Subgroups — #27299019

Order 13568.

570 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810661	C₂	1	1	yes	maximal, characteristic
3	#20272824	C₂	1	4	yes	maximal, complemented
4	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	—	2	1	no	—
6	C₂×#13133176	—	2	1	no	—
7	C₂×#13133153	—	2	2	no	—
8	C₂×#13133153	—	2	1	no	—
9	C₂×#13133153	C₄	1	4	yes	complemented
10	C₂×#13133153	C₂²	1	2	yes	—
11	#16810661	—	2	1	no	—
12	#16810661	—	2	1	no	—
13	#16810661	C₄	1	2	yes	—
14	#16810661	C₄	1	2	yes	—
15	C₂×#13133151	C₂²	1	2	yes	—
16	C₂×#13133149	—	2	1	no	—
17	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×M₃₂×C₅₃	—	2	2	no	—
19	C₂×M₃₂×C₅₃	—	2	1	no	—
20	C₂×M₃₂×C₅₃	C₄	1	4	yes	complemented
21	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
22	C₂²×C₁₆×C₅₃	—	2	1	no	—
23	C₂×#11719917	—	2	4	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	—	4	1	no	—
26	C₂×#11719917	—	4	2	no	—
27	C₂²×#88322	—	4	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	#13133153	—	2	4	no	—
36	#13133153	—	2	2	no	—
37	#13133153	C₂×C₄	1	4	yes	—
38	#13133151	—	2	2	no	—
39	#13133151	C₂×C₄	1	4	yes	—
40	#13133149	—	2	2	no	—
41	#13133149	—	2	2	no	—
42	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	D₄	1	2	yes	—
50	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	M₃₂×C₅₃	—	2	4	no	—
54	M₃₂×C₅₃	—	2	2	no	—
55	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
56	C₂×C₁₆×C₅₃	—	2	4	no	—
57	C₂×C₁₆×C₅₃	—	2	2	no	—
58	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	C₄²	1	2	yes	—
71	C₂×C₈×C₅₃	C₄²	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
88	C₂²⋊Dic₅₃	—	4	2	no	—
89	C₂²⋊Dic₅₃	—	4	2	no	—
90	C₄⋊Dic₅₃	—	2	4	no	—
91	C₄⋊Dic₅₃	—	4	2	no	—
92	C₄×Dic₅₃	—	4	2	no	—
93	C₄×Dic₅₃	—	4	2	no	—
94	#11719917	—	2	4	no	—
95	#11719917	—	4	4	no	—
96	#11719917	—	4	2	no	—
97	C₂×#88322	—	4	4	no	—
98	C₂×#88322	—	4	2	no	—
99	C₁₆×C₅₃	—	2	4	no	—
100	C₂³×C₅₃	—	4	2	no	—
101	C₂³×C₅₃	#102	1	1	yes	characteristic
102	C₂³×C₅₃	#102	1	1	yes	characteristic
103	C₂³×C₅₃	#103	1	2	yes	—
104	C₂³×C₅₃	#106	1	2	yes	—
105	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#95	1	2	yes	—
107	C₂×C₄×C₅₃	#95	1	2	yes	—
108	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#102	1	2	yes	—
110	C₂×C₄×C₅₃	#102	1	2	yes	—
111	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#103	1	2	yes	—
113	C₂×C₄×C₅₃	#103	1	2	yes	—
114	C₂×C₄×C₅₃	#103	1	2	yes	—
115	C₂×C₄×C₅₃	#106	1	2	yes	—
116	C₂×C₄×C₅₃	#107	1	2	yes	—
117	C₂×C₄×C₅₃	#107	1	2	yes	—
118	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
124	C₂×Dic₅₃	—	4	4	no	—
125	C₂×Dic₅₃	—	8	4	no	—
126	C₈×C₅₃	#95	1	2	yes	—
127	C₈×C₅₃	#95	1	2	yes	—
128	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
129	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	#88322	—	4	4	no	—
133	#12569	—	53	1	no	maximal, Sylow
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	—	2	2	no	—
137	C₂²×C₅₃	—	4	2	no	—
138	C₂²×C₅₃	—	4	2	no	—
139	C₂²×C₅₃	#340	1	2	yes	—
140	C₂²×C₅₃	#340	1	2	yes	—
141	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
142	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
143	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
144	C₄×C₅₃	#340	1	2	yes	—
145	C₄×C₅₃	#340	1	2	yes	—
146	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
147	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
148	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
149	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
150	Dic₅₃	—	8	4	no	—
151	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
152	C₂²×#108	—	53	1	no	—
153	C₂×#427	—	53	1	no	—
154	#1387	—	53	4	no	—
155	C₂×C₅₃	—	2	2	no	—
156	C₂×C₅₃	—	2	2	no	—
157	C₂×C₅₃	—	2	1	no	—
158	C₂×C₅₃	—	2	1	no	—
159	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
160	C₂×C₅₃	#2148	1	2	yes	—
161	C₂²×M₁₆	—	106	1	no	—
162	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
163	C₂×#117	—	106	1	no	—
164	C₂×M₃₂	—	2	1	no	—
165	C₂×M₃₂	—	2	2	no	—
166	C₂×M₃₂	Dic₅₃	1	4	yes	complemented
167	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
168	C₂²×C₁₆	—	2	1	no	—
169	C₂×#108	—	53	4	no	—
170	C₂×#108	—	53	2	no	—
171	C₂×#108	—	106	1	no	—
172	C₂×#108	—	106	2	no	—
173	#427	—	53	2	no	—
174	#427	—	53	2	no	—
175	#427	—	106	1	no	—
176	#427	—	106	1	no	—
177	C₂×#107	—	53	2	no	—
178	C₂×#106	—	106	1	no	—
179	C₅₃	#12569	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
181	C₂×M₁₆	—	106	2	no	—
182	C₂×M₁₆	—	106	4	no	—
183	C₂×M₁₆	—	212	2	no	—
184	C₂×M₁₆	—	212	1	no	—
185	C₂²×C₈	—	212	1	no	—
186	C₂²×C₈	Dic₁₀₆	1	2	yes	—
187	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
188	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
189	C₂²×C₈	C₄×D₅₃	1	2	yes	—
