Subgroups — #27354936

Order 13568.

624 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	#20285294	C₂	1	1	yes	maximal, characteristic
2	#20283790	C₂	1	2	yes	maximal, complemented
3	#20273581	C₂	1	4	yes	maximal, complemented
4	#20273546	C₂	1	4	yes	maximal, complemented
5	#20273478	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273185	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811173	C₂²	1	2	yes	—
9	#16811028	C₄	1	1	yes	characteristic
10	#16811000	C₄	1	1	yes	characteristic
11	#16810970	C₄	1	2	yes	—
12	#16810859	C₄	1	2	yes	—
13	#16810859	C₄	1	2	yes	—
14	C₄×D₄₂₄	C₂²	1	2	yes	—
15	C₄×D₄₂₄	C₂²	1	2	yes	—
16	#16810702	C₂²	1	4	yes	—
17	#16810693	—	2	4	no	—
18	C₈×D₂₁₂	C₂²	1	4	yes	—
19	#16810680	C₂²	1	2	yes	—
20	#16810588	—	2	4	no	—
21	#16810376	C₂²	1	2	yes	—
22	#16810374	C₂²	1	2	yes	—
23	#16810352	C₂²	1	8	yes	—
24	C₅₃×#431	C₂²	1	1	yes	characteristic
25	C₅₃×#431	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#322	C₂²	1	2	yes	—
28	C₅₃×#322	C₂²	1	2	yes	—
29	C₂×#11719945	—	2	2	no	—
30	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
31	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
32	C₂×#11719912	—	2	4	no	—
33	C₂×C₈×D₅₃	—	2	4	no	—
34	#13133217	C₂×C₄	1	2	yes	—
35	#13133214	C₂×C₄	1	2	yes	—
36	#13133205	—	2	2	no	—
37	#13133191	—	2	2	no	—
38	#13133191	C₂×C₄	1	4	yes	—
39	C₄×D₂₁₂	—	2	2	no	—
40	C₄×D₂₁₂	C₂³	1	4	yes	—
41	#13133151	C₂×C₄	1	2	yes	—
42	#13133151	C₂×C₄	1	2	yes	—
43	C₄⋊#88322	C₂³	1	4	yes	—
44	#13133139	C₂×C₄	1	4	yes	—
45	#13133139	C₂×C₄	1	4	yes	—
46	C₂²⋊#88322	—	2	2	no	—
47	#13133129	—	2	4	no	—
48	#13133129	—	2	4	no	—
49	#13133128	—	2	2	no	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₅₃	—	2	1	no	—
54	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
55	C₂²⋊C₈×C₅₃	—	2	1	no	—
56	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
57	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₅₃	C₂³	1	2	yes	—
60	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₅₃	C₂³	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
66	C₂×C₂²⋊D₅₃	—	2	4	no	—
67	#11719945	—	8	4	no	—
68	C₂×D₂₁₂	—	2	2	no	—
69	C₂×D₂₁₂	—	2	2	no	—
70	C₂×D₂₁₂	C₂²×C₄	1	4	yes	—
71	C₂×C₄×D₅₃	—	2	4	no	—
72	C₂×Dic₁₀₆	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×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	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₃	#41	1	1	yes	characteristic
84	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
85	#11719921	—	2	4	no	—
86	#11719921	—	2	4	no	—
87	C₄⋊Dic₅₃	—	2	2	no	—
88	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
89	C₂×#88322	—	2	4	no	—
90	D₄₂₄	—	4	2	no	—
91	D₄₂₄	—	4	2	no	—
92	#11719912	—	8	4	no	—
93	C₈×D₅₃	—	8	4	no	—
94	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
95	C₂²×D₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	4	1	no	—
100	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
106	C₂²⋊D₅₃	—	8	4	no	—
107	C₂×Dic₅₃	—	2	4	no	—
108	D₂₁₂	—	2	4	no	—
109	D₂₁₂	—	2	4	no	—
110	D₂₁₂	—	8	2	no	—
111	D₂₁₂	—	8	2	no	—
112	C₄×D₅₃	—	4	4	no	—
113	C₄×D₅₃	—	4	4	no	—
114	Dic₁₀₆	—	8	2	no	—
115	C₈×C₅₃	—	2	1	no	—
116	C₈×C₅₃	—	2	1	no	—
117	C₈×C₅₃	—	2	1	no	—
118	C₈×C₅₃	—	2	1	no	—
119	C₈×C₅₃	—	2	4	no	—
120	C₈×C₅₃	—	4	2	no	—
121	#88322	—	4	4	no	—
122	#17989	—	53	1	no	maximal, Sylow
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
127	C₂×D₅₃	—	4	4	no	—
128	C₂×D₅₃	—	4	4	no	—
129	C₂×D₅₃	—	4	4	no	—
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	#568	1	1	yes	characteristic
135	C₄×C₅₃	#568	1	1	yes	characteristic
136	C₄×C₅₃	#576	1	2	yes	—
137	Dic₅₃	—	4	4	no	—
138	#3169	—	53	1	no	—
139	#2981	—	53	2	no	—
140	#1586	—	53	4	no	—
141	#1581	—	53	4	no	—
142	#1568	—	53	1	no	—
143	#1496	—	53	2	no	—
144	#1452	D₅₃	1	1	yes	characteristic, complemented
145	C₂×C₅₃	—	4	1	no	—
146	C₂×C₅₃	#2932	1	1	yes	characteristic
147	C₂×C₅₃	#2948	1	1	yes	characteristic
148	C₂×C₅₃	#2957	1	1	yes	characteristic
149	D₅₃	—	8	4	no	—
150	#532	—	53	2	no	—
151	#500	—	53	1	no	—
152	#493	—	53	1	no	—
153	#486	—	53	2	no	—
154	#466	—	53	2	no	—
