Subgroups — #27537465

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#20283578	C₂	1	2	yes	maximal
3	#20276584	C₂	1	2	yes	maximal, complemented
4	#20276583	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276573	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276496	C₂	1	1	yes	maximal, characteristic
9	#20276433	C₂	1	2	yes	maximal
10	#20276310	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₅₃	C₂²	1	2	yes	—
13	#16811099	—	2	2	no	—
14	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic
15	C₂×#13133163	C₂²	1	2	yes	—
16	Q₃₂×D₅₃	C₄	1	8	yes	complemented
17	C₂×#13133156	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
19	#16810728	—	2	2	no	—
20	#16810727	—	2	2	no	—
21	C₄×Dic₂₁₂	C₂²	1	2	yes	—
22	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
23	#16810717	C₂²	1	2	yes	—
24	#16810660	C₂²	1	2	yes	—
25	#16810630	—	2	2	no	—
26	#16810577	C₂²	1	1	yes	characteristic
27	#16810450	C₂²	1	2	yes	—
28	#16810449	C₂²	1	1	yes	characteristic
29	#16810428	C₂²	1	2	yes	—
30	#16810427	C₂²	1	2	yes	—
31	#16810426	C₂²	1	2	yes	—
32	#16810399	C₂²	1	1	yes	characteristic
33	#16810398	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
37	C₅₃×#366	C₂²	1	1	yes	characteristic
38	C₅₃×#358	C₂²	1	2	yes	—
39	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×Q₈×D₅₃	—	2	2	no	—
41	C₂×Dic₂₁₂	C₂³	1	2	yes	—
42	C₂×#11719925	—	2	2	no	—
43	Q₁₆×D₅₃	—	2	4	no	—
44	Q₁₆×D₅₃	C₂×C₄	1	8	yes	—
45	C₂×#11719912	D₄	1	1	yes	characteristic
46	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
47	Q₈×Dic₅₃	—	2	2	no	—
48	C₄×Dic₁₀₆	—	2	2	no	—
49	C₄×Dic₁₀₆	—	4	2	no	—
50	#13133182	—	2	2	no	—
51	#13133182	—	4	2	no	—
52	#13133175	—	2	2	no	—
53	#13133175	—	4	2	no	—
54	#13133174	—	4	2	no	—
55	#13133174	D₄	1	1	yes	characteristic
56	Dic₄₂₄	C₂×C₄	1	4	yes	—
57	#13133163	C₂×C₄	1	8	yes	—
58	#13133156	—	2	2	no	—
59	C₂×#11719908	C₂³	1	1	yes	characteristic
60	C₂×#11719908	C₂³	1	1	yes	characteristic
61	C₁₆×D₅₃	C₂×C₄	1	4	yes	—
62	#13133151	C₂³	1	2	yes	—
63	#13133148	—	2	2	no	—
64	#13133142	—	2	2	no	—
65	#13133141	—	2	2	no	—
66	#13133140	—	2	2	no	—
67	#13133128	D₄	1	1	yes	characteristic
68	#13133127	D₄	1	1	yes	characteristic
69	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
71	C₄×Q₈×C₅₃	—	2	2	no	—
72	Q₃₂×C₅₃	C₂×C₄	1	4	yes	—
73	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
75	#107×C₅₃	C₂³	1	2	yes	—
76	#103×C₅₃	—	2	2	no	—
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×Q₈×C₅₃	—	2	2	no	—
79	Q₈×D₅₃	—	2	8	no	—
80	Q₈×D₅₃	—	4	4	no	—
81	C₂×C₄×D₅₃	—	4	2	no	—
82	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₁₀₆	—	2	2	no	—
84	C₂×Dic₁₀₆	—	4	2	no	—
85	Q₁₆×C₅₃	—	2	2	no	—
86	Q₁₆×C₅₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₅₃	—	2	2	no	—
90	C₄⋊C₄×C₅₃	—	4	2	no	—
91	C₄²×C₅₃	—	4	2	no	—
92	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
93	#11719925	—	2	8	no	—
94	#11719921	—	4	2	no	—
95	#11719921	D₈	1	1	yes	characteristic
96	#11719921	D₈	1	1	yes	characteristic
97	C₄⋊Dic₅₃	—	2	2	no	—
98	C₄⋊Dic₅₃	—	4	2	no	—
99	#11719919	—	4	2	no	—
100	#11719919	—	4	2	no	—
101	#11719919	D₈	1	1	yes	characteristic
102	#11719919	D₈	1	1	yes	characteristic
103	C₄×Dic₅₃	—	4	2	no	—
104	C₄×Dic₅₃	—	4	2	no	—
105	C₄×Dic₅₃	—	4	2	no	—
106	C₄×Dic₅₃	—	4	2	no	—
