Subgroups — #20283558

Order 6784.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133248	C₂	1	1	yes	maximal, characteristic
2	#16811103	C₂	1	1	yes	maximal, characteristic
3	#16810728	C₂	1	2	yes	maximal, complemented
4	#16810721	C₂	1	2	yes	maximal, complemented
5	C₄×#11719925	C₂	1	2	yes	maximal, complemented
6	C₄×#11719924	C₂	1	2	yes	maximal, complemented
7	#16810666	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810617	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133141	C₂	1	1	yes	maximal, characteristic, complemented
10	#16810579	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	1	yes	characteristic
13	#13133248	C₄	1	8	yes	complemented
14	C₂×#11719925	C₂²	1	2	yes	—
15	C₂×#11719924	C₂²	1	2	yes	—
16	C₂×#11719917	C₂²	1	1	yes	characteristic
17	C₂²×#88322	C₂²	1	1	yes	characteristic
18	C₄×Dic₁₀₆	C₂²	1	2	yes	—
19	C₄×D₂₁₂	C₂²	1	2	yes	—
20	#13133177	—	2	2	no	—
21	#13133175	—	2	1	no	—
22	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×D₅₃	—	2	1	no	—
24	#13133150	C₂²	1	2	yes	—
25	#13133148	C₂²	1	2	yes	—
26	#13133141	C₂²	1	2	yes	—
27	#13133141	C₂²	1	2	yes	—
28	#13133140	C₂²	1	1	yes	characteristic
29	#13133140	C₂²	1	1	yes	characteristic
30	#13133137	C₂²	1	1	yes	characteristic
31	#13133137	C₂²	1	1	yes	characteristic
32	#13133128	C₂²	1	2	yes	—
33	C₄×#88322	C₂²	1	2	yes	—
34	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₅₃	—	2	1	no	—
36	C₄×Q₈×C₅₃	—	2	1	no	—
37	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
38	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
39	#117×C₅₃	—	2	2	no	—
40	#117×C₅₃	C₂²	1	1	yes	characteristic
41	#117×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×Q₈×C₅₃	—	2	1	no	—
43	C₂×Q₈×C₅₃	—	2	2	no	—
44	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	complemented
45	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₅₃	—	2	1	no	—
49	C₂²×Dic₅₃	—	2	1	no	—
50	#11719945	—	2	2	no	—
51	#11719945	C₂×C₄	1	4	yes	—
52	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₅₃	—	2	1	no	—
54	C₂×C₄×D₅₃	—	2	2	no	—
55	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂²⋊C₄×C₅₃	—	2	1	no	—
63	C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
64	C₄²×C₅₃	—	2	2	no	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	2	1	no	—
67	C₄²×C₅₃	C₂³	1	2	yes	—
68	#11719925	C₂×C₄	1	8	yes	complemented
69	#11719924	C₂×C₄	1	8	yes	complemented
70	#11719921	—	2	2	no	—
71	C₄⋊Dic₅₃	—	2	1	no	—
72	C₄⋊Dic₅₃	C₂³	1	2	yes	—
73	#11719919	—	2	2	no	—
74	C₄×Dic₅₃	—	2	2	no	—
75	#11719917	C₂×C₄	1	4	yes	complemented
76	C₂×#88322	—	2	2	no	—
77	C₂×#88322	C₂³	1	1	yes	characteristic
78	C₂×#88322	C₂³	1	1	yes	characteristic
79	C₂×#88322	C₂³	1	2	yes	—
80	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×D₅₃	—	2	1	no	—
82	Q₈×C₅₃	—	2	4	no	—
83	Q₈×C₅₃	—	2	2	no	—
84	Q₈×C₅₃	C₂²×C₄	1	4	yes	—
85	C₂×C₄×C₅₃	—	2	4	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
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	1	yes	characteristic
93	C₂×C₄×C₅₃	#41	1	2	yes	—
94	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₂²⋊D₅₃	—	2	4	no	—
96	C₂×Dic₅₃	—	2	1	no	—
97	C₂×Dic₅₃	—	2	2	no	—
98	C₂×Dic₅₃	—	8	1	no	—
99	D₂₁₂	—	2	1	no	—
100	D₂₁₂	C₂²×C₄	1	2	yes	—
101	C₄×D₅₃	—	2	4	no	—
102	C₄×D₅₃	—	8	2	no	—
103	Dic₁₀₆	—	2	1	no	—
104	Dic₁₀₆	C₂²×C₄	1	2	yes	—
105	#88322	—	2	1	no	—
106	#88322	—	2	1	no	—
107	#88322	C₂²×C₄	1	4	yes	—
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	—	2	1	no	—
110	C₂²×C₅₃	C₄×D₄	1	2	yes	—
111	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×D₅₃	—	2	2	no	—
113	C₂×D₅₃	—	4	1	no	—
114	C₂×D₅₃	—	4	1	no	—
115	C₄×C₅₃	—	2	4	no	—
116	C₄×C₅₃	—	2	2	no	—
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	—	4	1	no	—
119	C₄×C₅₃	—	4	1	no	—
120	C₄×C₅₃	C₄×D₄	1	2	yes	—
121	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
122	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
123	Dic₅₃	—	2	2	no	—
124	Dic₅₃	—	4	2	no	—
125	#2947	—	53	1	no	maximal, Sylow
126	C₂×C₅₃	—	2	2	no	—
127	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂×C₅₃	#577	1	1	yes	characteristic
129	C₂×C₅₃	#578	1	1	yes	characteristic
130	D₅₃	—	4	2	no	—
131	C₂×#137	—	53	1	no	—
132	#520	—	53	1	no	—
