Subgroups — #27541890

Order 13568.

520 conjugacy classes of subgroups.

The subgroup lattice has 346 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	#20276938	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276862	C₂	1	2	yes	maximal, complemented
5	#20276857	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276767	C₂	1	2	yes	maximal, complemented
7	#20276747	C₂	1	1	yes	maximal, characteristic
8	#20276624	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276589	C₂	1	2	yes	maximal, complemented
10	C₄×#13133160	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3396	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×D₂₁₂	—	2	1	no	—
13	C₂×#13133160	C₂²	1	1	yes	characteristic
14	C₂×#13133158	C₂²	1	2	yes	—
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	#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	2	yes	—
26	#16810427	C₂²	1	1	yes	characteristic
27	#16810424	C₂²	1	2	yes	—
28	#16810424	C₂²	1	2	yes	—
29	#16810424	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	C₅₃×#559	—	2	1	no	—
34	C₅₃×#495	C₂²	1	2	yes	—
35	C₅₃×#493	C₂²	1	1	yes	characteristic
36	C₅₃×#463	—	2	2	no	—
37	C₅₃×#460	—	2	1	no	—
38	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₅₃	C₂²	1	2	yes	—
41	C₂×Q₈×D₅₃	—	4	1	no	—
42	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
43	#13133217	—	2	2	no	—
44	#13133217	—	2	1	no	—
45	#13133197	—	4	1	no	—
46	#13133197	—	4	2	no	—
47	C₄×D₂₁₂	—	2	2	no	—
48	C₄×D₂₁₂	—	2	1	no	—
49	#13133160	—	2	2	no	—
50	#13133158	—	2	4	no	—
51	C₂×#11719908	C₂³	1	2	yes	—
52	C₂×#11719908	C₂³	1	1	yes	characteristic
53	C₂×#11719908	C₂³	1	1	yes	characteristic
54	#13133151	—	2	2	no	—
55	#13133151	—	2	1	no	—
56	#13133151	C₂³	1	2	yes	—
57	#13133151	C₂³	1	1	yes	characteristic
58	#13133139	—	2	2	no	—
59	#13133139	—	2	1	no	—
60	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×SD₁₆×C₅₃	—	2	2	no	—
62	C₂×D₈×C₅₃	—	2	1	no	—
63	C₂×D₈×C₅₃	C₂³	1	2	yes	—
64	#127×C₅₃	—	2	1	no	—
65	#124×C₅₃	—	2	2	no	—
66	#124×C₅₃	—	4	1	no	—
67	#123×C₅₃	—	4	2	no	—
68	C₄×Q₈×C₅₃	—	2	1	no	—
69	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
70	C₄×D₄×C₅₃	—	2	2	no	—
71	C₄×D₄×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	1	no	—
76	#102×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	D₈	1	1	yes	characteristic
81	C₂×D₄×C₅₃	—	2	2	no	—
82	C₂×D₄×C₅₃	—	4	1	no	—
83	C₂×D₄×C₅₃	—	4	1	no	—
84	C₂²×C₄×C₅₃	—	4	2	no	—
85	Q₈×D₅₃	—	8	4	no	—
86	C₂×D₂₁₂	—	2	1	no	—
87	C₂×C₄×D₅₃	—	4	2	no	—
88	C₂×C₄×D₅₃	—	4	1	no	—
89	C₂×Dic₁₀₆	—	4	2	no	—
90	C₂×Dic₁₀₆	—	4	1	no	—
91	Q₁₆×C₅₃	—	2	1	no	—
92	Q₁₆×C₅₃	—	2	1	no	—
93	SD₁₆×C₅₃	—	8	2	no	—
94	D₈×C₅₃	—	2	2	no	—
95	D₈×C₅₃	—	2	2	no	—
96	D₈×C₅₃	—	8	1	no	—
97	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₅₃	—	2	1	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	2	no	—
106	C₂²⋊C₄×C₅₃	—	4	2	no	—
107	C₄²×C₅₃	—	4	1	no	—
108	C₄²×C₅₃	C₂×D₄	1	2	yes	—
109	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
110	#11719921	—	4	2	no	—
111	#11719921	—	4	1	no	—
112	C₄⋊Dic₅₃	—	2	2	no	—
113	C₄⋊Dic₅₃	—	2	1	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	2	no	—
119	#11719908	—	2	1	no	—
120	#11719908	—	2	1	no	—
121	C₂³×C₅₃	—	4	2	no	—
122	C₂²×D₅₃	—	4	1	no	—
