Subgroups — #27289896

Order 13568.

842 conjugacy classes of subgroups.

The subgroup lattice has 358 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×#16810379	C₂	1	4	yes	maximal
4	C₅₃×#2285	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	—	2	1	no	—
10	C₄×#11719917	—	2	1	no	—
11	C₂×#13133128	C₄	1	2	yes	complemented
12	C₂×C₄×#88322	C₄	1	2	yes	complemented
13	#16810493	—	2	1	no	—
14	#16810489	—	2	1	no	—
15	#16810483	—	2	2	no	—
16	C₄×C₄⋊Dic₅₃	C₄	1	4	yes	complemented
17	#16810379	—	2	8	no	—
18	C₂×#117×C₅₃	C₂²	1	2	yes	—
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#384	—	2	1	no	—
21	C₅₃×#382	—	2	1	no	—
22	C₅₃×#380	—	2	2	no	—
23	C₄×C₄⋊C₄×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	—	2	2	no	—
28	C₂×#11719917	D₄	1	4	yes	—
29	C₂×#11719917	Q₈	1	4	yes	—
30	C₂²×#88322	C₂×C₄	1	4	yes	—
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	—	2	2	no	—
36	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
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	D₄	1	2	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	#117×C₅₃	—	2	4	no	—
45	#117×C₅₃	—	2	4	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂×C₄	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	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
59	#95×C₅₃	—	2	4	no	—
60	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	—	2	4	no	—
62	C₂²×C₄×C₅₃	—	4	4	no	—
63	C₂²×C₄×C₅₃	C₂²⋊C₄	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	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₅₃	#41	1	2	yes	—
78	C₂²×C₄×C₅₃	#41	1	2	yes	—
79	C₂²×Dic₅₃	—	2	4	no	—
80	C₄⋊C₄×C₅₃	—	4	8	no	—
81	C₄⋊C₄×C₅₃	—	4	4	no	—
82	C₂²⋊C₄×C₅₃	—	4	2	no	—
83	C₂²⋊C₄×C₅₃	—	4	2	no	—
84	C₄²×C₅₃	—	2	8	no	—
85	C₄²×C₅₃	C₄²	1	2	yes	—
86	C₄²×C₅₃	C₄²	1	2	yes	—
87	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
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	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
95	C₂²⋊Dic₅₃	—	4	2	no	—
96	C₂²⋊Dic₅₃	—	4	2	no	—
97	C₄⋊Dic₅₃	—	4	8	no	—
98	C₄⋊Dic₅₃	—	4	4	no	—
99	C₄×Dic₅₃	—	2	8	no	—
100	#11719917	—	2	4	no	—
101	#11719917	—	2	4	no	—
102	#11719917	—	2	4	no	—
103	#11719917	—	2	4	no	—
104	C₂×#88322	—	2	4	no	—
105	C₂×#88322	—	2	4	no	—
106	C₂×#88322	C₄⋊C₄	1	8	yes	complemented
107	C₂³×C₅₃	—	4	2	no	—
108	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂³×C₅₃	#124	1	1	yes	characteristic
112	C₂³×C₅₃	#124	1	1	yes	characteristic
113	C₂³×C₅₃	#127	1	1	yes	characteristic
114	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	—	2	4	no	—
116	C₂×C₄×C₅₃	—	2	4	no	—
117	C₂×C₄×C₅₃	—	2	4	no	—
118	C₂×C₄×C₅₃	—	4	2	no	—
119	C₂×C₄×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	—	4	8	no	—
122	C₂×C₄×C₅₃	—	4	4	no	—
123	C₂×C₄×C₅₃	#95	1	2	yes	—
124	C₂×C₄×C₅₃	#95	1	2	yes	—
125	C₂×C₄×C₅₃	#95	1	2	yes	—
126	C₂×C₄×C₅₃	#95	1	2	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×C₄⋊C₄	1	2	yes	—
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	#117	1	2	yes	—
133	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
136	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
137	C₂×C₄×C₅₃	#121	1	2	yes	—
138	C₂×C₄×C₅₃	#121	1	2	yes	—
139	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
140	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
141	C₂×C₄×C₅₃	#125	1	2	yes	—
142	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
143	C₂×Dic₅₃	—	4	8	no	—
144	C₂×Dic₅₃	—	4	4	no	—
145	#88322	—	2	8	no	—
