Subgroups — #20288384

Order 6784.

1658 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133263	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811292	C₂	1	2	yes	maximal, complemented
3	#16811284	C₂	1	2	yes	maximal, complemented
4	#16811261	C₂	1	6	yes	maximal, complemented
5	D₄×C₂²⋊D₅₃	C₂	1	6	yes	maximal, complemented
6	#16811201	C₂	1	3	yes	maximal, complemented
7	#16811187	C₂	1	3	yes	maximal, complemented
8	#16811134	C₂	1	3	yes	maximal, complemented
9	C₂×#13133191	C₂	1	3	yes	maximal, complemented
10	#16811106	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#13133263	—	2	8	no	—
13	C₂×#11719949	C₂²	1	2	yes	complemented
14	C₂×#11719945	C₂²	1	3	yes	complemented
15	C₂²×D₂₁₂	C₂²	1	3	yes	complemented
16	C₂²×C₂²⋊D₅₃	C₂²	1	6	yes	complemented
17	C₂×D₄×D₅₃	C₂²	1	6	yes	complemented
18	#13133213	C₂²	1	6	yes	complemented
19	#13133206	C₂²	1	6	yes	complemented
20	#13133200	C₂²	1	6	yes	complemented
21	#13133196	C₂²	1	1	yes	characteristic, complemented
22	#13133196	C₂²	1	3	yes	complemented
23	#13133193	C₂²	1	6	yes	complemented
24	#13133191	C₂²	1	6	yes	complemented
25	#13133191	C₂²	1	6	yes	complemented
26	#13133191	C₂²	1	3	yes	complemented
27	#13133191	C₂²	1	6	yes	complemented
28	#13133191	C₂²	1	3	yes	complemented
29	#13133189	C₂²	1	12	yes	complemented
30	#13133188	C₂²	1	6	yes	complemented
31	#13133186	C₂²	1	6	yes	complemented
32	#13133185	C₂²	1	12	yes	complemented
33	Q₈×Dic₅₃	C₂²	1	2	yes	complemented
34	D₄×Dic₅₃	C₂²	1	6	yes	complemented
35	C₄×C₂²⋊D₅₃	C₂²	1	6	yes	complemented
36	C₄×C₂²⋊D₅₃	C₂²	1	6	yes	complemented
37	#13133176	C₂²	1	3	yes	complemented
38	C₂×C₄⋊Dic₅₃	C₂²	1	3	yes	complemented
39	C₂×#11719921	C₂²	1	6	yes	complemented
40	C₂×#41×C₅₃	—	2	8	no	—
41	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	C₂×#41×C₅₃	C₂²	1	3	yes	complemented
44	C₂×#41×C₅₃	C₂²	1	3	yes	complemented
45	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
46	C₂²×D₄×C₅₃	C₂²	1	3	yes	complemented
47	C₂³×C₄×C₅₃	C₂²	1	3	yes	complemented
48	C₂⁴×C₅₃	C₂³	1	3	yes	—
49	C₂³×D₅₃	C₂³	1	6	yes	—
50	#41×C₅₃	—	2	4	no	—
51	#41×C₅₃	—	2	12	no	—
52	#41×C₅₃	—	2	12	no	—
53	#41×C₅₃	D₄	1	8	yes	complemented
54	C₂×Q₈×C₅₃	—	2	4	no	—
55	C₂×Q₈×C₅₃	C₂³	1	3	yes	—
56	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×D₄×C₅₃	—	2	12	no	—
58	C₂×D₄×C₅₃	C₂³	1	6	yes	—
59	C₂×D₄×C₅₃	C₂³	1	3	yes	—
60	C₂×D₄×C₅₃	C₂³	1	3	yes	—
