Subgroups — #27289962

Order 13568.

802 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
2	#20277122	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810394	C₂	1	4	yes	maximal
4	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₂²	1	2	yes	—
6	C₂×#13133176	C₂²	1	2	yes	—
7	C₄×#11719917	—	2	1	no	—
8	C₄×#11719917	—	2	1	no	—
9	C₄×#11719917	C₄	1	2	yes	complemented
10	C₄×#11719917	C₄	1	2	yes	complemented
11	C₂×#13133128	—	2	1	no	—
12	C₂×C₄×#88322	—	2	1	no	—
13	#16810493	—	2	1	no	—
14	#16810489	—	2	1	no	—
15	#16810483	C₄	1	4	yes	—
16	C₄×C₄⋊Dic₅₃	—	2	2	no	—
17	#16810394	—	2	8	no	—
18	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#423	—	2	2	no	—
21	C₅₃×#423	—	2	1	no	—
22	C₅₃×#423	—	2	1	no	—
23	C₂×C₄⋊C₈×C₅₃	C₄	1	4	yes	complemented
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	—	2	2	no	—
27	C₂×#11719917	C₂×C₄	1	4	yes	—
28	C₂×#11719917	D₄	1	4	yes	—
29	C₂×#11719917	Q₈	1	4	yes	—
30	C₂²×#88322	—	2	2	no	—
31	#13133176	—	2	4	no	—
32	#13133176	—	2	4	no	—
33	C₂×C₄⋊Dic₅₃	—	2	4	no	—
34	C₂×C₄⋊Dic₅₃	—	2	2	no	—
35	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₅₃	—	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	#13133124	—	2	4	no	—
42	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₅₃	—	2	4	no	—
45	C₂×M₁₆×C₅₃	—	2	2	no	—
46	C₂×M₁₆×C₅₃	—	2	2	no	—
47	C₂×M₁₆×C₅₃	—	2	4	no	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	D₄	1	2	yes	—
55	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
58	C₄⋊C₈×C₅₃	—	2	4	no	—
59	C₄⋊C₈×C₅₃	—	2	4	no	—
60	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	—	4	4	no	—
62	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
63	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂²×C₄	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	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₅₃	#41	1	2	yes	—
77	C₂²×C₄×C₅₃	#41	1	2	yes	—
78	C₂²×Dic₅₃	—	2	4	no	—
79	M₁₆×C₅₃	—	4	4	no	—
80	M₁₆×C₅₃	—	4	2	no	—
81	M₁₆×C₅₃	—	4	2	no	—
82	C₂×C₈×C₅₃	—	2	4	no	—
83	C₂×C₈×C₅₃	—	2	8	no	—
84	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
91	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
92	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
93	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
94	C₂²⋊Dic₅₃	—	2	8	no	—
95	C₄⋊Dic₅₃	—	4	8	no	—
96	C₄⋊Dic₅₃	—	4	4	no	—
97	C₄×Dic₅₃	—	4	2	no	—
98	C₄×Dic₅₃	—	4	2	no	—
99	#11719917	—	2	4	no	—
100	#11719917	—	2	4	no	—
101	#11719917	—	2	4	no	—
102	#11719917	C₄⋊C₄	1	8	yes	complemented
103	C₂×#88322	—	2	4	no	—
104	C₂×#88322	—	2	4	no	—
105	C₂×#88322	—	2	4	no	—
106	C₂³×C₅₃	—	4	2	no	—
107	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
108	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂³×C₅₃	C₄×D₄	1	2	yes	—
111	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₅₃	—	2	4	no	—
113	C₂×C₄×C₅₃	—	2	4	no	—
114	C₂×C₄×C₅₃	—	2	4	no	—
115	C₂×C₄×C₅₃	—	4	2	no	—
116	C₂×C₄×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	—	4	2	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	#95	1	2	yes	—
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	2	yes	—
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	#117	1	2	yes	—
128	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₅₃	#121	1	2	yes	—
131	C₂×C₄×C₅₃	#121	1	2	yes	—
132	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
