Subgroups — #27537130

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810661	C₂	1	1	yes	maximal, characteristic
3	#20276504	C₂	1	4	yes	maximal, complemented
4	#20276501	C₂	1	4	yes	maximal, complemented
5	#20276497	C₂	1	4	yes	maximal, complemented
6	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133176	—	2	1	no	—
8	C₂×#13133157	C₄	1	8	yes	complemented
9	C₂×#13133157	C₂²	1	2	yes	—
10	C₂×#13133157	C₂²	1	2	yes	—
11	C₂²×#11719908	C₂²	1	2	yes	—
12	#16810661	—	2	1	no	—
13	#16810661	—	2	1	no	—
14	#16810661	C₂²	1	2	yes	—
15	#16810661	C₂²	1	2	yes	—
16	C₂×#13133151	C₂²	1	2	yes	—
17	C₂×#13133149	—	2	1	no	—
18	#16810450	C₂²	1	8	yes	complemented
19	#16810448	C₂²	1	8	yes	complemented
20	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#117×C₅₃	—	2	1	no	—
22	C₅₃×#427	—	2	1	no	—
23	C₅₃×#427	—	2	1	no	—
24	C₅₃×#427	C₂²	1	2	yes	—
25	C₅₃×#427	C₂²	1	2	yes	—
26	C₂×#107×C₅₃	C₂²	1	2	yes	—
27	C₂×#106×C₅₃	—	2	1	no	—
28	#13133176	—	2	2	no	—
29	#13133176	—	2	2	no	—
30	#13133176	—	4	1	no	—
31	#13133176	—	4	1	no	—
32	C₂×C₄⋊Dic₅₃	—	2	2	no	—
33	C₂×C₂²⋊Dic₅₃	—	4	1	no	—
34	C₂×C₄×Dic₅₃	—	4	1	no	—
35	#13133157	C₂×C₄	1	8	yes	complemented
36	#13133157	C₂×C₄	1	8	yes	complemented
37	C₂×#11719908	C₂×C₄	1	8	yes	complemented
38	C₂×#11719908	C₂³	1	4	yes	—
39	#13133151	—	2	2	no	—
40	#13133151	C₂³	1	4	yes	—
41	#13133149	—	2	2	no	—
42	#13133149	—	2	2	no	—
43	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	#117×C₅₃	—	2	2	no	—
53	#117×C₅₃	—	2	2	no	—
54	#117×C₅₃	—	4	1	no	—
55	#117×C₅₃	—	4	1	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
58	C₂×C₄²×C₅₃	—	4	1	no	—
59	#107×C₅₃	—	2	2	no	—
60	#107×C₅₃	C₂³	1	4	yes	—
61	#106×C₅₃	—	2	2	no	—
62	#106×C₅₃	—	2	2	no	—
63	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	—	4	2	no	—
65	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×Dic₅₃	—	4	2	no	—
74	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
75	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
76	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₅₃	—	2	4	no	—
88	C₄⋊C₄×C₅₃	—	4	2	no	—
89	C₂²⋊C₄×C₅₃	—	4	2	no	—
90	C₂²⋊C₄×C₅₃	—	4	2	no	—
91	C₄²×C₅₃	—	4	2	no	—
92	C₄²×C₅₃	—	4	2	no	—
93	C₂²⋊Dic₅₃	—	4	2	no	—
94	C₂²⋊Dic₅₃	—	4	2	no	—
95	C₄⋊Dic₅₃	—	2	4	no	—
96	C₄⋊Dic₅₃	—	4	2	no	—
97	C₄×Dic₅₃	—	4	2	no	—
98	C₄×Dic₅₃	—	4	2	no	—
99	#11719908	C₂²×C₄	1	8	yes	—
100	C₂³×C₅₃	—	4	2	no	—
101	C₂³×C₅₃	#107	1	4	yes	—
102	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
103	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
104	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	—	4	4	no	—
106	C₂×C₄×C₅₃	—	8	4	no	—
107	C₂×C₄×C₅₃	#107	1	4	yes	—
108	C₂×C₄×C₅₃	#107	1	4	yes	—
109	C₂×C₄×C₅₃	#107	1	4	yes	—
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
117	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
118	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
119	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×Dic₅₃	—	4	4	no	—
121	C₂×Dic₅₃	—	8	4	no	—
122	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
127	#33409	—	53	1	no	maximal, Sylow
128	C₂²×C₅₃	—	2	2	no	—
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	—	4	4	no	—
132	C₂²×C₅₃	C₂×#107	1	2	yes	—
133	C₂²×C₅₃	C₂×#107	1	2	yes	—
134	C₂²×C₅₃	C₂×#107	1	2	yes	—
135	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
136	C₄×C₅₃	—	8	4	no	—
137	C₄×C₅₃	C₂×#107	1	2	yes	—
138	C₄×C₅₃	C₂×#107	1	2	yes	—
139	C₄×C₅₃	C₂×#107	1	2	yes	—
140	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
142	Dic₅₃	—	8	4	no	—
143	C₂²×M₃₂	—	53	1	no	—
144	C₂×#427	—	53	1	no	—
145	C₂×#427	D₅₃	1	1	yes	characteristic
146	#2159	—	53	4	no	—
147	#2159	—	53	4	no	—
148	#2158	—	53	4	no	—
149	C₂×C₅₃	—	2	2	no	—
150	C₂×C₅₃	—	2	2	no	—
151	C₂×C₅₃	—	2	2	no	—
152	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
153	C₂×C₅₃	#3414	1	2	yes	—
154	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
155	C₂×#117	—	2	1	no	—
156	C₂×#117	—	106	1	no	—
157	C₂×M₃₂	—	53	8	no	—
158	C₂×M₃₂	—	53	2	no	—
159	C₂×M₃₂	—	53	2	no	—
160	C₂²×C₁₆	—	53	2	no	—
161	#427	—	2	1	no	—
162	#427	—	2	1	no	—
163	#427	—	53	2	no	—
164	#427	—	53	2	no	—
165	#427	—	106	1	no	—
166	#427	—	106	1	no	—
167	#427	C₂×D₅₃	1	2	yes	—
168	#427	C₂×D₅₃	1	2	yes	—
169	C₂×#107	—	53	2	no	—
170	C₂×#107	C₂×D₅₃	1	2	yes	—
171	C₂×#106	—	2	1	no	—
