Subgroups — #27353311

Order 13568.

572 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283834	C₂	1	2	yes	maximal
2	#20283357	C₂	1	1	yes	maximal, characteristic
3	#20273502	C₂	1	2	yes	maximal, complemented
4	#20273497	C₂	1	2	yes	maximal, complemented
5	#20273494	C₂	1	2	yes	maximal, complemented
6	#20273488	C₂	1	2	yes	maximal, complemented
7	#20273483	C₂	1	2	yes	maximal, complemented
8	#20273474	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#11719945	C₂²	1	1	yes	characteristic
11	#16810751	—	2	2	no	—
12	#16810698	—	2	2	no	—
13	#16810697	—	2	2	no	—
14	#16810696	—	2	2	no	—
15	#16810682	C₄	1	4	yes	—
16	#16810679	C₂²	1	2	yes	—
17	#16810675	C₂²	1	2	yes	—
18	#16810674	C₂²	1	2	yes	—
19	#16810674	C₂²	1	2	yes	—
20	#16810649	C₄	1	2	yes	—
21	#16810647	C₂²	1	2	yes	—
22	#16810645	C₂²	1	2	yes	—
23	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
24	#16810589	C₄	1	2	yes	—
25	C₄×#11719917	C₂²	1	1	yes	characteristic
26	#16810376	C₂²	1	2	yes	—
27	#16810376	C₂²	1	1	yes	characteristic
28	#16810376	C₂²	1	1	yes	characteristic
29	#16810375	C₂²	1	2	yes	—
30	#16810375	C₂²	1	2	yes	—
31	#16810373	C₂²	1	2	yes	—
32	#16810373	C₂²	1	2	yes	—
33	C₅₃×#432	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#335	C₂²	1	1	yes	characteristic
36	C₅₃×#335	C₂²	1	1	yes	characteristic
37	C₅₃×#334	C₂²	1	2	yes	—
38	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
39	C₂×#11719917	—	2	2	no	—
40	C₂×#11719917	C₂×C₄	1	2	yes	—
41	C₂×#11719912	—	2	2	no	—
42	C₂×C₈×D₅₃	—	2	2	no	—
43	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
44	C₄×Dic₁₀₆	Q₈	1	1	yes	characteristic
45	C₄×D₂₁₂	D₄	1	1	yes	characteristic
46	C₄×D₂₁₂	Q₈	1	1	yes	characteristic
47	C₄×C₂²⋊D₅₃	D₄	1	2	yes	—
48	C₄×C₂²⋊D₅₃	Q₈	1	2	yes	—
49	#13133176	C₂×C₄	1	1	yes	characteristic
50	#13133174	C₂×C₄	1	2	yes	—
51	C₄²×D₅₃	C₂³	1	2	yes	—
52	C₄⋊#88322	—	2	2	no	—
53	C₄⋊#88322	C₂³	1	2	yes	—
54	C₄⋊#88322	C₂³	1	2	yes	—
55	#13133146	—	2	2	no	—
56	#13133146	—	2	2	no	—
57	#13133146	C₂×C₄	1	4	yes	—
58	#13133146	C₂×C₄	1	4	yes	—
59	#13133129	—	2	2	no	—
60	#13133129	—	2	2	no	—
61	#13133129	C₂×C₄	1	4	yes	—
62	#13133129	C₂×C₄	1	4	yes	—
63	#13133128	C₂×C₄	1	2	yes	—
64	#13133127	—	2	2	no	—
65	C₄×#88322	C₂³	1	2	yes	—
66	C₈×Dic₅₃	—	2	2	no	—
67	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
70	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
71	C₂²⋊C₈×C₅₃	—	2	2	no	—
72	#97×C₅₃	—	2	2	no	—
73	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₂²⋊D₅₃	C₄⋊C₄	1	2	yes	—
80	#11719945	—	4	2	no	—
81	C₂×D₂₁₂	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
84	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
85	C₂×Dic₁₀₆	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	4	2	no	—
89	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
91	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
94	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₂²⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
96	#11719921	C₄⋊C₄	1	2	yes	—
97	#11719921	C₄⋊C₄	1	2	yes	—
98	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
100	#11719919	C₄⋊C₄	1	2	yes	—
101	#11719919	C₄⋊C₄	1	2	yes	—
102	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