135	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
136	C₂×C₄×C₅₃	#125	1	2	yes	—
137	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
140	C₂×Dic₅₃	—	4	8	no	—
141	C₂×Dic₅₃	—	4	4	no	—
142	C₈×C₅₃	—	4	4	no	—
143	#88322	—	2	8	no	—
144	#10855	—	53	1	no	maximal, Sylow
145	C₂²×C₅₃	—	2	2	no	—
146	C₂²×C₅₃	—	2	2	no	—
147	C₂²×C₅₃	—	2	2	no	—
148	C₂²×C₅₃	—	4	2	no	—
149	C₂²×C₅₃	—	4	2	no	—
150	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
151	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
152	C₂²×C₅₃	#389	1	2	yes	—
153	C₂²×C₅₃	#390	1	1	yes	characteristic
154	C₂²×C₅₃	#391	1	1	yes	characteristic
155	C₄×C₅₃	—	2	2	no	—
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	—	2	2	no	—
158	C₄×C₅₃	#344	1	4	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	Dic₅₃	—	8	4	no	—
166	C₂×#423	D₅₃	1	1	yes	characteristic
167	C₂×C₄×M₁₆	—	53	1	no	—
168	#2285	—	53	1	no	—
169	C₂×#344	—	53	4	no	—
170	C₂×C₅₃	—	2	1	no	—
171	C₂×C₅₃	—	2	2	no	—
172	C₂×C₅₃	—	2	2	no	—
173	C₂×C₅₃	—	2	1	no	—
174	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
175	C₂×C₅₃	C₂×#344	1	2	yes	—
176	C₂²×M₁₆	—	53	2	no	—
177	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
178	C₂×#117	—	53	2	no	—
179	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
180	#423	—	2	2	no	—
181	#423	—	2	1	no	—
182	#423	—	2	1	no	—
183	C₂×C₄⋊C₈	Dic₅₃	1	4	yes	complemented
184	C₄×M₁₆	—	53	2	no	—
185	C₄×M₁₆	—	53	2	no	—
186	C₄×M₁₆	—	106	1	no	—
187	C₄×M₁₆	—	106	1	no	—
188	C₂×#97	—	106	1	no	—
189	C₂×C₄×C₈	—	106	1	no	—
190	#384	—	106	1	no	—
191	#382	—	106	1	no	—
192	#380	—	53	4	no	—
193	C₄×C₄⋊C₄	—	106	2	no	—
194	#344	—	106	8	no	—
195	C₅₃	#10855	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₄	C₄×D₅₃	1	2	yes	—
197	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
198	C₂×M₁₆	—	2	4	no	—
199	C₂×M₁₆	—	2	4	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	53	4	no	—
203	C₂×M₁₆	—	53	4	no	—
204	C₂×M₁₆	—	53	4	no	—
205	C₂×M₁₆	—	106	2	no	—
206	C₂×M₁₆	—	106	2	no	—
207	C₂×M₁₆	—	106	2	no	—
208	C₂²×C₈	—	106	2	no	—
209	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
210	#117	—	106	4	no	—
211	#117	—	106	4	no	—
212	C₂×C₄⋊C₄	—	106	2	no	—
213	C₂×C₄⋊C₄	—	106	4	no	—
214	C₂×C₂²⋊C₄	—	53	4	no	—
215	C₂×C₄²	—	106	2	no	—
216	C₂×C₄²	Dic₁₀₆	1	2	yes	—
217	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
218	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
219	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
220	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
221	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
222	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
223	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
225	C₄⋊C₈	—	2	4	no	—
226	C₄⋊C₈	—	2	4	no	—
227	#97	—	106	2	no	—
228	#97	—	106	2	no	—
229	C₄×C₈	—	106	2	no	—
230	C₄×C₈	—	106	2	no	—
231	#95	—	106	4	no	—
232	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
233	C₂²×C₄	—	4	4	no	—
234	C₂²×C₄	—	106	4	no	—
235	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
236	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
237	C₂²×C₄	#11719919	1	2	yes	—
238	C₂²×C₄	#11719919	1	2	yes	—
239	C₂²×C₄	#11719921	1	2	yes	—
240	C₂²×C₄	#11719921	1	2	yes	—
241	C₂²×C₄	C₂×Dic₁₀₆	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	C₂²×Dic₅₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
