Subgroups — #27537347

Order 13568.

648 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287818	C₂	1	2	yes	maximal, complemented
2	#20276557	C₂	1	4	yes	maximal, complemented
3	#20276552	C₂	1	4	yes	maximal, complemented
4	C₂×#16810444	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276343	C₂	1	2	yes	maximal, complemented
6	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#443	C₂	1	1	yes	maximal, characteristic
8	#16811040	—	2	4	no	—
9	#16811040	C₂²	1	8	yes	complemented
10	C₂×#13133157	—	2	4	no	—
11	C₂×#13133157	C₂²	1	2	yes	—
12	C₂×#13133157	C₂²	1	1	yes	characteristic
13	C₂×#13133157	C₂²	1	1	yes	characteristic
14	#16810444	C₂²	1	2	yes	complemented
15	#16810444	C₂²	1	2	yes	complemented
16	#16810407	C₂²	1	8	yes	complemented
17	#16810399	C₂²	1	2	yes	complemented
18	C₄×#11719908	C₂²	1	2	yes	complemented
19	C₂×#135×C₅₃	C₄	1	1	yes	characteristic
20	C₂×#131×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#443	C₄	1	2	yes	—
22	C₅₃×#443	C₄	1	2	yes	—
23	C₅₃×#443	C₂²	1	2	yes	—
24	C₅₃×#443	C₂²	1	2	yes	—
25	C₂×#108×C₅₃	C₄	1	1	yes	characteristic
26	C₂×#104×C₅₃	C₄	1	2	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	#13133157	—	2	2	no	—
29	#13133157	—	2	1	no	—
30	#13133157	—	2	1	no	—
31	#13133157	—	2	8	no	—
32	#13133157	D₄	1	2	yes	—
33	#13133157	D₄	1	2	yes	—
34	#13133157	C₂³	1	4	yes	—
35	C₂×#11719908	—	2	2	no	—
36	C₂×#11719908	—	2	8	no	—
37	C₂×#11719908	C₂³	1	2	yes	—
38	C₂×#11719908	C₂³	1	2	yes	—
39	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
40	#135×C₅₃	C₈	1	2	yes	—
41	#135×C₅₃	C₈	1	2	yes	—
42	#135×C₅₃	C₂×C₄	1	2	yes	—
43	#135×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×Q₁₆×C₅₃	C₈	1	1	yes	characteristic
45	C₂×SD₁₆×C₅₃	C₈	1	2	yes	—
46	C₂×D₈×C₅₃	C₈	1	1	yes	characteristic
47	#131×C₅₃	—	2	2	no	—
48	#131×C₅₃	—	2	2	no	—
49	#131×C₅₃	C₂×C₄	1	4	yes	—
50	#131×C₅₃	C₂³	1	4	yes	—
51	C₂×M₁₆×C₅₃	—	2	2	no	—
52	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₅₃	—	2	2	no	—
54	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
57	#108×C₅₃	C₂×C₄	1	2	yes	—
58	#108×C₅₃	C₂×C₄	1	2	yes	—
59	#104×C₅₃	C₂×C₄	1	4	yes	—
60	#104×C₅₃	C₂×C₄	1	4	yes	—
61	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	#41×C₅₃	—	2	2	no	—
66	#41×C₅₃	—	2	2	no	—
67	#41×C₅₃	C₂×C₈	1	4	yes	—
68	#41×C₅₃	C₂²×C₄	1	4	yes	—
69	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
70	C₂×D₄×C₅₃	—	2	2	no	—
71	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
72	C₂²×C₄×C₅₃	—	2	2	no	—
73	C₂²×C₄×C₅₃	—	2	1	no	—
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	Q₁₆×C₅₃	C₂×C₈	1	2	yes	—
76	Q₁₆×C₅₃	C₂×C₈	1	2	yes	—
77	SD₁₆×C₅₃	C₂×C₈	1	4	yes	—
78	SD₁₆×C₅₃	C₂×C₈	1	4	yes	—
79	D₈×C₅₃	C₂×C₈	1	2	yes	—
80	D₈×C₅₃	C₂×C₈	1	2	yes	—
81	M₁₆×C₅₃	—	2	2	no	—
82	M₁₆×C₅₃	—	2	4	no	—
83	M₁₆×C₅₃	—	2	4	no	—
84	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	2	4	no	—
87	C₂×C₈×C₅₃	—	2	4	no	—
88	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
98	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₄²×C₅₃	—	2	1	no	—
101	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	#11719908	—	2	4	no	—
104	#11719908	—	2	2	no	—
105	#11719908	C₂×D₄	1	4	yes	—
106	C₂³×C₅₃	—	2	2	no	—
107	C₂³×C₅₃	#131	1	1	yes	characteristic
108	Q₈×C₅₃	—	2	2	no	—
109	Q₈×C₅₃	C₂²×C₈	1	4	yes	—
110	D₄×C₅₃	—	2	4	no	—
