Subgroups — #28326755

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291509	C₂	1	2	yes	maximal
2	#20290891	C₂	1	2	yes	maximal
3	#20290824	C₂	1	2	yes	maximal
4	#20290783	C₂	1	4	yes	maximal
5	C₄×#13133287	C₂	1	1	yes	maximal, characteristic
6	#20290522	C₂	1	2	yes	maximal
7	#20290509	C₂	1	1	yes	maximal, characteristic
8	#20285226	C₂	1	1	yes	maximal, characteristic
9	#16811675	—	2	4	no	—
10	#16811674	—	2	2	no	—
11	Q₈×#88324	—	2	2	no	—
12	C₂×#13133287	C₂²	1	1	yes	characteristic
13	C₂×#13133287	C₂²	1	1	yes	characteristic
14	#16811630	C₂²	1	4	yes	—
15	D₅₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
16	#16811588	C₂²	1	4	yes	—
17	C₄×#11719936	C₂²	1	2	yes	—
18	C₂×#13133271	C₂²	1	2	yes	—
19	C₄×C₄⋊#6157	C₂²	1	1	yes	characteristic
20	#16811499	C₂²	1	2	yes	—
21	#16811493	C₂²	1	4	yes	—
22	#16811493	C₂²	1	4	yes	—
23	(C₂×#6157)⋊C₈	C₂²	1	2	yes	—
24	#16811178	C₂²	1	2	yes	—
25	#16811028	C₄	1	1	yes	characteristic, complemented
26	#16811024	C₄	1	2	yes	complemented
27	#16811024	C₄	1	2	yes	complemented
28	#16811019	C₄	1	2	yes	complemented
29	#16810660	C₂²	1	2	yes	—
30	#16810660	C₂²	1	2	yes	—
31	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#500	C₄	1	1	yes	characteristic, complemented
33	C₂×#11719936	C₂³	1	2	yes	—
34	C₂×#11719936	C₂³	1	2	yes	—
35	C₂×D₅₃⋊C₈	C₂³	1	2	yes	—
36	C₂×C₄⋊#6157	—	2	1	no	—
37	C₂×C₄×#6157	C₂³	1	2	yes	—
38	#13133287	—	2	2	no	—
39	#13133287	Q₈	1	4	yes	complemented
40	#13133284	—	2	4	no	—
41	#13133284	D₄	1	2	yes	—
42	C₄⋊#88324	—	2	4	no	—
43	C₄⋊#88324	D₄	1	2	yes	—
44	#13133272	—	2	2	no	—
45	#13133272	—	2	4	no	—
46	#13133271	—	2	4	no	—
47	C₄×#88324	—	2	2	no	—
48	C₄×#88324	—	2	4	no	—
49	(C₂×#6157)⋊C₄	—	2	1	no	—
50	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
52	#13133217	C₂×C₄	1	2	yes	—
53	#13133208	C₂×C₄	1	4	yes	—
54	#13133197	—	2	2	no	—
55	#13133193	—	2	2	no	—
56	Q₈×Dic₅₃	—	2	2	no	—
57	#13133182	—	2	2	no	—
58	#13133175	C₂³	1	4	yes	—
59	C₄²×D₅₃	C₂³	1	1	yes	characteristic
60	#13133151	C₂×C₄	1	2	yes	—
61	#13133151	C₂×C₄	1	2	yes	—
62	#13133148	C₂×C₄	1	4	yes	—
63	#13133148	C₂×C₄	1	4	yes	—
64	C₄×#88322	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
67	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
68	#107×C₅₃	C₂×C₄	1	2	yes	—
69	#107×C₅₃	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂²×#6157	—	2	2	no	—
72	C₂×Q₈×C₅₃	—	2	2	no	—
73	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
74	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
75	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
76	C₂×Dic₁₀₆	—	2	2	no	—
77	C₂×#88324	—	2	2	no	—
78	C₂×#88324	—	2	2	no	—
79	C₂×#88324	C₂×D₄	1	4	yes	—
80	C₄⋊#6157	—	4	2	no	—
81	C₄×#6157	—	2	2	no	—
82	C₄×#6157	—	2	2	no	—
83	#11719936	—	2	2	no	—
84	#11719936	—	2	2	no	—
85	#11719936	—	2	2	no	—
86	#11719936	C₂×Q₈	1	4	yes	—
87	D₅₃⋊C₈	—	2	2	no	—
88	D₅₃⋊C₈	—	2	2	no	—
89	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
93	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	#11719921	—	2	4	no	—
95	C₄⋊Dic₅₃	—	2	2	no	—
96	C₄⋊Dic₅₃	—	2	4	no	—
97	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
98	#11719919	—	2	4	no	—
99	C₄×Dic₅₃	—	2	4	no	—
100	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
