Subgroups — #27541842

Order 13568.

474 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287704	C₂	1	1	yes	maximal, characteristic
2	#20276954	C₂	1	2	yes	maximal
3	#20276944	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276859	C₂	1	1	yes	maximal, characteristic
5	#20276854	C₂	1	2	yes	maximal, complemented
6	#20276777	C₂	1	2	yes	maximal
7	#20276765	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276624	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276584	C₂	1	2	yes	maximal, complemented
10	C₄×#13133160	C₂	1	1	yes	maximal, characteristic
11	Q₈×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811346	—	2	1	no	—
13	C₂×#13133163	C₂²	1	2	yes	—
14	C₂×#13133160	C₂²	1	1	yes	characteristic
15	#16811007	C₂²	1	2	yes	—
16	#16811003	C₂²	1	1	yes	characteristic
17	#16810846	—	2	2	no	—
18	#16810829	—	2	1	no	—
19	C₄×Dic₂₁₂	C₂²	1	2	yes	—
20	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
21	#16810450	C₂²	1	1	yes	characteristic
22	#16810450	C₂²	1	2	yes	—
23	C₄⋊#11719908	C₂²	1	1	yes	characteristic
24	C₄⋊#11719908	C₂²	1	2	yes	—
25	#16810427	C₂²	1	1	yes	characteristic
26	#16810426	C₂²	1	2	yes	—
27	#16810426	C₂²	1	2	yes	—
28	#16810426	C₂²	1	2	yes	—
29	#16810423	C₂²	1	2	yes	—
30	#16810423	C₂²	1	1	yes	characteristic
31	#16810399	C₂²	1	2	yes	—
32	C₄×#11719908	C₂²	1	1	yes	characteristic
33	Q₈²×C₅₃	—	2	1	no	—
34	C₅₃×#500	C₂²	1	2	yes	—
35	C₅₃×#500	C₂²	1	1	yes	characteristic
36	C₅₃×#477	—	2	1	no	—
37	C₅₃×#477	—	2	2	no	—
38	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
41	C₂×Dic₂₁₂	—	2	1	no	—
42	C₂×Dic₂₁₂	C₂³	1	2	yes	—
43	C₂×#11719913	—	2	2	no	—
44	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
45	#13133217	—	2	1	no	—
46	#13133212	—	4	2	no	—
47	#13133205	—	2	2	no	—
48	#13133193	—	4	1	no	—
49	C₄×Dic₁₀₆	—	2	2	no	—
50	C₄×D₂₁₂	—	2	1	no	—
51	#13133175	—	4	1	no	—
52	#13133163	—	2	4	no	—
53	#13133160	—	2	2	no	—
54	C₂×#11719908	C₂³	1	1	yes	characteristic
55	C₂×#11719908	C₂³	1	2	yes	—
56	C₂×#11719908	C₂³	1	1	yes	characteristic
57	#13133151	C₂³	1	1	yes	characteristic
58	#13133142	—	2	2	no	—
59	#13133139	—	2	1	no	—
60	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₅₃	—	2	1	no	—
62	C₄⋊Q₈×C₅₃	—	2	2	no	—
63	C₄⋊Q₈×C₅₃	—	4	1	no	—
64	C₄⋊Q₈×C₅₃	—	4	2	no	—
65	C₄×Q₈×C₅₃	—	2	1	no	—
66	C₄×Q₈×C₅₃	—	2	2	no	—
67	C₄×Q₈×C₅₃	—	4	1	no	—
68	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
69	#107×C₅₃	—	2	1	no	—
70	#107×C₅₃	—	2	2	no	—
71	#107×C₅₃	C₂³	1	1	yes	characteristic
72	#107×C₅₃	C₂³	1	2	yes	—
73	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
75	#103×C₅₃	—	2	1	no	—
76	#103×C₅₃	—	2	2	no	—
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂³	1	2	yes	—
79	C₂×Q₈×C₅₃	—	2	1	no	—
80	C₂×Q₈×C₅₃	—	4	2	no	—
81	C₂×Q₈×C₅₃	—	4	1	no	—
82	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
83	C₂×D₂₁₂	—	2	1	no	—
84	C₂×C₄×D₅₃	—	4	1	no	—
85	C₂×Dic₁₀₆	—	2	2	no	—
86	Q₁₆×C₅₃	—	2	1	no	—
87	Q₁₆×C₅₃	#41	1	2	yes	—
88	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₅₃	—	2	1	no	—
92	C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₄⋊C₄×C₅₃	—	4	2	no	—
94	C₄⋊C₄×C₅₃	—	4	1	no	—
95	C₄⋊C₄×C₅₃	—	4	2	no	—
96	C₄⋊C₄×C₅₃	—	4	1	no	—
97	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
98	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
