Subgroups — #27541789

Order 13568.

472 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287704	C₂	1	1	yes	maximal, characteristic
2	#20287607	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276947	C₂	1	2	yes	maximal
4	#20276943	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276880	C₂	1	2	yes	maximal, complemented
6	#20276853	C₂	1	1	yes	maximal, characteristic
7	#20276778	C₂	1	2	yes	maximal
8	#20276766	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×Dic₄₂₄	C₂	1	2	yes	maximal, complemented
10	C₄×#13133162	C₂	1	1	yes	maximal, characteristic
11	Q₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811351	—	2	1	no	—
13	#16811346	—	2	1	no	—
14	C₂×Dic₄₂₄	C₂²	1	2	yes	—
15	C₂×#13133162	C₂²	1	1	yes	characteristic
16	#16811028	C₂²	1	1	yes	characteristic
17	#16811025	C₂²	1	2	yes	—
18	#16811007	C₂²	1	2	yes	—
19	#16811003	C₂²	1	1	yes	characteristic
20	#16810931	—	2	2	no	—
21	#16810909	—	2	1	no	—
22	#16810846	—	2	2	no	—
23	#16810829	—	2	1	no	—
24	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
25	C₄×Dic₂₁₂	C₂²	1	2	yes	—
26	C₄×Dic₂₁₂	C₂²	1	2	yes	—
27	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
28	#16810451	C₂²	1	1	yes	characteristic
29	#16810449	C₂²	1	2	yes	—
30	#16810428	C₂²	1	2	yes	—
31	#16810428	C₂²	1	2	yes	—
32	#16810428	C₂²	1	1	yes	characteristic
33	#16810428	C₂²	1	2	yes	—
34	#16810425	C₂²	1	2	yes	—
35	#16810425	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
39	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
41	C₂×Dic₂₁₂	—	2	1	no	—
42	C₂×Dic₂₁₂	C₂³	1	2	yes	—
43	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic
44	C₂×#11719913	—	2	2	no	—
45	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
46	#13133217	—	2	1	no	—
47	#13133217	—	2	1	no	—
48	#13133215	—	4	1	no	—
49	#13133212	—	4	2	no	—
50	#13133209	—	2	2	no	—
51	#13133207	—	4	2	no	—
52	#13133205	—	2	2	no	—
53	#13133193	—	4	1	no	—
54	C₄×Dic₁₀₆	—	2	2	no	—
55	C₄×Dic₁₀₆	—	2	1	no	—
56	C₄×Dic₁₀₆	—	2	2	no	—
57	#13133182	—	4	1	no	—
58	C₄×D₂₁₂	—	2	1	no	—
59	#13133175	—	4	1	no	—
60	Dic₄₂₄	—	2	4	no	—
61	#13133162	—	2	2	no	—
62	#13133151	—	2	1	no	—
63	#13133151	C₂³	1	1	yes	characteristic
64	#13133151	C₂³	1	2	yes	—
65	#13133151	C₂³	1	1	yes	characteristic
66	#13133149	—	2	2	no	—
67	#13133142	—	2	2	no	—
68	#13133142	—	2	2	no	—
69	#13133142	—	2	1	no	—
70	#13133139	—	2	1	no	—
71	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
72	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
74	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
76	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
77	C₄×C₈×C₅₃	C₂³	1	2	yes	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
80	C₂×D₂₁₂	—	2	1	no	—
81	C₂×C₄×D₅₃	—	4	1	no	—
82	C₂×Dic₁₀₆	—	2	1	no	—
83	C₂×Dic₁₀₆	—	2	2	no	—
84	C₂×Dic₁₀₆	—	4	1	no	—
85	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
89	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₃	C₂×D₄	1	2	yes	—
92	#11719921	—	4	2	no	—
93	#11719921	—	4	1	no	—
94	#11719921	—	4	2	no	—
95	C₄⋊Dic₅₃	—	2	1	no	—
96	C₄⋊Dic₅₃	—	2	1	no	—
97	C₄⋊Dic₅₃	—	2	2	no	—
98	C₄⋊Dic₅₃	—	4	1	no	—
99	C₄⋊Dic₅₃	—	4	1	no	—
100	C₄⋊Dic₅₃	—	4	1	no	—
101	#11719919	—	4	2	no	—
102	#11719919	—	4	2	no	—
103	#11719919	—	4	1	no	—
104	#11719919	—	4	2	no	—
105	#11719919	—	4	2	no	—
