Subgroups — #27273612

Order 13568.

636 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	#20272552	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₄	1	2	yes	—
6	#16810680	—	2	1	no	—
7	#16810680	C₄	1	2	yes	—
8	#16810676	—	2	1	no	—
9	#16810676	C₂²	1	2	yes	—
10	#16810647	C₄	1	2	yes	—
11	#16810587	—	2	1	no	—
12	C₄×#11719917	—	2	1	no	—
13	C₄×#11719917	—	2	1	no	—
14	C₄×#11719917	—	2	1	no	—
15	C₄×#11719917	C₄	1	2	yes	—
16	C₄×#11719917	C₂²	1	2	yes	—
17	C₂×#13133128	—	2	1	no	—
18	C₂×C₄×#88322	—	2	1	no	—
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#431	—	2	1	no	—
21	C₅₃×#431	—	2	1	no	—
22	C₅₃×#431	C₄	1	2	yes	—
23	C₅₃×#431	C₄	1	2	yes	—
24	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
25	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
27	C₂×#11719917	—	2	8	no	—
28	C₂×#11719917	—	2	2	no	—
29	C₂×#11719917	—	2	2	no	—
30	C₂×#11719917	C₂×C₄	1	4	yes	—
31	C₂²×#88322	—	2	2	no	—
32	C₄⋊#88322	—	2	2	no	—
33	C₄⋊#88322	C₂×C₄	1	4	yes	—
34	C₂²⋊#88322	—	2	2	no	—
35	C₂²⋊#88322	—	2	2	no	—
36	#13133128	—	2	2	no	—
37	#13133128	—	2	2	no	—
38	#13133128	—	2	2	no	—
39	C₄×#88322	—	2	2	no	—
40	C₄×#88322	—	2	2	no	—
41	C₄×#88322	—	2	2	no	—
42	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₅₃	—	2	2	no	—
59	C₄⋊C₈×C₅₃	—	2	2	no	—
60	C₂²⋊C₈×C₅₃	—	2	2	no	—
61	C₂²⋊C₈×C₅₃	—	2	2	no	—
62	C₄×C₈×C₅₃	—	2	2	no	—
63	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
64	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	—	2	4	no	—
66	C₂²×C₄×C₅₃	—	2	4	no	—
67	C₂²×C₄×C₅₃	—	4	2	no	—
68	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₅₃	—	2	4	no	—
87	C₂×C₈×C₅₃	—	4	4	no	—
88	C₄²×C₅₃	C₄²	1	1	yes	characteristic
89	C₄²×C₅₃	C₄²	1	1	yes	characteristic
90	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
104	#11719917	—	2	8	no	—
105	#11719917	—	2	4	no	—
106	#11719917	—	4	4	no	—
107	#11719917	M₁₆	1	8	yes	complemented
108	C₂×#88322	—	2	8	no	—
109	C₂×#88322	—	2	4	no	—
110	C₂×#88322	—	2	4	no	—
111	C₂³×C₅₃	—	2	2	no	—
112	C₂³×C₅₃	—	2	2	no	—
113	C₂³×C₅₃	—	4	1	no	—
114	C₂³×C₅₃	#95	1	1	yes	characteristic
115	C₂³×C₅₃	#95	1	1	yes	characteristic
116	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
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	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
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	2	no	—
130	C₂×C₄×C₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	—	4	1	no	—
132	C₂×C₄×C₅₃	—	4	1	no	—
133	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
135	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
136	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
137	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
138	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
139	C₂×C₄×C₅₃	#108	1	4	yes	—
140	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
147	C₈×C₅₃	—	4	4	no	—
148	#88322	—	2	8	no	—
149	#88322	—	4	4	no	—
150	#7888	—	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	—	2	1	no	—
155	C₂²×C₅₃	—	2	1	no	—
156	C₂²×C₅₃	—	2	1	no	—
157	C₂²×C₅₃	—	2	1	no	—
158	C₂²×C₅₃	—	4	4	no	—
159	C₂²×C₅₃	#341	1	2	yes	—
160	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
161	C₂²×C₅₃	#423	1	2	yes	—
162	C₄×C₅₃	—	2	1	no	—
163	C₄×C₅₃	—	2	1	no	—
164	C₄×C₅₃	—	2	1	no	—
165	C₄×C₅₃	—	2	1	no	—
166	C₄×C₅₃	—	2	1	no	—
167	C₄×C₅₃	—	2	1	no	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	#341	1	2	yes	—
170	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
171	C₄×C₅₃	#407	1	2	yes	—
172	C₄×C₅₃	#421	1	2	yes	—
173	C₄×C₅₃	C₂×#108	1	2	yes	—
174	#2920	—	53	1	no	—
175	C₂×#431	D₅₃	1	1	yes	characteristic
176	C₂×C₄×M₁₆	—	53	1	no	—
177	#1278	—	53	4	no	—
178	C₂×C₅₃	—	2	2	no	—
179	C₂×C₅₃	—	2	2	no	—
180	C₂×C₅₃	—	4	1	no	—
181	C₂×C₅₃	#1728	1	1	yes	characteristic
182	C₂×C₅₃	#1733	1	1	yes	characteristic
183	C₂×C₅₃	C₂×#341	1	1	yes	characteristic
184	C₂²×M₁₆	—	53	2	no	—
185	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
186	#432	—	53	2	no	—
187	#432	—	106	1	no	—
188	#431	—	2	1	no	—
189	#431	—	2	1	no	—
190	#431	—	53	2	no	—
191	#431	—	106	1	no	—
192	#431	Dic₅₃	1	2	yes	—
193	#431	Dic₅₃	1	2	yes	—
194	#423	—	53	2	no	—
195	C₂×C₄⋊C₈	—	2	1	no	—
196	#407	—	106	1	no	—
197	C₂×C₂²⋊C₈	—	2	1	no	—
198	C₄×M₁₆	—	53	2	no	—
199	C₄×M₁₆	—	53	2	no	—
200	C₄×M₁₆	—	106	1	no	—
201	C₄×M₁₆	—	106	1	no	—
202	C₄×M₁₆	—	106	1	no	—
203	C₂×#97	—	106	1	no	—
204	C₂×C₄×C₈	—	106	1	no	—
205	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
206	C₅₃	#7888	1	1	yes	characteristic, complemented, Sylow
207	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
208	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
209	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
210	C₂×M₁₆	—	53	4	no	—
211	C₂×M₁₆	—	106	8	no	—
