Subgroups — #27541879

Order 13568.

476 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	#20287583	C₂	1	1	yes	maximal, characteristic
2	#20276960	C₂	1	2	yes	maximal, complemented
3	#20276942	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276879	C₂	1	2	yes	maximal, complemented
5	#20276867	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276757	C₂	1	1	yes	maximal, characteristic
7	#20276757	C₂	1	2	yes	maximal
8	#20276624	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276577	C₂	1	2	yes	maximal
10	C₄×#13133160	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#3395	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811333	—	2	1	no	—
13	C₂×#13133160	C₂²	1	1	yes	characteristic
14	C₂×#13133160	C₂²	1	2	yes	—
15	#16811028	C₂²	1	1	yes	characteristic
16	#16811025	C₂²	1	2	yes	—
17	#16810924	—	2	2	no	—
18	#16810914	—	2	1	no	—
19	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
20	C₄×D₄₂₄	C₂²	1	2	yes	—
21	#16810450	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	#16810427	C₂²	1	1	yes	characteristic
29	#16810427	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×#557	—	2	1	no	—
34	C₅₃×#494	C₂²	1	1	yes	characteristic
35	C₅₃×#494	C₂²	1	2	yes	—
36	C₅₃×#462	—	2	1	no	—
37	C₅₃×#462	—	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×D₄₂₄	C₂³	1	1	yes	characteristic
42	#13133217	—	2	1	no	—
43	#13133209	—	2	2	no	—
44	#13133195	—	4	2	no	—
45	#13133193	—	4	1	no	—
46	C₄×D₂₁₂	—	2	1	no	—
47	C₄×D₂₁₂	—	2	2	no	—
48	C₄⋊C₄×D₅₃	—	4	1	no	—
49	#13133160	—	2	2	no	—
50	#13133160	—	2	4	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	C₂³	1	1	yes	characteristic
56	#13133151	C₂³	1	2	yes	—
57	#13133149	—	2	2	no	—
58	#13133139	—	2	2	no	—
59	#13133139	—	2	1	no	—
60	C₂×Q₁₆×C₅₃	—	2	2	no	—
61	C₂×Q₁₆×C₅₃	—	2	1	no	—
62	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
63	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₅₃	—	2	2	no	—
65	C₄⋊Q₈×C₅₃	—	2	1	no	—
66	#125×C₅₃	—	4	1	no	—
67	#125×C₅₃	—	4	2	no	—
68	C₄×Q₈×C₅₃	—	2	2	no	—
69	C₄×Q₈×C₅₃	—	2	1	no	—
70	C₄×Q₈×C₅₃	—	4	1	no	—
71	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
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	#103×C₅₃	—	2	1	no	—
77	C₄×C₈×C₅₃	C₂³	1	2	yes	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	—	2	2	no	—
80	C₂×Q₈×C₅₃	—	2	1	no	—
81	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
82	C₂×D₂₁₂	—	2	1	no	—
83	C₂×C₄×D₅₃	—	4	1	no	—
84	C₂×C₄×D₅₃	—	4	2	no	—
85	C₂×Dic₁₀₆	—	4	1	no	—
86	Q₁₆×C₅₃	—	2	2	no	—
87	Q₁₆×C₅₃	—	2	1	no	—
88	Q₁₆×C₅₃	—	2	2	no	—
89	Q₁₆×C₅₃	—	2	1	no	—
90	Q₁₆×C₅₃	—	8	2	no	—
91	Q₁₆×C₅₃	—	8	1	no	—
92	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₅₃	—	2	1	no	—
96	C₄⋊C₄×C₅₃	—	4	1	no	—
97	C₄⋊C₄×C₅₃	—	4	2	no	—
98	C₄⋊C₄×C₅₃	—	4	2	no	—
99	C₄⋊C₄×C₅₃	—	4	1	no	—
100	C₄⋊C₄×C₅₃	—	4	1	no	—
101	C₄⋊C₄×C₅₃	—	4	2	no	—
102	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
103	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
104	C₄²×C₅₃	—	4	2	no	—
105	C₄²×C₅₃	—	4	1	no	—
106	C₄²×C₅₃	C₂×D₄	1	2	yes	—
107	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
108	#11719921	—	4	2	no	—
109	#11719921	—	4	1	no	—
110	C₄⋊Dic₅₃	—	2	1	no	—
111	C₄⋊Dic₅₃	—	2	2	no	—
112	C₄⋊Dic₅₃	—	4	1	no	—
113	#11719919	—	4	2	no	—
114	#11719919	—	4	2	no	—
115	D₄₂₄	—	2	1	no	—
116	D₄₂₄	#41	1	2	yes	—
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	—	4	2	no	—
123	Q₈×C₅₃	—	4	1	no	—
124	Q₈×C₅₃	—	4	2	no	—
125	Q₈×C₅₃	—	4	1	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	—	4	1	no	—
129	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
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	—	2	2	no	—
137	D₂₁₂	—	4	1	no	—
138	C₄×D₅₃	—	8	2	no	—
139	C₄×D₅₃	—	16	2	no	—
140	Dic₁₀₆	—	16	1	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	#33939	—	53	1	no	maximal, Sylow
146	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₅₃	—	4	2	no	—
148	C₂×D₅₃	—	8	1	no	—
149	C₂×D₅₃	—	8	1	no	—
150	C₄×C₅₃	—	2	2	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	—	2	2	no	—
153	C₄×C₅₃	—	8	2	no	—
154	C₄×C₅₃	—	8	1	no	—
155	C₄×C₅₃	—	8	1	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	1	no	—
160	Dic₅₃	—	8	2	no	—
161	#3395	D₅₃	1	1	yes	characteristic, complemented
162	#3380	—	53	1	no	—
163	#2245	—	53	2	no	—
164	#2241	—	53	1	no	—
165	#2229	—	53	2	no	—
166	#2227	—	53	1	no	—
167	#2208	—	53	2	no	—
168	#2208	—	53	1	no	—
169	#2183	—	53	1	no	—
