Subgroups — #27301041

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133151	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810427	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic
4	#20272806	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#467	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
7	C₂²×#11719908	C₂²	1	2	yes	—
8	C₂×#13133151	—	2	4	no	—
9	C₂×#13133151	—	2	1	no	—
10	C₂×#13133151	C₂²	1	2	yes	—
11	#16810427	—	2	4	no	—
12	C₂²⋊#11719908	—	2	2	no	—
13	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
16	C₅₃×#467	—	2	2	no	—
17	C₅₃×#467	C₄	1	4	yes	complemented
18	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#103×C₅₃	—	2	1	no	—
20	C₂³×Dic₅₃	—	4	1	no	—
21	C₂×C₄⋊Dic₅₃	—	2	4	no	—
22	C₂×C₄⋊Dic₅₃	—	2	2	no	—
23	C₂×C₄⋊Dic₅₃	—	4	2	no	—
24	C₂×C₄⋊Dic₅₃	—	4	1	no	—
25	C₂×#11719908	—	2	2	no	—
26	C₂×#11719908	—	2	2	no	—
27	C₂×#11719908	—	2	2	no	—
28	#13133151	—	2	4	no	—
29	#13133151	—	2	2	no	—
30	#13133151	D₄	1	4	yes	—
31	C₂²×Q₈×C₅₃	—	2	1	no	—
32	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
33	C₂×Q₁₆×C₅₃	—	2	1	no	—
34	C₂×Q₁₆×C₅₃	—	2	2	no	—
35	C₂×Q₁₆×C₅₃	—	2	2	no	—
36	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
37	C₂²×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	—	2	2	no	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₅₃	D₄	1	2	yes	—
41	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²⋊Q₈×C₅₃	—	2	4	no	—
44	C₂²⋊Q₈×C₅₃	—	4	2	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
46	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
47	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
48	#107×C₅₃	—	2	1	no	—
49	#107×C₅₃	C₂×C₄	1	2	yes	complemented
50	#103×C₅₃	—	2	4	no	—
51	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
52	C₂×Q₈×C₅₃	—	2	2	no	—
53	C₂×Q₈×C₅₃	—	4	1	no	—
54	C₂×Q₈×C₅₃	—	4	1	no	—
55	C₂×Q₈×C₅₃	—	4	1	no	—
56	C₂×Q₈×C₅₃	—	4	1	no	—
57	C₂×Q₈×C₅₃	—	4	1	no	—
58	C₂²×C₄×C₅₃	—	2	1	no	—
59	C₂²×C₄×C₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	—	2	1	no	—
61	C₂²×C₄×C₅₃	—	2	1	no	—
62	C₂²×C₄×C₅₃	—	4	1	no	—
63	C₂²×C₄×C₅₃	—	4	1	no	—
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₅₃	—	4	4	no	—
69	C₂²×Dic₅₃	—	4	2	no	—
70	C₂²×Dic₅₃	—	8	2	no	—
71	C₂²×Dic₅₃	—	8	2	no	—
72	Q₁₆×C₅₃	—	2	4	no	—
73	Q₁₆×C₅₃	—	4	2	no	—
74	C₂×C₈×C₅₃	—	2	1	no	—
75	C₂×C₈×C₅₃	—	2	1	no	—
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	2	4	no	—
78	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	#41	1	2	yes	—
83	C₄⋊C₄×C₅₃	—	2	2	no	—
84	C₄⋊C₄×C₅₃	—	4	4	no	—
85	C₄⋊C₄×C₅₃	—	4	1	no	—
86	C₂²⋊C₄×C₅₃	—	4	4	no	—
87	C₄⋊Dic₅₃	—	2	4	no	—
88	C₄⋊Dic₅₃	—	4	4	no	—
89	C₄⋊Dic₅₃	—	4	2	no	—
90	#11719908	—	4	2	no	—
91	C₂³×C₅₃	—	2	1	no	—
92	C₂³×C₅₃	—	2	1	no	—
93	C₂³×C₅₃	—	2	1	no	—
94	C₂³×C₅₃	—	2	1	no	—
95	C₂³×C₅₃	#102	1	2	yes	—
96	C₂³×C₅₃	SD₃₂	1	2	yes	—
97	C₂³×C₅₃	Q₃₂	1	2	yes	—
98	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
99	Q₈×C₅₃	—	4	2	no	—
100	Q₈×C₅₃	—	4	2	no	—
101	Q₈×C₅₃	—	8	1	no	—
102	Q₈×C₅₃	—	8	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	—	8	1	no	—
112	C₂×C₄×C₅₃	—	8	1	no	—
113	C₂×C₄×C₅₃	—	8	1	no	—