146	#10837	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	—	2	2	no	—
148	C₂²×C₅₃	—	2	2	no	—
149	C₂²×C₅₃	—	2	2	no	—
150	C₂²×C₅₃	—	4	1	no	—
151	C₂²×C₅₃	—	4	1	no	—
152	C₂²×C₅₃	—	4	1	no	—
153	C₂²×C₅₃	—	4	1	no	—
154	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
155	C₂²×C₅₃	#382	1	1	yes	characteristic
156	C₂²×C₅₃	#384	1	1	yes	characteristic
157	C₂²×C₅₃	#390	1	1	yes	characteristic
158	C₂²×C₅₃	#390	1	1	yes	characteristic
159	C₂²×C₅₃	#391	1	1	yes	characteristic
160	C₂²×C₅₃	#391	1	1	yes	characteristic
161	C₄×C₅₃	—	2	2	no	—
162	C₄×C₅₃	—	2	2	no	—
163	C₄×C₅₃	—	2	2	no	—
164	C₄×C₅₃	—	8	4	no	—
165	C₄×C₅₃	H(3,ℤ/4)	1	4	yes	complemented
166	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
167	C₄×C₅₃	#380	1	2	yes	—
168	C₄×C₅₃	#389	1	2	yes	—
169	C₄×C₅₃	#389	1	2	yes	—
170	Dic₅₃	—	8	4	no	—
171	C₂×C₄×M₁₆	—	53	1	no	—
172	#2285	—	53	1	no	—
173	#2285	D₅₃	1	1	yes	characteristic
174	C₂×H(3,ℤ/4)	—	53	4	no	—
175	C₂×C₅₃	—	2	1	no	—
176	C₂×C₅₃	—	2	1	no	—
177	C₂×C₅₃	—	2	1	no	—
178	C₂×C₅₃	—	2	1	no	—
179	C₂×C₅₃	—	2	1	no	—
180	C₂×C₅₃	—	2	1	no	—
181	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
182	C₂×C₅₃	C₂×H(3,ℤ/4)	1	2	yes	—
183	C₂²×M₁₆	—	53	2	no	—
184	C₂×#117	—	53	2	no	—
185	C₂×#117	C₂×D₅₃	1	2	yes	—
186	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
187	C₄×M₁₆	—	106	1	no	—
188	C₄×M₁₆	—	106	1	no	—
189	C₄×M₁₆	—	106	1	no	—
190	C₄×M₁₆	—	106	1	no	—
191	C₂×#97	—	53	2	no	—
192	C₂×C₄×C₈	—	53	2	no	—
193	#384	—	2	1	no	—
194	#384	—	106	1	no	—
195	#382	—	2	1	no	—
196	#382	—	106	1	no	—
197	#380	—	2	2	no	—
198	#380	—	106	2	no	—
199	C₄×C₄⋊C₄	—	53	4	no	—
200	C₄×C₄⋊C₄	Dic₅₃	1	4	yes	complemented
201	H(3,ℤ/4)	—	106	8	no	—
202	C₅₃	#10837	1	1	yes	characteristic, complemented, Sylow
203	C₂³×C₄	C₂²⋊D₅₃	1	2	yes	—
204	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
205	C₂×M₁₆	—	53	4	no	—
206	C₂×M₁₆	—	53	4	no	—
207	C₂×M₁₆	—	106	2	no	—
208	C₂×M₁₆	—	106	2	no	—
209	C₂×M₁₆	—	106	2	no	—
210	C₂×M₁₆	—	106	2	no	—
211	C₂²×C₈	—	53	4	no	—
212	#117	—	2	4	no	—
213	#117	—	2	4	no	—
214	#117	—	106	4	no	—
215	#117	—	106	4	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	4	no	—
218	C₂×C₄⋊C₄	—	106	2	no	—
219	C₂×C₄⋊C₄	—	106	4	no	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₂²⋊C₄	—	106	2	no	—
222	C₂×C₄²	—	53	4	no	—
223	C₂×C₄²	Dic₁₀₆	1	2	yes	—
224	C₂×C₄²	C₄×D₅₃	1	2	yes	—
225	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
226	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
227	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
228	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
229	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
230	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
231	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
232	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
233	#97	—	106	2	no	—
234	#97	—	106	2	no	—
235	C₄×C₈	—	106	2	no	—
236	C₄×C₈	—	106	2	no	—
237	#95	—	2	4	no	—
238	#95	—	106	4	no	—
239	C₂⁴	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	4	4	no	—
242	C₂²×C₄	—	106	4	no	—
243	C₂²×C₄	#11719919	1	2	yes	—
244	C₂²×C₄	#11719919	1	2	yes	—
245	C₂²×C₄	#11719921	1	2	yes	—
246	C₂²×C₄	#11719921	1	2	yes	—
247	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
248	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
249	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
253	C₂²×C₄	D₄×D₅₃	1	2	yes	—
254	C₂²×C₄	#11719947	1	2	yes	—
