Subgroups — #27273629

Order 13568.

802 conjugacy classes of subgroups.

The subgroup lattice has 322 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×#16810676	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	#16810676	—	2	2	no	—
6	#16810676	—	2	2	no	—
7	#16810676	C₄	1	4	yes	complemented
8	#16810676	C₄	1	4	yes	complemented
9	C₂×C₄⋊#88322	—	2	1	no	—
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	C₂×C₂²⋊#88322	—	2	1	no	—
12	C₂×C₂²⋊#88322	—	2	1	no	—
13	C₂×C₄×#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	C₂²×#88322	—	2	2	no	—
24	C₂²×#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	4	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	C₂×C₄	1	4	yes	complemented
28	C₂²⋊#88322	—	2	4	no	—
29	C₂²⋊#88322	—	2	4	no	—
30	C₄×#88322	—	2	4	no	—
31	C₄×#88322	—	2	2	no	—
32	C₄×#88322	C₂×C₄	1	4	yes	complemented
33	C₂³×C₄×C₅₃	—	2	2	no	—
34	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₅₃	C₈	1	8	yes	complemented
40	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₅₃	—	2	2	no	—
42	C₂×C₄²×C₅₃	—	2	2	no	—
43	C₂×C₄²×C₅₃	—	4	4	no	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	2	yes	—
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	#95×C₅₃	—	2	2	no	—
53	#95×C₅₃	—	2	2	no	—
54	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	2	8	no	—
58	C₂²×C₄×C₅₃	—	2	4	no	—
59	C₂²×C₄×C₅₃	—	4	2	no	—
60	C₂²×C₄×C₅₃	—	4	4	no	—
61	C₂²×C₄×C₅₃	—	4	4	no	—
62	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₄⋊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	C₂×C₈	1	4	yes	—
72	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
73	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₅₃	—	2	8	no	—
77	C₄⋊C₄×C₅₃	—	2	4	no	—
78	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
79	C₄²×C₅₃	—	4	4	no	—
80	C₄²×C₅₃	—	4	4	no	—
81	C₄²×C₅₃	C₄²	1	2	yes	—
82	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
85	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
86	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
87	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	—	2	4	no	—
93	C₂×#88322	—	4	4	no	—
94	C₂×#88322	—	4	2	no	—
95	C₂×#88322	—	4	2	no	—
96	C₂³×C₅₃	—	2	1	no	—
97	C₂³×C₅₃	—	2	1	no	—
98	C₂³×C₅₃	—	2	2	no	—
99	C₂³×C₅₃	#95	1	2	yes	—
100	C₂³×C₅₃	C₂²⋊C₈	1	4	yes	—
101	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	—	2	8	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	—	4	16	no	—
112	C₂×C₄×C₅₃	—	4	4	no	—
113	C₂×C₄×C₅₃	—	4	4	no	—
114	C₂×C₄×C₅₃	#95	1	2	yes	—
115	C₂×C₄×C₅₃	#95	1	2	yes	—
116	C₂×C₄×C₅₃	#95	1	2	yes	—
117	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
118	C₂×C₄×C₅₃	#97	1	4	yes	—
119	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
120	C₂×C₄×C₅₃	C₄⋊C₈	1	8	yes	—
121	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
130	C₂×C₄×C₅₃	#137	1	2	yes	—
131	#88322	—	4	4	no	—
132	#88322	—	4	4	no	—
133	#7892	—	53	1	no	maximal, Sylow
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	—	2	4	no	—
136	C₂²×C₅₃	—	2	4	no	—
137	C₂²×C₅₃	—	2	2	no	—
138	C₂²×C₅₃	#336	1	4	yes	—
139	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
140	C₂²×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
141	C₂²×C₅₃	#418	1	1	yes	characteristic
142	C₂²×C₅₃	#419	1	1	yes	characteristic
143	C₂²×C₅₃	#421	1	2	yes	—
144	C₄×C₅₃	—	2	1	no	—
145	C₄×C₅₃	—	2	2	no	—
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	—	4	8	no	—
149	C₄×C₅₃	#336	1	4	yes	—
150	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
151	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
152	C₄×C₅₃	#417	1	2	yes	—
153	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
154	C₄×C₅₃	#428	1	2	yes	—
155	C₂×#431	—	53	1	no	—
156	C₂×#431	—	53	1	no	—
157	C₂×C₄×C₄⋊C₄	D₅₃	1	1	yes	characteristic
158	#1279	—	53	4	no	—
159	C₂×C₅₃	—	2	4	no	—
160	C₂×C₅₃	#1478	1	2	yes	—
161	C₂×C₅₃	#1479	1	2	yes	—
162	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
163	C₂×C₅₃	#1733	1	1	yes	characteristic
164	C₂×C₅₃	#1736	1	1	yes	characteristic
165	C₂²×C₄⋊C₄	Dic₅₃	1	2	yes	—
166	C₂²×C₄²	—	2	1	no	—
167	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
168	#431	—	53	4	no	—
169	#431	—	53	4	no	—
170	#431	—	106	2	no	—
171	#431	—	106	2	no	—
172	C₂×C₄⋊C₈	—	53	2	no	—
173	C₂×C₄⋊C₈	—	106	1	no	—
174	C₂×C₂²⋊C₈	—	106	1	no	—
175	C₂×C₂²⋊C₈	—	106	1	no	—
176	C₂×C₄×C₈	—	53	2	no	—
177	C₂×C₄×C₈	—	106	1	no	—
178	C₄×C₄⋊C₄	—	2	1	no	—
179	C₄×C₄⋊C₄	—	2	1	no	—
180	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