114	C₂×C₄×C₅₃	—	8	1	no	—
115	C₂×C₄×C₅₃	#102	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
122	C₂×Dic₅₃	—	4	4	no	—
123	C₂×Dic₅₃	—	8	8	no	—
124	C₂×Dic₅₃	—	8	2	no	—
125	C₂×Dic₅₃	—	8	2	no	—
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	C₄×D₄	1	2	yes	—
128	C₈×C₅₃	#121	1	2	yes	—
129	#12993	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	—	2	2	no	—
132	C₂²×C₅₃	—	2	2	no	—
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	#358	1	4	yes	complemented
137	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
138	C₂²×C₅₃	#447	1	1	yes	characteristic
139	C₂²×C₅₃	#480	1	1	yes	characteristic
140	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
141	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
142	C₂²×C₅₃	#509	1	1	yes	characteristic
143	C₂²×C₅₃	#510	1	1	yes	characteristic
144	C₄×C₅₃	—	2	2	no	—
145	C₄×C₅₃	—	8	2	no	—
146	C₄×C₅₃	—	8	2	no	—
147	C₄×C₅₃	#386	1	1	yes	characteristic
148	C₄×C₅₃	#417	1	1	yes	characteristic
149	C₄×C₅₃	#448	1	1	yes	characteristic
150	C₄×C₅₃	#481	1	1	yes	characteristic
151	Dic₅₃	—	8	4	no	—
152	C₂×#467	D₅₃	1	1	yes	characteristic
153	C₂²×#107	—	53	1	no	—
154	C₂×#358	—	53	2	no	—
155	C₂×C₂²⋊C₁₆	—	53	1	no	—
156	#1380	—	53	2	no	—
157	C₂×C₅₃	—	2	4	no	—
158	C₂×C₅₃	#1874	1	1	yes	characteristic
159	C₂×C₅₃	C₂×#358	1	1	yes	characteristic
160	C₂×C₅₃	#2139	1	1	yes	characteristic
161	C₂×C₅₃	#2187	1	1	yes	characteristic
162	C₂×C₅₃	#2188	1	1	yes	characteristic
163	C₂×C₅₃	#2235	1	1	yes	characteristic
164	C₂×C₅₃	#2236	1	1	yes	characteristic
165	C₂²×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
166	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
167	C₂×C₂²⋊Q₈	—	2	1	no	—
168	C₂²×C₄⋊C₄	—	106	1	no	—
169	C₂²×C₁₆	—	53	2	no	—
170	#467	—	2	2	no	—
171	#467	Dic₅₃	1	4	yes	complemented
172	C₂×#107	—	53	2	no	—
173	C₂×#107	—	106	4	no	—
174	C₂×#107	—	106	1	no	—
175	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
176	C₂×#103	—	2	1	no	—
177	#358	—	106	4	no	—
178	C₂²⋊C₁₆	—	106	2	no	—
179	C₅₃	#12993	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	—	2	1	no	—
181	C₂³×C₄	—	212	1	no	—
182	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	2	2	no	—
185	C₂×Q₁₆	—	2	1	no	—
186	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	complemented
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
190	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
191	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
193	C₂²⋊Q₈	—	2	4	no	—
194	C₂²⋊Q₈	—	4	2	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	4	1	no	—
197	C₂×C₄⋊C₄	—	106	4	no	—
198	C₂×C₄⋊C₄	—	106	2	no	—
199	C₂×C₄⋊C₄	—	212	2	no	—
200	C₂×C₄⋊C₄	—	212	1	no	—
201	C₂×C₂²⋊C₄	—	4	1	no	—
202	C₂×C₁₆	—	106	2	no	—
203	C₂×C₁₆	—	106	2	no	—
204	C₂×C₁₆	—	106	2	no	—
205	#107	—	2	1	no	—
206	#107	—	53	4	no	—
207	#107	—	106	4	no	—
208	#107	—	106	2	no	—
209	#107	C₂×Dic₅₃	1	2	yes	complemented
210	#103	—	2	4	no	—
211	C₂⁴	#11719922	1	1	yes	characteristic
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	4	1	no	—
214	C₂×Q₈	—	4	1	no	—
215	C₂×Q₈	—	4	1	no	—
216	C₂×Q₈	—	4	1	no	—
