Subgroups — #20291550

Order 6784.

370 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811675	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811672	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811657	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811656	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811656	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811653	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811652	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811650	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811648	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811647	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811641	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811588	C₂	1	1	yes	maximal, characteristic, complemented
13	#16811587	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#11719936	C₂	1	1	yes	maximal, characteristic, complemented
15	#16811156	C₂	1	1	yes	maximal, characteristic
16	C₂×#11719936	C₂²	1	1	yes	characteristic
17	C₂×#11719936	C₂²	1	1	yes	characteristic
18	C₂×#11719936	C₂²	1	1	yes	characteristic
19	C₂×#11719940	C₂²	1	1	yes	characteristic
20	C₂×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₂²×#88324	C₂²	1	1	yes	characteristic
22	#13133284	C₂²	1	1	yes	characteristic
23	#13133284	C₂²	1	1	yes	characteristic
24	#13133284	C₂²	1	1	yes	characteristic
25	#13133284	C₂²	1	1	yes	characteristic
26	C₄⋊#88324	C₂²	1	1	yes	characteristic
27	C₄⋊#88324	C₂²	1	1	yes	characteristic
28	#13133274	C₂²	1	1	yes	characteristic
29	#13133274	C₂²	1	1	yes	characteristic
30	#13133274	C₂²	1	1	yes	characteristic
31	#13133274	C₂²	1	1	yes	characteristic
32	C₂²⋊#88324	C₂²	1	1	yes	characteristic
33	C₂²⋊#88324	C₂²	1	1	yes	characteristic
34	#13133272	C₂²	1	1	yes	characteristic
35	#13133272	C₂²	1	1	yes	characteristic
36	#13133272	C₂²	1	1	yes	characteristic
37	#13133272	C₂²	1	1	yes	characteristic
38	C₄×#88324	C₂²	1	1	yes	characteristic
39	C₄×#88324	C₂²	1	1	yes	characteristic
40	C₄×#88324	C₂²	1	1	yes	characteristic
41	C₄×#88324	C₂²	1	1	yes	characteristic
42	C₄×#88324	C₂²	1	1	yes	characteristic
43	C₄×#88324	C₂²	1	1	yes	characteristic
44	#13133211	C₄	1	1	yes	characteristic
45	#13133207	C₄	1	1	yes	characteristic
46	#13133203	C₄	1	1	yes	characteristic
47	#13133202	C₄	1	1	yes	characteristic
48	#13133199	C₄	1	1	yes	characteristic
49	#13133195	C₄	1	1	yes	characteristic
50	#13133190	C₄	1	1	yes	characteristic
51	#13133182	C₂²	1	1	yes	characteristic
52	#13133179	C₂²	1	1	yes	characteristic
53	#13133177	C₂²	1	1	yes	characteristic
54	#13133177	C₂²	1	1	yes	characteristic
55	#13133175	C₂²	1	1	yes	characteristic
56	#13133173	C₂²	1	1	yes	characteristic
57	C₄²×D₅₃	C₂²	1	1	yes	characteristic
58	#126×C₅₃	C₄	1	1	yes	characteristic
59	C₂×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
62	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
67	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
68	#11719940	—	2	2	no	—
69	C₂×#88324	—	2	2	no	—
70	C₂×#88324	C₂³	1	1	yes	characteristic
71	C₂×#88324	C₂³	1	1	yes	characteristic
72	C₂×#88324	C₂³	1	1	yes	characteristic
73	C₂×#88324	C₂³	1	1	yes	characteristic
74	C₂×#88324	C₂³	1	1	yes	characteristic
75	C₂×#88324	C₂³	1	1	yes	characteristic
76	C₂×#88324	C₂³	1	1	yes	characteristic
77	C₂×#88324	C₂³	1	1	yes	characteristic
78	#11719936	—	2	2	no	—
79	#11719936	—	2	2	no	—
80	#11719936	—	2	2	no	—
81	D₅₃⋊C₈	—	2	2	no	—
82	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
89	C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
