Subgroups — #27274785

Order 13568.

412 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283360	C₂	1	1	yes	maximal, characteristic
2	#20273098	C₂	1	1	yes	maximal, characteristic
3	#20273095	C₂	1	1	yes	maximal, characteristic
4	#20273008	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810345	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272553	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11719945	C₂²	1	1	yes	characteristic
9	#16810685	—	2	2	no	—
10	#16810676	C₂²	1	1	yes	characteristic
11	#16810674	—	2	1	no	—
12	#16810674	—	2	1	no	—
13	#16810651	C₄	1	2	yes	—
14	#16810591	C₄	1	2	yes	—
15	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
16	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
17	#16810355	C₄	1	2	yes	—
18	#16810353	C₄	1	2	yes	—
19	#16810352	C₄	1	2	yes	—
20	#16810350	C₄	1	2	yes	—
21	#16810345	—	2	2	no	—
22	#16810345	—	2	1	no	—
23	#16810344	—	2	1	no	—
24	C₅₃×#431	C₂²	1	1	yes	characteristic
25	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
28	C₂²×#88322	—	2	1	no	—
29	C₂²×#88322	C₂×C₄	1	2	yes	—
30	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
32	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
33	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
34	C₄×C₂²⋊D₅₃	—	2	1	no	—
35	C₄×C₂²⋊D₅₃	—	2	1	no	—
36	#13133176	C₂×C₄	1	1	yes	characteristic
37	#13133174	—	2	1	no	—
38	C₄²×D₅₃	—	2	1	no	—
39	C₄⋊#88322	C₂×C₄	1	2	yes	—
40	#13133146	—	2	2	no	—
41	#13133146	—	2	2	no	—
42	C₂²⋊#88322	—	2	1	no	—
43	#13133129	—	2	2	no	—
44	#13133129	—	2	2	no	—
45	C₄×#88322	—	2	1	no	—
46	C₄×#88322	—	2	1	no	—
47	C₄×#88322	C₂×C₄	1	2	yes	—
48	C₄×#88322	D₄	1	1	yes	characteristic
49	C₄×#88322	D₄	1	1	yes	characteristic
50	C₄×#88322	D₄	1	2	yes	—
51	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₅₃	—	2	1	no	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
55	C₂²⋊C₈×C₅₃	—	2	1	no	—
56	#97×C₅₃	—	2	1	no	—
57	C₄×C₈×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	—	2	1	no	—
59	C₄×C₈×C₅₃	—	2	1	no	—
60	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
64	C₂×C₂²⋊D₅₃	—	2	1	no	—
65	#11719945	—	2	2	no	—
66	#11719945	—	2	2	no	—
67	C₂×D₂₁₂	C₄²	1	1	yes	characteristic
68	C₂×C₄×D₅₃	—	2	1	no	—
69	C₂×C₄×D₅₃	—	2	1	no	—
70	C₂×C₄×D₅₃	—	2	1	no	—
71	C₂×Dic₁₀₆	C₄²	1	1	yes	characteristic
72	M₁₆×C₅₃	—	2	2	no	—
73	M₁₆×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	4	2	no	—
78	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂²⋊Dic₅₃	—	2	1	no	—
83	#11719921	—	2	1	no	—
84	#11719921	—	2	1	no	—
85	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
86	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
87	#11719919	—	2	1	no	—
88	#11719919	—	2	1	no	—
89	C₄×Dic₅₃	—	2	1	no	—
90	C₄×Dic₅₃	—	2	1	no	—
91	C₄×Dic₅₃	—	2	1	no	—
92	C₂×#88322	—	2	2	no	—
93	C₂×#88322	—	2	2	no	—
94	C₂×#88322	—	2	2	no	—
95	C₂×#88322	—	2	2	no	—
96	C₂×#88322	—	4	2	no	—
97	C₂×#88322	C₂²⋊C₄	1	2	yes	—
