Subgroups — #27541816

Order 13568.

508 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276951	C₂	1	2	yes	maximal, complemented
3	#20276942	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276867	C₂	1	2	yes	maximal, complemented
5	#20276852	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276767	C₂	1	2	yes	maximal
7	#20276757	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276624	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276577	C₂	1	2	yes	maximal, complemented
10	C₄×#13133158	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#3393	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×D₂₁₂	—	2	1	no	—
13	C₂×#13133160	C₂²	1	2	yes	—
14	C₂×#13133158	C₂²	1	1	yes	characteristic
15	#16811028	C₂²	1	1	yes	characteristic
16	#16811028	C₂²	1	2	yes	—
17	#16810904	—	2	1	no	—
18	#16810904	—	2	2	no	—
19	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
20	C₄×D₄₂₄	C₂²	1	2	yes	—
21	#16810448	C₂²	1	1	yes	characteristic
22	#16810448	C₂²	1	2	yes	—
23	C₄⋊#11719908	C₂²	1	1	yes	characteristic
24	C₄⋊#11719908	C₂²	1	2	yes	—
25	#16810427	C₂²	1	2	yes	—
26	#16810427	C₂²	1	2	yes	—
27	#16810427	C₂²	1	2	yes	—
28	#16810424	C₂²	1	2	yes	—
29	#16810424	C₂²	1	1	yes	characteristic
30	#16810423	C₂²	1	1	yes	characteristic
31	#16810399	C₂²	1	2	yes	—
32	C₄×#11719908	C₂²	1	1	yes	characteristic
33	C₅₃×#556	—	2	1	no	—
34	C₅₃×#495	C₂²	1	2	yes	—
35	C₅₃×#494	C₂²	1	1	yes	characteristic
36	C₅₃×#464	—	2	2	no	—
37	C₅₃×#461	—	2	1	no	—
38	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
40	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₅₃	—	4	1	no	—
42	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
43	#13133217	—	2	1	no	—
44	#13133217	—	2	2	no	—
45	#13133197	—	4	1	no	—
46	#13133197	—	4	2	no	—
47	C₄×D₂₁₂	—	2	1	no	—
48	C₄×D₂₁₂	—	2	2	no	—
49	#13133160	—	2	4	no	—
50	#13133158	—	2	2	no	—
51	C₂×#11719908	C₂³	1	1	yes	characteristic
52	C₂×#11719908	C₂³	1	2	yes	—
53	C₂×#11719908	C₂³	1	1	yes	characteristic
54	#13133151	—	2	1	no	—
55	#13133151	—	2	2	no	—
56	#13133151	C₂³	1	2	yes	—
57	#13133151	C₂³	1	1	yes	characteristic
58	#13133139	—	2	1	no	—
59	#13133139	—	2	2	no	—
60	C₂×Q₁₆×C₅₃	—	2	1	no	—
61	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
62	C₂×SD₁₆×C₅₃	—	2	2	no	—
63	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₅₃	—	2	1	no	—
65	#126×C₅₃	—	4	2	no	—
66	#124×C₅₃	—	2	2	no	—
67	C₂²⋊Q₈×C₅₃	—	4	1	no	—
68	C₄×Q₈×C₅₃	—	2	2	no	—
69	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
70	C₄×D₄×C₅₃	—	2	1	no	—
71	#117×C₅₃	—	4	1	no	—
72	#107×C₅₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
75	#103×C₅₃	—	2	2	no	—
76	#102×C₅₃	—	2	1	no	—
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂³	1	2	yes	—
79	C₂×Q₈×C₅₃	—	2	2	no	—
80	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
81	C₂×D₄×C₅₃	—	2	1	no	—
82	C₂²×C₄×C₅₃	—	4	1	no	—
83	Q₈×D₅₃	—	8	4	no	—
84	C₂×D₂₁₂	—	2	1	no	—
85	C₂×C₄×D₅₃	—	4	2	no	—
86	C₂×C₄×D₅₃	—	4	1	no	—
87	C₂×Dic₁₀₆	—	4	2	no	—
88	C₂×Dic₁₀₆	—	4	1	no	—
89	Q₁₆×C₅₃	—	2	2	no	—
90	Q₁₆×C₅₃	—	2	2	no	—
91	Q₁₆×C₅₃	—	8	1	no	—
92	SD₁₆×C₅₃	—	8	2	no	—
93	D₈×C₅₃	—	2	1	no	—
94	D₈×C₅₃	—	2	1	no	—
95	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₄⋊C₄×C₅₃	—	4	1	no	—
100	C₄⋊C₄×C₅₃	—	4	2	no	—
101	C₄⋊C₄×C₅₃	—	4	1	no	—
102	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
103	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
104	C₂²⋊C₄×C₅₃	—	4	2	no	—
105	C₂²⋊C₄×C₅₃	—	4	1	no	—
106	C₂²⋊C₄×C₅₃	—	4	2	no	—