99	C₄²×C₅₃	—	4	2	no	—
100	C₄²×C₅₃	—	4	1	no	—
101	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂×D₄	1	2	yes	—
103	#11719921	—	4	2	no	—
104	#11719921	—	4	1	no	—
105	#11719921	—	4	2	no	—
106	C₄⋊Dic₅₃	—	2	1	no	—
107	C₄⋊Dic₅₃	—	4	1	no	—
108	C₄⋊Dic₅₃	—	4	1	no	—
109	#11719919	—	4	2	no	—
110	C₄×Dic₅₃	—	4	2	no	—
111	Dic₂₁₂	—	2	2	no	—
112	Dic₂₁₂	—	2	2	no	—
113	Dic₂₁₂	—	8	1	no	—
114	D₄₂₄	—	2	1	no	—
115	D₄₂₄	—	2	1	no	—
116	#11719913	—	8	2	no	—
117	#11719908	—	2	1	no	—
118	#11719908	—	2	2	no	—
119	#11719908	—	2	1	no	—
120	C₂²×D₅₃	—	4	1	no	—
121	Q₈×C₅₃	—	2	2	no	—
122	Q₈×C₅₃	—	2	2	no	—
123	Q₈×C₅₃	—	4	1	no	—
124	Q₈×C₅₃	—	16	2	no	—
125	Q₈×C₅₃	—	16	1	no	—
126	C₂×C₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	—	4	2	no	—
129	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×Dic₅₃	—	4	2	no	—
135	C₂×Dic₅₃	—	4	1	no	—
136	D₂₁₂	—	4	1	no	—
137	D₂₁₂	—	4	1	no	—
138	C₄×D₅₃	—	8	2	no	—
139	Dic₁₀₆	—	4	2	no	—
140	Dic₁₀₆	—	4	2	no	—
141	C₈×C₅₃	—	2	2	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
144	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
145	#33936	—	53	1	no	maximal, Sylow
146	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₅₃	—	8	1	no	—
148	C₂×D₅₃	—	8	1	no	—
149	C₂×D₅₃	—	8	1	no	—
150	C₄×C₅₃	—	2	2	no	—
151	C₄×C₅₃	—	2	2	no	—
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	—	4	2	no	—
154	C₄×C₅₃	—	8	1	no	—
155	C₄×C₅₃	—	8	2	no	—
156	C₄×C₅₃	#510	1	2	yes	—
157	C₄×C₅₃	#548	1	1	yes	characteristic
158	C₄×C₅₃	#577	1	1	yes	characteristic
159	Dic₅₃	—	8	2	no	—
160	Dic₅₃	—	8	1	no	—
161	#3393	—	53	1	no	—
162	Q₈×Q₁₆	D₅₃	1	1	yes	characteristic, complemented
163	#2243	—	53	2	no	—
164	#2241	—	53	1	no	—
165	#2226	—	53	1	no	—
166	#2225	—	53	2	no	—
167	#2211	—	53	2	no	—
168	#2209	—	53	1	no	—
169	#2183	—	53	1	no	—
170	#2176	—	53	2	no	—
171	C₄×SD₃₂	—	53	1	no	—
172	C₂×C₅₃	#3358	1	1	yes	characteristic
173	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
174	C₂×C₅₃	#3416	1	1	yes	characteristic
175	D₅₃	—	16	1	no	—
176	Q₈²	—	2	1	no	—
177	#556	—	106	1	no	—
178	C₂×Q₃₂	—	53	2	no	—
179	C₂×SD₃₂	—	53	1	no	—
180	#500	C₂×D₅₃	1	2	yes	—
181	#500	C₂×D₅₃	1	1	yes	characteristic
182	#495	—	53	2	no	—
183	#494	—	53	1	no	—
184	#477	—	2	1	no	—
185	#477	—	2	2	no	—
186	#464	—	106	2	no	—
187	#461	—	106	1	no	—
188	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
189	C₄×Q₁₆	—	53	2	no	—
190	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
191	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
192	C₄×D₈	—	53	1	no	—
193	#367	—	53	1	no	—
194	#367	—	53	2	no	—
195	C₄⋊C₁₆	—	53	1	no	—
196	C₄⋊C₁₆	—	53	2	no	—
197	#358	—	53	1	no	—
198	#358	—	53	2	no	—
199	#358	—	53	2	no	—
200	#358	—	53	2	no	—
201	#357	—	53	1	no	—
202	#357	—	53	2	no	—
203	#346	—	53	2	no	—
204	C₄×C₁₆	—	53	1	no	—
205	C₅₃	#33936	1	1	yes	characteristic, complemented, Sylow
206	C₂×Q₁₆	—	53	2	no	—
207	C₂×Q₁₆	—	106	1	no	—
208	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
209	C₂×SD₁₆	—	106	2	no	—
210	C₂×D₈	—	53	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	2	no	—
213	C₄⋊Q₈	—	4	1	no	—
214	C₄⋊Q₈	—	4	2	no	—
215	C₄⋊Q₈	—	106	1	no	—
216	#126	—	212	2	no	—
217	#124	—	106	2	no	—
218	C₂²⋊Q₈	—	212	1	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×Q₈	—	4	1	no	—
