Subgroups — #27355850

Order 13568.

722 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283459	C₂	1	2	yes	maximal, complemented
2	#20273600	C₂	1	2	yes	maximal, complemented
3	#20273545	C₂	1	4	yes	maximal, complemented
4	D₈×#88322	C₂	1	2	yes	maximal, complemented
5	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273114	C₂	1	2	yes	maximal, complemented
7	C₈×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133224	—	2	2	no	—
10	#16810701	C₂²	1	4	yes	—
11	#16810695	—	2	4	no	—
12	D₄×#88322	C₂²	1	4	yes	—
13	#16810676	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#88322	C₂²	1	2	yes	—
15	#16810590	—	2	2	no	—
16	C₄×#11719917	C₂²	1	1	yes	characteristic
17	#16810376	C₂²	1	1	yes	characteristic
18	#16810376	C₂²	1	2	yes	—
19	#16810376	C₂²	1	1	yes	characteristic
20	D₄⋊#88322	C₂²	1	4	yes	—
21	D₄⋊#88322	C₂²	1	4	yes	—
22	#16810345	C₂²	1	2	yes	—
23	C₈×#88322	C₂²	1	2	yes	—
24	C₅₃×#532	C₂²	1	2	yes	—
25	C₅₃×#500	C₄	1	1	yes	characteristic
26	C₅₃×#493	C₄	1	1	yes	characteristic
27	C₅₃×#486	C₄	1	2	yes	—
28	C₅₃×#466	C₄	1	2	yes	—
29	C₅₃×#466	C₄	1	2	yes	—
30	C₄×D₈×C₅₃	C₂²	1	2	yes	—
31	C₄×D₈×C₅₃	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	#13133224	—	4	8	no	—
34	C₂×#11719917	—	2	4	no	—
35	C₂×#11719917	D₄	1	1	yes	characteristic
36	C₂×#11719917	D₄	1	1	yes	characteristic
37	C₂²×#88322	—	2	4	no	—
38	C₂²×#88322	—	2	2	no	—
39	C₄⋊#88322	—	2	2	no	—
40	C₄⋊#88322	—	2	1	no	—
41	C₄⋊#88322	C₂³	1	2	yes	—
42	C₄⋊#88322	C₂³	1	2	yes	—
43	C₂²⋊#88322	—	2	4	no	—
44	C₂²⋊#88322	—	2	4	no	—
45	C₂²⋊#88322	—	2	1	no	—
46	#13133128	D₄	1	2	yes	—
47	C₄×#88322	C₂³	1	2	yes	—
48	C₄×#88322	C₂³	1	2	yes	—
49	C₂×#41×C₅₃	—	2	2	no	—
50	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
55	#127×C₅₃	C₂×C₄	1	2	yes	—
56	#124×C₅₃	—	2	2	no	—
57	#121×C₅₃	—	2	2	no	—
58	#121×C₅₃	C₂×C₄	1	4	yes	—
59	C₄×D₄×C₅₃	—	2	2	no	—
60	C₄×D₄×C₅₃	C₂³	1	4	yes	—
61	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
62	#107×C₅₃	C₂×C₄	1	2	yes	—
63	#107×C₅₃	C₂×C₄	1	2	yes	—
64	#102×C₅₃	C₂×C₄	1	4	yes	—
65	#102×C₅₃	C₂×C₄	1	4	yes	—
66	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	#41×C₅₃	—	4	4	no	—
71	#41×C₅₃	—	4	4	no	—
72	C₂×Q₈×C₅₃	—	2	2	no	—
73	C₂×D₄×C₅₃	—	2	4	no	—
74	C₂×D₄×C₅₃	—	2	2	no	—
75	C₂×D₄×C₅₃	—	2	2	no	—
76	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₅₃	—	2	4	no	—
78	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
81	D₈×C₅₃	—	2	4	no	—
82	D₈×C₅₃	—	4	2	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	4	1	no	—
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₄⋊C₄×C₅₃	—	2	2	no	—
90	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
91	C₂²⋊C₄×C₅₃	—	2	4	no	—
92	C₂²⋊C₄×C₅₃	—	2	4	no	—
93	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅₃	#41	1	1	yes	characteristic
96	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
97	#11719917	—	2	2	no	—
98	#11719917	—	4	8	no	—
99	#11719917	D₈	1	4	yes	complemented
100	C₂×#88322	—	2	2	no	—
101	C₂×#88322	—	2	2	no	—
102	C₂×#88322	—	2	2	no	—
103	C₂×#88322	—	4	4	no	—
104	C₂×#88322	—	4	8	no	—
105	C₂×#88322	C₂×D₄	1	2	yes	—
106	C₂×#88322	C₂×D₄	1	2	yes	—
107	C₂³×C₅₃	—	2	4	no	—
108	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
