Subgroups — #27539641

Order 13568.

526 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285283	C₂	1	2	yes	maximal, complemented
2	#20276992	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276973	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276945	C₂	1	2	yes	maximal, complemented
5	#20276810	C₂	1	2	yes	maximal, complemented
6	#20276807	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276798	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13133161	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811155	—	2	2	no	—
13	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic
14	C₂×#13133161	C₂²	1	2	yes	—
15	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
17	#16811024	C₂²	1	2	yes	—
18	#16811006	C₂²	1	2	yes	—
19	#16810976	—	2	2	no	—
20	#16810865	C₂²	1	2	yes	—
21	#16810856	C₂²	1	2	yes	—
22	#16810851	—	2	2	no	—
23	C₄×Dic₂₁₂	C₂²	1	2	yes	—
24	C₄×#11719923	—	2	2	no	—
25	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
26	#16810450	C₂²	1	2	yes	—
27	#16810449	C₂²	1	1	yes	characteristic
28	#16810428	C₂²	1	2	yes	—
29	#16810426	C₂²	1	2	yes	—
30	#16810424	C₂²	1	2	yes	—
31	C₄×#11719908	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
34	C₂×D₁₆×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×D₈×C₅₃	C₂²	1	2	yes	—
36	C₅₃×#366	C₂²	1	1	yes	characteristic
37	C₅₃×#357	C₂²	1	2	yes	—
38	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×Dic₂₁₂	C₂³	1	2	yes	—
40	C₂×#11719923	—	2	2	no	—
41	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
42	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
43	#13133210	—	4	2	no	—
44	#13133208	—	2	2	no	—
45	#13133204	—	2	2	no	—
46	#13133199	—	4	2	no	—
47	#13133197	—	2	2	no	—
48	#13133188	—	2	2	no	—
49	C₄×Dic₁₀₆	—	2	2	no	—
50	C₄×C₂²⋊D₅₃	—	4	2	no	—
51	#13133176	—	4	2	no	—
52	C₄²×D₅₃	D₄	1	1	yes	characteristic
53	Dic₄₂₄	—	2	2	no	—
54	#13133161	—	2	4	no	—
55	C₂×#11719908	C₂³	1	1	yes	characteristic
56	C₂×#11719908	C₂³	1	1	yes	characteristic
57	C₁₆×D₅₃	—	2	2	no	—
58	C₁₆×D₅₃	—	2	2	no	—
59	#13133151	C₂³	1	2	yes	—
60	#13133148	—	2	2	no	—
61	#13133142	—	2	2	no	—
62	#13133140	—	2	2	no	—
63	#13133138	—	2	2	no	—
64	C₄×#88322	D₄	1	1	yes	characteristic
65	C₈×Dic₅₃	D₄	1	1	yes	characteristic
66	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
67	C₂×D₈×C₅₃	C₂³	1	2	yes	—
68	C₄×D₄×C₅₃	—	2	2	no	—
69	D₁₆×C₅₃	—	2	2	no	—
70	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
71	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
72	#107×C₅₃	C₂³	1	2	yes	—
73	#102×C₅₃	—	2	2	no	—
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×D₄×C₅₃	—	2	2	no	—
76	C₂²×C₄×C₅₃	—	4	2	no	—
77	C₂×C₂²⋊D₅₃	—	4	2	no	—
78	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
79	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
80	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂×Dic₁₀₆	—	2	2	no	—
82	D₈×C₅₃	—	2	2	no	—
83	D₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₅₃	—	2	2	no	—
87	C₂²⋊C₄×C₅₃	—	4	2	no	—
88	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²⋊Dic₅₃	—	4	2	no	—
90	C₂²⋊Dic₅₃	—	4	2	no	—
91	C₂²⋊Dic₅₃	—	4	2	no	—
92	#11719923	—	4	4	no	—
93	#11719921	—	4	2	no	—
94	C₄⋊Dic₅₃	—	2	2	no	—
95	C₄⋊Dic₅₃	—	4	2	no	—
96	C₄⋊Dic₅₃	—	4	2	no	—
97	#11719919	—	4	2	no	—
98	#11719919	—	4	2	no	—
99	C₄×Dic₅₃	—	4	2	no	—
100	C₄×Dic₅₃	D₈	1	1	yes	characteristic
101	C₄×Dic₅₃	D₈	1	1	yes	characteristic
