Subgroups — #20283651

Order 6784.

348 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	#16811177	C₂	1	1	yes	maximal, characteristic
2	#16810754	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810753	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810752	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810752	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810751	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810751	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810750	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810648	C₂	1	1	yes	maximal, characteristic
10	#16810648	C₂	1	1	yes	maximal, characteristic
11	#16810645	C₂	1	1	yes	maximal, characteristic
12	#16810580	C₂	1	1	yes	maximal, characteristic
13	#16810577	C₂	1	1	yes	maximal, characteristic
14	#97×D₅₃	C₂	1	1	yes	maximal, characteristic
15	C₅₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11719912	C₂²	1	1	yes	characteristic
17	C₂×#11719912	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
20	#13133216	C₄	1	1	yes	characteristic
21	#13133215	C₄	1	1	yes	characteristic
22	#13133207	C₄	1	1	yes	characteristic
23	#13133207	C₄	1	1	yes	characteristic
24	#13133197	C₄	1	1	yes	characteristic
25	#13133195	C₄	1	1	yes	characteristic
26	#13133195	C₄	1	1	yes	characteristic
27	#13133193	C₄	1	1	yes	characteristic
28	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
31	#13133174	C₂²	1	1	yes	characteristic
32	#13133174	C₂²	1	1	yes	characteristic
33	C₄²×D₅₃	C₂²	1	1	yes	characteristic
34	C₄⋊#88322	C₂²	1	1	yes	characteristic
35	C₄⋊#88322	C₂²	1	1	yes	characteristic
36	C₄⋊#88322	C₂²	1	1	yes	characteristic
37	#13133146	C₂²	1	1	yes	characteristic
38	#13133146	C₂²	1	1	yes	characteristic
39	#13133146	C₂²	1	1	yes	characteristic
40	#13133146	C₂²	1	1	yes	characteristic
41	#13133146	C₂²	1	1	yes	characteristic
42	#13133146	C₂²	1	1	yes	characteristic
43	#13133128	C₂²	1	1	yes	characteristic
44	#13133128	C₂²	1	1	yes	characteristic
45	#13133127	C₂²	1	1	yes	characteristic
46	#13133127	C₂²	1	1	yes	characteristic
47	#13133127	C₂²	1	1	yes	characteristic
48	#13133127	C₂²	1	1	yes	characteristic
49	C₄×#88322	C₂²	1	1	yes	characteristic
50	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
52	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
56	#97×C₅₃	C₂²	1	1	yes	characteristic
57	#97×C₅₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
63	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
64	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
65	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	C₄²×C₅₃	C₂³	1	1	yes	characteristic
76	#11719921	C₂×C₄	1	1	yes	characteristic
77	#11719921	C₂×C₄	1	1	yes	characteristic
78	#11719921	C₂×C₄	1	1	yes	characteristic
79	#11719921	C₂×C₄	1	1	yes	characteristic
80	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
83	#11719919	C₂×C₄	1	1	yes	characteristic
84	#11719919	C₂×C₄	1	1	yes	characteristic
85	#11719919	C₂×C₄	1	1	yes	characteristic
86	#11719919	C₂×C₄	1	1	yes	characteristic
87	#11719919	C₂×C₄	1	1	yes	characteristic
88	#11719919	C₂×C₄	1	1	yes	characteristic
