Subgroups — #27402620

Order 13568.

808 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	#20283631	C₂	1	2	yes	maximal
2	C₂×#16810746	C₂	1	1	yes	maximal, characteristic
3	#20283323	C₂	1	2	yes	maximal, complemented
4	#20275503	C₂	1	2	yes	maximal, complemented
5	#20274954	C₂	1	4	yes	maximal, complemented
6	#20274770	C₂	1	2	yes	maximal, complemented
7	#20274599	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1849	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	C₂²	1	1	yes	characteristic
10	#16811389	C₂²	1	4	yes	—
11	C₂×#13133223	C₂²	1	2	yes	—
12	#16810746	C₄	1	8	yes	complemented
13	#16810744	—	2	4	no	—
14	#16810739	—	2	4	no	—
15	#16810671	C₂²	1	4	yes	—
16	#16810670	—	2	4	no	—
17	C₂×#13133153	C₂²	1	1	yes	characteristic
18	C₂×#13133153	C₂²	1	1	yes	characteristic
19	#108×D₅₃	—	2	4	no	—
20	#16810661	C₂²	1	1	yes	characteristic
21	#16810639	C₂²	1	4	yes	—
22	C₂×#13133145	C₂²	1	2	yes	—
23	#16810578	C₂²	1	2	yes	—
24	#16810394	C₂²	1	4	yes	—
25	#16810383	C₂²	1	2	yes	—
26	C₂×#108×C₅₃	C₂²	1	2	yes	—
27	C₂×#108×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#423	C₂²	1	2	yes	—
29	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#11719945	C₂³	1	2	yes	—
31	#13133239	C₂×C₄	1	4	yes	—
32	#13133239	D₄	1	4	yes	—
33	C₂×Dic₂₁₂	Q₈	1	1	yes	characteristic
34	C₂×#11719913	Q₈	1	2	yes	—
35	C₂×D₄₂₄	Q₈	1	1	yes	characteristic
36	#13133225	—	2	8	no	—
37	#13133223	—	2	2	no	—
38	#13133223	—	2	2	no	—
39	#13133223	—	4	4	no	—
40	#13133223	C₂×C₄	1	4	yes	—
41	#13133223	C₂×C₄	1	4	yes	—
42	C₂×#11719917	C₂³	1	2	yes	—
43	C₂×#11719917	C₂³	1	2	yes	—
44	C₂×#11719912	—	2	2	no	—
45	M₁₆×D₅₃	—	2	8	no	—
46	C₂×C₈×D₅₃	—	2	2	no	—
47	#13133176	C₂³	1	2	yes	—
48	#13133153	C₂×C₄	1	4	yes	—
49	#13133153	D₄	1	4	yes	—
50	#13133152	—	2	8	no	—
51	#13133151	—	2	1	no	—
52	#13133149	—	2	1	no	—
53	#13133145	C₂×C₄	1	8	yes	—
54	#13133143	—	2	8	no	—
55	#13133127	—	2	2	no	—
56	C₈×Dic₅₃	—	2	2	no	—
57	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
58	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	#108×C₅₃	—	2	2	no	—
62	#108×C₅₃	—	2	2	no	—
63	#108×C₅₃	—	2	2	no	—
64	#108×C₅₃	—	2	2	no	—
65	C₄⋊C₈×C₅₃	—	4	2	no	—
66	#97×C₅₃	C₂×C₄	1	4	yes	—
67	C₂²×C₄×C₅₃	—	2	1	no	—
68	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₂×C₂²⋊D₅₃	—	2	2	no	—
70	#11719945	—	2	2	no	—
71	#11719945	—	2	2	no	—
72	#11719945	C₂²×C₄	1	4	yes	—
73	#11719945	C₂×D₄	1	4	yes	—
74	C₂×D₂₁₂	C₂×Q₈	1	2	yes	—
75	C₂×C₄×D₅₃	—	2	2	no	—
76	C₂×Dic₁₀₆	C₂×Q₈	1	2	yes	—
77	M₁₆×C₅₃	—	2	4	no	—
78	M₁₆×C₅₃	—	2	4	no	—
79	C₂×C₈×C₅₃	—	2	4	no	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
87	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
88	C₄²×C₅₃	—	2	1	no	—
89	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂²⋊Dic₅₃	—	2	2	no	—
91	C₄⋊Dic₅₃	—	2	2	no	—
92	C₄×Dic₅₃	—	2	2	no	—
93	#11719917	—	2	2	no	—
94	#11719917	—	2	2	no	—
95	#11719917	C₂²×C₄	1	4	yes	—
96	#11719917	C₂×D₄	1	4	yes	—
97	C₂×#88322	—	2	2	no	—
98	C₂×#88322	—	2	2	no	—
99	Dic₂₁₂	C₄⋊C₄	1	4	yes	—
100	D₄₂₄	C₄⋊C₄	1	4	yes	—
101	#11719913	C₄⋊C₄	1	8	yes	—
102	#11719912	—	2	4	no	—
103	#11719912	—	2	4	no	—
104	#11719912	—	4	8	no	—
105	C₈×D₅₃	—	2	4	no	—
