Subgroups — #27537277

Order 13568.

690 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811041	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811040	C₂	1	1	yes	maximal, characteristic, complemented
3	#20283425	C₂	1	1	yes	maximal, characteristic
4	#20276544	C₂	1	4	yes	maximal, complemented
5	#20276532	C₂	1	4	yes	maximal, complemented
6	#20276470	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276328	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2113	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133223	C₂²	1	1	yes	characteristic
11	#16811169	C₄	1	1	yes	characteristic
12	#16811041	—	2	4	no	—
13	#16811040	—	2	4	no	—
14	C₂×#13133157	C₂²	1	2	yes	—
15	C₂×#13133157	C₂²	1	1	yes	characteristic
16	C₂×#13133156	C₂²	1	2	yes	—
17	C₂²×#11719908	C₂²	1	2	yes	—
18	C₂²×#11719908	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
20	#16810699	C₄	1	4	yes	—
21	C₈×C₂²⋊D₅₃	C₂²	1	4	yes	—
22	#16810646	C₄	1	1	yes	characteristic
23	#16810591	C₄	1	1	yes	characteristic
24	#16810590	C₄	1	1	yes	characteristic
25	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
26	#16810441	C₂²	1	4	yes	—
27	#16810402	C₂²	1	4	yes	—
28	C₁₆×Dic₅₃	C₂²	1	2	yes	—
29	C₁₆×Dic₅₃	C₂²	1	2	yes	—
30	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
32	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
34	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
35	#13133223	—	2	2	no	—
36	#13133223	—	2	2	no	—
37	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
38	C₂×#11719912	C₂×C₄	1	2	yes	—
39	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₅₃	C₂³	1	2	yes	—
41	#13133215	C₈	1	1	yes	characteristic
42	#13133213	C₈	1	1	yes	characteristic
43	#13133206	C₈	1	1	yes	characteristic
44	#13133204	C₈	1	1	yes	characteristic
45	#13133189	C₈	1	4	yes	—
46	C₄×C₂²⋊D₅₃	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
48	#13133157	—	2	2	no	—
49	#13133157	—	2	4	no	—
50	#13133156	—	2	4	no	—
51	C₂×#11719908	—	2	4	no	—
52	C₂×#11719908	D₄	1	4	yes	complemented
53	C₂×#11719908	C₂³	1	2	yes	—
54	C₂×#11719908	C₂³	1	1	yes	characteristic
55	C₂×#11719908	C₂³	1	1	yes	characteristic
56	C₁₆×D₅₃	—	2	4	no	—
57	#13133146	C₂×C₄	1	4	yes	—
58	C₂²⋊#88322	C₂×C₄	1	4	yes	—
59	#13133129	C₂×C₄	1	4	yes	—
60	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
61	C₈×Dic₅₃	C₂³	1	2	yes	—
62	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
63	#131×C₅₃	—	2	2	no	—
64	#131×C₅₃	—	2	2	no	—
65	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
67	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
68	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	M₃₂×C₅₃	—	2	2	no	—
70	C₂×C₁₆×C₅₃	—	2	2	no	—
71	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
72	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
73	#41×C₅₃	—	2	2	no	—
74	#41×C₅₃	—	2	2	no	—
75	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
76	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
77	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊D₅₃	C₂×C₈	1	4	yes	—
81	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
82	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
83	#11719945	—	2	2	no	—
84	#11719945	—	2	2	no	—
85	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
87	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
88	M₁₆×C₅₃	—	2	2	no	—
89	M₁₆×C₅₃	—	2	4	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₅₃	#41	1	2	yes	—
96	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊Dic₅₃	C₂×C₈	1	4	yes	—
98	#11719921	C₂×C₈	1	4	yes	—
99	#11719919	C₂×C₈	1	4	yes	—
100	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