61	C₂²×C₄×C₅₃	—	2	12	no	—
62	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
64	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
65	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
66	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
67	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
68	C₂×C₂²⋊D₅₃	—	4	12	no	—
69	C₂×C₂²⋊D₅₃	C₂³	1	6	yes	—
70	C₂×C₂²⋊D₅₃	C₂³	1	6	yes	—
71	C₂×C₂²⋊D₅₃	C₂³	1	6	yes	—
72	C₂×C₂²⋊D₅₃	C₂³	1	12	yes	—
73	C₂²×Dic₅₃	C₂³	1	6	yes	—
74	#11719949	—	2	8	no	—
75	D₄×D₅₃	—	2	24	no	—
76	#11719945	—	2	12	no	—
77	C₂×D₂₁₂	—	2	12	no	—
78	C₂×D₂₁₂	C₂³	1	3	yes	—
79	C₂×D₂₁₂	C₂³	1	6	yes	—
80	C₂×C₄×D₅₃	C₂³	1	6	yes	—
81	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
82	C₂²⋊Dic₅₃	C₂³	1	6	yes	—
83	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
84	C₂²⋊Dic₅₃	C₂³	1	6	yes	—
85	#11719921	C₂³	1	6	yes	—
86	#11719921	C₂³	1	12	yes	—
87	#11719921	C₂³	1	6	yes	—
88	C₄⋊Dic₅₃	C₂³	1	3	yes	—
89	C₄⋊Dic₅₃	C₂³	1	6	yes	—
90	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
91	#11719919	C₂³	1	2	yes	—
92	#11719919	C₂³	1	6	yes	—
93	C₄×Dic₅₃	C₂³	1	6	yes	—
94	C₂³×C₅₃	—	2	6	no	—
95	C₂³×C₅₃	—	2	6	no	—
96	C₂³×C₅₃	—	4	3	no	—
97	C₂³×C₅₃	C₂⁴	1	3	yes	—
98	C₂³×C₅₃	C₂⁴	1	3	yes	—
99	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₂²×D₅₃	—	2	12	no	—
101	C₂²×D₅₃	—	4	6	no	—
102	C₂²×D₅₃	—	4	6	no	—
103	C₂²×D₅₃	C₂⁴	1	6	yes	—
104	Q₈×C₅₃	—	2	6	no	—
105	Q₈×C₅₃	C₂×D₄	1	4	yes	complemented
106	D₄×C₅₃	—	2	12	no	—
107	D₄×C₅₃	—	2	6	no	—
108	D₄×C₅₃	C₂×D₄	1	12	yes	complemented
109	C₂×C₄×C₅₃	—	2	6	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	6	no	—
112	C₂×C₄×C₅₃	—	2	6	no	—
113	C₂×C₄×C₅₃	—	2	6	no	—
114	C₂×C₄×C₅₃	C₂×D₄	1	12	yes	complemented
115	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂⁴	1	3	yes	—
117	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂⁴	1	3	yes	—
119	C₂²⋊D₅₃	—	2	12	no	—
120	C₂²⋊D₅₃	—	2	12	no	—
121	C₂²⋊D₅₃	—	4	12	no	—
122	C₂²⋊D₅₃	—	4	6	no	—
123	C₂×Dic₅₃	—	4	6	no	—
124	C₂×Dic₅₃	C₂⁴	1	2	yes	—
125	C₂×Dic₅₃	C₂⁴	1	6	yes	—
126	C₂×Dic₅₃	C₂⁴	1	2	yes	—
127	D₂₁₂	—	2	6	no	—
128	D₂₁₂	—	2	12	no	—
129	C₄×D₅₃	—	2	12	no	—
130	Dic₁₀₆	—	2	2	no	—
131	C₂²×C₅₃	—	2	1	no	—
132	C₂²×C₅₃	—	2	3	no	—
133	C₂²×C₅₃	—	2	3	no	—
134	C₂²×C₅₃	—	2	1	no	—
135	C₂²×C₅₃	—	2	3	no	—
136	C₂²×C₅₃	—	2	3	no	—
137	C₂²×C₅₃	—	2	3	no	—
138	C₂²×C₅₃	—	4	12	no	—
139	C₂²×C₅₃	C₂²×D₄	1	6	yes	—
