Subgroups — #27299359

Order 13568.

428 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810741	C₂	1	1	yes	maximal, characteristic
2	#20276462	C₂	1	1	yes	maximal, characteristic
3	#20276450	C₂	1	1	yes	maximal, characteristic
4	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272831	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272828	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133239	C₂²	1	1	yes	characteristic
9	C₂×#13133223	—	2	1	no	—
10	C₂×#13133157	C₂²	1	1	yes	characteristic
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	#16810741	—	2	1	no	—
13	#16810741	—	2	1	no	—
14	#16810741	C₄	1	2	yes	complemented
15	#16810741	C₄	1	2	yes	complemented
16	C₂×#13133153	C₂²	1	1	yes	characteristic
17	C₂×#13133145	—	2	1	no	—
18	#16810438	C₄	1	2	yes	complemented
19	#16810435	C₄	1	2	yes	complemented
20	#16810432	C₄	1	4	yes	complemented
21	#16810399	—	2	1	no	—
22	C₄×#11719908	—	2	1	no	—
23	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#427	C₂²	1	1	yes	characteristic
25	C₅₃×#405	—	2	1	no	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#339	—	2	1	no	—
28	C₂×#11719945	—	2	1	no	—
29	#13133239	—	2	1	no	—
30	#13133239	—	2	1	no	—
31	#13133239	C₂×C₄	1	2	yes	—
32	#13133239	C₂×C₄	1	2	yes	—
33	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
34	C₂×#11719913	—	2	1	no	—
35	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
36	#13133223	—	2	2	no	—
37	#13133223	—	2	2	no	—
38	#13133223	—	4	1	no	—
39	#13133223	—	4	1	no	—
40	C₂×#11719917	—	2	1	no	—
41	C₂×#11719912	—	4	1	no	—
42	C₂×C₈×D₅₃	—	4	1	no	—
43	#13133157	—	2	1	no	—
44	#13133157	—	2	1	no	—
45	#13133157	C₂×C₄	1	2	yes	—
46	#13133157	C₂×C₄	1	2	yes	—
47	C₂×#11719908	—	2	1	no	—
48	C₂×#11719908	—	2	1	no	—
49	#13133153	—	2	1	no	—
50	#13133153	C₂×C₄	1	2	yes	—
51	#13133145	—	2	2	no	—
52	#13133145	—	2	2	no	—
53	C₂×M₁₆×C₅₃	—	2	1	no	—
54	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	#117×C₅₃	—	2	1	no	—
57	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
58	#108×C₅₃	—	2	1	no	—
59	#108×C₅₃	C₂×C₄	1	2	yes	—
60	#107×C₅₃	—	2	1	no	—
61	#106×C₅₃	C₂×C₄	1	2	yes	—
62	#97×C₅₃	—	4	1	no	—
63	C₄×C₈×C₅₃	—	4	1	no	—
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
67	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	—	2	1	no	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₂²⋊D₅₃	—	4	1	no	—
71	#11719945	—	2	2	no	—
72	#11719945	—	2	2	no	—
73	#11719945	—	4	1	no	—
74	#11719945	—	4	1	no	—
75	C₂×D₂₁₂	—	2	1	no	—
76	C₂×C₄×D₅₃	—	4	1	no	—
77	C₂×Dic₁₀₆	—	2	1	no	—
78	M₁₆×C₅₃	—	2	2	no	—
79	M₁₆×C₅₃	—	4	1	no	—
80	C₂×C₈×C₅₃	—	4	1	no	—
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
84	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
85	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
86	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
92	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
93	C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₂²⋊C₄×C₅₃	—	4	1	no	—
95	C₄²×C₅₃	—	2	1	no	—
96	C₄²×C₅₃	—	4	1	no	—
97	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
99	#11719917	—	2	2	no	—
100	#11719917	—	4	1	no	—
101	C₂×#88322	—	4	1	no	—
102	Dic₂₁₂	—	2	1	no	—
103	Dic₂₁₂	C₄²	1	2	yes	—
104	D₄₂₄	—	2	1	no	—
105	D₄₂₄	C₄²	1	2	yes	—
106	#11719913	—	2	2	no	—
107	#11719913	—	2	2	no	—
108	#11719912	—	4	2	no	—
109	#11719912	—	4	2	no	—
110	C₈×D₅₃	—	4	2	no	—
111	C₈×D₅₃	—	4	2	no	—
112	#11719908	—	2	2	no	—
113	#11719908	—	2	2	no	—
114	C₂³×C₅₃	#135	1	1	yes	characteristic