106	C₄×Dic₅₃	—	4	2	no	—
107	C₄×Dic₅₃	—	4	2	no	—
108	C₄×Dic₅₃	—	4	1	no	—
109	Dic₂₁₂	—	2	2	no	—
110	Dic₂₁₂	—	2	1	no	—
111	Dic₂₁₂	—	2	2	no	—
112	Dic₂₁₂	—	8	1	no	—
113	Dic₂₁₂	#41	1	2	yes	—
114	D₄₂₄	—	2	1	no	—
115	D₄₂₄	—	2	1	no	—
116	#11719913	—	8	2	no	—
117	C₁₆×C₅₃	—	2	1	no	—
118	C₁₆×C₅₃	—	2	2	no	—
119	C₁₆×C₅₃	—	2	1	no	—
120	C₂²×D₅₃	—	4	1	no	—
121	Q₈×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₅₃	—	4	1	no	—
128	C₂×Dic₅₃	—	4	2	no	—
129	C₂×Dic₅₃	—	4	2	no	—
130	C₂×Dic₅₃	—	4	1	no	—
131	C₂×Dic₅₃	—	4	1	no	—
132	D₂₁₂	—	4	1	no	—
133	D₂₁₂	—	4	1	no	—
134	C₄×D₅₃	—	8	2	no	—
135	Dic₁₀₆	—	2	2	no	—
136	Dic₁₀₆	—	4	1	no	—
137	Dic₁₀₆	—	4	2	no	—
138	Dic₁₀₆	—	4	2	no	—
139	Dic₁₀₆	—	8	2	no	—
140	C₈×C₅₃	—	2	2	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
143	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
144	#33931	—	53	1	no	maximal, Sylow
145	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
146	C₂×D₅₃	—	8	1	no	—
147	C₂×D₅₃	—	8	1	no	—
148	C₂×D₅₃	—	8	1	no	—
149	C₄×C₅₃	—	2	2	no	—
150	C₄×C₅₃	—	2	2	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	#508	1	2	yes	—
153	C₄×C₅₃	#548	1	1	yes	characteristic
154	C₄×C₅₃	#577	1	1	yes	characteristic
155	Dic₅₃	—	4	2	no	—
156	Dic₅₃	—	8	2	no	—
157	Dic₅₃	—	8	2	no	—
158	Dic₅₃	—	8	1	no	—
159	Dic₅₃	—	8	1	no	—
160	#3393	—	53	1	no	—
161	#3383	—	53	1	no	—
162	#2242	—	53	2	no	—
163	#2241	—	53	1	no	—
164	#2229	—	53	2	no	—
165	#2224	—	53	1	no	—
166	#2211	—	53	2	no	—
167	#2209	—	53	1	no	—
168	Q₈×C₁₆	D₅₃	1	1	yes	characteristic, complemented
169	C₄×Q₃₂	—	53	2	no	—
170	C₄×SD₃₂	—	53	1	no	—
171	C₂×C₅₃	#3358	1	1	yes	characteristic
172	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
173	C₂×C₅₃	#3417	1	1	yes	characteristic
174	D₅₃	—	16	1	no	—
175	#557	—	106	1	no	—
176	#556	—	106	1	no	—
177	C₂×Q₃₂	—	53	2	no	—
178	C₂×SD₃₂	—	53	1	no	—
179	#500	—	53	1	no	—
180	#499	—	53	2	no	—
181	#495	—	53	2	no	—
182	#494	—	53	1	no	—
183	#479	—	106	2	no	—
184	#475	—	106	1	no	—
185	#464	—	106	2	no	—
186	#461	—	106	1	no	—
187	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
188	C₄×Q₁₆	—	53	1	no	—
189	C₄×Q₁₆	—	53	2	no	—
190	C₄×Q₁₆	—	53	2	no	—
191	C₄×D₈	—	53	1	no	—
192	#367	—	53	1	no	—
193	#366	—	53	2	no	—
194	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
195	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
196	#358	—	53	1	no	—
197	#358	—	53	2	no	—
198	#358	—	53	2	no	—
199	#358	—	53	2	no	—
200	#357	—	53	1	no	—
201	#357	—	53	2	no	—
202	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
203	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
204	C₅₃	#33931	1	1	yes	characteristic, complemented, Sylow
205	C₂×Q₁₆	—	53	1	no	—
206	C₂×Q₁₆	—	53	2	no	—
207	C₂×Q₁₆	—	106	1	no	—
208	C₂×SD₁₆	—	106	2	no	—
209	C₂×D₈	—	53	1	no	—
210	C₄⋊Q₈	—	106	1	no	—
211	C₄⋊Q₈	—	106	1	no	—
212	C₄⋊Q₈	—	212	1	no	—
213	#126	—	212	2	no	—
214	#125	—	106	2	no	—
215	#125	—	212	2	no	—
216	#124	—	106	2	no	—
217	C₂²⋊Q₈	—	212	1	no	—
218	C₄×Q₈	—	106	2	no	—
219	C₄×Q₈	—	106	1	no	—
220	C₄×Q₈	—	106	2	no	—
221	C₄×Q₈	—	212	1	no	—
222	C₄×Q₈	D₂₁₂	1	1	yes	characteristic
223	C₄×D₄	—	106	1	no	—
224	#117	—	212	1	no	—
225	Q₃₂	—	106	4	no	—
226	SD₃₂	—	106	2	no	—
227	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
228	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