140	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	complemented
141	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
142	C₂×D₅₃	—	2	6	no	—
143	C₂×D₅₃	—	2	6	no	—
144	C₂×D₅₃	—	2	6	no	—
145	C₂×D₅₃	—	8	12	no	—
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	—	2	3	no	—
148	C₄×C₅₃	C₂²×D₄	1	2	yes	—
149	C₄×C₅₃	C₂²×D₄	1	6	yes	—
150	Dic₅₃	—	2	2	no	—
151	Dic₅₃	—	2	6	no	—
152	Dic₅₃	—	2	2	no	—
153	#3488	—	53	1	no	maximal, Sylow
154	C₂×C₅₃	—	2	6	no	—
155	C₂×C₅₃	—	2	2	no	—
156	C₂×C₅₃	—	4	3	no	—
157	C₂×C₅₃	C₂³×D₄	1	1	yes	characteristic
158	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×C₅₃	#585	1	1	yes	characteristic
160	D₅₃	—	4	6	no	—
161	C₂×H(5,𝔽₂)	—	53	1	no	—
162	C₂²×#41	D₅₃	1	1	yes	characteristic, complemented
163	#550	—	53	2	no	—
164	#548	—	53	2	no	—
165	#546	—	53	6	no	—
166	D₄²	—	53	6	no	—
167	#537	—	53	3	no	—
168	#535	—	53	3	no	—
169	#525	—	53	3	no	—
170	C₂×#121	—	53	3	no	—
171	#520	—	53	1	no	—
172	C₅₃	#3488	1	1	yes	characteristic, complemented, Sylow
173	H(5,𝔽₂)	—	106	8	no	—
174	C₂×#41	—	2	8	no	—
175	C₂×#41	—	53	2	no	—
176	C₂×#41	—	53	3	no	—
177	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
178	C₂×#41	C₂×D₅₃	1	3	yes	complemented
179	C₂×#41	C₂×D₅₃	1	3	yes	complemented
180	C₂×#41	C₂×D₅₃	1	1	yes	characteristic, complemented
181	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
182	C₂²×D₄	—	53	6	no	—
183	C₂²×D₄	—	53	6	no	—
184	C₂²×D₄	—	53	3	no	—
185	C₂²×D₄	C₂×D₅₃	1	3	yes	complemented
186	C₂³×C₄	C₂×D₅₃	1	3	yes	complemented
187	#127	—	53	6	no	—
188	#124	—	53	6	no	—
189	#123	—	53	6	no	—
190	C₂²⋊Q₈	—	53	6	no	—
191	C₂²⋊Q₈	—	53	1	no	—
192	C₂²⋊Q₈	—	53	3	no	—
193	#121	—	53	12	no	—
194	#121	—	53	6	no	—
195	#121	—	53	3	no	—
196	#121	—	53	6	no	—
197	#121	—	53	6	no	—
198	#121	—	53	6	no	—
199	#121	—	53	3	no	—
200	#120	—	53	6	no	—
201	#120	—	53	12	no	—
202	C₄×Q₈	—	53	2	no	—
203	C₄×D₄	—	53	6	no	—
204	C₄×D₄	—	53	6	no	—
205	C₄×D₄	—	53	6	no	—
206	#117	—	53	3	no	—
207	C₂×C₄⋊C₄	—	53	3	no	—
208	C₂×C₂²⋊C₄	—	53	6	no	—
209	C₂⁴	—	53	6	no	—
210	C₂⁴	C₂²×D₅₃	1	3	yes	—
211	#41	—	2	4	no	—
212	#41	—	2	12	no	—
213	#41	—	2	12	no	—
214	#41	—	106	8	no	—
215	#41	—	106	12	no	—
216	#41	C₂²⋊D₅₃	1	8	yes	complemented
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	—	53	1	no	—
219	C₂×Q₈	C₂²×D₅₃	1	3	yes	—
220	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
221	C₂×D₄	—	2	12	no	—
