Subgroups — #27355789

Order 13568.

656 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283847	C₂	1	2	yes	maximal, complemented
2	#20273600	C₂	1	2	yes	maximal, complemented
3	#20273594	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#88322	C₂	1	2	yes	maximal, complemented
5	#20273474	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273154	C₂	1	2	yes	maximal, complemented
7	C₈×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
9	Q₈×#88322	C₂²	1	4	yes	—
10	#16810701	—	2	4	no	—
11	#16810701	C₂²	1	4	yes	—
12	#16810676	—	2	2	no	—
13	#16810676	C₂²	1	1	yes	characteristic
14	#16810647	—	2	2	no	—
15	#16810647	C₂²	1	2	yes	—
16	C₄×#11719917	C₂²	1	1	yes	characteristic
17	#16810376	C₂²	1	2	yes	—
18	#16810376	C₂²	1	2	yes	—
19	Q₈⋊#88322	C₂²	1	4	yes	—
20	D₄⋊#88322	C₂²	1	4	yes	—
21	#16810345	C₂²	1	2	yes	—
22	C₈×#88322	C₂²	1	2	yes	—
23	C₂×#135×C₅₃	C₄	1	1	yes	characteristic
24	C₄×#41×C₅₃	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
26	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×SD₁₆×C₅₃	C₄	1	2	yes	—
28	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
29	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic
30	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#427	C₄	1	1	yes	characteristic
32	C₅₃×#416	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#11719917	—	2	4	no	—
35	C₂×#11719917	—	2	2	no	—
36	C₂×#11719917	—	2	1	no	—
37	C₄⋊#88322	—	2	2	no	—
38	C₄⋊#88322	—	2	4	no	—
39	C₄⋊#88322	D₄	1	1	yes	characteristic
40	C₄⋊#88322	D₄	1	1	yes	characteristic
41	C₄⋊#88322	C₂³	1	4	yes	—
42	C₂²⋊#88322	—	2	4	no	—
43	C₂²⋊#88322	D₄	1	2	yes	—
44	#13133128	—	2	1	no	—
45	C₄×#88322	—	2	4	no	—
46	C₄×#88322	C₂³	1	2	yes	—
47	C₄×#88322	C₂³	1	2	yes	—
48	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
49	#135×C₅₃	—	2	4	no	—
50	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
56	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
57	C₄×D₄×C₅₃	—	2	2	no	—
58	C₄×D₄×C₅₃	—	2	2	no	—
59	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×D₄×C₅₃	C₂³	1	2	yes	—
61	#117×C₅₃	—	2	2	no	—
62	#117×C₅₃	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₅₃	—	2	2	no	—
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
66	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
67	#106×C₅₃	C₂×C₄	1	2	yes	—
68	#103×C₅₃	C₂×C₄	1	2	yes	—
69	#103×C₅₃	C₂×C₄	1	2	yes	—
70	#102×C₅₃	C₂×C₄	1	2	yes	—
71	#102×C₅₃	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
76	#41×C₅₃	—	2	4	no	—
77	#41×C₅₃	—	4	2	no	—
78	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×D₄×C₅₃	—	2	2	no	—
80	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅₃	—	2	2	no	—
82	C₂²×C₄×C₅₃	—	2	2	no	—
83	C₂²×C₄×C₅₃	—	2	2	no	—
84	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
87	Q₁₆×C₅₃	M₁₆	1	4	yes	complemented
88	SD₁₆×C₅₃	—	2	4	no	—
89	D₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	4	1	no	—
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
99	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
100	C₂²⋊C₄×C₅₃	—	2	2	no	—
101	C₂²⋊C₄×C₅₃	—	2	2	no	—
102	C₂²⋊C₄×C₅₃	—	2	2	no	—
103	C₄²×C₅₃	—	2	2	no	—
104	C₄²×C₅₃	—	2	2	no	—
105	C₄²×C₅₃	—	2	2	no	—
106	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅₃	#41	1	1	yes	characteristic
109	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
110	#11719917	—	2	4	no	—
111	#11719917	—	8	4	no	—