101	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
102	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
103	#11719917	—	2	2	no	—
104	C₂×#88322	—	2	2	no	—
105	C₂×#88322	C₂²×C₄	1	2	yes	—
106	C₂×#88322	C₂²×C₄	1	2	yes	—
107	#11719912	—	2	4	no	—
108	C₈×D₅₃	—	2	2	no	—
109	C₈×D₅₃	—	2	2	no	—
110	C₁₆×C₅₃	—	2	2	no	—
111	C₁₆×C₅₃	—	2	2	no	—
112	#11719908	—	2	2	no	—
113	#11719908	C₂×D₄	1	2	yes	—
114	#11719908	C₂×D₄	1	2	yes	—
115	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
116	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
117	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
118	Q₈×C₅₃	—	2	2	no	—
119	D₄×C₅₃	—	2	4	no	—
120	D₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	#131	1	2	yes	—
127	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
128	C₂²⋊D₅₃	—	2	4	no	—
129	C₂²⋊D₅₃	—	2	8	no	—
130	C₂×Dic₅₃	—	2	4	no	—
131	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
132	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
133	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
134	D₂₁₂	—	2	2	no	—
135	C₄×D₅₃	—	2	2	no	—
136	C₄×D₅₃	—	2	2	no	—
137	Dic₁₀₆	—	2	2	no	—
138	C₈×C₅₃	—	2	2	no	—
139	C₈×C₅₃	C₄×D₄	1	2	yes	—
140	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
142	#88322	—	2	2	no	—
143	#88322	—	2	2	no	—
144	#33434	—	53	1	no	maximal, Sylow
145	C₂²×C₅₃	—	2	2	no	—
146	C₂²×C₅₃	—	2	4	no	—
147	C₂²×C₅₃	—	2	2	no	—
148	C₂²×C₅₃	C₈×D₄	1	2	yes	—
149	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
150	C₂×D₅₃	—	2	4	no	—
151	C₂×D₅₃	—	2	2	no	—
152	C₄×C₅₃	—	2	2	no	—
153	C₄×C₅₃	C₈×D₄	1	2	yes	—
154	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
156	Dic₅₃	—	2	2	no	—
157	Dic₅₃	—	2	2	no	—
158	Dic₅₃	#504	1	4	yes	complemented
159	C₂×#504	—	53	1	no	—
160	C₂×#504	—	53	1	no	—
161	#2931	—	53	1	no	—
162	#2169	—	53	4	no	—
163	D₄×C₁₆	—	53	4	no	—
164	#2150	—	53	1	no	—
165	#2113	—	53	1	no	—
166	#2113	D₅₃	1	1	yes	characteristic, complemented
167	C₂×C₄×C₁₆	—	53	1	no	—
168	C₂×C₅₃	—	2	2	no	—
169	C₂×C₅₃	—	4	2	no	—
170	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
171	C₂×C₅₃	C₂×#504	1	2	yes	—
172	D₅₃	—	4	2	no	—
173	C₂×#131	—	53	1	no	—
174	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
175	#532	—	53	1	no	—
176	#504	—	106	4	no	—
177	#504	—	106	4	no	—
178	C₂×M₃₂	—	53	1	no	—
179	C₂×M₃₂	—	53	2	no	—
180	C₂×M₃₂	—	53	2	no	—
181	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
182	C₂²×C₁₆	—	53	1	no	—
183	C₂²×C₁₆	—	53	2	no	—
184	C₂²×C₁₆	—	53	2	no	—
185	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
186	#436	—	53	4	no	—
187	C₈×D₄	—	53	4	no	—
188	#423	—	53	1	no	—
189	#408	—	53	1	no	—
190	#408	—	53	1	no	—
191	C₂×C₄×C₈	—	53	1	no	—
192	C₄⋊C₁₆	—	53	4	no	—
193	C₂²⋊C₁₆	—	53	4	no	—
194	C₂²⋊C₁₆	C₂×D₅₃	1	4	yes	—
195	C₄×C₁₆	—	53	2	no	—
196	C₄×C₁₆	—	53	2	no	—
197	C₅₃	#33434	1	1	yes	characteristic, complemented, Sylow
198	C₂×#41	—	53	1	no	—
199	C₂×#41	C₄×D₅₃	1	1	yes	characteristic
200	#131	—	2	2	no	—
201	#131	—	2	2	no	—
202	#131	—	106	2	no	—
203	#131	—	106	2	no	—
204	C₂×M₁₆	—	53	1	no	—
205	C₂×M₁₆	—	53	2	no	—
206	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
207	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
208	C₂²×C₈	—	53	1	no	—
209	C₂²×C₈	—	53	2	no	—
210	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
211	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
212	C₄⋊Q₈	—	53	1	no	—
213	#127	—	53	1	no	—
214	#124	—	53	1	no	—
215	#124	—	53	1	no	—
216	#121	—	53	4	no	—
217	C₄×D₄	—	53	4	no	—
218	C₂×C₄²	—	53	1	no	—
219	M₃₂	—	2	2	no	—
220	M₃₂	—	106	4	no	—
221	M₃₂	—	106	4	no	—
222	M₃₂	—	106	2	no	—
223	C₂×C₁₆	—	2	2	no	—
