Subgroups — #20287118

Order 6784.

366 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811319	C₂	1	2	yes	maximal, complemented
2	#16811025	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810948	C₂	1	2	yes	maximal, complemented
4	#16810942	C₂	1	2	yes	maximal, complemented
5	#16810924	C₂	1	2	yes	maximal, complemented
6	#16810859	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810859	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810661	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133151	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
13	#13133212	—	2	2	no	—
14	#13133209	C₂²	1	2	yes	—
15	#13133199	—	2	2	no	—
16	#13133191	C₂²	1	2	yes	—
17	#13133191	C₂²	1	2	yes	—
18	#13133189	—	2	2	no	—
19	C₄×D₂₁₂	C₂²	1	2	yes	—
20	#13133177	—	2	2	no	—
21	#13133176	C₂²	1	2	yes	—
22	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
23	#13133151	C₂²	1	1	yes	characteristic
24	#13133151	C₂²	1	1	yes	characteristic
25	#13133151	C₂²	1	1	yes	characteristic
26	#13133151	C₂²	1	2	yes	—
27	#13133149	C₂²	1	2	yes	—
28	#13133149	C₂²	1	2	yes	—
29	#13133139	C₂²	1	2	yes	—
30	#13133139	C₂²	1	2	yes	—
31	#13133139	C₂²	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₂²⋊D₅₃	—	2	2	no	—
43	C₂×C₂²⋊D₅₃	—	2	2	no	—
44	C₂²×Dic₅₃	—	2	2	no	—
45	C₂×D₂₁₂	C₂³	1	2	yes	—
46	C₂×C₄×D₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
54	C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
55	C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₂²⋊Dic₅₃	—	2	2	no	—
58	#11719921	—	2	2	no	—
59	#11719921	—	2	2	no	—
60	#11719921	—	2	2	no	—
61	C₄⋊Dic₅₃	—	2	2	no	—
62	C₄⋊Dic₅₃	C₂³	1	2	yes	—
63	C₄⋊Dic₅₃	C₂³	1	2	yes	—
64	C₄⋊Dic₅₃	C₂³	1	2	yes	—
65	#11719919	—	2	2	no	—
66	#11719919	—	2	2	no	—
67	C₄×Dic₅₃	—	2	2	no	—
68	D₄₂₄	—	2	2	no	—
69	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×D₅₃	—	2	2	no	—
72	D₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
82	C₂²⋊D₅₃	—	4	4	no	—
83	C₂²⋊D₅₃	—	8	2	no	—
84	C₂×Dic₅₃	—	2	2	no	—
85	C₂×Dic₅₃	—	2	2	no	—
86	C₂×Dic₅₃	—	2	2	no	—
87	C₂×Dic₅₃	—	8	2	no	—
88	D₂₁₂	—	2	2	no	—
89	D₂₁₂	—	2	2	no	—
90	C₄×D₅₃	—	8	2	no	—
91	C₈×C₅₃	—	2	1	no	—
92	C₈×C₅₃	—	2	1	no	—
93	C₈×C₅₃	—	2	1	no	—
94	C₈×C₅₃	#41	1	2	yes	—
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	—	2	1	no	—
98	C₂²×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	—	2	1	no	—
100	C₂²×C₅₃	#135	1	2	yes	—
101	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×D₅₃	—	4	2	no	—
103	C₂×D₅₃	—	4	2	no	—
104	C₂×D₅₃	—	4	2	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
110	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
111	Dic₅₃	—	4	2	no	—
112	Dic₅₃	—	4	2	no	—
113	Dic₅₃	—	4	2	no	—
114	#3339	—	53	1	no	maximal, Sylow
115	C₂×C₅₃	—	2	2	no	—
116	C₂×C₅₃	—	4	1	no	—
117	C₂×C₅₃	#547	1	1	yes	characteristic
118	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
119	C₂×C₅₃	#576	1	1	yes	characteristic
120	D₅₃	—	8	2	no	—
121	#553	—	53	2	no	—
122	#499	—	53	1	no	—
123	#482	—	53	2	no	—
124	#481	—	53	2	no	—
125	#478	—	53	2	no	—
126	#466	—	53	1	no	—
127	#466	—	53	1	no	—
128	C₄×D₈	—	53	1	no	—
129	C₈×D₄	D₅₃	1	1	yes	characteristic, complemented
130	#427	—	53	1	no	—
131	C₂×#107	—	53	1	no	—
132	C₅₃	#3339	1	1	yes	characteristic, complemented, Sylow
133	C₂×D₈	—	53	1	no	—
134	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
135	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