190	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
191	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
192	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
193	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
194	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
195	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
196	#117	—	106	2	no	—
197	#117	—	106	2	no	—
198	#117	—	212	1	no	—
199	#117	—	212	1	no	—
200	C₂×C₄⋊C₄	—	106	2	no	—
201	C₂×C₂²⋊C₄	—	212	1	no	—
202	C₂×C₄²	—	212	1	no	—
203	M₃₂	—	2	2	no	—
204	M₃₂	—	2	4	no	—
205	M₃₂	C₂×Dic₅₃	1	4	yes	—
206	C₂×C₁₆	—	2	2	no	—
207	C₂×C₁₆	—	2	4	no	—
208	#108	—	53	4	no	—
209	#108	—	106	2	no	—
210	#108	—	106	4	no	—
211	#107	—	53	4	no	—
212	#107	—	106	2	no	—
213	#106	—	106	2	no	—
214	#106	—	106	2	no	—
215	C₂⁴	#11719913	1	1	yes	characteristic
216	C₂²×C₄	—	212	2	no	—
217	C₂²×C₄	#11719913	1	2	yes	—
218	C₂²×C₄	#11719913	1	1	yes	characteristic
219	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
220	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
221	C₂²×C₄	Dic₂₁₂	1	2	yes	—
222	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
223	C₂²×C₄	#11719921	1	2	yes	—
224	C₂²×C₄	#11719921	1	1	yes	characteristic
225	C₂²×C₄	#11719921	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
227	M₁₆	—	106	4	no	—
228	M₁₆	—	212	4	no	—
229	M₁₆	—	212	2	no	—
230	C₂×C₈	—	212	4	no	—
231	C₂×C₈	—	212	2	no	—
232	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
233	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
234	C₂×C₈	#11719919	1	2	yes	—
235	C₂×C₈	#11719919	1	2	yes	—
236	C₂×C₈	#11719919	1	2	yes	—
237	C₂×C₈	#11719919	1	2	yes	—
238	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
239	C₂×C₈	#11719921	1	1	yes	characteristic
240	C₂×C₈	#11719921	1	1	yes	characteristic
241	C₂×C₈	#11719921	1	2	yes	—
242	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
243	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
244	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
245	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
246	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
247	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
248	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
250	C₄⋊C₄	—	106	4	no	—
251	C₄⋊C₄	—	212	2	no	—
252	C₂²⋊C₄	—	212	2	no	—
253	C₂²⋊C₄	—	212	2	no	—
254	C₄²	—	212	2	no	—
255	C₄²	—	212	2	no	—
256	C₁₆	—	2	4	no	—
257	C₂³	—	4	2	no	—
258	C₂³	#13133139	1	1	yes	characteristic
259	C₂³	#13133139	1	1	yes	characteristic
260	C₂³	#13133142	1	2	yes	—
261	C₂³	#13133149	1	2	yes	—
262	C₂³	C₂×#11719913	1	1	yes	characteristic
263	C₂×C₄	—	424	4	no	—
264	C₂×C₄	—	212	4	no	—
265	C₂×C₄	#13133124	1	2	yes	—
266	C₂×C₄	#13133124	1	2	yes	—
267	C₂×C₄	#13133139	1	2	yes	—
268	C₂×C₄	#13133139	1	2	yes	—
269	C₂×C₄	#13133139	1	1	yes	characteristic
270	C₂×C₄	#13133139	1	1	yes	characteristic
271	C₂×C₄	#13133142	1	2	yes	—
272	C₂×C₄	#13133142	1	2	yes	—
273	C₂×C₄	#13133142	1	2	yes	—
274	C₂×C₄	#13133149	1	2	yes	—
275	C₂×C₄	#13133151	1	2	yes	—
276	C₂×C₄	#13133151	1	2	yes	—
277	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
278	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
280	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
281	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
282	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
283	C₈	—	212	4	no	—
284	C₈	#13133124	1	2	yes	—
285	C₈	#13133124	1	2	yes	—
286	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
287	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
288	C₈	C₂×#11719919	1	1	yes	characteristic
289	C₈	C₂×#11719919	1	1	yes	characteristic
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	#16810386	1	2	yes	—
296	C₂²	#16810386	1	2	yes	—
297	C₂²	C₂×#13133139	1	1	yes	characteristic
298	C₂²	C₂×#13133142	1	1	yes	characteristic
299	C₂²	C₂×#13133149	1	1	yes	characteristic
300	C₄	—	424	4	no	—
301	C₄	#16810386	1	2	yes	—
302	C₄	#16810386	1	2	yes	—
303	C₄	C₂×#13133124	1	1	yes	characteristic
304	C₄	C₂×#13133139	1	1	yes	characteristic
305	C₄	C₂×#13133142	1	1	yes	characteristic
306	C₄	C₂×#13133151	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	—	2	1	no	—
311	C₂	C₂×#16810386	1	1	yes	characteristic
312	C₂	#20276465	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133139
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	#12569
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20272824	C₂	semidirect
9. C₂×#13133153	C₄	semidirect
20. C₂×M₃₂×C₅₃	C₄	semidirect
166. C₂×M₃₂	Dic₅₃	semidirect
179. C₅₃	#12569	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133153	2
C₂×#16810661	2
#20272824	2
C₂²×M₃₂×C₅₃	2
#12569	53