98	C₂×#88322	C₂²⋊C₄	1	2	yes	—
99	C₂³×C₅₃	#117	1	1	yes	characteristic
100	C₂²×D₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#131	1	2	yes	—
110	C₂×C₄×C₅₃	#131	1	2	yes	—
111	C₂²⋊D₅₃	—	4	2	no	—
112	C₂×Dic₅₃	—	2	1	no	—
113	C₂×Dic₅₃	—	2	1	no	—
114	C₂×Dic₅₃	—	2	1	no	—
115	D₂₁₂	—	2	1	no	—
116	D₂₁₂	—	2	1	no	—
117	C₄×D₅₃	—	4	2	no	—
118	C₄×D₅₃	—	4	2	no	—
119	C₄×D₅₃	—	4	2	no	—
120	Dic₁₀₆	—	2	1	no	—
121	Dic₁₀₆	—	2	1	no	—
122	C₈×C₅₃	—	4	2	no	—
123	C₈×C₅₃	—	4	2	no	—
124	C₈×C₅₃	—	4	2	no	—
125	#88322	—	2	2	no	—
126	#88322	—	2	2	no	—
127	#88322	—	4	2	no	—
128	#88322	—	4	2	no	—
129	#8188	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	—	2	1	no	—
131	C₂²×C₅₃	—	2	1	no	—
132	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₅₃	#433	1	2	yes	—
134	C₂×D₅₃	—	4	1	no	—
135	C₂×D₅₃	—	4	1	no	—
136	C₂×D₅₃	—	4	1	no	—
137	C₄×C₅₃	—	2	1	no	—
138	C₄×C₅₃	—	2	1	no	—
139	C₄×C₅₃	#405	1	1	yes	characteristic
140	C₄×C₅₃	#405	1	1	yes	characteristic
141	C₄×C₅₃	#408	1	2	yes	—
142	C₄×C₅₃	#444	1	2	yes	—
143	C₄×C₅₃	#444	1	2	yes	—
144	Dic₅₃	—	4	1	no	—
145	Dic₅₃	—	4	1	no	—
146	Dic₅₃	—	4	1	no	—
147	#2922	—	53	1	no	—
148	#1476	—	53	1	no	—
149	#1476	—	53	1	no	—
150	#1452	—	53	1	no	—
151	C₂×#322	—	53	1	no	—
152	#1278	—	53	1	no	—
153	#1278	D₅₃	1	1	yes	characteristic, complemented
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	#1750	1	1	yes	characteristic
156	C₂×C₅₃	#1765	1	1	yes	characteristic
157	C₂×C₅₃	#1765	1	1	yes	characteristic
158	D₅₃	—	8	1	no	—
159	C₄×#41	—	53	1	no	—
160	#433	—	106	2	no	—
161	#431	—	53	1	no	—
162	#431	—	106	1	no	—
163	#431	—	106	1	no	—
164	#431	C₂×D₅₃	1	1	yes	characteristic
165	#424	—	53	2	no	—
166	#408	—	53	2	no	—
167	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
168	C₂×C₄×C₈	—	53	1	no	—
169	C₂×C₄×C₈	—	53	1	no	—
170	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
171	Q₈⋊C₈	—	53	2	no	—
172	Q₈⋊C₈	—	53	2	no	—
173	D₄⋊C₈	—	53	2	no	—
174	D₄⋊C₈	—	53	2	no	—
175	#322	—	106	1	no	—
176	#322	—	106	2	no	—
177	#322	—	106	1	no	—
178	C₅₃	#8188	1	1	yes	characteristic, complemented, Sylow
179	C₂×#41	—	53	1	no	—
180	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	53	2	no	—
183	C₂²×C₈	—	106	1	no	—
184	C₄×Q₈	—	53	1	no	—
185	C₄×Q₈	—	53	1	no	—
186	C₄×D₄	—	53	1	no	—
187	C₄×D₄	—	53	1	no	—
188	C₄×D₄	—	106	1	no	—
189	C₄×D₄	—	106	1	no	—
190	#117	—	53	1	no	—
191	#117	—	106	1	no	—
192	C₂×C₄²	—	106	1	no	—
193	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
194	C₄⋊C₈	—	53	2	no	—
195	C₄⋊C₈	—	106	2	no	—
196	C₄⋊C₈	—	106	2	no	—
197	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
198	C₂²⋊C₈	—	2	1	no	—
199	C₂²⋊C₈	—	106	2	no	—
200	C₂²⋊C₈	—	106	2	no	—
201	C₂²⋊C₈	—	106	1	no	—
202	#97	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	53	2	no	—
207	C₄×C₈	—	53	2	no	—
208	C₄×C₈	—	53	1	no	—
209	C₄×C₈	—	53	1	no	—
210	C₄×C₈	—	106	1	no	—
211	C₄×C₈	—	106	1	no	—
212	C₄×C₈	C₄×D₅₃	1	2	yes	—
213	#41	—	106	2	no	—
