Subgroups — #20283354

Order 6784.

400 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11719945	C₂	1	1	yes	maximal, characteristic
2	#16810683	C₂	1	2	yes	maximal, complemented
3	#16810681	C₂	1	2	yes	maximal, complemented
4	#16810676	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810674	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810674	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810648	C₂	1	2	yes	maximal, complemented
8	#16810588	C₂	1	2	yes	maximal
9	C₄×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	1	yes	characteristic
13	C₂×#11719912	C₂²	1	2	yes	—
14	C₂×C₈×D₅₃	C₂²	1	2	yes	—
15	C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
16	C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
17	C₄×D₂₁₂	C₄	1	1	yes	characteristic
18	C₄×D₂₁₂	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
22	C₄×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
23	#13133176	C₂²	1	1	yes	characteristic
24	#13133174	C₂²	1	1	yes	characteristic
25	#13133174	C₂²	1	1	yes	characteristic
26	C₄²×D₅₃	C₂²	1	1	yes	characteristic
27	C₄²×D₅₃	C₂²	1	1	yes	characteristic
28	C₄⋊#88322	C₂²	1	2	yes	—
29	#13133146	C₂²	1	2	yes	—
30	#13133146	C₂²	1	2	yes	—
31	C₂²⋊#88322	C₂²	1	2	yes	—
32	#13133129	C₂²	1	2	yes	—
33	#13133129	C₂²	1	2	yes	—
34	#13133127	C₂²	1	2	yes	—
35	C₄×#88322	C₂²	1	1	yes	characteristic
36	C₄×#88322	C₂²	1	1	yes	characteristic
37	C₈×Dic₅₃	C₂²	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
40	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
41	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
48	#11719945	—	2	4	no	—
49	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
56	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂³	1	2	yes	—
58	C₂×C₈×C₅₃	C₂³	1	2	yes	—
59	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂³	1	1	yes	characteristic
62	C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
65	#11719921	C₂×C₄	1	1	yes	characteristic
66	#11719921	C₂×C₄	1	1	yes	characteristic
67	#11719921	C₂×C₄	1	1	yes	characteristic
68	#11719921	C₂×C₄	1	1	yes	characteristic
69	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
71	#11719919	C₂×C₄	1	1	yes	characteristic
72	#11719919	C₂×C₄	1	1	yes	characteristic
73	#11719919	C₂×C₄	1	1	yes	characteristic
74	#11719919	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
76	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
80	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
81	C₂×#88322	C₂³	1	2	yes	—
82	C₂×#88322	C₂³	1	2	yes	—
83	#11719912	—	2	4	no	—
84	C₈×D₅₃	—	2	4	no	—
85	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₅₃	—	2	2	no	—
98	C₂²⋊D₅₃	—	2	2	no	—
99	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
105	D₂₁₂	—	2	2	no	—
106	C₄×D₅₃	—	2	2	no	—
107	C₄×D₅₃	—	2	2	no	—
108	C₄×D₅₃	—	2	2	no	—
109	C₄×D₅₃	—	2	2	no	—
110	C₄×D₅₃	—	2	2	no	—
111	C₄×D₅₃	#41	1	4	yes	—
112	Dic₁₀₆	—	2	2	no	—
113	C₈×C₅₃	—	2	2	no	—
114	C₈×C₅₃	—	2	2	no	—
115	#88322	—	2	2	no	—
116	#88322	—	2	2	no	—
117	C₂²×C₅₃	—	2	1	no	—
118	C₂²×C₅₃	—	2	1	no	—
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×D₅₃	—	2	1	no	—
122	C₂×D₅₃	—	2	1	no	—
123	C₂×D₅₃	—	2	1	no	—
124	C₂×D₅₃	—	2	1	no	—
125	C₂×D₅₃	—	2	1	no	—
126	C₂×D₅₃	#117	1	2	yes	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	#131	1	2	yes	—
131	C₄×C₅₃	#131	1	2	yes	—
132	C₄×C₅₃	C₂×#41	1	2	yes	—
133	Dic₅₃	—	2	1	no	—
134	Dic₅₃	—	2	1	no	—
135	Dic₅₃	—	2	1	no	—
136	Dic₅₃	—	2	1	no	—
137	Dic₅₃	—	2	1	no	—
138	Dic₅₃	#117	1	2	yes	—
139	#2922	—	53	1	no	maximal, Sylow
140	C₂×C₅₃	—	4	1	no	—
141	C₂×C₅₃	C₂×#117	1	1	yes	characteristic
142	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
143	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
144	D₅₃	—	2	2	no	—
145	D₅₃	—	4	1	no	—
146	C₄×#41	—	53	1	no	—
147	#433	—	53	2	no	—
148	#433	—	53	2	no	—
149	#431	—	53	1	no	—
150	#431	—	53	1	no	—