106	C₈×D₅₃	—	2	4	no	—
107	C₈×D₅₃	—	4	8	no	—
108	C₂³×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂²×D₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	8	1	no	—
112	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²⋊D₅₃	—	2	4	no	—
119	C₂²⋊D₅₃	—	2	4	no	—
120	C₂×Dic₅₃	—	2	2	no	—
121	C₂×Dic₅₃	—	4	2	no	—
122	D₂₁₂	—	2	2	no	—
123	D₂₁₂	C₂×C₄⋊C₄	1	4	yes	—
124	C₄×D₅₃	—	2	4	no	—
125	C₄×D₅₃	—	2	4	no	—
126	Dic₁₀₆	—	2	2	no	—
127	Dic₁₀₆	C₂×C₄⋊C₄	1	4	yes	—
128	C₈×C₅₃	—	4	4	no	—
129	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₈×C₅₃	C₄×D₄	1	2	yes	—
132	C₈×C₅₃	C₄×D₄	1	2	yes	—
133	#88322	—	2	4	no	—
134	#88322	—	2	4	no	—
135	#22695	—	53	1	no	maximal, Sylow
136	C₂²×C₅₃	—	4	1	no	—
137	C₂²×C₅₃	#520	1	1	yes	characteristic
138	C₂²×C₅₃	#547	1	1	yes	characteristic
139	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
140	C₂×D₅₃	—	2	4	no	—
141	C₂×D₅₃	—	8	2	no	—
142	C₄×C₅₃	—	4	2	no	—
143	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
145	C₄×C₅₃	D₄²	1	1	yes	characteristic
146	C₄×C₅₃	#554	1	1	yes	characteristic
147	Dic₅₃	—	2	4	no	—
148	Dic₅₃	—	8	2	no	—
149	#2957	—	53	2	no	—
150	C₂×#444	—	53	1	no	—
151	#2916	—	53	2	no	—
152	#1953	—	53	2	no	—
153	#1861	—	53	4	no	—
154	#1849	D₅₃	1	1	yes	characteristic, complemented
155	#1814	—	53	2	no	—
156	#1773	—	53	1	no	—
157	C₂×C₅₃	—	2	1	no	—
158	C₂×C₅₃	—	2	1	no	—
159	C₂×C₅₃	—	2	1	no	—
160	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
161	D₅₃	—	4	4	no	—
162	C₂×#135	—	53	1	no	—
163	#568	—	53	4	no	—
164	C₂×#131	—	53	2	no	—
165	#444	—	53	8	no	—
166	#444	—	106	4	no	—
167	#443	—	106	4	no	—
168	#430	—	53	4	no	—
169	#430	—	106	4	no	—
170	C₂×#108	—	53	1	no	—
171	C₂×#108	—	53	1	no	—
172	C₂×#108	—	106	4	no	—
173	C₂×#108	C₂×D₅₃	1	2	yes	—
174	C₂×#108	C₂×D₅₃	1	2	yes	—
175	#427	—	53	1	no	—
176	#423	C₂×D₅₃	1	2	yes	—
177	#421	—	53	4	no	—
178	C₂×#104	—	53	2	no	—
179	#405	—	53	2	no	—
180	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
181	#344	—	53	4	no	—
182	#339	—	53	2	no	—
183	C₅₃	#22695	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	53	2	no	—
185	#135	—	53	4	no	—
186	#135	—	53	4	no	—
187	C₂×Q₁₆	—	53	1	no	—
188	C₂×SD₁₆	—	53	2	no	—
189	C₂×D₈	—	53	1	no	—
190	#131	—	53	4	no	—
191	#131	—	53	4	no	—
192	#131	—	106	2	no	—
193	#131	—	106	2	no	—
194	#131	—	106	8	no	—
195	#131	—	212	4	no	—
196	C₂×M₁₆	—	53	2	no	—
197	C₂×M₁₆	—	53	2	no	—
198	C₂×M₁₆	—	106	2	no	—
199	C₂×M₁₆	—	106	8	no	—
200	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
201	C₂²×C₈	—	106	2	no	—
202	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
204	#117	—	53	2	no	—
205	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
206	#108	—	2	2	no	—
207	#108	—	2	2	no	—
208	#108	—	2	2	no	—
209	#108	—	2	2	no	—
210	#108	—	53	4	no	—
211	#108	—	53	4	no	—
212	#108	—	106	8	no	—
213	#107	—	106	1	no	—
214	#106	—	106	1	no	—
215	C₄⋊C₈	—	4	2	no	—
216	#104	—	53	8	no	—
217	#104	—	106	8	no	—
218	#97	—	106	2	no	—
219	#97	C₄×D₅₃	1	4	yes	—
220	C₄×C₈	—	106	2	no	—
221	#41	—	53	4	no	—
222	#41	—	53	4	no	—
223	#41	—	106	2	no	—
224	#41	—	106	2	no	—
225	C₂×Q₈	—	53	2	no	—
226	C₂×D₄	—	53	2	no	—
227	C₂×D₄	—	106	2	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	106	2	no	—
