Subgroups — #27402583

Order 13568.

846 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283599	C₂	1	2	yes	maximal
2	#20283315	C₂	1	2	yes	maximal, complemented
3	#20275496	C₂	1	2	yes	maximal, complemented
4	#20274950	C₂	1	4	yes	maximal, complemented
5	#20274879	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274779	C₂	1	2	yes	maximal, complemented
7	#20274594	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#443	C₂	1	1	yes	maximal, characteristic
9	#16811106	C₂²	1	4	yes	—
10	#16810735	—	2	2	no	—
11	#16810730	—	2	2	no	—
12	SD₁₆×Dic₅₃	—	2	4	no	—
13	C₂×#13133153	C₂²	1	1	yes	characteristic
14	#16810665	—	2	4	no	—
15	#16810665	C₂²	1	4	yes	—
16	#16810661	—	2	2	no	—
17	#16810661	C₂²	1	1	yes	characteristic
18	C₂×#13133151	—	2	2	no	—
19	C₂×#13133151	C₂²	1	2	yes	—
20	C₂×#13133149	C₂²	1	2	yes	—
21	#16810624	C₂²	1	4	yes	—
22	#16810578	C₂²	1	2	yes	—
23	#16810489	C₂²	1	2	yes	—
24	#16810391	C₂²	1	4	yes	—
25	#16810383	C₂²	1	2	yes	—
26	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#131×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#443	C₄	1	8	yes	complemented
29	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#104×C₅₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#11719917	C₂³	1	2	yes	—
33	Q₈×Dic₅₃	—	2	4	no	—
34	D₄×Dic₅₃	—	2	4	no	—
35	D₄×Dic₅₃	—	4	4	no	—
36	#13133176	—	2	4	no	—
37	#13133176	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₅₃	—	2	4	no	—
39	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
40	#13133153	—	2	2	no	—
41	#13133151	—	2	2	no	—
42	#13133151	—	2	2	no	—
43	#13133151	—	2	4	no	—
44	#13133151	D₄	1	2	yes	—
45	#13133149	—	2	2	no	—
46	#13133149	—	2	2	no	—
47	#13133149	—	2	4	no	—
48	#13133149	D₄	1	2	yes	—
49	#13133141	—	2	4	no	—
50	#13133138	—	2	4	no	—
51	#13133127	—	2	2	no	—
52	C₈×Dic₅₃	—	2	2	no	—
53	#13133124	—	4	2	no	—
54	C₂×#41×C₅₃	C₂³	1	2	yes	—
55	#135×C₅₃	C₂×C₄	1	4	yes	—
56	#135×C₅₃	Q₈	1	4	yes	—
57	C₂×Q₁₆×C₅₃	D₄	1	1	yes	characteristic
58	C₂×SD₁₆×C₅₃	D₄	1	2	yes	—
59	C₂×D₈×C₅₃	D₄	1	1	yes	characteristic
60	#131×C₅₃	—	2	4	no	—
61	#131×C₅₃	C₂×C₄	1	8	yes	complemented
62	C₂×M₁₆×C₅₃	—	2	2	no	—
63	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
64	C₂²×C₈×C₅₃	—	2	2	no	—
65	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	#108×C₅₃	C₂×C₄	1	4	yes	complemented
69	#104×C₅₃	C₂×C₄	1	8	yes	complemented
70	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
71	#41×C₅₃	—	2	2	no	—
72	#41×C₅₃	—	2	2	no	—
73	#41×C₅₃	C₂²×C₄	1	4	yes	—
74	#41×C₅₃	C₂×Q₈	1	4	yes	—
75	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
76	C₂×D₄×C₅₃	—	2	2	no	—
77	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₅₃	—	2	2	no	—
79	C₂²×C₄×C₅₃	—	2	1	no	—
80	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₅₃	—	2	4	no	—
82	C₂²×Dic₅₃	—	4	4	no	—
83	Q₁₆×C₅₃	C₄⋊C₄	1	4	yes	—
84	SD₁₆×C₅₃	C₄⋊C₄	1	8	yes	—
85	D₈×C₅₃	C₄⋊C₄	1	4	yes	—
86	M₁₆×C₅₃	—	2	2	no	—
87	M₁₆×C₅₃	—	2	8	no	—
88	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	2	8	no	—
91	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
98	C₂×C₈×C₅₃	#41	1	2	yes	—
99	C₄²×C₅₃	—	2	1	no	—
100	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
101	C₂²⋊Dic₅₃	—	2	2	no	—
102	C₂²⋊Dic₅₃	—	4	4	no	—
103	C₄⋊Dic₅₃	—	2	4	no	—
104	C₄⋊Dic₅₃	—	2	4	no	—
105	C₄⋊Dic₅₃	—	4	4	no	—
106	C₄⋊Dic₅₃	C₂×D₄	1	4	yes	—
107	C₄×Dic₅₃	—	2	2	no	—
108	C₄×Dic₅₃	—	4	4	no	—
109	#11719917	—	2	2	no	—
110	#11719917	—	2	2	no	—
111	C₂×#88322	—	2	2	no	—
112	C₂³×C₅₃	—	2	2	no	—