151	#431	—	53	1	no	—
152	#431	D₅₃	1	1	yes	characteristic, complemented
153	#424	—	53	2	no	—
154	#408	—	53	2	no	—
155	C₄×M₁₆	—	53	1	no	—
156	C₂×C₄×C₈	—	53	1	no	—
157	C₅₃	#2922	1	1	yes	characteristic, complemented, Sylow
158	C₂×#41	—	53	1	no	—
159	C₂×M₁₆	—	53	2	no	—
160	C₂²×C₈	—	53	2	no	—
161	C₄×Q₈	—	53	1	no	—
162	C₄×Q₈	—	53	1	no	—
163	C₄×D₄	—	53	1	no	—
164	C₄×D₄	—	53	1	no	—
165	C₄×D₄	—	53	1	no	—
166	C₄×D₄	—	53	1	no	—
167	C₄×D₄	—	53	1	no	—
168	C₄×D₄	—	53	1	no	—
169	#117	—	53	1	no	—
170	#117	—	53	1	no	—
171	#117	—	53	1	no	—
172	C₂×C₄²	—	53	1	no	—
173	C₂×C₄²	—	53	1	no	—
174	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
175	C₄⋊C₈	—	53	2	no	—
176	C₄⋊C₈	—	53	2	no	—
177	C₄⋊C₈	—	53	2	no	—
178	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
179	C₂²⋊C₈	—	53	2	no	—
180	C₂²⋊C₈	—	53	2	no	—
181	C₂²⋊C₈	—	53	2	no	—
182	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
183	#97	—	53	2	no	—
184	C₄×C₈	—	53	2	no	—
185	C₄×C₈	—	53	1	no	—
186	C₄×C₈	—	53	1	no	—
187	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
188	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
189	#41	—	106	4	no	—
190	C₂×Q₈	—	53	1	no	—
191	C₂×D₄	—	53	1	no	—
192	C₂×D₄	—	53	1	no	—
193	C₂×D₄	—	53	1	no	—
194	C₂²×C₄	—	53	1	no	—
195	C₂²×C₄	—	53	1	no	—
196	C₂²×C₄	—	53	1	no	—
197	C₂²×C₄	—	53	1	no	—
198	C₂²×C₄	—	53	1	no	—
199	C₂²×C₄	—	53	1	no	—
200	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
203	M₁₆	—	106	4	no	—
204	C₂×C₈	—	53	2	no	—
205	C₂×C₈	—	53	2	no	—
206	C₂×C₈	—	106	4	no	—
207	C₂×C₈	C₂²×D₅₃	1	2	yes	—
208	C₂×C₈	C₂²×D₅₃	1	2	yes	—
209	C₄⋊C₄	—	53	1	no	—
210	C₄⋊C₄	—	53	1	no	—
211	C₄⋊C₄	—	53	1	no	—
212	C₄⋊C₄	—	53	1	no	—
213	C₄⋊C₄	—	53	1	no	—
214	C₄⋊C₄	—	53	1	no	—
215	C₂²⋊C₄	—	53	1	no	—
216	C₂²⋊C₄	—	53	1	no	—
217	C₂²⋊C₄	—	53	1	no	—
218	C₂²⋊C₄	—	53	1	no	—
219	C₂²⋊C₄	—	53	1	no	—
220	C₂²⋊C₄	—	53	1	no	—
221	C₄²	—	53	1	no	—
222	C₄²	—	53	1	no	—
223	C₄²	—	53	1	no	—
224	C₄²	—	53	1	no	—
225	C₄²	—	53	1	no	—
226	C₄²	—	53	1	no	—
227	C₄²	C₄×D₅₃	1	1	yes	characteristic
228	C₄²	C₄×D₅₃	1	1	yes	characteristic
229	C₄²	C₂²×D₅₃	1	1	yes	characteristic
230	C₄²	C₂²×D₅₃	1	1	yes	characteristic
231	C₂³	—	53	1	no	—
232	C₂³	—	53	1	no	—
233	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
234	Q₈	—	106	2	no	—
235	D₄	—	106	2	no	—
236	D₄	—	106	2	no	—
237	D₄	—	106	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	53	1	no	—
242	C₂×C₄	—	53	4	no	—
243	C₂×C₄	—	53	1	no	—
244	C₂×C₄	—	53	1	no	—
245	C₂×C₄	—	53	1	no	—
246	C₂×C₄	—	53	1	no	—
247	C₂×C₄	—	53	1	no	—
248	C₂×C₄	—	106	2	no	—
249	C₂×C₄	—	106	2	no	—
250	C₂×C₄	—	106	2	no	—
251	C₂×C₄	—	106	2	no	—
252	C₂×C₄	—	106	2	no	—
253	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
258	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	2	2	no	—
261	C₈	—	106	2	no	—
262	C₈	—	106	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	53	2	no	—
267	C₂²	—	106	1	no	—
268	C₂²	—	106	1	no	—
269	C₂²	—	106	1	no	—
270	C₂²	—	106	1	no	—
271	C₂²	—	106	1	no	—
272	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	53	2	no	—
277	C₄	—	106	1	no	—
278	C₄	—	106	1	no	—
279	C₄	—	106	1	no	—
280	C₄	—	106	1	no	—
281	C₄	—	106	1	no	—
282	C₄	#13133223	1	2	yes	—
283	C₄	#13133225	1	2	yes	—
284	C₄	#41×D₅₃	1	2	yes	—
285	C₂	—	4	1	no	—
286	C₂	—	106	2	no	—
287	C₂	—	212	1	no	—
288	C₂	#117×D₅₃	1	1	yes	characteristic
289	C₂	C₂×#13133223	1	1	yes	characteristic
290	C₂	C₂×#13133225	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#431	C₂
hypercenter	#431	D₅₃
hypocenter	C₅₃	#2922
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
2. #16810683	C₂	semidirect
3. #16810681	C₂	semidirect
4. #16810676	C₂	semidirect
5. #16810674	C₂	semidirect
6. #16810674	C₂	semidirect
7. #16810648	C₂	semidirect
9. C₄×#11719912	C₂	semidirect
10. C₄×C₈×D₅₃	C₂	semidirect
11. C₅₃×#431	C₂	semidirect
152. #431	D₅₃	semidirect
157. C₅₃	#2922	semidirect