111	D₄×C₅₃	—	2	4	no	—
112	D₄×C₅₃	—	2	2	no	—
113	D₄×C₅₃	C₂²×C₈	1	4	yes	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	2	4	no	—
117	C₂×C₄×C₅₃	—	2	4	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	C₈×C₅₃	—	2	4	no	—
127	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
128	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
129	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
130	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
131	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
132	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
133	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
134	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
135	#33443	—	53	1	no	maximal, Sylow
136	C₂²×C₅₃	—	2	4	no	—
137	C₂²×C₅₃	—	4	1	no	—
138	C₂²×C₅₃	—	8	2	no	—
139	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
140	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
141	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
142	C₄×C₅₃	—	2	2	no	—
143	C₄×C₅₃	—	2	4	no	—
144	C₄×C₅₃	—	4	1	no	—
145	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
146	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
147	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
149	#3407	—	53	2	no	—
150	C₂×#443	D₅₃	1	1	yes	characteristic
151	#2171	—	53	4	no	—
152	#2170	—	53	4	no	—
153	C₂×#364	—	53	1	no	—
154	#2117	—	53	2	no	—
155	C₄×M₃₂	—	53	1	no	—
156	C₂×C₅₃	—	2	1	no	—
157	C₂×C₅₃	—	2	1	no	—
158	C₂×C₅₃	—	2	1	no	—
159	C₂×C₅₃	—	4	4	no	—
160	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
161	C₂×#135	Dic₅₃	1	1	yes	characteristic
162	C₂×#131	C₂×D₅₃	1	2	yes	—
163	#504	—	53	8	no	—
164	#504	—	106	4	no	—
165	C₂×M₃₂	—	53	1	no	—
166	C₂×M₃₂	—	53	1	no	—
167	C₂×M₃₂	—	53	2	no	—
168	C₂×M₃₂	—	106	4	no	—
169	#443	Dic₅₃	1	2	yes	—
170	#443	Dic₅₃	1	2	yes	—
171	#443	C₂×D₅₃	1	2	yes	—
172	#443	C₂×D₅₃	1	2	yes	—
173	C₂×#108	Dic₅₃	1	1	yes	characteristic
174	C₂×#104	Dic₅₃	1	2	yes	—
175	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
176	#364	—	53	2	no	—
177	#364	—	53	2	no	—
178	#350	—	53	8	no	—
179	#346	—	53	2	no	—
180	C₄×C₁₆	—	53	2	no	—
181	C₅₃	#33443	1	1	yes	characteristic, complemented, Sylow
182	C₂×#41	C₂×Dic₅₃	1	2	yes	—
183	#135	#88322	1	2	yes	—
184	#135	#88322	1	2	yes	—
185	#135	C₂×Dic₅₃	1	2	yes	—
186	#135	C₂×Dic₅₃	1	2	yes	—
187	C₂×Q₁₆	#88322	1	1	yes	characteristic
188	C₂×SD₁₆	#88322	1	2	yes	—
189	C₂×D₈	#88322	1	1	yes	characteristic
190	#131	—	2	2	no	—
191	#131	—	2	2	no	—
192	#131	C₂×Dic₅₃	1	4	yes	—
193	#131	C₂²×D₅₃	1	4	yes	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
199	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
200	M₃₂	—	53	2	no	—
201	M₃₂	—	53	2	no	—
202	M₃₂	—	53	4	no	—
203	M₃₂	—	106	8	no	—
204	M₃₂	—	106	2	no	—
205	M₃₂	—	106	1	no	—
206	M₃₂	—	106	1	no	—
207	C₂×C₁₆	—	53	2	no	—
208	C₂×C₁₆	—	53	2	no	—
209	C₂×C₁₆	—	106	8	no	—
210	C₂×C₁₆	—	106	2	no	—
211	#108	C₂×Dic₅₃	1	2	yes	—
212	#108	C₂×Dic₅₃	1	2	yes	—
213	#104	C₂×Dic₅₃	1	4	yes	—
214	#104	C₂×Dic₅₃	1	4	yes	—
215	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
216	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	#41	C₂×#88322	1	4	yes	—
222	#41	C₂²×Dic₅₃	1	4	yes	—
223	C₂×Q₈	C₂×#88322	1	2	yes	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂×#88322	1	2	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