102	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
104	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
105	C₈×D₅₃	—	2	1	no	—
106	C₈×D₅₃	—	2	1	no	—
107	C₈×D₅₃	C₂×Q₈	1	2	yes	—
108	C₈×D₅₃	#41	1	2	yes	—
109	C₂²×D₅₃	C₂×#41	1	1	yes	characteristic
110	C₂×#6157	—	4	2	no	—
111	C₂×#6157	—	4	2	no	—
112	C₂×#6157	—	4	2	no	—
113	Q₈×C₅₃	—	4	4	no	—
114	C₂×C₄×C₅₃	—	2	4	no	—
115	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×Dic₅₃	—	2	4	no	—
119	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
120	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
122	C₄×D₅₃	—	2	2	no	—
123	C₄×D₅₃	—	4	1	no	—
124	C₄×D₅₃	C₂²×Q₈	1	1	yes	characteristic
125	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
126	C₄×D₅₃	2₋¹⁺⁴	1	1	yes	characteristic
127	C₄×D₅₃	2₋¹⁺⁴	1	1	yes	characteristic
128	Dic₁₀₆	—	8	2	no	—
129	#88324	—	2	4	no	—
130	#88324	—	2	4	no	—
131	#88324	—	4	2	no	—
132	C₈×C₅₃	—	2	1	no	—
133	C₈×C₅₃	C₄×Q₈	1	2	yes	—
134	#88322	—	2	1	no	—
135	#88322	C₄×Q₈	1	2	yes	—
136	#23171	—	53	1	no	maximal, Sylow
137	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂×D₅₃	—	2	1	no	—
139	C₂×D₅₃	—	2	1	no	—
140	C₂×D₅₃	#556	1	1	yes	characteristic
141	C₂×D₅₃	#557	1	1	yes	characteristic
142	#6157	—	8	2	no	—
143	C₄×C₅₃	—	2	1	no	—
144	C₄×C₅₃	—	4	4	no	—
145	C₄×C₅₃	#444	1	2	yes	—
146	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
147	C₄×C₅₃	#519	1	1	yes	characteristic
148	Dic₅₃	—	2	1	no	—
149	Dic₅₃	—	2	1	no	—
150	Dic₅₃	—	4	4	no	—
151	Dic₅₃	#547	1	1	yes	characteristic
152	Dic₅₃	D₄×Q₈	1	1	yes	characteristic
153	#3160	—	53	1	no	—
154	#2962	—	53	2	no	—
155	#1986	—	53	2	no	—
156	#1946	—	53	2	no	—
157	#1922	—	53	4	no	—
158	C₄×#108	—	53	1	no	—
159	#1773	—	53	2	no	—
160	#1770	—	53	1	no	—
161	C₂×C₅₃	#2365	1	1	yes	characteristic
162	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
163	C₂×C₅₃	#2957	1	1	yes	characteristic
164	D₅₃	—	2	1	no	—
165	D₅₃	—	2	1	no	—
166	#533	—	53	2	no	—
167	#500	—	53	1	no	—
168	#500	#6157	1	1	yes	characteristic, complemented
169	#499	—	53	2	no	—
170	#499	—	53	2	no	—
171	#498	—	53	2	no	—
172	#446	—	106	2	no	—
173	#446	—	106	4	no	—
174	C₈×Q₈	—	106	2	no	—
175	C₂×#108	—	53	1	no	—
176	C₂×#108	—	53	1	no	—
177	#427	—	53	2	no	—
178	#427	—	53	2	no	—
179	#424	—	53	4	no	—
180	C₂×C₄⋊C₈	—	53	1	no	—
181	#405	—	53	4	no	—
182	C₄×M₁₆	—	53	2	no	—
183	C₂×#97	—	53	2	no	—
184	C₂×C₄×C₈	—	53	1	no	—
185	C₄×C₄⋊C₄	—	53	1	no	—
186	#341	—	53	2	no	—
187	#339	—	53	4	no	—
188	#339	—	53	4	no	—
189	#336	—	53	2	no	—
190	C₅₃	#23171	1	1	yes	characteristic, complemented, Sylow
191	C₂×M₁₆	—	53	2	no	—
192	C₂×M₁₆	—	53	2	no	—
193	C₂²×C₈	—	53	1	no	—
194	C₂²×C₈	—	53	2	no	—
195	C₂²×C₈	—	53	1	no	—
196	C₄⋊Q₈	—	53	2	no	—
197	C₄⋊Q₈	C₂×#6157	1	2	yes	—
198	#125	—	53	4	no	—
199	C₂²⋊Q₈	—	106	2	no	—
200	C₂²⋊Q₈	—	106	2	no	—
201	C₄×Q₈	—	106	2	no	—
202	C₄×Q₈	—	106	2	no	—
203	#117	—	53	4	no	—
204	C₂×C₄⋊C₄	—	106	1	no	—
205	C₂×C₄²	—	53	1	no	—
206	C₂×C₄²	—	53	2	no	—
207	#108	—	53	4	no	—
208	#108	—	106	2	no	—
209	#107	—	53	2	no	—
210	#107	—	53	2	no	—
211	#107	C₂×#6157	1	2	yes	—
212	#107	C₂×#6157	1	2	yes	—
213	#106	—	53	4	no	—
214	#106	—	53	4	no	—
215	C₄⋊C₈	—	53	2	no	—