107	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂×#88322	C₂×D₄	1	1	yes	characteristic
109	C₂×#88322	C₂×D₄	1	1	yes	characteristic
110	Dic₂₁₂	—	2	2	no	—
111	Dic₂₁₂	C₂²×C₄	1	4	yes	—
112	#11719912	—	2	2	no	—
113	C₈×D₅₃	C₂²×C₄	1	2	yes	—
114	C₈×D₅₃	#41	1	2	yes	—
115	C₁₆×C₅₃	—	2	1	no	—
116	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
117	#11719908	—	2	1	no	—
118	#11719908	C₂²×C₄	1	2	yes	—
119	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₅₃	—	2	4	no	—
121	Q₈×C₅₃	—	4	2	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	4	2	no	—
124	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₅₃	—	4	2	no	—
128	C₂×Dic₅₃	—	4	2	no	—
129	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
132	C₄×D₅₃	—	4	8	no	—
133	C₄×D₅₃	C₄×D₄	1	2	yes	—
134	C₄×D₅₃	#135	1	2	yes	—
135	Dic₁₀₆	—	2	4	no	—
136	Dic₁₀₆	—	4	2	no	—
137	Dic₁₀₆	—	4	8	no	—
138	C₈×C₅₃	—	2	1	no	—
139	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
140	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
141	#88322	—	2	1	no	—
142	#88322	C₄×D₄	1	2	yes	—
143	#33458	—	53	1	no	maximal, Sylow
144	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₅₃	—	2	1	no	—
146	C₂×D₅₃	—	2	1	no	—
147	C₂×D₅₃	C₄×D₈	1	2	yes	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	4	4	no	—
151	C₄×C₅₃	—	8	2	no	—
152	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
154	Dic₅₃	—	2	1	no	—
155	Dic₅₃	—	2	1	no	—
156	Dic₅₃	—	4	4	no	—
157	Dic₅₃	—	8	2	no	—
158	Dic₅₃	C₄×D₈	1	2	yes	—
159	C₂²×Q₃₂	—	53	1	no	—
160	#2950	—	53	2	no	—
161	#2176	—	53	2	no	—
162	#2176	—	53	2	no	—
163	C₄×Q₃₂	—	53	1	no	—
164	C₄×Q₃₂	—	53	1	no	—
165	C₄×Q₃₂	—	53	1	no	—
166	C₄×Q₃₂	D₅₃	1	1	yes	characteristic, complemented
167	#2158	—	53	1	no	—
168	#2141	—	53	2	no	—
169	#2108	—	53	1	no	—
170	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
171	C₂×C₅₃	#3417	1	1	yes	characteristic
172	C₂×C₅₃	#3417	1	1	yes	characteristic
173	D₅₃	—	2	2	no	—
174	C₂²×Q₁₆	—	53	2	no	—
175	#519	—	106	2	no	—
176	C₂×Q₃₂	—	53	2	no	—
177	C₂×Q₃₂	—	53	8	no	—
178	C₂×Q₃₂	—	53	1	no	—
179	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic
180	C₂×M₃₂	—	53	1	no	—
181	C₂²×C₁₆	—	53	1	no	—
182	#441	—	106	2	no	—
183	#441	—	106	2	no	—
184	C₄×Q₁₆	—	53	2	no	—
185	C₄×Q₁₆	—	53	2	no	—
186	C₄×Q₁₆	—	53	2	no	—
187	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
188	#427	—	53	2	no	—
189	#419	—	106	2	no	—
190	#405	—	53	1	no	—
191	#367	—	53	2	no	—
192	#366	—	53	1	no	—
193	#366	C₂×D₅₃	1	1	yes	characteristic
194	#358	—	53	2	no	—
195	#358	—	53	2	no	—
196	#358	—	53	2	no	—
197	#358	C₂×D₅₃	1	2	yes	—
198	#346	—	53	1	no	—
199	#346	—	53	1	no	—
200	C₄×C₁₆	—	53	1	no	—
201	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
202	C₅₃	#33458	1	1	yes	characteristic, complemented, Sylow
203	C₂²×Q₈	—	106	2	no	—
204	C₂×Q₁₆	—	53	8	no	—
205	C₂×Q₁₆	—	53	2	no	—
206	C₂×Q₁₆	—	106	4	no	—
207	C₂×Q₁₆	—	106	2	no	—
208	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
209	C₂×M₁₆	—	53	1	no	—
210	C₂²×C₈	—	53	1	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	106	2	no	—
213	C₄×Q₈	—	106	2	no	—
214	C₄×Q₈	—	106	2	no	—
215	C₄×Q₈	—	212	2	no	—
216	C₄×Q₈	—	212	2	no	—
217	#117	—	53	1	no	—
218	#117	—	106	2	no	—
219	#117	—	212	2	no	—
220	#117	—	212	2	no	—
221	Q₃₂	—	53	8	no	—
222	Q₃₂	—	53	4	no	—
