Subgroups — #20291490

Order 6784.

458 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811703	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811702	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811669	C₂	1	2	yes	maximal, complemented
4	#16811668	C₂	1	2	yes	maximal, complemented
5	#16811667	C₂	1	2	yes	maximal, complemented
6	#16811666	C₂	1	2	yes	maximal, complemented
7	#16811638	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133282	C₂	1	1	yes	maximal, characteristic, complemented
9	D₅₃⋊#104	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×#6157	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811422	C₂	1	1	yes	maximal, characteristic
12	#13133305	—	2	1	no	—
13	#13133305	—	2	1	no	—
14	#13133305	C₂²	1	2	yes	complemented
15	#13133305	C₂²	1	2	yes	complemented
16	#13133305	C₂²	1	2	yes	complemented
17	#13133304	—	2	2	no	—
18	#13133304	C₂²	1	2	yes	complemented
19	C₂×#11719936	—	2	1	no	—
20	C₂×#11719936	C₂²	1	1	yes	characteristic
21	D₅₃⋊M₁₆	—	2	2	no	—
22	D₅₃⋊M₁₆	—	2	1	no	—
23	D₅₃⋊M₁₆	C₂²	1	1	yes	characteristic
24	D₅₃⋊M₁₆	C₂²	1	1	yes	characteristic
25	C₂×C₄×#6157	C₂²	1	1	yes	characteristic
26	#13133288	C₂²	1	2	yes	complemented
27	#13133287	C₂²	1	2	yes	complemented
28	#13133282	C₂²	1	1	yes	characteristic, complemented
29	#13133282	C₂²	1	1	yes	characteristic, complemented
30	#13133282	C₂²	1	2	yes	complemented
31	#13133282	C₂²	1	2	yes	complemented
32	#13133281	C₂²	1	1	yes	characteristic, complemented
33	#13133281	C₂²	1	1	yes	characteristic, complemented
34	#13133271	C₂²	1	2	yes	complemented
35	C₈×#6157	C₂²	1	2	yes	complemented
36	C₂×#11719949	C₂²	1	1	yes	characteristic
37	#41×D₅₃	C₂²	1	1	yes	characteristic
38	#13133249	C₄	1	2	yes	complemented
39	#13133248	C₄	1	1	yes	characteristic, complemented
40	#13133245	C₄	1	1	yes	characteristic, complemented
41	#13133244	C₄	1	2	yes	complemented
42	#13133242	C₄	1	1	yes	characteristic, complemented
43	#13133240	C₂²	1	2	yes	—
44	#13133238	C₂²	1	2	yes	—
45	M₁₆×D₅₃	C₂²	1	1	yes	characteristic
46	#137×C₅₃	C₄	1	1	yes	characteristic, complemented
47	C₂²×#6157	—	2	1	no	—
48	#41×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	#11719949	—	2	1	no	—
51	#11719949	—	2	1	no	—
52	#11719949	C₂×C₄	1	2	yes	—
53	#11719949	C₂³	1	2	yes	—
54	#11719949	C₂³	1	1	yes	characteristic
55	Q₈×D₅₃	C₂×C₄	1	1	yes	characteristic
56	#11719947	—	2	1	no	—
57	#11719947	C₂³	1	1	yes	characteristic
58	D₄×D₅₃	—	2	1	no	—
59	D₄×D₅₃	C₂×C₄	1	1	yes	characteristic
60	#11719945	—	2	1	no	—
61	#11719945	C₂×C₄	1	1	yes	characteristic
62	C₂×D₂₁₂	—	2	1	no	—
63	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₅₃	—	2	1	no	—
65	C₂×C₄×D₅₃	—	2	1	no	—
66	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
67	#11719940	—	2	1	no	—
68	#11719940	—	2	2	no	—
69	#11719940	D₄	1	2	yes	—
70	C₂×#88324	—	2	1	no	—
