Subgroups — #27542138

Order 13568.

498 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287719	C₂	1	1	yes	maximal, characteristic, complemented
2	#20287705	C₂	1	1	yes	maximal, characteristic
3	#20276983	C₂	1	2	yes	maximal
4	#20276779	C₂	1	2	yes	maximal
5	#20276763	C₂	1	2	yes	maximal, complemented
6	#20276762	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276757	C₂	1	1	yes	maximal, characteristic
8	#20276623	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276586	C₂	1	2	yes	maximal, complemented
10	C₄×#13133162	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811348	—	2	1	no	—
13	#16811347	—	2	1	no	—
14	C₂×#13133163	C₂²	1	2	yes	—
15	C₂×#13133162	C₂²	1	1	yes	characteristic
16	C₂×#13133160	C₂²	1	2	yes	—
17	#16811008	C₂²	1	2	yes	—
18	#16811001	C₂²	1	2	yes	—
19	#16810847	—	2	2	no	—
20	#16810831	—	2	2	no	—
21	#16810830	—	2	1	no	—
22	#16810825	—	2	1	no	—
23	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
24	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
25	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
26	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
27	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
28	#16810451	C₂²	1	1	yes	characteristic
29	C₄⋊#11719908	C₂²	1	1	yes	characteristic
30	#16810441	C₂²	1	2	yes	—
31	#16810428	C₂²	1	1	yes	characteristic
32	#16810427	C₂²	1	2	yes	—
33	#16810427	C₂²	1	2	yes	—
34	#16810425	C₂²	1	1	yes	characteristic
35	#16810398	C₂²	1	2	yes	—
36	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
37	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
38	C₅₃×#494	C₂²	1	1	yes	characteristic
39	C₅₃×#494	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic
42	C₂×#11719925	—	2	2	no	—
43	C₂×#11719924	—	2	2	no	—
44	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
45	#13133215	—	2	2	no	—
46	#13133212	—	4	2	no	—
47	#13133208	—	4	1	no	—
48	#13133207	—	4	2	no	—
49	#13133206	—	2	2	no	—
50	#13133197	—	4	1	no	—
51	Q₈×Dic₅₃	—	2	2	no	—
52	Q₈×Dic₅₃	—	2	2	no	—
53	C₄×Dic₁₀₆	—	2	1	no	—
54	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
55	#13133182	—	4	1	no	—
56	C₄×D₂₁₂	—	2	1	no	—
57	#13133175	—	4	1	no	—
58	#13133163	—	4	2	no	—
59	#13133162	—	2	2	no	—
60	#13133160	—	4	2	no	—
61	C₂×#11719908	C₂³	1	2	yes	—
62	#13133151	C₂³	1	1	yes	characteristic
63	#13133151	C₂³	1	1	yes	characteristic
64	C₄⋊#88322	D₄	1	1	yes	characteristic
65	#13133146	D₄	1	2	yes	—
66	#13133142	—	2	1	no	—
67	#13133141	—	2	2	no	—
68	#13133141	—	2	2	no	—
69	#13133139	—	2	1	no	—
70	C₈×Dic₅₃	C₂³	1	2	yes	—
71	C₂×Q₁₆×C₅₃	—	2	1	no	—
72	C₂×Q₁₆×C₅₃	—	2	1	no	—
73	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
74	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₅₃	—	2	1	no	—
76	C₄⋊Q₈×C₅₃	—	2	1	no	—
77	Q₃₂×C₅₃	—	2	4	no	—
78	Q₃₂×C₅₃	—	4	2	no	—
79	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
80	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₅₃	—	2	2	no	—
83	C₂×Q₈×C₅₃	—	2	2	no	—
84	C₂×D₂₁₂	—	2	1	no	—
85	C₂×C₄×D₅₃	—	4	1	no	—
86	C₂×Dic₁₀₆	—	2	1	no	—
87	C₂×Dic₁₀₆	D₈	1	1	yes	characteristic
88	Q₁₆×C₅₃	—	2	2	no	—
89	Q₁₆×C₅₃	—	2	2	no	—
90	Q₁₆×C₅₃	—	2	2	no	—
91	Q₁₆×C₅₃	—	2	2	no	—
92	Q₁₆×C₅₃	—	8	1	no	—
93	Q₁₆×C₅₃	—	8	1	no	—
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₅₃	—	4	1	no	—
97	C₄⋊C₄×C₅₃	—	4	1	no	—
98	C₄⋊C₄×C₅₃	—	4	1	no	—
99	C₄⋊C₄×C₅₃	—	4	1	no	—
100	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
101	#11719925	—	8	2	no	—
102	#11719924	—	8	2	no	—
