Subgroups — #20284170

Order 6784.

496 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133237	C₂	1	2	yes	maximal, complemented
2	#16811173	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810845	C₂	1	2	yes	maximal, complemented
4	#16810838	C₂	1	2	yes	maximal, complemented
5	#16810833	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810828	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810823	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810818	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
10	#16810617	C₂	1	2	yes	maximal, complemented
11	C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	2	yes	—
13	#13133237	—	2	8	no	—
14	C₂×#11719925	C₂²	1	2	yes	—
15	C₂×#11719923	C₂²	1	2	yes	—
16	C₂×#11719924	C₂²	1	2	yes	—
17	C₂×#11719922	C₂²	1	2	yes	—
18	C₂²×#88322	C₂²	1	2	yes	—
19	#13133217	C₂²	1	1	yes	characteristic
20	#13133214	C₂²	1	1	yes	characteristic
21	#13133205	C₂²	1	2	yes	—
22	#13133191	—	2	1	no	—
23	#13133191	—	2	1	no	—
24	C₄×D₂₁₂	—	2	1	no	—
25	C₄×D₂₁₂	—	2	1	no	—
26	C₄⋊#88322	C₂²	1	1	yes	characteristic
27	C₄⋊#88322	C₂²	1	1	yes	characteristic
28	C₄⋊#88322	C₂²	1	2	yes	—
29	#13133140	C₂²	1	2	yes	—
30	#13133140	C₂²	1	2	yes	—
31	#13133137	C₂²	1	2	yes	—
32	#13133137	C₂²	1	2	yes	—
33	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₅₃	—	2	1	no	—
37	C₄×D₄×C₅₃	—	2	1	no	—
38	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
40	#117×C₅₃	—	2	1	no	—
41	#117×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	—	2	1	no	—
43	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
44	#41×C₅₃	—	2	2	no	—
45	#41×C₅₃	D₄	1	4	yes	complemented
46	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₅₃	—	2	1	no	—
48	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₂²⋊D₅₃	—	2	2	no	—
56	#11719945	—	2	4	no	—
57	#11719945	—	2	4	no	—
58	C₂×D₂₁₂	—	2	1	no	—
59	C₂×D₂₁₂	—	2	1	no	—
60	C₂×D₂₁₂	C₂³	1	2	yes	—
61	C₂×C₄×D₅₃	—	2	2	no	—
62	C₂×Dic₁₀₆	C₂³	1	2	yes	—
63	C₄⋊C₄×C₅₃	—	2	1	no	—
64	C₄⋊C₄×C₅₃	—	2	1	no	—
65	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂²⋊C₄×C₅₃	—	2	1	no	—
68	C₂²⋊C₄×C₅₃	—	2	1	no	—
69	C₂²⋊C₄×C₅₃	—	2	1	no	—
70	C₄²×C₅₃	—	2	1	no	—
71	C₄²×C₅₃	—	2	1	no	—
72	C₄²×C₅₃	—	2	1	no	—
73	C₄²×C₅₃	D₄	1	1	yes	characteristic
74	C₄²×C₅₃	D₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂³	1	1	yes	characteristic
76	C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	#11719925	—	2	4	no	—
78	#11719924	—	2	4	no	—
79	#11719923	—	2	4	no	—
80	#11719922	—	2	4	no	—
81	#11719921	—	2	2	no	—
82	#11719921	—	2	2	no	—
83	C₄⋊Dic₅₃	—	2	1	no	—
84	C₄⋊Dic₅₃	—	2	1	no	—
85	C₂×#88322	—	2	4	no	—
86	C₂×#88322	C₂³	1	2	yes	—
87	C₂×#88322	C₂³	1	2	yes	—
88	C₂³×C₅₃	—	2	1	no	—
89	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₅₃	—	2	2	no	—
91	Q₈×C₅₃	—	2	1	no	—
92	Q₈×C₅₃	C₂×D₄	1	2	yes	—
93	D₄×C₅₃	—	2	4	no	—
94	D₄×C₅₃	—	2	1	no	—
95	D₄×C₅₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	4	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	4	1	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	4	2	no	—
104	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#41	1	2	yes	—
111	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
112	C₂²⋊D₅₃	—	4	4	no	—
113	C₂×Dic₅₃	—	2	2	no	—
114	D₂₁₂	—	2	2	no	—
115	D₂₁₂	—	2	2	no	—
116	D₂₁₂	—	4	2	no	—
117	D₂₁₂	—	4	2	no	—
118	C₄×D₅₃	—	4	4	no	—
119	Dic₁₀₆	—	2	2	no	—
120	Dic₁₀₆	—	2	2	no	—
121	#88322	—	2	2	no	—
122	#88322	—	2	2	no	—
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	2	no	—
126	C₂²×C₅₃	—	4	1	no	—
127	C₂²×C₅₃	—	4	1	no	—
128	C₂²×C₅₃	#121	1	2	yes	—
129	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×D₅₃	—	4	2	no	—
131	C₂×D₅₃	—	4	2	no	—
132	C₂×D₅₃	—	4	2	no	—
133	C₄×C₅₃	—	2	2	no	—
134	C₄×C₅₃	—	2	1	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	2	1	no	—
137	C₄×C₅₃	—	4	1	no	—
138	C₄×C₅₃	—	4	1	no	—
139	C₄×C₅₃	#121	1	2	yes	—
140	C₄×C₅₃	#135	1	2	yes	complemented
141	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
143	Dic₅₃	—	4	2	no	—
144	#3035	—	53	1	no	maximal, Sylow
145	C₂×C₅₃	—	2	2	no	—
146	C₂×C₅₃	—	4	2	no	—
147	C₂×C₅₃	C₂×#121	1	1	yes	characteristic
148	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
