Subgroups — #27354770

Order 13568.

560 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285301	C₂	1	1	yes	maximal, characteristic
2	#20283812	C₂	1	2	yes	maximal
3	#20273623	C₂	1	2	yes	maximal, complemented
4	#20273611	C₂	1	2	yes	maximal, complemented
5	#20273533	C₂	1	2	yes	maximal, complemented
6	#20273521	C₂	1	2	yes	maximal, complemented
7	#20273187	C₂	1	2	yes	maximal
8	#20272998	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic
10	#16811180	C₂²	1	2	yes	—
11	#16811028	C₄	1	1	yes	characteristic
12	#16811003	C₄	1	1	yes	characteristic
13	#16810974	C₄	1	2	yes	—
14	#16810864	C₄	1	2	yes	—
15	#16810864	C₄	1	2	yes	—
16	#16810751	—	2	4	no	—
17	#16810750	C₂²	1	4	yes	—
18	#16810747	C₂²	1	4	yes	—
19	C₄×Dic₂₁₂	C₂²	1	2	yes	—
20	C₄×Dic₂₁₂	C₂²	1	2	yes	—
21	#16810681	—	2	4	no	—
22	#16810676	C₂²	1	1	yes	characteristic
23	#16810676	C₂²	1	1	yes	characteristic
24	#16810353	C₂²	1	4	yes	—
25	#16810353	C₂²	1	4	yes	—
26	#16810344	C₂²	1	1	yes	characteristic
27	#16810344	C₂²	1	1	yes	characteristic
28	C₈×#88322	C₂²	1	1	yes	characteristic
29	C₈×#88322	C₂²	1	1	yes	characteristic
30	C₅₃×#432	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#335	C₂²	1	2	yes	—
33	C₅₃×#334	C₂²	1	2	yes	—
34	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
35	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
36	#13133217	C₂×C₄	1	2	yes	—
37	#13133217	C₂×C₄	1	2	yes	—
38	#13133209	—	2	2	no	—
39	#13133196	—	2	2	no	—
40	#13133196	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₀₆	—	2	2	no	—
42	C₄×Dic₁₀₆	C₂³	1	4	yes	—
43	#13133176	—	2	2	no	—
44	#13133151	C₂×C₄	1	2	yes	—
45	#13133151	C₂×C₄	1	2	yes	—
46	C₄⋊#88322	—	2	1	no	—
47	C₄⋊#88322	D₄	1	2	yes	—
48	#13133146	—	2	4	no	—
49	#13133146	—	2	4	no	—
50	#13133142	C₂×C₄	1	4	yes	—
51	#13133142	C₂×C₄	1	4	yes	—
52	C₂²⋊#88322	—	2	1	no	—
53	C₂²⋊#88322	D₄	1	2	yes	—
54	#13133127	—	2	4	no	—
55	C₄×#88322	C₂³	1	1	yes	characteristic
56	C₄×#88322	C₂³	1	1	yes	characteristic
57	C₄×#88322	C₂³	1	1	yes	characteristic
58	C₄×#88322	C₂³	1	1	yes	characteristic
59	C₈×Dic₅₃	—	2	4	no	—
60	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
64	C₂²⋊C₈×C₅₃	—	2	2	no	—
65	#97×C₅₃	—	2	2	no	—
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	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×Dic₁₀₆	—	2	2	no	—
74	C₂×Dic₁₀₆	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	4	no	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₃	#41	1	1	yes	characteristic
85	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₂²⋊Dic₅₃	—	2	4	no	—
87	C₄⋊Dic₅₃	—	2	2	no	—
88	C₄⋊Dic₅₃	—	2	2	no	—
89	C₄⋊Dic₅₃	—	2	2	no	—
90	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
91	#11719919	—	2	4	no	—
92	#11719919	—	2	4	no	—
93	C₄×Dic₅₃	—	2	4	no	—
94	C₂×#88322	—	2	1	no	—
95	C₂×#88322	—	2	1	no	—
96	C₂×#88322	C₂×D₄	1	2	yes	—
97	C₂×#88322	C₂×D₄	1	2	yes	—
98	Dic₂₁₂	—	2	4	no	—
99	Dic₂₁₂	—	2	4	no	—
100	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	4	1	no	—
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	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
111	C₂×Dic₅₃	—	2	4	no	—
112	C₂×Dic₅₃	—	2	4	no	—
113	Dic₁₀₆	—	2	4	no	—
114	Dic₁₀₆	—	2	4	no	—
115	Dic₁₀₆	—	8	2	no	—
116	C₈×C₅₃	—	2	1	no	—
117	C₈×C₅₃	—	2	1	no	—
118	C₈×C₅₃	—	4	4	no	—
119	C₈×C₅₃	#131	1	2	yes	—
120	C₈×C₅₃	#131	1	2	yes	—
121	#88322	—	2	2	no	—
122	#88322	—	2	2	no	—
123	#88322	—	4	1	no	—
124	#88322	—	4	1	no	—
125	#17972	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	2	1	no	—
127	C₂²×C₅₃	—	2	1	no	—
128	C₂²×C₅₃	—	2	1	no	—
129	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
135	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
136	C₄×C₅₃	#578	1	2	yes	—
137	Dic₅₃	—	4	4	no	—
138	Dic₅₃	—	4	4	no	—
139	#3170	—	53	1	no	—
140	#2985	—	53	2	no	—
141	#1593	—	53	2	no	—
142	#1591	—	53	2	no	—
143	#1579	—	53	2	no	—
144	C₈×Q₁₆	—	53	2	no	—
145	#1568	D₅₃	1	1	yes	characteristic
146	#1497	—	53	2	no	—
147	#1450	—	53	1	no	—
148	C₂×C₅₃	—	4	1	no	—
149	C₂×C₅₃	#2931	1	1	yes	characteristic
150	C₂×C₅₃	#2950	1	1	yes	characteristic
