Subgroups — #27300573

Order 13568.

370 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287817	C₂	1	1	yes	maximal, characteristic
2	#20276481	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276480	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276328	C₂	1	1	yes	maximal, characteristic
5	#20274516	C₂	1	1	yes	maximal, characteristic
6	#20272803	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133223	C₂²	1	1	yes	characteristic
9	#16811041	—	2	2	no	—
10	#16811039	—	2	2	no	—
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	C₂×#13133157	C₂²	1	1	yes	characteristic
13	M₃₂×D₅₃	—	2	2	no	—
14	#16810605	C₄	1	1	yes	characteristic
15	#16810596	C₄	1	1	yes	characteristic
16	#16810578	C₂²	1	1	yes	characteristic
17	#16810444	—	2	2	no	—
18	#16810443	—	2	2	no	—
19	#16810402	—	2	2	no	—
20	#16810395	C₄	1	1	yes	characteristic
21	#16810393	C₄	1	1	yes	characteristic
22	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#405	C₂²	1	1	yes	characteristic
25	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
26	#13133223	—	2	1	no	—
27	#13133223	—	2	1	no	—
28	#13133223	—	2	1	no	—
29	#13133223	—	2	1	no	—
30	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
31	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
32	C₂×#11719912	Q₈	1	2	yes	—
33	C₂×C₈×D₅₃	D₄	1	2	yes	—
34	#13133176	C₂×C₄	1	1	yes	characteristic
35	#13133157	—	2	1	no	—
36	#13133157	—	2	1	no	—
37	#13133157	—	2	1	no	—
38	#13133157	D₄	1	2	yes	—
39	#13133156	—	2	4	no	—
40	C₂×#11719908	—	2	1	no	—
41	C₂×#11719908	—	2	1	no	—
42	C₁₆×D₅₃	—	2	4	no	—
43	#13133135	C₈	1	2	yes	—
44	#13133133	C₈	1	2	yes	—
45	#13133131	—	2	2	no	—
46	#13133127	—	2	1	no	—
47	C₈×Dic₅₃	—	2	1	no	—
48	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
51	M₃₂×C₅₃	—	2	1	no	—
52	M₃₂×C₅₃	D₄	1	2	yes	—
53	C₂×C₁₆×C₅₃	—	2	1	no	—
54	C₂×C₁₆×C₅₃	—	2	1	no	—
55	C₂×C₁₆×C₅₃	—	2	1	no	—
56	C₂×C₁₆×C₅₃	—	2	1	no	—
57	#97×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	—	2	1	no	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₅₃	M₁₆	1	2	yes	—
61	#11719945	—	2	1	no	—
62	#11719945	—	2	1	no	—
63	#11719945	—	2	1	no	—
64	#11719945	—	2	1	no	—
65	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
66	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
67	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
68	M₁₆×C₅₃	—	2	1	no	—
69	M₁₆×C₅₃	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₅₃	—	2	1	no	—
71	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
76	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
77	C₄⋊C₄×C₅₃	—	2	1	no	—
78	C₂²⋊C₄×C₅₃	—	2	1	no	—
79	C₄²×C₅₃	—	2	1	no	—
80	C₂²⋊Dic₅₃	—	2	1	no	—
81	C₄⋊Dic₅₃	—	2	1	no	—
82	C₄×Dic₅₃	—	2	1	no	—
83	#11719917	—	2	1	no	—
84	#11719917	—	2	1	no	—
85	#11719917	—	2	1	no	—
86	#11719917	C₂×C₈	1	2	yes	—
87	C₂×#88322	—	2	1	no	—
88	C₂×#88322	C₄⋊C₄	1	2	yes	—
89	#11719912	—	2	2	no	—
90	#11719912	—	2	2	no	—
91	C₈×D₅₃	—	2	2	no	—
92	C₈×D₅₃	—	2	2	no	—
93	C₁₆×C₅₃	—	2	2	no	—
94	C₁₆×C₅₃	—	4	1	no	—
