Subgroups — #27354753

Order 13568.

640 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	#20285294	C₂	1	1	yes	maximal, characteristic
2	#20283790	C₂	1	2	yes	maximal, complemented
3	#20273630	C₂	1	2	yes	maximal, complemented
4	#20273601	C₂	1	2	yes	maximal, complemented
5	#20273540	C₂	1	2	yes	maximal, complemented
6	C₈×D₄₂₄	C₂	1	2	yes	maximal, complemented
7	#20273478	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273185	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811173	C₂²	1	2	yes	—
11	#16811028	C₄	1	1	yes	characteristic
12	#16811000	C₄	1	1	yes	characteristic
13	#16810970	C₄	1	2	yes	—
14	#16810859	C₄	1	2	yes	—
15	#16810859	C₄	1	2	yes	—
16	C₄×D₄₂₄	C₂²	1	2	yes	—
17	C₄×D₄₂₄	C₂²	1	2	yes	—
18	#16810702	C₂²	1	4	yes	—
19	#16810693	—	2	4	no	—
20	C₈×D₂₁₂	C₂²	1	4	yes	—
21	#16810680	C₂²	1	2	yes	—
22	#16810588	—	2	4	no	—
23	#16810376	C₂²	1	2	yes	—
24	#16810374	C₂²	1	2	yes	—
25	#16810352	C₂²	1	4	yes	—
26	#16810352	C₂²	1	4	yes	—
27	C₅₃×#431	C₂²	1	1	yes	characteristic
28	C₅₃×#431	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#322	C₂²	1	1	yes	characteristic
31	C₅₃×#322	C₂²	1	1	yes	characteristic
32	C₈²×C₅₃	C₂²	1	1	yes	characteristic
33	C₈²×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#11719945	—	2	2	no	—
35	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
36	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
37	C₂×#11719912	—	2	4	no	—
38	C₂×C₈×D₅₃	—	2	4	no	—
39	#13133217	C₂×C₄	1	2	yes	—
40	#13133214	C₂×C₄	1	2	yes	—
41	#13133205	—	2	2	no	—
42	#13133191	—	2	2	no	—
43	#13133191	C₂×C₄	1	4	yes	—
44	C₄×D₂₁₂	—	2	2	no	—
45	C₄×D₂₁₂	C₂³	1	4	yes	—
46	#13133151	C₂×C₄	1	2	yes	—
47	#13133151	C₂×C₄	1	2	yes	—
48	C₄⋊#88322	C₂³	1	4	yes	—
49	#13133139	C₂×C₄	1	4	yes	—
50	#13133139	C₂×C₄	1	4	yes	—
51	C₂²⋊#88322	—	2	2	no	—
52	#13133129	—	2	4	no	—
53	#13133129	—	2	4	no	—
54	#13133128	—	2	2	no	—
55	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₅₃	—	2	1	no	—
59	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
60	C₂²⋊C₈×C₅₃	—	2	1	no	—
61	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
62	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
73	C₂×C₂²⋊D₅₃	—	2	4	no	—
74	#11719945	—	8	4	no	—
75	C₂×D₂₁₂	—	2	2	no	—
76	C₂×D₂₁₂	—	2	2	no	—
77	C₂×D₂₁₂	C₂²×C₄	1	4	yes	—
78	C₂×C₄×D₅₃	—	2	4	no	—
79	C₂×Dic₁₀₆	—	2	2	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	1	no	—
83	C₂×C₈×C₅₃	—	2	1	no	—
84	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
90	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅₃	#41	1	1	yes	characteristic
93	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
94	#11719921	—	2	4	no	—
95	#11719921	—	2	4	no	—
96	C₄⋊Dic₅₃	—	2	2	no	—
97	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
98	C₂×#88322	—	2	4	no	—
99	D₄₂₄	—	2	4	no	—
100	D₄₂₄	—	2	4	no	—
101	#11719912	—	8	4	no	—
102	C₈×D₅₃	—	8	4	no	—
103	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
104	C₂²×D₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	4	1	no	—