230	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
231	Q₁₆	—	53	4	no	—
232	SD₁₆	—	53	8	no	—
233	D₈	—	53	4	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	53	4	no	—
237	M₁₆	—	53	4	no	—
238	M₁₆	—	106	2	no	—
239	M₁₆	—	106	2	no	—
240	M₁₆	—	106	4	no	—
241	M₁₆	—	106	4	no	—
242	M₁₆	—	212	8	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	106	2	no	—
245	C₂×C₈	—	106	2	no	—
246	C₂×C₈	—	106	4	no	—
247	C₂×C₈	—	106	4	no	—
248	C₂×C₈	—	212	8	no	—
249	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
251	C₂×C₈	D₄×D₅₃	1	2	yes	—
252	C₂×C₈	D₄×D₅₃	1	2	yes	—
253	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
254	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
255	C₂×C₈	#11719949	1	1	yes	characteristic
256	C₂×C₈	#11719949	1	1	yes	characteristic
257	C₄⋊C₄	—	106	2	no	—
258	C₂²⋊C₄	—	106	2	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	106	2	no	—
261	C₄²	C₂×C₄×D₅₃	1	2	yes	—
262	C₂³	—	106	2	no	—
263	C₂³	#13133265	1	1	yes	characteristic
264	Q₈	—	53	4	no	—
265	Q₈	—	106	2	no	—
266	D₄	—	53	4	no	—
267	D₄	—	106	4	no	—
268	D₄	—	106	4	no	—
269	D₄	—	106	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	106	2	no	—
273	C₂×C₄	—	106	4	no	—
274	C₂×C₄	—	106	4	no	—
275	C₂×C₄	—	212	2	no	—
276	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
278	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
279	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
280	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
281	C₂×C₄	#13133262	1	1	yes	characteristic
282	C₈	—	4	4	no	—
283	C₈	—	106	4	no	—
284	C₈	—	106	4	no	—
285	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
286	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
287	C₈	#13133179	1	2	yes	—
288	C₈	#13133179	1	2	yes	—
289	C₂²	—	4	1	no	—
290	C₂²	—	106	4	no	—
291	C₂²	—	424	2	no	—
292	C₂²	#16811104	1	1	yes	characteristic
293	C₂²	#16811276	1	1	yes	characteristic
294	C₂²	#16811288	1	1	yes	characteristic
295	C₄	—	4	2	no	—
296	C₄	—	106	4	no	—
297	C₄	—	424	2	no	—
298	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
299	C₄	C₂×#13133179	1	1	yes	characteristic
300	C₄	#16811258	1	1	yes	characteristic
301	C₄	#16811334	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	212	4	no	—
306	C₂	#20277615	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16811258
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1849	C₂
socle	C₂×C₅₃	D₄×C₄⋊C₄
prime radical	C₂×#13133239	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1849	D₅₃
hypocenter	C₅₃	#22695
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1849

Complements

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

N	H ≅ G/N	type
3. #20283323	C₂	semidirect
4. #20275503	C₂	semidirect
5. #20274954	C₂	semidirect
6. #20274770	C₂	semidirect
7. #20274599	C₂	semidirect
8. C₅₃×#1849	C₂	semidirect
12. #16810746	C₄	semidirect
154. #1849	D₅₃	semidirect
183. C₅₃	#22695	semidirect

Maximal subgroups

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

subgroup	index
#20283631	2
C₂×#16810746	2
#20283323	2
#20275503	2
#20274954	2
#20274770	2
#20274599	2
C₅₃×#1849	2
#22695	53