112	#11719917	—	8	2	no	—
113	C₂×#88322	—	2	2	no	—
114	C₂×#88322	—	2	4	no	—
115	C₂×#88322	C₂×D₄	1	2	yes	—
116	C₂×#88322	C₂×D₄	1	2	yes	—
117	C₂³×C₅₃	—	2	2	no	—
118	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
119	Q₈×C₅₃	—	2	2	no	—
120	Q₈×C₅₃	C₂×M₁₆	1	4	yes	—
121	D₄×C₅₃	—	2	2	no	—
122	D₄×C₅₃	—	2	2	no	—
123	D₄×C₅₃	—	4	4	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	2	1	no	—
127	C₂×C₄×C₅₃	—	2	1	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	4	4	no	—
133	C₂×C₄×C₅₃	—	4	2	no	—
134	C₂×C₄×C₅₃	—	4	2	no	—
135	C₂×C₄×C₅₃	—	4	4	no	—
136	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
146	#88322	—	2	4	no	—
147	#88322	—	2	2	no	—
148	#88322	—	2	2	no	—
149	#88322	—	4	4	no	—
150	#18059	—	53	1	no	maximal, Sylow
151	C₂²×C₅₃	—	2	1	no	—
152	C₂²×C₅₃	—	2	1	no	—
153	C₂²×C₅₃	—	2	1	no	—
154	C₂²×C₅₃	—	4	2	no	—
155	C₂²×C₅₃	—	4	2	no	—
156	C₂²×C₅₃	—	4	2	no	—
157	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	2	4	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	2	1	no	—
162	C₄×C₅₃	—	4	2	no	—
163	C₄×C₅₃	—	4	2	no	—
164	C₄×C₅₃	#443	1	2	yes	—
165	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
166	C₄×C₅₃	#568	1	1	yes	characteristic
167	C₄×C₅₃	#576	1	1	yes	characteristic
168	C₄×C₅₃	#578	1	1	yes	characteristic
169	#2991	—	53	2	no	—
170	C₄×#135	D₅₃	1	1	yes	characteristic
171	#1589	—	53	2	no	—
172	#1588	—	53	4	no	—
173	C₈×Q₁₆	—	53	2	no	—
174	#1567	—	53	1	no	—
175	#1489	—	53	2	no	—
176	C₈×M₁₆	—	53	1	no	—
177	C₂×C₅₃	—	4	1	no	—
178	C₂×C₅₃	—	8	2	no	—
179	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
180	C₂×C₅₃	#2951	1	1	yes	characteristic
181	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
182	C₂×#135	Dic₅₃	1	1	yes	characteristic
183	C₄×#41	C₂×D₅₃	1	2	yes	—
184	C₈×Q₈	—	53	4	no	—
185	C₄×Q₁₆	Dic₅₃	1	1	yes	characteristic
186	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
187	C₄×SD₁₆	Dic₅₃	1	2	yes	—
188	C₄×SD₁₆	C₂×D₅₃	1	2	yes	—
189	C₄×D₈	Dic₅₃	1	1	yes	characteristic
190	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
191	#436	—	53	4	no	—
192	#436	—	106	4	no	—
193	#431	—	53	1	no	—
194	#431	—	106	2	no	—
195	#427	Dic₅₃	1	1	yes	characteristic
196	#423	—	53	2	no	—
197	#423	—	106	2	no	—
198	#416	Dic₅₃	1	2	yes	—
199	C₄×M₁₆	—	53	1	no	—
200	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
201	#335	—	53	2	no	—
202	#335	—	53	2	no	—
203	Q₈⋊C₈	—	53	4	no	—
204	D₄⋊C₈	—	53	4	no	—
205	#322	—	53	2	no	—
206	C₈²	—	53	2	no	—
207	C₅₃	#18059	1	1	yes	characteristic, complemented, Sylow
208	C₂×#41	C₂×Dic₅₃	1	2	yes	—
209	#135	—	2	4	no	—
210	C₂×Q₁₆	C₂×Dic₅₃	1	1	yes	characteristic
211	C₂×SD₁₆	C₂×Dic₅₃	1	2	yes	—
212	C₂×D₈	C₂×Dic₅₃	1	1	yes	characteristic
213	C₂×M₁₆	—	106	4	no	—
214	C₂×M₁₆	—	106	1	no	—
215	C₂×M₁₆	—	106	2	no	—
216	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
217	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
218	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
219	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	—	2	2	no	—
222	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
223	C₄×D₄	C₂²×D₅₃	1	2	yes	—
224	#117	—	2	2	no	—
225	#117	C₂×Dic₅₃	1	2	yes	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
228	#107	C₂×Dic₅₃	1	1	yes	characteristic
229	#107	C₂×Dic₅₃	1	1	yes	characteristic
230	#106	C₂×Dic₅₃	1	2	yes	—
231	C₄⋊C₈	—	53	1	no	—
232	C₄⋊C₈	—	53	1	no	—