115	C₂²×D₅₃	—	4	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	4	2	no	—
119	C₂×C₄×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
126	C₂²⋊D₅₃	—	4	2	no	—
127	C₂²⋊D₅₃	—	4	2	no	—
128	C₂×Dic₅₃	—	4	1	no	—
129	D₂₁₂	—	2	2	no	—
130	D₂₁₂	—	4	1	no	—
131	C₄×D₅₃	—	4	2	no	—
132	C₄×D₅₃	—	4	2	no	—
133	Dic₁₀₆	—	2	2	no	—
134	Dic₁₀₆	—	4	1	no	—
135	C₈×C₅₃	—	4	2	no	—
136	C₈×C₅₃	#102	1	1	yes	characteristic
137	C₈×C₅₃	#102	1	1	yes	characteristic
138	C₈×C₅₃	#102	1	1	yes	characteristic
139	C₈×C₅₃	#102	1	1	yes	characteristic
140	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
141	C₈×C₅₃	#117	1	1	yes	characteristic
142	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
143	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
144	#88322	—	4	2	no	—
145	#12646	—	53	1	no	maximal, Sylow
146	C₂²×C₅₃	—	4	1	no	—
147	C₂²×C₅₃	#416	1	1	yes	characteristic
148	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
149	C₂²×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
150	C₂×D₅₃	—	4	2	no	—
151	C₂×D₅₃	—	8	2	no	—
152	C₄×C₅₃	—	2	2	no	—
153	C₄×C₅₃	—	4	1	no	—
154	C₄×C₅₃	—	8	2	no	—
155	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
156	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
157	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
158	C₄×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
159	Dic₅₃	—	4	2	no	—
160	C₂×#443	—	53	1	no	—
161	#2148	—	53	1	no	—
162	#2145	—	53	1	no	—
163	C₄×M₃₂	—	53	1	no	—
164	#1772	D₅₃	1	1	yes	characteristic, complemented
165	#1389	—	53	1	no	—
166	#1388	—	53	1	no	—
167	C₂×C₅₃	—	2	1	no	—
168	C₂×C₅₃	—	2	1	no	—
169	C₂×C₅₃	—	2	1	no	—
170	C₂×C₅₃	C₄×#102	1	1	yes	characteristic
171	D₅₃	—	8	2	no	—
172	C₂×#135	—	53	1	no	—
173	C₂×#131	—	106	1	no	—
174	C₂×M₃₂	—	53	1	no	—
175	C₂×M₃₂	—	53	1	no	—
176	#443	—	53	2	no	—
177	#443	—	53	2	no	—
178	#443	—	106	1	no	—
179	#443	—	106	1	no	—
180	C₂×#108	—	53	1	no	—
181	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
182	#427	C₂×D₅₃	1	1	yes	characteristic
183	C₂×#104	—	106	1	no	—
184	#405	—	2	1	no	—
185	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
186	#362	—	53	2	no	—
187	#361	—	53	2	no	—
188	#360	—	53	4	no	—
189	#346	—	106	1	no	—
190	C₄×C₁₆	—	106	1	no	—
191	#339	—	2	1	no	—
192	C₅₃	#12646	1	1	yes	characteristic, complemented, Sylow
193	C₂×#41	—	106	1	no	—
194	#135	—	53	2	no	—
195	#135	—	53	2	no	—
196	#135	—	106	1	no	—
197	#135	—	106	1	no	—
198	C₂×Q₁₆	—	53	1	no	—
199	C₂×SD₁₆	—	106	1	no	—
200	C₂×D₈	—	53	1	no	—
201	#131	—	106	2	no	—
202	#131	—	106	2	no	—
203	#131	—	212	1	no	—
204	#131	—	212	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	106	1	no	—
207	C₂×M₁₆	—	212	1	no	—
208	C₂²×C₈	—	212	1	no	—
209	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
211	#117	—	2	1	no	—
212	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
213	M₃₂	—	53	2	no	—
214	M₃₂	—	53	2	no	—
215	M₃₂	—	106	1	no	—
216	M₃₂	—	106	1	no	—
217	C₂×C₁₆	—	106	1	no	—
218	C₂×C₁₆	—	106	1	no	—
219	#108	—	2	1	no	—
220	#108	—	53	2	no	—
221	#108	—	106	1	no	—
222	#108	C₄×D₅₃	1	2	yes	—
223	#107	—	2	1	no	—
224	#106	C₄×D₅₃	1	2	yes	—
225	#104	—	106	2	no	—
226	#104	—	106	2	no	—
227	#97	—	4	1	no	—
228	C₄×C₈	—	4	1	no	—
229	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
230	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
231	C₄×C₈	D₂₁₂	1	1	yes	characteristic
232	C₄×C₈	D₂₁₂	1	1	yes	characteristic
233	#41	—	106	2	no	—
234	#41	—	106	2	no	—
235	#41	—	212	1	no	—
236	#41	—	212	1	no	—