181	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
182	C₂×#95	—	2	1	no	—
183	C₅₃	#7892	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	—	2	2	no	—
185	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
186	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
187	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
188	C₂²×C₈	—	106	2	no	—
189	C₂²×C₈	—	106	2	no	—
190	C₂×C₄⋊C₄	—	2	4	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	#88322	1	8	yes	complemented
193	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	4	4	no	—
197	C₂×C₄²	Dic₁₀₆	1	2	yes	—
198	C₂×C₄²	C₄×D₅₃	1	2	yes	—
199	C₂×C₄²	C₄×D₅₃	1	2	yes	—
200	C₂×C₄²	D₂₁₂	1	2	yes	—
201	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
202	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
204	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
205	C₄⋊C₈	—	53	4	no	—
206	C₄⋊C₈	—	106	2	no	—
207	C₄⋊C₈	—	106	4	no	—
208	C₂²⋊C₈	—	106	4	no	—
209	C₂²⋊C₈	—	106	4	no	—
210	C₄×C₈	—	53	4	no	—
211	C₄×C₈	—	106	2	no	—
212	C₄×C₈	—	106	4	no	—
213	#95	—	2	2	no	—
214	#95	—	2	2	no	—
215	C₂⁴	C₂²⋊Dic₅₃	1	1	yes	characteristic
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	8	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	C₂×#88322	1	4	yes	—
224	C₂²×C₄	#11719917	1	4	yes	—
225	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
226	C₂²×C₄	#11719919	1	2	yes	—
227	C₂²×C₄	#11719919	1	2	yes	—
228	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
229	C₂²×C₄	#11719921	1	2	yes	—
230	C₂²×C₄	#11719921	1	2	yes	—
231	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
233	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
234	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
237	C₂×C₈	—	106	4	no	—
238	C₂×C₈	—	106	4	no	—
239	C₂×C₈	—	212	4	no	—
240	C₂×C₈	—	212	2	no	—
241	C₂×C₈	—	212	2	no	—
242	C₄⋊C₄	—	2	8	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	C₂×#88322	1	8	yes	—
245	C₄²	—	4	4	no	—
246	C₄²	—	4	4	no	—
247	C₄²	C₄×Dic₅₃	1	2	yes	—
248	C₄²	#11719919	1	2	yes	—
249	C₄²	#11719919	1	2	yes	—
250	C₄²	C₄⋊Dic₅₃	1	2	yes	—
251	C₄²	#11719921	1	2	yes	—
252	C₄²	#11719921	1	2	yes	—
253	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
254	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
255	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
256	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
257	C₂³	—	2	1	no	—
258	C₂³	—	2	1	no	—
259	C₂³	—	2	2	no	—
260	C₂³	#13133124	1	2	yes	—
261	C₂³	C₂²⋊#88322	1	4	yes	—
262	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	16	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	#13133124	1	2	yes	—
276	C₂×C₄	#13133124	1	2	yes	—
277	C₂×C₄	#13133124	1	2	yes	—
278	C₂×C₄	C₄×#88322	1	4	yes	—
279	C₂×C₄	#13133128	1	4	yes	—
280	C₂×C₄	C₂²⋊#88322	1	4	yes	—
281	C₂×C₄	C₄⋊#88322	1	8	yes	—
282	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
283	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
284	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
285	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
286	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
288	C₂×C₄	C₂²×#88322	1	2	yes	—
289	C₂×C₄	C₂×#11719917	1	2	yes	—
290	C₂×C₄	#13133243	1	2	yes	—
291	C₂×C₄	#13133248	1	2	yes	—
292	C₈	—	212	4	no	—
293	C₈	—	212	4	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	#16810378	1	4	yes	—
299	C₂²	C₂×#13133124	1	1	yes	characteristic
300	C₂²	C₂×C₂²⋊#88322	1	2	yes	—
301	C₂²	#16810628	1	1	yes	characteristic
302	C₂²	#16810632	1	1	yes	characteristic
303	C₂²	#16810640	1	2	yes	—
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	1	no	—
308	C₄	—	4	8	no	—
309	C₄	#16810378	1	4	yes	—
310	C₄	C₂×C₄×#88322	1	1	yes	characteristic
311	C₄	C₂×#13133128	1	1	yes	characteristic
312	C₄	#16810622	1	2	yes	—
313	C₄	C₂×C₄⋊#88322	1	2	yes	—
314	C₄	#16810663	1	2	yes	—
315	C₂	—	2	4	no	—
316	C₂	#20273106	1	2	yes	—
317	C₂	#20273110	1	2	yes	—
318	C₂	C₂×#16810378	1	1	yes	characteristic
319	C₂	#20274424	1	1	yes	characteristic
320	C₂	#20274434	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
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	#7892
LCS	G ▷ 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
8. #16810676	C₄	semidirect
27. C₄⋊#88322	C₂×C₄	semidirect
32. C₄×#88322	C₂×C₄	semidirect
39. C₂×C₄⋊C₄×C₅₃	C₈	semidirect
183. C₅₃	#7892	semidirect
192. C₂×C₄⋊C₄	#88322	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
#20272556	2
C₂×C₄×C₄⋊C₄×C₅₃	2
#7892	53