Subgroups — #27354752

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	#20283791	C₂	1	2	yes	maximal, complemented
2	#20273629	C₂	1	2	yes	maximal, complemented
3	#20273600	C₂	1	2	yes	maximal, complemented
4	#20273539	C₂	1	2	yes	maximal, complemented
5	D₈×#88322	C₂	1	2	yes	maximal, complemented
6	#20273478	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273184	C₂	1	2	yes	maximal, complemented
8	#20272998	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
10	#16810701	C₂²	1	4	yes	—
11	#16810695	—	2	4	no	—
12	D₄×#88322	C₂²	1	4	yes	—
13	#16810680	C₂²	1	2	yes	—
14	#16810676	C₂²	1	1	yes	characteristic
15	#16810676	C₂²	1	1	yes	characteristic
16	#16810590	—	2	4	no	—
17	#16810376	C₂²	1	2	yes	—
18	#16810374	C₂²	1	2	yes	—
19	D₄⋊#88322	C₂²	1	4	yes	—
20	D₄⋊#88322	C₂²	1	4	yes	—
21	#16810344	C₂²	1	1	yes	characteristic
22	#16810344	C₂²	1	1	yes	characteristic
23	C₈×#88322	C₂²	1	1	yes	characteristic
24	C₈×#88322	C₂²	1	1	yes	characteristic
25	C₅₃×#532	C₂²	1	2	yes	—
26	C₅₃×#500	C₄	1	1	yes	characteristic
27	C₅₃×#493	C₄	1	1	yes	characteristic
28	C₅₃×#486	C₄	1	2	yes	—
29	C₅₃×#466	C₄	1	2	yes	—
30	C₅₃×#466	C₄	1	2	yes	—
31	C₄×D₈×C₅₃	C₂²	1	2	yes	—
32	C₄×D₈×C₅₃	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#11719917	—	2	4	no	—
35	C₂²×#88322	—	2	4	no	—
36	C₄⋊#88322	—	2	1	no	—
37	C₄⋊#88322	D₄	1	2	yes	—
38	C₄⋊#88322	C₂³	1	4	yes	—
39	C₂²⋊#88322	—	2	4	no	—
40	C₂²⋊#88322	—	2	4	no	—
41	C₂²⋊#88322	—	2	2	no	—
42	C₂²⋊#88322	—	2	1	no	—
43	C₂²⋊#88322	D₄	1	2	yes	—
44	#13133128	—	2	2	no	—
45	C₄×#88322	C₂³	1	1	yes	characteristic
46	C₄×#88322	C₂³	1	1	yes	characteristic
47	C₄×#88322	C₂³	1	1	yes	characteristic
48	C₄×#88322	C₂³	1	1	yes	characteristic
49	C₂×#41×C₅₃	—	2	2	no	—
50	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
55	#127×C₅₃	C₂×C₄	1	2	yes	—
56	#124×C₅₃	—	2	2	no	—
57	#121×C₅₃	—	2	2	no	—
58	#121×C₅₃	C₂×C₄	1	4	yes	—
59	C₄×D₄×C₅₃	—	2	2	no	—
60	C₄×D₄×C₅₃	C₂³	1	4	yes	—
61	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
62	#107×C₅₃	C₂×C₄	1	2	yes	—
63	#107×C₅₃	C₂×C₄	1	2	yes	—
64	#102×C₅₃	C₂×C₄	1	4	yes	—
65	#102×C₅₃	C₂×C₄	1	4	yes	—
66	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	#41×C₅₃	—	8	4	no	—
71	C₂×Q₈×C₅₃	—	2	2	no	—
72	C₂×D₄×C₅₃	—	2	4	no	—
73	C₂×D₄×C₅₃	—	2	2	no	—
74	C₂×D₄×C₅₃	—	2	2	no	—
75	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₅₃	—	2	4	no	—
77	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
80	D₈×C₅₃	—	2	4	no	—
81	D₈×C₅₃	—	2	4	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	2	2	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	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
90	C₂²⋊C₄×C₅₃	—	2	4	no	—
91	C₂²⋊C₄×C₅₃	—	2	4	no	—
92	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅₃	#41	1	1	yes	characteristic
95	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
96	#11719917	—	8	4	no	—
97	C₂×#88322	—	2	4	no	—
98	C₂×#88322	—	2	1	no	—
99	C₂×#88322	—	2	1	no	—
100	C₂×#88322	—	8	4	no	—
101	C₂×#88322	C₂×D₄	1	2	yes	—
102	C₂×#88322	C₂×D₄	1	2	yes	—
103	C₂³×C₅₃	—	2	4	no	—
104	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
105	Q₈×C₅₃	—	8	2	no	—
106	D₄×C₅₃	—	2	4	no	—
107	D₄×C₅₃	—	2	4	no	—
108	D₄×C₅₃	—	8	2	no	—
109	D₄×C₅₃	—	8	2	no	—
110	D₄×C₅₃	—	8	4	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	4	no	—
115	C₂×C₄×C₅₃	—	4	1	no	—
116	C₂×C₄×C₅₃	—	4	4	no	—
117	C₂×C₄×C₅₃	—	4	4	no	—
118	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₄×D₄	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	C₂×#41	1	1	yes	characteristic
124	C₈×C₅₃	—	2	1	no	—
125	C₈×C₅₃	—	2	1	no	—
126	C₈×C₅₃	#131	1	2	yes	—
127	C₈×C₅₃	#131	1	2	yes	—
128	#88322	—	2	2	no	—
129	#88322	—	2	2	no	—
130	#88322	—	4	1	no	—
131	#88322	—	4	1	no	—
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	—	4	4	no	—
138	C₂²×C₅₃	—	4	4	no	—
139	C₂²×C₅₃	—	4	4	no	—
140	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
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	—	4	4	no	—
146	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
147	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
148	C₄×C₅₃	#576	1	2	yes	—
149	#3169	D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
158	C₂×C₅₃	—	4	1	no	—
159	C₂×C₅₃	—	8	4	no	—
160	C₂×C₅₃	#2931	1	1	yes	characteristic
161	C₂×C₅₃	#2948	1	1	yes	characteristic
162	C₂×C₅₃	#2957	1	1	yes	characteristic