216	C₄⋊C₈	—	53	2	no	—
217	C₄⋊C₈	—	106	4	no	—
218	C₄⋊C₈	—	106	4	no	—
219	#97	—	106	2	no	—
220	#97	—	106	4	no	—
221	#97	—	106	4	no	—
222	C₄×C₈	—	53	1	no	—
223	C₄×C₈	—	53	1	no	—
224	C₄×C₈	—	53	1	no	—
225	C₄×C₈	—	106	2	no	—
226	C₄×C₈	—	106	4	no	—
227	C₄×C₈	C₂×#6157	1	1	yes	characteristic
228	#95	—	106	1	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	106	2	no	—
231	C₂²×C₄	—	53	1	no	—
232	C₂²×C₄	—	53	1	no	—
233	C₂²×C₄	—	53	1	no	—
234	C₂²×C₄	—	106	2	no	—
235	M₁₆	—	53	4	no	—
236	M₁₆	—	106	2	no	—
237	M₁₆	—	106	2	no	—
238	M₁₆	—	106	2	no	—
239	C₂×C₈	—	53	1	no	—
240	C₂×C₈	—	53	2	no	—
241	C₂×C₈	—	53	2	no	—
242	C₂×C₈	—	53	4	no	—
243	C₂×C₈	—	53	1	no	—
244	C₂×C₈	—	106	1	no	—
245	C₂×C₈	—	106	1	no	—
246	C₂×C₈	—	106	2	no	—
247	C₂×C₈	—	106	2	no	—
248	C₂×C₈	—	106	2	no	—
249	C₂×C₈	—	106	2	no	—
250	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
251	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	53	4	no	—
254	C₄⋊C₄	—	106	4	no	—
255	C₄⋊C₄	—	106	4	no	—
256	C₄⋊C₄	—	106	2	no	—
257	C₄⋊C₄	—	212	2	no	—
258	C₄⋊C₄	C₂²×#6157	1	4	yes	—
259	C₂²⋊C₄	—	106	4	no	—
260	C₄²	—	53	1	no	—
261	C₄²	—	53	1	no	—
262	C₄²	—	53	1	no	—
263	C₄²	—	106	4	no	—
264	C₄²	—	106	2	no	—
265	C₄²	—	106	2	no	—
266	C₄²	C₂²×#6157	1	1	yes	characteristic
267	C₂³	—	53	1	no	—
268	Q₈	—	4	4	no	—
269	Q₈	—	424	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	53	1	no	—
272	C₂×C₄	—	53	1	no	—
273	C₂×C₄	—	53	1	no	—
274	C₂×C₄	—	53	1	no	—
275	C₂×C₄	—	53	1	no	—
276	C₂×C₄	—	53	1	no	—
277	C₂×C₄	—	53	1	no	—
278	C₂×C₄	—	106	2	no	—
279	C₂×C₄	—	106	4	no	—
280	C₂×C₄	—	212	1	no	—
281	C₂×C₄	—	212	2	no	—
282	C₂×C₄	—	212	2	no	—
283	C₂×C₄	—	212	2	no	—
284	C₂×C₄	D₄×#6157	1	1	yes	characteristic
285	C₂×C₄	D₄×#6157	1	1	yes	characteristic
286	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
287	C₈	—	2	1	no	—
288	C₈	—	53	2	no	—
289	C₈	—	106	4	no	—
290	C₈	—	106	4	no	—
291	C₈	—	106	1	no	—
292	C₈	—	212	2	no	—
293	C₈	Q₈×#6157	1	2	yes	—
294	C₂²	—	53	1	no	—
295	C₂²	—	53	1	no	—
296	C₂²	—	106	1	no	—
297	C₂²	—	106	1	no	—
298	C₂²	C₂×D₄×#6157	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	4	4	no	—
301	C₄	—	53	1	no	—
302	C₄	—	53	1	no	—
303	C₄	—	106	1	no	—
304	C₄	—	106	1	no	—
305	C₄	—	424	2	no	—
306	C₄	—	212	4	no	—
307	C₄	#16811670	1	2	yes	—
308	C₄	C₂×Q₈×#6157	1	1	yes	characteristic
309	C₄	#16811691	1	1	yes	characteristic
310	C₂	—	106	1	no	—
311	C₂	—	106	1	no	—
312	C₂	#20291244	1	1	yes	characteristic
313	C₂	C₂×#16811670	1	1	yes	characteristic
314	C₂	#20291495	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#500	C₄
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₅₃	C₂×#41
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	#500	#6157
hypocenter	C₅₃	#23171
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
25. #16811028	C₄	semidirect
26. #16811024	C₄	semidirect
27. #16811024	C₄	semidirect
28. #16811019	C₄	semidirect
32. C₅₃×#500	C₄	semidirect
39. #13133287	Q₈	semidirect
168. #500	#6157	semidirect
190. C₅₃	#23171	semidirect

Maximal subgroups

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

subgroup	index
#20291509	2
#20290891	2
#20290824	2
#20290783	2
C₄×#13133287	2
#20290522	2
#20290509	2
#20285226	2
#23171	53