Subgroups — #27289886

Order 13568.

1062 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810378	C₂	1	4	yes	maximal
4	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₂²	1	2	yes	—
6	C₂³×#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	—	2	2	no	—
8	C₂×C₄⋊#88322	—	2	2	no	—
9	C₂×C₄⋊#88322	C₄	1	4	yes	complemented
10	C₂×C₄×#88322	—	2	2	no	—
11	C₂×C₄×#88322	—	2	2	no	—
12	C₂×C₄×#88322	C₄	1	4	yes	complemented
13	#16810378	—	2	8	no	—
14	C₂²×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
18	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
19	C₂×#95×C₅₃	C₄	1	2	yes	—
20	C₂²×#88322	—	2	2	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₂²×#88322	—	2	4	no	—
23	C₂²×#88322	—	2	2	no	—
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	—	2	2	no	—
26	C₂²×#88322	—	2	2	no	—
27	C₂²×#88322	—	2	4	no	—
28	C₂²×#88322	C₂×C₄	1	4	yes	—
29	C₂²×#88322	C₂×C₄	1	4	yes	—
30	C₂²×#88322	D₄	1	4	yes	—
31	C₂²×#88322	Q₈	1	4	yes	—
32	C₄⋊#88322	—	2	8	no	—
33	C₄×#88322	—	2	8	no	—
34	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
35	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
40	C₂×C₄⋊C₄×C₅₃	C₈	1	8	yes	complemented
41	C₂×C₄²×C₅₃	—	2	4	no	—
42	C₂×C₄²×C₅₃	—	2	2	no	—
43	C₂×C₄²×C₅₃	—	2	2	no	—
44	C₂×C₄²×C₅₃	—	2	4	no	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₅₃	D₄	1	2	yes	—
48	C₂×C₄²×C₅₃	D₄	1	2	yes	—
49	C₂×C₄²×C₅₃	D₄	1	2	yes	—
50	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
51	#95×C₅₃	—	2	4	no	—
52	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	—	2	4	no	—
54	C₂²×C₄×C₅₃	—	2	4	no	—
55	C₂²×C₄×C₅₃	—	2	8	no	—
56	C₂²×C₄×C₅₃	—	2	4	no	—
57	C₂²×C₄×C₅₃	—	2	8	no	—
58	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
59	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
65	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
66	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₅₃	#41	1	2	yes	—
76	C₂²×C₄×C₅₃	#41	1	2	yes	—
77	C₄⋊C₄×C₅₃	—	2	16	no	—
78	C₄²×C₅₃	—	4	4	no	—
79	C₄²×C₅₃	—	4	2	no	—
80	C₄²×C₅₃	—	4	2	no	—
81	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
82	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
83	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
84	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
85	C₂×#88322	—	2	4	no	—
86	C₂×#88322	—	2	4	no	—
87	C₂×#88322	—	2	4	no	—
88	C₂×#88322	—	2	4	no	—
89	C₂×#88322	—	2	4	no	—
90	C₂×#88322	—	2	4	no	—
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	—	2	8	no	—
93	C₂×#88322	—	4	4	no	—
94	C₂×#88322	—	4	2	no	—
95	C₂×#88322	—	4	2	no	—
96	C₂×#88322	C₄⋊C₄	1	8	yes	complemented
97	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
98	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂³×C₅₃	C₄×D₄	1	2	yes	—
101	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
102	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
103	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
104	C₂³×C₅₃	#131	1	4	yes	—
105	C₂×C₄×C₅₃	—	2	4	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	—	2	16	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	—	4	4	no	—
112	C₂×C₄×C₅₃	—	4	8	no	—
113	C₂×C₄×C₅₃	#95	1	4	yes	—
114	C₂×C₄×C₅₃	#95	1	4	yes	—
115	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
116	C₂×C₄×C₅₃	#97	1	4	yes	—
117	C₂×C₄×C₅₃	C₂²⋊C₈	1	8	yes	—
118	C₂×C₄×C₅₃	C₄⋊C₈	1	8	yes	—
119	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#117	1	2	yes	—
125	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₅₃	#121	1	2	yes	—
128	C₂×C₄×C₅₃	#121	1	2	yes	—
129	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
133	C₂×C₄×C₅₃	#125	1	2	yes	—
134	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
137	#88322	—	2	8	no	—
138	#88322	—	4	4	no	—