133	#519	D₅₃	1	1	yes	characteristic, complemented
134	#441	—	53	2	no	—
135	#440	—	53	2	no	—
136	C₄×Q₁₆	—	53	2	no	—
137	C₄×SD₁₆	—	53	2	no	—
138	#428	—	53	1	no	—
139	#416	—	53	1	no	—
140	C₂×#103	—	53	1	no	—
141	#405	—	53	1	no	—
142	C₅₃	#2947	1	1	yes	characteristic, complemented, Sylow
143	C₂×#41	—	53	1	no	—
144	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
145	#137	—	53	8	no	—
146	C₂×Q₁₆	—	53	2	no	—
147	C₂×SD₁₆	—	53	2	no	—
148	C₂×M₁₆	—	53	1	no	—
149	C₂²×C₈	—	53	1	no	—
150	C₄×Q₈	—	2	1	no	—
151	C₄×Q₈	—	2	1	no	—
152	C₄×Q₈	—	53	2	no	—
153	C₄×Q₈	C₂×D₅₃	1	2	yes	—
154	C₄×Q₈	C₂×D₅₃	1	2	yes	—
155	C₄×D₄	—	53	2	no	—
156	C₄×D₄	—	106	2	no	—
157	#117	—	2	2	no	—
158	#117	—	106	1	no	—
159	#117	C₂×D₅₃	1	1	yes	characteristic
160	#117	C₂×D₅₃	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	53	1	no	—
162	C₂×C₄⋊C₄	—	106	1	no	—
163	#107	—	53	2	no	—
164	#106	—	53	2	no	—
165	#103	—	53	2	no	—
166	#103	—	53	2	no	—
167	#103	—	53	1	no	—
168	#103	—	53	1	no	—
169	#102	—	53	1	no	—
170	#102	—	53	1	no	—
171	#97	—	53	2	no	—
172	C₄×C₈	—	53	2	no	—
173	#41	—	53	4	no	—
174	#41	—	106	2	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	1	no	—
177	C₂×Q₈	—	53	1	no	—
178	C₂×Q₈	C₄×D₅₃	1	4	yes	complemented
179	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
180	C₂×D₄	—	53	1	no	—
181	C₂×D₄	—	106	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	106	1	no	—
184	C₂²×C₄	—	106	1	no	—
185	C₂²×C₄	—	106	2	no	—
186	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
187	Q₁₆	—	53	8	no	—
188	SD₁₆	—	53	8	no	—
189	M₁₆	—	53	4	no	—
190	C₂×C₈	—	53	2	no	—
191	C₂×C₈	—	53	1	no	—
192	C₂×C₈	—	53	1	no	—
193	C₂×C₈	—	106	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	53	2	no	—
197	C₄⋊C₄	—	106	1	no	—
198	C₄⋊C₄	—	106	2	no	—
199	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
200	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
201	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
202	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	106	2	no	—
205	C₂²⋊C₄	C₂²⋊D₅₃	1	2	yes	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	106	2	no	—
210	C₄²	C₂²×D₅₃	1	2	yes	—
211	C₂³	—	106	1	no	—
212	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
213	Q₈	—	2	4	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	53	2	no	—
216	Q₈	—	106	1	no	—
217	Q₈	C₂×C₄×D₅₃	1	4	yes	—
218	D₄	—	53	2	no	—
219	D₄	—	106	4	no	—
220	D₄	—	106	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	106	1	no	—
226	C₂×C₄	—	106	4	no	—
227	C₂×C₄	—	106	2	no	—
228	C₂×C₄	—	424	2	no	—
229	C₂×C₄	—	424	1	no	—
230	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
231	C₂×C₄	#11719945	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
233	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
234	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
236	C₈	—	53	4	no	—
237	C₈	—	106	1	no	—
238	C₈	—	106	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	106	2	no	—
242	C₂²	—	212	1	no	—
243	C₂²	—	212	1	no	—
244	C₂²	C₄×C₂²⋊D₅₃	1	2	yes	—
245	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	1	no	—
251	C₄	—	106	2	no	—
252	C₄	—	212	2	no	—
253	C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
254	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
255	C₄	C₂×#11719945	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	212	2	no	—
258	C₂	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
259	C₂	#16811431	1	1	yes	characteristic
260	C₂	#16811438	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#519	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#519	D₅₃
hypocenter	C₅₃	#2947
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #519

Complements

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

N	H ≅ G/N	type
3. #16810728	C₂	semidirect
4. #16810721	C₂	semidirect
5. C₄×#11719925	C₂	semidirect
6. C₄×#11719924	C₂	semidirect
7. #16810666	C₂	semidirect
9. C₂×#13133141	C₂	semidirect
10. #16810579	C₂	semidirect
11. C₅₃×#519	C₂	semidirect
13. #13133248	C₄	semidirect
44. C₂×Q₈×C₅₃	C₂×C₄	semidirect
68. #11719925	C₂×C₄	semidirect
69. #11719924	C₂×C₄	semidirect
75. #11719917	C₂×C₄	semidirect
133. #519	D₅₃	semidirect
142. C₅₃	#2947	semidirect
178. C₂×Q₈	C₄×D₅₃	semidirect