109	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
115	C₂²⋊D₅₃	—	8	4	no	—
116	C₂×Dic₅₃	—	2	4	no	—
117	D₂₁₂	—	2	4	no	—
118	D₂₁₂	—	2	4	no	—
119	D₂₁₂	—	8	2	no	—
120	D₂₁₂	—	8	2	no	—
121	C₄×D₅₃	—	4	4	no	—
122	C₄×D₅₃	—	4	4	no	—
123	Dic₁₀₆	—	8	2	no	—
124	C₈×C₅₃	—	2	1	no	—
125	C₈×C₅₃	—	2	1	no	—
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	—	2	2	no	—
128	C₈×C₅₃	—	4	1	no	—
129	C₈×C₅₃	—	4	1	no	—
130	C₈×C₅₃	#131	1	2	yes	—
131	C₈×C₅₃	#131	1	2	yes	—
132	#88322	—	4	4	no	—
133	#17970	—	53	1	no	maximal, Sylow
134	C₂²×C₅₃	—	2	1	no	—
135	C₂²×C₅₃	—	2	1	no	—
136	C₂²×C₅₃	—	2	1	no	—
137	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂×D₅₃	—	4	4	no	—
139	C₂×D₅₃	—	4	4	no	—
140	C₂×D₅₃	—	4	4	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	2	1	no	—
144	C₄×C₅₃	—	2	1	no	—
145	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
146	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
147	C₄×C₅₃	#576	1	2	yes	—
148	Dic₅₃	—	4	4	no	—
149	#3169	—	53	1	no	—
150	#2981	—	53	2	no	—
151	#1594	—	53	2	no	—
152	#1589	—	53	2	no	—
153	#1580	—	53	2	no	—
154	C₈×D₈	—	53	2	no	—
155	#1568	—	53	1	no	—
156	#1496	—	53	2	no	—
157	#1450	D₅₃	1	1	yes	characteristic, complemented
158	C₂×C₅₃	—	4	1	no	—
159	C₂×C₅₃	#2931	1	1	yes	characteristic
160	C₂×C₅₃	#2948	1	1	yes	characteristic
161	C₂×C₅₃	#2957	1	1	yes	characteristic
162	D₅₃	—	8	4	no	—
163	#532	—	53	2	no	—
164	#500	—	53	1	no	—
165	#493	—	53	1	no	—
166	#486	—	53	2	no	—
167	#466	—	53	2	no	—
168	#466	—	53	2	no	—
169	C₄×D₈	—	53	2	no	—
170	C₄×D₈	—	53	2	no	—
171	#436	—	53	4	no	—
172	#435	—	106	4	no	—
173	C₈×D₄	—	53	4	no	—
174	#432	—	53	2	no	—
175	#431	C₂×D₅₃	1	1	yes	characteristic
176	#431	C₂×D₅₃	1	1	yes	characteristic
177	#408	—	106	4	no	—
178	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
179	#335	—	53	2	no	—
180	#334	—	53	2	no	—
181	D₄⋊C₈	—	53	4	no	—
182	D₄⋊C₈	—	53	4	no	—
183	#322	C₂×D₅₃	1	1	yes	characteristic
184	#322	C₂×D₅₃	1	1	yes	characteristic
185	C₈²	C₂×D₅₃	1	1	yes	characteristic
186	C₈²	C₂×D₅₃	1	1	yes	characteristic
187	C₅₃	#17970	1	1	yes	characteristic, complemented, Sylow
188	C₂×#41	—	106	2	no	—
189	C₂×D₈	—	53	2	no	—
190	C₂×D₈	—	53	2	no	—
191	C₂×M₁₆	—	106	4	no	—
192	C₂²×C₈	—	106	4	no	—
193	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
194	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
195	C₄⋊Q₈	—	53	2	no	—
196	#127	—	53	2	no	—
197	#124	—	106	2	no	—
198	#121	—	53	4	no	—
199	#121	—	106	2	no	—
200	C₄×D₄	—	53	4	no	—
201	C₄×D₄	—	106	2	no	—
202	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
203	#107	—	53	2	no	—
204	#107	—	53	2	no	—
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	—	53	4	no	—
207	C₄⋊C₈	D₂₁₂	1	2	yes	—
208	#102	—	53	4	no	—
209	#102	—	53	4	no	—
210	C₂²⋊C₈	—	2	1	no	—
211	C₂²⋊C₈	—	106	2	no	—
212	C₂²⋊C₈	—	106	4	no	—
213	C₂²⋊C₈	—	106	4	no	—
214	C₂²⋊C₈	D₂₁₂	1	2	yes	—
215	#97	—	106	2	no	—
216	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
217	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
220	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