103	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
104	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
105	#11719917	—	4	2	no	—
106	#11719917	—	8	2	no	—
107	C₂×#88322	—	2	2	no	—
108	C₂×#88322	—	2	2	no	—
109	C₂×#88322	C₂²×C₄	1	4	yes	—
110	#11719912	—	8	2	no	—
111	C₈×D₅₃	—	8	2	no	—
112	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	4	1	no	—
118	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂²⋊D₅₃	—	4	2	no	—
125	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
128	D₂₁₂	—	4	1	no	—
129	C₄×D₅₃	—	2	4	no	—
130	C₄×D₅₃	—	2	2	no	—
131	C₄×D₅₃	—	2	2	no	—
132	C₄×D₅₃	—	4	2	no	—
133	Dic₁₀₆	—	4	1	no	—
134	C₈×C₅₃	—	2	2	no	—
135	C₈×C₅₃	—	2	2	no	—
136	C₈×C₅₃	—	4	2	no	—
137	C₈×C₅₃	—	4	2	no	—
138	#88322	—	2	4	no	—
139	#88322	—	4	2	no	—
140	#88322	—	4	2	no	—
141	#17825	—	53	1	no	maximal, Sylow
142	C₂²×C₅₃	—	2	1	no	—
143	C₂²×C₅₃	—	2	1	no	—
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₂×D₅₃	—	2	2	no	—
147	C₂×D₅₃	—	2	2	no	—
148	C₂×D₅₃	—	2	2	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	#568	1	2	yes	—
154	C₄×C₅₃	#576	1	1	yes	characteristic
155	C₄×C₅₃	#578	1	1	yes	characteristic
156	Dic₅₃	—	2	2	no	—
157	Dic₅₃	—	2	2	no	—
158	Dic₅₃	—	2	2	no	—
159	#2989	—	53	2	no	—
160	#2922	—	53	1	no	—
161	#1574	—	53	2	no	—
162	#1573	—	53	2	no	—
163	#1572	—	53	2	no	—
164	#1571	—	53	2	no	—
165	#1569	—	53	2	no	—
166	#1568	D₅₃	1	1	yes	characteristic, complemented
167	#1567	—	53	1	no	—
168	C₂×C₅₃	—	4	1	no	—
169	C₂×C₅₃	#2905	1	1	yes	characteristic
170	C₂×C₅₃	#2913	1	1	yes	characteristic
171	C₂×C₅₃	#2916	1	1	yes	characteristic
172	D₅₃	—	4	2	no	—
173	C₄×#41	—	53	1	no	—
174	#446	—	106	2	no	—
175	#436	—	106	2	no	—
176	#435	—	106	2	no	—
177	#435	—	106	2	no	—
178	#433	—	53	4	no	—
179	#432	—	53	2	no	—
180	#432	C₂×D₅₃	1	2	yes	—
181	#431	—	53	2	no	—
182	#431	—	53	2	no	—
183	#431	—	53	2	no	—
184	#424	—	53	2	no	—
185	#423	—	53	2	no	—
186	#423	—	53	2	no	—
187	C₂×C₄⋊C₈	—	53	2	no	—
188	#408	—	53	2	no	—
189	C₄×M₁₆	—	53	1	no	—
190	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
191	#335	—	53	2	no	—
192	#335	—	53	2	no	—
193	#335	—	53	2	no	—
194	#335	—	53	1	no	—
195	#335	—	53	1	no	—
196	#335	C₂×D₅₃	1	1	yes	characteristic
197	#335	C₂×D₅₃	1	1	yes	characteristic
198	#334	—	53	2	no	—
199	#334	—	53	2	no	—
200	#334	C₂×D₅₃	1	2	yes	—
201	C₅₃	#17825	1	1	yes	characteristic, complemented, Sylow
202	C₂×#41	—	53	1	no	—
203	C₂×M₁₆	—	53	2	no	—
204	C₂×M₁₆	—	106	2	no	—
205	C₂×M₁₆	—	106	2	no	—
206	C₂²×C₈	—	106	2	no	—
207	C₂²×C₈	C₄×D₅₃	1	2	yes	—
208	C₄×Q₈	—	53	1	no	—
209	C₄×Q₈	—	53	1	no	—
210	C₄×D₄	—	53	2	no	—
211	C₄×D₄	—	53	2	no	—
212	C₄×D₄	—	53	1	no	—
213	C₄×D₄	—	53	1	no	—
214	#117	—	53	1	no	—
215	#117	—	53	2	no	—
216	C₂×C₄²	—	53	2	no	—
217	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
218	C₄⋊C₈	—	53	4	no	—
219	C₄⋊C₈	—	53	4	no	—
220	C₄⋊C₈	—	53	2	no	—
221	C₄⋊C₈	—	53	2	no	—
222	C₄⋊C₈	—	106	2	no	—
223	C₄⋊C₈	—	106	2	no	—
224	C₄⋊C₈	—	106	2	no	—
225	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
226	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
227	C₂²⋊C₈	—	2	2	no	—
228	C₂²⋊C₈	—	53	4	no	—
229	C₂²⋊C₈	—	53	4	no	—