229	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
230	#107	—	53	2	no	—
231	#107	—	53	1	no	—
232	#107	—	53	1	no	—
233	#107	—	106	1	no	—
234	#106	—	106	2	no	—
235	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
236	C₄⋊C₈	D₂₁₂	1	2	yes	—
237	#103	—	106	2	no	—
238	#103	—	106	2	no	—
239	#103	—	106	1	no	—
240	#102	—	106	1	no	—
241	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₅₃	1	2	yes	—
243	C₂×Q₈	—	106	1	no	—
244	C₂×Q₈	—	106	2	no	—
245	C₂×Q₈	—	212	1	no	—
246	C₂×Q₈	D₄₂₄	1	1	yes	characteristic
247	C₂×D₄	—	106	1	no	—
248	C₂²×C₄	—	212	1	no	—
249	Q₁₆	—	53	2	no	—
250	Q₁₆	—	106	2	no	—
251	Q₁₆	—	106	1	no	—
252	Q₁₆	—	106	2	no	—
253	Q₁₆	—	424	1	no	—
254	SD₁₆	—	424	2	no	—
255	D₈	—	106	1	no	—
256	D₈	—	106	1	no	—
257	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
258	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
259	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
260	C₄⋊C₄	—	106	1	no	—
261	C₄⋊C₄	—	106	1	no	—
262	C₄⋊C₄	—	106	2	no	—
263	C₄⋊C₄	—	212	1	no	—
264	C₄⋊C₄	—	212	2	no	—
265	C₄⋊C₄	—	212	1	no	—
266	C₄⋊C₄	—	212	2	no	—
267	C₄⋊C₄	—	212	2	no	—
268	C₄⋊C₄	—	212	2	no	—
269	C₄⋊C₄	—	212	1	no	—
270	C₄⋊C₄	—	212	1	no	—
271	C₄⋊C₄	D₄₂₄	1	1	yes	characteristic
272	C₄⋊C₄	D₄₂₄	1	2	yes	—
273	C₂²⋊C₄	—	212	2	no	—
274	C₂²⋊C₄	—	212	1	no	—
275	C₂²⋊C₄	—	212	2	no	—
276	C₄²	—	212	2	no	—
277	C₄²	—	212	2	no	—
278	C₄²	—	212	1	no	—
279	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
280	C₄²	C₂×D₂₁₂	1	2	yes	—
281	C₁₆	—	2	1	no	—
282	C₁₆	—	2	2	no	—
283	C₁₆	—	2	1	no	—
284	C₂³	—	212	1	no	—
285	Q₈	—	2	2	no	—
286	Q₈	—	106	2	no	—
287	Q₈	—	424	2	no	—
288	Q₈	—	212	2	no	—
289	Q₈	—	212	2	no	—
290	Q₈	—	212	1	no	—
291	D₄	—	212	1	no	—
292	D₄	—	212	1	no	—
293	C₂×C₄	—	424	2	no	—
294	C₂×C₄	—	212	1	no	—
295	C₂×C₄	—	212	2	no	—
296	C₂×C₄	—	212	1	no	—
297	C₂×C₄	—	212	2	no	—
298	C₂×C₄	—	212	1	no	—
299	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
300	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
302	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	1	no	—
306	C₈	#41×D₅₃	1	1	yes	characteristic
307	C₈	#13133265	1	1	yes	characteristic
308	C₂²	—	424	1	no	—
309	C₂²	—	424	1	no	—
310	C₂²	—	424	1	no	—
311	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	424	1	no	—
316	C₄	—	424	2	no	—
317	C₄	—	424	2	no	—
318	C₄	—	424	1	no	—
319	C₄	—	212	2	no	—
320	C₄	#16811055	1	2	yes	—
321	C₄	#16811285	1	1	yes	characteristic
322	C₄	#16811435	1	1	yes	characteristic
323	C₂	—	848	1	no	—
324	C₂	#20287351	1	1	yes	characteristic
325	C₂	C₂×#16811055	1	1	yes	characteristic
326	C₂	#20287868	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	Q₈×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₄₂₄	C₂³
hypercenter	Q₈×C₁₆	D₅₃
hypocenter	C₅₃	#33931
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20287607	C₂	semidirect
4. #20276943	C₂	semidirect
5. #20276880	C₂	semidirect
8. #20276766	C₂	semidirect
9. C₄×Dic₄₂₄	C₂	semidirect
11. Q₈×C₁₆×C₅₃	C₂	semidirect
168. Q₈×C₁₆	D₅₃	semidirect
204. C₅₃	#33931	semidirect

Maximal subgroups

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

subgroup	index
#20287704	2
#20287607	2
#20276947	2
#20276943	2
#20276880	2
#20276853	2
#20276778	2
#20276766	2
C₄×Dic₄₂₄	2
C₄×#13133162	2
Q₈×C₁₆×C₅₃	2
#33931	53