103	#11719921	—	4	1	no	—
104	#11719921	—	4	2	no	—
105	#11719921	—	4	2	no	—
106	C₄⋊Dic₅₃	—	2	1	no	—
107	C₄⋊Dic₅₃	—	2	1	no	—
108	C₄⋊Dic₅₃	—	4	2	no	—
109	C₄⋊Dic₅₃	—	4	2	no	—
110	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
111	#11719919	—	4	1	no	—
112	#11719919	—	4	2	no	—
113	#11719919	—	4	2	no	—
114	#11719919	D₈	1	2	yes	—
115	C₄×Dic₅₃	—	4	1	no	—
116	C₄×Dic₅₃	—	4	2	no	—
117	C₄×Dic₅₃	—	4	2	no	—
118	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
119	C₂×#88322	C₂×D₄	1	2	yes	—
120	Dic₂₁₂	—	2	1	no	—
121	Dic₂₁₂	—	2	1	no	—
122	D₄₂₄	—	2	1	no	—
123	D₄₂₄	—	2	1	no	—
124	C₁₆×C₅₃	—	2	1	no	—
125	C₁₆×C₅₃	#41	1	2	yes	—
126	#11719908	—	2	2	no	—
127	C₂²×D₅₃	—	4	1	no	—
128	Q₈×C₅₃	—	4	2	no	—
129	Q₈×C₅₃	—	4	2	no	—
130	Q₈×C₅₃	—	4	2	no	—
131	Q₈×C₅₃	—	4	2	no	—
132	C₂×C₄×C₅₃	—	4	2	no	—
133	C₂×C₄×C₅₃	—	4	2	no	—
134	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₅₃	—	4	1	no	—
138	C₂×Dic₅₃	—	4	1	no	—
139	C₂×Dic₅₃	—	4	1	no	—
140	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
141	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
142	D₂₁₂	—	4	1	no	—
143	D₂₁₂	—	4	1	no	—
144	C₄×D₅₃	—	8	2	no	—
145	Dic₁₀₆	—	2	2	no	—
146	Dic₁₀₆	—	4	1	no	—
147	Dic₁₀₆	—	4	1	no	—
148	C₈×C₅₃	—	2	1	no	—
149	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
150	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
151	#88322	—	2	2	no	—
152	#33965	—	53	1	no	maximal, Sylow
153	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
154	C₂×D₅₃	—	8	1	no	—
155	C₂×D₅₃	—	8	1	no	—
156	C₂×D₅₃	—	8	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	8	2	no	—
159	C₄×C₅₃	—	8	2	no	—
160	C₄×C₅₃	#510	1	2	yes	—
161	C₄×C₅₃	#550	1	1	yes	characteristic
162	C₄×C₅₃	#576	1	1	yes	characteristic
163	Dic₅₃	—	2	2	no	—
164	Dic₅₃	—	2	2	no	—
165	Dic₅₃	—	8	1	no	—
166	Dic₅₃	—	8	1	no	—
167	Dic₅₃	—	8	1	no	—
168	#3395	—	53	1	no	—
169	#3393	—	53	1	no	—
170	#2251	—	53	2	no	—
171	#2247	D₅₃	1	1	yes	characteristic, complemented
172	#2211	—	53	2	no	—
173	#2209	—	53	2	no	—
174	#2209	—	53	1	no	—
175	#2208	—	53	1	no	—
176	#2183	—	53	1	no	—
177	#2176	—	53	2	no	—
178	C₄×SD₃₂	—	53	1	no	—
179	C₂×C₅₃	#3391	1	1	yes	characteristic
180	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
181	C₂×C₅₃	#3417	1	1	yes	characteristic
182	D₅₃	—	16	1	no	—
183	#557	—	106	1	no	—
184	#556	—	106	1	no	—
185	C₂×Q₃₂	—	53	2	no	—
186	C₂×Q₃₂	C₂×D₅₃	1	2	yes	—
187	C₂×Q₃₂	C₂×D₅₃	1	2	yes	—
188	C₂×SD₃₂	—	53	2	no	—
189	C₂×SD₃₂	—	53	1	no	—
190	#495	—	53	2	no	—
191	#494	—	53	2	no	—
192	#494	C₂×D₅₃	1	1	yes	characteristic
193	#494	C₂×D₅₃	1	1	yes	characteristic
194	#464	—	106	2	no	—
195	#462	—	106	1	no	—
196	#462	—	106	2	no	—
197	#461	—	106	1	no	—
198	C₈×Q₈	—	53	1	no	—
199	C₄×Q₁₆	—	53	2	no	—
200	C₄×Q₁₆	—	53	2	no	—
201	C₄×Q₁₆	—	53	1	no	—
202	C₄×D₈	—	53	1	no	—
203	#367	—	53	1	no	—
204	C₄⋊C₁₆	—	53	2	no	—
205	C₄⋊C₁₆	—	53	1	no	—
206	#358	—	53	2	no	—
207	#358	—	53	2	no	—
208	#358	—	53	1	no	—
209	#357	—	53	1	no	—
210	#346	—	53	2	no	—
211	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
212	C₅₃	#33965	1	1	yes	characteristic, complemented, Sylow
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	—	2	1	no	—
215	C₂×Q₁₆	—	53	1	no	—
216	C₂×Q₁₆	—	106	2	no	—
217	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
218	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
219	C₂×SD₁₆	—	106	2	no	—
220	C₂×D₈	—	53	1	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	—	106	2	no	—
224	#126	—	212	2	no	—
225	#125	—	212	2	no	—
