Subgroups — #20283436

Order 6784.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133223	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811134	C₂	1	1	yes	maximal, characteristic
4	#16810698	C₂	1	2	yes	maximal, complemented
5	#16810697	C₂	1	2	yes	maximal, complemented
6	#16810691	C₂	1	2	yes	maximal, complemented
7	#16810686	C₂	1	2	yes	maximal, complemented
8	#16810649	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810589	C₂	1	1	yes	maximal, characteristic, complemented
10	#16810578	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	1	yes	characteristic
13	#13133223	—	2	4	no	—
14	C₂×#11719917	—	2	4	no	—
15	C₂×#11719917	C₂²	1	1	yes	characteristic
16	C₂×#11719917	C₂²	1	2	yes	—
17	C₂×#11719917	C₂²	1	1	yes	characteristic
18	C₂×#11719912	C₂²	1	2	yes	complemented
19	C₂²×#88322	C₂²	1	2	yes	—
20	C₂×C₈×D₅₃	C₂²	1	2	yes	complemented
21	#13133200	C₄	1	2	yes	—
22	#13133196	C₄	1	1	yes	characteristic
23	#13133196	C₄	1	1	yes	characteristic
24	#13133191	C₄	1	1	yes	characteristic
25	#13133191	C₄	1	1	yes	characteristic
26	#13133186	C₄	1	2	yes	—
27	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
28	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
29	#13133176	C₂²	1	1	yes	characteristic
30	#13133146	C₂²	1	2	yes	complemented
31	#13133146	C₂²	1	2	yes	complemented
32	#13133129	C₂²	1	2	yes	complemented
33	#13133129	C₂²	1	2	yes	complemented
34	#13133127	C₂²	1	2	yes	complemented
35	C₈×Dic₅₃	C₂²	1	2	yes	complemented
36	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
42	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
43	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	—	2	2	no	—
45	C₂²×C₄×C₅₃	—	2	2	no	—
46	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
55	#11719945	—	2	2	no	—
56	#11719945	—	2	2	no	—
57	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₅₃	C₂³	1	2	yes	—
59	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
67	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
68	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
69	#11719921	C₂×C₄	1	2	yes	—
70	#11719921	C₂×C₄	1	2	yes	—
71	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
73	#11719919	C₂×C₄	1	2	yes	—
74	#11719919	C₂×C₄	1	2	yes	—
75	C₄×Dic₅₃	C₂³	1	2	yes	—
76	#11719917	—	2	2	no	—
77	#11719917	—	2	4	no	—
78	#11719917	D₄	1	4	yes	complemented
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	C₂³	1	2	yes	—
81	C₂×#88322	C₂³	1	2	yes	—
82	#11719912	—	2	4	no	—
83	C₈×D₅₃	—	2	4	no	—
84	C₂³×C₅₃	—	2	2	no	—
85	C₂³×C₅₃	—	2	2	no	—
86	C₂³×C₅₃	—	4	1	no	—
87	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₅₃	#41	1	2	yes	—
103	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₅₃	—	2	4	no	—
105	C₂²⋊D₅₃	—	2	4	no	—
106	C₂²⋊D₅₃	—	4	2	no	—
107	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
108	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
109	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
110	D₂₁₂	—	2	2	no	—
111	C₄×D₅₃	—	2	2	no	—
112	C₄×D₅₃	—	2	2	no	—
113	Dic₁₀₆	—	2	2	no	—
114	C₈×C₅₃	—	2	1	no	—
115	C₈×C₅₃	—	2	1	no	—
116	C₈×C₅₃	—	2	1	no	—
117	C₈×C₅₃	—	2	1	no	—
118	#88322	—	2	2	no	—
119	#88322	C₂×D₄	1	4	yes	—
120	C₂²×C₅₃	—	2	1	no	—
121	C₂²×C₅₃	—	2	1	no	—
122	C₂²×C₅₃	—	2	1	no	—
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	—	2	1	no	—
127	C₂²×C₅₃	—	4	4	no	—
128	C₂²×C₅₃	C₄×D₄	1	2	yes	—
129	C₂²×C₅₃	#131	1	2	yes	complemented
130	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
131	C₂×D₅₃	—	2	2	no	—
132	C₂×D₅₃	—	2	2	no	—
133	C₂×D₅₃	—	2	2	no	—
134	C₄×C₅₃	—	2	1	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	C₄×D₄	1	2	yes	—
137	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
138	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
139	Dic₅₃	—	2	2	no	—
140	Dic₅₃	—	2	2	no	—
141	Dic₅₃	—	2	2	no	—
142	#2933	—	53	1	no	maximal, Sylow
143	C₂×C₅₃	—	2	2	no	—
144	C₂×C₅₃	—	2	2	no	—
145	C₂×C₅₃	—	4	1	no	—
146	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
148	C₂×C₅₃	#568	1	1	yes	characteristic
149	D₅₃	—	4	2	no	—
150	C₂×#131	—	53	1	no	—
151	C₂²×M₁₆	—	53	1	no	—
152	#525	—	53	1	no	—
153	#436	—	53	2	no	—
154	#435	—	53	2	no	—
155	#435	—	53	2	no	—
156	C₈×D₄	—	53	2	no	—