139	#10834	—	53	1	no	maximal, Sylow
140	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
141	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
142	C₂²×C₅₃	#389	1	2	yes	—
143	C₂²×C₅₃	#390	1	1	yes	characteristic
144	C₂²×C₅₃	#391	1	1	yes	characteristic
145	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
146	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
147	C₂²×C₅₃	#405	1	2	yes	—
148	C₂²×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
149	C₂²×C₅₃	#408	1	2	yes	—
150	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
151	C₂²×C₅₃	#424	1	2	yes	—
152	C₂²×C₅₃	C₈×D₄	1	4	yes	—
153	C₂²×C₅₃	#436	1	4	yes	—
154	C₂²×C₅₃	C₈×Q₈	1	4	yes	—
155	C₂²×C₅₃	#446	1	4	yes	—
156	C₄×C₅₃	—	2	4	no	—
157	C₄×C₅₃	—	4	8	no	—
158	C₄×C₅₃	#336	1	8	yes	complemented
159	C₄×C₅₃	#380	1	2	yes	—
160	C₄×C₅₃	#382	1	1	yes	characteristic
161	C₄×C₅₃	#384	1	1	yes	characteristic
162	C₄×C₅₃	#389	1	2	yes	—
163	C₄×C₅₃	#390	1	1	yes	characteristic
164	C₄×C₅₃	#391	1	1	yes	characteristic
165	C₂²×C₄⋊C₈	—	53	1	no	—
166	C₂²×C₄×C₈	—	53	1	no	—
167	C₂×C₄×C₄⋊C₄	D₅₃	1	1	yes	characteristic
168	C₂×#336	—	53	4	no	—
169	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
170	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
171	C₂×C₅₃	#1729	1	1	yes	characteristic
172	C₂×C₅₃	C₈×C₄⋊C₄	1	2	yes	—
173	C₂×C₅₃	#1770	1	2	yes	—
174	C₂×C₅₃	#1939	1	4	yes	—
175	C₂×C₅₃	#1954	1	2	yes	—
176	C₂×C₅₃	#1984	1	2	yes	—
177	C₂³×C₈	—	53	2	no	—
178	C₂³×C₈	—	53	2	no	—
179	C₂²×C₄⋊C₄	Dic₅₃	1	2	yes	—
180	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
181	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
182	C₂×C₄⋊C₈	—	53	4	no	—
183	C₂×C₄⋊C₈	—	106	2	no	—
184	C₂×C₄⋊C₈	—	106	2	no	—
185	C₂×C₄×C₈	—	53	4	no	—
186	C₂×C₄×C₈	—	106	2	no	—
187	C₂×C₄×C₈	—	106	2	no	—
188	C₄×C₄⋊C₄	—	2	2	no	—
189	C₄×C₄⋊C₄	—	2	2	no	—
190	C₂×#95	Dic₅₃	1	2	yes	—
191	#336	—	106	8	no	—
192	C₅₃	#10834	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₄	C₂×Dic₅₃	1	4	yes	—
194	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
195	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
196	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
197	C₂²×C₈	—	53	4	no	—
198	C₂²×C₈	—	53	4	no	—
199	C₂²×C₈	—	53	4	no	—
200	C₂²×C₈	—	53	4	no	—
201	C₂²×C₈	—	106	2	no	—
202	C₂²×C₈	—	106	2	no	—
203	C₂²×C₈	—	106	2	no	—
204	C₂²×C₈	—	106	2	no	—
205	C₂²×C₈	—	106	2	no	—
206	C₂²×C₈	—	106	4	no	—
207	C₂²×C₈	—	106	4	no	—
208	C₂²×C₈	—	106	2	no	—
209	C₂×C₄⋊C₄	—	2	4	no	—
210	C₂×C₄⋊C₄	—	2	4	no	—
211	C₂×C₄⋊C₄	#88322	1	8	yes	complemented
212	C₂×C₄²	—	2	4	no	—
213	C₂×C₄²	—	2	4	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	Dic₁₀₆	1	2	yes	—
217	C₂×C₄²	C₄×D₅₃	1	2	yes	—
218	C₂×C₄²	C₄×D₅₃	1	2	yes	—
219	C₂×C₄²	D₂₁₂	1	2	yes	—
220	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
221	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
222	C₄⋊C₈	—	106	8	no	—
223	C₄×C₈	—	106	8	no	—
224	#95	—	2	4	no	—
225	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	8	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	8	no	—
231	C₂²×C₄	C₂×#88322	1	8	yes	—
232	C₂²×C₄	C₂×#88322	1	4	yes	—
233	C₂²×C₄	#11719917	1	4	yes	—
234	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
235	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
236	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
237	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
238	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
239	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
240	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
244	C₂²×C₄	D₄×D₅₃	1	2	yes	—
245	C₂²×C₄	#11719947	1	2	yes	—
246	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