71	C₂×#88324	—	2	2	no	—
72	C₄×#6157	—	2	1	no	—
73	C₄×#6157	C₂³	1	1	yes	characteristic
74	C₄×#6157	C₂³	1	1	yes	characteristic
75	#11719936	—	2	1	no	—
76	#11719936	—	2	2	no	—
77	#11719936	—	2	2	no	—
78	#11719936	—	2	2	no	—
79	#11719936	D₄	1	1	yes	characteristic
80	#11719936	D₄	1	1	yes	characteristic
81	#11719936	C₂³	1	1	yes	characteristic
82	#11719936	C₂³	1	1	yes	characteristic
83	#11719936	C₂³	1	2	yes	—
84	D₅₃⋊C₈	—	2	1	no	—
85	D₅₃⋊C₈	—	2	2	no	—
86	D₅₃⋊C₈	—	2	2	no	—
87	D₅₃⋊C₈	—	2	2	no	—
88	D₅₃⋊C₈	C₂³	1	2	yes	—
89	Q₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
90	SD₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
91	M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
92	#11719925	C₂×C₄	1	2	yes	complemented
93	#11719924	C₂×C₄	1	2	yes	complemented
94	#11719924	C₂×C₄	1	2	yes	complemented
95	#11719922	C₂×C₄	1	2	yes	complemented
96	#11719917	C₂×C₄	1	1	yes	characteristic
97	D₄₂₄	C₂×C₄	1	2	yes	complemented
98	#11719913	C₂×C₄	1	2	yes	complemented
99	#11719912	C₂×C₄	1	2	yes	—
100	C₈×D₅₃	C₂³	1	2	yes	—
101	C₂²×D₅₃	—	2	1	no	—
102	C₂²×D₅₃	—	2	1	no	—
103	C₂²×D₅₃	#41	1	1	yes	characteristic
104	C₂×#6157	—	2	1	no	—
105	C₂×#6157	—	2	1	no	—
106	C₂×#6157	—	4	2	no	—
107	Q₈×C₅₃	—	2	1	no	—
108	Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	Q₈×C₅₃	C₂²×C₄	1	2	yes	—
110	D₄×C₅₃	—	2	1	no	—
111	D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂²⋊D₅₃	—	2	1	no	—
116	C₂²⋊D₅₃	—	2	1	no	—
117	C₂²⋊D₅₃	—	2	1	no	—
118	C₂×Dic₅₃	—	2	1	no	—
119	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
120	D₂₁₂	—	2	2	no	—
121	D₂₁₂	—	2	2	no	—
122	D₂₁₂	—	2	1	no	—
123	D₂₁₂	—	2	1	no	—
124	D₂₁₂	C₂²×C₄	1	2	yes	—
125	D₂₁₂	C₂²×C₄	1	1	yes	characteristic
126	C₄×D₅₃	—	2	1	no	—
127	C₄×D₅₃	—	2	1	no	—
128	C₄×D₅₃	—	2	2	no	—
129	C₄×D₅₃	—	2	1	no	—
130	C₄×D₅₃	—	2	2	no	—
131	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
133	C₄×D₅₃	#41	1	1	yes	characteristic
134	C₄×D₅₃	C₂⁴	1	1	yes	characteristic
135	Dic₁₀₆	—	2	1	no	—
136	Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
137	#88324	—	2	2	no	—
138	#88324	—	2	1	no	—
139	#88324	—	2	1	no	—
140	#88324	C₂×D₄	1	2	yes	—
141	#88324	C₂×D₄	1	2	yes	—
142	C₈×C₅₃	C₂²×C₄	1	2	yes	—
143	#88322	C₂²×C₄	1	2	yes	—
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
146	C₂×D₅₃	—	2	2	no	—
147	C₂×D₅₃	—	2	1	no	—
148	C₂×D₅₃	—	4	1	no	—
149	C₂×D₅₃	—	8	1	no	—
150	C₂×D₅₃	—	8	1	no	—
151	C₂×D₅₃	C₄×D₄	1	1	yes	characteristic
152	C₂×D₅₃	C₂×#41	1	1	yes	characteristic
153	#6157	—	2	2	no	—
154	#6157	—	4	1	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
158	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
159	Dic₅₃	—	2	1	no	—
160	Dic₅₃	C₄×D₄	1	1	yes	characteristic
161	Dic₅₃	C₂²×D₄	1	1	yes	characteristic
162	#2956	—	53	1	no	maximal, Sylow
163	C₂×C₅₃	—	2	1	no	—
164	C₂×C₅₃	—	4	1	no	—