251	M₁₆	—	4	4	no	—
252	M₁₆	—	4	2	no	—
253	M₁₆	—	4	2	no	—
254	M₁₆	—	53	8	no	—
255	M₁₆	—	106	4	no	—
256	M₁₆	—	106	4	no	—
257	M₁₆	—	106	4	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	8	no	—
260	C₂×C₈	—	106	4	no	—
261	C₂×C₈	—	106	4	no	—
262	C₂×C₈	—	106	4	no	—
263	C₄⋊C₄	—	212	8	no	—
264	C₄⋊C₄	—	212	4	no	—
265	C₂²⋊C₄	—	106	8	no	—
266	C₄²	—	212	2	no	—
267	C₄²	—	212	2	no	—
268	C₄²	#11719919	1	2	yes	—
269	C₄²	#11719919	1	2	yes	—
270	C₄²	C₄⋊Dic₅₃	1	2	yes	—
271	C₄²	C₄⋊Dic₅₃	1	2	yes	—
272	C₄²	#11719921	1	1	yes	characteristic
273	C₄²	#11719921	1	1	yes	characteristic
274	C₄²	#11719921	1	1	yes	characteristic
275	C₄²	#11719921	1	1	yes	characteristic
276	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
277	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
278	C₂³	—	4	2	no	—
279	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
280	C₂³	C₂×#11719921	1	1	yes	characteristic
281	C₂³	C₂×#11719919	1	1	yes	characteristic
282	C₂³	#13133179	1	2	yes	—
283	C₂³	#13133182	1	2	yes	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	212	8	no	—
291	C₂×C₄	—	212	4	no	—
292	C₂×C₄	#13133124	1	2	yes	—
293	C₂×C₄	#13133124	1	2	yes	—
294	C₂×C₄	#13133124	1	2	yes	—
295	C₂×C₄	#13133124	1	2	yes	—
296	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
298	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
299	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
300	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
301	C₂×C₄	#13133175	1	2	yes	—
302	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
303	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
304	C₂×C₄	#13133188	1	2	yes	—
305	C₂×C₄	#13133192	1	2	yes	—
306	C₂×C₄	#13133193	1	2	yes	—
307	C₂×C₄	#13133197	1	2	yes	—
308	C₂×C₄	#13133204	1	1	yes	characteristic
309	C₂×C₄	#13133206	1	1	yes	characteristic
310	C₂×C₄	#13133208	1	2	yes	—
311	C₂×C₄	#13133213	1	1	yes	characteristic
312	C₂×C₄	#13133215	1	1	yes	characteristic
313	C₂×C₄	#13133216	1	2	yes	—
314	C₈	—	4	4	no	—
315	C₈	—	106	8	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	C₂×#13133124	1	1	yes	characteristic
322	C₂²	C₄⋊C₄×Dic₅₃	1	2	yes	—
323	C₂²	#16810519	1	2	yes	—
324	C₂²	#16810524	1	1	yes	characteristic
325	C₂²	#16810528	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	424	4	no	—
330	C₄	#16810394	1	4	yes	complemented
331	C₄	#16810480	1	2	yes	—
332	C₄	#16810488	1	1	yes	characteristic
333	C₄	#16810492	1	1	yes	characteristic
334	C₄	#16810522	1	2	yes	—
335	C₄	#16810526	1	1	yes	characteristic
336	C₄	#16810530	1	1	yes	characteristic
337	C₂	—	2	1	no	—
338	C₂	—	2	2	no	—
339	C₂	—	2	1	no	—
340	C₂	—	2	2	no	—
341	C₂	#20272944	1	1	yes	characteristic
342	C₂	C₂×#16810394	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#423	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#423	D₅₃
hypocenter	C₅₃	#10855
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
9. C₄×#11719917	C₄	semidirect
10. C₄×#11719917	C₄	semidirect
23. C₂×C₄⋊C₈×C₅₃	C₄	semidirect
102. #11719917	C₄⋊C₄	semidirect
158. C₄×C₅₃	#344	semidirect
183. C₂×C₄⋊C₈	Dic₅₃	semidirect
195. C₅₃	#10855	semidirect
330. C₄	#16810394	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11719917	2
#20277122	2
C₂×#16810394	2
C₂×C₅₃×#423	2
#10855	53