247	C₂²×C₄	#11719949	1	2	yes	—
248	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
250	C₂×C₈	—	53	8	no	—
251	C₂×C₈	—	106	4	no	—
252	C₂×C₈	—	106	4	no	—
253	C₂×C₈	—	106	4	no	—
254	C₂×C₈	—	106	4	no	—
255	C₂×C₈	—	106	4	no	—
256	C₂×C₈	—	106	4	no	—
257	C₂×C₈	—	106	4	no	—
258	C₂×C₈	—	106	8	no	—
259	C₂×C₈	—	212	4	no	—
260	C₂×C₈	—	212	2	no	—
261	C₂×C₈	—	212	2	no	—
262	C₄⋊C₄	—	2	16	no	—
263	C₄²	—	4	4	no	—
264	C₄²	—	4	2	no	—
265	C₄²	—	4	2	no	—
266	C₄²	#11719919	1	4	yes	—
267	C₄²	#11719919	1	4	yes	—
268	C₄²	#11719921	1	4	yes	—
269	C₄²	#11719921	1	4	yes	—
270	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
271	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
272	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
273	C₂³	D₄×Dic₅₃	1	2	yes	—
274	C₂³	Q₈×Dic₅₃	1	2	yes	—
275	C₂³	C₂²×#88322	1	2	yes	—
276	C₂³	C₂×#11719917	1	2	yes	—
277	C₂³	#13133224	1	4	yes	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	16	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	8	no	—
286	C₂×C₄	#13133124	1	4	yes	—
287	C₂×C₄	#13133124	1	4	yes	—
288	C₂×C₄	C₄×#88322	1	4	yes	—
289	C₂×C₄	#13133128	1	4	yes	—
290	C₂×C₄	C₂²⋊#88322	1	8	yes	—
291	C₂×C₄	C₄⋊#88322	1	8	yes	—
292	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
293	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
294	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
295	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
296	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
297	C₂×C₄	#13133175	1	2	yes	—
298	C₂×C₄	#13133179	1	2	yes	—
299	C₂×C₄	#13133182	1	2	yes	—
300	C₂×C₄	#13133188	1	2	yes	—
301	C₂×C₄	#13133192	1	2	yes	—
302	C₂×C₄	#13133193	1	2	yes	—
303	C₂×C₄	#13133197	1	2	yes	—
304	C₂×C₄	#13133204	1	1	yes	characteristic
305	C₂×C₄	#13133206	1	1	yes	characteristic
306	C₂×C₄	#13133208	1	2	yes	—
307	C₂×C₄	#13133213	1	1	yes	characteristic
308	C₂×C₄	#13133215	1	1	yes	characteristic
309	C₂×C₄	#13133216	1	2	yes	—
310	C₈	—	106	8	no	—
311	C₈	—	212	4	no	—
312	C₂²	C₂×#13133124	1	1	yes	characteristic
313	C₂²	C₄⋊C₄×Dic₅₃	1	2	yes	—
314	C₂²	#16810522	1	2	yes	—
315	C₂²	#16810526	1	1	yes	characteristic
316	C₂²	#16810530	1	1	yes	characteristic
317	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
318	C₂²	C₂×#13133128	1	1	yes	characteristic
319	C₂²	#16810579	1	2	yes	—
320	C₂²	C₂×C₂²⋊#88322	1	2	yes	—
321	C₂²	#16810590	1	2	yes	—
322	C₂²	C₂×C₄⋊#88322	1	2	yes	—
323	C₂²	#16810650	1	2	yes	—
324	C₂²	D₄×#88322	1	4	yes	—
325	C₂²	#16810701	1	4	yes	—
326	C₂²	Q₈×#88322	1	4	yes	—
327	C₂²	#16810754	1	4	yes	—
328	C₄	—	2	4	no	—
329	C₄	—	4	8	no	—
330	C₄	#16810378	1	8	yes	complemented
331	C₄	#16810480	1	2	yes	—
332	C₄	#16810488	1	1	yes	characteristic
333	C₄	#16810492	1	1	yes	characteristic
334	C₄	#16810519	1	2	yes	—
335	C₄	#16810524	1	1	yes	characteristic
336	C₄	#16810528	1	1	yes	characteristic
337	C₂	#20272944	1	1	yes	characteristic
338	C₂	C₂×#16810378	1	1	yes	characteristic
339	C₂	#20274412	1	1	yes	characteristic
340	C₂	C₄⋊C₄×#88322	1	2	yes	—
341	C₂	#20274584	1	2	yes	—
342	C₂	#20275434	1	4	yes	—
343	C₂	#20275510	1	2	yes	—
344	C₂	#20275652	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₄×C₄⋊C₄×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×C₄×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#10834
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. C₂×C₄⋊#88322	C₄	semidirect
12. C₂×C₄×#88322	C₄	semidirect
40. C₂×C₄⋊C₄×C₅₃	C₈	semidirect
96. C₂×#88322	C₄⋊C₄	semidirect
158. C₄×C₅₃	#336	semidirect
192. C₅₃	#10834	semidirect
211. C₂×C₄⋊C₄	#88322	semidirect
330. C₄	#16810378	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#88322	2
C₂²×C₄×#88322	2
C₂×#16810378	2
C₂×C₄×C₄⋊C₄×C₅₃	2
#10834	53