Subgroups — #27384363

Order 13568.

684 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274864	C₂	1	8	yes	maximal, complemented
2	C₂×#16810394	C₂	1	2	yes	maximal
3	#20274425	C₂	1	2	yes	maximal
4	C₂×#16810381	C₂	1	1	yes	maximal, characteristic
5	C₂×#16810380	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#2873	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	2	yes	—
8	C₂×#13133176	C₂²	1	2	yes	—
9	#16810394	C₂²	1	8	yes	—
10	#16810383	C₂²	1	8	yes	—
11	#16810381	C₂²	1	4	yes	—
12	#16810380	C₂²	1	4	yes	—
13	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#405	C₄	1	4	yes	complemented
17	C₅₃×#405	C₄	1	4	yes	complemented
18	C₄×M₁₆×C₅₃	C₂²	1	4	yes	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	C₂³	1	4	yes	—
25	C₂²×#88322	—	2	2	no	—
26	#13133176	—	2	2	no	—
27	#13133176	—	2	2	no	—
28	#13133176	—	2	2	no	—
29	#13133176	C₂³	1	4	yes	—
30	C₂×C₄⋊Dic₅₃	—	2	2	no	—
31	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
32	C₂×C₄×Dic₅₃	—	2	2	no	—
33	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₅₃	—	2	2	no	—
35	C₂×M₁₆×C₅₃	—	2	2	no	—
36	C₂×M₁₆×C₅₃	—	2	2	no	—
37	C₂×M₁₆×C₅₃	—	2	2	no	—
38	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
41	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	#117×C₅₃	—	2	1	no	—
45	#117×C₅₃	—	2	1	no	—
46	#117×C₅₃	C₂×C₄	1	2	yes	—
47	#117×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
50	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
51	#97×C₅₃	C₂×C₄	1	8	yes	complemented
52	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
53	C₂⁴×C₅₃	#41	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	#41	1	2	yes	—
59	C₂²×C₄×C₅₃	#41	1	2	yes	—
60	C₂²×C₄×C₅₃	#41	1	2	yes	—
61	C₂²×C₄×C₅₃	#41	1	2	yes	—
62	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₅₃	—	4	2	no	—
66	M₁₆×C₅₃	—	2	4	no	—
67	M₁₆×C₅₃	—	2	2	no	—
68	M₁₆×C₅₃	—	2	2	no	—
69	M₁₆×C₅₃	—	2	4	no	—
70	M₁₆×C₅₃	—	2	4	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
76	C₄⋊C₄×C₅₃	—	2	2	no	—
77	C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₂²⋊C₄×C₅₃	—	2	2	no	—
79	C₂²⋊C₄×C₅₃	—	2	2	no	—
80	C₄²×C₅₃	—	2	2	no	—
81	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
82	C₂²⋊Dic₅₃	—	2	4	no	—
83	C₂²⋊Dic₅₃	—	4	2	no	—
84	C₄⋊Dic₅₃	—	2	4	no	—
85	C₄⋊Dic₅₃	—	4	2	no	—
86	C₄×Dic₅₃	—	2	4	no	—
87	C₄×Dic₅₃	—	4	2	no	—
88	#11719917	—	2	4	no	—
89	#11719917	—	2	4	no	—
90	#11719917	—	4	2	no	—
91	#11719917	—	4	2	no	—
92	C₂×#88322	—	2	4	no	—
93	C₂×#88322	—	4	2	no	—
94	C₂×#88322	—	4	2	no	—
95	C₂³×C₅₃	—	8	1	no	—
96	C₂³×C₅₃	#117	1	1	yes	characteristic
97	C₂³×C₅₃	#117	1	1	yes	characteristic
98	C₂³×C₅₃	#126	1	2	yes	—
99	C₂³×C₅₃	#126	1	2	yes	—
100	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	8	2	no	—
103	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#117	1	2	yes	—
106	C₂×C₄×C₅₃	#117	1	2	yes	—
107	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#126	1	2	yes	—
112	C₂×C₄×C₅₃	#126	1	2	yes	—
113	C₂×C₄×C₅₃	#126	1	2	yes	—
114	C₂×C₄×C₅₃	#126	1	2	yes	—
115	C₂×C₄×C₅₃	#126	1	2	yes	—
116	C₂×C₄×C₅₃	#126	1	2	yes	—
117	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