212	C₂×M₁₆	—	106	2	no	—
213	C₂×M₁₆	—	106	2	no	—
214	C₂²×C₈	—	2	2	no	—
215	C₂²×C₈	—	106	2	no	—
216	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
217	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
218	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
219	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
220	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
221	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
222	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
223	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
225	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
226	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
227	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	53	4	no	—
231	C₄⋊C₈	—	106	2	no	—
232	C₂²⋊C₈	—	2	2	no	—
233	C₂²⋊C₈	—	2	2	no	—
234	C₂²⋊C₈	—	106	2	no	—
235	C₂²⋊C₈	—	106	2	no	—
236	#97	—	106	2	no	—
237	#97	—	106	2	no	—
238	#97	—	106	2	no	—
239	C₄×C₈	—	2	2	no	—
240	C₄×C₈	—	106	2	no	—
241	C₄×C₈	—	106	2	no	—
242	C₄×C₈	—	106	2	no	—
243	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
244	C₂⁴	#11719919	1	1	yes	characteristic
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
249	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
250	C₂²×C₄	#11719919	1	1	yes	characteristic
251	C₂²×C₄	#11719919	1	1	yes	characteristic
252	C₂²×C₄	#11719919	1	1	yes	characteristic
253	C₂²×C₄	#11719919	1	1	yes	characteristic
254	C₂²×C₄	#11719919	1	1	yes	characteristic
255	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
256	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
257	C₂²×C₄	#11719921	1	1	yes	characteristic
258	C₂²×C₄	#11719921	1	1	yes	characteristic
259	C₂²×C₄	#11719921	1	1	yes	characteristic
260	C₂²×C₄	#11719921	1	1	yes	characteristic
261	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
262	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
263	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
266	M₁₆	—	53	8	no	—
267	M₁₆	—	106	8	no	—
268	M₁₆	—	106	4	no	—
269	M₁₆	—	212	4	no	—
270	C₂×C₈	—	2	4	no	—
271	C₂×C₈	—	4	4	no	—
272	C₂×C₈	—	106	8	no	—
273	C₂×C₈	—	106	4	no	—
274	C₂×C₈	—	106	4	no	—
275	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
276	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
277	C₄²	#11719919	1	1	yes	characteristic
278	C₄²	#11719919	1	1	yes	characteristic
279	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
280	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
281	C₄²	#11719921	1	1	yes	characteristic
282	C₄²	#11719921	1	1	yes	characteristic
283	C₄²	#11719921	1	1	yes	characteristic
284	C₄²	#11719921	1	1	yes	characteristic
285	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
286	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
287	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
288	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
289	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
290	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	—	4	1	no	—
294	C₂³	#13133124	1	1	yes	characteristic
295	C₂³	#13133124	1	1	yes	characteristic
296	C₂³	C₂×#11719919	1	1	yes	characteristic
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	#13133124	1	1	yes	characteristic
314	C₂×C₄	#13133124	1	1	yes	characteristic
315	C₂×C₄	#13133124	1	1	yes	characteristic
316	C₂×C₄	#13133124	1	1	yes	characteristic
317	C₂×C₄	#13133124	1	1	yes	characteristic
318	C₂×C₄	#13133124	1	1	yes	characteristic
319	C₂×C₄	#13133153	1	4	yes	—
320	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
321	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
322	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
323	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
324	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
325	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
326	C₂×C₄	M₁₆×D₅₃	1	4	yes	—
327	C₈	—	4	4	no	—
328	C₈	—	106	8	no	—
329	C₈	—	212	4	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	4	4	no	—
338	C₂²	#16810388	1	2	yes	—
339	C₂²	C₂×#13133124	1	1	yes	characteristic
340	C₂²	#16810646	1	2	yes	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	#16810388	1	2	yes	—
349	C₄	M₁₆×Dic₅₃	1	2	yes	—
350	C₄	#16810586	1	2	yes	—
351	C₄	#16810641	1	2	yes	—
352	C₄	C₂×#13133153	1	2	yes	—
353	C₂	—	2	2	no	—
354	C₂	—	2	2	no	—
355	C₂	—	4	1	no	—
356	C₂	#20274408	1	1	yes	characteristic
357	C₂	#20274425	1	1	yes	characteristic
358	C₂	C₂×#16810388	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#7888
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20272552	C₂	semidirect
107. #11719917	M₁₆	semidirect
206. C₅₃	#7888	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
C₂×C₄×#11719917	2
#20272552	2
C₂×C₅₃×#431	2
#7888	53