Subgroups — #27273665

Order 13568.

818 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
3	#20272556	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	—	2	1	no	—
6	#16810676	—	2	2	no	—
7	#16810676	C₄	1	4	yes	complemented
8	#16810647	—	2	2	no	—
9	#16810647	C₄	1	4	yes	complemented
10	C₂×C₄⋊#88322	—	2	1	no	—
11	C₂×C₄⋊#88322	—	2	1	no	—
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×C₂²⋊#88322	—	2	1	no	—
14	C₂×C₄×#88322	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
16	C₂²×C₄²×C₅₃	—	2	1	no	—
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
19	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
20	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×#95×C₅₃	—	2	1	no	—
23	C₂×#11719917	—	2	4	no	—
24	C₂×#11719917	—	4	2	no	—
25	C₂×#11719917	—	4	2	no	—
26	C₂²×#88322	—	2	2	no	—
27	C₂²×#88322	—	2	2	no	—
28	C₄⋊#88322	—	2	4	no	—
29	C₄⋊#88322	—	2	4	no	—
30	C₄⋊#88322	—	2	2	no	—
31	C₄⋊#88322	C₂×C₄	1	4	yes	complemented
32	C₂²⋊#88322	—	2	4	no	—
33	C₄×#88322	—	2	2	no	—
34	C₄×#88322	C₂×C₄	1	4	yes	complemented
35	C₂³×C₄×C₅₃	—	2	2	no	—
36	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₅₃	—	2	2	no	—
43	C₂×C₄²×C₅₃	—	2	2	no	—
44	C₂×C₄²×C₅₃	—	4	2	no	—
45	C₂×C₄²×C₅₃	—	4	2	no	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	D₄	1	2	yes	—
50	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	#95×C₅₃	—	2	2	no	—
55	#95×C₅₃	—	2	2	no	—
56	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	2	8	no	—
59	C₂²×C₄×C₅₃	—	2	4	no	—
60	C₂²×C₄×C₅₃	—	4	2	no	—
61	C₂²×C₄×C₅₃	—	4	4	no	—
62	C₂²×C₄×C₅₃	—	4	4	no	—
63	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅₃	D₈	1	2	yes	—
73	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
74	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
75	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₄⋊C₄×C₅₃	—	2	8	no	—
80	C₄⋊C₄×C₅₃	—	2	4	no	—
81	C₄⋊C₄×C₅₃	M₁₆	1	8	yes	complemented
82	C₄²×C₅₃	—	4	4	no	—
83	C₄²×C₅₃	—	4	2	no	—
84	C₄²×C₅₃	—	4	2	no	—
85	C₄²×C₅₃	C₄²	1	2	yes	—
86	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
89	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
90	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
91	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
95	#11719917	—	4	8	no	—
96	C₂×#88322	—	2	4	no	—
97	C₂×#88322	—	2	4	no	—
98	C₂×#88322	—	4	4	no	—
99	C₂×#88322	—	4	2	no	—
100	C₂×#88322	—	4	2	no	—
101	C₂³×C₅₃	—	2	2	no	—
102	C₂³×C₅₃	—	2	2	no	—
103	C₂³×C₅₃	#95	1	2	yes	—
104	C₂³×C₅₃	#102	1	2	yes	—
105	C₂³×C₅₃	#103	1	2	yes	—
106	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	4	no	—
111	C₂×C₄×C₅₃	—	2	4	no	—
112	C₂×C₄×C₅₃	—	2	8	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	—	4	16	no	—
116	C₂×C₄×C₅₃	—	4	4	no	—
117	C₂×C₄×C₅₃	—	4	4	no	—
118	C₂×C₄×C₅₃	#95	1	2	yes	—
119	C₂×C₄×C₅₃	#95	1	2	yes	—
120	C₂×C₄×C₅₃	#95	1	2	yes	—
121	C₂×C₄×C₅₃	#102	1	4	yes	—
122	C₂×C₄×C₅₃	#102	1	2	yes	—
123	C₂×C₄×C₅₃	#103	1	4	yes	—
124	C₂×C₄×C₅₃	#103	1	2	yes	—
125	C₂×C₄×C₅₃	#106	1	4	yes	—
126	C₂×C₄×C₅₃	#107	1	4	yes	—
127	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
134	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
138	#88322	—	4	4	no	—
139	#88322	—	4	4	no	—
140	#7901	—	53	1	no	maximal, Sylow
141	C₂²×C₅₃	—	2	2	no	—
142	C₂²×C₅₃	—	2	2	no	—
