Subgroups — #20283865

Order 6784.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811155	C₂	1	1	yes	maximal, characteristic
2	#16810750	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810701	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810698	C₂	1	2	yes	maximal, complemented
6	#16810694	C₂	1	2	yes	maximal, complemented
7	#16810681	C₂	1	2	yes	maximal, complemented
8	#16810674	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810649	C₂	1	2	yes	maximal, complemented
10	#16810645	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719917	C₂²	1	2	yes	—
13	C₂×#11719912	C₂²	1	2	yes	—
14	#13133210	C₄	1	2	yes	—
15	#13133208	C₄	1	1	yes	characteristic
16	#13133204	C₄	1	1	yes	characteristic
17	#13133199	C₄	1	2	yes	—
18	#13133197	C₄	1	1	yes	characteristic
19	#13133188	C₄	1	1	yes	characteristic
20	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
22	#13133176	C₂²	1	2	yes	—
23	C₄²×D₅₃	C₂²	1	1	yes	characteristic
24	C₄⋊#88322	C₂²	1	1	yes	characteristic
25	#13133146	C₂²	1	2	yes	—
26	#13133146	C₂²	1	1	yes	characteristic
27	#13133146	C₂²	1	2	yes	—
28	#13133146	C₂²	1	2	yes	—
29	#13133146	C₂²	1	1	yes	characteristic
30	C₂²⋊#88322	C₂²	1	2	yes	—
31	#13133129	C₂²	1	1	yes	characteristic
32	#13133129	C₂²	1	1	yes	characteristic
33	#13133127	C₂²	1	2	yes	—
34	C₄×#88322	C₂²	1	1	yes	characteristic
35	C₈×Dic₅₃	C₂²	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
37	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
40	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
47	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	—	4	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	2	yes	—
52	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
54	C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
56	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
58	#11719921	C₂×C₄	1	2	yes	—
59	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
62	#11719919	C₂×C₄	1	2	yes	—
63	#11719919	C₂×C₄	1	2	yes	—
64	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₅₃	C₂³	1	2	yes	—
67	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
68	#11719917	—	4	2	no	—
69	C₂×#88322	C₂³	1	2	yes	—
70	C₂×#88322	C₂³	1	1	yes	characteristic
71	C₂×#88322	C₂³	1	1	yes	characteristic
72	#11719912	—	4	2	no	—
73	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
75	D₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
82	C₂²⋊D₅₃	—	4	2	no	—
83	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
84	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
88	C₄×D₅₃	—	2	2	no	—
89	C₄×D₅₃	—	2	1	no	—
90	C₄×D₅₃	—	2	1	no	—
91	C₄×D₅₃	—	4	1	no	—
92	Dic₁₀₆	—	2	2	no	—
93	C₈×C₅₃	—	2	2	no	—
94	C₈×C₅₃	—	2	1	no	—
95	C₈×C₅₃	—	2	1	no	—
96	#88322	—	2	2	no	—
97	#88322	—	2	1	no	—
98	#88322	—	2	1	no	—
99	C₂²×C₅₃	—	2	2	no	—
100	C₂²×C₅₃	—	2	2	no	—
101	C₂²×C₅₃	—	2	2	no	—
102	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₅₃	—	2	1	no	—
104	C₂×D₅₃	—	2	1	no	—
105	C₂×D₅₃	—	2	1	no	—
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	#131	1	2	yes	—
109	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
110	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
111	Dic₅₃	—	2	1	no	—
112	Dic₅₃	—	2	2	no	—
113	Dic₅₃	—	2	1	no	—
114	Dic₅₃	—	2	1	no	—
115	Dic₅₃	—	2	2	no	—
116	#2993	—	53	1	no	maximal, Sylow
117	C₂×C₅₃	—	4	2	no	—
118	C₂×C₅₃	#520	1	1	yes	characteristic
119	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
120	C₂×C₅₃	#568	1	1	yes	characteristic
121	D₅₃	—	4	1	no	—
122	#529	—	53	1	no	—
123	#446	—	53	1	no	—
124	C₈×Q₈	—	53	1	no	—
