Subgroups — #27299320

Order 13568.

442 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	#20283486	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810431	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810429	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810398	C₂	1	1	yes	maximal, characteristic, complemented
5	#108×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272810	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133239	C₂²	1	1	yes	characteristic
9	#16811091	—	2	1	no	—
10	C₂²×#11719908	C₂²	1	1	yes	characteristic
11	#16810717	C₄	1	2	yes	—
12	#16810710	—	2	2	no	—
13	#16810705	C₄	1	2	yes	—
14	C₂×#13133153	C₂²	1	1	yes	characteristic
15	#16810617	—	2	1	no	—
16	M₁₆×Dic₅₃	—	2	1	no	—
17	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	#16810431	C₄	1	4	yes	complemented
19	#16810429	C₄	1	4	yes	complemented
20	#16810398	—	2	2	no	—
21	#16810387	—	2	1	no	—
22	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#427	C₂²	1	1	yes	characteristic
25	C₂×#11719945	—	2	1	no	—
26	#13133239	—	2	1	no	—
27	#13133239	—	2	1	no	—
28	#13133239	C₂×C₄	1	2	yes	—
29	#13133239	C₂×C₄	1	2	yes	—
30	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
31	C₂×#11719913	—	2	1	no	—
32	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
33	C₂×#11719917	—	2	1	no	—
34	C₂²×#88322	D₄	1	1	yes	characteristic
35	#13133182	—	2	2	no	—
36	#13133179	—	2	2	no	—
37	#13133177	—	4	2	no	—
38	#13133174	—	4	1	no	—
39	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
40	C₄²×D₅₃	—	4	1	no	—
41	C₂×#11719908	—	2	1	no	—
42	C₂×#11719908	—	2	1	no	—
43	C₂×#11719908	C₂×C₄	1	2	yes	complemented
44	#13133153	—	2	1	no	—
45	#13133153	C₂×C₄	1	2	yes	complemented
46	#13133140	—	2	2	no	—
47	#13133137	—	2	2	no	—
48	#13133127	—	4	1	no	—
49	C₈×Dic₅₃	—	4	1	no	—
50	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
51	C₈×Dic₅₃	D₄	1	2	yes	—
52	C₂×M₁₆×C₅₃	—	2	1	no	—
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	#117×C₅₃	—	2	1	no	—
55	C₂×C₁₆×C₅₃	—	2	1	no	—
56	C₂×C₁₆×C₅₃	—	2	1	no	—
57	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
58	#108×C₅₃	—	2	1	no	—
59	#108×C₅₃	C₂×C₄	1	2	yes	complemented
60	#107×C₅₃	C₂×C₄	1	2	yes	—
61	#106×C₅₃	—	2	1	no	—
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₅₃	—	4	1	no	—
64	C₂²×Dic₅₃	SD₁₆	1	2	yes	—
65	#11719945	—	2	2	no	—
66	#11719945	—	2	2	no	—
67	#11719945	—	4	1	no	—
68	#11719945	—	4	1	no	—
69	C₂×D₂₁₂	—	2	1	no	—
70	C₂×C₄×D₅₃	—	4	1	no	—
71	C₂×C₄×D₅₃	—	4	2	no	—
72	C₂×Dic₁₀₆	—	2	1	no	—
73	M₁₆×C₅₃	—	2	2	no	—
74	M₁₆×C₅₃	—	4	1	no	—
75	C₂×C₈×C₅₃	—	4	1	no	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
81	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
82	C₄⋊C₄×C₅₃	—	2	2	no	—
83	C₂²⋊C₄×C₅₃	—	4	1	no	—
84	C₄²×C₅₃	—	4	1	no	—
85	#11719921	—	4	2	no	—
86	#11719919	—	4	2	no	—
87	C₄×Dic₅₃	—	4	2	no	—
88	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
89	C₄×Dic₅₃	D₈	1	1	yes	characteristic
90	C₄×Dic₅₃	D₈	1	1	yes	characteristic
91	#11719917	—	2	2	no	—
92	#11719917	—	4	1	no	—
93	C₂×#88322	—	2	1	no	—
94	C₂×#88322	—	2	1	no	—
95	C₂×#88322	—	4	1	no	—
96	C₂×#88322	C₂²⋊C₄	1	2	yes	—
97	Dic₂₁₂	—	2	1	no	—
98	Dic₂₁₂	C₄²	1	2	yes	—
99	D₄₂₄	—	2	1	no	—
100	D₄₂₄	C₄²	1	2	yes	—
101	#11719913	—	2	2	no	—
102	#11719913	—	2	2	no	—
103	C₁₆×C₅₃	—	2	2	no	—
