Subgroups — #20279301

Order 6784.

496 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133212	C₂	1	2	yes	maximal, complemented
2	#16810550	C₂	1	2	yes	maximal, complemented
3	#16810545	C₂	1	2	yes	maximal, complemented
4	#16810525	C₂	1	1	yes	maximal, characteristic
5	#16810525	C₂	1	1	yes	maximal, characteristic
6	#16810525	C₂	1	1	yes	maximal, characteristic
7	#16810521	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810515	C₂	1	2	yes	maximal, complemented
9	#16810493	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
13	#13133212	—	2	8	no	—
14	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
19	C₂×#11719919	C₂²	1	2	yes	—
20	C₂×#11719921	C₂²	1	2	yes	—
21	C₂×#11719921	C₂²	1	2	yes	—
22	C₂×#11719921	C₂²	1	2	yes	—
23	C₂×#11719921	C₂²	1	2	yes	—
24	C₂×#11719921	C₂²	1	2	yes	—
25	C₂×#11719921	C₂²	1	2	yes	—
26	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
27	#13133124	C₂²	1	2	yes	—
28	#13133124	C₂²	1	2	yes	—
29	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
34	#95×C₅₃	C₂²	1	1	yes	characteristic
35	#95×C₅₃	C₂²	1	1	yes	characteristic
36	C₂³×D₅₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₂³	1	2	yes	—
45	C₂²×Dic₅₃	C₂³	1	2	yes	—
46	C₂²×Dic₅₃	C₂³	1	2	yes	—
47	C₂×D₂₁₂	—	2	2	no	—
48	C₂×D₂₁₂	—	2	2	no	—
49	C₂×D₂₁₂	—	2	2	no	—
50	C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₄⋊C₄×C₅₃	D₄	1	4	yes	complemented
52	C₄²×C₅₃	—	2	2	no	—
53	C₄²×C₅₃	—	2	2	no	—
54	C₄²×C₅₃	—	2	2	no	—
55	#11719921	—	2	4	no	—
56	#11719921	—	2	4	no	—
57	#11719921	—	2	4	no	—
58	#11719921	—	2	4	no	—
59	#11719921	—	4	2	no	—
60	#11719921	—	4	2	no	—
61	C₄⋊Dic₅₃	—	2	4	no	—
62	C₄⋊Dic₅₃	—	2	2	no	—
63	C₄⋊Dic₅₃	—	2	2	no	—
64	C₄⋊Dic₅₃	—	2	2	no	—
65	C₄⋊Dic₅₃	—	4	2	no	—
66	#11719919	—	2	4	no	—
67	C₄×Dic₅₃	—	2	4	no	—
68	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₂²×D₅₃	—	2	2	no	—
70	C₂²×D₅₃	—	2	2	no	—
71	C₂²×D₅₃	—	2	2	no	—
72	C₂²×D₅₃	—	2	2	no	—
73	C₂²×D₅₃	—	2	2	no	—
74	C₂²×D₅₃	—	2	2	no	—
75	C₂²×D₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₅₃	#41	1	2	yes	—
94	C₂×C₄×C₅₃	#41	1	2	yes	—
95	C₂×C₄×C₅₃	#41	1	2	yes	—
96	C₂×C₄×C₅₃	#41	1	2	yes	—
97	C₂×Dic₅₃	—	2	2	no	—
98	C₂×Dic₅₃	—	2	2	no	—
99	C₂×Dic₅₃	—	2	2	no	—
100	C₂×Dic₅₃	—	2	2	no	—
101	C₂×Dic₅₃	—	2	2	no	—
102	C₂×Dic₅₃	—	2	2	no	—
103	C₂×Dic₅₃	—	2	2	no	—
104	C₂×Dic₅₃	—	2	2	no	—
105	C₂×Dic₅₃	—	2	2	no	—
106	D₂₁₂	—	4	4	no	—
107	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
108	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
109	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
110	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
111	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
112	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
114	C₂×D₅₃	—	4	2	no	—
115	C₂×D₅₃	—	4	2	no	—
116	C₂×D₅₃	—	4	2	no	—
117	C₂×D₅₃	—	4	2	no	—
118	C₂×D₅₃	—	4	2	no	—
119	C₂×D₅₃	—	4	2	no	—
120	C₂×D₅₃	—	4	2	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	—	2	2	no	—
124	C₄×C₅₃	—	2	2	no	—
125	C₄×C₅₃	—	4	1	no	—
126	C₄×C₅₃	—	4	1	no	—
127	C₄×C₅₃	#126	1	4	yes	complemented
128	Dic₅₃	—	4	2	no	—
129	Dic₅₃	—	4	2	no	—
130	Dic₅₃	—	4	2	no	—
131	#2512	—	53	1	no	maximal, Sylow
132	C₂×C₅₃	C₂×#126	1	1	yes	characteristic
133	C₂×C₅₃	#532	1	1	yes	characteristic
134	C₂×C₅₃	#542	1	1	yes	characteristic
135	C₂×C₅₃	#547	1	1	yes	characteristic
136	C₂×C₅₃	#550	1	1	yes	characteristic
137	C₂×C₅₃	#553	1	1	yes	characteristic
138	C₂×C₅₃	#556	1	1	yes	characteristic
139	D₅₃	—	8	2	no	—
140	C₂×#126	—	53	2	no	—
141	#397	—	53	2	no	—
142	#396	—	53	2	no	—
143	#390	—	53	1	no	—
144	#390	—	53	1	no	—
145	#390	—	53	1	no	—
146	#389	—	53	1	no	—
147	#388	—	53	2	no	—
148	#384	—	53	1	no	—
149	C₄×C₄⋊C₄	—	53	1	no	—
150	C₄×C₄⋊C₄	D₅₃	1	1	yes	characteristic, complemented
151	C₅₃	#2512	1	1	yes	characteristic, complemented, Sylow
152	C₂²×D₄	—	53	1	no	—
