Subgroups — #27292777

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810672	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810671	C₂	1	2	yes	maximal
3	C₂×#16810381	C₂	1	1	yes	maximal, characteristic
4	#20274409	C₂	1	2	yes	maximal
5	C₅₃×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂²×#11719917	C₂²	1	2	yes	—
8	#16810672	—	2	4	no	—
9	#16810671	—	2	4	no	—
10	#16810671	—	2	4	no	—
11	C₂×#13133153	—	2	2	no	—
12	C₂×#13133152	—	2	4	no	—
13	#16810381	C₄	1	4	yes	complemented
14	#16810378	—	2	4	no	—
15	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#336	—	2	2	no	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	—	2	2	no	—
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719917	C₂×C₄	1	4	yes	—
29	C₂×#11719917	D₄	1	4	yes	—
30	C₂×#11719917	Q₈	1	4	yes	—
31	C₂²×#88322	—	2	2	no	—
32	C₂²×#88322	D₄	1	4	yes	—
33	#13133153	—	2	8	no	—
34	#13133152	—	2	16	no	—
35	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
36	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₅₃	—	2	2	no	—
38	C₂×M₁₆×C₅₃	—	2	2	no	—
39	C₂×M₁₆×C₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	—	2	2	no	—
41	C₂×M₁₆×C₅₃	—	2	1	no	—
42	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
44	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
45	C₂²×C₈×C₅₃	—	2	1	no	—
46	C₂²×C₈×C₅₃	D₄	1	2	yes	—
47	C₂×C₄²×C₅₃	—	2	1	no	—
48	C₂×C₄²×C₅₃	—	2	2	no	—
49	C₂×C₄²×C₅₃	—	2	1	no	—
50	C₂×C₄²×C₅₃	—	2	2	no	—
51	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	—	4	4	no	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	#41	1	2	yes	—
64	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
65	M₁₆×C₅₃	—	2	4	no	—
66	M₁₆×C₅₃	—	2	4	no	—
67	M₁₆×C₅₃	—	2	4	no	—
68	M₁₆×C₅₃	—	4	2	no	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	4	1	no	—
73	C₂×C₈×C₅₃	—	4	1	no	—
74	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
75	C₄²×C₅₃	—	4	1	no	—
76	C₄²×C₅₃	—	4	2	no	—
77	C₄²×C₅₃	—	4	1	no	—
78	#11719917	—	2	4	no	—
79	#11719917	—	2	4	no	—
80	#11719917	—	2	4	no	—
81	#11719917	—	2	4	no	—
82	#11719917	—	2	4	no	—
83	#11719917	—	2	4	no	—
84	#11719917	—	4	2	no	—
85	#11719917	—	4	2	no	—
86	C₂×#88322	—	2	4	no	—
87	C₂×#88322	—	2	4	no	—
88	C₂×#88322	—	2	4	no	—
89	C₂×#88322	—	4	2	no	—
90	C₂×#88322	—	4	2	no	—
91	C₂×#88322	C₄⋊C₄	1	8	yes	—
92	C₂³×C₅₃	—	4	2	no	—
93	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
94	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
95	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
97	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	—	2	8	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#120	1	2	yes	—
109	C₂×C₄×C₅₃	#121	1	2	yes	—
110	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#123	1	2	yes	—
116	C₂×C₄×C₅₃	#125	1	2	yes	—
117	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
121	C₈×C₅₃	—	2	4	no	—
122	C₈×C₅₃	—	4	2	no	—
123	#88322	—	2	8	no	—
124	#88322	—	4	4	no	—
125	#11338	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	2	4	no	—
127	C₂²×C₅₃	—	2	2	no	—
128	C₂²×C₅₃	—	4	1	no	—
129	C₂²×C₅₃	—	4	2	no	—
130	C₂²×C₅₃	—	4	1	no	—
131	C₂²×C₅₃	#389	1	2	yes	—
132	C₂²×C₅₃	#389	1	1	yes	characteristic
133	C₂²×C₅₃	#395	1	1	yes	characteristic
134	C₂²×C₅₃	#395	1	2	yes	—
135	C₂²×C₅₃	#399	1	1	yes	characteristic
136	C₄×C₅₃	—	4	4	no	—
137	C₄×C₅₃	#384	1	1	yes	characteristic
138	C₄×C₅₃	#385	1	2	yes	—
139	C₄×C₅₃	#385	1	1	yes	characteristic
140	C₄×C₅₃	#392	1	1	yes	characteristic
141	C₄×C₅₃	#398	1	1	yes	characteristic
142	C₄×C₅₃	#398	1	2	yes	—
143	C₂×#430	—	53	2	no	—
144	C₂×#430	—	53	1	no	—
145	C₂×#338	—	53	1	no	—
146	#1728	—	53	2	no	—
147	#1728	D₅₃	1	1	yes	characteristic
148	C₂×C₅₃	—	2	2	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	—	2	1	no	—
151	C₂×C₅₃	—	2	2	no	—
152	C₂×C₅₃	#1444	1	1	yes	characteristic
153	C₂×C₅₃	#2091	1	2	yes	—
154	C₂²×M₁₆	—	53	2	no	—
155	C₂²×M₁₆	—	53	2	no	—
156	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
157	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