217	C₂×Q₈	—	4	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	424	2	no	—
225	C₂²×C₄	—	424	2	no	—
226	C₂²×C₄	—	212	4	no	—
227	C₂²×C₄	—	212	2	no	—
228	C₂²×C₄	#11719922	1	1	yes	characteristic
229	C₂²×C₄	#11719923	1	2	yes	—
230	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
231	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
232	Q₁₆	—	2	4	no	—
233	Q₁₆	—	4	2	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
239	C₂×C₈	D₄×D₅₃	1	2	yes	—
240	C₂×C₈	#11719947	1	2	yes	—
241	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
242	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₄⋊C₄	—	4	1	no	—
246	C₄⋊C₄	—	106	4	no	—
247	C₄⋊C₄	—	212	4	no	—
248	C₄⋊C₄	—	212	2	no	—
249	C₂²⋊C₄	—	4	4	no	—
250	C₁₆	—	212	2	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	#13133138	1	2	yes	—
256	C₂³	#13133160	1	2	yes	—
257	C₂³	#13133163	1	2	yes	—
258	C₂³	C₂×#11719922	1	1	yes	characteristic
259	Q₈	—	4	2	no	—
260	Q₈	—	4	2	no	—
261	Q₈	—	8	1	no	—
262	Q₈	—	8	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	424	8	no	—
276	C₂×C₄	—	424	2	no	—
277	C₂×C₄	—	424	2	no	—
278	C₂×C₄	—	212	4	no	—
279	C₂×C₄	#13133138	1	2	yes	—
280	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
281	C₂×C₄	#13133185	1	1	yes	characteristic
282	C₂×C₄	#13133198	1	1	yes	characteristic
283	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
284	C₂×C₄	#13133242	1	1	yes	characteristic
285	C₂×C₄	#13133247	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	D₄×Dic₅₃	1	2	yes	—
288	C₈	#13133189	1	2	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	#16810427	1	4	yes	complemented
296	C₂²	C₂×#13133138	1	1	yes	characteristic
297	C₂²	#16810756	1	1	yes	characteristic
298	C₂²	#16810934	1	1	yes	characteristic
299	C₂²	C₂×#13133160	1	1	yes	characteristic
300	C₂²	C₂×#13133163	1	1	yes	characteristic
301	C₂²	#16811058	1	1	yes	characteristic
302	C₂²	#16811063	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	—	424	4	no	—
307	C₄	#16810503	1	1	yes	characteristic
308	C₄	#16810624	1	1	yes	characteristic
309	C₄	#16810761	1	1	yes	characteristic
310	C₄	#16810939	1	1	yes	characteristic
311	C₂	—	2	4	no	—
312	C₂	#20275033	1	1	yes	characteristic
313	C₂	C₂×#16810427	1	1	yes	characteristic
314	C₂	#20276427	1	1	yes	characteristic
315	C₂	#20276643	1	1	yes	characteristic
316	C₂	#20276648	1	1	yes	characteristic
317	C₂	#20276910	1	1	yes	characteristic
318	C₂	#20276915	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#467	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#467	D₅₃
hypocenter	C₅₃	#12993
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
2. C₂×#16810427	C₂	semidirect
4. #20272806	C₂	semidirect
17. C₅₃×#467	C₄	semidirect
36. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
49. #107×C₅₃	C₂×C₄	semidirect
136. C₂²×C₅₃	#358	semidirect
171. #467	Dic₅₃	semidirect
179. C₅₃	#12993	semidirect
186. C₂×Q₁₆	C₂×Dic₅₃	semidirect
209. #107	C₂×Dic₅₃	semidirect
295. C₂²	#16810427	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133151	2
C₂×#16810427	2
C₂×C₂²⋊#11719908	2
#20272806	2
C₂×C₅₃×#467	2
#12993	53