Subgroups — #20283585

Order 6784.

484 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133236	C₂	1	1	yes	maximal, characteristic
2	#16811102	C₂	1	2	yes	maximal, complemented
3	#16810723	C₂	1	2	yes	maximal, complemented
4	#16810720	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810717	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#107×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810621	C₂	1	2	yes	maximal, complemented
10	#16810577	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719947	C₂²	1	2	yes	—
13	#13133236	C₄	1	8	yes	complemented
14	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
15	C₂×#11719923	C₂²	1	2	yes	—
16	C₂×#11719912	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
18	C₄×Dic₁₀₆	C₂²	1	2	yes	—
19	#13133182	C₂²	1	2	yes	—
20	D₄×Dic₅₃	C₂²	1	2	yes	—
21	C₄×C₂²⋊D₅₃	—	2	2	no	—
22	#13133177	—	2	2	no	—
23	#13133174	C₂²	1	1	yes	characteristic
24	#13133173	—	2	2	no	—
25	C₂×#11719919	—	2	2	no	—
26	C₄⋊C₄×D₅₃	C₂²	1	2	yes	—
27	#13133151	C₂²	1	1	yes	characteristic
28	#13133150	C₂²	1	2	yes	—
29	#13133142	C₂²	1	2	yes	—
30	#13133140	C₂²	1	2	yes	—
31	#13133138	C₂²	1	2	yes	—
32	#13133128	C₂²	1	1	yes	characteristic
33	#13133127	C₂²	1	1	yes	characteristic
34	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
35	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₅₃	C₂²	1	2	yes	—
37	#107×C₅₃	C₂²	1	1	yes	characteristic
38	#102×C₅₃	C₂²	1	2	yes	—
39	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×D₄×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₅₃	—	2	2	no	—
42	C₂×C₂²⋊D₅₃	—	2	2	no	—
43	C₂²×Dic₅₃	—	2	2	no	—
44	C₂²×Dic₅₃	—	2	2	no	—
45	#11719947	—	2	4	no	—
46	#11719947	C₂×C₄	1	8	yes	complemented
47	C₂×C₄×D₅₃	—	2	2	no	—
48	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
49	C₂×Dic₁₀₆	C₂³	1	2	yes	—
50	D₈×C₅₃	C₂×C₄	1	4	yes	complemented
51	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
54	C₂²⋊C₄×C₅₃	—	2	2	no	—
55	C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₂²⋊Dic₅₃	—	2	2	no	—
57	#11719923	C₂×C₄	1	8	yes	complemented
58	#11719921	D₄	1	1	yes	characteristic
59	#11719921	D₄	1	1	yes	characteristic
60	C₄⋊Dic₅₃	C₂³	1	2	yes	—
61	#11719919	—	2	2	no	—
62	#11719919	—	2	2	no	—
63	#11719919	—	2	2	no	—
64	#11719919	D₄	1	1	yes	characteristic
65	#11719919	D₄	1	1	yes	characteristic
66	C₄×Dic₅₃	—	2	2	no	—
67	C₄×Dic₅₃	—	2	2	no	—
68	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
69	C₂×#88322	C₂³	1	1	yes	characteristic
70	C₂×#88322	C₂³	1	1	yes	characteristic
71	Dic₂₁₂	C₂×C₄	1	4	yes	complemented
72	#11719912	—	2	2	no	—
73	C₈×D₅₃	C₂×C₄	1	4	yes	complemented
74	C₂³×C₅₃	—	2	2	no	—
75	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
76	D₄×C₅₃	—	2	2	no	—
77	D₄×C₅₃	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	8	2	no	—
80	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₅₃	—	2	8	no	—
84	C₂×Dic₅₃	—	2	2	no	—
85	C₂×Dic₅₃	—	2	8	no	—
86	C₂×Dic₅₃	—	2	2	no	—
87	C₂×Dic₅₃	—	8	2	no	—
88	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄×D₅₃	—	8	2	no	—
92	C₄×D₅₃	C₂²×C₄	1	2	yes	—
93	C₄×D₅₃	#41	1	2	yes	—
94	Dic₁₀₆	—	2	2	no	—
95	Dic₁₀₆	C₂²×C₄	1	4	yes	—
96	C₈×C₅₃	—	2	1	no	—
97	C₈×C₅₃	C₂²×C₄	1	2	yes	—
98	#88322	—	2	1	no	—
99	#88322	C₂²×C₄	1	2	yes	—
100	C₂²×C₅₃	—	2	4	no	—
101	C₂²×C₅₃	—	4	2	no	—
102	C₂²×C₅₃	—	4	2	no	—
103	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×D₅₃	—	2	1	no	—
105	C₂×D₅₃	—	2	1	no	—
106	C₂×D₅₃	C₄×D₄	1	2	yes	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	4	2	no	—
110	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
111	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
112	Dic₅₃	—	2	4	no	—
113	Dic₅₃	—	2	1	no	—
114	Dic₅₃	—	2	1	no	—
115	Dic₅₃	—	4	2	no	—
116	Dic₅₃	C₄×D₄	1	2	yes	—
117	#2951	—	53	1	no	maximal, Sylow
118	C₂×C₅₃	—	4	4	no	—
119	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
120	C₂×C₅₃	#576	1	1	yes	characteristic
121	C₂×C₅₃	#578	1	1	yes	characteristic