102	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×#88322	C₂×D₄	1	1	yes	characteristic
104	C₂×#88322	C₂×D₄	1	1	yes	characteristic
105	Dic₂₁₂	—	2	2	no	—
106	Dic₂₁₂	—	2	2	no	—
107	C₈×D₅₃	—	2	1	no	—
108	C₈×D₅₃	—	2	1	no	—
109	C₈×D₅₃	—	2	2	no	—
110	C₁₆×C₅₃	—	2	1	no	—
111	C₁₆×C₅₃	#41	1	2	yes	—
112	#11719908	—	2	1	no	—
113	#11719908	#41	1	2	yes	—
114	C₂³×C₅₃	—	4	2	no	—
115	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
116	D₄×C₅₃	—	4	2	no	—
117	D₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	4	2	no	—
119	C₂×C₄×C₅₃	—	8	4	no	—
120	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	C₂²⋊D₅₃	—	8	4	no	—
124	C₂×Dic₅₃	—	4	2	no	—
125	C₂×Dic₅₃	—	4	2	no	—
126	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
127	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
128	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
129	C₄×D₅₃	—	2	2	no	—
130	C₄×D₅₃	—	2	1	no	—
131	C₄×D₅₃	—	2	1	no	—
132	C₄×D₅₃	—	2	2	no	—
133	Dic₁₀₆	—	4	2	no	—
134	Dic₁₀₆	—	4	2	no	—
135	C₈×C₅₃	—	2	1	no	—
136	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
137	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
138	#88322	—	2	1	no	—
139	#88322	#135	1	2	yes	—
140	#33735	—	53	1	no	maximal, Sylow
141	C₂²×C₅₃	—	8	2	no	—
142	C₂²×C₅₃	—	8	2	no	—
143	C₂²×C₅₃	—	8	2	no	—
144	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₅₃	—	2	1	no	—
146	C₂×D₅₃	—	2	1	no	—
147	C₂×D₅₃	#508	1	2	yes	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	8	2	no	—
150	C₄×C₅₃	#508	1	2	yes	—
151	C₄×C₅₃	#532	1	1	yes	characteristic
152	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
153	Dic₅₃	—	2	1	no	—
154	Dic₅₃	—	2	1	no	—
155	Dic₅₃	—	2	1	no	—
156	Dic₅₃	—	8	2	no	—
157	Dic₅₃	—	8	2	no	—
158	#3168	—	53	2	no	—
159	#2253	—	53	1	no	—
160	#2249	—	53	1	no	—
161	#2242	—	53	2	no	—
162	#2217	—	53	2	no	—
163	#2216	—	53	1	no	—
164	#2215	—	53	1	no	—
165	C₄×Q₃₂	—	53	1	no	—
166	C₄×SD₃₂	—	53	2	no	—
167	C₄×D₁₆	D₅₃	1	1	yes	characteristic, complemented
168	C₂×C₄×C₁₆	—	53	1	no	—
169	C₂×C₅₃	—	16	2	no	—
170	C₂×C₅₃	#3169	1	1	yes	characteristic
171	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
172	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
173	D₅₃	—	2	2	no	—
174	#529	—	106	2	no	—
175	C₂×Q₃₂	—	53	1	no	—
176	C₂×SD₃₂	—	53	2	no	—
177	C₂×D₁₆	C₂×D₅₃	1	1	yes	characteristic
178	C₂²×C₁₆	—	53	1	no	—
179	C₂²×C₁₆	—	53	1	no	—
180	#499	—	53	2	no	—
181	#495	—	53	2	no	—
182	#488	—	106	2	no	—
183	#467	—	53	2	no	—
184	#466	—	53	2	no	—
185	#465	—	106	2	no	—
186	C₄×Q₁₆	—	53	2	no	—
187	C₄×SD₁₆	—	106	2	no	—
188	C₄×D₈	C₂×D₅₃	1	2	yes	—
189	C₂×C₄×C₈	—	53	1	no	—
190	#367	—	53	2	no	—
191	#366	—	53	1	no	—
192	#366	C₂×D₅₃	1	1	yes	characteristic
193	#358	—	53	2	no	—
194	#358	—	53	2	no	—
195	#357	—	53	2	no	—
196	#357	C₂×D₅₃	1	2	yes	—
197	C₄×C₁₆	—	53	1	no	—
198	C₄×C₁₆	—	53	1	no	—
199	C₄×C₁₆	—	53	1	no	—
200	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
201	C₅₃	#33735	1	1	yes	characteristic, complemented, Sylow
202	C₂×Q₁₆	—	53	2	no	—
203	C₂×SD₁₆	—	106	2	no	—
204	C₂×D₈	C₂²×D₅₃	1	2	yes	—
205	C₂²×C₈	—	53	1	no	—
206	C₂²×C₈	—	53	1	no	—
207	#126	—	212	2	no	—
208	#125	—	106	2	no	—
209	#124	—	106	2	no	—
210	#123	—	212	2	no	—
211	C₂²⋊Q₈	—	106	2	no	—
212	#121	—	106	2	no	—
213	C₄×Q₈	—	106	2	no	—
214	C₄×D₄	—	2	2	no	—
215	C₄×D₄	—	212	2	no	—
216	#117	—	212	2	no	—
217	C₂×C₄²	—	53	1	no	—
218	Q₃₂	—	106	2	no	—
219	SD₃₂	—	106	4	no	—
220	D₁₆	—	2	2	no	—
221	C₂×C₁₆	—	53	1	no	—
222	C₂×C₁₆	—	53	1	no	—
