Subgroups — #28303054

Order 13568.

462 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	#20290262	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811484	C₂	1	1	yes	maximal, characteristic, complemented
3	#20290010	C₂	1	2	yes	maximal
4	#20289561	C₂	1	2	yes	maximal, complemented
5	#131×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811639	C₂²	1	2	yes	—
7	C₂×C₄⋊#88324	C₂²	1	2	yes	—
8	#16811486	—	2	1	no	—
9	#16811484	—	2	1	no	—
10	#16811484	—	2	1	no	—
11	#16811484	—	2	1	no	—
12	Q₈⋊#88324	—	2	2	no	—
13	D₄⋊#88324	—	2	2	no	—
14	C₂×#13133224	C₄	1	1	yes	characteristic
15	#41×Dic₅₃	C₂²	1	1	yes	characteristic
16	#16810581	—	2	1	no	—
17	#16810581	C₄	1	1	yes	characteristic
18	#16810578	—	2	1	no	—
19	#16810578	C₄	1	1	yes	characteristic
20	M₁₆×Dic₅₃	—	2	1	no	—
21	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₅₃	—	2	1	no	—
23	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
24	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
25	C₂²×#88324	—	2	2	no	—
26	C₄⋊#88324	—	2	2	no	—
27	C₄⋊#88324	—	2	2	no	—
28	C₄⋊#88324	—	2	2	no	—
29	C₂²⋊#88324	—	2	2	no	—
30	C₄×#88324	—	2	2	no	—
31	#13133224	—	2	2	no	—
32	#13133224	—	2	2	no	—
33	C₂×#11719917	—	2	1	no	—
34	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
35	C₂²×#88322	—	2	1	no	—
36	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
37	Q₈×Dic₅₃	D₄	1	1	yes	characteristic
38	Q₈×Dic₅₃	Q₈	1	1	yes	characteristic
39	D₄×Dic₅₃	—	2	1	no	—
40	D₄×Dic₅₃	—	2	1	no	—
41	D₄×Dic₅₃	D₄	1	1	yes	characteristic
42	D₄×Dic₅₃	Q₈	1	1	yes	characteristic
43	#13133176	—	2	1	no	—
44	#13133176	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×Dic₅₃	—	2	1	no	—
46	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
47	#13133127	—	2	1	no	—
48	#13133127	—	2	1	no	—
49	#13133127	—	2	1	no	—
50	C₈×Dic₅₃	—	2	1	no	—
51	C₈×Dic₅₃	—	2	1	no	—
52	C₈×Dic₅₃	—	2	1	no	—
53	C₈×Dic₅₃	D₄	1	1	yes	characteristic
54	C₈×Dic₅₃	D₄	1	1	yes	characteristic
55	C₈×Dic₅₃	D₄	1	1	yes	characteristic
56	C₈×Dic₅₃	D₄	1	1	yes	characteristic
57	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
58	#131×C₅₃	—	2	2	no	—
59	#131×C₅₃	C₈	1	4	yes	complemented
60	C₂×M₁₆×C₅₃	—	2	1	no	—
61	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₅₃	—	2	1	no	—
63	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	#41×C₅₃	—	2	1	no	—
65	#41×C₅₃	—	2	1	no	—
66	#41×C₅₃	C₂×C₈	1	2	yes	—
67	#41×C₅₃	M₁₆	1	2	yes	—
68	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×D₄×C₅₃	—	2	1	no	—
70	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	—	2	1	no	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×Dic₅₃	—	2	1	no	—
74	C₂²×Dic₅₃	—	2	1	no	—
75	C₂²×Dic₅₃	#41	1	1	yes	characteristic
76	C₂²×Dic₅₃	#41	1	1	yes	characteristic
77	C₂×#88324	—	4	2	no	—
78	C₂×#88324	—	4	2	no	—
79	C₂×#88324	—	8	2	no	—
80	M₁₆×C₅₃	—	2	1	no	—
81	M₁₆×C₅₃	—	2	4	no	—
82	M₁₆×C₅₃	C₂×C₈	1	2	yes	—
83	C₂×C₈×C₅₃	—	2	1	no	—
84	C₂×C₈×C₅₃	—	2	4	no	—
85	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
88	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
89	C₂²⋊Dic₅₃	—	2	1	no	—
90	C₂²⋊Dic₅₃	—	2	1	no	—
91	C₂²⋊Dic₅₃	—	2	1	no	—
92	C₄⋊Dic₅₃	—	2	1	no	—
93	C₄⋊Dic₅₃	—	2	1	no	—
94	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
95	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
96	C₄×Dic₅₃	—	2	1	no	—
97	C₄×Dic₅₃	—	2	1	no	—
98	C₄×Dic₅₃	—	2	1	no	—
99	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
100	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
101	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