125	#436	—	53	1	no	—
126	#436	—	53	2	no	—
127	#435	—	53	2	no	—
128	C₈×D₄	D₅₃	1	1	yes	characteristic, complemented
129	#433	—	53	2	no	—
130	#431	—	53	1	no	—
131	#424	—	53	2	no	—
132	#423	—	53	1	no	—
133	C₅₃	#2993	1	1	yes	characteristic, complemented, Sylow
134	C₂×M₁₆	—	53	2	no	—
135	C₂×M₁₆	—	53	2	no	—
136	C₂²×C₈	C₂×D₅₃	1	2	yes	—
137	#126	—	53	2	no	—
138	#125	—	53	1	no	—
139	#124	—	53	1	no	—
140	#123	—	53	2	no	—
141	C₂²⋊Q₈	—	53	1	no	—
142	#121	—	53	1	no	—
143	C₄×Q₈	—	53	1	no	—
144	C₄×D₄	—	53	2	no	—
145	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
146	#117	—	53	2	no	—
147	C₂×C₄²	—	53	1	no	—
148	C₄⋊C₈	—	53	2	no	—
149	C₄⋊C₈	—	53	1	no	—
150	C₄⋊C₈	—	53	2	no	—
151	C₄⋊C₈	—	53	2	no	—
152	C₄⋊C₈	—	53	1	no	—
153	C₄⋊C₈	—	53	1	no	—
154	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
155	C₂²⋊C₈	—	53	2	no	—
156	C₂²⋊C₈	—	53	1	no	—
157	C₂²⋊C₈	—	53	1	no	—
158	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
159	#97	—	53	2	no	—
160	C₄×C₈	—	53	2	no	—
161	C₄×C₈	—	53	1	no	—
162	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
163	C₂×Q₈	—	53	1	no	—
164	C₂×D₄	—	53	2	no	—
165	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
166	C₂²×C₄	—	53	1	no	—
167	C₂²×C₄	—	53	1	no	—
168	C₂²×C₄	—	53	1	no	—
169	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
170	M₁₆	—	212	2	no	—
171	M₁₆	—	212	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	53	1	no	—
174	C₂×C₈	—	53	2	no	—
175	C₂×C₈	—	53	1	no	—
176	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₅₃	1	2	yes	—
179	C₄⋊C₄	—	53	2	no	—
180	C₄⋊C₄	—	53	2	no	—
181	C₄⋊C₄	—	53	2	no	—
182	C₄⋊C₄	—	53	1	no	—
183	C₄⋊C₄	—	53	2	no	—
184	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
185	C₂²⋊C₄	—	53	2	no	—
186	C₂²⋊C₄	—	53	2	no	—
187	C₂²⋊C₄	—	53	2	no	—
188	C₂²⋊C₄	—	53	2	no	—
189	C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
190	C₄²	—	53	1	no	—
191	C₄²	—	53	1	no	—
192	C₄²	—	53	2	no	—
193	C₄²	—	53	1	no	—
194	C₄²	C₂²×D₅₃	1	1	yes	characteristic
195	C₂³	—	53	1	no	—
196	C₂³	C₂×C₄×D₅₃	1	2	yes	—
197	Q₈	—	106	2	no	—
198	D₄	—	2	2	no	—
199	D₄	—	212	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	53	1	no	—
203	C₂×C₄	—	53	1	no	—
204	C₂×C₄	—	53	1	no	—
205	C₂×C₄	—	53	2	no	—
206	C₂×C₄	—	53	2	no	—
207	C₂×C₄	—	106	2	no	—
208	C₂×C₄	—	106	1	no	—
209	C₂×C₄	—	106	1	no	—
210	C₂×C₄	—	212	1	no	—
211	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
213	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	2	no	—
217	C₈	—	2	1	no	—
218	C₈	—	106	2	no	—
219	C₈	—	106	1	no	—
220	C₈	—	106	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	106	1	no	—
225	C₂²	—	106	1	no	—
226	C₂²	—	106	1	no	—
227	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	106	1	no	—
231	C₄	—	106	1	no	—
232	C₄	—	106	2	no	—
233	C₄	—	106	1	no	—
234	C₄	—	106	2	no	—
235	C₄	#13133223	1	2	yes	—
236	C₄	#13133262	1	1	yes	characteristic
237	C₄	#13133264	1	1	yes	characteristic
238	C₂	—	4	2	no	—
239	C₂	—	212	1	no	—
240	C₂	#16811102	1	1	yes	characteristic
241	C₂	C₂×#13133223	1	1	yes	characteristic
242	C₂	#16811388	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₈×D₄×C₅₃	C₂
hypercenter	C₈×D₄	D₅₃
hypocenter	C₅₃	#2993
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #16810750	C₂	semidirect
3. C₈×Dic₁₀₆	C₂	semidirect
4. #16810701	C₂	semidirect
5. #16810698	C₂	semidirect
6. #16810694	C₂	semidirect
7. #16810681	C₂	semidirect
8. #16810674	C₂	semidirect
9. #16810649	C₂	semidirect
10. #16810645	C₂	semidirect
11. C₈×D₄×C₅₃	C₂	semidirect
128. C₈×D₄	D₅₃	semidirect
133. C₅₃	#2993	semidirect