237	C₂×Q₈	—	106	1	no	—
238	C₂×D₄	—	106	1	no	—
239	C₂×D₄	—	212	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	212	1	no	—
242	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
243	Q₁₆	—	53	2	no	—
244	Q₁₆	—	106	1	no	—
245	SD₁₆	—	106	2	no	—
246	SD₁₆	—	106	2	no	—
247	D₈	—	53	2	no	—
248	D₈	—	106	1	no	—
249	M₁₆	—	2	2	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	106	2	no	—
252	M₁₆	—	212	1	no	—
253	M₁₆	—	212	2	no	—
254	M₁₆	—	212	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	212	1	no	—
257	C₂×C₈	—	212	2	no	—
258	C₂×C₈	—	212	2	no	—
259	C₂×C₈	#11719921	1	1	yes	characteristic
260	C₂×C₈	#11719921	1	1	yes	characteristic
261	C₂×C₈	#11719922	1	1	yes	characteristic
262	C₂×C₈	#11719922	1	1	yes	characteristic
263	C₂×C₈	#11719924	1	1	yes	characteristic
264	C₂×C₈	#11719924	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
268	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
269	C₂×C₈	#11719945	1	1	yes	characteristic
270	C₂×C₈	#11719945	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	4	1	no	—
275	C₄²	#11719921	1	1	yes	characteristic
276	C₄²	#11719921	1	1	yes	characteristic
277	C₁₆	—	106	2	no	—
278	C₁₆	—	106	2	no	—
279	C₂³	—	212	1	no	—
280	C₂³	#13133237	1	1	yes	characteristic
281	Q₈	—	106	2	no	—
282	Q₈	—	212	1	no	—
283	D₄	—	106	2	no	—
284	D₄	—	212	2	no	—
285	D₄	—	212	2	no	—
286	D₄	—	212	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	212	1	no	—
292	C₂×C₄	—	212	2	no	—
293	C₂×C₄	—	212	2	no	—
294	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
295	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
296	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
297	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
298	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
299	C₂×C₄	#13133237	1	1	yes	characteristic
300	C₈	—	4	2	no	—
301	C₈	—	212	2	no	—
302	C₈	#13133137	1	1	yes	characteristic
303	C₈	#13133137	1	1	yes	characteristic
304	C₈	#13133137	1	1	yes	characteristic
305	C₈	#13133137	1	1	yes	characteristic
306	C₈	C₄²×D₅₃	1	1	yes	characteristic
307	C₈	#13133174	1	1	yes	characteristic
308	C₈	C₄×D₂₁₂	1	1	yes	characteristic
309	C₈	C₄×D₂₁₂	1	1	yes	characteristic
310	C₂²	—	4	1	no	—
311	C₂²	—	424	2	no	—
312	C₂²	—	212	2	no	—
313	C₂²	#16810617	1	1	yes	characteristic
314	C₂²	C₄×#11719922	1	1	yes	characteristic
315	C₂²	C₄×#11719924	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	4	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	212	2	no	—
320	C₄	C₄×#11719921	1	1	yes	characteristic
321	C₄	C₂×#13133137	1	1	yes	characteristic
322	C₄	C₄×#11719922	1	1	yes	characteristic
323	C₄	C₄×#11719924	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	424	2	no	—
328	C₂	C₄×#13133137	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133137
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1772	C₂
socle	C₂×C₅₃	C₄×#102
prime radical	C₂×#13133239	C₂²
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1772	D₅₃
hypocenter	C₅₃	#12646
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
4. C₄×#13133157	C₂	semidirect
5. #20272831	C₂	semidirect
6. #20272828	C₂	semidirect
7. C₅₃×#1772	C₂	semidirect
14. #16810741	C₄	semidirect
15. #16810741	C₄	semidirect
18. #16810438	C₄	semidirect
19. #16810435	C₄	semidirect
20. #16810432	C₄	semidirect
164. #1772	D₅₃	semidirect
192. C₅₃	#12646	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810741	2
#20276462	2
#20276450	2
C₄×#13133157	2
#20272831	2
#20272828	2
C₅₃×#1772	2
#12646	53