223	Q₃₂	C₄×D₅₃	1	4	yes	—
224	M₃₂	—	106	2	no	—
225	C₂×C₁₆	—	53	1	no	—
226	C₂×C₁₆	—	53	4	no	—
227	C₂×C₁₆	—	53	1	no	—
228	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
229	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
230	#107	—	53	2	no	—
231	#107	C₂²×D₅₃	1	2	yes	—
232	#106	—	106	2	no	—
233	#103	—	2	2	no	—
234	#103	—	106	2	no	—
235	#103	—	106	2	no	—
236	#103	—	106	2	no	—
237	#97	—	53	1	no	—
238	#97	—	53	1	no	—
239	C₄×C₈	—	53	1	no	—
240	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	106	8	no	—
243	C₂×Q₈	—	106	2	no	—
244	C₂×Q₈	—	212	2	no	—
245	C₂×Q₈	—	212	4	no	—
246	C₂²×C₄	—	53	1	no	—
247	C₂²×C₄	—	212	2	no	—
248	Q₁₆	—	2	2	no	—
249	Q₁₆	—	53	4	no	—
250	Q₁₆	—	106	8	no	—
251	Q₁₆	—	106	2	no	—
252	Q₁₆	C₂×C₄×D₅₃	1	4	yes	—
253	M₁₆	—	106	2	no	—
254	C₂×C₈	—	53	1	no	—
255	C₂×C₈	—	53	2	no	—
256	C₂×C₈	—	53	2	no	—
257	C₂×C₈	—	53	1	no	—
258	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
259	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	53	1	no	—
263	C₄⋊C₄	—	53	1	no	—
264	C₄⋊C₄	—	106	2	no	—
265	C₄⋊C₄	—	212	2	no	—
266	C₄⋊C₄	—	212	2	no	—
267	C₄⋊C₄	—	212	2	no	—
268	C₂²⋊C₄	—	53	1	no	—
269	C₂²⋊C₄	—	53	1	no	—
270	C₂²⋊C₄	—	212	2	no	—
271	C₄²	—	4	2	no	—
272	C₄²	—	53	1	no	—
273	C₄²	—	212	2	no	—
274	C₄²	—	212	2	no	—
275	C₄²	—	212	2	no	—
276	C₄²	—	212	2	no	—
277	C₄²	D₄×D₅₃	1	1	yes	characteristic
278	C₁₆	—	2	1	no	—
279	C₁₆	—	53	2	no	—
280	C₁₆	—	106	1	no	—
281	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
282	C₂³	—	53	1	no	—
283	Q₈	—	2	4	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	106	4	no	—
286	Q₈	—	212	8	no	—
287	Q₈	—	212	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	53	1	no	—
291	C₂×C₄	—	53	2	no	—
292	C₂×C₄	—	53	2	no	—
293	C₂×C₄	—	53	1	no	—
294	C₂×C₄	—	53	1	no	—
295	C₂×C₄	—	212	8	no	—
296	C₂×C₄	—	212	2	no	—
297	C₂×C₄	—	212	2	no	—
298	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
299	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	53	2	no	—
303	C₈	—	106	1	no	—
304	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
305	C₈	#41×D₅₃	1	1	yes	characteristic
306	C₂²	—	53	2	no	—
307	C₂²	—	106	1	no	—
308	C₂²	—	106	1	no	—
309	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	4	no	—
313	C₄	—	8	2	no	—
314	C₄	—	53	2	no	—
315	C₄	—	106	1	no	—
316	C₄	—	106	1	no	—
317	C₄	—	424	2	no	—
318	C₄	—	212	4	no	—
319	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
320	C₄	#135×D₅₃	1	1	yes	characteristic
321	C₂	—	106	2	no	—
322	C₂	C₄×D₈×D₅₃	1	1	yes	characteristic
323	C₂	#20287864	1	1	yes	characteristic
324	C₂	#20287865	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×Q₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₅₃	C₂×D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₄×Q₃₂	D₅₃
hypocenter	C₅₃	#33458
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
3. #20276584	C₂	semidirect
4. #20276583	C₂	semidirect
5. C₄×Dic₄₂₄	C₂	semidirect
6. Q₃₂×Dic₅₃	C₂	semidirect
7. #20276573	C₂	semidirect
11. C₄×Q₃₂×C₅₃	C₂	semidirect
16. Q₃₂×D₅₃	C₄	semidirect
166. C₄×Q₃₂	D₅₃	semidirect
202. C₅₃	#33458	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₅₃	2
#20283578	2
#20276584	2
#20276583	2
C₄×Dic₄₂₄	2
Q₃₂×Dic₅₃	2
#20276573	2
#20276496	2
#20276433	2
#20276310	2
C₄×Q₃₂×C₅₃	2
#33458	53