109	Q₈×C₅₃	—	4	4	no	—
110	D₄×C₅₃	—	2	4	no	—
111	D₄×C₅₃	—	2	4	no	—
112	D₄×C₅₃	—	4	8	no	—
113	D₄×C₅₃	—	4	4	no	—
114	D₄×C₅₃	—	8	2	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	4	no	—
118	C₂×C₄×C₅₃	—	4	1	no	—
119	C₂×C₄×C₅₃	—	4	4	no	—
120	C₂×C₄×C₅₃	—	4	4	no	—
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₅₃	#135	1	2	yes	—
126	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
129	C₈×C₅₃	—	2	1	no	—
130	C₈×C₅₃	—	2	1	no	—
131	C₈×C₅₃	—	2	1	no	—
132	C₈×C₅₃	#131	1	2	yes	complemented
133	#88322	—	2	2	no	—
134	#88322	—	4	2	no	—
135	#88322	—	4	2	no	—
136	#88322	—	4	2	no	—
137	#88322	C₂×D₈	1	4	yes	—
138	#18066	—	53	1	no	maximal, Sylow
139	C₂²×C₅₃	—	2	1	no	—
140	C₂²×C₅₃	—	2	1	no	—
141	C₂²×C₅₃	—	4	4	no	—
142	C₂²×C₅₃	—	4	4	no	—
143	C₂²×C₅₃	—	4	4	no	—
144	C₂²×C₅₃	C₄×D₈	1	2	yes	—
145	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	—	2	1	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	4	4	no	—
150	C₄×C₅₃	C₄×D₈	1	2	yes	—
151	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
152	C₄×C₅₃	#568	1	1	yes	characteristic
153	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
154	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
155	#3169	D₅₃	1	1	yes	characteristic
156	#2935	—	53	2	no	—
157	#1589	—	53	2	no	—
158	#1581	—	53	4	no	—
159	C₈×D₈	—	53	2	no	—
160	C₂×#335	—	53	1	no	—
161	#1480	—	53	2	no	—
162	C₈×M₁₆	—	53	1	no	—
163	C₂×C₅₃	—	2	2	no	—
164	C₂×C₅₃	—	8	4	no	—
165	C₂×C₅₃	#2935	1	1	yes	characteristic
166	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
167	C₂×C₅₃	#2957	1	1	yes	characteristic
168	C₂×#131	—	106	2	no	—
169	#532	C₂×D₅₃	1	2	yes	—
170	#500	Dic₅₃	1	1	yes	characteristic
171	#493	Dic₅₃	1	1	yes	characteristic
172	#486	Dic₅₃	1	2	yes	—
173	#466	Dic₅₃	1	2	yes	—
174	#466	Dic₅₃	1	2	yes	—
175	C₄×D₈	C₂×D₅₃	1	2	yes	—
176	C₄×D₈	C₂×D₅₃	1	2	yes	—
177	#436	—	53	4	no	—
178	#435	—	106	4	no	—
179	C₈×D₄	—	53	4	no	—
180	#431	—	53	1	no	—
181	C₂×C₄⋊C₈	—	53	2	no	—
182	#408	—	106	2	no	—
183	C₄×M₁₆	—	53	1	no	—
184	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
185	#335	—	53	1	no	—
186	#335	—	53	2	no	—
187	#335	—	53	1	no	—
188	D₄⋊C₈	—	53	4	no	—
189	D₄⋊C₈	—	53	4	no	—
190	#322	—	53	2	no	—
191	C₈²	—	53	2	no	—
192	C₅₃	#18066	1	1	yes	characteristic, complemented, Sylow
193	C₂×#41	—	2	2	no	—
194	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
195	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
196	#131	—	212	8	no	—
197	C₂×M₁₆	—	53	1	no	—
198	C₂×M₁₆	—	53	1	no	—
199	C₂×M₁₆	—	106	4	no	—
200	C₂²×C₈	—	106	4	no	—
201	C₂²×C₈	—	106	2	no	—
202	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
203	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
204	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
205	#127	C₂×Dic₅₃	1	2	yes	—
206	#124	—	2	2	no	—
207	#121	—	2	2	no	—
208	#121	C₂×Dic₅₃	1	4	yes	—
209	C₄×D₄	—	2	2	no	—
210	C₄×D₄	C₂²×D₅₃	1	4	yes	—
211	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
212	#107	C₂×Dic₅₃	1	2	yes	—
213	#107	C₂×Dic₅₃	1	2	yes	—
214	C₄⋊C₈	—	53	2	no	—
215	C₄⋊C₈	—	53	2	no	—
216	C₄⋊C₈	—	106	2	no	—
217	C₄⋊C₈	—	106	1	no	—
218	#102	C₂×Dic₅₃	1	4	yes	—
219	#102	C₂×Dic₅₃	1	4	yes	—
220	C₂²⋊C₈	—	106	4	no	—