172	C₂×#106	—	106	1	no	—
173	#367	—	53	8	no	—
174	#366	—	53	8	no	—
175	C₅₃	#33409	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
177	C₂²×C₈	Dic₁₀₆	1	4	yes	—
178	C₂²×C₈	C₄×D₅₃	1	4	yes	—
179	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
181	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	—	4	1	no	—
188	#117	—	4	1	no	—
189	#117	—	106	2	no	—
190	#117	—	106	2	no	—
191	#117	—	212	1	no	—
192	#117	—	212	1	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	106	2	no	—
195	C₂×C₂²⋊C₄	—	4	1	no	—
196	C₂×C₂²⋊C₄	—	212	1	no	—
197	C₂×C₄²	—	4	1	no	—
198	C₂×C₄²	—	212	1	no	—
199	M₃₂	—	53	8	no	—
200	M₃₂	—	53	8	no	—
201	C₂×C₁₆	—	53	8	no	—
202	C₂×C₁₆	—	53	4	no	—
203	#107	—	2	2	no	—
204	#107	—	53	4	no	—
205	#107	—	106	2	no	—
206	#107	C₂²×D₅₃	1	4	yes	—
207	#106	—	2	2	no	—
208	#106	—	2	2	no	—
209	#106	—	106	2	no	—
210	#106	—	106	2	no	—
211	C₂⁴	#11719922	1	1	yes	characteristic
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	—	212	2	no	—
214	C₂²×C₄	#11719919	1	4	yes	—
215	C₂²×C₄	#11719922	1	1	yes	characteristic
216	C₂²×C₄	#11719922	1	1	yes	characteristic
217	C₂²×C₄	#11719922	1	1	yes	characteristic
218	C₂²×C₄	#11719925	1	4	yes	—
219	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
222	C₂×C₈	#11719919	1	4	yes	—
223	C₂×C₈	#11719919	1	4	yes	—
224	C₂×C₈	#11719919	1	4	yes	—
225	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
226	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
227	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
228	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
229	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
230	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
231	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
232	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
233	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
234	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	106	4	no	—
238	C₄⋊C₄	—	212	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	212	2	no	—
242	C₂²⋊C₄	—	212	2	no	—
243	C₄²	—	4	2	no	—
244	C₄²	—	4	2	no	—
245	C₄²	—	212	2	no	—
246	C₄²	—	212	2	no	—
247	C₁₆	—	53	8	no	—
248	C₂³	—	4	2	no	—
249	C₂³	#13133150	1	4	yes	—
250	C₂³	C₂×#11719922	1	1	yes	characteristic
251	C₂³	C₂×#11719922	1	1	yes	characteristic
252	C₂³	C₂×#11719922	1	1	yes	characteristic
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	—	424	4	no	—
256	C₂×C₄	—	212	4	no	—
257	C₂×C₄	#13133150	1	4	yes	—
258	C₂×C₄	#13133150	1	4	yes	—
259	C₂×C₄	#13133150	1	4	yes	—
260	C₂×C₄	C₂×#11719919	1	2	yes	—
261	C₂×C₄	C₂×#11719919	1	2	yes	—
262	C₂×C₄	C₂×#11719919	1	2	yes	—
263	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
264	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
265	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
266	C₂×C₄	C₂×#11719925	1	2	yes	—
267	C₂×C₄	C₂×#11719925	1	2	yes	—
268	C₂×C₄	C₂×#11719925	1	2	yes	—
269	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
270	C₈	C₂×#11719919	1	2	yes	—
271	C₈	C₂×#11719919	1	2	yes	—
272	C₈	C₂×#11719919	1	2	yes	—
273	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
274	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	4	no	—
279	C₂²	C₂×#13133150	1	2	yes	—
280	C₂²	C₂×#13133150	1	2	yes	—
281	C₂²	C₂×#13133150	1	2	yes	—
282	C₂²	C₂²×#11719922	1	1	yes	characteristic
283	C₄	—	8	4	no	—
284	C₄	—	424	4	no	—
285	C₄	C₂×#13133150	1	2	yes	—
286	C₄	C₂×#13133150	1	2	yes	—
287	C₄	C₂×#13133150	1	2	yes	—
288	C₄	C₂²×#11719919	1	1	yes	characteristic
289	C₄	C₂²×#11719925	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	2	2	no	—
292	C₂	—	2	2	no	—
293	C₂	C₂²×#13133150	1	1	yes	characteristic
294	C₂	#20287849	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#427	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#427	D₅₃
hypocenter	C₅₃	#33409
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #20276504	C₂	semidirect
4. #20276501	C₂	semidirect
5. #20276497	C₂	semidirect
8. C₂×#13133157	C₄	semidirect
18. #16810450	C₂²	semidirect
19. #16810448	C₂²	semidirect
35. #13133157	C₂×C₄	semidirect
36. #13133157	C₂×C₄	semidirect
37. C₂×#11719908	C₂×C₄	semidirect
175. C₅₃	#33409	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133157	2
C₂×#16810661	2
#20276504	2
#20276501	2
#20276497	2
C₂×C₅₃×#427	2
#33409	53