158	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
159	#430	—	106	4	no	—
160	#430	—	106	4	no	—
161	#430	—	106	4	no	—
162	C₂×#108	—	106	4	no	—
163	C₂×#108	—	106	2	no	—
164	#338	—	53	4	no	—
165	#336	—	2	2	no	—
166	#336	—	106	4	no	—
167	C₅₃	#11338	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₄	Dic₁₀₆	1	2	yes	—
169	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	53	4	no	—
176	C₂×M₁₆	—	53	4	no	—
177	C₂×M₁₆	—	53	4	no	—
178	C₂×M₁₆	—	106	2	no	—
179	C₂×M₁₆	—	106	2	no	—
180	C₂×M₁₆	—	106	2	no	—
181	C₂×M₁₆	—	106	2	no	—
182	C₂×M₁₆	—	106	2	no	—
183	C₂×M₁₆	—	106	2	no	—
184	C₂×M₁₆	—	106	2	no	—
185	C₂×M₁₆	—	106	2	no	—
186	C₂×M₁₆	—	106	2	no	—
187	C₂×M₁₆	Dic₁₀₆	1	2	yes	—
188	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
189	C₂×M₁₆	D₂₁₂	1	2	yes	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	53	4	no	—
192	C₂²×C₈	—	106	2	no	—
193	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	2	1	no	—
198	#108	—	106	8	no	—
199	#108	—	106	16	no	—
200	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
207	C₂²×C₄	#11719945	1	1	yes	characteristic
208	C₂²×C₄	#11719945	1	1	yes	characteristic
209	C₂²×C₄	D₄×D₅₃	1	2	yes	—
210	C₂²×C₄	D₄×D₅₃	1	2	yes	—
211	C₂²×C₄	#11719947	1	2	yes	—
212	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
213	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	106	4	no	—
219	M₁₆	—	106	4	no	—
220	M₁₆	—	106	4	no	—
221	M₁₆	—	106	4	no	—
222	M₁₆	—	106	4	no	—
223	M₁₆	—	106	4	no	—
224	M₁₆	—	212	2	no	—
225	M₁₆	—	212	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	53	8	no	—
232	C₂×C₈	—	106	4	no	—
233	C₂×C₈	—	106	4	no	—
234	C₂×C₈	—	106	4	no	—
235	C₂×C₈	—	212	2	no	—
236	C₂×C₈	—	212	2	no	—
237	C₂×C₈	#11719919	1	4	yes	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	2	no	—
240	C₄²	—	4	1	no	—
241	C₂³	—	4	2	no	—
242	C₂³	C₄×Dic₁₀₆	1	1	yes	characteristic
243	C₂³	#13133194	1	2	yes	—
244	C₂³	#13133196	1	1	yes	characteristic
245	C₂³	#13133216	1	2	yes	—
246	C₂³	#13133217	1	1	yes	characteristic
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
253	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
254	C₂×C₄	#13133177	1	2	yes	—
255	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
256	C₂×C₄	#13133185	1	1	yes	characteristic
257	C₂×C₄	#13133187	1	2	yes	—
258	C₂×C₄	#13133190	1	2	yes	—
259	C₂×C₄	#13133191	1	1	yes	characteristic
260	C₂×C₄	#13133194	1	2	yes	—
261	C₂×C₄	#13133195	1	2	yes	—
262	C₂×C₄	#13133197	1	2	yes	—
263	C₂×C₄	#13133198	1	1	yes	characteristic
264	C₂×C₄	#13133199	1	2	yes	—
265	C₂×C₄	#13133207	1	2	yes	—
266	C₂×C₄	#13133209	1	1	yes	characteristic
267	C₂×C₄	#13133213	1	1	yes	characteristic
268	C₂×C₄	#13133215	1	1	yes	characteristic
269	C₂×C₄	#13133216	1	2	yes	—
270	C₈	—	2	4	no	—
271	C₈	—	4	2	no	—
272	C₈	—	106	8	no	—
273	C₈	—	212	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	1	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	1	no	—
279	C₂²	#16810520	1	2	yes	—
280	C₂²	#16810521	1	1	yes	characteristic
281	C₂²	#16810543	1	1	yes	characteristic
282	C₂²	#16810544	1	2	yes	—
283	C₂²	#16810559	1	1	yes	characteristic
284	C₄	—	4	4	no	—
285	C₄	#16810492	1	1	yes	characteristic
286	C₄	#16810495	1	1	yes	characteristic
287	C₄	#16810497	1	2	yes	—
288	C₄	#16810532	1	1	yes	characteristic
289	C₄	#16810553	1	1	yes	characteristic
290	C₄	#16810557	1	2	yes	—
291	C₂	—	2	2	no	—
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	—	2	2	no	—
295	C₂	#20272979	1	1	yes	characteristic
296	C₂	#20276231	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133213
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1728	C₂
socle	C₂²×C₅₃	#399
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1728	D₅₃
hypocenter	C₅₃	#11338
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1728

Complements

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

N	H ≅ G/N	type
13. #16810381	C₄	semidirect
167. C₅₃	#11338	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810672	2
C₂×#16810671	2
C₂×#16810381	2
#20274409	2
C₅₃×#1728	2
#11338	53