151	C₂×C₅₃	#2957	1	1	yes	characteristic
152	#533	—	53	2	no	—
153	#500	—	53	1	no	—
154	#494	—	53	1	no	—
155	#487	—	53	2	no	—
156	#467	—	53	2	no	—
157	#467	—	53	2	no	—
158	#446	—	53	4	no	—
159	#446	—	106	4	no	—
160	C₈×Q₈	—	53	4	no	—
161	C₄×Q₁₆	—	53	2	no	—
162	C₄×Q₁₆	—	53	2	no	—
163	#433	—	106	4	no	—
164	#432	C₂×D₅₃	1	2	yes	—
165	#431	—	53	1	no	—
166	#431	—	53	1	no	—
167	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
168	#335	C₂×D₅₃	1	2	yes	—
169	#334	C₂×D₅₃	1	2	yes	—
170	Q₈⋊C₈	—	53	4	no	—
171	Q₈⋊C₈	—	53	4	no	—
172	#322	—	53	1	no	—
173	#322	—	53	1	no	—
174	C₈²	—	53	1	no	—
175	C₈²	—	53	1	no	—
176	C₅₃	#17972	1	1	yes	characteristic, complemented, Sylow
177	C₂×Q₁₆	—	53	2	no	—
178	C₂×Q₁₆	—	53	2	no	—
179	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
180	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
181	C₄⋊Q₈	—	53	2	no	—
182	C₄⋊Q₈	—	53	2	no	—
183	#125	—	106	2	no	—
184	C₂²⋊Q₈	—	53	4	no	—
185	C₂²⋊Q₈	—	106	2	no	—
186	C₄×Q₈	—	53	4	no	—
187	C₄×Q₈	—	106	2	no	—
188	#117	—	106	2	no	—
189	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
190	#107	—	53	2	no	—
191	#107	—	53	2	no	—
192	C₄⋊C₈	—	53	2	no	—
193	C₄⋊C₈	—	106	4	no	—
194	C₄⋊C₈	—	106	4	no	—
195	C₄⋊C₈	—	106	1	no	—
196	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
197	#103	—	53	4	no	—
198	#103	—	53	4	no	—
199	C₂²⋊C₈	—	2	2	no	—
200	C₂²⋊C₈	—	53	2	no	—
201	C₂²⋊C₈	—	106	1	no	—
202	#97	—	2	2	no	—
203	#97	—	106	4	no	—
204	C₄×C₈	—	53	1	no	—
205	C₄×C₈	—	53	1	no	—
206	C₄×C₈	—	53	1	no	—
207	C₄×C₈	—	53	1	no	—
208	C₄×C₈	—	106	4	no	—
209	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
210	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
211	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
213	C₂×Q₈	—	53	4	no	—
214	C₂×Q₈	—	106	2	no	—
215	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
217	C₂²×C₄	#11719949	1	1	yes	characteristic
218	Q₁₆	—	106	4	no	—
219	Q₁₆	—	106	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	53	2	no	—
224	C₂×C₈	—	53	2	no	—
225	C₂×C₈	—	106	1	no	—
226	C₂×C₈	—	106	1	no	—
227	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
231	C₄⋊C₄	—	53	4	no	—
232	C₄⋊C₄	—	106	2	no	—
233	C₄⋊C₄	—	106	2	no	—
234	C₄⋊C₄	—	106	4	no	—
235	C₄⋊C₄	—	106	2	no	—
236	C₄⋊C₄	—	106	4	no	—
237	C₂²⋊C₄	—	106	4	no	—
238	C₄²	—	106	4	no	—
239	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
240	C₄²	D₄×D₅₃	1	1	yes	characteristic
241	C₄²	#11719949	1	1	yes	characteristic
242	C₄²	C₂³×D₅₃	1	1	yes	characteristic
243	C₂³	#13133179	1	1	yes	characteristic
244	Q₈	—	106	4	no	—
245	Q₈	—	106	4	no	—
246	Q₈	—	424	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	106	4	no	—
252	C₂×C₄	—	106	4	no	—
253	C₂×C₄	#13133179	1	1	yes	characteristic
254	C₂×C₄	#13133179	1	1	yes	characteristic
255	C₂×C₄	#13133179	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
257	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
258	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	4	4	no	—
262	C₈	—	106	2	no	—
263	C₈	—	106	2	no	—
264	C₈	—	212	1	no	—
265	C₈	—	212	1	no	—
266	C₈	#13133225	1	2	yes	—
267	C₈	#13133225	1	2	yes	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×#13133179	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	212	4	no	—
277	C₄	—	212	4	no	—
278	C₄	C₂×#13133225	1	1	yes	characteristic
279	C₄	C₂×#13133225	1	1	yes	characteristic
280	C₄	#16811437	1	2	yes	—
281	C₂	—	4	1	no	—
282	C₂	#20283426	1	1	yes	characteristic
283	C₂	#20283582	1	1	yes	characteristic
284	C₂	#20283631	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1568	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1568	D₅₃
hypocenter	C₅₃	#17972
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
3. #20273623	C₂	semidirect
4. #20273611	C₂	semidirect
5. #20273533	C₂	semidirect
6. #20273521	C₂	semidirect
176. C₅₃	#17972	semidirect

Maximal subgroups

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

subgroup	index
#20285301	2
#20283812	2
#20273623	2
#20273611	2
#20273533	2
#20273521	2
#20273187	2
#20272998	2
C₅₃×#1568	2
#17972	53