230	C₂²⋊C₈	—	106	2	no	—
231	C₂²⋊C₈	—	106	2	no	—
232	#97	—	2	2	no	—
233	#97	—	53	2	no	—
234	#97	—	106	2	no	—
235	C₄×C₈	—	53	2	no	—
236	C₄×C₈	—	106	2	no	—
237	C₄×C₈	C₄×D₅₃	1	2	yes	—
238	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
240	#41	—	212	2	no	—
241	C₂×Q₈	—	53	1	no	—
242	C₂×D₄	—	53	2	no	—
243	C₂×D₄	—	53	1	no	—
244	C₂²×C₄	—	53	2	no	—
245	C₂²×C₄	—	53	2	no	—
246	C₂²×C₄	—	53	2	no	—
247	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
248	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
249	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
250	M₁₆	—	424	2	no	—
251	M₁₆	—	424	2	no	—
252	M₁₆	—	212	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	53	4	no	—
257	C₂×C₈	—	106	2	no	—
258	C₂×C₈	—	106	2	no	—
259	C₂×C₈	—	424	2	no	—
260	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
261	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
262	C₄⋊C₄	—	53	2	no	—
263	C₄⋊C₄	—	53	2	no	—
264	C₄⋊C₄	—	53	1	no	—
265	C₄⋊C₄	—	53	1	no	—
266	C₂²⋊C₄	—	53	2	no	—
267	C₂²⋊C₄	—	53	2	no	—
268	C₂²⋊C₄	—	53	2	no	—
269	C₄²	—	53	2	no	—
270	C₄²	—	53	2	no	—
271	C₄²	—	53	2	no	—
272	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
273	C₄²	D₄×D₅₃	1	1	yes	characteristic
274	C₄²	Q₈×D₅₃	1	1	yes	characteristic
275	C₄²	C₂³×D₅₃	1	1	yes	characteristic
276	C₂³	—	53	2	no	—
277	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
278	Q₈	—	212	1	no	—
279	D₄	—	212	1	no	—
280	D₄	—	212	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	53	2	no	—
286	C₂×C₄	—	53	2	no	—
287	C₂×C₄	—	53	2	no	—
288	C₂×C₄	—	106	4	no	—
289	C₂×C₄	—	106	2	no	—
290	C₂×C₄	—	106	2	no	—
291	C₂×C₄	—	212	2	no	—
292	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
293	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
294	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
295	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
297	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	2	no	—
300	C₈	—	4	2	no	—
301	C₈	—	4	2	no	—
302	C₈	—	106	4	no	—
303	C₈	—	212	2	no	—
304	C₈	—	212	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	106	2	no	—
309	C₂²	—	106	2	no	—
310	C₂²	—	106	2	no	—
311	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	106	2	no	—
317	C₄	—	106	2	no	—
318	C₄	—	106	2	no	—
319	C₄	#16811389	1	2	yes	—
320	C₄	#16811425	1	1	yes	characteristic
321	C₄	#16811437	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	212	2	no	—
324	C₂	#20283277	1	1	yes	characteristic
325	C₂	#20283312	1	1	yes	characteristic
326	C₂	#20283323	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1568	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1568	D₅₃
hypocenter	C₅₃	#17825
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
3. #20273502	C₂	semidirect
4. #20273497	C₂	semidirect
5. #20273494	C₂	semidirect
6. #20273488	C₂	semidirect
7. #20273483	C₂	semidirect
8. #20273474	C₂	semidirect
9. C₅₃×#1568	C₂	semidirect
166. #1568	D₅₃	semidirect
201. C₅₃	#17825	semidirect

Maximal subgroups

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

subgroup	index
#20283834	2
#20283357	2
#20273502	2
#20273497	2
#20273494	2
#20273488	2
#20273483	2
#20273474	2
C₅₃×#1568	2
#17825	53