Subgroups — #27402978

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20277672	C₂	1	2	yes	maximal
2	#20275485	C₂	1	2	yes	maximal
3	#20275485	C₂	1	2	yes	maximal
4	#20275474	C₂	1	2	yes	maximal
5	#20274883	C₂	1	2	yes	maximal
6	#20274879	C₂	1	1	yes	maximal, characteristic
7	#20274612	C₂	1	2	yes	maximal
8	C₄×#13133151	C₂	1	1	yes	maximal, characteristic
9	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic
10	C₂×#13133151	C₂²	1	1	yes	characteristic
11	C₂×#13133151	C₂²	1	1	yes	characteristic
12	C₂×#13133151	C₂²	1	2	yes	—
13	C₂×#13133151	C₂²	1	1	yes	characteristic
14	C₂×#13133151	C₂²	1	1	yes	characteristic
15	C₂×#13133149	C₂²	1	2	yes	—
16	C₂×#13133149	C₂²	1	2	yes	—
17	#16810530	—	2	2	no	—
18	#16810509	—	2	2	no	—
19	#16810509	—	2	2	no	—
20	#16810493	C₂²	1	2	yes	—
21	#16810489	C₂²	1	2	yes	—
22	#16810489	C₂²	1	2	yes	—
23	C₄⋊C₄×Dic₅₃	—	2	2	no	—
24	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
25	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
26	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
27	#16810386	C₂²	1	2	yes	—
28	#16810386	C₂²	1	2	yes	—
29	#16810386	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#446	C₄	1	8	yes	complemented
32	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
33	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#97×C₅₃	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊Dic₅₃	—	2	2	no	—
37	C₂×C₄⋊Dic₅₃	—	2	2	no	—
38	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
42	C₂×C₄×Dic₅₃	—	2	2	no	—
43	C₂×C₄×Dic₅₃	—	2	2	no	—
44	C₂×C₄×Dic₅₃	—	2	2	no	—
45	#13133151	—	2	2	no	—
46	#13133151	—	4	1	no	—
47	#13133151	—	4	2	no	—
48	#13133151	—	4	1	no	—
49	#13133151	Q₈	1	4	yes	complemented
50	#13133149	—	2	4	no	—
51	#13133149	—	4	2	no	—
52	#13133124	—	2	2	no	—
53	#13133124	—	2	2	no	—
54	#13133124	—	2	2	no	—
55	#13133124	—	2	2	no	—
56	#13133124	—	2	2	no	—
57	#13133124	—	2	2	no	—
58	#13133124	—	2	2	no	—
59	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
61	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
64	C₄×Q₈×C₅₃	Q₈	1	4	yes	—
65	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
69	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
70	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
71	#97×C₅₃	C₂×C₄	1	8	yes	—
72	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
73	C₂×Q₈×C₅₃	—	4	1	no	—
74	C₂×Q₈×C₅₃	—	4	1	no	—
75	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₅₃	—	2	2	no	—
82	C₂²×Dic₅₃	—	2	2	no	—
83	C₂²×Dic₅₃	—	2	2	no	—
84	C₂²×Dic₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	2	1	no	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	2	1	no	—
89	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₅₃	#41	1	2	yes	—
94	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
95	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
96	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
97	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
98	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
99	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
100	C₄⋊Dic₅₃	—	2	2	no	—
101	C₄⋊Dic₅₃	—	2	2	no	—
102	C₄⋊Dic₅₃	—	2	2	no	—
103	C₄⋊Dic₅₃	—	2	2	no	—
104	C₄⋊Dic₅₃	—	2	2	no	—
105	C₄⋊Dic₅₃	—	2	2	no	—
106	C₄⋊Dic₅₃	—	2	2	no	—
107	C₄⋊Dic₅₃	—	4	4	no	—
108	C₄⋊Dic₅₃	—	8	2	no	—
109	C₄⋊Dic₅₃	C₂×Q₈	1	4	yes	—
110	C₄×Dic₅₃	—	4	4	no	—
111	C₄×Dic₅₃	—	8	2	no	—
112	C₄×Dic₅₃	—	8	2	no	—
113	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₅₃	—	4	4	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
125	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