89	#11719919	C₂×C₄	1	1	yes	characteristic
90	#11719919	C₂×C₄	1	1	yes	characteristic
91	#11719919	C₂×C₄	1	1	yes	characteristic
92	#11719919	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
96	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
97	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
98	C₂×#88322	C₂³	1	1	yes	characteristic
99	C₂×#88322	C₂³	1	1	yes	characteristic
100	C₂×#88322	C₂³	1	1	yes	characteristic
101	C₂×#88322	C₂³	1	1	yes	characteristic
102	#11719912	—	2	2	no	—
103	#11719912	—	2	2	no	—
104	C₈×D₅₃	—	2	2	no	—
105	C₈×D₅₃	Q₈	1	4	yes	complemented
106	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
107	Q₈×C₅₃	—	4	1	no	—
108	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	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	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₄×D₅₃	—	4	1	no	—
123	C₄×D₅₃	—	4	1	no	—
124	C₄×D₅₃	C₂×Q₈	1	2	yes	—
125	C₄×D₅₃	#41	1	2	yes	—
126	Dic₁₀₆	—	4	1	no	—
127	Dic₁₀₆	—	4	1	no	—
128	Dic₁₀₆	—	4	1	no	—
129	C₈×C₅₃	—	2	1	no	—
130	C₈×C₅₃	—	2	1	no	—
131	C₈×C₅₃	—	2	1	no	—
132	C₈×C₅₃	C₂×Q₈	1	2	yes	—
133	#88322	—	2	1	no	—
134	#88322	—	2	1	no	—
135	#88322	—	2	1	no	—
136	#88322	C₂×Q₈	1	2	yes	—
137	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×D₅₃	—	2	1	no	—
139	C₂×D₅₃	—	2	1	no	—
140	C₂×D₅₃	C₄×Q₈	1	2	yes	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	2	1	no	—
144	C₄×C₅₃	—	2	1	no	—
145	C₄×C₅₃	—	2	1	no	—
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
148	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
149	Dic₅₃	—	2	1	no	—
150	Dic₅₃	—	2	1	no	—
151	Dic₅₃	—	2	1	no	—
152	Dic₅₃	—	2	1	no	—
153	Dic₅₃	—	2	1	no	—
154	Dic₅₃	—	2	1	no	—
155	Dic₅₃	C₄×Q₈	1	2	yes	—
156	#2961	—	53	1	no	maximal, Sylow
157	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
158	C₂×C₅₃	#568	1	1	yes	characteristic
159	C₂×C₅₃	#568	1	1	yes	characteristic
160	D₅₃	—	2	2	no	—
161	#533	—	53	1	no	—
162	#446	—	53	1	no	—
163	#446	—	53	1	no	—
164	#446	—	53	1	no	—
165	#446	—	53	1	no	—
166	#446	—	53	1	no	—
167	#446	—	53	1	no	—
168	#446	—	53	1	no	—
169	#446	D₅₃	1	1	yes	characteristic, complemented
170	#424	—	53	1	no	—
171	#424	—	53	1	no	—
172	#423	—	53	1	no	—
173	#405	—	53	1	no	—
174	#405	—	53	1	no	—
175	C₂×#97	—	53	1	no	—
176	C₅₃	#2961	1	1	yes	characteristic, complemented, Sylow
177	C₂×M₁₆	—	53	1	no	—
178	C₂×M₁₆	—	53	1	no	—
179	C₂²×C₈	—	53	1	no	—
180	C₂²×C₈	—	53	1	no	—
181	C₄⋊Q₈	—	53	1	no	—
182	C₄⋊Q₈	—	53	1	no	—
183	#125	—	53	1	no	—
184	#125	—	53	1	no	—
185	C₂²⋊Q₈	—	53	1	no	—
186	C₂²⋊Q₈	—	53	1	no	—
187	C₂²⋊Q₈	—	53	1	no	—
188	C₂²⋊Q₈	—	53	1	no	—
189	C₄×Q₈	—	53	1	no	—
190	C₄×Q₈	—	53	1	no	—
191	C₄×Q₈	—	53	1	no	—
192	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
193	#117	—	53	1	no	—
194	#117	—	53	1	no	—
195	C₂×C₄²	—	53	1	no	—
196	C₄⋊C₈	—	53	1	no	—
197	C₄⋊C₈	—	53	1	no	—
198	C₄⋊C₈	—	53	1	no	—
199	C₄⋊C₈	—	53	1	no	—
200	C₄⋊C₈	—	53	1	no	—
201	C₄⋊C₈	—	53	1	no	—
202	C₄⋊C₈	—	53	1	no	—
203	C₄⋊C₈	—	53	1	no	—
204	C₄⋊C₈	—	53	1	no	—
205	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
206	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
207	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
