Subgroups — #27277291

Order 13568.

386 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
2	#20273474	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273154	C₂	1	2	yes	maximal
4	#20272606	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2965	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133153	—	2	1	no	—
7	#16810647	C₂²	1	2	yes	—
8	C₄×#11719917	C₂²	1	2	yes	—
9	#16810388	—	2	1	no	—
10	#16810376	—	2	1	no	—
11	#16810376	—	2	1	no	—
12	Q₈⋊#88322	—	2	2	no	—
13	D₄⋊#88322	—	2	2	no	—
14	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#446	C₄	1	1	yes	characteristic
16	C₅₃×#446	C₄	1	1	yes	characteristic
17	C₅₃×#436	C₄	1	1	yes	characteristic
18	C₅₃×#436	C₄	1	1	yes	characteristic
19	C₅₃×#435	—	2	1	no	—
20	C₅₃×#435	—	2	1	no	—
21	C₅₃×#431	—	2	1	no	—
22	C₅₃×#431	C₂²	1	1	yes	characteristic
23	C₄×M₁₆×C₅₃	—	2	1	no	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	—	2	2	no	—
27	#13133153	—	2	4	no	—
28	C₄⋊#88322	—	2	2	no	—
29	C₄⋊#88322	—	2	2	no	—
30	#13133128	—	2	2	no	—
31	C₄×#88322	—	2	2	no	—
32	C₂×#41×C₅₃	Q₈	1	1	yes	characteristic
33	C₂×M₁₆×C₅₃	—	2	1	no	—
34	C₂×M₁₆×C₅₃	—	2	1	no	—
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
39	C₄×D₄×C₅₃	—	2	1	no	—
40	C₄×D₄×C₅₃	—	2	1	no	—
41	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
43	#117×C₅₃	—	2	1	no	—
44	#117×C₅₃	Q₈	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	—	2	1	no	—
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₅₃	—	2	1	no	—
48	C₄⋊C₈×C₅₃	—	2	1	no	—
49	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²⋊C₈×C₅₃	—	2	1	no	—
52	C₂²⋊C₈×C₅₃	—	2	1	no	—
53	C₂²⋊C₈×C₅₃	—	2	1	no	—
54	#97×C₅₃	—	2	1	no	—
55	#97×C₅₃	—	2	1	no	—
56	C₄×C₈×C₅₃	—	2	1	no	—
57	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
61	#41×C₅₃	—	4	2	no	—
62	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂×D₄×C₅₃	—	2	1	no	—
64	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	—	2	1	no	—
66	C₂²×C₄×C₅₃	—	2	1	no	—
67	C₂²×C₄×C₅₃	—	2	1	no	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
71	M₁₆×C₅₃	—	4	2	no	—
72	M₁₆×C₅₃	—	4	2	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	1	no	—
75	C₂×C₈×C₅₃	—	2	1	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₅₃	—	2	1	no	—
82	C₄⋊C₄×C₅₃	—	2	1	no	—
83	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
84	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂²⋊C₄×C₅₃	—	2	1	no	—
86	C₂²⋊C₄×C₅₃	—	2	1	no	—
87	C₂²⋊C₄×C₅₃	—	2	1	no	—
88	C₄²×C₅₃	—	2	1	no	—
89	C₄²×C₅₃	—	2	1	no	—
90	C₄²×C₅₃	—	2	1	no	—
91	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅₃	#41	1	1	yes	characteristic
95	#11719917	—	2	4	no	—
96	#11719917	—	4	2	no	—
97	#11719917	—	8	2	no	—
98	C₂×#88322	—	4	2	no	—
99	C₂×#88322	—	4	2	no	—
100	C₂³×C₅₃	—	2	1	no	—
101	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
102	Q₈×C₅₃	—	2	2	no	—
103	D₄×C₅₃	—	2	2	no	—
104	D₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	4	1	no	—
115	C₂×C₄×C₅₃	—	4	1	no	—
116	C₂×C₄×C₅₃	—	4	1	no	—
117	C₂×C₄×C₅₃	—	4	1	no	—
118	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
124	C₈×C₅₃	—	2	1	no	—
125	C₈×C₅₃	—	2	1	no	—
126	C₈×C₅₃	—	2	1	no	—
127	C₈×C₅₃	—	4	1	no	—
128	C₈×C₅₃	—	4	1	no	—
129	C₈×C₅₃	#104	1	2	yes	—
130	#88322	—	4	4	no	—
131	#88322	—	8	2	no	—
132	#8673	—	53	1	no	maximal, Sylow
133	C₂²×C₅₃	—	2	1	no	—
134	C₂²×C₅₃	—	2	1	no	—
135	C₂²×C₅₃	—	2	1	no	—
136	C₂²×C₅₃	—	4	1	no	—
137	C₂²×C₅₃	—	4	1	no	—
138	C₂²×C₅₃	—	4	1	no	—
139	C₂²×C₅₃	#380	1	1	yes	characteristic
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	4	1	no	—
144	C₄×C₅₃	—	4	1	no	—
145	C₄×C₅₃	—	4	1	no	—
146	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
147	C₄×C₅₃	#421	1	1	yes	characteristic
148	C₄×C₅₃	#430	1	2	yes	—
149	C₄×C₅₃	#472	1	1	yes	characteristic
150	C₄×C₅₃	#473	1	1	yes	characteristic
151	#2965	D₅₃	1	1	yes	characteristic
152	C₄×#108	—	53	1	no	—
153	#1567	—	53	1	no	—
154	#1489	—	53	2	no	—
155	#1299	—	53	2	no	—
156	C₂×C₅₃	—	4	1	no	—
157	C₂×C₅₃	—	8	1	no	—
158	C₂×C₅₃	#1806	1	1	yes	characteristic
159	C₂×C₅₃	#1811	1	1	yes	characteristic
160	C₂×C₅₃	#1816	1	1	yes	characteristic
161	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
162	#446	Dic₅₃	1	1	yes	characteristic
163	#446	Dic₅₃	1	1	yes	characteristic