113	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
114	Q₈×C₅₃	—	2	2	no	—
115	Q₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
116	D₄×C₅₃	—	2	4	no	—
117	D₄×C₅₃	—	2	4	no	—
118	D₄×C₅₃	—	2	2	no	—
119	D₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	4	no	—
122	C₂×C₄×C₅₃	—	2	4	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	4	2	no	—
125	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
131	C₂×Dic₅₃	—	2	4	no	—
132	C₂×Dic₅₃	—	4	4	no	—
133	C₂×Dic₅₃	—	8	4	no	—
134	C₂×Dic₅₃	—	16	4	no	—
135	C₈×C₅₃	—	2	4	no	—
136	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
138	C₈×C₅₃	C₄×D₄	1	2	yes	—
139	C₈×C₅₃	C₄×D₄	1	2	yes	—
140	#88322	—	4	2	no	—
141	#22691	—	53	1	no	maximal, Sylow
142	C₂²×C₅₃	—	2	2	no	—
143	C₂²×C₅₃	—	2	4	no	—
144	C₂²×C₅₃	—	4	2	no	—
145	C₂²×C₅₃	—	4	2	no	—
146	C₂²×C₅₃	#520	1	1	yes	characteristic
147	C₂²×C₅₃	D₄²	1	1	yes	characteristic
148	C₂²×C₅₃	#554	1	1	yes	characteristic
149	C₄×C₅₃	—	2	4	no	—
150	C₄×C₅₃	—	4	2	no	—
151	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₄×C₅₃	#547	1	1	yes	characteristic
154	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
155	Dic₅₃	—	4	4	no	—
156	Dic₅₃	—	8	4	no	—
157	C₂×#443	D₅₃	1	1	yes	characteristic
158	#2952	—	53	2	no	—
159	#2913	—	53	2	no	—
160	#1951	—	53	2	no	—
161	#1860	—	53	4	no	—
162	#1846	—	53	1	no	—
163	#1816	—	53	2	no	—
164	#1772	—	53	1	no	—
165	C₂×C₅₃	—	2	1	no	—
166	C₂×C₅₃	—	2	1	no	—
167	C₂×C₅₃	—	4	4	no	—
168	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₅₃	#3290	1	2	yes	—
170	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
171	C₂×#131	C₂×D₅₃	1	2	yes	—
172	#520	—	53	4	no	—
173	#443	Dic₅₃	1	8	yes	complemented
174	#442	—	106	2	no	—
175	#441	—	106	2	no	—
176	C₄×SD₁₆	—	106	4	no	—
177	C₂×#108	—	53	1	no	—
178	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
179	#428	—	53	4	no	—
180	#428	—	106	4	no	—
181	#427	—	53	1	no	—
182	#427	—	106	2	no	—
183	C₂×#107	—	53	2	no	—
184	C₂×#107	—	106	2	no	—
185	C₂×#106	—	53	2	no	—
186	C₂×#104	C₂×D₅₃	1	2	yes	—
187	#417	—	53	4	no	—
188	#405	—	53	2	no	—
189	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
190	#382	—	53	2	no	—
191	#343	—	53	4	no	—
192	#339	—	53	2	no	—
193	C₅₃	#22691	1	1	yes	characteristic, complemented, Sylow
194	C₂×#41	C₂²×D₅₃	1	2	yes	—
195	#135	Dic₁₀₆	1	4	yes	—
196	#135	C₂×Dic₅₃	1	4	yes	—
197	C₂×Q₁₆	D₂₁₂	1	1	yes	characteristic
198	C₂×SD₁₆	D₂₁₂	1	2	yes	—
199	C₂×D₈	D₂₁₂	1	1	yes	characteristic
200	#131	—	2	4	no	—
201	#131	C₂×Dic₅₃	1	8	yes	complemented
202	C₂×M₁₆	—	2	2	no	—
203	C₂×M₁₆	—	53	2	no	—
204	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
208	C₄×Q₈	—	106	4	no	—
209	C₄×D₄	—	106	4	no	—
210	C₄×D₄	—	212	4	no	—
211	#117	—	53	2	no	—
212	#117	—	106	4	no	—
213	C₂×C₄⋊C₄	—	53	4	no	—
214	C₂×C₄⋊C₄	—	106	4	no	—
215	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
216	#108	—	106	2	no	—
217	#108	C₂×Dic₅₃	1	4	yes	complemented
218	#107	—	53	2	no	—
219	#107	—	106	4	no	—
220	#107	—	106	2	no	—
221	#107	—	106	2	no	—
222	#106	—	53	2	no	—
223	#106	—	106	4	no	—
224	#106	—	106	2	no	—
225	#106	—	106	2	no	—
226	#104	C₂×Dic₅₃	1	8	yes	complemented
227	#103	—	106	4	no	—
228	#102	—	106	4	no	—
229	#97	—	106	2	no	—
230	C₄×C₈	—	106	2	no	—
231	C₄×C₈	C₂×Dic₅₃	1	4	yes	complemented
232	#95	—	212	2	no	—
233	#41	—	2	2	no	—
234	#41	—	2	2	no	—
235	#41	C₂×Dic₁₀₆	1	4	yes	—
236	#41	C₂²×Dic₅₃	1	4	yes	—
237	C₂×Q₈	C₂×D₂₁₂	1	2	yes	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	C₂×D₂₁₂	1	2	yes	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	106	4	no	—