208	#97	—	53	1	no	—
209	#97	—	53	1	no	—
210	#97	—	53	1	no	—
211	#97	—	53	1	no	—
212	#97	—	53	1	no	—
213	#97	—	53	1	no	—
214	#97	C₂×D₅₃	1	1	yes	characteristic
215	#97	C₂×D₅₃	1	1	yes	characteristic
216	C₄×C₈	—	53	1	no	—
217	C₄×C₈	—	53	1	no	—
218	C₄×C₈	—	53	1	no	—
219	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
220	C₂×Q₈	—	53	1	no	—
221	C₂×Q₈	—	53	1	no	—
222	C₂×Q₈	—	53	1	no	—
223	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
224	C₂²×C₄	—	53	1	no	—
225	C₂²×C₄	—	53	1	no	—
226	C₂²×C₄	—	53	1	no	—
227	M₁₆	—	106	2	no	—
228	M₁₆	—	106	2	no	—
229	C₂×C₈	—	53	1	no	—
230	C₂×C₈	—	53	4	no	—
231	C₂×C₈	—	53	1	no	—
232	C₂×C₈	—	53	1	no	—
233	C₂×C₈	—	53	1	no	—
234	C₂×C₈	—	106	2	no	—
235	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
236	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
237	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
239	C₄⋊C₄	—	53	1	no	—
240	C₄⋊C₄	—	53	1	no	—
241	C₄⋊C₄	—	53	1	no	—
242	C₄⋊C₄	—	53	1	no	—
243	C₄⋊C₄	—	53	1	no	—
244	C₄⋊C₄	—	53	1	no	—
245	C₄⋊C₄	—	53	1	no	—
246	C₄⋊C₄	—	53	1	no	—
247	C₄⋊C₄	—	53	1	no	—
248	C₄⋊C₄	—	53	1	no	—
249	C₄⋊C₄	—	53	1	no	—
250	C₄⋊C₄	—	53	1	no	—
251	C₄⋊C₄	—	53	1	no	—
252	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
253	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
254	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
255	C₂²⋊C₄	—	53	1	no	—
256	C₂²⋊C₄	—	53	1	no	—
257	C₂²⋊C₄	—	53	1	no	—
258	C₂²⋊C₄	—	53	1	no	—
259	C₄²	—	53	1	no	—
260	C₄²	—	53	1	no	—
261	C₄²	—	53	1	no	—
262	C₄²	—	53	1	no	—
263	C₄²	—	53	1	no	—
264	C₄²	C₂²×D₅₃	1	1	yes	characteristic
265	C₄²	C₂²×D₅₃	1	1	yes	characteristic
266	C₄²	C₂²×D₅₃	1	1	yes	characteristic
267	C₂³	—	53	1	no	—
268	Q₈	—	4	1	no	—
269	Q₈	—	212	1	no	—
270	Q₈	—	212	1	no	—
271	Q₈	—	212	1	no	—
272	C₂×C₄	—	53	1	no	—
273	C₂×C₄	—	53	2	no	—
274	C₂×C₄	—	53	2	no	—
275	C₂×C₄	—	53	1	no	—
276	C₂×C₄	—	53	1	no	—
277	C₂×C₄	—	53	1	no	—
278	C₂×C₄	—	53	1	no	—
279	C₂×C₄	—	53	1	no	—
280	C₂×C₄	—	53	1	no	—
281	C₂×C₄	—	212	1	no	—
282	C₂×C₄	—	212	1	no	—
283	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	53	2	no	—
294	C₈	—	106	1	no	—
295	C₈	—	106	1	no	—
296	C₈	—	106	1	no	—
297	C₈	Q₈×D₅₃	1	2	yes	—
298	C₂²	—	53	2	no	—
299	C₂²	—	106	1	no	—
300	C₂²	—	106	1	no	—
301	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	53	2	no	—
309	C₄	—	106	1	no	—
310	C₄	—	106	1	no	—
311	C₄	—	106	1	no	—
312	C₄	—	106	1	no	—
313	C₄	—	106	1	no	—
314	C₄	—	106	1	no	—
315	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
316	C₄	#41×D₅₃	1	1	yes	characteristic
317	C₂	—	106	2	no	—
318	C₂	C₄×Q₈×D₅₃	1	1	yes	characteristic
319	C₂	#16811388	1	1	yes	characteristic
320	C₂	#16811389	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#446	C₂
hypercenter	#446	D₅₃
hypocenter	C₅₃	#2961
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #16810754	C₂	semidirect
3. #16810753	C₂	semidirect
4. #16810752	C₂	semidirect
5. #16810752	C₂	semidirect
6. #16810751	C₂	semidirect
7. #16810751	C₂	semidirect
8. #16810750	C₂	semidirect
15. C₅₃×#446	C₂	semidirect
105. C₈×D₅₃	Q₈	semidirect
169. #446	D₅₃	semidirect
176. C₅₃	#2961	semidirect