222	C₄×Q₈	—	106	2	no	—
223	C₄×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
224	C₄×D₄	—	106	1	no	—
225	#117	—	212	1	no	—
226	Q₃₂	—	106	4	no	—
227	SD₃₂	—	106	2	no	—
228	C₂×C₁₆	—	53	1	no	—
229	C₂×C₁₆	—	53	1	no	—
230	C₂×C₁₆	—	53	2	no	—
231	#107	—	2	1	no	—
232	#107	—	2	2	no	—
233	#107	—	53	1	no	—
234	#107	C₂²×D₅₃	1	1	yes	characteristic
235	#107	C₂²×D₅₃	1	2	yes	—
236	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
237	C₄⋊C₈	C₂²⋊D₅₃	1	2	yes	—
238	#103	—	2	1	no	—
239	#103	—	2	2	no	—
240	#103	—	106	2	no	—
241	#102	—	106	1	no	—
242	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₅₃	1	2	yes	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	106	2	no	—
248	C₂×Q₈	#11719922	1	1	yes	characteristic
249	C₂×D₄	—	106	1	no	—
250	C₂²×C₄	—	212	1	no	—
251	Q₁₆	—	2	1	no	—
252	Q₁₆	—	106	2	no	—
253	Q₁₆	—	106	2	no	—
254	Q₁₆	—	424	1	no	—
255	Q₁₆	#11719949	1	2	yes	—
256	SD₁₆	—	424	2	no	—
257	D₈	—	106	1	no	—
258	D₈	—	106	1	no	—
259	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
261	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	106	1	no	—
269	C₄⋊C₄	—	212	1	no	—
270	C₄⋊C₄	—	212	2	no	—
271	C₄⋊C₄	—	212	1	no	—
272	C₄⋊C₄	#11719922	1	1	yes	characteristic
273	C₄⋊C₄	#11719922	1	2	yes	—
274	C₂²⋊C₄	—	212	2	no	—
275	C₂²⋊C₄	—	212	1	no	—
276	C₂²⋊C₄	—	212	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	212	2	no	—
280	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
281	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
282	C₁₆	—	106	1	no	—
283	C₁₆	—	106	2	no	—
284	C₁₆	—	106	1	no	—
285	C₂³	—	212	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	16	2	no	—
290	Q₈	—	16	1	no	—
291	Q₈	—	212	2	no	—
292	Q₈	—	212	2	no	—
293	D₄	—	212	1	no	—
294	D₄	—	212	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	424	2	no	—
299	C₂×C₄	—	212	1	no	—
300	C₂×C₄	—	212	2	no	—
301	C₂×C₄	C₂×#11719922	1	2	yes	—
302	C₂×C₄	C₂×#11719922	1	2	yes	—
303	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
304	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	C₂×#11719949	1	1	yes	characteristic
309	C₈	#13133265	1	1	yes	characteristic
310	C₂²	—	424	1	no	—
311	C₂²	—	424	1	no	—
312	C₂²	—	424	1	no	—
313	C₂²	C₂²×#11719922	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	1	no	—
320	C₄	—	424	1	no	—
321	C₄	—	424	2	no	—
322	C₄	#16811064	1	2	yes	—
323	C₄	#16811284	1	1	yes	characteristic
324	C₄	#16811431	1	1	yes	characteristic
325	C₂	—	848	1	no	—
326	C₂	#20287350	1	1	yes	characteristic
327	C₂	C₂×#16811064	1	1	yes	characteristic
328	C₂	#20287862	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	Q₈×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₄₂₄	C₂³
hypercenter	Q₈×Q₁₆	D₅₃
hypocenter	C₅₃	#33936
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #20276944	C₂	semidirect
5. #20276854	C₂	semidirect
7. #20276765	C₂	semidirect
8. #20276624	C₂	semidirect
9. #20276584	C₂	semidirect
11. Q₈×Q₁₆×C₅₃	C₂	semidirect
162. Q₈×Q₁₆	D₅₃	semidirect
205. C₅₃	#33936	semidirect

Maximal subgroups

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

subgroup	index
#20287704	2
#20276954	2
#20276944	2
#20276859	2
#20276854	2
#20276777	2
#20276765	2
#20276624	2
#20276584	2
C₄×#13133160	2
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