149	C₂×C₅₃	#575	1	1	yes	characteristic
150	D₅₃	—	8	2	no	—
151	C₂×#135	—	53	2	no	—
152	#532	—	53	1	no	—
153	C₄×#41	D₅₃	1	1	yes	characteristic, complemented
154	#464	—	53	2	no	—
155	#463	—	53	2	no	—
156	#462	—	53	1	no	—
157	#461	—	53	1	no	—
158	#460	—	53	1	no	—
159	#459	—	53	1	no	—
160	C₂×C₄⋊C₈	—	53	1	no	—
161	#416	—	53	2	no	—
162	C₅₃	#3035	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	—	53	2	no	—
164	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
165	#135	—	106	8	no	—
166	C₂×Q₁₆	—	53	2	no	—
167	C₂×SD₁₆	—	53	2	no	—
168	C₂×SD₁₆	—	53	2	no	—
169	C₂×D₈	—	53	2	no	—
170	C₂²×C₈	—	53	2	no	—
171	C₄⋊Q₈	—	53	1	no	—
172	#127	—	53	1	no	—
173	#124	—	53	2	no	—
174	#121	—	106	1	no	—
175	#121	—	106	1	no	—
176	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
177	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	2	1	no	—
180	C₄×D₄	—	106	1	no	—
181	C₄×D₄	—	106	1	no	—
182	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
183	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
184	#117	—	2	1	no	—
185	#117	C₂×D₅₃	1	1	yes	characteristic
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
188	C₄⋊C₈	—	53	1	no	—
189	C₄⋊C₈	—	53	1	no	—
190	C₄⋊C₈	—	53	2	no	—
191	#103	—	53	2	no	—
192	#103	—	53	2	no	—
193	#102	—	53	2	no	—
194	#102	—	53	2	no	—
195	#41	—	2	2	no	—
196	#41	—	106	4	no	—
197	#41	—	106	4	no	—
198	#41	D₂₁₂	1	4	yes	complemented
199	C₂×Q₈	—	53	2	no	—
200	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	53	2	no	—
203	C₂×D₄	—	106	2	no	—
204	C₂×D₄	—	106	1	no	—
205	C₂×D₄	—	106	1	no	—
206	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	106	2	no	—
211	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
214	Q₁₆	—	106	4	no	—
215	SD₁₆	—	106	4	no	—
216	SD₁₆	—	106	4	no	—
217	D₈	—	106	4	no	—
218	C₂×C₈	—	53	2	no	—
219	C₂×C₈	—	53	2	no	—
220	C₂×C₈	—	106	4	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	106	1	no	—
224	C₄⋊C₄	—	106	1	no	—
225	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
226	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
227	C₂²⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	106	2	no	—
231	C₂²⋊C₄	—	106	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
236	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
237	C₄²	C₂²×D₅₃	1	1	yes	characteristic
238	C₄²	C₂²×D₅₃	1	1	yes	characteristic
239	C₂³	—	2	1	no	—
240	C₂³	—	106	2	no	—
241	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
242	Q₈	—	2	1	no	—
243	Q₈	—	106	2	no	—
244	Q₈	—	106	2	no	—
245	Q₈	C₂×D₂₁₂	1	2	yes	—
246	D₄	—	2	4	no	—
247	D₄	—	2	1	no	—
248	D₄	—	106	2	no	—
249	D₄	—	106	2	no	—
250	D₄	—	212	2	no	—
251	D₄	—	212	4	no	—
252	D₄	—	212	2	no	—
253	D₄	C₂×D₂₁₂	1	2	yes	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	106	2	no	—
263	C₂×C₄	—	212	4	no	—
264	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
265	C₂×C₄	#11719945	1	2	yes	—
266	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
271	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
272	C₈	—	106	2	no	—
273	C₈	—	106	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	212	2	no	—
280	C₂²	—	212	2	no	—
281	C₂²	—	212	2	no	—
282	C₂²	#13133191	1	2	yes	—
283	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	212	2	no	—
291	C₄	#13133191	1	2	yes	—
292	C₄	#13133237	1	2	yes	complemented
293	C₄	C₂²×D₂₁₂	1	1	yes	characteristic
294	C₄	C₂×#11719945	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	4	2	no	—
297	C₂	—	424	2	no	—
298	C₂	C₂×#13133191	1	1	yes	characteristic
299	C₂	C₂×#13133237	1	1	yes	characteristic
300	C₂	#16811421	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	#3035
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. C₂×#13133237	C₂	semidirect
2. #16811173	C₂	semidirect
3. #16810845	C₂	semidirect
4. #16810838	C₂	semidirect
5. #16810833	C₂	semidirect
6. #16810828	C₂	semidirect
7. #16810823	C₂	semidirect
8. #16810818	C₂	semidirect
9. C₂×C₄⋊#88322	C₂	semidirect
10. #16810617	C₂	semidirect
11. C₄×#41×C₅₃	C₂	semidirect
45. #41×C₅₃	D₄	semidirect
140. C₄×C₅₃	#135	semidirect
153. C₄×#41	D₅₃	semidirect
162. C₅₃	#3035	semidirect
198. #41	D₂₁₂	semidirect
292. C₄	#13133237	semidirect