102	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
103	#11719917	—	2	4	no	—
104	#11719917	—	2	1	no	—
105	#11719917	M₁₆	1	2	yes	—
106	C₂×#88322	—	2	4	no	—
107	C₂×#88322	—	2	1	no	—
108	C₂×#88322	—	2	1	no	—
109	C₂×#88322	—	2	1	no	—
110	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
111	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
112	C₂³×C₅₃	—	2	1	no	—
113	C₂³×C₅₃	#117	1	1	yes	characteristic
114	Q₈×C₅₃	—	2	1	no	—
115	Q₈×C₅₃	C₄⋊C₈	1	2	yes	—
116	D₄×C₅₃	—	2	2	no	—
117	D₄×C₅₃	—	2	2	no	—
118	D₄×C₅₃	—	2	1	no	—
119	D₄×C₅₃	C₄⋊C₈	1	2	yes	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
129	C₂×Dic₅₃	—	2	2	no	—
130	C₂×Dic₅₃	—	2	1	no	—
131	C₂×Dic₅₃	—	2	1	no	—
132	C₂×Dic₅₃	—	4	1	no	—
133	C₂×Dic₅₃	—	4	2	no	—
134	C₂×Dic₅₃	—	4	2	no	—
135	C₂×Dic₅₃	#121	1	1	yes	characteristic
136	C₂×Dic₅₃	#121	1	1	yes	characteristic
137	C₂×Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
139	#88324	—	8	2	no	—
140	#88324	—	8	2	no	—
141	C₈×C₅₃	—	2	2	no	—
142	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
143	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
144	#88322	—	2	1	no	—
145	#88322	—	2	2	no	—
146	#88322	—	2	1	no	—
147	#8604	—	53	1	no	maximal, Sylow
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	2	no	—
151	C₂²×C₅₃	—	4	1	no	—
152	C₂²×C₅₃	—	4	1	no	—
153	C₂²×C₅₃	#380	1	1	yes	characteristic
154	C₂²×C₅₃	#431	1	1	yes	characteristic
155	C₂²×C₅₃	#432	1	1	yes	characteristic
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
158	C₄×C₅₃	#407	1	1	yes	characteristic
159	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
160	C₄×C₅₃	#423	1	1	yes	characteristic
161	Dic₅₃	—	2	1	no	—
162	Dic₅₃	—	2	1	no	—
163	Dic₅₃	—	2	1	no	—
164	Dic₅₃	—	4	1	no	—
165	Dic₅₃	—	4	1	no	—
166	Dic₅₃	#471	1	2	yes	—
167	C₄×#131	—	53	1	no	—
168	#1564	—	53	1	no	—
169	C₂×#334	—	53	1	no	—
170	#1477	—	53	2	no	—
171	#1283	—	53	2	no	—
172	C₂×C₅₃	—	2	1	no	—
173	C₂×C₅₃	—	2	1	no	—
174	C₂×C₅₃	—	4	2	no	—
175	C₂×C₅₃	#1797	1	1	yes	characteristic
176	C₂×C₅₃	#1814	1	1	yes	characteristic
177	C₂×C₅₃	#1815	1	1	yes	characteristic
178	C₂×#131	—	53	1	no	—
179	C₂×#131	#6157	1	1	yes	characteristic
180	C₄×#41	—	53	1	no	—
181	#431	—	53	2	no	—
182	C₂×C₄⋊C₈	—	53	2	no	—
183	#405	—	53	1	no	—
184	#405	—	53	1	no	—
185	#405	—	106	1	no	—
186	#405	—	106	1	no	—
187	C₄×M₁₆	—	53	1	no	—
188	C₄×M₁₆	—	106	1	no	—
189	C₂×C₄×C₈	—	53	1	no	—
190	C₂×C₄×C₈	—	106	1	no	—
191	#335	—	106	1	no	—
192	#334	—	106	1	no	—
193	#334	—	106	1	no	—
194	#334	—	106	1	no	—
195	Q₈⋊C₈	—	106	2	no	—
196	D₄⋊C₈	—	106	2	no	—
197	C₅₃	#8604	1	1	yes	characteristic, complemented, Sylow
198	C₂×#41	C₂×#6157	1	1	yes	characteristic
199	#131	—	2	2	no	—
200	#131	—	106	2	no	—
201	#131	—	106	2	no	—
202	#131	#88324	1	4	yes	complemented
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	53	1	no	—
205	C₂×M₁₆	—	106	1	no	—
206	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	—	53	1	no	—
209	C₂²×C₈	—	106	1	no	—
210	C₂²×C₈	—	106	2	no	—
211	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
212	C₄×Q₈	—	53	1	no	—
213	C₄×Q₈	—	53	1	no	—
214	C₄×D₄	—	53	1	no	—
215	C₄×D₄	—	53	1	no	—
216	C₄×D₄	—	106	1	no	—
217	C₄×D₄	—	106	1	no	—
218	#117	—	53	1	no	—
219	#117	—	106	1	no	—
220	C₂×C₄²	—	53	1	no	—
221	C₂×C₄²	—	106	1	no	—
222	C₄⋊C₈	—	106	2	no	—
223	C₄⋊C₈	—	106	2	no	—
224	C₄⋊C₈	—	106	2	no	—
225	C₂²⋊C₈	—	106	2	no	—
226	#97	—	106	1	no	—
227	#97	—	106	1	no	—
228	#97	—	106	1	no	—
229	C₄×C₈	—	53	1	no	—
230	C₄×C₈	—	53	1	no	—
231	C₄×C₈	—	53	1	no	—
232	C₄×C₈	—	53	1	no	—
233	C₄×C₈	—	106	1	no	—
234	C₄×C₈	—	106	2	no	—
235	C₄×C₈	—	106	1	no	—