155	#466	—	53	2	no	—
156	C₄×D₈	—	53	2	no	—
157	C₄×D₈	—	53	2	no	—
158	#436	—	53	4	no	—
159	#435	—	106	4	no	—
160	C₈×D₄	—	53	4	no	—
161	#432	—	53	2	no	—
162	#431	C₂×D₅₃	1	1	yes	characteristic
163	#431	C₂×D₅₃	1	1	yes	characteristic
164	#408	—	106	4	no	—
165	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
166	#335	—	53	2	no	—
167	#334	—	53	2	no	—
168	D₄⋊C₈	—	53	8	no	—
169	#322	C₂×D₅₃	1	2	yes	—
170	#322	C₂×D₅₃	1	2	yes	—
171	C₅₃	#17989	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	106	2	no	—
173	C₂×D₈	—	53	2	no	—
174	C₂×D₈	—	53	2	no	—
175	C₂×M₁₆	—	106	4	no	—
176	C₂²×C₈	—	106	4	no	—
177	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
178	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
179	C₄⋊Q₈	—	53	2	no	—
180	#127	—	53	2	no	—
181	#124	—	106	2	no	—
182	#121	—	53	4	no	—
183	#121	—	106	2	no	—
184	C₄×D₄	—	53	4	no	—
185	C₄×D₄	—	106	2	no	—
186	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
187	#107	—	53	2	no	—
188	#107	—	53	2	no	—
189	C₄⋊C₈	—	2	1	no	—
190	C₄⋊C₈	—	53	4	no	—
191	C₄⋊C₈	D₂₁₂	1	2	yes	—
192	#102	—	53	4	no	—
193	#102	—	53	4	no	—
194	C₂²⋊C₈	—	2	1	no	—
195	C₂²⋊C₈	—	106	2	no	—
196	C₂²⋊C₈	—	106	4	no	—
197	C₂²⋊C₈	—	106	4	no	—
198	C₂²⋊C₈	D₂₁₂	1	2	yes	—
199	#97	—	106	2	no	—
200	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
201	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
202	C₄×C₈	C₂²×D₅₃	1	2	yes	—
203	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
204	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
205	C₄×C₈	C₂²×D₅₃	1	2	yes	—
206	#41	—	424	4	no	—
207	C₂×Q₈	—	106	2	no	—
208	C₂×D₄	—	53	4	no	—
209	C₂×D₄	—	106	4	no	—
210	C₂×D₄	—	106	2	no	—
211	C₂×D₄	—	106	2	no	—
212	C₂²×C₄	—	106	4	no	—
213	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
215	C₂²×C₄	#11719945	1	1	yes	characteristic
216	D₈	—	212	2	no	—
217	D₈	—	212	2	no	—
218	M₁₆	—	424	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	106	4	no	—
223	C₂×C₈	—	424	4	no	—
224	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
228	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
229	C₄⋊C₄	—	53	4	no	—
230	C₄⋊C₄	—	106	2	no	—
231	C₂²⋊C₄	—	106	4	no	—
232	C₂²⋊C₄	—	106	4	no	—
233	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
234	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
235	C₄²	#11719945	1	1	yes	characteristic
236	C₄²	C₂³×D₅₃	1	1	yes	characteristic
237	C₂³	—	106	4	no	—
238	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
239	Q₈	—	424	2	no	—
240	D₄	—	106	4	no	—
241	D₄	—	106	4	no	—
242	D₄	—	424	2	no	—
243	D₄	—	424	2	no	—
244	D₄	—	424	4	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	106	4	no	—
250	C₂×C₄	—	212	4	no	—
251	C₂×C₄	—	212	4	no	—
252	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
253	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
254	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
256	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
257	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	4	no	—
263	C₈	—	4	2	no	—
264	C₈	—	212	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	212	4	no	—
269	C₂²	—	212	4	no	—
270	C₂²	—	212	4	no	—
271	C₂²	C₂×C₄×D₂₁₂	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	#16811389	1	1	yes	characteristic
278	C₄	#16811389	1	1	yes	characteristic
279	C₄	#16811428	1	2	yes	—
280	C₂	—	4	1	no	—
281	C₂	—	424	4	no	—
282	C₂	#20283434	1	1	yes	characteristic
283	C₂	#20283568	1	1	yes	characteristic
284	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×#1452	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#20285294	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1452	D₅₃
hypocenter	C₅₃	#17989
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
2. #20283790	C₂	semidirect
3. #20273581	C₂	semidirect
4. #20273546	C₂	semidirect
5. #20273478	C₂	semidirect
6. #20273185	C₂	semidirect
7. C₅₃×#1452	C₂	semidirect
144. #1452	D₅₃	semidirect
171. C₅₃	#17989	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
#20273581	2
#20273546	2
#20273478	2
#20273185	2
C₅₃×#1452	2
#17989	53