90	#11719921	C₂×C₄	1	1	yes	characteristic
91	#11719921	C₂×C₄	1	1	yes	characteristic
92	#11719921	C₂×C₄	1	1	yes	characteristic
93	#11719921	C₂×C₄	1	1	yes	characteristic
94	#11719921	C₂×C₄	1	1	yes	characteristic
95	#11719921	C₂×C₄	1	1	yes	characteristic
96	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
97	#11719919	C₂×C₄	1	1	yes	characteristic
98	#11719919	C₂×C₄	1	1	yes	characteristic
99	#11719919	C₂×C₄	1	1	yes	characteristic
100	#11719919	C₂×C₄	1	1	yes	characteristic
101	#11719919	C₂×C₄	1	1	yes	characteristic
102	#11719919	C₂×C₄	1	1	yes	characteristic
103	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
104	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
105	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
106	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
107	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
108	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂²⋊D₅₃	—	2	2	no	—
118	C₂²⋊D₅₃	—	4	1	no	—
119	C₂×Dic₅₃	—	2	1	no	—
120	C₂×Dic₅₃	—	2	1	no	—
121	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
127	D₂₁₂	—	4	1	no	—
128	C₄×D₅₃	—	2	2	no	—
129	C₄×D₅₃	—	2	1	no	—
130	C₄×D₅₃	—	2	1	no	—
131	C₄×D₅₃	—	2	1	no	—
132	C₄×D₅₃	—	2	1	no	—
133	C₄×D₅₃	—	4	1	no	—
134	Dic₁₀₆	—	2	2	no	—
135	#88324	—	2	1	no	—
136	#88324	—	2	1	no	—
137	#88324	—	2	1	no	—
138	#88324	—	2	1	no	—
139	#88324	—	2	1	no	—
140	#88324	—	2	1	no	—
141	#88324	#41	1	2	yes	—
142	#88324	#41	1	2	yes	—
143	C₂²×C₅₃	—	2	1	no	—
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
147	C₂×D₅₃	—	2	1	no	—
148	C₂×D₅₃	—	2	1	no	—
149	C₂×D₅₃	—	2	1	no	—
150	C₂×D₅₃	—	2	1	no	—
151	C₂×D₅₃	—	2	1	no	—
152	C₂×D₅₃	—	2	1	no	—
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	2	1	no	—
159	Dic₅₃	—	2	1	no	—
160	Dic₅₃	—	2	1	no	—
161	Dic₅₃	—	2	1	no	—
162	Dic₅₃	—	2	1	no	—
163	Dic₅₃	#131	1	2	yes	—
164	Dic₅₃	C₂×#41	1	1	yes	characteristic
165	Dic₅₃	C₂×#41	1	1	yes	characteristic
166	#2969	—	53	1	no	maximal, Sylow
167	C₂×C₅₃	—	4	1	no	—
168	C₂×C₅₃	C₄×#41	1	1	yes	characteristic
169	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
170	C₂×C₅₃	#568	1	1	yes	characteristic
171	D₅₃	—	4	1	no	—
172	D₅₃	—	4	1	no	—
173	#529	—	53	1	no	—
174	#446	—	53	1	no	—
175	C₈×Q₈	—	53	1	no	—
176	#436	—	53	1	no	—
177	#436	—	53	1	no	—
178	#436	—	53	1	no	—
179	#435	—	53	1	no	—
180	#435	—	53	1	no	—
181	C₈×D₄	—	53	1	no	—
182	#433	—	53	1	no	—
183	#433	—	53	1	no	—
184	#431	—	53	1	no	—
185	#405	—	53	1	no	—
186	#405	—	53	1	no	—
187	C₄×M₁₆	—	53	1	no	—
188	C₅₃	#2969	1	1	yes	characteristic, complemented, Sylow
189	C₂×M₁₆	—	53	1	no	—
190	C₂×M₁₆	—	53	1	no	—
191	C₂×M₁₆	—	53	1	no	—
192	C₂×M₁₆	—	53	1	no	—
193	C₂²×C₈	—	53	1	no	—
194	C₂²×C₈	—	53	1	no	—
195	#126	—	53	1	no	—
196	#126	#6157	1	1	yes	characteristic
197	#125	—	53	1	no	—
198	#124	—	53	1	no	—
199	#123	—	53	1	no	—
200	#123	—	53	1	no	—
201	C₂²⋊Q₈	—	53	1	no	—
202	#121	—	53	1	no	—
203	C₄×Q₈	—	53	1	no	—
204	C₄×D₄	—	53	1	no	—
205	C₄×D₄	—	53	1	no	—
206	C₄×D₄	—	53	1	no	—
207	#117	—	53	1	no	—
208	#117	—	53	1	no	—
209	C₂×C₄²	—	53	1	no	—
210	C₄⋊C₈	—	53	1	no	—
211	C₄⋊C₈	—	53	1	no	—
212	C₄⋊C₈	—	53	1	no	—
213	C₄⋊C₈	—	53	1	no	—
214	C₄⋊C₈	—	53	1	no	—
215	C₄⋊C₈	—	53	1	no	—
216	C₂²⋊C₈	—	53	1	no	—
217	C₂²⋊C₈	—	53	1	no	—
218	C₂²⋊C₈	—	53	1	no	—
219	C₂²⋊C₈	—	53	1	no	—
220	C₂²⋊C₈	—	53	1	no	—
221	C₂²⋊C₈	—	53	1	no	—
222	#97	—	53	1	no	—
223	#97	—	53	1	no	—
224	#97	—	53	1	no	—
225	#97	—	53	1	no	—
226	C₄×C₈	—	53	1	no	—
227	C₄×C₈	—	53	1	no	—
228	C₄×C₈	—	53	1	no	—
229	C₄×C₈	—	53	1	no	—
230	C₄×C₈	—	53	1	no	—