104	#11719908	—	2	2	no	—
105	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
106	C₂²×D₅₃	—	4	1	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
114	C₂²⋊D₅₃	—	4	2	no	—
115	C₂²⋊D₅₃	—	4	2	no	—
116	C₂×Dic₅₃	—	4	2	no	—
117	C₂×Dic₅₃	—	4	1	no	—
118	C₂×Dic₅₃	#102	1	2	yes	—
119	C₂×Dic₅₃	#102	1	2	yes	—
120	D₂₁₂	—	2	2	no	—
121	D₂₁₂	—	4	1	no	—
122	C₄×D₅₃	—	4	2	no	—
123	C₄×D₅₃	—	4	2	no	—
124	C₄×D₅₃	—	8	4	no	—
125	Dic₁₀₆	—	2	2	no	—
126	Dic₁₀₆	—	4	1	no	—
127	C₈×C₅₃	—	4	2	no	—
128	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
129	C₈×C₅₃	#117	1	1	yes	characteristic
130	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
131	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
132	#88322	—	2	2	no	—
133	#88322	—	4	2	no	—
134	#12640	—	53	1	no	maximal, Sylow
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
137	C₂²×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
138	C₂²×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
139	C₂×D₅₃	—	4	2	no	—
140	C₂×D₅₃	—	8	2	no	—
141	C₄×C₅₃	—	8	2	no	—
142	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₄×C₅₃	#416	1	1	yes	characteristic
144	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
145	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
146	Dic₅₃	—	2	2	no	—
147	Dic₅₃	—	2	2	no	—
148	Dic₅₃	—	4	2	no	—
149	C₄×#135	—	53	1	no	—
150	C₂×#359	—	53	1	no	—
151	C₂×#359	—	53	1	no	—
152	C₂×#346	—	53	1	no	—
153	C₄×#108	—	53	1	no	—
154	#1382	—	53	1	no	—
155	#1382	D₅₃	1	1	yes	characteristic, complemented
156	C₂×C₅₃	—	2	1	no	—
157	C₂×C₅₃	—	2	1	no	—
158	C₂×C₅₃	C₄×#102	1	1	yes	characteristic
159	C₂×C₅₃	#2143	1	2	yes	—
160	D₅₃	—	8	2	no	—
161	C₂×#135	—	53	1	no	—
162	C₄×#41	—	106	1	no	—
163	C₂²×C₁₆	—	53	1	no	—
164	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
165	C₄×Q₁₆	—	53	2	no	—
166	C₄×SD₁₆	—	106	2	no	—
167	C₄×D₈	—	53	2	no	—
168	C₂×#108	—	53	1	no	—
169	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
170	#427	C₂×D₅₃	1	1	yes	characteristic
171	#416	—	106	1	no	—
172	C₄×M₁₆	—	106	1	no	—
173	C₂×C₄×C₈	—	53	1	no	—
174	#359	—	53	4	no	—
175	#359	—	53	4	no	—
176	#346	—	106	2	no	—
177	#341	—	106	1	no	—
178	C₅₃	#12640	1	1	yes	characteristic, complemented, Sylow
179	C₂×#41	—	106	1	no	—
180	#135	—	53	2	no	—
181	#135	—	53	2	no	—
182	#135	—	106	1	no	—
183	#135	—	106	1	no	—
184	C₂×Q₁₆	—	53	1	no	—
185	C₂×SD₁₆	—	106	1	no	—
186	C₂×D₈	—	53	1	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	106	1	no	—
189	C₂²×C₈	—	53	1	no	—
190	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
191	C₄×Q₈	—	106	2	no	—
192	C₄×D₄	—	106	2	no	—
193	C₄×D₄	—	212	2	no	—
194	#117	—	2	1	no	—
195	#117	—	212	1	no	—
196	C₂×C₄²	—	53	1	no	—
197	C₂×C₄²	—	212	1	no	—
198	C₂×C₁₆	—	2	1	no	—
199	C₂×C₁₆	—	2	1	no	—
200	C₂×C₁₆	—	53	2	no	—
201	C₂×C₁₆	—	106	1	no	—
202	C₂×C₁₆	—	106	1	no	—
203	C₂×C₁₆	C₄×D₅₃	1	2	yes	complemented
204	#108	—	2	1	no	—
205	#108	—	53	2	no	—
206	#108	—	106	1	no	—
207	#108	C₄×D₅₃	1	2	yes	complemented
208	#107	C₄×D₅₃	1	2	yes	—
209	#106	—	2	1	no	—
210	#103	—	106	2	no	—
211	#102	—	106	2	no	—
212	#97	—	212	1	no	—
213	C₄×C₈	—	53	2	no	—
214	C₄×C₈	—	53	2	no	—
215	C₄×C₈	—	212	1	no	—
216	#41	—	106	2	no	—
217	#41	—	106	2	no	—
218	#41	—	212	1	no	—
219	#41	—	212	1	no	—
220	C₂×Q₈	—	106	1	no	—
221	C₂×D₄	—	106	1	no	—
222	C₂×D₄	—	212	1	no	—
223	C₂²×C₄	—	53	2	no	—
