Subgroups — #27276955

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810579	C₂	1	1	yes	maximal, characteristic
2	#20273461	C₂	1	2	yes	maximal, complemented
3	#20273029	C₂	1	1	yes	maximal, characteristic
4	#20272545	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1458	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₄	1	1	yes	characteristic
7	C₂³×#88322	C₄	1	1	yes	characteristic
8	#16810676	C₂²	1	2	yes	—
9	#16810579	—	2	2	no	—
10	#16810579	—	2	2	no	—
11	C₂×#13133128	C₄	1	1	yes	characteristic
12	C₂×#13133128	C₂²	1	1	yes	characteristic
13	C₂×C₄×#88322	C₄	1	1	yes	characteristic
14	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
15	#16810375	—	2	2	no	—
16	#16810373	—	2	2	no	—
17	#16810349	—	2	1	no	—
18	#16810347	—	2	1	no	—
19	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#431	C₂²	1	2	yes	—
21	C₅₃×#324	—	2	1	no	—
22	C₅₃×#323	—	2	1	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	C₈	1	4	yes	complemented
26	C₂²×#88322	—	2	1	no	—
27	C₂²×#88322	—	2	1	no	—
28	C₂²×#88322	—	2	2	no	—
29	C₂²×#88322	—	2	2	no	—
30	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
32	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
33	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
34	C₄⋊#88322	—	2	2	no	—
35	C₂²⋊#88322	—	2	2	no	—
36	#13133128	—	2	2	no	—
37	#13133128	—	2	2	no	—
38	C₄×#88322	—	2	1	no	—
39	C₄×#88322	—	2	1	no	—
40	C₄×#88322	—	2	2	no	—
41	C₄×#88322	D₄	1	2	yes	—
42	C₄×#88322	D₄	1	2	yes	—
43	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	#117×C₅₃	—	2	2	no	—
45	#117×C₅₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₅₃	—	2	2	no	—
53	C₂²⋊C₈×C₅₃	—	2	2	no	—
54	C₄×C₈×C₅₃	—	2	2	no	—
55	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	—	2	1	no	—
57	C₂²×C₄×C₅₃	—	2	1	no	—
58	C₂²×C₄×C₅₃	—	2	1	no	—
59	C₂²×C₄×C₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₅₃	—	4	2	no	—
69	C₄⋊C₄×C₅₃	—	2	2	no	—
70	C₄⋊C₄×C₅₃	—	2	2	no	—
71	C₂²⋊C₄×C₅₃	—	2	2	no	—
72	C₂²⋊C₄×C₅₃	—	2	2	no	—
73	C₄²×C₅₃	—	2	1	no	—
74	C₄²×C₅₃	—	2	1	no	—
75	C₄²×C₅₃	C₂×D₄	1	2	yes	—
76	C₄²×C₅₃	#41	1	2	yes	—
77	#11719917	—	2	8	no	—
78	C₂×#88322	—	2	4	no	—
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	—	2	1	no	—
81	C₂×#88322	—	2	1	no	—
82	C₂×#88322	—	2	1	no	—
83	C₂×#88322	—	2	1	no	—
84	C₂×#88322	—	2	1	no	—
85	C₂×#88322	—	2	1	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×#88322	C₂²⋊C₄	1	2	yes	—
90	C₂×#88322	C₂×C₈	1	2	yes	—
91	C₂×#88322	C₂×C₈	1	2	yes	—
92	C₂×#88322	M₁₆	1	2	yes	—
93	C₂×#88322	M₁₆	1	2	yes	—
94	C₂³×C₅₃	—	2	1	no	—
95	C₂³×C₅₃	—	2	1	no	—
96	C₂³×C₅₃	—	4	1	no	—
97	C₂³×C₅₃	—	4	1	no	—
98	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
99	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	4	1	no	—
109	C₂×C₄×C₅₃	—	4	1	no	—
110	C₂×C₄×C₅₃	—	4	1	no	—
111	C₂×C₄×C₅₃	—	4	1	no	—
112	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#121	1	2	yes	—
116	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
121	C₈×C₅₃	—	8	2	no	—
122	#88322	—	2	2	no	—
123	#88322	—	2	1	no	—
124	#88322	—	2	1	no	—
125	#88322	—	2	2	no	—
126	#88322	—	8	2	no	—
127	#88322	C₂²⋊C₈	1	4	yes	—
128	#8602	—	53	1	no	maximal, Sylow
129	C₂²×C₅₃	—	2	1	no	—
130	C₂²×C₅₃	—	2	1	no	—
131	C₂²×C₅₃	—	2	2	no	—
132	C₂²×C₅₃	—	4	2	no	—
133	C₂²×C₅₃	—	4	2	no	—
134	C₂²×C₅₃	—	4	1	no	—
135	C₂²×C₅₃	—	4	1	no	—
136	C₂²×C₅₃	#380	1	1	yes	characteristic
137	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
138	C₂²×C₅₃	#423	1	1	yes	characteristic
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	—	2	2	no	—
141	C₄×C₅₃	—	2	2	no	—
142	C₄×C₅₃	—	4	1	no	—
143	C₄×C₅₃	—	4	1	no	—
144	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₄×C₅₃	#407	1	1	yes	characteristic
146	C₄×C₅₃	#431	1	1	yes	characteristic
147	C₄×C₅₃	#432	1	1	yes	characteristic
148	C₂×#405	—	53	1	no	—
149	#1564	—	53	2	no	—
150	#1458	—	53	1	no	—
151	#1458	D₅₃	1	1	yes	characteristic
152	#1275	—	53	2	no	—
153	C₂×C₅₃	—	2	1	no	—
154	C₂×C₅₃	—	2	1	no	—
155	C₂×C₅₃	—	4	2	no	—
156	C₂×C₅₃	#1797	1	1	yes	characteristic