226	#125	—	212	1	no	—
227	#124	—	106	2	no	—
228	C₂²⋊Q₈	—	212	1	no	—
229	C₄×Q₈	—	53	1	no	—
230	C₄×Q₈	—	106	2	no	—
231	C₄×Q₈	—	106	2	no	—
232	C₄×Q₈	—	106	1	no	—
233	C₄×Q₈	—	212	1	no	—
234	C₄×D₄	—	106	1	no	—
235	#117	—	212	1	no	—
236	Q₃₂	—	2	4	no	—
237	Q₃₂	—	4	2	no	—
238	Q₃₂	—	212	2	no	—
239	SD₃₂	—	106	2	no	—
240	SD₃₂	—	212	2	no	—
241	C₂×C₁₆	—	53	2	no	—
242	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
243	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
244	#107	—	53	1	no	—
245	#107	—	53	1	no	—
246	C₄⋊C₈	—	53	2	no	—
247	C₄⋊C₈	—	53	1	no	—
248	#103	—	106	2	no	—
249	#103	—	106	2	no	—
250	#103	—	106	1	no	—
251	#102	—	106	1	no	—
252	C₄×C₈	—	53	2	no	—
253	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	53	1	no	—
257	C₂×Q₈	—	106	1	no	—
258	C₂×D₄	—	106	1	no	—
259	C₂²×C₄	—	212	1	no	—
260	Q₁₆	—	2	2	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	—	2	2	no	—
263	Q₁₆	—	2	2	no	—
264	Q₁₆	—	8	1	no	—
265	Q₁₆	—	8	1	no	—
266	Q₁₆	—	106	1	no	—
267	Q₁₆	—	106	1	no	—
268	Q₁₆	—	424	2	no	—
269	SD₁₆	—	424	2	no	—
270	D₈	—	106	1	no	—
271	D₈	—	106	1	no	—
272	C₂×C₈	—	53	2	no	—
273	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	53	1	no	—
280	C₄⋊C₄	—	53	2	no	—
281	C₄⋊C₄	—	106	1	no	—
282	C₄⋊C₄	—	106	1	no	—
283	C₄⋊C₄	—	212	2	no	—
284	C₄⋊C₄	—	212	2	no	—
285	C₄⋊C₄	—	212	2	no	—
286	C₄⋊C₄	—	212	2	no	—
287	C₄⋊C₄	—	212	1	no	—
288	C₂²⋊C₄	—	212	2	no	—
289	C₂²⋊C₄	—	212	2	no	—
290	C₂²⋊C₄	—	212	1	no	—
291	C₄²	—	53	2	no	—
292	C₄²	—	212	2	no	—
293	C₄²	—	212	2	no	—
294	C₄²	—	212	1	no	—
295	C₄²	D₄×D₅₃	1	1	yes	characteristic
296	C₁₆	—	2	1	no	—
297	C₁₆	—	106	2	no	—
298	C₁₆	#11719947	1	2	yes	—
299	C₂³	—	212	1	no	—
300	Q₈	—	4	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	106	2	no	—
305	Q₈	—	212	1	no	—
306	Q₈	—	212	1	no	—
307	D₄	—	212	1	no	—
308	D₄	—	212	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	53	2	no	—
312	C₂×C₄	—	53	2	no	—
313	C₂×C₄	—	424	2	no	—
314	C₂×C₄	—	212	1	no	—
315	C₂×C₄	—	212	1	no	—
316	C₂×C₄	—	212	1	no	—
317	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
318	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
320	C₈	—	2	1	no	—
321	C₈	—	106	2	no	—
322	C₈	C₂×#11719947	1	1	yes	characteristic
323	C₈	#13133262	1	1	yes	characteristic
324	C₂²	—	424	1	no	—
325	C₂²	—	424	1	no	—
326	C₂²	—	424	1	no	—
327	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	106	2	no	—
332	C₄	—	106	2	no	—
333	C₄	—	424	1	no	—
334	C₄	—	424	1	no	—
335	C₄	—	424	1	no	—
336	C₄	#16811065	1	2	yes	—
337	C₄	#16811293	1	1	yes	characteristic
338	C₄	#16811424	1	1	yes	characteristic
339	C₂	—	848	1	no	—
340	C₂	#20287688	1	1	yes	characteristic
341	C₂	C₂×#16811065	1	1	yes	characteristic
342	C₂	#20287864	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2247	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₄₂₄	C₂³
hypercenter	#2247	D₅₃
hypocenter	C₅₃	#33965
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

Complements

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

N	H ≅ G/N	type
1. #20287719	C₂	semidirect
5. #20276763	C₂	semidirect
6. #20276762	C₂	semidirect
8. #20276623	C₂	semidirect
9. #20276586	C₂	semidirect
11. C₅₃×#2247	C₂	semidirect
171. #2247	D₅₃	semidirect
212. C₅₃	#33965	semidirect

Maximal subgroups

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

subgroup	index
#20287719	2
#20287705	2
#20276983	2
#20276779	2
#20276763	2
#20276762	2
#20276757	2
#20276623	2
#20276586	2
C₄×#13133162	2
C₅₃×#2247	2
#33965	53