Subgroups — #27537265

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811039	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#20283411	C₂	1	1	yes	maximal, characteristic
4	#20276542	C₂	1	4	yes	maximal, complemented
5	C₁₆×D₂₁₂	C₂	1	4	yes	maximal, complemented
6	#20276474	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276330	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276325	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133223	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#16811135	C₄	1	1	yes	characteristic
13	#16811039	—	2	4	no	—
14	C₂×#13133157	C₂²	1	1	yes	characteristic
15	C₂×#13133156	C₂²	1	2	yes	complemented
16	C₂²×#11719908	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₅₃	—	2	4	no	—
18	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
19	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
20	C₂×C₁₆×D₅₃	C₂²	1	2	yes	complemented
21	#16810696	C₄	1	4	yes	—
22	C₈×D₂₁₂	C₂²	1	4	yes	—
23	#16810650	C₄	1	1	yes	characteristic
24	#16810591	C₄	1	1	yes	characteristic
25	#16810586	C₄	1	1	yes	characteristic
26	C₄⋊#11719908	C₂²	1	4	yes	complemented
27	#16810402	C₂²	1	4	yes	—
28	#16810402	C₂²	1	4	yes	complemented
29	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
30	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#405	C₂²	1	1	yes	characteristic
32	C₅₃×#346	C₂²	1	2	yes	complemented
33	C₄×C₁₆×C₅₃	C₂²	1	2	yes	complemented
34	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
36	#13133223	—	2	2	no	—
37	#13133223	—	2	2	no	—
38	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
39	C₂×#11719912	C₂×C₄	1	2	yes	—
40	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×D₅₃	—	2	2	no	—
42	C₂×C₈×D₅₃	—	2	2	no	—
43	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₅₃	C₂³	1	2	yes	—
45	C₂×C₈×D₅₃	C₂³	1	2	yes	—
46	#13133201	C₈	1	2	yes	—
47	#13133197	C₈	1	2	yes	—
48	#13133192	C₈	1	2	yes	—
49	#13133187	C₈	1	2	yes	—
50	C₄×D₂₁₂	C₂×C₄	1	4	yes	—
51	#13133157	—	2	2	no	—
52	#13133156	—	2	4	no	—
53	C₂×#11719908	—	2	2	no	—
54	C₂×#11719908	C₂³	1	2	yes	—
55	C₂×#11719908	C₂³	1	2	yes	—
56	C₁₆×D₅₃	—	2	4	no	—
57	C₁₆×D₅₃	—	2	4	no	—
58	C₁₆×D₅₃	—	2	4	no	—
59	C₄⋊#88322	C₂×C₄	1	4	yes	—
60	#13133129	C₂×C₄	1	4	yes	—
61	#13133129	C₂×C₄	1	4	yes	—
62	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
65	M₃₂×C₅₃	—	2	2	no	—
66	C₂×C₁₆×C₅₃	D₄	1	4	yes	complemented
67	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
68	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
70	#97×C₅₃	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₅₃	C₂³	1	2	yes	—
72	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
75	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
76	#11719945	—	2	2	no	—
77	#11719945	—	2	2	no	—
78	C₂×D₂₁₂	C₂×C₈	1	4	yes	—
79	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₅₃	—	2	2	no	—
81	C₂×C₄×D₅₃	—	2	2	no	—
82	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
83	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
86	M₁₆×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₅₃	#41	1	2	yes	—
91	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
93	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
94	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
95	#11719921	C₂×C₈	1	4	yes	—
96	#11719921	C₂×C₈	1	4	yes	—
97	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
98	#11719917	—	2	2	no	—
99	C₂×#88322	—	2	2	no	—
100	C₂×#88322	C₂²×C₄	1	2	yes	—
101	C₂×#88322	C₂²×C₄	1	2	yes	—
102	#11719912	—	2	4	no	—
103	C₈×D₅₃	—	2	4	no	—
104	C₈×D₅₃	—	2	2	no	—
105	C₈×D₅₃	—	2	2	no	—
106	C₈×D₅₃	—	2	2	no	—
107	C₈×D₅₃	—	2	2	no	—
108	C₁₆×C₅₃	—	2	2	no	—