107	C₄²×C₅₃	—	4	2	no	—
108	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂×D₄	1	2	yes	—
110	#11719921	—	4	2	no	—
111	#11719921	—	4	1	no	—
112	C₄⋊Dic₅₃	—	2	1	no	—
113	C₄⋊Dic₅₃	—	2	2	no	—
114	C₄⋊Dic₅₃	—	4	2	no	—
115	C₄⋊Dic₅₃	—	4	1	no	—
116	D₄₂₄	—	2	1	no	—
117	D₄₂₄	#41	1	2	yes	—
118	#11719908	—	2	1	no	—
119	#11719908	—	2	2	no	—
120	#11719908	—	2	1	no	—
121	C₂³×C₅₃	—	4	1	no	—
122	C₂²×D₅₃	—	4	1	no	—
123	Q₈×C₅₃	—	2	2	no	—
124	Q₈×C₅₃	—	4	2	no	—
125	Q₈×C₅₃	—	4	2	no	—
126	D₄×C₅₃	—	4	1	no	—
127	D₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	—	4	1	no	—
129	C₂×C₄×C₅₃	—	4	2	no	—
130	C₂×C₄×C₅₃	—	8	2	no	—
131	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₅₃	—	4	1	no	—
137	C₂×Dic₅₃	—	4	2	no	—
138	D₂₁₂	—	2	2	no	—
139	D₂₁₂	—	4	1	no	—
140	C₄×D₅₃	—	8	4	no	—
141	C₄×D₅₃	—	8	2	no	—
142	Dic₁₀₆	—	8	4	no	—
143	Dic₁₀₆	—	8	2	no	—
144	C₈×C₅₃	—	2	2	no	—
145	C₈×C₅₃	—	2	1	no	—
146	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
147	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
148	#33934	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	—	8	1	no	—
150	C₂²×C₅₃	—	8	1	no	—
151	C₂²×C₅₃	—	8	1	no	—
152	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
153	C₂×D₅₃	—	4	2	no	—
154	C₂×D₅₃	—	8	1	no	—
155	C₂×D₅₃	—	8	1	no	—
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	—	2	2	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	8	2	no	—
160	C₄×C₅₃	—	8	1	no	—
161	C₄×C₅₃	#509	1	2	yes	—
162	C₄×C₅₃	#548	1	1	yes	characteristic
163	C₄×C₅₃	#577	1	1	yes	characteristic
164	Dic₅₃	—	8	1	no	—
165	Dic₅₃	—	8	2	no	—
166	#3393	D₅₃	1	1	yes	characteristic, complemented
167	Q₈×D₈	—	53	1	no	—
168	#2243	—	53	2	no	—
169	#2241	—	53	1	no	—
170	#2227	—	53	2	no	—
171	#2224	—	53	1	no	—
172	#2210	—	53	2	no	—
173	#2208	—	53	1	no	—
174	#2183	—	53	1	no	—
175	#2175	—	53	2	no	—
176	C₄×D₁₆	—	53	1	no	—
177	C₂×C₅₃	—	16	1	no	—
178	C₂×C₅₃	#3358	1	1	yes	characteristic
179	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
180	C₂×C₅₃	#3417	1	1	yes	characteristic
181	D₅₃	—	8	2	no	—
182	#556	—	2	1	no	—
183	D₄×Q₈	—	106	1	no	—
184	C₂×SD₃₂	—	53	2	no	—
185	C₂×D₁₆	—	53	1	no	—
186	#500	—	53	1	no	—
187	#500	—	53	2	no	—
188	#495	C₂×D₅₃	1	2	yes	—
189	#494	C₂×D₅₃	1	1	yes	characteristic
190	#474	—	106	1	no	—
191	#474	—	106	2	no	—
192	#464	—	2	2	no	—
193	#461	—	2	1	no	—
194	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
195	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
196	C₄×D₈	—	53	1	no	—
197	C₄×D₈	—	53	2	no	—
198	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
199	#366	—	53	1	no	—
200	#366	—	53	2	no	—
201	C₄⋊C₁₆	—	53	1	no	—
202	C₄⋊C₁₆	—	53	2	no	—
203	#358	—	53	2	no	—
204	#358	—	53	2	no	—
205	#358	—	53	2	no	—
206	#357	—	53	1	no	—
207	#357	—	53	2	no	—
208	#357	—	53	1	no	—
209	#346	—	53	2	no	—
210	C₄×C₁₆	—	53	1	no	—
211	C₅₃	#33934	1	1	yes	characteristic, complemented, Sylow
212	C₂²×Q₈	—	212	1	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
215	C₂×SD₁₆	—	2	2	no	—
216	C₂×D₈	—	53	1	no	—
217	C₂×D₈	C₂²×D₅₃	1	1	yes	characteristic
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	106	1	no	—
220	C₄⋊Q₈	—	106	2	no	—
221	#126	—	4	2	no	—
222	#124	—	2	2	no	—
223	C₂²⋊Q₈	—	4	1	no	—
224	C₂²⋊Q₈	—	212	1	no	—
225	C₂²⋊Q₈	—	212	2	no	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
228	C₄×D₄	—	2	1	no	—
229	C₄×D₄	—	106	1	no	—
230	C₄×D₄	—	106	2	no	—
231	#117	—	4	1	no	—
232	SD₃₂	—	106	4	no	—
233	D₁₆	—	106	2	no	—
234	C₂×C₁₆	—	53	1	no	—