119	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
120	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
121	C₂×Dic₅₃	—	4	4	no	—
122	C₂×Dic₅₃	—	8	4	no	—
123	C₈×C₅₃	—	2	4	no	—
124	C₈×C₅₃	—	2	4	no	—
125	#88322	—	4	4	no	—
126	#20776	—	53	1	no	maximal, Sylow
127	C₂²×C₅₃	—	4	2	no	—
128	C₂²×C₅₃	—	4	1	no	—
129	C₂²×C₅₃	—	4	1	no	—
130	C₂²×C₅₃	—	4	1	no	—
131	C₂²×C₅₃	—	4	2	no	—
132	C₂²×C₅₃	#383	1	1	yes	characteristic
133	C₂²×C₅₃	#383	1	2	yes	—
134	C₂²×C₅₃	#383	1	1	yes	characteristic
135	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
136	C₂²×C₅₃	C₂×#126	1	2	yes	—
137	C₄×C₅₃	—	4	4	no	—
138	C₄×C₅₃	#383	1	1	yes	characteristic
139	C₄×C₅₃	#383	1	2	yes	—
140	C₄×C₅₃	#383	1	1	yes	characteristic
141	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
142	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
143	C₄×C₅₃	C₂×#126	1	2	yes	—
144	Dic₅₃	—	8	4	no	—
145	#2873	D₅₃	1	1	yes	characteristic
146	#1841	—	53	8	no	—
147	C₂×#344	—	53	2	no	—
148	#1733	—	53	2	no	—
149	C₂×#338	—	53	1	no	—
150	C₂×H(3,ℤ/4)	—	53	1	no	—
151	C₂×C₅₃	—	2	2	no	—
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	—	2	1	no	—
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	—	2	1	no	—
156	C₂×C₅₃	C₂×#383	1	1	yes	characteristic
157	C₂²×M₁₆	—	53	2	no	—
158	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
159	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
160	C₂×#117	—	53	2	no	—
161	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
162	#405	Dic₅₃	1	4	yes	complemented
163	#405	Dic₅₃	1	4	yes	complemented
164	C₄×M₁₆	C₂×D₅₃	1	4	yes	—
165	#344	—	53	8	no	—
166	#339	—	53	8	no	—
167	#338	—	53	4	no	—
168	H(3,ℤ/4)	—	53	4	no	—
169	C₅₃	#20776	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	53	4	no	—
176	C₂×M₁₆	—	106	2	no	—
177	C₂×M₁₆	—	106	2	no	—
178	C₂×M₁₆	—	106	2	no	—
179	C₂×M₁₆	—	106	2	no	—
180	C₂×M₁₆	—	106	2	no	—
181	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
182	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
183	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
184	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
185	C₂²×C₈	—	106	2	no	—
186	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
187	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
188	#117	—	2	1	no	—
189	#117	—	2	1	no	—
190	#117	—	53	4	no	—
191	#117	—	106	2	no	—
192	#117	—	106	2	no	—
193	#117	—	106	2	no	—
194	#117	C₂×Dic₅₃	1	2	yes	—
195	#117	C₂×Dic₅₃	1	2	yes	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	106	2	no	—
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₂²⋊C₄	—	106	2	no	—
200	C₂×C₄²	—	106	2	no	—
201	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
202	#97	C₂×Dic₅₃	1	8	yes	complemented
203	C₄×C₈	C₂×Dic₅₃	1	8	yes	complemented
204	C₂⁴	#11719945	1	1	yes	characteristic
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	212	2	no	—
207	C₂²×C₄	#11719945	1	1	yes	characteristic
208	C₂²×C₄	#11719945	1	2	yes	—
209	C₂²×C₄	#11719945	1	2	yes	—
210	C₂²×C₄	#11719945	1	2	yes	—
211	C₂²×C₄	#11719945	1	2	yes	—
212	C₂²×C₄	#11719945	1	1	yes	characteristic
213	C₂²×C₄	#11719945	1	1	yes	characteristic