170	#2175	—	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	—	8	2	no	—
176	#557	—	2	1	no	—
177	#554	—	106	1	no	—
178	C₂×SD₃₂	—	53	1	no	—
179	C₂×SD₃₂	—	53	2	no	—
180	#500	—	53	1	no	—
181	#499	—	53	2	no	—
182	#494	C₂×D₅₃	1	1	yes	characteristic
183	#494	C₂×D₅₃	1	2	yes	—
184	#478	—	106	2	no	—
185	#476	—	106	1	no	—
186	#462	—	2	1	no	—
187	#462	—	2	2	no	—
188	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
189	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
190	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
191	C₄×D₈	—	53	1	no	—
192	C₄×D₈	—	53	2	no	—
193	#367	—	53	1	no	—
194	#366	—	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	#358	—	53	2	no	—
202	#357	—	53	1	no	—
203	#346	—	53	2	no	—
204	C₄×C₁₆	—	53	1	no	—
205	C₅₃	#33939	1	1	yes	characteristic, complemented, Sylow
206	C₂×Q₁₆	—	2	2	no	—
207	C₂×Q₁₆	—	2	1	no	—
208	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
209	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
210	C₂×D₈	—	53	1	no	—
211	C₄⋊Q₈	—	2	2	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	106	1	no	—
214	#125	—	4	1	no	—
215	#125	—	4	2	no	—
216	#125	—	106	2	no	—
217	C₂²⋊Q₈	—	212	1	no	—
218	C₂²⋊Q₈	—	212	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	—	4	1	no	—
222	C₄×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
223	C₄×D₄	—	106	1	no	—
224	C₄×D₄	—	106	2	no	—
225	C₂×C₄⋊C₄	—	212	1	no	—
226	SD₃₂	—	106	2	no	—
227	SD₃₂	—	106	4	no	—
228	C₂×C₁₆	—	53	1	no	—
229	C₂×C₁₆	—	53	1	no	—
230	C₂×C₁₆	—	53	2	no	—
231	#107	—	53	2	no	—
232	#107	—	53	1	no	—
233	#107	—	106	1	no	—
234	#107	C₂²×D₅₃	1	1	yes	characteristic
235	#106	—	106	2	no	—
236	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
237	C₄⋊C₈	C₂²⋊D₅₃	1	2	yes	—
238	#103	—	2	2	no	—
239	#103	—	2	1	no	—
240	#102	—	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	2	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	212	1	no	—
247	C₂×Q₈	#11719922	1	1	yes	characteristic
248	C₂×D₄	—	106	1	no	—
249	C₂²×C₄	—	212	2	no	—
250	C₂²×C₄	—	212	1	no	—
251	Q₁₆	—	2	1	no	—
252	Q₁₆	—	2	2	no	—
253	Q₁₆	—	2	1	no	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	8	2	no	—
256	Q₁₆	—	8	1	no	—
257	D₈	—	53	2	no	—
258	D₈	—	106	1	no	—
259	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
260	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
261	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	106	1	no	—
270	C₄⋊C₄	—	106	2	no	—
271	C₄⋊C₄	—	212	2	no	—
272	C₄⋊C₄	—	212	1	no	—
273	C₄⋊C₄	—	212	2	no	—
274	C₄⋊C₄	#11719922	1	1	yes	characteristic
275	C₄⋊C₄	#11719922	1	2	yes	—
276	C₂²⋊C₄	—	212	1	no	—
277	C₂²⋊C₄	—	212	2	no	—
278	C₄²	—	4	2	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
281	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
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	—	4	1	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	848	1	no	—
292	D₄	—	106	2	no	—
293	D₄	—	212	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	848	2	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	1	yes	characteristic
303	C₂×C₄	C₂×#11719922	1	2	yes	—
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	#41×D₅₃	1	1	yes	characteristic
309	C₈	#13133264	1	1	yes	characteristic
310	C₂²	—	424	1	no	—
311	C₂²	—	424	1	no	—
312	C₂²	—	212	2	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	—	8	2	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	424	1	no	—
321	C₄	—	424	2	no	—
322	C₄	#16811064	1	2	yes	—
323	C₄	#16811280	1	1	yes	characteristic
324	C₄	#16811434	1	1	yes	characteristic
325	C₂	—	424	2	no	—
326	C₂	#20287346	1	1	yes	characteristic
327	C₂	C₂×#16811064	1	1	yes	characteristic
328	C₂	#20287859	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3395	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₄₂₄	C₂³
hypercenter	#3395	D₅₃
hypocenter	C₅₃	#33939
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3395

Complements

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

N	H ≅ G/N	type
2. #20276960	C₂	semidirect
3. #20276942	C₂	semidirect
4. #20276879	C₂	semidirect
5. #20276867	C₂	semidirect
8. #20276624	C₂	semidirect
11. C₅₃×#3395	C₂	semidirect
161. #3395	D₅₃	semidirect
205. C₅₃	#33939	semidirect

Maximal subgroups

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

subgroup	index
#20287583	2
#20276960	2
#20276942	2
#20276879	2
#20276867	2
#20276757	2
#20276624	2
#20276577	2
C₄×#13133160	2
C₅₃×#3395	2
#33939	53