95	#11719908	—	2	2	no	—
96	#11719908	—	4	1	no	—
97	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
99	C₂×C₄×C₅₃	—	4	1	no	—
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
106	C₂²⋊D₅₃	—	2	2	no	—
107	C₂²⋊D₅₃	—	2	2	no	—
108	C₂×Dic₅₃	—	4	1	no	—
109	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
110	D₂₁₂	—	2	1	no	—
111	D₂₁₂	—	2	1	no	—
112	C₄×D₅₃	—	2	2	no	—
113	C₄×D₅₃	—	2	2	no	—
114	Dic₁₀₆	—	2	1	no	—
115	Dic₁₀₆	—	2	1	no	—
116	C₈×C₅₃	—	2	2	no	—
117	C₈×C₅₃	#120	1	1	yes	characteristic
118	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₈×C₅₃	#123	1	1	yes	characteristic
121	#88322	—	2	2	no	—
122	#88322	—	2	2	no	—
123	#12916	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	—	4	1	no	—
125	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
126	C₂²×C₅₃	#435	1	1	yes	characteristic
127	C₂²×C₅₃	#435	1	1	yes	characteristic
128	C₂×D₅₃	—	2	2	no	—
129	C₂×D₅₃	—	4	2	no	—
130	C₄×C₅₃	—	8	1	no	—
131	C₄×C₅₃	#385	1	1	yes	characteristic
132	C₄×C₅₃	#424	1	1	yes	characteristic
133	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
134	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
135	Dic₅₃	—	2	2	no	—
136	Dic₅₃	—	8	1	no	—
137	#3407	—	53	1	no	—
138	#2153	—	53	1	no	—
139	#2153	—	53	1	no	—
140	#2113	—	53	1	no	—
141	#1757	—	53	1	no	—
142	#1379	—	53	1	no	—
143	#1379	D₅₃	1	1	yes	characteristic, complemented
144	C₂×C₅₃	—	2	1	no	—
145	C₂×C₅₃	—	2	1	no	—
146	C₂×C₅₃	—	2	1	no	—
147	C₂×C₅₃	#1850	1	1	yes	characteristic
148	D₅₃	—	4	2	no	—
149	C₂×#131	—	53	1	no	—
150	#504	—	106	2	no	—
151	#504	—	106	2	no	—
152	C₂×M₃₂	—	53	1	no	—
153	C₂×M₃₂	—	53	1	no	—
154	C₂×M₃₂	—	106	2	no	—
155	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
156	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
157	#413	—	53	1	no	—
158	#410	—	53	1	no	—
159	#405	—	53	1	no	—
160	#405	C₂×D₅₃	1	1	yes	characteristic
161	#364	—	106	2	no	—
162	#364	—	106	2	no	—
163	C₂²⋊C₁₆	—	106	2	no	—
164	#344	—	53	1	no	—
165	#344	—	53	1	no	—
166	C₅₃	#12916	1	1	yes	characteristic, complemented, Sylow
167	C₂×#41	—	53	1	no	—
168	#131	—	106	1	no	—
169	#131	—	106	1	no	—
170	#131	—	106	1	no	—
171	#131	—	106	1	no	—
172	C₂×M₁₆	—	53	1	no	—
173	C₂×M₁₆	—	53	2	no	—
174	C₂×M₁₆	—	53	1	no	—
175	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
176	C₂²×C₈	—	53	2	no	—
177	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
178	#117	—	53	1	no	—
179	#117	C₄×D₅₃	1	1	yes	characteristic
180	M₃₂	—	2	1	no	—
181	M₃₂	—	53	2	no	—
182	M₃₂	—	106	4	no	—
183	M₃₂	—	106	1	no	—
184	M₃₂	—	106	1	no	—
185	M₃₂	—	106	1	no	—
186	M₃₂	D₂₁₂	1	2	yes	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	106	4	no	—
192	C₂×C₁₆	—	106	1	no	—
193	C₂×C₁₆	—	106	1	no	—
194	#101	—	53	2	no	—
195	#100	—	53	2	no	—
196	#99	—	106	2	no	—
197	#97	—	2	1	no	—
198	#97	—	106	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	106	1	no	—
201	#41	—	106	1	no	—
202	#41	—	106	1	no	—
203	#41	—	106	1	no	—
204	#41	—	106	1	no	—
205	C₂×Q₈	—	53	1	no	—
