Subgroups — #20284775

Order 6784.

434 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811173	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810947	C₂	1	2	yes	maximal, complemented
3	#16810918	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810913	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810823	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810818	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810768	C₂	1	2	yes	maximal, complemented
8	C₄×#11719924	C₂	1	2	yes	maximal, complemented
9	C₄×#11719922	C₂	1	2	yes	maximal, complemented
10	#16810676	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	—	2	1	no	—
13	C₂×#11719924	C₂²	1	2	yes	—
14	C₂×#11719922	C₂²	1	2	yes	—
15	#13133217	C₂²	1	1	yes	characteristic
16	#13133214	C₂²	1	1	yes	characteristic
17	#13133205	—	2	1	no	—
18	#13133191	—	2	1	no	—
19	#13133191	C₂²	1	2	yes	—
20	C₄×D₂₁₂	—	2	1	no	—
21	C₄×D₂₁₂	C₂²	1	2	yes	—
22	#13133150	C₂²	1	2	yes	—
23	#13133148	C₂²	1	2	yes	—
24	C₄⋊#88322	C₂²	1	1	yes	characteristic
25	C₄⋊#88322	C₂²	1	1	yes	characteristic
26	#13133141	C₂²	1	2	yes	—
27	#13133138	C₂²	1	2	yes	—
28	#13133137	C₂²	1	2	yes	—
29	#13133137	C₂²	1	2	yes	—
30	C₂²⋊#88322	C₂²	1	2	yes	—
31	C₄×#88322	C₂²	1	2	yes	—
32	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
33	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×D₄×C₅₃	—	2	1	no	—
36	C₄×D₄×C₅₃	—	2	1	no	—
37	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
39	#117×C₅₃	—	2	1	no	—
40	#117×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	—	2	1	no	—
42	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
43	#41×C₅₃	—	2	2	no	—
44	#41×C₅₃	—	2	2	no	—
45	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₅₃	—	2	1	no	—
47	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₂²⋊D₅₃	—	2	2	no	—
55	#11719945	—	4	4	no	—
56	C₂×D₂₁₂	—	2	1	no	—
57	C₂×D₂₁₂	—	2	1	no	—
58	C₂×D₂₁₂	C₂³	1	2	yes	—
59	C₂×C₄×D₅₃	—	2	2	no	—
60	C₂×Dic₁₀₆	—	2	1	no	—
61	C₄⋊C₄×C₅₃	—	2	1	no	—
62	C₄⋊C₄×C₅₃	—	2	1	no	—
63	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
65	C₂²⋊C₄×C₅₃	—	2	1	no	—
66	C₂²⋊C₄×C₅₃	—	2	1	no	—
67	C₂²⋊C₄×C₅₃	—	2	1	no	—
68	C₄²×C₅₃	—	2	1	no	—
69	C₄²×C₅₃	—	2	1	no	—
70	C₄²×C₅₃	—	2	1	no	—
71	C₄²×C₅₃	D₄	1	1	yes	characteristic
72	C₄²×C₅₃	D₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	#11719924	—	2	4	no	—
76	#11719922	—	2	4	no	—
77	#11719921	—	2	2	no	—
78	#11719921	—	2	2	no	—
79	C₄⋊Dic₅₃	—	2	1	no	—
80	C₄⋊Dic₅₃	C₂³	1	2	yes	—
81	C₂×#88322	C₂³	1	2	yes	—
82	C₂×#88322	C₂³	1	2	yes	—
83	C₂³×C₅₃	—	2	1	no	—
84	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×D₅₃	—	2	2	no	—
86	Q₈×C₅₃	—	2	1	no	—
87	Q₈×C₅₃	#41	1	2	yes	complemented
88	D₄×C₅₃	—	2	4	no	—
89	D₄×C₅₃	—	2	1	no	—
90	D₄×C₅₃	#41	1	2	yes	complemented
91	C₂×C₄×C₅₃	—	2	4	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	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	—	2	1	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
107	C₂²⋊D₅₃	—	4	4	no	—
108	C₂×Dic₅₃	—	2	2	no	—
109	D₂₁₂	—	2	2	no	—
110	D₂₁₂	—	2	2	no	—
111	D₂₁₂	—	4	2	no	—
112	D₂₁₂	—	4	2	no	—
113	C₄×D₅₃	—	4	4	no	—
114	Dic₁₀₆	—	4	2	no	—
115	#88322	—	2	2	no	—
116	#88322	—	2	2	no	—
117	C₂²×C₅₃	—	2	1	no	—
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	—	2	1	no	—
121	C₂²×C₅₃	—	4	1	no	—
122	C₂²×C₅₃	—	4	1	no	—
123	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×D₅₃	—	4	2	no	—
125	C₂×D₅₃	—	4	2	no	—
126	C₂×D₅₃	—	4	2	no	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	4	1	no	—
132	C₄×C₅₃	—	4	1	no	—
133	C₄×C₅₃	#135	1	2	yes	—
134	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
135	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
136	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
137	Dic₅₃	—	4	2	no	—
138	#3106	—	53	1	no	maximal, Sylow
139	C₂×C₅₃	—	4	1	no	—
140	C₂×C₅₃	—	4	2	no	—
141	C₂×C₅₃	#525	1	1	yes	characteristic
142	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