109	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
110	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
111	#11719908	—	2	2	no	—
112	#11719908	—	2	2	no	—
113	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
114	C₂²×D₅₃	—	2	2	no	—
115	C₂²×D₅₃	—	2	4	no	—
116	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
117	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
120	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#131	1	2	yes	—
122	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
123	C₂²⋊D₅₃	—	2	4	no	—
124	C₂×Dic₅₃	—	2	2	no	—
125	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
126	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
127	D₂₁₂	—	2	8	no	—
128	D₂₁₂	—	2	2	no	—
129	C₄×D₅₃	—	2	2	no	—
130	C₄×D₅₃	—	2	2	no	—
131	C₄×D₅₃	—	2	2	no	—
132	C₄×D₅₃	—	2	2	no	—
133	C₄×D₅₃	—	2	4	no	—
134	Dic₁₀₆	—	2	2	no	—
135	C₈×C₅₃	—	2	2	no	—
136	C₈×C₅₃	C₄×D₄	1	2	yes	—
137	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
139	#88322	—	2	2	no	—
140	#88322	—	2	2	no	—
141	#33432	—	53	1	no	maximal, Sylow
142	C₂²×C₅₃	—	2	2	no	—
143	C₂²×C₅₃	C₈×D₄	1	2	yes	—
144	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
145	C₂×D₅₃	—	2	8	no	—
146	C₂×D₅₃	—	2	2	no	—
147	C₂×D₅₃	—	2	2	no	—
148	C₂×D₅₃	—	2	4	no	—
149	C₂×D₅₃	—	2	2	no	—
150	C₂×D₅₃	#504	1	4	yes	complemented
151	C₄×C₅₃	—	2	4	no	—
152	C₄×C₅₃	C₈×D₄	1	2	yes	—
153	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
154	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
155	Dic₅₃	—	2	2	no	—
156	Dic₅₃	—	2	2	no	—
157	C₂×#504	—	53	1	no	—
158	C₂³×C₁₆	—	53	1	no	—
159	#2929	—	53	1	no	—
160	#2168	—	53	4	no	—
161	D₄×C₁₆	—	53	4	no	—
162	#2151	—	53	1	no	—
163	#2113	—	53	1	no	—
164	#2112	—	53	1	no	—
165	#2108	D₅₃	1	1	yes	characteristic, complemented
166	C₂×C₅₃	—	2	2	no	—
167	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
168	C₂×C₅₃	C₂×#504	1	2	yes	—
169	D₅₃	—	2	4	no	—
170	D₅₃	—	4	2	no	—
171	C₂×#131	—	53	1	no	—
172	C₂³×C₈	—	53	1	no	—
173	#525	—	53	1	no	—
174	#504	—	106	4	no	—
175	C₂×M₃₂	—	53	1	no	—
176	C₂×M₃₂	—	53	2	no	—
177	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
178	C₂²×C₁₆	—	53	1	no	—
179	C₂²×C₁₆	—	53	2	no	—
180	C₂²×C₁₆	—	53	2	no	—
181	C₂²×C₁₆	—	53	2	no	—
182	C₂²×C₁₆	—	106	4	no	—
183	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
184	#435	—	53	4	no	—
185	C₈×D₄	—	53	4	no	—
186	#424	—	53	1	no	—
187	#408	—	53	1	no	—
188	#407	—	53	1	no	—
189	#405	C₂×D₅₃	1	1	yes	characteristic
190	C₄⋊C₁₆	—	53	4	no	—
191	C₂²⋊C₁₆	—	53	4	no	—
192	C₂²⋊C₁₆	—	53	4	no	—
193	#346	C₂×D₅₃	1	2	yes	complemented
194	C₄×C₁₆	C₂×D₅₃	1	2	yes	complemented
195	C₅₃	#33432	1	1	yes	characteristic, complemented, Sylow
196	C₂×#41	—	53	1	no	—
197	C₂³×C₄	—	53	1	no	—
198	#131	—	106	2	no	—
199	#131	—	106	2	no	—
200	C₂×M₁₆	—	53	1	no	—
201	C₂×M₁₆	—	53	2	no	—
202	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
203	C₂²×C₈	—	53	1	no	—
204	C₂²×C₈	—	53	2	no	—
205	C₂²×C₈	—	53	2	no	—
206	C₂²×C₈	—	53	2	no	—
207	C₂²×C₈	—	106	2	no	—
208	C₂²×C₈	—	106	2	no	—
209	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
210	#123	—	53	2	no	—
211	C₂²⋊Q₈	—	53	2	no	—
212	#121	—	53	2	no	—
213	#120	—	53	2	no	—
214	C₄×D₄	—	53	4	no	—
215	#117	C₄×D₅₃	1	1	yes	characteristic
216	M₃₂	—	2	2	no	—
217	M₃₂	—	106	2	no	—
218	M₃₂	—	106	4	no	—
219	C₂×C₁₆	—	53	2	no	—
220	C₂×C₁₆	—	53	2	no	—
221	C₂×C₁₆	—	106	2	no	—
222	C₂×C₁₆	—	106	4	no	—
223	C₂×C₁₆	—	106	4	no	—
224	C₂×C₁₆	—	106	4	no	—
225	C₂×C₁₆	D₂₁₂	1	4	yes	complemented
226	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
227	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
228	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