165	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
166	D₅₃	—	2	1	no	—
167	D₅₃	—	2	1	no	—
168	D₅₃	—	4	2	no	—
169	D₅₃	—	4	1	no	—
170	#575	—	53	1	no	—
171	#568	—	53	1	no	—
172	#568	—	53	1	no	—
173	#444	—	53	2	no	—
174	#444	—	53	2	no	—
175	#443	—	53	2	no	—
176	#443	—	53	2	no	—
177	#430	—	53	1	no	—
178	C₂×#104	—	53	1	no	—
179	C₂×#104	—	53	1	no	—
180	C₄×M₁₆	—	53	1	no	—
181	C₅₃	#2956	1	1	yes	characteristic, complemented, Sylow
182	C₂×#41	—	53	1	no	—
183	C₂×#41	—	53	1	no	—
184	#137	—	53	1	no	—
185	#137	—	53	2	no	—
186	#137	#6157	1	1	yes	characteristic, complemented
187	AGL(1,ℤ/8)	—	53	1	no	—
188	AGL(1,ℤ/8)	—	53	2	no	—
189	AGL(1,ℤ/8)	—	53	1	no	—
190	#135	—	53	2	no	—
191	#135	—	53	2	no	—
192	#131	—	53	2	no	—
193	#131	—	53	2	no	—
194	#131	—	53	2	no	—
195	#131	—	53	2	no	—
196	#131	—	106	2	no	—
197	#131	—	106	1	no	—
198	#131	—	106	1	no	—
199	C₂×M₁₆	—	53	1	no	—
200	C₂×M₁₆	—	53	1	no	—
201	C₂×M₁₆	—	53	1	no	—
202	C₂×M₁₆	—	53	1	no	—
203	C₂×M₁₆	—	106	1	no	—
204	C₂×M₁₆	—	106	2	no	—
205	C₂×M₁₆	—	106	1	no	—
206	C₂×C₄²	—	53	1	no	—
207	#108	—	53	2	no	—
208	#108	—	53	2	no	—
209	#104	—	53	1	no	—
210	#104	—	53	1	no	—
211	#104	—	53	2	no	—
212	#104	—	53	1	no	—
213	#104	—	53	1	no	—
214	#104	—	53	2	no	—
215	#97	—	53	2	no	—
216	C₄×C₈	—	53	2	no	—
217	#41	—	53	2	no	—
218	#41	—	53	1	no	—
219	#41	—	53	2	no	—
220	#41	—	53	1	no	—
221	#41	—	53	1	no	—
222	#41	—	106	1	no	—
223	#41	—	106	1	no	—
224	#41	—	106	1	no	—
225	#41	—	106	1	no	—
226	#41	C₂×#6157	1	1	yes	characteristic
227	C₂×Q₈	—	53	1	no	—
228	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
229	C₂×D₄	—	53	1	no	—
230	C₂×D₄	—	53	1	no	—
231	C₂×D₄	—	106	1	no	—
232	C₂×D₄	—	106	1	no	—
233	C₂²×C₄	—	53	1	no	—
234	C₂²×C₄	—	106	1	no	—
235	C₂²×C₄	—	106	1	no	—
236	C₂²×C₄	—	106	1	no	—
237	Q₁₆	—	53	2	no	—
238	Q₁₆	C₂×#6157	1	2	yes	complemented
239	SD₁₆	—	53	2	no	—
240	SD₁₆	—	53	2	no	—
241	SD₁₆	—	53	2	no	—
242	SD₁₆	C₂×#6157	1	2	yes	complemented
243	D₈	—	53	2	no	—
244	D₈	—	53	2	no	—
245	M₁₆	—	53	1	no	—
246	M₁₆	—	53	1	no	—
247	M₁₆	—	53	2	no	—
248	M₁₆	—	53	1	no	—
249	M₁₆	—	53	2	no	—
250	M₁₆	—	53	1	no	—
251	M₁₆	—	53	1	no	—
252	M₁₆	—	53	2	no	—
253	M₁₆	—	106	2	no	—
254	M₁₆	—	106	2	no	—
255	M₁₆	—	106	2	no	—
256	M₁₆	—	106	2	no	—
257	M₁₆	—	106	1	no	—
258	M₁₆	—	106	1	no	—
259	M₁₆	C₂×#6157	1	1	yes	characteristic
260	C₂×C₈	—	53	2	no	—
261	C₂×C₈	—	53	2	no	—
262	C₂×C₈	—	106	2	no	—
263	C₂×C₈	—	106	2	no	—
264	C₂×C₈	—	106	2	no	—
265	C₂×C₈	—	106	2	no	—
266	C₂×C₈	—	106	1	no	—
267	C₂×C₈	—	106	1	no	—
268	C₄²	—	53	1	no	—
269	C₄²	—	53	1	no	—
270	C₄²	—	106	1	no	—
271	C₂³	—	53	1	no	—
272	C₂³	—	106	1	no	—
273	C₂³	—	106	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	53	1	no	—
276	Q₈	—	106	1	no	—
277	Q₈	C₂²×#6157	1	1	yes	characteristic
278	Q₈	C₂²×#6157	1	2	yes	—
279	D₄	—	2	1	no	—
280	D₄	—	53	2	no	—
281	D₄	—	53	1	no	—
