Subgroups — #27299291

Order 13568.

452 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276400	C₂	1	1	yes	maximal, characteristic
2	#20276380	C₂	1	1	yes	maximal, characteristic
3	#20274515	C₂	1	2	yes	maximal, complemented
4	#20272844	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2932	C₂	1	1	yes	maximal, characteristic
6	#16810607	C₂²	1	2	yes	—
7	#16810597	C₂²	1	2	yes	—
8	#16810578	—	2	2	no	—
9	#16810422	C₄	1	2	yes	complemented
10	#16810420	C₄	1	2	yes	complemented
11	#16810418	C₄	1	2	yes	—
12	#16810416	C₄	1	2	yes	—
13	#16810394	—	2	2	no	—
14	C₂×#131×C₅₃	C₄	1	1	yes	characteristic, complemented
15	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#532	C₂²	1	1	yes	characteristic
17	C₅₃×#435	—	2	2	no	—
18	C₅₃×#435	—	2	2	no	—
19	C₅₃×#423	C₄	1	1	yes	characteristic, complemented
20	C₅₃×#408	C₄	1	1	yes	characteristic, complemented
21	C₅₃×#408	C₂²	1	1	yes	characteristic
22	C₂×#97×C₅₃	C₄	1	1	yes	characteristic, complemented
23	C₂×#11719917	—	2	2	no	—
24	#13133176	—	2	2	no	—
25	#13133136	C₂×C₄	1	4	yes	—
26	#13133134	—	2	2	no	—
27	#13133132	—	2	2	no	—
28	#13133132	C₂×C₄	1	4	yes	—
29	#13133127	—	4	2	no	—
30	C₈×Dic₅₃	—	4	2	no	—
31	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
33	#131×C₅₃	—	4	2	no	—
34	#131×C₅₃	—	4	2	no	—
35	C₂×M₁₆×C₅₃	—	2	1	no	—
36	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
39	C₂²×C₈×C₅₃	—	2	1	no	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	D₄	1	2	yes	—
43	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
45	#124×C₅₃	C₂×C₄	1	1	yes	characteristic
46	#124×C₅₃	C₂×C₄	1	1	yes	characteristic
47	#121×C₅₃	—	2	2	no	—
48	C₄×D₄×C₅₃	—	2	2	no	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₈×C₅₃	—	2	2	no	—
51	C₂²⋊C₈×C₅₃	—	2	2	no	—
52	C₂²⋊C₈×C₅₃	—	2	2	no	—
53	#97×C₅₃	—	2	1	no	—
54	#97×C₅₃	—	2	1	no	—
55	#41×C₅₃	—	4	1	no	—
56	#41×C₅₃	—	4	1	no	—
57	#41×C₅₃	—	4	1	no	—
58	#41×C₅₃	—	4	1	no	—
59	C₂×Q₈×C₅₃	C₄²	1	1	yes	characteristic
60	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×D₄×C₅₃	—	2	1	no	—
62	C₂×D₄×C₅₃	—	2	2	no	—
63	C₂×D₄×C₅₃	C₄²	1	1	yes	characteristic
64	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×D₄×C₅₃	#41	1	2	yes	—
66	C₂²×C₄×C₅₃	—	2	1	no	—
67	C₂²×C₄×C₅₃	—	2	1	no	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
70	M₁₆×C₅₃	—	4	1	no	—
71	M₁₆×C₅₃	—	4	2	no	—
72	M₁₆×C₅₃	—	4	2	no	—
73	M₁₆×C₅₃	—	4	1	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	1	no	—
78	C₂×C₈×C₅₃	—	2	1	no	—
79	C₂×C₈×C₅₃	—	4	2	no	—
80	C₂×C₈×C₅₃	—	4	1	no	—
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₄⋊C₄×C₅₃	—	2	2	no	—
85	C₂²⋊C₄×C₅₃	—	2	2	no	—
86	C₂²⋊C₄×C₅₃	—	2	2	no	—
87	C₄²×C₅₃	—	2	1	no	—
88	C₄²×C₅₃	C₄²	1	1	yes	characteristic
89	C₄²×C₅₃	C₄²	1	1	yes	characteristic
90	C₂²⋊Dic₅₃	—	2	4	no	—
91	C₄⋊Dic₅₃	—	4	2	no	—
92	C₄×Dic₅₃	—	4	2	no	—
93	#11719917	—	2	4	no	—
94	#11719917	—	4	2	no	—