223	C₂×C₁₆	—	106	2	no	—
224	C₂×C₁₆	—	106	2	no	—
225	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
226	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
227	#107	—	53	2	no	—
228	#107	C₂²×D₅₃	1	2	yes	—
229	#106	—	106	2	no	—
230	#103	—	106	2	no	—
231	#103	—	106	2	no	—
232	#102	—	2	2	no	—
233	#102	—	106	2	no	—
234	C₄×C₈	—	53	1	no	—
235	C₄×C₈	—	53	1	no	—
236	C₄×C₈	—	53	1	no	—
237	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
238	C₂×Q₈	—	106	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	212	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	53	1	no	—
243	C₂²×C₄	—	53	1	no	—
244	C₂²×C₄	—	53	1	no	—
245	Q₁₆	—	106	2	no	—
246	Q₁₆	—	106	2	no	—
247	SD₁₆	—	212	4	no	—
248	D₈	—	2	2	no	—
249	D₈	—	2	2	no	—
250	C₂×C₈	—	53	1	no	—
251	C₂×C₈	—	53	1	no	—
252	C₂×C₈	—	106	2	no	—
253	C₂×C₈	—	106	1	no	—
254	C₂×C₈	—	106	1	no	—
255	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
256	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	106	2	no	—
259	C₄⋊C₄	—	212	2	no	—
260	C₄⋊C₄	—	212	2	no	—
261	C₄⋊C₄	—	212	2	no	—
262	C₄⋊C₄	—	212	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	212	2	no	—
265	C₂²⋊C₄	—	212	2	no	—
266	C₂²⋊C₄	—	212	2	no	—
267	C₂²⋊C₄	—	212	2	no	—
268	C₄²	—	53	1	no	—
269	C₄²	—	53	1	no	—
270	C₄²	—	53	1	no	—
271	C₄²	—	212	2	no	—
272	C₄²	D₄×D₅₃	1	1	yes	characteristic
273	C₁₆	—	2	1	no	—
274	C₁₆	—	53	2	no	—
275	C₁₆	—	106	1	no	—
276	C₁₆	#11719945	1	2	yes	—
277	C₂³	—	4	2	no	—
278	C₂³	—	53	1	no	—
279	Q₈	—	212	2	no	—
280	Q₈	—	212	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	424	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	4	no	—
286	C₂×C₄	—	53	1	no	—
287	C₂×C₄	—	53	1	no	—
288	C₂×C₄	—	53	1	no	—
289	C₂×C₄	—	106	2	no	—
290	C₂×C₄	—	106	2	no	—
291	C₂×C₄	—	106	1	no	—
292	C₂×C₄	—	106	1	no	—
293	C₂×C₄	—	212	2	no	—
294	C₂×C₄	—	212	2	no	—
295	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
296	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	53	2	no	—
300	C₈	—	106	1	no	—
301	C₈	#41×D₅₃	1	1	yes	characteristic
302	C₈	C₂×#11719945	1	1	yes	characteristic
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	53	2	no	—
307	C₂²	—	106	1	no	—
308	C₂²	—	106	1	no	—
309	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	8	2	no	—
312	C₄	—	106	1	no	—
313	C₄	—	106	1	no	—
314	C₄	—	106	1	no	—
315	C₄	—	424	2	no	—
316	C₄	—	424	2	no	—
317	C₄	#16811052	1	2	yes	—
318	C₄	#16811170	1	1	yes	characteristic
319	C₄	#135×D₅₃	1	1	yes	characteristic
320	C₂	—	16	2	no	—
321	C₂	—	106	2	no	—
322	C₂	#20285291	1	1	yes	characteristic
323	C₂	#508×D₅₃	1	1	yes	characteristic
324	C₂	C₂×#16811052	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×D₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20276798	C₂
hypercenter	C₄×D₁₆	D₅₃
hypocenter	C₅₃	#33735
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
1. #20285283	C₂	semidirect
2. #20276992	C₂	semidirect
3. #20276973	C₂	semidirect
4. #20276945	C₂	semidirect
5. #20276810	C₂	semidirect
6. #20276807	C₂	semidirect
8. C₄×Dic₄₂₄	C₂	semidirect
9. C₄×#13133161	C₂	semidirect
10. C₄×C₁₆×D₅₃	C₂	semidirect
11. C₄×D₁₆×C₅₃	C₂	semidirect
167. C₄×D₁₆	D₅₃	semidirect
201. C₅₃	#33735	semidirect

Maximal subgroups

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

subgroup	index
#20285283	2
#20276992	2
#20276973	2
#20276945	2
#20276810	2
#20276807	2
#20276798	2
C₄×Dic₄₂₄	2
C₄×#13133161	2
C₄×C₁₆×D₅₃	2
C₄×D₁₆×C₅₃	2
#33735	53