122	D₅₃	—	2	2	no	—
123	C₂×#135	—	53	1	no	—
124	#520	—	53	2	no	—
125	#440	—	53	2	no	—
126	#440	—	53	2	no	—
127	C₄×Q₁₆	—	53	1	no	—
128	C₄×Q₁₆	—	53	1	no	—
129	C₄×D₈	—	53	1	no	—
130	C₄×D₈	D₅₃	1	1	yes	characteristic, complemented
131	C₂×#107	—	53	1	no	—
132	#417	—	53	2	no	—
133	#405	—	53	1	no	—
134	C₅₃	#2951	1	1	yes	characteristic, complemented, Sylow
135	C₂×#41	—	53	2	no	—
136	#135	—	53	8	no	—
137	C₂×Q₁₆	—	53	1	no	—
138	C₂×SD₁₆	—	53	2	no	—
139	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
140	C₂×M₁₆	—	53	1	no	—
141	C₂²×C₈	—	53	1	no	—
142	C₄×Q₈	—	53	2	no	—
143	C₄×Q₈	—	53	2	no	—
144	C₄×D₄	—	53	2	no	—
145	C₄×D₄	—	106	2	no	—
146	C₄×D₄	—	106	2	no	—
147	C₄×D₄	C₂×D₅₃	1	2	yes	—
148	#117	—	53	1	no	—
149	#117	—	106	2	no	—
150	C₂×C₄⋊C₄	—	53	2	no	—
151	C₂×C₄⋊C₄	—	106	2	no	—
152	#107	—	53	1	no	—
153	#107	—	53	2	no	—
154	#107	C₂×D₅₃	1	1	yes	characteristic
155	#103	—	53	2	no	—
156	#103	—	53	2	no	—
157	#102	—	53	2	no	—
158	#102	C₂×D₅₃	1	2	yes	—
159	#97	—	53	1	no	—
160	#97	—	53	1	no	—
161	C₄×C₈	—	53	1	no	—
162	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
163	#41	—	53	8	no	—
164	#41	—	106	4	no	—
165	C₂×Q₈	—	53	2	no	—
166	C₂×D₄	—	106	2	no	—
167	C₂×D₄	C₂²×D₅₃	1	2	yes	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	53	1	no	—
170	C₂²×C₄	—	106	2	no	—
171	C₂²×C₄	—	106	2	no	—
172	C₂²×C₄	—	106	2	no	—
173	Q₁₆	—	53	4	no	—
174	SD₁₆	—	53	8	no	—
175	D₈	C₄×D₅₃	1	4	yes	complemented
176	M₁₆	—	106	2	no	—
177	C₂×C₈	—	53	1	no	—
178	C₂×C₈	—	53	4	no	—
179	C₂×C₈	—	53	1	no	—
180	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
181	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
182	C₄⋊C₄	—	53	1	no	—
183	C₄⋊C₄	—	53	1	no	—
184	C₄⋊C₄	—	53	2	no	—
185	C₄⋊C₄	—	106	2	no	—
186	C₄⋊C₄	—	106	2	no	—
187	C₄⋊C₄	—	106	2	no	—
188	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	53	1	no	—
191	C₂²⋊C₄	—	53	1	no	—
192	C₂²⋊C₄	—	106	2	no	—
193	C₄²	—	53	1	no	—
194	C₄²	—	106	2	no	—
195	C₄²	—	106	2	no	—
196	C₄²	C₂²×D₅₃	1	1	yes	characteristic
197	C₂³	—	2	2	no	—
198	C₂³	—	53	1	no	—
199	Q₈	—	53	4	no	—
200	Q₈	—	106	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	106	8	no	—
203	D₄	C₂×C₄×D₅₃	1	4	yes	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	—	53	1	no	—
207	C₂×C₄	—	53	2	no	—
208	C₂×C₄	—	53	2	no	—
209	C₂×C₄	—	53	1	no	—
210	C₂×C₄	—	53	1	no	—
211	C₂×C₄	—	106	8	no	—
212	C₂×C₄	—	106	2	no	—
213	C₂×C₄	—	106	2	no	—
214	C₂×C₄	—	424	2	no	—
215	C₂×C₄	—	424	2	no	—
216	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
217	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
218	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	53	2	no	—
221	C₈	—	106	1	no	—
222	C₈	C₂×C₄×D₅₃	1	2	yes	—
223	C₂²	—	2	4	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	53	2	no	—
227	C₂²	—	106	1	no	—
228	C₂²	—	106	1	no	—
229	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	2	no	—
233	C₄	—	53	2	no	—
234	C₄	—	106	4	no	—
235	C₄	—	106	1	no	—
236	C₄	—	106	1	no	—
237	C₄	—	212	2	no	—
238	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
239	C₄	#41×D₅₃	1	1	yes	characteristic
240	C₂	—	4	4	no	—
241	C₂	—	106	2	no	—
242	C₂	C₄×D₄×D₅₃	1	1	yes	characteristic
243	C₂	#16811424	1	1	yes	characteristic
244	C₂	#16811437	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₄×D₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₄×D₈	D₅₃
hypocenter	C₅₃	#2951
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #16811102	C₂	semidirect
3. #16810723	C₂	semidirect
4. #16810720	C₂	semidirect
5. C₄×Dic₂₁₂	C₂	semidirect
6. #16810717	C₂	semidirect
7. D₈×Dic₅₃	C₂	semidirect
8. #107×D₅₃	C₂	semidirect
9. #16810621	C₂	semidirect
10. #16810577	C₂	semidirect
11. C₄×D₈×C₅₃	C₂	semidirect
13. #13133236	C₄	semidirect
46. #11719947	C₂×C₄	semidirect
50. D₈×C₅₃	C₂×C₄	semidirect
57. #11719923	C₂×C₄	semidirect
71. Dic₂₁₂	C₂×C₄	semidirect
73. C₈×D₅₃	C₂×C₄	semidirect
130. C₄×D₈	D₅₃	semidirect
134. C₅₃	#2951	semidirect
175. D₈	C₄×D₅₃	semidirect