243	C₂²×C₄	—	212	4	no	—
244	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
245	Q₁₆	C₄⋊Dic₅₃	1	4	yes	—
246	SD₁₆	C₄⋊Dic₅₃	1	8	yes	—
247	D₈	C₄⋊Dic₅₃	1	4	yes	—
248	M₁₆	—	2	2	no	—
249	M₁₆	—	2	8	no	—
250	M₁₆	—	106	2	no	—
251	M₁₆	—	106	2	no	—
252	M₁₆	C₂²×Dic₅₃	1	4	yes	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	8	no	—
255	C₂×C₈	—	106	2	no	—
256	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
257	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
258	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
259	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
260	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
261	C₂×C₈	#11719947	1	2	yes	—
262	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
263	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
264	C₄⋊C₄	—	53	4	no	—
265	C₄⋊C₄	—	106	4	no	—
266	C₄⋊C₄	—	106	4	no	—
267	C₄⋊C₄	—	212	4	no	—
268	C₂²⋊C₄	—	106	2	no	—
269	C₂²⋊C₄	—	212	4	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	106	2	no	—
272	C₄²	—	212	4	no	—
273	C₄²	C₂²×Dic₅₃	1	2	yes	—
274	C₂³	—	2	2	no	—
275	C₂³	#13133263	1	1	yes	characteristic
276	Q₈	—	2	2	no	—
277	Q₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
278	D₄	—	2	4	no	—
279	D₄	—	2	4	no	—
280	D₄	—	2	2	no	—
281	D₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	848	4	no	—
288	C₂×C₄	—	106	4	no	—
289	C₂×C₄	—	424	4	no	—
290	C₂×C₄	—	212	4	no	—
291	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
294	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
295	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
296	C₂×C₄	#13133264	1	1	yes	characteristic
297	C₈	—	2	4	no	—
298	C₈	—	212	2	no	—
299	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
300	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
301	C₈	D₄×Dic₅₃	1	2	yes	—
302	C₈	D₄×Dic₅₃	1	2	yes	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	4	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	2	no	—
307	C₂²	#16811106	1	1	yes	characteristic
308	C₂²	D₄×D₂₁₂	1	1	yes	characteristic
309	C₂²	#16811330	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	4	2	no	—
312	C₄	—	424	4	no	—
313	C₄	—	212	4	no	—
314	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
315	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
316	C₄	#16811277	1	1	yes	characteristic
317	C₄	D₄×Dic₁₀₆	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	4	4	no	—
321	C₂	D₄×C₄⋊Dic₅₃	1	1	yes	characteristic
322	C₂	#20286502	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#443	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#135×C₅₃	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#443	D₅₃
hypocenter	C₅₃	#22691
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
2. #20283315	C₂	semidirect
3. #20275496	C₂	semidirect
4. #20274950	C₂	semidirect
5. #20274879	C₂	semidirect
6. #20274779	C₂	semidirect
7. #20274594	C₂	semidirect
28. C₅₃×#443	C₄	semidirect
61. #131×C₅₃	C₂×C₄	semidirect
68. #108×C₅₃	C₂×C₄	semidirect
69. #104×C₅₃	C₂×C₄	semidirect
70. C₄×C₈×C₅₃	C₂×C₄	semidirect
173. #443	Dic₅₃	semidirect
193. C₅₃	#22691	semidirect
201. #131	C₂×Dic₅₃	semidirect
217. #108	C₂×Dic₅₃	semidirect
226. #104	C₂×Dic₅₃	semidirect
231. C₄×C₈	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20283599	2
#20283315	2
#20275496	2
#20274950	2
#20274879	2
#20274779	2
#20274594	2
C₂×C₅₃×#443	2
#22691	53