136	#126	—	106	2	no	—
137	#125	—	53	2	no	—
138	#123	—	106	2	no	—
139	#121	—	53	2	no	—
140	#121	—	53	2	no	—
141	#121	—	106	2	no	—
142	C₄×D₄	—	53	2	no	—
143	C₄×D₄	—	106	2	no	—
144	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
145	#117	—	53	2	no	—
146	C₂×C₄⋊C₄	—	53	2	no	—
147	#107	—	53	1	no	—
148	#107	—	53	1	no	—
149	#107	—	53	2	no	—
150	#107	—	53	1	no	—
151	#106	—	53	2	no	—
152	#106	—	53	2	no	—
153	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
154	#102	—	53	2	no	—
155	#102	—	53	2	no	—
156	#102	—	53	2	no	—
157	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
158	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
159	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
160	C₂×D₄	—	53	2	no	—
161	C₂×D₄	—	106	2	no	—
162	C₂×D₄	—	106	2	no	—
163	C₂×D₄	D₂₁₂	1	1	yes	characteristic
164	C₂²×C₄	—	106	2	no	—
165	C₂²×C₄	—	106	2	no	—
166	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
167	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
168	D₈	—	106	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
174	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
175	C₄⋊C₄	—	53	2	no	—
176	C₄⋊C₄	—	53	2	no	—
177	C₄⋊C₄	—	53	2	no	—
178	C₄⋊C₄	—	106	2	no	—
179	C₄⋊C₄	—	106	2	no	—
180	C₄⋊C₄	—	106	2	no	—
181	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
182	C₂²⋊C₄	—	106	2	no	—
183	C₂²⋊C₄	—	106	2	no	—
184	C₂²⋊C₄	—	106	2	no	—
185	C₂²⋊C₄	—	106	2	no	—
186	C₂²⋊C₄	D₂₁₂	1	1	yes	characteristic
187	C₂²⋊C₄	D₂₁₂	1	1	yes	characteristic
188	C₄²	—	106	2	no	—
189	C₄²	C₂²×D₅₃	1	1	yes	characteristic
190	C₂³	—	106	2	no	—
191	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
192	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
193	D₄	—	2	2	no	—
194	D₄	—	106	2	no	—
195	D₄	—	106	2	no	—
196	D₄	—	424	2	no	—
197	D₄	—	212	4	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	106	2	no	—
203	C₂×C₄	—	106	2	no	—
204	C₂×C₄	—	106	2	no	—
205	C₂×C₄	—	424	2	no	—
206	C₂×C₄	—	424	2	no	—
207	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
208	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
209	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
210	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
211	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	#11719947	1	2	yes	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	212	2	no	—
222	C₂²	—	212	2	no	—
223	C₂²	—	212	2	no	—
224	C₂²	#13133239	1	2	yes	—
225	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	212	2	no	—
231	C₄	—	212	2	no	—
232	C₄	—	212	2	no	—
233	C₄	C₂×#11719947	1	1	yes	characteristic
234	C₄	#13133264	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	—	4	1	no	—
237	C₂	—	424	2	no	—
238	C₂	#16811277	1	1	yes	characteristic
239	C₂	C₂×#13133239	1	1	yes	characteristic
240	C₂	#16811428	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₈×D₄×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₈×D₄	D₅₃
hypocenter	C₅₃	#3339
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #16811319	C₂	semidirect
2. #16811025	C₂	semidirect
3. #16810948	C₂	semidirect
4. #16810942	C₂	semidirect
5. #16810924	C₂	semidirect
6. #16810859	C₂	semidirect
7. #16810859	C₂	semidirect
8. C₄×D₄₂₄	C₂	semidirect
9. #16810661	C₂	semidirect
10. C₂×#13133151	C₂	semidirect
11. C₈×D₄×C₅₃	C₂	semidirect
129. C₈×D₄	D₅₃	semidirect
132. C₅₃	#3339	semidirect