229	C₄⋊C₈	—	53	4	no	—
230	C₂²⋊C₈	—	53	4	no	—
231	C₂²⋊C₈	—	53	4	no	—
232	#97	C₄×D₅₃	1	2	yes	—
233	C₄×C₈	C₂²×D₅₃	1	2	yes	—
234	C₂⁴	—	53	1	no	—
235	#41	—	106	2	no	—
236	#41	—	106	2	no	—
237	C₂×Q₈	—	53	1	no	—
238	C₂×D₄	—	53	2	no	—
239	C₂×D₄	—	53	4	no	—
240	C₂×D₄	—	53	1	no	—
241	C₂²×C₄	—	53	1	no	—
242	C₂²×C₄	—	53	2	no	—
243	C₂²×C₄	—	53	2	no	—
244	C₂²×C₄	—	53	2	no	—
245	C₂²×C₄	—	106	2	no	—
246	C₂²×C₄	—	106	2	no	—
247	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
248	M₁₆	—	2	2	no	—
249	M₁₆	—	106	2	no	—
250	M₁₆	—	106	4	no	—
251	C₂×C₈	—	53	2	no	—
252	C₂×C₈	—	53	2	no	—
253	C₂×C₈	—	106	2	no	—
254	C₂×C₈	—	106	4	no	—
255	C₂×C₈	—	106	2	no	—
256	C₂×C₈	—	106	2	no	—
257	C₂×C₈	—	106	2	no	—
258	C₂×C₈	—	106	2	no	—
259	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
260	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
261	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
262	C₂×C₈	#11719945	1	2	yes	—
263	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
264	C₄⋊C₄	—	53	4	no	—
265	C₄⋊C₄	C₈×D₅₃	1	2	yes	—
266	C₂²⋊C₄	—	53	4	no	—
267	C₂²⋊C₄	—	53	4	no	—
268	C₂²⋊C₄	C₈×D₅₃	1	2	yes	—
269	C₄²	C₂×C₄×D₅₃	1	2	yes	—
270	C₁₆	—	2	2	no	—
271	C₁₆	—	106	2	no	—
272	C₁₆	—	106	2	no	—
273	C₁₆	C₂×D₂₁₂	1	2	yes	—
274	C₁₆	C₂×D₂₁₂	1	2	yes	—
275	C₂³	—	53	2	no	—
276	C₂³	—	53	2	no	—
277	C₂³	—	106	4	no	—
278	C₂³	—	106	2	no	—
279	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
280	Q₈	—	106	2	no	—
281	D₄	—	106	8	no	—
282	D₄	—	106	4	no	—
283	D₄	—	106	2	no	—
284	C₂×C₄	—	53	2	no	—
285	C₂×C₄	—	53	2	no	—
286	C₂×C₄	—	106	2	no	—
287	C₂×C₄	—	106	2	no	—
288	C₂×C₄	—	106	2	no	—
289	C₂×C₄	—	106	2	no	—
290	C₂×C₄	—	106	2	no	—
291	C₂×C₄	—	106	4	no	—
292	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
293	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
294	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
295	C₂×C₄	#13133223	1	2	yes	—
296	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	106	2	no	—
299	C₈	—	106	2	no	—
300	C₈	C₄×D₂₁₂	1	2	yes	—
301	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
302	C₈	C₂×#11719945	1	1	yes	characteristic
303	C₂²	—	2	2	no	—
304	C₂²	—	53	4	no	—
305	C₂²	—	106	8	no	—
306	C₂²	—	106	2	no	—
307	C₂²	—	106	2	no	—
308	C₂²	—	106	4	no	—
309	C₂²	—	106	2	no	—
310	C₂²	C₈×D₂₁₂	1	2	yes	—
311	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
312	C₄	—	2	4	no	—
313	C₄	—	106	2	no	—
314	C₄	—	106	2	no	—
315	C₄	C₈×D₂₁₂	1	2	yes	—
316	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
317	C₄	C₂×#13133223	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	106	4	no	—
320	C₂	—	212	2	no	—
321	C₂	C₂×C₈×D₂₁₂	1	1	yes	characteristic
322	C₂	#504×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2108	C₂
prime radical	#16811135	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2108	D₅₃
hypocenter	C₅₃	#33432
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
1. C₂×#16811039	C₂	semidirect
2. C₂²×C₁₆×D₅₃	C₂	semidirect
4. #20276542	C₂	semidirect
5. C₁₆×D₂₁₂	C₂	semidirect
6. #20276474	C₂	semidirect
7. #20276330	C₂	semidirect
8. #20276325	C₂	semidirect
9. C₅₃×#2108	C₂	semidirect
15. C₂×#13133156	C₂²	semidirect
20. C₂×C₁₆×D₅₃	C₂²	semidirect
26. C₄⋊#11719908	C₂²	semidirect
28. #16810402	C₂²	semidirect
32. C₅₃×#346	C₂²	semidirect
33. C₄×C₁₆×C₅₃	C₂²	semidirect
66. C₂×C₁₆×C₅₃	D₄	semidirect
150. C₂×D₅₃	#504	semidirect
165. #2108	D₅₃	semidirect
193. #346	C₂×D₅₃	semidirect
194. C₄×C₁₆	C₂×D₅₃	semidirect
195. C₅₃	#33432	semidirect
225. C₂×C₁₆	D₂₁₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811039	2
C₂²×C₁₆×D₅₃	2
#20283411	2
#20276542	2
C₁₆×D₂₁₂	2
#20276474	2
#20276330	2
#20276325	2
C₅₃×#2108	2
#33432	53