214	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
216	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	106	4	no	—
223	M₁₆	—	106	4	no	—
224	M₁₆	—	212	2	no	—
225	M₁₆	—	212	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	106	4	no	—
230	C₂×C₈	—	212	2	no	—
231	C₂×C₈	—	212	2	no	—
232	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
233	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	106	4	no	—
237	C₄⋊C₄	—	212	2	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	106	4	no	—
241	C₂²⋊C₄	—	212	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	106	4	no	—
244	C₄²	—	212	2	no	—
245	C₄²	C₂²×Dic₅₃	1	4	yes	—
246	C₂³	—	8	1	no	—
247	C₂³	#13133176	1	1	yes	characteristic
248	C₂³	#13133176	1	1	yes	characteristic
249	C₂³	#13133211	1	2	yes	—
250	C₂³	#13133211	1	2	yes	—
251	C₂³	C₂×#11719945	1	1	yes	characteristic
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	—	424	4	no	—
255	C₂×C₄	—	212	4	no	—
256	C₂×C₄	#13133176	1	1	yes	characteristic
257	C₂×C₄	#13133176	1	1	yes	characteristic
258	C₂×C₄	#13133176	1	2	yes	—
259	C₂×C₄	#13133176	1	2	yes	—
260	C₂×C₄	#13133176	1	1	yes	characteristic
261	C₂×C₄	#13133176	1	1	yes	characteristic
262	C₂×C₄	#13133176	1	1	yes	characteristic
263	C₂×C₄	#13133176	1	1	yes	characteristic
264	C₂×C₄	#13133211	1	2	yes	—
265	C₂×C₄	#13133211	1	2	yes	—
266	C₂×C₄	#13133211	1	2	yes	—
267	C₂×C₄	#13133211	1	2	yes	—
268	C₂×C₄	#13133211	1	2	yes	—
269	C₂×C₄	#13133211	1	2	yes	—
270	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
271	C₂×C₄	C₂×#11719945	1	2	yes	—
272	C₂×C₄	C₂×#11719945	1	2	yes	—
273	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
274	C₈	—	2	4	no	—
275	C₈	—	2	4	no	—
276	C₈	—	212	4	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	2	no	—
282	C₂²	#16810491	1	1	yes	characteristic
283	C₂²	#16810491	1	1	yes	characteristic
284	C₂²	#16810491	1	2	yes	—
285	C₂²	C₂×#13133176	1	1	yes	characteristic
286	C₂²	C₂×#13133211	1	2	yes	—
287	C₄	—	4	4	no	—
288	C₄	—	424	4	no	—
289	C₄	#16810491	1	1	yes	characteristic
290	C₄	#16810491	1	1	yes	characteristic
291	C₄	#16810491	1	2	yes	—
292	C₄	C₂×#13133176	1	1	yes	characteristic
293	C₄	C₂×#13133176	1	1	yes	characteristic
294	C₄	C₂×#13133211	1	2	yes	—
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	—	2	1	no	—
300	C₂	C₂×#16810491	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13133176
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2873	C₂
socle	C₂×C₅₃	C₂×#383
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#2873	D₅₃
hypocenter	C₅₃	#20776
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ #2873

Complements

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

N	H ≅ G/N	type
1. #20274864	C₂	semidirect
16. C₅₃×#405	C₄	semidirect
17. C₅₃×#405	C₄	semidirect
51. #97×C₅₃	C₂×C₄	semidirect
52. C₄×C₈×C₅₃	C₂×C₄	semidirect
162. #405	Dic₅₃	semidirect
163. #405	Dic₅₃	semidirect
169. C₅₃	#20776	semidirect
202. #97	C₂×Dic₅₃	semidirect
203. C₄×C₈	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20274864	2
C₂×#16810394	2
#20274425	2
C₂×#16810381	2
C₂×#16810380	2
C₅₃×#2873	2
#20776	53