143	C₂²×C₅₃	—	2	4	no	—
144	C₂²×C₅₃	—	2	4	no	—
145	C₂²×C₅₃	#340	1	4	yes	—
146	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
147	C₂²×C₅₃	#407	1	2	yes	—
148	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
149	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
150	C₂²×C₅₃	#421	1	2	yes	—
151	C₄×C₅₃	—	2	2	no	—
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	—	2	2	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	4	8	no	—
156	C₄×C₅₃	#340	1	4	yes	—
157	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
158	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
159	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
160	C₄×C₅₃	#423	1	2	yes	—
161	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
162	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
163	C₂×#431	—	53	1	no	—
164	C₂×#423	—	53	1	no	—
165	C₂×C₄×C₄⋊C₄	D₅₃	1	1	yes	characteristic
166	#1279	—	53	4	no	—
167	C₂×C₅₃	—	2	4	no	—
168	C₂×C₅₃	#1486	1	2	yes	—
169	C₂×C₅₃	#1487	1	2	yes	—
170	C₂×C₅₃	#1728	1	1	yes	characteristic
171	C₂×C₅₃	#1733	1	1	yes	characteristic
172	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
173	C₂²×M₁₆	—	106	1	no	—
174	C₂²×C₄⋊C₄	Dic₅₃	1	2	yes	—
175	C₂²×C₄²	—	2	1	no	—
176	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
177	#431	—	53	4	no	—
178	#431	—	106	2	no	—
179	#423	—	53	4	no	—
180	#423	—	106	2	no	—
181	C₂×C₄⋊C₈	—	53	2	no	—
182	C₂×C₄⋊C₈	—	106	1	no	—
183	C₂×C₄⋊C₈	—	106	1	no	—
184	C₂×C₂²⋊C₈	—	106	1	no	—
185	C₂×C₄×C₈	—	53	2	no	—
186	C₄×C₄⋊C₄	—	2	1	no	—
187	C₄×C₄⋊C₄	—	2	1	no	—
188	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
189	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
190	C₂×#95	—	2	1	no	—
191	C₅₃	#7901	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₄	—	2	2	no	—
193	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
194	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
195	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
196	C₂×M₁₆	—	106	4	no	—
197	C₂×M₁₆	—	212	2	no	—
198	C₂×M₁₆	—	212	2	no	—
199	C₂²×C₈	—	106	2	no	—
200	C₂²×C₈	—	106	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	8	no	—
203	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	4	2	no	—
207	C₂×C₄²	—	4	2	no	—
208	C₂×C₄²	Dic₁₀₆	1	2	yes	—
209	C₂×C₄²	C₄×D₅₃	1	2	yes	—
210	C₂×C₄²	C₄×D₅₃	1	2	yes	—
211	C₂×C₄²	D₂₁₂	1	2	yes	—
212	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
216	C₄⋊C₈	—	53	4	no	—
217	C₄⋊C₈	—	106	2	no	—
218	C₄⋊C₈	—	106	4	no	—
219	C₄⋊C₈	—	106	4	no	—
220	C₂²⋊C₈	—	106	4	no	—
221	C₄×C₈	—	53	4	no	—
222	C₄×C₈	—	106	2	no	—
223	#95	—	2	2	no	—
224	#95	—	2	2	no	—
225	C₂⁴	C₂²⋊Dic₅₃	1	1	yes	characteristic
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	—	4	4	no	—
232	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
233	C₂²×C₄	#11719919	1	2	yes	—
234	C₂²×C₄	#11719919	1	2	yes	—
235	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
236	C₂²×C₄	#11719921	1	2	yes	—
237	C₂²×C₄	#11719921	1	2	yes	—
238	C₂²×C₄	#11719922	1	2	yes	—
239	C₂²×C₄	#11719923	1	2	yes	—
240	C₂²×C₄	#11719924	1	2	yes	—
241	C₂²×C₄	#11719925	1	2	yes	—
242	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
243	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
244	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