157	#424	—	53	1	no	—
158	#408	—	53	1	no	—
159	C₂×C₂²⋊C₈	D₅₃	1	1	yes	characteristic, complemented
160	#405	—	53	1	no	—
161	C₅₃	#2933	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	—	53	1	no	—
163	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
164	#131	—	106	4	no	—
165	C₂×M₁₆	—	53	1	no	—
166	C₂×M₁₆	—	53	1	no	—
167	C₂×M₁₆	—	53	2	no	—
168	C₂×M₁₆	—	53	2	no	—
169	C₂×M₁₆	—	106	4	no	—
170	C₂²×C₈	—	53	2	no	—
171	C₂²×C₈	—	53	2	no	—
172	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
173	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
174	#123	—	53	2	no	—
175	C₂²⋊Q₈	—	53	1	no	—
176	C₂²⋊Q₈	—	53	1	no	—
177	#121	—	53	1	no	—
178	#121	—	53	1	no	—
179	#120	—	53	2	no	—
180	C₄×D₄	—	53	2	no	—
181	C₄×D₄	—	53	2	no	—
182	#117	—	53	1	no	—
183	C₄⋊C₈	—	53	2	no	—
184	C₄⋊C₈	—	53	2	no	—
185	C₂²⋊C₈	—	53	2	no	—
186	C₂²⋊C₈	—	53	2	no	—
187	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
188	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
189	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
190	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
191	#97	—	53	2	no	—
192	C₄×C₈	—	53	2	no	—
193	C₂⁴	C₄×D₅₃	1	1	yes	characteristic
194	#41	—	106	2	no	—
195	#41	—	106	2	no	—
196	C₂×Q₈	—	53	1	no	—
197	C₂×D₄	—	53	2	no	—
198	C₂×D₄	—	53	2	no	—
199	C₂×D₄	—	53	2	no	—
200	C₂×D₄	—	53	1	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	53	2	no	—
204	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
210	M₁₆	—	53	4	no	—
211	M₁₆	—	106	4	no	—
212	M₁₆	—	106	4	no	—
213	M₁₆	—	106	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	53	2	no	—
217	C₂×C₈	—	53	2	no	—
218	C₂×C₈	—	106	4	no	—
219	C₂×C₈	—	106	4	no	—
220	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
221	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
224	C₄⋊C₄	—	53	1	no	—
225	C₄⋊C₄	—	53	1	no	—
226	C₄⋊C₄	—	53	2	no	—
227	C₄⋊C₄	—	53	2	no	—
228	C₂²⋊C₄	—	53	2	no	—
229	C₂²⋊C₄	—	53	2	no	—
230	C₂²⋊C₄	—	53	2	no	—
231	C₂²⋊C₄	—	53	2	no	—
232	C₂²⋊C₄	—	53	2	no	—
233	C₄²	—	53	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	1	no	—
237	C₂³	—	53	2	no	—
238	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
239	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
240	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
241	Q₈	—	106	2	no	—
242	D₄	—	106	4	no	—
243	D₄	—	106	4	no	—
244	D₄	—	106	2	no	—
245	D₄	—	212	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	53	2	no	—
254	C₂×C₄	—	53	2	no	—
255	C₂×C₄	—	53	2	no	—
256	C₂×C₄	—	106	2	no	—
257	C₂×C₄	—	106	2	no	—
258	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
261	C₂×C₄	D₄×D₅₃	1	2	yes	—
262	C₂×C₄	#11719947	1	2	yes	—
263	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	53	4	no	—
269	C₈	—	106	2	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	106	2	no	—
279	C₂²	—	106	2	no	—
280	C₂²	—	106	2	no	—
281	C₂²	#13133177	1	2	yes	—
282	C₂²	#13133223	1	2	yes	complemented
283	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	106	2	no	—
287	C₄	—	106	2	no	—
288	C₄	—	106	2	no	—
289	C₄	#13133177	1	2	yes	—
290	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
291	C₄	C₂×#11719947	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	—	4	1	no	—
295	C₂	—	212	2	no	—
296	C₂	C₂×#13133177	1	1	yes	characteristic
297	C₂	C₂×#13133223	1	1	yes	characteristic
298	C₂	#16811389	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₂×C₂²⋊C₈×C₅₃	C₂
hypercenter	C₂×C₂²⋊C₈	D₅₃
hypocenter	C₅₃	#2933
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#13133223	C₂	semidirect
2. C₂²×#11719917	C₂	semidirect
4. #16810698	C₂	semidirect
5. #16810697	C₂	semidirect
6. #16810691	C₂	semidirect
7. #16810686	C₂	semidirect
8. #16810649	C₂	semidirect
9. #16810589	C₂	semidirect
10. #16810578	C₂	semidirect
11. C₂×C₂²⋊C₈×C₅₃	C₂	semidirect
18. C₂×#11719912	C₂²	semidirect
20. C₂×C₈×D₅₃	C₂²	semidirect
30. #13133146	C₂²	semidirect
31. #13133146	C₂²	semidirect
32. #13133129	C₂²	semidirect
33. #13133129	C₂²	semidirect
34. #13133127	C₂²	semidirect
35. C₈×Dic₅₃	C₂²	semidirect
78. #11719917	D₄	semidirect
129. C₂²×C₅₃	#131	semidirect
159. C₂×C₂²⋊C₈	D₅₃	semidirect
161. C₅₃	#2933	semidirect
282. C₂²	#13133223	semidirect