153	#126	—	106	8	no	—
154	C₂×C₄⋊C₄	—	53	2	no	—
155	C₂×C₄⋊C₄	—	53	2	no	—
156	C₂×C₄⋊C₄	—	53	1	no	—
157	C₂×C₄⋊C₄	—	53	1	no	—
158	C₂×C₄⋊C₄	—	53	1	no	—
159	C₂×C₄⋊C₄	—	53	2	no	—
160	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
161	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
162	C₂×C₂²⋊C₄	—	53	2	no	—
163	C₂×C₂²⋊C₄	—	53	2	no	—
164	C₂×C₂²⋊C₄	—	53	2	no	—
165	C₂×C₂²⋊C₄	—	53	2	no	—
166	C₂×C₂²⋊C₄	—	53	2	no	—
167	C₂×C₂²⋊C₄	—	53	2	no	—
168	C₂×C₄²	—	53	2	no	—
169	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
170	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
171	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
172	#95	—	53	2	no	—
173	#95	—	53	2	no	—
174	#95	C₂×D₅₃	1	1	yes	characteristic
175	#95	C₂×D₅₃	1	1	yes	characteristic
176	C₂⁴	—	53	2	no	—
177	C₂×D₄	—	106	2	no	—
178	C₂×D₄	—	106	2	no	—
179	C₂×D₄	—	106	2	no	—
180	C₂²×C₄	—	53	2	no	—
181	C₂²×C₄	—	53	2	no	—
182	C₂²×C₄	—	53	2	no	—
183	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
184	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
187	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
188	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	106	4	no	—
192	C₄⋊C₄	—	106	4	no	—
193	C₄⋊C₄	—	106	2	no	—
194	C₄⋊C₄	—	106	2	no	—
195	C₄⋊C₄	—	106	2	no	—
196	C₄⋊C₄	—	212	2	no	—
197	C₄⋊C₄	D₂₁₂	1	4	yes	complemented
198	C₂²⋊C₄	—	106	4	no	—
199	C₂²⋊C₄	—	106	4	no	—
200	C₂²⋊C₄	—	106	4	no	—
201	C₂²⋊C₄	—	106	4	no	—
202	C₂²⋊C₄	—	212	2	no	—
203	C₂²⋊C₄	—	212	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	106	4	no	—
208	C₂³	—	106	2	no	—
209	C₂³	—	106	2	no	—
210	C₂³	—	106	2	no	—
211	C₂³	—	106	2	no	—
212	C₂³	—	106	2	no	—
213	C₂³	—	106	2	no	—
214	C₂³	—	106	2	no	—
215	C₂³	C₂³×D₅₃	1	1	yes	characteristic
216	D₄	—	212	4	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	106	2	no	—
232	C₂×C₄	—	106	2	no	—
233	C₂×C₄	—	106	2	no	—
234	C₂×C₄	—	106	2	no	—
235	C₂×C₄	—	106	2	no	—
236	C₂×C₄	—	106	2	no	—
237	C₂×C₄	—	106	2	no	—
238	C₂×C₄	—	106	2	no	—
239	C₂×C₄	—	106	2	no	—
240	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
241	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
242	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
243	C₂×C₄	#11719945	1	2	yes	—
244	C₂×C₄	#11719945	1	2	yes	—
245	C₂×C₄	#11719947	1	2	yes	—
246	C₂×C₄	#11719949	1	2	yes	—
247	C₂²	—	212	2	no	—
248	C₂²	—	212	2	no	—
249	C₂²	—	212	2	no	—
250	C₂²	—	212	2	no	—
251	C₂²	—	212	2	no	—
252	C₂²	—	212	2	no	—
253	C₂²	—	212	2	no	—
254	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
255	C₂²	C₂×#11719947	1	1	yes	characteristic
256	C₂²	C₂×#11719945	1	1	yes	characteristic
257	C₂²	C₂×#11719945	1	1	yes	characteristic
258	C₂²	C₂×#11719949	1	1	yes	characteristic
259	C₂²	#13133263	1	1	yes	characteristic
260	C₂²	#13133264	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	4	1	no	—
266	C₄	—	4	1	no	—
267	C₄	—	212	2	no	—
268	C₄	—	212	2	no	—
269	C₄	—	212	2	no	—
270	C₄	#13133212	1	4	yes	complemented
271	C₂	—	424	2	no	—
272	C₂	C₂×#13133212	1	1	yes	characteristic
273	C₂	#16811173	1	1	yes	characteristic
274	C₂	#16811244	1	1	yes	characteristic
275	C₂	#16811277	1	1	yes	characteristic
276	C₂	#16811295	1	1	yes	characteristic
277	C₂	#16811319	1	1	yes	characteristic
278	C₂	#16811346	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₄×C₄⋊C₄×C₅₃	C₂
hypercenter	C₄×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#2512
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#13133212	C₂	semidirect
2. #16810550	C₂	semidirect
3. #16810545	C₂	semidirect
7. #16810521	C₂	semidirect
8. #16810515	C₂	semidirect
9. #16810493	C₂	semidirect
10. C₄×C₄⋊Dic₅₃	C₂	semidirect
11. C₄×C₄⋊C₄×C₅₃	C₂	semidirect
51. C₄⋊C₄×C₅₃	D₄	semidirect
127. C₄×C₅₃	#126	semidirect
150. C₄×C₄⋊C₄	D₅₃	semidirect
151. C₅₃	#2512	semidirect
197. C₄⋊C₄	D₂₁₂	semidirect
270. C₄	#13133212	semidirect