229	Q₁₆	C₂×#88322	1	2	yes	—
230	Q₁₆	C₂×#88322	1	2	yes	—
231	SD₁₆	C₂×#88322	1	4	yes	—
232	SD₁₆	C₂×#88322	1	4	yes	—
233	D₈	C₂×#88322	1	2	yes	—
234	D₈	C₂×#88322	1	2	yes	—
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	C₂²×Dic₅₃	1	4	yes	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	C₂×#88322	1	1	yes	characteristic
243	C₂×C₈	C₂×#88322	1	1	yes	characteristic
244	C₂×C₈	C₂×#88322	1	1	yes	characteristic
245	C₂×C₈	C₂×#88322	1	1	yes	characteristic
246	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
247	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
248	C₂×C₈	#11719947	1	1	yes	characteristic
249	C₂×C₈	#11719947	1	1	yes	characteristic
250	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
251	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
252	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
254	C₄²	—	2	1	no	—
255	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
256	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
257	C₁₆	—	53	4	no	—
258	C₁₆	—	106	4	no	—
259	C₁₆	—	106	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	#13133224	1	1	yes	characteristic
262	Q₈	—	2	2	no	—
263	Q₈	C₂²×#88322	1	4	yes	—
264	D₄	—	2	4	no	—
265	D₄	—	2	4	no	—
266	D₄	—	2	2	no	—
267	D₄	C₂²×#88322	1	4	yes	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
275	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
277	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
278	C₂×C₄	#13133224	1	1	yes	characteristic
279	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
280	C₈	—	2	4	no	—
281	C₈	D₄×Dic₅₃	1	1	yes	characteristic
282	C₈	D₄×Dic₅₃	1	1	yes	characteristic
283	C₈	C₂²×#88322	1	1	yes	characteristic
284	C₈	C₂²×#88322	1	1	yes	characteristic
285	C₈	C₂²×#88322	1	1	yes	characteristic
286	C₈	C₂²×#88322	1	1	yes	characteristic
287	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
288	C₈	C₂×#11719947	1	1	yes	characteristic
289	C₂²	—	2	4	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	D₄×#88322	1	1	yes	characteristic
293	C₂²	D₄×#88322	1	1	yes	characteristic
294	C₂²	C₂×#13133224	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	4	no	—
297	C₄	—	4	1	no	—
298	C₄	D₄×#88322	1	1	yes	characteristic
299	C₄	D₄×#88322	1	1	yes	characteristic
300	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
301	C₄	C₂³×#88322	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	4	4	no	—
306	C₂	C₂×D₄×#88322	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#443	C₂
socle	C₂×C₅₃	C₂×C₈×D₄
prime radical	C₂×#135×C₅₃	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#443	D₅₃
hypocenter	C₅₃	#33443
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. #20287818	C₂	semidirect
2. #20276557	C₂	semidirect
3. #20276552	C₂	semidirect
4. C₂×#16810444	C₂	semidirect
5. #20276343	C₂	semidirect
6. C₄×#13133157	C₂	semidirect
9. #16811040	C₂²	semidirect
14. #16810444	C₂²	semidirect
15. #16810444	C₂²	semidirect
16. #16810407	C₂²	semidirect
17. #16810399	C₂²	semidirect
18. C₄×#11719908	C₂²	semidirect
181. C₅₃	#33443	semidirect

Maximal subgroups

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

subgroup	index
#20287818	2
#20276557	2
#20276552	2
C₂×#16810444	2
#20276343	2
C₄×#13133157	2
C₂×C₅₃×#443	2
#33443	53