127	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×Dic₅₃	—	2	4	no	—
134	C₂×Dic₅₃	—	4	2	no	—
135	C₂×Dic₅₃	—	4	2	no	—
136	C₂×Dic₅₃	—	4	2	no	—
137	C₂×Dic₅₃	—	4	2	no	—
138	C₂×Dic₅₃	—	4	2	no	—
139	C₂×Dic₅₃	—	4	2	no	—
140	C₂×Dic₅₃	—	4	2	no	—
141	C₂×Dic₅₃	—	4	2	no	—
142	C₂×Dic₅₃	—	4	2	no	—
143	C₂×Dic₅₃	—	4	2	no	—
144	C₂×Dic₅₃	—	4	2	no	—
145	C₈×C₅₃	—	2	4	no	—
146	C₈×C₅₃	—	2	2	no	—
147	C₈×C₅₃	C₄×Q₈	1	4	yes	—
148	#22731	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
151	C₂²×C₅₃	#550	1	1	yes	characteristic
152	C₂²×C₅₃	#575	1	1	yes	characteristic
153	C₂²×C₅₃	#575	1	1	yes	characteristic
154	C₂²×C₅₃	#576	1	1	yes	characteristic
155	C₂²×C₅₃	#578	1	1	yes	characteristic
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	—	2	4	no	—
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	—	2	4	no	—
160	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
161	C₄×C₅₃	#520	1	1	yes	characteristic
162	C₄×C₅₃	#557	1	1	yes	characteristic
163	C₄×C₅₃	Q₈²	1	1	yes	characteristic
164	Dic₅₃	—	4	4	no	—
165	Dic₅₃	—	8	2	no	—
166	Dic₅₃	—	8	2	no	—
167	Dic₅₃	—	8	2	no	—
168	C₂×#446	D₅₃	1	1	yes	characteristic
169	#2355	—	53	2	no	—
170	#1949	—	53	2	no	—
171	#1949	—	53	2	no	—
172	#1947	—	53	2	no	—
173	#1847	—	53	2	no	—
174	#1846	—	53	1	no	—
175	#1776	—	53	2	no	—
176	C₄×#107	—	53	1	no	—
177	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₅₃	#2911	1	1	yes	characteristic
179	C₂×C₅₃	#2913	1	1	yes	characteristic
180	C₂×C₅₃	#3389	1	1	yes	characteristic
181	C₂×C₅₃	#3390	1	1	yes	characteristic
182	C₂×C₅₃	#3402	1	1	yes	characteristic
183	C₂×C₅₃	#3403	1	1	yes	characteristic
184	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
185	#446	Dic₅₃	1	8	yes	complemented
186	C₂×#107	—	53	1	no	—
187	C₂×#107	—	53	1	no	—
188	C₂×#107	—	53	1	no	—
189	C₂×#107	—	53	1	no	—
190	C₂×#107	—	53	2	no	—
191	C₂×#106	—	53	2	no	—
192	C₂×#106	—	53	2	no	—
193	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
194	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
195	C₂×#97	C₂×D₅₃	1	2	yes	—
196	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
197	#391	—	106	2	no	—
198	#387	—	106	2	no	—
199	#387	—	106	2	no	—
200	#384	—	53	2	no	—
201	#382	—	53	2	no	—
202	#382	—	53	2	no	—
203	C₄×C₄⋊C₄	—	53	2	no	—
204	C₄×C₄⋊C₄	—	53	2	no	—
205	C₄×C₄⋊C₄	—	53	2	no	—
206	C₄×C₄⋊C₄	—	106	2	no	—
207	#340	—	53	2	no	—
208	#340	—	53	2	no	—
209	#340	—	53	2	no	—
210	C₅₃	#22731	1	1	yes	characteristic, complemented, Sylow
211	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
214	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
215	C₄×Q₈	Dic₁₀₆	1	4	yes	—
216	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
217	C₂×C₄⋊C₄	—	53	2	no	—
218	C₂×C₄⋊C₄	—	53	2	no	—
219	C₂×C₄⋊C₄	—	53	2	no	—
220	C₂×C₄⋊C₄	—	53	2	no	—
221	C₂×C₄⋊C₄	—	106	2	no	—
222	C₂×C₄⋊C₄	—	106	2	no	—
223	C₂×C₄⋊C₄	D₂₁₂	1	2	yes	—
224	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
225	C₂×C₄²	—	106	2	no	—
226	C₂×C₄²	—	106	2	no	—
227	C₂×C₄²	—	106	2	no	—
228	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
229	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
230	#107	—	53	4	no	—
231	#107	—	106	2	no	—
232	#107	—	212	1	no	—
233	#107	—	212	2	no	—
234	#107	—	212	1	no	—
235	#106	—	106	4	no	—
236	#106	—	212	2	no	—
237	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
238	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
239	#97	C₂×Dic₅₃	1	8	yes	—
240	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
241	#95	—	106	2	no	—
242	#95	—	106	2	no	—
243	#95	—	106	2	no	—
244	#95	—	106	2	no	—
245	#95	—	106	2	no	—
246	#95	—	106	2	no	—