164	#436	Dic₅₃	1	1	yes	characteristic
165	#436	Dic₅₃	1	1	yes	characteristic
166	#435	—	2	1	no	—
167	#435	—	2	1	no	—
168	#431	—	2	1	no	—
169	#431	C₂×D₅₃	1	1	yes	characteristic
170	C₂×#108	—	106	1	no	—
171	#423	—	53	2	no	—
172	C₄×M₁₆	—	2	1	no	—
173	C₄×M₁₆	—	53	2	no	—
174	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
175	#341	—	106	1	no	—
176	#335	—	106	1	no	—
177	#335	—	106	1	no	—
178	Q₈⋊C₈	—	106	2	no	—
179	D₄⋊C₈	—	106	2	no	—
180	C₅₃	#8673	1	1	yes	characteristic, complemented, Sylow
181	C₂×#41	Dic₁₀₆	1	1	yes	characteristic
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	106	2	no	—
185	C₂×M₁₆	—	106	2	no	—
186	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
187	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
188	C₄×Q₈	D₂₁₂	1	1	yes	characteristic
189	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
190	C₄×D₄	—	2	1	no	—
191	C₄×D₄	—	2	1	no	—
192	C₄×D₄	D₂₁₂	1	1	yes	characteristic
193	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
194	#117	—	2	1	no	—
195	#117	Dic₁₀₆	1	1	yes	characteristic
196	C₂×C₄²	—	2	1	no	—
197	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
198	#108	—	106	4	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	106	2	no	—
202	C₄⋊C₈	—	106	2	no	—
203	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
204	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
205	C₂²⋊C₈	—	2	1	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	C₂²⋊C₈	—	2	1	no	—
208	#97	—	2	1	no	—
209	#97	—	2	1	no	—
210	#97	—	106	2	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	106	2	no	—
213	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
214	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
215	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
217	#41	—	4	2	no	—
218	C₂×Q₈	C₄⋊Dic₅₃	1	1	yes	characteristic
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	C₄⋊Dic₅₃	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
225	C₂²×C₄	#11719945	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	106	4	no	—
230	M₁₆	—	424	2	no	—
231	M₁₆	—	212	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	212	2	no	—
237	C₂×C₈	—	212	2	no	—
238	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
239	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
240	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
241	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
245	C₄⋊C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	2	1	no	—
248	C₂²⋊C₄	—	2	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
253	C₄²	#11719945	1	1	yes	characteristic
254	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
255	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
256	C₂³	—	2	1	no	—
257	C₂³	#13133196	1	1	yes	characteristic
258	Q₈	—	2	2	no	—
259	D₄	—	2	2	no	—
260	D₄	—	4	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
276	C₂×C₄	#13133176	1	1	yes	characteristic
277	C₂×C₄	#13133191	1	1	yes	characteristic
278	C₂×C₄	#13133191	1	1	yes	characteristic
279	C₂×C₄	#13133196	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	4	1	no	—
284	C₈	—	4	1	no	—
285	C₈	—	424	2	no	—
286	C₈	—	212	4	no	—
287	C₈	#13133144	1	2	yes	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	#16810483	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	C₂×#13133144	1	1	yes	characteristic
302	C₄	#16810640	1	1	yes	characteristic
303	C₄	#16810672	1	2	yes	—
304	C₄	#16810893	1	1	yes	characteristic
305	C₄	#16810898	1	1	yes	characteristic
306	C₂	—	4	1	no	—
307	C₂	—	8	1	no	—
308	C₂	#20274735	1	1	yes	characteristic
309	C₂	#20274757	1	1	yes	characteristic
310	C₂	#20274779	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2965	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂×#41×C₅₃	Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2965	D₅₃
hypocenter	C₅₃	#8673
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2965

Complements

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

N	H ≅ G/N	type
1. C₄×#13133153	C₂	semidirect
2. #20273474	C₂	semidirect
4. #20272606	C₂	semidirect
180. C₅₃	#8673	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133153	2
#20273474	2
#20273154	2
#20272606	2
C₅₃×#2965	2
#8673	53