245	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
248	M₁₆	—	212	8	no	—
249	C₂×C₈	—	106	4	no	—
250	C₂×C₈	—	106	4	no	—
251	C₂×C₈	—	212	4	no	—
252	C₂×C₈	—	212	2	no	—
253	C₂×C₈	—	212	2	no	—
254	C₄⋊C₄	—	2	8	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	#11719917	1	8	yes	complemented
257	C₄²	—	4	4	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	C₄×Dic₅₃	1	2	yes	—
261	C₄²	#11719919	1	2	yes	—
262	C₄²	#11719919	1	2	yes	—
263	C₄²	C₄⋊Dic₅₃	1	2	yes	—
264	C₄²	#11719921	1	2	yes	—
265	C₄²	#11719921	1	2	yes	—
266	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
267	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
268	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
269	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	#13133124	1	2	yes	—
273	C₂³	#13133138	1	2	yes	—
274	C₂³	#13133141	1	2	yes	—
275	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	16	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	#13133124	1	2	yes	—
288	C₂×C₄	#13133124	1	2	yes	—
289	C₂×C₄	#13133124	1	2	yes	—
290	C₂×C₄	#13133137	1	4	yes	—
291	C₂×C₄	#13133138	1	2	yes	—
292	C₂×C₄	#13133140	1	4	yes	—
293	C₂×C₄	#13133141	1	2	yes	—
294	C₂×C₄	#13133148	1	4	yes	—
295	C₂×C₄	#13133150	1	4	yes	—
296	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
297	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
298	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
299	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
300	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
301	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
302	C₂×C₄	C₂×#11719917	1	4	yes	—
303	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
304	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
305	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
306	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
307	C₈	—	212	4	no	—
308	C₈	—	212	4	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	2	2	no	—
313	C₂²	#16810385	1	4	yes	—
314	C₂²	C₂×#13133124	1	1	yes	characteristic
315	C₂²	#16810587	1	2	yes	—
316	C₂²	C₂×#13133138	1	1	yes	characteristic
317	C₂²	C₂×#13133141	1	1	yes	characteristic
318	C₂²	#16810640	1	2	yes	—
319	C₄	—	2	2	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	1	no	—
323	C₄	—	4	8	no	—
324	C₄	#16810385	1	4	yes	—
325	C₄	C₄×#11719917	1	2	yes	—
326	C₄	C₂×#13133137	1	1	yes	characteristic
327	C₄	C₂×#13133140	1	1	yes	characteristic
328	C₄	#16810647	1	2	yes	—
329	C₄	C₂×#13133148	1	1	yes	characteristic
330	C₄	C₂×#13133150	1	1	yes	characteristic
331	C₂	—	2	4	no	—
332	C₂	#20273140	1	2	yes	—
333	C₂	#20273144	1	2	yes	—
334	C₂	#20274409	1	1	yes	characteristic
335	C₂	#20274424	1	1	yes	characteristic
336	C₂	C₂×#16810385	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₄×C₄⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₄×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#7901
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #20272556	C₂	semidirect
7. #16810676	C₄	semidirect
9. #16810647	C₄	semidirect
31. C₄⋊#88322	C₂×C₄	semidirect
34. C₄×#88322	C₂×C₄	semidirect
81. C₄⋊C₄×C₅₃	M₁₆	semidirect
191. C₅₃	#7901	semidirect
256. C₄⋊C₄	#11719917	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
C₂×#16810647	2
#20272556	2
C₂×C₄×C₄⋊C₄×C₅₃	2
#7901	53