235	C₂×C₁₆	—	53	1	no	—
236	C₂×C₁₆	—	53	2	no	—
237	#107	—	53	1	no	—
238	#107	—	53	2	no	—
239	#107	—	106	1	no	—
240	#107	—	106	2	no	—
241	#107	C₂²×D₅₃	1	1	yes	characteristic
242	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
243	C₄⋊C₈	C₂²⋊D₅₃	1	2	yes	—
244	#103	—	2	2	no	—
245	#102	—	2	1	no	—
246	#102	—	106	1	no	—
247	#102	—	106	2	no	—
248	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₅₃	1	2	yes	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	424	4	no	—
252	C₂×Q₈	—	212	2	no	—
253	C₂×Q₈	—	212	1	no	—
254	C₂×Q₈	#11719922	1	1	yes	characteristic
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	106	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	212	2	no	—
259	C₂²×C₄	—	212	1	no	—
260	Q₁₆	—	2	2	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	—	8	1	no	—
263	SD₁₆	—	8	2	no	—
264	D₈	—	2	1	no	—
265	D₈	—	2	1	no	—
266	D₈	—	53	2	no	—
267	D₈	—	106	1	no	—
268	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
269	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
270	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	106	1	no	—
276	C₄⋊C₄	—	106	2	no	—
277	C₄⋊C₄	—	212	2	no	—
278	C₄⋊C₄	—	212	1	no	—
279	C₄⋊C₄	#11719922	1	1	yes	characteristic
280	C₄⋊C₄	#11719922	1	2	yes	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	212	2	no	—
285	C₂²⋊C₄	—	212	1	no	—
286	C₄²	—	4	2	no	—
287	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
289	C₁₆	—	106	1	no	—
290	C₁₆	—	106	2	no	—
291	C₁₆	—	106	1	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	212	1	no	—
294	Q₈	—	2	2	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	4	2	no	—
297	Q₈	—	424	4	no	—
298	Q₈	—	424	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	106	2	no	—
302	D₄	—	212	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	424	4	no	—
307	C₂×C₄	—	424	2	no	—
308	C₂×C₄	—	212	2	no	—
309	C₂×C₄	—	212	1	no	—
310	C₂×C₄	C₂×#11719922	1	2	yes	—
311	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
312	C₂×C₄	C₂×#11719922	1	2	yes	—
313	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	2	1	no	—
317	C₈	#41×D₅₃	1	1	yes	characteristic
318	C₈	#13133264	1	1	yes	characteristic
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	—	424	1	no	—
323	C₂²	—	424	1	no	—
324	C₂²	—	212	2	no	—
325	C₂²	C₂²×#11719922	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	2	no	—
331	C₄	—	424	2	no	—
332	C₄	—	424	1	no	—
333	C₄	#16811058	1	2	yes	—
334	C₄	#16811280	1	1	yes	characteristic
335	C₄	#16811434	1	1	yes	characteristic
336	C₂	—	16	1	no	—
337	C₂	—	424	2	no	—
338	C₂	#20287346	1	1	yes	characteristic
339	C₂	C₂×#16811058	1	1	yes	characteristic
340	C₂	#20287866	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3393	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#13133158	C₂²
hypercenter	#3393	D₅₃
hypocenter	C₅₃	#33934
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3393

Complements

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

N	H ≅ G/N	type
1. Q₈×D₄₂₄	C₂	semidirect
2. #20276951	C₂	semidirect
3. #20276942	C₂	semidirect
4. #20276867	C₂	semidirect
5. #20276852	C₂	semidirect
7. #20276757	C₂	semidirect
8. #20276624	C₂	semidirect
9. #20276577	C₂	semidirect
11. C₅₃×#3393	C₂	semidirect
166. #3393	D₅₃	semidirect
211. C₅₃	#33934	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₄₂₄	2
#20276951	2
#20276942	2
#20276867	2
#20276852	2
#20276767	2
#20276757	2
#20276624	2
#20276577	2
C₄×#13133158	2
C₅₃×#3393	2
#33934	53