222	C₂×D₄	—	53	6	no	—
223	C₂×D₄	—	53	6	no	—
224	C₂×D₄	—	53	6	no	—
225	C₂×D₄	—	53	12	no	—
226	C₂×D₄	—	53	3	no	—
227	C₂×D₄	—	53	6	no	—
228	C₂×D₄	—	106	24	no	—
229	C₂×D₄	—	106	12	no	—
230	C₂×D₄	—	212	12	no	—
231	C₂×D₄	C₂²×D₅₃	1	6	yes	—
232	C₂×D₄	C₂²×D₅₃	1	3	yes	—
233	C₂×D₄	C₂²×D₅₃	1	3	yes	—
234	C₂²×C₄	—	2	12	no	—
235	C₂²×C₄	—	53	6	no	—
236	C₂²×C₄	—	53	6	no	—
237	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
238	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
239	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
240	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
241	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
242	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
243	C₄⋊C₄	—	53	3	no	—
244	C₄⋊C₄	—	53	6	no	—
245	C₄⋊C₄	—	53	1	no	—
246	C₄⋊C₄	—	53	2	no	—
247	C₄⋊C₄	—	53	6	no	—
248	C₂²⋊C₄	—	53	6	no	—
249	C₂²⋊C₄	—	53	2	no	—
250	C₂²⋊C₄	—	53	6	no	—
251	C₂²⋊C₄	—	53	6	no	—
252	C₂²⋊C₄	—	53	12	no	—
253	C₂²⋊C₄	—	53	6	no	—
254	C₄²	—	53	6	no	—
255	C₂³	—	2	6	no	—
256	C₂³	—	2	6	no	—
257	C₂³	—	4	3	no	—
258	C₂³	—	53	6	no	—
259	C₂³	—	106	12	no	—
260	C₂³	—	212	6	no	—
261	C₂³	—	212	6	no	—
262	C₂³	C₂³×D₅₃	1	3	yes	—
263	C₂³	C₂³×D₅₃	1	3	yes	—
264	C₂³	C₂³×D₅₃	1	1	yes	characteristic
265	Q₈	—	2	6	no	—
266	Q₈	—	106	2	no	—
267	Q₈	C₂×C₂²⋊D₅₃	1	4	yes	complemented
268	D₄	—	2	12	no	—
269	D₄	—	2	6	no	—
270	D₄	—	106	12	no	—
271	D₄	—	106	12	no	—
272	D₄	—	106	6	no	—
273	D₄	—	106	12	no	—
274	D₄	—	212	6	no	—
275	D₄	—	212	12	no	—
276	D₄	C₂×C₂²⋊D₅₃	1	12	yes	complemented
277	C₂×C₄	—	2	6	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	6	no	—
280	C₂×C₄	—	2	6	no	—
281	C₂×C₄	—	2	6	no	—
282	C₂×C₄	—	53	2	no	—
283	C₂×C₄	—	53	6	no	—
284	C₂×C₄	—	53	2	no	—
285	C₂×C₄	—	106	12	no	—
286	C₂×C₄	—	212	6	no	—
287	C₂×C₄	C₂×C₂²⋊D₅₃	1	12	yes	complemented
288	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₅₃	1	3	yes	—
290	C₂×C₄	C₂³×D₅₃	1	3	yes	—
291	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
292	C₂²	—	2	1	no	—
293	C₂²	—	2	3	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	3	no	—
296	C₂²	—	2	3	no	—
297	C₂²	—	2	3	no	—
298	C₂²	—	2	3	no	—
299	C₂²	—	4	12	no	—
300	C₂²	—	106	6	no	—
301	C₂²	—	106	6	no	—
302	C₂²	—	106	6	no	—
303	C₂²	—	424	12	no	—
304	C₂²	C₂²×C₂²⋊D₅₃	1	6	yes	—
305	C₂²	#13133263	1	2	yes	complemented
306	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	3	no	—