143	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
144	D₅₃	—	8	2	no	—
145	#532	—	53	1	no	—
146	C₄×#41	D₅₃	1	1	yes	characteristic, complemented
147	#482	—	53	2	no	—
148	#477	—	53	1	no	—
149	#476	—	53	1	no	—
150	#460	—	53	1	no	—
151	#459	—	53	1	no	—
152	#449	—	53	2	no	—
153	C₄×SD₁₆	—	53	2	no	—
154	C₄×D₈	—	53	2	no	—
155	#431	—	53	1	no	—
156	C₅₃	#3106	1	1	yes	characteristic, complemented, Sylow
157	C₂×#41	—	106	1	no	—
158	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
159	C₂×SD₁₆	—	53	2	no	—
160	C₂×D₈	—	53	2	no	—
161	C₄⋊Q₈	—	53	1	no	—
162	#127	—	53	1	no	—
163	#124	—	106	1	no	—
164	#121	—	53	2	no	—
165	#121	—	106	1	no	—
166	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
167	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	—	53	2	no	—
171	C₄×D₄	—	106	1	no	—
172	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
173	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
174	#117	—	2	1	no	—
175	#117	C₂×D₅₃	1	1	yes	characteristic
176	C₂×C₄²	—	2	1	no	—
177	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
178	#107	—	53	2	no	—
179	#106	—	53	2	no	—
180	C₄⋊C₈	—	53	1	no	—
181	C₄⋊C₈	—	53	1	no	—
182	#103	—	53	2	no	—
183	#102	—	53	2	no	—
184	#102	—	53	2	no	—
185	#102	—	53	2	no	—
186	C₂²⋊C₈	—	53	2	no	—
187	C₄×C₈	—	53	2	no	—
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	#41	—	212	4	no	—
191	C₂×Q₈	—	106	1	no	—
192	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	—	53	2	no	—
195	C₂×D₄	—	106	2	no	—
196	C₂×D₄	—	106	1	no	—
197	C₂×D₄	—	106	1	no	—
198	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	106	2	no	—
203	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
206	SD₁₆	—	106	4	no	—
207	D₈	—	106	4	no	—
208	C₂×C₈	—	53	2	no	—
209	C₂×C₈	—	53	2	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	53	2	no	—
213	C₄⋊C₄	—	106	1	no	—
214	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
215	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	106	2	no	—
220	C₂²⋊C₄	—	106	2	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
225	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
226	C₄²	C₂²×D₅₃	1	1	yes	characteristic
227	C₄²	C₂²×D₅₃	1	1	yes	characteristic
228	C₂³	—	2	1	no	—
229	C₂³	—	106	2	no	—
230	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
231	Q₈	—	2	1	no	—
232	Q₈	—	212	2	no	—
233	Q₈	#11719945	1	2	yes	complemented
234	D₄	—	2	4	no	—
235	D₄	—	2	1	no	—
236	D₄	—	106	2	no	—
237	D₄	—	106	2	no	—
238	D₄	—	212	2	no	—
239	D₄	—	212	4	no	—
240	D₄	—	212	2	no	—
241	D₄	#11719945	1	2	yes	complemented
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	106	2	no	—
253	C₂×C₄	—	212	4	no	—
254	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
255	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
260	C₈	—	106	2	no	—
261	C₈	—	106	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	212	2	no	—
269	C₂²	—	212	2	no	—
270	C₂²	—	212	2	no	—
271	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	212	2	no	—
279	C₄	#13133237	1	2	yes	—
280	C₄	#13133242	1	2	yes	—
281	C₄	C₂×#11719945	1	1	yes	characteristic
282	C₄	C₂×#11719945	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	4	2	no	—
285	C₂	—	424	2	no	—
286	C₂	#16811134	1	1	yes	characteristic
287	C₂	C₂×#13133237	1	1	yes	characteristic
288	C₂	C₂×#13133242	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₄×#41×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₄×#41	D₅₃
hypocenter	C₅₃	#3106
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #16811173	C₂	semidirect
2. #16810947	C₂	semidirect
3. #16810918	C₂	semidirect
4. #16810913	C₂	semidirect
5. #16810823	C₂	semidirect
6. #16810818	C₂	semidirect
7. #16810768	C₂	semidirect
8. C₄×#11719924	C₂	semidirect
9. C₄×#11719922	C₂	semidirect
10. #16810676	C₂	semidirect
11. C₄×#41×C₅₃	C₂	semidirect
87. Q₈×C₅₃	#41	semidirect
90. D₄×C₅₃	#41	semidirect
146. C₄×#41	D₅₃	semidirect
156. C₅₃	#3106	semidirect
233. Q₈	#11719945	semidirect
241. D₄	#11719945	semidirect