214	#41	—	106	2	no	—
215	C₂×Q₈	—	53	1	no	—
216	C₂×D₄	—	53	1	no	—
217	C₂×D₄	—	106	1	no	—
218	C₂²×C₄	—	106	1	no	—
219	C₂²×C₄	—	106	1	no	—
220	C₂²×C₄	—	106	1	no	—
221	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
222	C₂²×C₄	#11719947	1	1	yes	characteristic
223	C₂²×C₄	#11719949	1	1	yes	characteristic
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	53	2	no	—
231	C₂×C₈	—	53	2	no	—
232	C₂×C₈	—	106	2	no	—
233	C₂×C₈	—	106	2	no	—
234	C₂×C₈	—	106	2	no	—
235	C₂×C₈	—	106	2	no	—
236	C₂×C₈	—	212	2	no	—
237	C₄⋊C₄	—	53	1	no	—
238	C₄⋊C₄	—	53	1	no	—
239	C₄⋊C₄	—	106	1	no	—
240	C₄⋊C₄	—	106	1	no	—
241	C₂²⋊C₄	—	106	1	no	—
242	C₂²⋊C₄	—	106	1	no	—
243	C₂²⋊C₄	—	106	1	no	—
244	C₄²	—	106	1	no	—
245	C₄²	—	106	1	no	—
246	C₄²	—	106	1	no	—
247	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
249	C₄²	D₄×D₅₃	1	1	yes	characteristic
250	C₄²	D₄×D₅₃	1	1	yes	characteristic
251	C₂³	—	106	1	no	—
252	C₂³	#13133175	1	1	yes	characteristic
253	Q₈	—	106	1	no	—
254	Q₈	—	106	1	no	—
255	D₄	—	106	1	no	—
256	D₄	—	106	1	no	—
257	D₄	—	212	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	106	1	no	—
261	C₂×C₄	—	106	1	no	—
262	C₂×C₄	—	106	1	no	—
263	C₂×C₄	—	212	2	no	—
264	C₂×C₄	—	212	2	no	—
265	C₂×C₄	—	212	2	no	—
266	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
267	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
268	C₂×C₄	#13133177	1	1	yes	characteristic
269	C₂×C₄	#13133177	1	1	yes	characteristic
270	C₂×C₄	#13133179	1	1	yes	characteristic
271	C₂×C₄	#13133179	1	1	yes	characteristic
272	C₂×C₄	#13133223	1	2	yes	—
273	C₂×C₄	#13133225	1	2	yes	—
274	C₈	—	4	2	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	2	no	—
277	C₈	—	106	2	no	—
278	C₈	—	106	2	no	—
279	C₈	—	212	2	no	—
280	C₈	—	212	2	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	212	1	no	—
284	C₂²	—	212	1	no	—
285	C₂²	—	212	1	no	—
286	C₂²	#16810470	1	1	yes	characteristic
287	C₂²	#16810683	1	2	yes	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	212	1	no	—
291	C₄	—	212	1	no	—
292	C₄	—	212	1	no	—
293	C₄	#16810577	1	1	yes	characteristic
294	C₄	#16810580	1	1	yes	characteristic
295	C₄	#16810588	1	2	yes	—
296	C₄	#16810742	1	2	yes	—
297	C₄	#16810744	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	424	1	no	—
300	C₂	#20274480	1	1	yes	characteristic
301	C₂	#20274556	1	1	yes	characteristic
302	C₂	#20274558	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1278	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1278	D₅₃
hypocenter	C₅₃	#8188
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
4. #20273008	C₂	semidirect
5. C₂×#16810345	C₂	semidirect
6. #20272553	C₂	semidirect
7. C₅₃×#1278	C₂	semidirect
153. #1278	D₅₃	semidirect
178. C₅₃	#8188	semidirect

Maximal subgroups

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

subgroup	index
#20283360	2
#20273098	2
#20273095	2
#20273008	2
C₂×#16810345	2
#20272553	2
C₅₃×#1278	2
#8188	53