309	C₄	—	106	2	no	—
310	C₄	—	106	6	no	—
311	C₄	—	106	2	no	—
312	C₄	C₂²×C₂²⋊D₅₃	1	2	yes	—
313	C₄	C₂²×C₂²⋊D₅₃	1	6	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	2	6	no	—
316	C₂	—	4	3	no	—
317	C₂	—	212	6	no	—
318	C₂	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
319	C₂	C₂×#13133263	1	1	yes	characteristic
320	C₂	#16811459	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₂²×#41×C₅₃	C₂
hypercenter	C₂²×#41	D₅₃
hypocenter	C₅₃	#3488
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#13133263	C₂	semidirect
2. #16811292	C₂	semidirect
3. #16811284	C₂	semidirect
4. #16811261	C₂	semidirect
5. D₄×C₂²⋊D₅₃	C₂	semidirect
6. #16811201	C₂	semidirect
7. #16811187	C₂	semidirect
8. #16811134	C₂	semidirect
9. C₂×#13133191	C₂	semidirect
10. #16811106	C₂	semidirect
11. C₂²×#41×C₅₃	C₂	semidirect
13. C₂×#11719949	C₂²	semidirect
14. C₂×#11719945	C₂²	semidirect
15. C₂²×D₂₁₂	C₂²	semidirect
16. C₂²×C₂²⋊D₅₃	C₂²	semidirect
17. C₂×D₄×D₅₃	C₂²	semidirect
18. #13133213	C₂²	semidirect
19. #13133206	C₂²	semidirect
20. #13133200	C₂²	semidirect
21. #13133196	C₂²	semidirect
22. #13133196	C₂²	semidirect
23. #13133193	C₂²	semidirect
24. #13133191	C₂²	semidirect
25. #13133191	C₂²	semidirect
26. #13133191	C₂²	semidirect
27. #13133191	C₂²	semidirect
28. #13133191	C₂²	semidirect
29. #13133189	C₂²	semidirect
30. #13133188	C₂²	semidirect
31. #13133186	C₂²	semidirect
32. #13133185	C₂²	semidirect
33. Q₈×Dic₅₃	C₂²	semidirect
34. D₄×Dic₅₃	C₂²	semidirect
35. C₄×C₂²⋊D₅₃	C₂²	semidirect
36. C₄×C₂²⋊D₅₃	C₂²	semidirect
37. #13133176	C₂²	semidirect
38. C₂×C₄⋊Dic₅₃	C₂²	semidirect
39. C₂×#11719921	C₂²	semidirect
42. C₂×#41×C₅₃	C₂²	semidirect
43. C₂×#41×C₅₃	C₂²	semidirect
44. C₂×#41×C₅₃	C₂²	semidirect
45. C₂²×Q₈×C₅₃	C₂²	semidirect
46. C₂²×D₄×C₅₃	C₂²	semidirect
47. C₂³×C₄×C₅₃	C₂²	semidirect
53. #41×C₅₃	D₄	semidirect
105. Q₈×C₅₃	C₂×D₄	semidirect
108. D₄×C₅₃	C₂×D₄	semidirect
114. C₂×C₄×C₅₃	C₂×D₄	semidirect
140. C₂²×C₅₃	H(5,𝔽₂)	semidirect
162. C₂²×#41	D₅₃	semidirect
172. C₅₃	#3488	semidirect
178. C₂×#41	C₂×D₅₃	semidirect
179. C₂×#41	C₂×D₅₃	semidirect
180. C₂×#41	C₂×D₅₃	semidirect
181. C₂²×Q₈	C₂×D₅₃	semidirect
185. C₂²×D₄	C₂×D₅₃	semidirect
186. C₂³×C₄	C₂×D₅₃	semidirect
216. #41	C₂²⋊D₅₃	semidirect
267. Q₈	C₂×C₂²⋊D₅₃	semidirect
276. D₄	C₂×C₂²⋊D₅₃	semidirect
287. C₂×C₄	C₂×C₂²⋊D₅₃	semidirect
305. C₂²	#13133263	semidirect