157	C₂×C₅₃	#1812	1	1	yes	characteristic
158	C₂×C₅₃	#1813	1	1	yes	characteristic
159	C₂²×M₁₆	—	53	1	no	—
160	C₂³×C₈	—	53	1	no	—
161	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
162	#431	—	53	2	no	—
163	#431	C₂×D₅₃	1	2	yes	—
164	#405	—	106	2	no	—
165	#405	—	106	2	no	—
166	C₂×#97	—	53	1	no	—
167	C₂×#97	—	53	1	no	—
168	C₂×C₄×C₈	—	53	1	no	—
169	C₂×C₄×C₈	—	53	1	no	—
170	#335	—	106	2	no	—
171	#334	—	106	2	no	—
172	#324	—	2	1	no	—
173	#324	—	106	1	no	—
174	#323	—	2	1	no	—
175	#323	—	106	1	no	—
176	C₅₃	#8602	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
178	C₂×M₁₆	—	53	4	no	—
179	C₂×M₁₆	—	106	2	no	—
180	C₂×M₁₆	—	106	2	no	—
181	C₂²×C₈	—	53	1	no	—
182	C₂²×C₈	—	53	1	no	—
183	C₂²×C₈	—	53	1	no	—
184	C₂²×C₈	—	53	1	no	—
185	C₂²×C₈	—	106	1	no	—
186	C₂²×C₈	—	106	1	no	—
187	C₂²×C₈	—	106	2	no	—
188	C₂²×C₈	—	106	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	C₂×C₄⋊C₄	Dic₁₀₆	1	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
193	C₂×C₂²⋊C₄	Dic₁₀₆	1	1	yes	characteristic
194	C₂×C₂²⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
196	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	106	2	no	—
199	C₂²⋊C₈	—	2	2	no	—
200	C₂²⋊C₈	—	106	2	no	—
201	#97	—	106	2	no	—
202	#97	—	106	2	no	—
203	C₄×C₈	—	2	2	no	—
204	C₄×C₈	—	53	2	no	—
205	C₄×C₈	—	53	2	no	—
206	C₄×C₈	—	106	1	no	—
207	C₄×C₈	—	106	1	no	—
208	C₄×C₈	—	106	2	no	—
209	C₂⁴	#11719919	1	1	yes	characteristic
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₈×D₅₃	1	2	yes	—
215	C₂²×C₄	#11719912	1	2	yes	—
216	C₂²×C₄	#11719919	1	1	yes	characteristic
217	C₂²×C₄	#11719919	1	1	yes	characteristic
218	C₂²×C₄	#11719919	1	1	yes	characteristic
219	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
222	M₁₆	—	106	8	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	53	2	no	—
225	C₂×C₈	—	53	2	no	—
226	C₂×C₈	—	53	2	no	—
227	C₂×C₈	—	53	1	no	—
228	C₂×C₈	—	53	1	no	—
229	C₂×C₈	—	53	2	no	—
230	C₂×C₈	—	53	2	no	—
231	C₂×C₈	—	106	4	no	—
232	C₂×C₈	—	106	4	no	—
233	C₂×C₈	—	106	1	no	—
234	C₂×C₈	—	106	1	no	—
235	C₂×C₈	—	106	1	no	—
236	C₂×C₈	—	106	1	no	—
237	C₂×C₈	—	106	1	no	—
238	C₂×C₈	—	106	1	no	—
239	C₂×C₈	—	212	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₄²	D₄×D₅₃	1	2	yes	—
247	C₄²	#11719947	1	2	yes	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	#13133146	1	2	yes	—
253	C₂³	C₂×#11719919	1	1	yes	characteristic
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	#13133146	1	2	yes	—
267	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
268	C₂×C₄	#13133173	1	1	yes	characteristic
269	C₂×C₄	#13133189	1	2	yes	—
270	C₂×C₄	#13133194	1	2	yes	—
271	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
272	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
273	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
274	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
275	C₈	—	8	2	no	—
276	C₈	—	53	4	no	—
277	C₈	—	106	2	no	—
278	C₈	—	106	1	no	—
279	C₈	—	106	1	no	—
280	C₈	—	106	2	no	—
281	C₈	—	424	2	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	#16810481	1	1	yes	characteristic
290	C₂²	C₂×#13133146	1	1	yes	characteristic
291	C₂²	#16810646	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
298	C₄	#16810585	1	1	yes	characteristic
299	C₄	#16810673	1	1	yes	characteristic
300	C₄	#16810677	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	4	2	no	—
304	C₂	#20274693	1	1	yes	characteristic
305	C₂	#20274759	1	1	yes	characteristic
306	C₂	#20274763	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133173
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1458	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1458	D₅₃
hypocenter	C₅₃	#8602
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1458

Complements

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

N	H ≅ G/N	type
2. #20273461	C₂	semidirect
4. #20272545	C₂	semidirect
25. C₂×#11719917	C₈	semidirect
176. C₅₃	#8602	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810579	2
#20273461	2
#20273029	2
#20272545	2
C₅₃×#1458	2
#8602	53