123	Q₈×C₅₃	—	2	2	no	—
124	Q₈×C₅₃	—	4	1	no	—
125	Q₈×C₅₃	—	4	1	no	—
126	D₄×C₅₃	—	4	2	no	—
127	D₄×C₅₃	—	4	2	no	—
128	D₄×C₅₃	—	8	2	no	—
129	D₄×C₅₃	—	16	1	no	—
130	C₂×C₄×C₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	—	4	1	no	—
132	C₂×C₄×C₅₃	—	16	2	no	—
133	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₅₃	—	4	1	no	—
139	C₂×Dic₅₃	—	4	2	no	—
140	D₂₁₂	—	2	2	no	—
141	D₂₁₂	—	4	1	no	—
142	C₄×D₅₃	—	8	4	no	—
143	C₄×D₅₃	—	8	2	no	—
144	Dic₁₀₆	—	8	4	no	—
145	Dic₁₀₆	—	8	2	no	—
146	C₈×C₅₃	—	2	1	no	—
147	C₈×C₅₃	—	2	2	no	—
148	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
149	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
150	#33940	—	53	1	no	maximal, Sylow
151	C₂²×C₅₃	—	8	2	no	—
152	C₂²×C₅₃	—	8	2	no	—
153	C₂²×C₅₃	—	8	2	no	—
154	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₅₃	—	4	2	no	—
156	C₂×D₅₃	—	8	1	no	—
157	C₂×D₅₃	—	8	1	no	—
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	8	1	no	—
162	C₄×C₅₃	—	8	1	no	—
163	C₄×C₅₃	#509	1	2	yes	—
164	C₄×C₅₃	#548	1	1	yes	characteristic
165	C₄×C₅₃	#577	1	1	yes	characteristic
166	Dic₅₃	—	8	1	no	—
167	Dic₅₃	—	8	2	no	—
168	#3396	D₅₃	1	1	yes	characteristic, complemented
169	Q₈×D₈	—	53	1	no	—
170	#2243	—	53	2	no	—
171	#2240	—	53	1	no	—
172	#2226	—	53	2	no	—
173	#2225	—	53	1	no	—
174	#2210	—	53	2	no	—
175	#2206	—	53	1	no	—
176	#2183	—	53	1	no	—
177	#2177	—	53	2	no	—
178	C₄×SD₃₂	—	53	1	no	—
179	C₂×C₅₃	—	16	2	no	—
180	C₂×C₅₃	#3358	1	1	yes	characteristic
181	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
182	C₂×C₅₃	#3416	1	1	yes	characteristic
183	D₅₃	—	8	2	no	—
184	#559	—	2	1	no	—
185	D₄×Q₈	—	106	1	no	—
186	C₂×SD₃₂	—	53	1	no	—
187	C₂×D₁₆	—	53	2	no	—
188	#500	—	53	1	no	—
189	#500	—	53	2	no	—
190	#495	C₂×D₅₃	1	2	yes	—
191	#493	C₂×D₅₃	1	1	yes	characteristic
192	#474	—	106	1	no	—
193	#474	—	106	2	no	—
194	#463	—	2	2	no	—
195	#460	—	2	1	no	—
196	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
197	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
198	C₄×D₈	—	53	1	no	—
199	C₄×D₈	—	53	2	no	—
200	C₄×D₈	C₂×D₅₃	1	2	yes	—
201	#367	—	53	1	no	—
202	#367	—	53	2	no	—
203	C₄⋊C₁₆	—	53	1	no	—
204	C₄⋊C₁₆	—	53	2	no	—
205	#358	—	53	1	no	—
206	#358	—	53	2	no	—
207	#357	—	53	2	no	—
208	#357	—	53	2	no	—
209	#357	—	53	2	no	—
210	#357	—	53	1	no	—
211	#346	—	53	2	no	—
212	C₄×C₁₆	—	53	1	no	—
213	C₅₃	#33940	1	1	yes	characteristic, complemented, Sylow
214	C₂²×Q₈	—	212	1	no	—
215	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
216	C₂×SD₁₆	—	2	2	no	—
217	C₂×D₈	—	2	1	no	—
218	C₂×D₈	—	53	1	no	—
219	C₂×D₈	C₂²×D₅₃	1	2	yes	—
220	C₄⋊Q₈	—	106	2	no	—
221	C₄⋊Q₈	—	106	1	no	—
222	#127	—	2	1	no	—
223	#124	—	2	2	no	—
224	#124	—	4	1	no	—
225	#123	—	4	2	no	—
226	C₂²⋊Q₈	—	212	1	no	—
227	C₂²⋊Q₈	—	212	2	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
230	C₄×D₄	—	2	2	no	—
231	C₄×D₄	—	4	1	no	—
232	C₄×D₄	—	106	2	no	—
233	C₄×D₄	—	106	1	no	—
234	SD₃₂	—	106	2	no	—
235	D₁₆	—	106	4	no	—
236	C₂×C₁₆	—	53	1	no	—
237	C₂×C₁₆	—	53	1	no	—
238	C₂×C₁₆	—	53	2	no	—
239	#107	—	53	2	no	—
240	#107	—	53	1	no	—
241	#107	—	106	2	no	—
242	#107	—	106	1	no	—
243	#107	C₂²×D₅₃	1	1	yes	characteristic
244	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