224	C₂²×C₄	—	212	2	no	—
225	C₂²×C₄	—	212	1	no	—
226	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
227	Q₁₆	—	53	2	no	—
228	Q₁₆	—	106	1	no	—
229	SD₁₆	—	106	2	no	—
230	SD₁₆	—	106	2	no	—
231	D₈	—	53	2	no	—
232	D₈	—	106	1	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	4	1	no	—
235	M₁₆	—	106	2	no	—
236	M₁₆	—	212	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	53	2	no	—
239	C₂×C₈	—	106	1	no	—
240	C₂×C₈	—	106	1	no	—
241	C₂×C₈	—	212	1	no	—
242	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
245	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
246	C₂×C₈	#11719947	1	1	yes	characteristic
247	C₂×C₈	#11719949	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	212	2	no	—
250	C₂²⋊C₄	—	4	1	no	—
251	C₂²⋊C₄	—	212	2	no	—
252	C₄²	—	4	1	no	—
253	C₄²	—	53	2	no	—
254	C₄²	—	53	1	no	—
255	C₄²	—	53	1	no	—
256	C₄²	—	212	2	no	—
257	C₁₆	—	2	2	no	—
258	C₁₆	—	106	2	no	—
259	C₂³	—	212	1	no	—
260	C₂³	SD₁₆×D₅₃	1	1	yes	characteristic
261	Q₈	—	106	2	no	—
262	Q₈	—	212	1	no	—
263	D₄	—	106	2	no	—
264	D₄	—	212	2	no	—
265	D₄	—	212	2	no	—
266	D₄	—	212	1	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	53	2	no	—
269	C₂×C₄	—	53	2	no	—
270	C₂×C₄	—	424	4	no	—
271	C₂×C₄	—	212	2	no	—
272	C₂×C₄	—	212	1	no	—
273	C₂×C₄	—	212	2	no	—
274	C₂×C₄	—	212	2	no	—
275	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
276	C₂×C₄	#13133177	1	1	yes	characteristic
277	C₂×C₄	#13133179	1	1	yes	characteristic
278	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
279	C₂×C₄	#13133238	1	1	yes	characteristic
280	C₂×C₄	#13133240	1	1	yes	characteristic
281	C₈	—	4	2	no	—
282	C₈	—	106	2	no	—
283	C₈	—	212	2	no	—
284	C₈	C₄²×D₅₃	1	1	yes	characteristic
285	C₈	#13133175	1	1	yes	characteristic
286	C₈	#13133177	1	1	yes	characteristic
287	C₈	#13133179	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	424	2	no	—
290	C₂²	—	212	2	no	—
291	C₂²	#102×D₅₃	1	1	yes	characteristic
292	C₂²	#16810708	1	1	yes	characteristic
293	C₂²	#16810710	1	1	yes	characteristic
294	C₄	—	8	2	no	—
295	C₄	—	106	2	no	—
296	C₄	—	106	2	no	—
297	C₄	—	212	2	no	—
298	C₄	#16810470	1	1	yes	characteristic
299	C₄	#16810618	1	1	yes	characteristic
300	C₄	#16810703	1	1	yes	characteristic
301	C₄	#16810705	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	424	2	no	—
305	C₂	#20274529	1	1	yes	characteristic
306	C₂	#20276441	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1382	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂×#13133239	C₂²
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1382	D₅₃
hypocenter	C₅₃	#12640
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1382

Complements

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

N	H ≅ G/N	type
4. C₂×#16810398	C₂	semidirect
5. #108×Dic₅₃	C₂	semidirect
6. #20272810	C₂	semidirect
7. C₅₃×#1382	C₂	semidirect
18. #16810431	C₄	semidirect
19. #16810429	C₄	semidirect
43. C₂×#11719908	C₂×C₄	semidirect
45. #13133153	C₂×C₄	semidirect
57. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
59. #108×C₅₃	C₂×C₄	semidirect
155. #1382	D₅₃	semidirect
178. C₅₃	#12640	semidirect
203. C₂×C₁₆	C₄×D₅₃	semidirect
207. #108	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20283486	2
C₂×#16810431	2
C₂×#16810429	2
C₂×#16810398	2
#108×Dic₅₃	2
#20272810	2
C₅₃×#1382	2
#12640	53