Subgroups — #27290010

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133152	C₂	1	2	yes	maximal
2	C₂×#16810579	C₂	1	1	yes	maximal, characteristic
3	#20274454	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810381	C₂	1	2	yes	maximal
5	C₅₃×#1742	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂²×#11719917	C₂²	1	1	yes	characteristic
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	C₂×#13133152	—	2	4	no	—
10	C₂×#13133152	—	2	4	no	—
11	C₂×#13133152	C₄	1	8	yes	complemented
12	#16810579	—	2	4	no	—
13	C₂×#13133128	C₄	1	2	yes	—
14	C₂×C₄×#88322	C₄	1	2	yes	—
15	#16810392	—	2	2	no	—
16	#16810381	—	2	4	no	—
17	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
18	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#343	—	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	C₂×C₄	1	4	yes	—
28	C₂×#11719917	D₄	1	2	yes	—
29	C₂×#11719917	Q₈	1	2	yes	—
30	C₂²×#88322	—	2	2	no	—
31	C₂²×#88322	C₂×C₄	1	2	yes	—
32	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
33	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
34	C₂²×#88322	C₂×C₄	1	4	yes	—
35	C₂²×#88322	D₄	1	4	yes	—
36	C₂²×#88322	D₄	1	2	yes	—
37	C₂²×#88322	Q₈	1	2	yes	—
38	#13133152	—	2	16	no	—
39	#13133128	—	2	4	no	—
40	C₄×#88322	—	2	4	no	—
41	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₅₃	—	2	2	no	—
43	C₂×M₁₆×C₅₃	—	2	2	no	—
44	C₂×M₁₆×C₅₃	—	2	2	no	—
45	C₂×M₁₆×C₅₃	—	2	2	no	—
46	C₂×M₁₆×C₅₃	—	2	2	no	—
47	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₅₃	—	2	2	no	—
49	#117×C₅₃	—	2	2	no	—
50	#117×C₅₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₅₃	Q₈	1	2	yes	—
53	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
54	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	#41	1	2	yes	—
65	C₂²×C₄×C₅₃	#41	1	2	yes	—
66	M₁₆×C₅₃	—	2	4	no	—
67	M₁₆×C₅₃	—	2	4	no	—
68	M₁₆×C₅₃	—	4	2	no	—
69	M₁₆×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	2	4	no	—
71	C₂×C₈×C₅₃	—	4	2	no	—
72	C₂×C₈×C₅₃	—	4	2	no	—
73	C₄⋊C₄×C₅₃	—	2	4	no	—
74	C₂²⋊C₄×C₅₃	—	2	4	no	—
75	C₄²×C₅₃	—	2	2	no	—
76	C₄²×C₅₃	—	2	2	no	—
77	#11719917	—	2	4	no	—
78	#11719917	—	2	4	no	—
79	#11719917	—	2	4	no	—
80	#11719917	—	2	4	no	—
81	#11719917	—	4	2	no	—
82	#11719917	—	4	2	no	—
83	C₂×#88322	—	2	2	no	—
84	C₂×#88322	—	2	2	no	—
85	C₂×#88322	—	2	4	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂×#88322	—	4	2	no	—
88	C₂×#88322	C₄²	1	2	yes	—
89	C₂×#88322	C₄²	1	2	yes	—
90	C₂×#88322	C₂²⋊C₄	1	2	yes	—
91	C₂×#88322	C₂²⋊C₄	1	2	yes	—
92	C₂×#88322	C₂²⋊C₄	1	8	yes	—
93	C₂×#88322	C₄⋊C₄	1	4	yes	—
94	C₂×#88322	C₄⋊C₄	1	2	yes	—
95	C₂×#88322	C₄⋊C₄	1	2	yes	—
96	C₂×#88322	C₄⋊C₄	1	8	yes	—
97	C₂³×C₅₃	—	4	2	no	—
98	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
100	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
102	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	4	4	no	—
105	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₅₃	#121	1	2	yes	—
117	C₂×C₄×C₅₃	#121	1	2	yes	—
118	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₅₃	#124	1	2	yes	—
120	C₂×C₄×C₅₃	#125	1	2	yes	—
121	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
123	C₈×C₅₃	—	4	4	no	—
124	#88322	—	2	4	no	—
125	#88322	—	4	4	no	—
126	#88322	#95	1	4	yes	—
127	#88322	#95	1	4	yes	—
128	#10867	—	53	1	no	maximal, Sylow
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	—	2	4	no	—
131	C₂²×C₅₃	—	4	2	no	—
132	C₂²×C₅₃	—	4	1	no	—
133	C₂²×C₅₃	—	4	1	no	—
134	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₅₃	#380	1	1	yes	characteristic
136	C₂²×C₅₃	#384	1	1	yes	characteristic
137	C₂²×C₅₃	#389	1	2	yes	—
138	C₂²×C₅₃	#391	1	2	yes	—
139	C₄×C₅₃	—	4	4	no	—
140	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
141	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
142	C₄×C₅₃	#380	1	1	yes	characteristic
143	C₄×C₅₃	#382	1	1	yes	characteristic
144	C₄×C₅₃	#389	1	2	yes	—
145	C₄×C₅₃	#390	1	2	yes	—
146	C₂²×#108	—	53	2	no	—
147	C₂×#405	—	53	1	no	—
148	#1742	—	53	1	no	—
149	#1742	D₅₃	1	1	yes	characteristic
150	C₂×#338	—	53	2	no	—
151	C₂×C₅₃	—	2	1	no	—
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	—	2	2	no	—
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
156	C₂×C₅₃	#1813	1	2	yes	—
157	C₂²×M₁₆	—	53	1	no	—
158	C₂²×M₁₆	—	53	2	no	—