224	C₂×C₁₆	—	53	4	no	—
225	C₂×C₁₆	—	53	1	no	—
226	C₂×C₁₆	—	53	1	no	—
227	C₂×C₁₆	—	53	2	no	—
228	C₂×C₁₆	—	106	4	no	—
229	C₂×C₁₆	—	106	4	no	—
230	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
231	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
232	C₄⋊C₈	—	53	4	no	—
233	C₂²⋊C₈	—	53	4	no	—
234	C₂²⋊C₈	—	53	4	no	—
235	C₄×C₈	—	53	2	no	—
236	C₄×C₈	—	53	2	no	—
237	#41	—	2	2	no	—
238	#41	—	2	2	no	—
239	#41	—	106	2	no	—
240	#41	—	106	2	no	—
241	C₂×Q₈	—	53	1	no	—
242	C₂×Q₈	C₈×D₅₃	1	1	yes	characteristic
243	C₂×D₄	—	53	4	no	—
244	C₂×D₄	—	53	2	no	—
245	C₂×D₄	—	53	1	no	—
246	C₂×D₄	C₈×D₅₃	1	2	yes	—
247	C₂×D₄	C₈×D₅₃	1	1	yes	characteristic
248	C₂²×C₄	—	53	2	no	—
249	C₂²×C₄	—	53	2	no	—
250	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
251	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	4	no	—
254	M₁₆	—	106	2	no	—
255	M₁₆	—	106	4	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	53	2	no	—
259	C₂×C₈	—	53	2	no	—
260	C₂×C₈	—	106	2	no	—
261	C₂×C₈	—	106	2	no	—
262	C₂×C₈	—	106	2	no	—
263	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
264	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
265	C₂×C₈	D₄×D₅₃	1	2	yes	—
266	C₂×C₈	#11719947	1	2	yes	—
267	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
268	C₄⋊C₄	—	53	4	no	—
269	C₂²⋊C₄	—	53	4	no	—
270	C₂²⋊C₄	—	53	4	no	—
271	C₄²	—	53	1	no	—
272	C₄²	—	53	1	no	—
273	C₄²	—	53	2	no	—
274	C₁₆	—	2	2	no	—
275	C₁₆	—	2	2	no	—
276	C₁₆	—	53	2	no	—
277	C₁₆	—	53	2	no	—
278	C₁₆	—	106	2	no	—
279	C₂³	—	53	2	no	—
280	C₂³	C₂×C₈×D₅₃	1	2	yes	—
281	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
282	Q₈	—	2	2	no	—
283	Q₈	—	106	2	no	—
284	D₄	—	2	4	no	—
285	D₄	—	2	2	no	—
286	D₄	—	106	8	no	—
287	D₄	—	106	4	no	—
288	D₄	—	106	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	53	2	no	—
292	C₂×C₄	—	53	2	no	—
293	C₂×C₄	—	53	2	no	—
294	C₂×C₄	—	106	4	no	—
295	C₂×C₄	—	106	2	no	—
296	C₂×C₄	—	106	2	no	—
297	C₂×C₄	#13133177	1	2	yes	—
298	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
299	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
300	C₂×C₄	#13133225	1	2	yes	—
301	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	106	2	no	—
304	C₈	—	106	2	no	—
305	C₈	#13133177	1	2	yes	—
306	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
307	C₈	C₂×#11719947	1	1	yes	characteristic
308	C₂²	—	2	2	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	106	4	no	—
312	C₂²	—	106	2	no	—
313	C₂²	#16810686	1	2	yes	—
314	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	53	4	no	—
317	C₄	—	106	2	no	—
318	C₄	—	106	2	no	—
319	C₄	#16810686	1	2	yes	—
320	C₄	C₂×#13133177	1	1	yes	characteristic
321	C₄	C₂×#13133225	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	—	212	2	no	—
325	C₂	C₂×#16810686	1	1	yes	characteristic
326	C₂	#504×D₅₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2113	C₂
prime radical	#16811169	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2113	D₅₃
hypocenter	C₅₃	#33434
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2113

Complements

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

N	H ≅ G/N	type
1. C₂×#16811041	C₂	semidirect
2. C₂×#16811040	C₂	semidirect
4. #20276544	C₂	semidirect
5. #20276532	C₂	semidirect
6. #20276470	C₂	semidirect
7. #20276328	C₂	semidirect
8. C₂×C₁₆×Dic₅₃	C₂	semidirect
9. C₅₃×#2113	C₂	semidirect
52. C₂×#11719908	D₄	semidirect
158. Dic₅₃	#504	semidirect
166. #2113	D₅₃	semidirect
197. C₅₃	#33434	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811041	2
C₂×#16811040	2
#20283425	2
#20276544	2
#20276532	2
#20276470	2
#20276328	2
C₂×C₁₆×Dic₅₃	2
C₅₃×#2113	2
#33434	53