247	#95	—	106	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	C₄⋊Dic₅₃	1	4	yes	—
251	C₂²×C₄	—	106	2	no	—
252	C₂²×C₄	—	106	2	no	—
253	C₂²×C₄	—	106	2	no	—
254	C₂²×C₄	—	106	2	no	—
255	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
258	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
259	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	Q₈×D₅₃	1	2	yes	—
265	C₂×C₈	#11719949	1	2	yes	—
266	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
267	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
268	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
269	C₄⋊C₄	—	53	4	no	—
270	C₄⋊C₄	—	106	2	no	—
271	C₄⋊C₄	—	106	2	no	—
272	C₄⋊C₄	—	106	2	no	—
273	C₄⋊C₄	—	106	2	no	—
274	C₄⋊C₄	—	106	2	no	—
275	C₄⋊C₄	—	106	2	no	—
276	C₄⋊C₄	—	106	2	no	—
277	C₄⋊C₄	—	424	2	no	—
278	C₄⋊C₄	—	212	4	no	—
279	C₄⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
280	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
281	C₄²	—	424	2	no	—
282	C₄²	—	424	2	no	—
283	C₄²	—	212	4	no	—
284	C₄²	C₂×Dic₁₀₆	1	2	yes	—
285	C₄²	C₂×Dic₁₀₆	1	4	yes	—
286	C₄²	C₂²×Dic₅₃	1	2	yes	—
287	C₄²	C₂²×Dic₅₃	1	4	yes	—
288	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
289	Q₈	—	4	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	106	4	no	—
299	C₂×C₄	—	212	2	no	—
300	C₂×C₄	—	212	2	no	—
301	C₂×C₄	—	212	2	no	—
302	C₂×C₄	—	212	2	no	—
303	C₂×C₄	—	212	2	no	—
304	C₂×C₄	—	212	2	no	—
305	C₂×C₄	—	212	2	no	—
306	C₂×C₄	—	212	2	no	—
307	C₂×C₄	—	212	2	no	—
308	C₂×C₄	—	212	2	no	—
309	C₂×C₄	—	212	2	no	—
310	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
311	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
312	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
313	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
314	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
315	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
316	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
317	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
318	C₂×C₄	#13133263	1	1	yes	characteristic
319	C₂×C₄	#13133264	1	1	yes	characteristic
320	C₈	—	2	4	no	—
321	C₈	—	2	2	no	—
322	C₈	Q₈×Dic₅₃	1	4	yes	—
323	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
324	C₂²	Q₈×D₂₁₂	1	1	yes	characteristic
325	C₂²	#16811295	1	1	yes	characteristic
326	C₂²	#16811423	1	1	yes	characteristic
327	C₂²	#16811423	1	1	yes	characteristic
328	C₂²	#16811428	1	1	yes	characteristic
329	C₂²	#16811440	1	1	yes	characteristic
330	C₄	—	2	4	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	4	no	—
334	C₄	—	424	2	no	—
335	C₄	—	424	2	no	—
336	C₄	—	424	2	no	—
337	C₄	—	212	4	no	—
338	C₄	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
339	C₄	#16811106	1	1	yes	characteristic
340	C₄	#16811352	1	1	yes	characteristic
341	C₄	Q₈×Dic₁₀₆	1	1	yes	characteristic
342	C₂	Q₈×C₄⋊Dic₅₃	1	1	yes	characteristic
343	C₂	#20283305	1	1	yes	characteristic
344	C₂	#20283315	1	1	yes	characteristic
345	C₂	#20287675	1	1	yes	characteristic
346	C₂	#20287684	1	1	yes	characteristic
347	C₂	#20287796	1	1	yes	characteristic
348	C₂	#20287805	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#446	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#446	D₅₃
hypocenter	C₅₃	#22731
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
31. C₅₃×#446	C₄	semidirect
49. #13133151	Q₈	semidirect
185. #446	Dic₅₃	semidirect
210. C₅₃	#22731	semidirect

Maximal subgroups

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

subgroup	index
#20277672	2
#20275485	2
#20275474	2
#20274883	2
#20274879	2
#20274612	2
C₄×#13133151	2
C₂×C₅₃×#446	2
#22731	53