163	#532	C₂×D₅₃	1	2	yes	—
164	#500	Dic₅₃	1	1	yes	characteristic
165	#493	Dic₅₃	1	1	yes	characteristic
166	#486	Dic₅₃	1	2	yes	—
167	#466	Dic₅₃	1	2	yes	—
168	#466	Dic₅₃	1	2	yes	—
169	C₄×D₈	C₂×D₅₃	1	2	yes	—
170	C₄×D₈	C₂×D₅₃	1	2	yes	—
171	#436	—	53	4	no	—
172	#435	—	106	4	no	—
173	C₈×D₄	—	53	4	no	—
174	#432	—	53	2	no	—
175	#431	—	53	1	no	—
176	#431	—	53	1	no	—
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	—	53	1	no	—
184	#322	—	53	1	no	—
185	C₈²	—	53	1	no	—
186	C₈²	—	53	1	no	—
187	C₅₃	#17970	1	1	yes	characteristic, complemented, Sylow
188	C₂×#41	—	2	2	no	—
189	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
190	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
191	C₂×M₁₆	—	106	4	no	—
192	C₂²×C₈	—	106	4	no	—
193	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
194	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
195	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
196	#127	C₂×Dic₅₃	1	2	yes	—
197	#124	—	2	2	no	—
198	#121	—	2	2	no	—
199	#121	C₂×Dic₅₃	1	4	yes	—
200	C₄×D₄	—	2	2	no	—
201	C₄×D₄	C₂²×D₅₃	1	4	yes	—
202	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
203	#107	C₂×Dic₅₃	1	2	yes	—
204	#107	C₂×Dic₅₃	1	2	yes	—
205	C₄⋊C₈	—	53	2	no	—
206	C₄⋊C₈	—	53	4	no	—
207	C₄⋊C₈	—	106	1	no	—
208	#102	C₂×Dic₅₃	1	4	yes	—
209	#102	C₂×Dic₅₃	1	4	yes	—
210	C₂²⋊C₈	—	53	2	no	—
211	C₂²⋊C₈	—	106	4	no	—
212	C₂²⋊C₈	—	106	4	no	—
213	C₂²⋊C₈	—	106	2	no	—
214	C₂²⋊C₈	—	106	1	no	—
215	#97	—	106	2	no	—
216	C₄×C₈	—	53	1	no	—
217	C₄×C₈	—	53	1	no	—
218	C₄×C₈	—	53	1	no	—
219	C₄×C₈	—	53	1	no	—
220	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
221	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
223	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
224	#41	—	8	4	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×Dic₅₃	1	4	yes	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
232	C₂²×C₄	#11719947	1	1	yes	characteristic
233	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
234	D₈	—	2	4	no	—
235	D₈	—	2	4	no	—
236	M₁₆	—	424	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	53	2	no	—
240	C₂×C₈	—	53	2	no	—
241	C₂×C₈	—	106	4	no	—
242	C₂×C₈	—	106	1	no	—
243	C₂×C₈	—	106	1	no	—
244	C₂×C₈	—	424	4	no	—
245	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
246	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
247	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
248	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₄²	D₄×D₅₃	1	1	yes	characteristic
254	C₄²	#11719947	1	1	yes	characteristic
255	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
256	C₄²	C₂³×D₅₃	1	1	yes	characteristic
257	C₂³	—	2	4	no	—
258	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
259	Q₈	—	8	2	no	—
260	D₄	—	2	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	8	2	no	—
263	D₄	—	8	2	no	—
264	D₄	—	8	4	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
273	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
274	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
275	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
276	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
277	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	106	2	no	—
281	C₈	—	106	2	no	—
282	C₈	—	212	1	no	—
283	C₈	—	212	1	no	—
284	C₈	—	212	4	no	—
285	C₈	#13133224	1	2	yes	—
286	C₈	#13133224	1	2	yes	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	4	no	—
292	C₂²	—	4	4	no	—
293	C₂²	C₂×D₄×Dic₅₃	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	—	4	4	no	—
299	C₄	C₂×#13133224	1	1	yes	characteristic
300	C₄	C₂×#13133224	1	1	yes	characteristic
301	C₄	#16811424	1	2	yes	—
302	C₂	—	4	1	no	—
303	C₂	—	8	4	no	—
304	C₂	#20283427	1	1	yes	characteristic
305	C₂	#20283567	1	1	yes	characteristic
306	C₂	#20283632	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3169	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#3169	D₅₃
hypocenter	C₅₃	#17970
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #20283791	C₂	semidirect
2. #20273629	C₂	semidirect
3. #20273600	C₂	semidirect
4. #20273539	C₂	semidirect
5. D₈×#88322	C₂	semidirect
6. #20273478	C₂	semidirect
7. #20273184	C₂	semidirect
8. #20272998	C₂	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
#20283791	2
#20273629	2
#20273600	2
#20273539	2
D₈×#88322	2
#20273478	2
#20273184	2
#20272998	2
C₅₃×#3169	2
#17970	53