159	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
160	C₂³×C₈	—	53	1	no	—
161	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
162	C₂×#108	—	53	8	no	—
163	C₂×#108	—	106	4	no	—
164	C₂×#108	—	106	4	no	—
165	#405	—	106	4	no	—
166	C₂×#97	—	53	2	no	—
167	C₂×C₄×C₈	—	53	2	no	—
168	#343	—	2	2	no	—
169	#343	—	106	2	no	—
170	#338	—	106	4	no	—
171	C₅₃	#10867	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	53	2	no	—
179	C₂×M₁₆	—	53	2	no	—
180	C₂×M₁₆	—	53	4	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	—	106	2	no	—
188	C₂×M₁₆	C₄×D₅₃	1	4	yes	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	53	1	no	—
191	C₂²×C₈	—	53	1	no	—
192	C₂²×C₈	—	53	4	no	—
193	C₂²×C₈	—	53	4	no	—
194	C₂²×C₈	—	53	2	no	—
195	C₂²×C₈	—	53	2	no	—
196	C₂²×C₈	—	53	2	no	—
197	C₂²×C₈	—	106	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
201	C₂×C₂²⋊C₄	Dic₁₀₆	1	2	yes	—
202	C₂×C₄²	C₄×D₅₃	1	2	yes	—
203	#108	—	106	16	no	—
204	#97	—	106	4	no	—
205	C₄×C₈	—	106	4	no	—
206	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
207	C₂²×C₄	#11719919	1	4	yes	—
208	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
210	C₂²×C₄	#11719945	1	2	yes	—
211	C₂²×C₄	D₄×D₅₃	1	2	yes	—
212	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
213	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
214	C₂²×C₄	#11719947	1	2	yes	—
215	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
216	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	106	4	no	—
223	M₁₆	—	106	4	no	—
224	M₁₆	—	106	4	no	—
225	M₁₆	—	106	4	no	—
226	M₁₆	—	212	2	no	—
227	M₁₆	—	212	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	53	2	no	—
232	C₂×C₈	—	53	2	no	—
233	C₂×C₈	—	53	2	no	—
234	C₂×C₈	—	53	2	no	—
235	C₂×C₈	—	53	8	no	—
236	C₂×C₈	—	53	8	no	—
237	C₂×C₈	—	53	2	no	—
238	C₂×C₈	—	53	2	no	—
239	C₂×C₈	—	53	4	no	—
240	C₂×C₈	—	106	2	no	—
241	C₂×C₈	—	106	2	no	—
242	C₂×C₈	—	106	4	no	—
243	C₂×C₈	—	212	2	no	—
244	C₂×C₈	—	212	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	—	2	4	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	C₂×#11719919	1	1	yes	characteristic
251	C₂³	C₄×Dic₁₀₆	1	2	yes	—
252	C₂³	#13133194	1	2	yes	—
253	C₂³	#13133216	1	1	yes	characteristic
254	C₂³	#13133216	1	1	yes	characteristic
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
258	C₂×C₄	C₂²⋊C₄×D₅₃	1	2	yes	—
259	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
260	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
261	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
262	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
263	C₂×C₄	#13133173	1	1	yes	characteristic
264	C₂×C₄	#13133174	1	1	yes	characteristic
265	C₂×C₄	#13133177	1	2	yes	—
266	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
267	C₂×C₄	#13133182	1	2	yes	—
268	C₂×C₄	#13133189	1	2	yes	—
269	C₂×C₄	#13133190	1	2	yes	—
270	C₂×C₄	#13133195	1	2	yes	—
271	C₂×C₄	#13133203	1	2	yes	—
272	C₂×C₄	#13133207	1	2	yes	—
273	C₂×C₄	#13133213	1	1	yes	characteristic
274	C₂×C₄	#13133215	1	1	yes	characteristic
275	C₈	—	4	4	no	—
276	C₈	—	53	4	no	—
277	C₈	—	53	4	no	—
278	C₈	—	106	4	no	—
279	C₈	—	212	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	C₄×#11719919	1	1	yes	characteristic
286	C₂²	#16810481	1	1	yes	characteristic
287	C₂²	#16810492	1	1	yes	characteristic
288	C₂²	#16810520	1	2	yes	—
289	C₂²	#16810527	1	2	yes	—
290	C₄	—	4	4	no	—
291	C₄	#95×D₅₃	1	1	yes	characteristic
292	C₄	#16810474	1	1	yes	characteristic
293	C₄	#16810482	1	1	yes	characteristic
294	C₄	#16810487	1	1	yes	characteristic
295	C₄	#16810518	1	2	yes	—
296	C₄	#16810523	1	2	yes	—
297	C₂	—	2	2	no	—
298	C₂	—	2	2	no	—
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	#20272943	1	1	yes	characteristic
302	C₂	#20274763	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133173
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1742	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1742	D₅₃
hypocenter	C₅₃	#10867
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1742

Complements

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

N	H ≅ G/N	type
11. C₂×#13133152	C₄	semidirect
171. C₅₃	#10867	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133152	2
C₂×#16810579	2
#20274454	2
C₂×#16810381	2
C₅₃×#1742	2
#10867	53