282	D₄	—	106	1	no	—
283	D₄	—	106	2	no	—
284	D₄	—	106	2	no	—
285	D₄	—	106	1	no	—
286	D₄	—	106	1	no	—
287	D₄	—	106	1	no	—
288	D₄	—	106	1	no	—
289	D₄	C₂²×#6157	1	1	yes	characteristic
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	53	1	no	—
293	C₂×C₄	—	53	1	no	—
294	C₂×C₄	—	53	1	no	—
295	C₂×C₄	—	53	1	no	—
296	C₂×C₄	—	53	1	no	—
297	C₂×C₄	—	106	1	no	—
298	C₂×C₄	—	106	2	no	—
299	C₂×C₄	—	106	2	no	—
300	C₂×C₄	—	106	1	no	—
301	C₂×C₄	—	106	1	no	—
302	C₂×C₄	—	106	1	no	—
303	C₂×C₄	—	106	1	no	—
304	C₂×C₄	—	106	1	no	—
305	C₂×C₄	—	212	2	no	—
306	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
307	C₈	—	53	2	no	—
308	C₈	—	53	2	no	—
309	C₈	—	53	2	no	—
310	C₈	—	106	2	no	—
311	C₈	—	106	1	no	—
312	C₈	—	106	1	no	—
313	C₈	C₂²×#6157	1	2	yes	—
314	C₂²	—	2	1	no	—
315	C₂²	—	53	1	no	—
316	C₂²	—	53	1	no	—
317	C₂²	—	106	2	no	—
318	C₂²	—	106	1	no	—
319	C₂²	—	424	1	no	—
320	C₂²	—	424	1	no	—
321	C₂²	—	212	1	no	—
322	C₂²	D₄×#6157	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	2	no	—
325	C₄	—	53	1	no	—
326	C₄	—	53	1	no	—
327	C₄	—	106	2	no	—
328	C₄	—	106	1	no	—
329	C₄	—	212	1	no	—
330	C₄	D₄×#6157	1	1	yes	characteristic
331	C₄	C₂³×#6157	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	4	1	no	—
334	C₂	—	106	1	no	—
335	C₂	—	106	1	no	—
336	C₂	—	212	2	no	—
337	C₂	—	212	1	no	—
338	C₂	C₂×D₄×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#6157
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₄	C₂³×#6157
Fitting	#137×C₅₃	C₄
socle	C₂×C₅₃	C₂×C₄×D₄
hypercenter	#137	#6157
hypocenter	C₅₃	#2956
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. #16811703	C₂	semidirect
2. #16811702	C₂	semidirect
3. #16811669	C₂	semidirect
4. #16811668	C₂	semidirect
5. #16811667	C₂	semidirect
6. #16811666	C₂	semidirect
7. #16811638	C₂	semidirect
8. C₂×#13133282	C₂	semidirect
9. D₅₃⋊#104	C₂	semidirect
10. M₁₆×#6157	C₂	semidirect
14. #13133305	C₂²	semidirect
15. #13133305	C₂²	semidirect
16. #13133305	C₂²	semidirect
18. #13133304	C₂²	semidirect
26. #13133288	C₂²	semidirect
27. #13133287	C₂²	semidirect
28. #13133282	C₂²	semidirect
29. #13133282	C₂²	semidirect
30. #13133282	C₂²	semidirect
31. #13133282	C₂²	semidirect
32. #13133281	C₂²	semidirect
33. #13133281	C₂²	semidirect
34. #13133271	C₂²	semidirect
35. C₈×#6157	C₂²	semidirect
38. #13133249	C₄	semidirect
39. #13133248	C₄	semidirect
40. #13133245	C₄	semidirect
41. #13133244	C₄	semidirect
42. #13133242	C₄	semidirect
46. #137×C₅₃	C₄	semidirect
89. Q₁₆×C₅₃	C₂×C₄	semidirect
90. SD₁₆×C₅₃	C₂×C₄	semidirect
92. #11719925	C₂×C₄	semidirect
93. #11719924	C₂×C₄	semidirect
94. #11719924	C₂×C₄	semidirect
95. #11719922	C₂×C₄	semidirect
97. D₄₂₄	C₂×C₄	semidirect
98. #11719913	C₂×C₄	semidirect
181. C₅₃	#2956	semidirect
186. #137	#6157	semidirect
238. Q₁₆	C₂×#6157	semidirect
242. SD₁₆	C₂×#6157	semidirect