221	C₂²⋊C₈	—	106	4	no	—
222	C₂²⋊C₈	—	106	1	no	—
223	#97	—	53	2	no	—
224	C₄×C₈	—	53	2	no	—
225	C₄×C₈	—	53	2	no	—
226	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
227	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
230	#41	—	4	4	no	—
231	#41	—	4	4	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	C₂²×Dic₅₃	1	4	yes	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
239	C₂²×C₄	#11719947	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
241	D₈	—	2	4	no	—
242	D₈	—	4	2	no	—
243	M₁₆	—	53	4	no	—
244	M₁₆	—	106	2	no	—
245	M₁₆	—	212	8	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	53	2	no	—
249	C₂×C₈	—	53	2	no	—
250	C₂×C₈	—	106	2	no	—
251	C₂×C₈	—	106	2	no	—
252	C₂×C₈	—	106	2	no	—
253	C₂×C₈	—	212	8	no	—
254	C₂×C₈	—	212	4	no	—
255	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
256	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
257	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
258	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₂²⋊C₄	—	2	4	no	—
263	C₄²	D₄×D₅₃	1	1	yes	characteristic
264	C₄²	#11719947	1	1	yes	characteristic
265	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
266	C₄²	C₂³×D₅₃	1	1	yes	characteristic
267	C₂³	—	2	4	no	—
268	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
269	Q₈	—	4	4	no	—
270	D₄	—	2	4	no	—
271	D₄	—	2	4	no	—
272	D₄	—	4	8	no	—
273	D₄	—	4	4	no	—
274	D₄	—	8	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
282	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
283	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
284	C₂×C₄	D₈×D₅₃	1	2	yes	—
285	C₂×C₄	#13133236	1	2	yes	—
286	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
287	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
288	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	53	4	no	—
293	C₈	—	106	2	no	—
294	C₈	—	212	2	no	—
295	C₈	—	212	2	no	—
296	C₈	—	212	2	no	—
297	C₈	#13133224	1	2	yes	complemented
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	4	4	no	—
303	C₂²	D₈×Dic₅₃	1	2	yes	—
304	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	4	4	no	—
309	C₄	D₈×Dic₅₃	1	2	yes	—
310	C₄	C₂×#13133224	1	1	yes	characteristic
311	C₄	#16811390	1	1	yes	characteristic
312	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
313	C₄	C₂×#13133236	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	8	4	no	—
316	C₂	#20283459	1	1	yes	characteristic
317	C₂	C₂×D₈×Dic₅₃	1	1	yes	characteristic
318	C₂	#20283632	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3169	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#3169	D₅₃
hypocenter	C₅₃	#18066
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #20283459	C₂	semidirect
2. #20273600	C₂	semidirect
3. #20273545	C₂	semidirect
4. D₈×#88322	C₂	semidirect
5. C₂×#16810376	C₂	semidirect
6. #20273114	C₂	semidirect
7. C₈×#11719917	C₂	semidirect
99. #11719917	D₈	semidirect
132. C₈×C₅₃	#131	semidirect
192. C₅₃	#18066	semidirect
297. C₈	#13133224	semidirect

Maximal subgroups

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

subgroup	index
#20283459	2
#20273600	2
#20273545	2
D₈×#88322	2
C₂×#16810376	2
#20273114	2
C₈×#11719917	2
C₅₃×#3169	2
#18066	53