221	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
223	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
224	#41	—	424	4	no	—
225	C₂×Q₈	—	106	2	no	—
226	C₂×D₄	—	53	4	no	—
227	C₂×D₄	—	106	4	no	—
228	C₂×D₄	—	106	2	no	—
229	C₂×D₄	—	106	2	no	—
230	C₂²×C₄	—	106	4	no	—
231	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
233	C₂²×C₄	#11719945	1	1	yes	characteristic
234	D₈	—	106	4	no	—
235	D₈	—	106	4	no	—
236	M₁₆	—	424	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	106	4	no	—
242	C₂×C₈	—	424	4	no	—
243	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
245	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
246	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
247	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
248	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
249	C₄⋊C₄	—	53	4	no	—
250	C₄⋊C₄	—	106	2	no	—
251	C₂²⋊C₄	—	106	4	no	—
252	C₂²⋊C₄	—	106	4	no	—
253	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
254	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
255	C₄²	#11719945	1	1	yes	characteristic
256	C₄²	C₂³×D₅₃	1	1	yes	characteristic
257	C₂³	—	106	4	no	—
258	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
259	Q₈	—	424	2	no	—
260	D₄	—	106	4	no	—
261	D₄	—	106	4	no	—
262	D₄	—	424	2	no	—
263	D₄	—	424	2	no	—
264	D₄	—	424	4	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	106	4	no	—
270	C₂×C₄	—	212	4	no	—
271	C₂×C₄	—	212	4	no	—
272	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
273	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
274	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
277	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	2	no	—
281	C₈	—	2	2	no	—
282	C₈	—	4	1	no	—
283	C₈	—	4	1	no	—
284	C₈	—	212	4	no	—
285	C₈	#13133223	1	2	yes	—
286	C₈	#13133223	1	2	yes	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	212	4	no	—
291	C₂²	—	212	4	no	—
292	C₂²	—	212	4	no	—
293	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	212	4	no	—
299	C₄	C₂×#13133223	1	1	yes	characteristic
300	C₄	C₂×#13133223	1	1	yes	characteristic
301	C₄	#16811428	1	2	yes	—
302	C₂	—	4	1	no	—
303	C₂	—	424	4	no	—
304	C₂	#20283428	1	1	yes	characteristic
305	C₂	#20283568	1	1	yes	characteristic
306	C₂	#20283633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1450	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#20285294	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1450	D₅₃
hypocenter	C₅₃	#17970
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
2. #20283790	C₂	semidirect
3. #20273630	C₂	semidirect
4. #20273601	C₂	semidirect
5. #20273540	C₂	semidirect
6. C₈×D₄₂₄	C₂	semidirect
7. #20273478	C₂	semidirect
8. #20273185	C₂	semidirect
9. C₅₃×#1450	C₂	semidirect
157. #1450	D₅₃	semidirect
187. C₅₃	#17970	semidirect

Maximal subgroups

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

subgroup	index
#20285294	2
#20283790	2
#20273630	2
#20273601	2
#20273540	2
C₈×D₄₂₄	2
#20273478	2
#20273185	2
C₅₃×#1450	2
#17970	53