233	C₄⋊C₈	—	53	4	no	—
234	C₄⋊C₈	—	106	4	no	—
235	C₄⋊C₈	—	106	2	no	—
236	#103	C₂×Dic₅₃	1	2	yes	—
237	#103	C₂×Dic₅₃	1	2	yes	—
238	#102	C₂×Dic₅₃	1	2	yes	—
239	#102	C₂×Dic₅₃	1	2	yes	—
240	C₂²⋊C₈	—	53	2	no	—
241	C₂²⋊C₈	—	106	4	no	—
242	#97	—	106	1	no	—
243	C₄×C₈	—	53	2	no	—
244	C₄×C₈	—	53	2	no	—
245	C₄×C₈	—	106	4	no	—
246	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
247	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
250	#41	—	2	4	no	—
251	#41	—	4	2	no	—
252	C₂×Q₈	C₂²×Dic₅₃	1	2	yes	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
259	C₂²×C₄	#11719947	1	1	yes	characteristic
260	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
261	Q₁₆	#11719917	1	4	yes	complemented
262	SD₁₆	—	2	4	no	—
263	D₈	—	2	2	no	—
264	M₁₆	—	106	4	no	—
265	M₁₆	—	424	4	no	—
266	M₁₆	—	424	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	53	2	no	—
270	C₂×C₈	—	53	2	no	—
271	C₂×C₈	—	106	2	no	—
272	C₂×C₈	—	106	4	no	—
273	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
274	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
275	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
276	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
280	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₂²⋊C₄	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	D₄×D₅₃	1	1	yes	characteristic
288	C₄²	#11719947	1	1	yes	characteristic
289	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
290	C₄²	C₂³×D₅₃	1	1	yes	characteristic
291	C₂³	—	2	2	no	—
292	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
293	Q₈	—	2	2	no	—
294	Q₈	C₂×#11719917	1	4	yes	—
295	D₄	—	2	2	no	—
296	D₄	—	2	2	no	—
297	D₄	—	4	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
311	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
312	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
313	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
314	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
315	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	106	4	no	—
320	C₈	—	106	2	no	—
321	C₈	—	106	2	no	—
322	C₈	—	212	4	no	—
323	C₈	C₂×#11719917	1	2	yes	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	4	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	4	2	no	—
336	C₄	—	4	2	no	—
337	C₄	#16810740	1	2	yes	—
338	C₄	C₂²×#11719917	1	1	yes	characteristic
339	C₄	#16811390	1	1	yes	characteristic
340	C₄	#16811425	1	1	yes	characteristic
341	C₄	#16811437	1	1	yes	characteristic
342	C₂	—	4	1	no	—
343	C₂	—	8	2	no	—
344	C₂	D₄×#11719917	1	1	yes	characteristic
345	C₂	#20283591	1	1	yes	characteristic
346	C₂	C₂×#16810740	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₄×#135×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#135×C₅₃	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₄×#135	D₅₃
hypocenter	C₅₃	#18059
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
1. #20283847	C₂	semidirect
2. #20273600	C₂	semidirect
3. #20273594	C₂	semidirect
4. Q₁₆×#88322	C₂	semidirect
5. #20273474	C₂	semidirect
6. #20273154	C₂	semidirect
7. C₈×#11719917	C₂	semidirect
87. Q₁₆×C₅₃	M₁₆	semidirect
207. C₅₃	#18059	semidirect
261. Q₁₆	#11719917	semidirect

Maximal subgroups

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

subgroup	index
#20283847	2
#20273600	2
#20273594	2
Q₁₆×#88322	2
#20273474	2
#20273154	2
C₈×#11719917	2
C₄×#135×C₅₃	2
#18059	53