231	C₄×C₈	—	53	1	no	—
232	C₂×Q₈	—	53	1	no	—
233	C₂×D₄	—	53	1	no	—
234	C₂×D₄	—	53	1	no	—
235	C₂×D₄	—	53	1	no	—
236	C₂²×C₄	—	53	1	no	—
237	C₂²×C₄	—	53	1	no	—
238	C₂²×C₄	—	53	1	no	—
239	C₂²×C₄	—	53	1	no	—
240	C₂²×C₄	—	53	1	no	—
241	M₁₆	—	106	2	no	—
242	M₁₆	—	106	2	no	—
243	M₁₆	—	106	2	no	—
244	M₁₆	—	106	2	no	—
245	C₂×C₈	—	53	1	no	—
246	C₂×C₈	—	53	1	no	—
247	C₂×C₈	—	53	1	no	—
248	C₂×C₈	—	53	1	no	—
249	C₂×C₈	—	53	1	no	—
250	C₂×C₈	—	53	1	no	—
251	C₂×C₈	—	53	1	no	—
252	C₂×C₈	—	53	1	no	—
253	C₂×C₈	—	106	2	no	—
254	C₂×C₈	—	106	2	no	—
255	C₄⋊C₄	—	53	1	no	—
256	C₄⋊C₄	—	53	1	no	—
257	C₄⋊C₄	—	53	1	no	—
258	C₄⋊C₄	—	53	1	no	—
259	C₄⋊C₄	—	53	1	no	—
260	C₄⋊C₄	—	53	1	no	—
261	C₄⋊C₄	—	53	1	no	—
262	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
263	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
264	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
265	C₂²⋊C₄	—	53	1	no	—
266	C₂²⋊C₄	—	53	1	no	—
267	C₂²⋊C₄	—	53	1	no	—
268	C₂²⋊C₄	—	53	1	no	—
269	C₂²⋊C₄	—	53	1	no	—
270	C₂²⋊C₄	—	53	1	no	—
271	C₂²⋊C₄	—	53	1	no	—
272	C₂²⋊C₄	C₂×#6157	1	1	yes	characteristic
273	C₂²⋊C₄	C₂×#6157	1	1	yes	characteristic
274	C₂²⋊C₄	C₂×#6157	1	1	yes	characteristic
275	C₄²	—	53	1	no	—
276	C₄²	—	53	1	no	—
277	C₄²	—	53	1	no	—
278	C₄²	—	53	1	no	—
279	C₄²	—	53	1	no	—
280	C₄²	C₂×#6157	1	1	yes	characteristic
281	C₂³	—	53	1	no	—
282	C₂³	—	53	1	no	—
283	C₂³	C₂²×#6157	1	1	yes	characteristic
284	Q₈	—	106	2	no	—
285	D₄	—	106	2	no	—
286	D₄	—	212	1	no	—
287	D₄	—	212	1	no	—
288	C₂×C₄	—	53	1	no	—
289	C₂×C₄	—	53	1	no	—
290	C₂×C₄	—	53	1	no	—
291	C₂×C₄	—	53	1	no	—
292	C₂×C₄	—	53	1	no	—
293	C₂×C₄	—	53	1	no	—
294	C₂×C₄	—	106	1	no	—
295	C₂×C₄	—	106	1	no	—
296	C₂×C₄	—	106	1	no	—
297	C₂×C₄	—	106	1	no	—
298	C₂×C₄	—	106	2	no	—
299	C₂×C₄	—	106	1	no	—
300	C₂×C₄	—	106	1	no	—
301	C₂×C₄	—	212	1	no	—
302	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
303	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
304	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
305	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
306	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
307	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
308	C₈	—	53	2	no	—
309	C₈	—	53	2	no	—
310	C₈	—	106	1	no	—
311	C₈	—	106	1	no	—
312	C₈	—	106	1	no	—
313	C₈	—	106	1	no	—
314	C₈	—	106	1	no	—
315	C₈	—	106	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	106	1	no	—
320	C₂²	—	106	1	no	—
321	C₂²	—	106	1	no	—
322	C₂²	—	106	1	no	—
323	C₂²	—	106	1	no	—
324	C₂²	—	106	1	no	—
325	C₂²	C₂³×#6157	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	53	1	no	—
333	C₄	—	53	1	no	—
334	C₄	—	53	2	no	—
335	C₄	—	106	1	no	—
336	C₄	—	106	1	no	—
337	C₄	—	106	1	no	—
338	C₄	—	106	1	no	—
339	C₂	—	4	1	no	—
340	C₂	—	212	1	no	—
341	C₂	—	212	1	no	—
342	C₂	#41×#6157	1	1	yes	characteristic
343	C₂	D₅₃⋊#131	1	1	yes	characteristic
344	C₂	#16811702	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	#126×C₅₃	C₄
hypercenter	#126	#6157
hypocenter	C₅₃	#2969
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #126

Complements

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

N	H ≅ G/N	type
1. #16811675	C₂	semidirect
2. #16811672	C₂	semidirect
3. #16811657	C₂	semidirect
4. #16811656	C₂	semidirect
5. #16811656	C₂	semidirect
6. #16811653	C₂	semidirect
7. #16811652	C₂	semidirect
8. #16811650	C₂	semidirect
9. #16811648	C₂	semidirect
10. #16811647	C₂	semidirect
11. #16811641	C₂	semidirect
12. #16811588	C₂	semidirect
13. #16811587	C₂	semidirect
14. C₄×#11719936	C₂	semidirect
188. C₅₃	#2969	semidirect