95	C₂×#88322	—	4	2	no	—
96	C₂³×C₅₃	—	2	1	no	—
97	C₂³×C₅₃	#117	1	1	yes	characteristic
98	C₂³×C₅₃	C₄×D₄	1	2	yes	—
99	Q₈×C₅₃	—	4	1	no	—
100	Q₈×C₅₃	—	4	1	no	—
101	D₄×C₅₃	—	4	1	no	—
102	D₄×C₅₃	—	4	2	no	—
103	D₄×C₅₃	—	4	1	no	—
104	D₄×C₅₃	—	4	2	no	—
105	D₄×C₅₃	—	8	2	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	4	1	no	—
114	C₂×C₄×C₅₃	—	4	1	no	—
115	C₂×C₄×C₅₃	—	8	1	no	—
116	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
119	C₂×Dic₅₃	—	4	4	no	—
120	C₈×C₅₃	—	2	1	no	—
121	C₈×C₅₃	—	2	2	no	—
122	C₈×C₅₃	—	4	1	no	—
123	C₈×C₅₃	—	4	1	no	—
124	C₈×C₅₃	#99	1	2	yes	—
125	#88322	—	4	4	no	—
126	#12634	—	53	1	no	maximal, Sylow
127	C₂²×C₅₃	—	2	2	no	—
128	C₂²×C₅₃	—	2	1	no	—
129	C₂²×C₅₃	—	4	1	no	—
130	C₂²×C₅₃	—	4	2	no	—
131	C₂²×C₅₃	—	4	2	no	—
132	C₂²×C₅₃	—	4	1	no	—
133	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₅₃	—	2	2	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	4	1	no	—
137	C₄×C₅₃	—	4	1	no	—
138	C₄×C₅₃	—	4	1	no	—
139	C₄×C₅₃	C₂×#99	1	1	yes	characteristic
140	C₄×C₅₃	#410	1	1	yes	characteristic
141	Dic₅₃	—	8	4	no	—
142	#2932	D₅₃	1	1	yes	characteristic
143	#2132	—	53	1	no	—
144	#2126	—	53	1	no	—
145	#1757	—	53	2	no	—
146	#1395	—	53	2	no	—
147	C₂×C₅₃	—	2	1	no	—
148	C₂×C₅₃	—	4	2	no	—
149	C₂×C₅₃	—	4	1	no	—
150	C₂×C₅₃	—	8	1	no	—
151	C₂×C₅₃	C₄×#99	1	1	yes	characteristic
152	C₂×#131	Dic₅₃	1	1	yes	characteristic, complemented
153	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
154	#532	C₂×D₅₃	1	1	yes	characteristic
155	#435	—	2	2	no	—
156	#435	—	2	2	no	—
157	#423	Dic₅₃	1	1	yes	characteristic, complemented
158	#413	—	53	2	no	—
159	#410	—	53	2	no	—
160	#408	Dic₅₃	1	1	yes	characteristic, complemented
161	#408	C₂×D₅₃	1	1	yes	characteristic
162	#405	—	106	2	no	—
163	C₂×#97	Dic₅₃	1	1	yes	characteristic, complemented
164	#356	—	53	2	no	—
165	#355	—	53	2	no	—
166	#354	—	53	2	no	—
167	#353	—	53	2	no	—
168	#344	—	106	2	no	—
169	C₅₃	#12634	1	1	yes	characteristic, complemented, Sylow
170	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
171	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
172	#131	—	4	2	no	—
173	#131	—	4	2	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	—	106	2	no	—
176	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
177	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
178	C₂×M₁₆	C₂²⋊D₅₃	1	2	yes	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
181	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
183	C₄⋊Q₈	C₄×D₅₃	1	1	yes	characteristic
184	#127	C₄×D₅₃	1	1	yes	characteristic
185	#124	C₄×D₅₃	1	1	yes	characteristic
186	#124	C₄×D₅₃	1	1	yes	characteristic
187	#121	—	2	2	no	—
188	C₄×D₄	—	2	2	no	—
189	#117	—	106	2	no	—
190	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
191	C₄⋊C₈	—	2	2	no	—
192	#101	—	53	4	no	—
193	#100	—	106	2	no	—
194	#99	—	53	4	no	—
195	#99	—	106	2	no	—
196	C₂²⋊C₈	—	2	2	no	—