245	C₄⋊C₈	C₂²⋊D₅₃	1	2	yes	—
246	#103	—	2	1	no	—
247	#102	—	2	2	no	—
248	#102	—	106	2	no	—
249	#102	—	106	1	no	—
250	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
251	C₄×C₈	C₂²×D₅₃	1	2	yes	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	424	4	no	—
254	C₂×Q₈	—	212	2	no	—
255	C₂×Q₈	—	212	1	no	—
256	C₂×Q₈	#11719922	1	1	yes	characteristic
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	—	106	1	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	212	2	no	—
263	C₂²×C₄	—	212	1	no	—
264	Q₁₆	—	2	1	no	—
265	Q₁₆	—	2	1	no	—
266	SD₁₆	—	8	2	no	—
267	D₈	—	2	2	no	—
268	D₈	—	2	2	no	—
269	D₈	—	8	1	no	—
270	D₈	—	53	2	no	—
271	D₈	—	106	1	no	—
272	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
273	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	106	2	no	—
278	C₄⋊C₄	—	106	1	no	—
279	C₄⋊C₄	—	212	2	no	—
280	C₄⋊C₄	—	212	1	no	—
281	C₄⋊C₄	#11719922	1	1	yes	characteristic
282	C₄⋊C₄	#11719922	1	2	yes	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	212	2	no	—
287	C₂²⋊C₄	—	212	1	no	—
288	C₄²	—	4	1	no	—
289	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
290	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
291	C₁₆	—	106	2	no	—
292	C₁₆	—	106	1	no	—
293	C₁₆	—	106	1	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	212	1	no	—
296	Q₈	—	2	2	no	—
297	Q₈	—	4	1	no	—
298	Q₈	—	4	1	no	—
299	Q₈	—	424	4	no	—
300	Q₈	—	424	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	8	2	no	—
304	D₄	—	16	1	no	—
305	D₄	—	106	2	no	—
306	D₄	—	212	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	16	2	no	—
310	C₂×C₄	—	424	4	no	—
311	C₂×C₄	—	424	2	no	—
312	C₂×C₄	—	212	2	no	—
313	C₂×C₄	—	212	1	no	—
314	C₂×C₄	C₂×#11719922	1	2	yes	—
315	C₂×C₄	C₂×#11719922	1	2	yes	—
316	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
317	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
318	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	2	no	—
321	C₈	#41×D₅₃	1	1	yes	characteristic
322	C₈	#13133264	1	1	yes	characteristic
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	424	1	no	—
327	C₂²	—	424	1	no	—
328	C₂²	—	212	2	no	—
329	C₂²	C₂²×#11719922	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	424	2	no	—
336	C₄	—	424	1	no	—
337	C₄	#16811058	1	2	yes	—
338	C₄	#16811280	1	1	yes	characteristic
339	C₄	#16811434	1	1	yes	characteristic
340	C₂	—	16	2	no	—
341	C₂	—	424	2	no	—
342	C₂	#20287346	1	1	yes	characteristic
343	C₂	C₂×#16811058	1	1	yes	characteristic
344	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	C₅₃×#3396	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20276747	C₂
hypercenter	#3396	D₅₃
hypocenter	C₅₃	#33940
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3396

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. #20276938	C₂	semidirect
4. #20276862	C₂	semidirect
5. #20276857	C₂	semidirect
6. #20276767	C₂	semidirect
8. #20276624	C₂	semidirect
9. #20276589	C₂	semidirect
10. C₄×#13133160	C₂	semidirect
11. C₅₃×#3396	C₂	semidirect
168. #3396	D₅₃	semidirect
213. C₅₃	#33940	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
#20276938	2
#20276862	2
#20276857	2
#20276767	2
#20276747	2
#20276624	2
#20276589	2
C₄×#13133160	2
C₅₃×#3396	2
#33940	53