236	C₄×C₈	—	106	1	no	—
237	#41	—	2	1	no	—
238	#41	—	2	1	no	—
239	#41	#11719936	1	2	yes	—
240	#41	C₂×#88324	1	2	yes	—
241	C₂×Q₈	C₄⋊#6157	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	C₄⋊#6157	1	1	yes	characteristic
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	53	1	no	—
246	C₂²×C₄	—	53	1	no	—
247	C₂²×C₄	—	106	1	no	—
248	C₂²×C₄	—	106	1	no	—
249	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
250	M₁₆	—	2	1	no	—
251	M₁₆	—	2	4	no	—
252	M₁₆	—	53	2	no	—
253	M₁₆	—	106	4	no	—
254	M₁₆	—	106	1	no	—
255	M₁₆	C₂×#88324	1	2	yes	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	53	1	no	—
259	C₂×C₈	—	53	1	no	—
260	C₂×C₈	—	106	4	no	—
261	C₂×C₈	—	106	1	no	—
262	C₂×C₈	—	106	1	no	—
263	C₂×C₈	—	106	1	no	—
264	C₂×C₈	—	424	2	no	—
265	C₂×C₈	—	212	2	no	—
266	C₂×C₈	—	212	2	no	—
267	C₂×C₈	C₂×#88324	1	2	yes	—
268	C₂×C₈	#11719940	1	2	yes	—
269	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
270	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
271	C₄⋊C₄	—	53	1	no	—
272	C₄⋊C₄	—	53	1	no	—
273	C₄⋊C₄	—	106	1	no	—
274	C₄⋊C₄	—	106	1	no	—
275	C₂²⋊C₄	—	106	1	no	—
276	C₂²⋊C₄	—	106	1	no	—
277	C₂²⋊C₄	—	106	1	no	—
278	C₄²	—	53	1	no	—
279	C₄²	—	53	1	no	—
280	C₄²	—	53	1	no	—
281	C₄²	—	53	1	no	—
282	C₄²	—	106	1	no	—
283	C₄²	—	106	1	no	—
284	C₄²	—	106	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	#13133296	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	C₄⋊#88324	1	2	yes	—
289	D₄	—	2	2	no	—
290	D₄	—	2	2	no	—
291	D₄	—	2	1	no	—
292	D₄	C₄⋊#88324	1	2	yes	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	53	1	no	—
297	C₂×C₄	—	53	1	no	—
298	C₂×C₄	—	53	1	no	—
299	C₂×C₄	—	53	1	no	—
300	C₂×C₄	—	106	2	no	—
301	C₂×C₄	—	106	1	no	—
302	C₂×C₄	—	106	1	no	—
303	C₂×C₄	—	212	2	no	—
304	C₂×C₄	—	212	1	no	—
305	C₂×C₄	—	212	2	no	—
306	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
307	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
308	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
309	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
310	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
311	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	106	1	no	—
314	C₈	—	106	1	no	—
315	C₈	—	106	2	no	—
316	C₈	—	424	2	no	—
317	C₈	—	424	2	no	—
318	C₈	C₂²⋊#88324	1	2	yes	—
319	C₈	C₂²⋊#88324	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#16811565	1	1	yes	characteristic
326	C₂²	#16811639	1	1	yes	characteristic
327	C₂²	#16811644	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	53	2	no	—
330	C₄	—	106	1	no	—
331	C₄	—	106	1	no	—
332	C₄	—	106	1	no	—
333	C₄	—	212	1	no	—
334	C₄	—	212	1	no	—
335	C₄	C₂×C₂²⋊#88324	1	1	yes	characteristic
336	C₄	#16811594	1	1	yes	characteristic
337	C₄	C₂×C₄⋊#88324	1	1	yes	characteristic
338	C₄	#16811626	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	4	2	no	—
342	C₂	#20290569	1	1	yes	characteristic
343	C₂	#20290601	1	1	yes	characteristic
344	C₂	#20290603	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16811565
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×#131×C₅₃	C₄
socle	C₂²×C₅₃	#380
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#131	#6157
hypocenter	C₅₃	#8604
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. #20290262	C₂	semidirect
2. C₂×#16811484	C₂	semidirect
4. #20289561	C₂	semidirect
59. #131×C₅₃	C₈	semidirect
197. C₅₃	#8604	semidirect
202. #131	#88324	semidirect

Maximal subgroups

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

subgroup	index
#20290262	2
C₂×#16811484	2
#20290010	2
#20289561	2
#131×Dic₅₃	2
#8604	53