206	C₂×D₄	—	53	2	no	—
207	C₂×D₄	—	53	1	no	—
208	C₂²×C₄	—	53	2	no	—
209	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
210	M₁₆	—	2	1	no	—
211	M₁₆	—	53	2	no	—
212	M₁₆	—	106	1	no	—
213	M₁₆	—	106	1	no	—
214	M₁₆	—	106	1	no	—
215	M₁₆	—	106	2	no	—
216	M₁₆	—	106	2	no	—
217	M₁₆	C₈×D₅₃	1	2	yes	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	53	2	no	—
220	C₂×C₈	—	106	1	no	—
221	C₂×C₈	—	106	2	no	—
222	C₂×C₈	—	106	2	no	—
223	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
224	C₂×C₈	#11719945	1	1	yes	characteristic
225	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
226	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
227	C₂×C₈	#11719947	1	1	yes	characteristic
228	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	106	1	no	—
231	C₂²⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	106	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	106	1	no	—
235	C₁₆	—	2	2	no	—
236	C₁₆	—	4	1	no	—
237	C₁₆	—	106	2	no	—
238	C₁₆	—	212	1	no	—
239	C₂³	—	53	2	no	—
240	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
241	Q₈	—	106	1	no	—
242	Q₈	—	106	1	no	—
243	D₄	—	106	2	no	—
244	D₄	—	106	2	no	—
245	D₄	—	106	1	no	—
246	D₄	—	106	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	53	2	no	—
249	C₂×C₄	—	106	2	no	—
250	C₂×C₄	—	106	2	no	—
251	C₂×C₄	—	212	1	no	—
252	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
253	C₂×C₄	#13133177	1	1	yes	characteristic
254	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
255	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
256	C₂×C₄	#13133223	1	1	yes	characteristic
257	C₂×C₄	#13133225	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	106	2	no	—
260	C₈	—	106	2	no	—
261	C₈	#13133187	1	1	yes	characteristic
262	C₈	#13133195	1	1	yes	characteristic
263	C₈	#13133197	1	1	yes	characteristic
264	C₈	#13133199	1	1	yes	characteristic
265	C₂²	—	4	1	no	—
266	C₂²	—	106	2	no	—
267	C₂²	—	212	2	no	—
268	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
269	C₂²	#16810696	1	1	yes	characteristic
270	C₂²	#16810697	1	1	yes	characteristic
271	C₄	—	8	1	no	—
272	C₄	—	106	2	no	—
273	C₄	—	424	1	no	—
274	C₄	#16810497	1	1	yes	characteristic
275	C₄	#16810648	1	1	yes	characteristic
276	C₄	#16810686	1	1	yes	characteristic
277	C₄	C₈×D₂₁₂	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	212	2	no	—
282	C₂	#20274894	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133187
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1379	C₂
socle	C₂×C₅₃	#1850
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1379	D₅₃
hypocenter	C₅₃	#12916
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
2. #20276481	C₂	semidirect
3. #20276480	C₂	semidirect
6. #20272803	C₂	semidirect
7. C₅₃×#1379	C₂	semidirect
143. #1379	D₅₃	semidirect
166. C₅₃	#12916	semidirect

Maximal subgroups

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

subgroup	index
#20287817	2
#20276481	2
#20276480	2
#20276328	2
#20274516	2
#20272803	2
C₅₃×#1379	2
#12916	53