255	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
256	C₂²×C₄	#11719949	1	2	yes	—
257	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
259	M₁₆	—	106	4	no	—
260	M₁₆	—	106	4	no	—
261	M₁₆	—	106	4	no	—
262	M₁₆	—	106	4	no	—
263	C₂×C₈	—	53	8	no	—
264	C₂×C₈	—	106	4	no	—
265	C₂×C₈	—	106	4	no	—
266	C₄⋊C₄	—	4	8	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄⋊C₄	—	212	8	no	—
269	C₄⋊C₄	—	212	4	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	212	2	no	—
273	C₂²⋊C₄	—	212	2	no	—
274	C₄²	—	2	8	no	—
275	C₄²	—	106	8	no	—
276	C₄²	C₄×Dic₅₃	1	2	yes	—
277	C₄²	C₄×Dic₅₃	1	2	yes	—
278	C₄²	#11719919	1	2	yes	—
279	C₄²	#11719919	1	2	yes	—
280	C₄²	C₄⋊Dic₅₃	1	2	yes	—
281	C₄²	C₄⋊Dic₅₃	1	2	yes	—
282	C₄²	#11719921	1	1	yes	characteristic
283	C₄²	#11719921	1	1	yes	characteristic
284	C₄²	#11719921	1	1	yes	characteristic
285	C₄²	#11719921	1	1	yes	characteristic
286	C₂³	—	4	2	no	—
287	C₂³	C₂×#11719921	1	1	yes	characteristic
288	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
289	C₂³	C₂×#11719919	1	1	yes	characteristic
290	C₂³	#13133204	1	1	yes	characteristic
291	C₂³	#13133206	1	1	yes	characteristic
292	C₂³	#13133213	1	1	yes	characteristic
293	C₂³	#13133215	1	1	yes	characteristic
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	8	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	212	8	no	—
303	C₂×C₄	—	212	4	no	—
304	C₂×C₄	#13133124	1	2	yes	—
305	C₂×C₄	#13133124	1	2	yes	—
306	C₂×C₄	#13133124	1	2	yes	—
307	C₂×C₄	#13133124	1	2	yes	—
308	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
309	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
310	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
311	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
313	C₂×C₄	#13133175	1	2	yes	—
314	C₂×C₄	#13133179	1	2	yes	—
315	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
316	C₂×C₄	#13133182	1	2	yes	—
317	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
318	C₂×C₄	#13133188	1	2	yes	—
319	C₂×C₄	#13133192	1	2	yes	—
320	C₂×C₄	#13133193	1	2	yes	—
321	C₂×C₄	#13133197	1	2	yes	—
322	C₂×C₄	#13133208	1	2	yes	—
323	C₂×C₄	#13133216	1	2	yes	—
324	C₈	—	106	8	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	C₂×#13133124	1	1	yes	characteristic
333	C₂²	#16810488	1	1	yes	characteristic
334	C₂²	#16810492	1	1	yes	characteristic
335	C₂²	#16810524	1	1	yes	characteristic
336	C₂²	#16810526	1	1	yes	characteristic
337	C₂²	#16810528	1	1	yes	characteristic
338	C₂²	#16810530	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	8	4	no	—
343	C₄	—	424	4	no	—
344	C₄	#16810379	1	4	yes	complemented
345	C₄	C₄⋊C₄×Dic₅₃	1	2	yes	—
346	C₄	#16810480	1	2	yes	—
347	C₄	#16810519	1	2	yes	—
348	C₄	#16810522	1	2	yes	—
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	#20272944	1	1	yes	characteristic
356	C₂	C₂×#16810379	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#2285	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#2285	D₅₃
hypocenter	C₅₃	#10837
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #2285

Complements

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

N	H ≅ G/N	type
11. C₂×#13133128	C₄	semidirect
12. C₂×C₄×#88322	C₄	semidirect
16. C₄×C₄⋊Dic₅₃	C₄	semidirect
23. C₄×C₄⋊C₄×C₅₃	C₄	semidirect
106. C₂×#88322	C₄⋊C₄	semidirect
165. C₄×C₅₃	H(3,ℤ/4)	semidirect
200. C₄×C₄⋊C₄	Dic₅₃	semidirect
202. C₅₃	#10837	semidirect
344. C₄	#16810379	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×#16810379	2
C₅₃×#2285	2
#10837	53