	yes	—
170	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
171	C₂×C₅₃	C₄×#104	1	1	yes	characteristic
172	C₂×C₅₃	C₄×#103	1	1	yes	characteristic
173	C₂×C₄×Q₈	—	53	2	no	—
174	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
175	C₂×C₄⋊C₈	—	53	1	no	—
176	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
177	C₂×C₄×C₈	—	53	1	no	—
178	C₂×C₄×C₈	—	106	2	no	—
179	C₂×C₄×C₈	—	106	1	no	—
180	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
181	C₄×C₄⋊C₄	—	53	2	no	—
182	C₄×C₄⋊C₄	—	53	1	no	—
183	C₄×C₄⋊C₄	—	106	1	no	—
184	C₄×C₄⋊C₄	—	106	2	no	—
185	C₄³	—	53	1	no	—
186	C₄³	—	106	1	no	—
187	#336	—	106	1	no	—
188	Q₈⋊C₈	—	53	4	no	—
189	Q₈⋊C₈	Dic₅₃	1	4	yes	direct product factor
190	C₅₃	C₄×Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
191	C₂²×Q₈	C₄×D₅₃	1	1	yes	characteristic
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	1	no	—
194	C₂²×C₈	—	106	1	no	—
195	C₂²×C₈	—	106	1	no	—
196	C₄×Q₈	—	2	1	no	—
197	C₄×Q₈	—	2	1	no	—
198	C₄×Q₈	—	53	8	no	—
199	C₄×Q₈	—	106	4	no	—
200	C₄×Q₈	C₄×D₅₃	1	2	yes	—
201	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	53	2	no	—
204	C₂×C₄⋊C₄	—	106	2	no	—
205	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	—	53	2	no	—
208	C₂×C₄²	—	53	1	no	—
209	C₂×C₄²	—	53	2	no	—
210	C₂×C₄²	—	53	1	no	—
211	C₂×C₄²	—	106	2	no	—
212	C₂×C₄²	—	106	2	no	—
213	C₂×C₄²	—	106	1	no	—
214	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	53	2	no	—
217	C₄⋊C₈	—	106	1	no	—
218	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	53	2	no	—
221	C₄×C₈	—	106	8	no	—
222	C₄×C₈	—	106	1	no	—
223	C₄×C₈	—	212	2	no	—
224	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
225	#95	—	106	2	no	—
226	#95	—	106	1	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	C₈×D₅₃	1	4	yes	—
230	C₂×Q₈	C₄×Dic₅₃	1	2	yes	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	53	2	no	—
234	C₂²×C₄	—	53	2	no	—
235	C₂²×C₄	—	106	2	no	—
236	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
237	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
238	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	106	4	no	—
245	C₂×C₈	—	106	2	no	—
246	C₂×C₈	—	106	2	no	—
247	C₂×C₈	—	212	1	no	—
248	C₂×C₈	—	212	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	53	4	no	—
253	C₄⋊C₄	—	106	8	no	—
254	C₄⋊C₄	—	106	2	no	—
255	C₄⋊C₄	C₄×Dic₅₃	1	2	yes	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	53	4	no	—
259	C₄²	—	53	4	no	—
260	C₄²	—	53	2	no	—
261	C₄²	—	53	2	no	—
262	C₄²	—	106	4	no	—
263	C₄²	—	106	8	no	—
264	C₄²	—	106	1	no	—
265	C₄²	—	106	1	no	—
266	C₄²	—	212	4	no	—
267	C₄²	—	212	1	no	—
268	C₄²	—	212	1	no	—
269	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
270	C₄²	#11719947	1	1	yes	characteristic
271	C₄²	#11719947	1	1	yes	characteristic
272	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
273	C₂³	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
274	Q₈	—	2	2	no	—
275	Q₈	—	2	4	no	—
276	Q₈	C₈×Dic₅₃	1	4	yes	complemented
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	53	4	no	—
283	C₂×C₄	—	53	4	no	—
284	C₂×C₄	—	53	4	no	—
285	C₂×C₄	—	53	4	no	—
286	C₂×C₄	—	106	2	no	—
287	C₂×C₄	—	106	2	no	—
288	C₂×C₄	—	106	4	no	—
289	C₂×C₄	—	212	2	no	—
290	C₂×C₄	—	212	2	no	—
291	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
292	C₂×C₄	#13133173	1	1	yes	characteristic
293	C₂×C₄	#13133177	1	1	yes	characteristic
294	C₂×C₄	#13133177	1	1	yes	characteristic
295	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
296	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
297	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
298	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
299	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
300	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
301	C₂×C₄	#13133238	1	1	yes	characteristic
302	C₂×C₄	#13133240	1	1	yes	characteristic
303	C₈	—	2	4	no	—
304	C₈	—	4	2	no	—
305	C₈	—	106	4	no	—
306	C₈	—	212	2	no	—
307	C₂²	C₂²⋊C₄×Dic₅₃	1	1	yes	characteristic
308	C₂²	C₂²⋊C₈×D₅₃	1	2	yes	—
309	C₂²	#103×D₅₃	1	1	yes	characteristic
310	C₂²	#16810618	1	1	yes	characteristic
311	C₂²	#104×D₅₃	1	2	yes	—
312	C₄	—	2	4	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	53	8	no	—
317	C₄	—	106	4	no	—
318	C₄	—	212	4	no	—
319	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
320	C₄	M₁₆×Dic₅₃	1	1	yes	characteristic
321	C₄	#16810673	1	2	yes	—
322	C₄	#16810708	1	1	yes	characteristic
323	C₄	SD₁₆×Dic₅₃	1	1	yes	characteristic
324	C₄	#16810715	1	1	yes	characteristic
325	C₄	Q₁₆×Dic₅₃	1	1	yes	characteristic
326	C₂	Q₈⋊C₈×D₅₃	1	2	yes	—
327	C₂	#20273096	1	2	yes	—
328	C₂	C₂²⋊C₈×Dic₅₃	1	1	yes	characteristic
329	C₂	#104×Dic₅₃	1	1	yes	characteristic
330	C₂	#103×Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×Q₈⋊C₈	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×Q₈⋊C₈	D₅₃
hypocenter	C₅₃	C₄×Q₈⋊C₈
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
20. Q₈⋊#88322	C₄	semidirect
24. C₅₃×Q₈⋊C₈	C₄	semidirect
28. Q₈×Dic₅₃	C₈	semidirect
110. Q₈×C₅₃	C₄×C₈	semidirect
160. Dic₅₃	Q₈⋊C₈	direct (N × H)
189. Q₈⋊C₈	Dic₅₃	direct (N × H)
190. C₅₃	C₄×Q₈⋊C₈	semidirect
276. Q₈	C₈×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×Dic₅₃	2
C₄⋊C₈×Dic₅₃	2
C₄×C₈×Dic₅₃	2
C₂×Q₈⋊#88322	2
#20272555	2
#20272554	2
C₂×C₅₃×Q₈⋊C₈	2
C₄×Q₈⋊C₈	53

Subgroups — Q8⋊C8×Dic53 #27274691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈⋊C₈×Dic₅₃ #27274691