197	C₂²⋊C₈	—	2	2	no	—
198	#97	—	2	1	no	—
199	#97	—	2	1	no	—
200	#97	—	212	2	no	—
201	C₄×C₈	—	212	2	no	—
202	#41	—	4	1	no	—
203	#41	—	4	1	no	—
204	#41	—	4	1	no	—
205	#41	—	4	1	no	—
206	C₂×Q₈	C₄×Dic₅₃	1	1	yes	characteristic
207	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	C₄×Dic₅₃	1	1	yes	characteristic
211	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
212	C₂×D₄	#11719945	1	2	yes	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
217	M₁₆	—	4	1	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	106	4	no	—
222	M₁₆	—	212	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	212	2	no	—
231	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
233	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	212	2	no	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	106	4	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	212	2	no	—
241	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
242	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
243	C₂³	—	2	1	no	—
244	C₂³	#13133176	1	1	yes	characteristic
245	C₂³	C₄×C₂²⋊D₅₃	1	2	yes	—
246	Q₈	—	4	1	no	—
247	Q₈	—	4	1	no	—
248	D₄	—	4	1	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	1	no	—
251	D₄	—	4	2	no	—
252	D₄	—	8	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	8	1	no	—
263	C₂×C₄	—	212	4	no	—
264	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
266	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
267	C₈	—	2	1	no	—
268	C₈	—	2	2	no	—
269	C₈	—	4	1	no	—
270	C₈	—	4	1	no	—
271	C₈	—	212	4	no	—
272	C₈	#13133132	1	2	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	1	no	—
279	C₂²	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	1	no	—
285	C₄	—	424	4	no	—
286	C₄	C₂×#13133132	1	1	yes	characteristic
287	C₄	#16810597	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	4	2	no	—
290	C₂	—	4	1	no	—
291	C₂	—	8	1	no	—
292	C₂	C₄×#13133132	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13133132
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×#41	C₂²×D₅₃
Fitting	C₅₃×#2932	C₂
socle	C₂×C₅₃	C₄×#99
prime radical	C₅₃×#532	C₂²
cosocle	C₂×#41×C₅₃	C₂³
hypercenter	#2932	D₅₃
hypocenter	C₅₃	#12634
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×#131 ◁ #2932

Complements

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

N	H ≅ G/N	type
3. #20274515	C₂	semidirect
4. #20272844	C₂	semidirect
9. #16810422	C₄	semidirect
10. #16810420	C₄	semidirect
14. C₂×#131×C₅₃	C₄	semidirect
19. C₅₃×#423	C₄	semidirect
20. C₅₃×#408	C₄	semidirect
22. C₂×#97×C₅₃	C₄	semidirect
152. C₂×#131	Dic₅₃	semidirect
157. #423	Dic₅₃	semidirect
160. #408	Dic₅₃	semidirect
163. C₂×#97	Dic₅₃	semidirect
169. C₅₃	#12634	semidirect

Maximal subgroups

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

subgroup	index
#20276400	2
#20276380	2
#20274515	2
#20272844	2
C₅₃×#2932	2
#12634	53