Subgroups — #20283843

Order 6784.

394 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11719947	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810747	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810701	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810699	C₂	1	2	yes	maximal, complemented
6	#16810698	C₂	1	2	yes	maximal, complemented
7	#16810674	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810673	C₂	1	2	yes	maximal, complemented
9	#16810646	C₂	1	2	yes	maximal, complemented
10	#16810645	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719947	C₄	1	1	yes	characteristic
13	C₂×#11719917	C₂²	1	2	yes	—
14	C₂×#11719912	C₂²	1	2	yes	—
15	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
16	#13133182	C₄	1	1	yes	characteristic
17	D₄×Dic₅₃	C₄	1	1	yes	characteristic
18	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
19	#13133177	C₄	1	2	yes	—
20	#13133175	C₄	1	1	yes	characteristic
21	#13133173	C₄	1	2	yes	—
22	C₂×C₄×Dic₅₃	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	C₄×#88322	C₂²	1	1	yes	characteristic
34	C₈×Dic₅₃	C₂²	1	2	yes	—
35	C₈×Dic₅₃	C₂²	1	2	yes	—
36	C₂×M₁₆×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	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
45	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
46	#11719947	—	2	4	no	—
47	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
50	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
51	M₁₆×C₅₃	—	4	2	no	—
52	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂³	1	2	yes	—
55	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
57	C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
59	#11719921	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
61	#11719919	C₂×C₄	1	2	yes	—
62	#11719919	C₂×C₄	1	2	yes	—
63	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
66	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
67	C₄×Dic₅₃	C₂³	1	2	yes	—
68	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
69	#11719917	—	4	2	no	—
70	C₂×#88322	C₂³	1	1	yes	characteristic
71	C₂×#88322	C₂³	1	2	yes	—
72	C₂×#88322	C₂³	1	1	yes	characteristic
73	#11719912	—	4	2	no	—
74	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
76	D₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₅₃	—	2	4	no	—
84	C₂×Dic₅₃	—	2	4	no	—
85	C₂×Dic₅₃	—	2	4	no	—
86	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
89	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
90	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₄×D₅₃	—	2	2	no	—
92	C₄×D₅₃	—	2	1	no	—
93	C₄×D₅₃	—	2	1	no	—
94	C₄×D₅₃	—	2	2	no	—
95	Dic₁₀₆	M₁₆	1	4	yes	complemented
96	C₈×C₅₃	—	2	2	no	—
97	C₈×C₅₃	—	2	1	no	—
98	C₈×C₅₃	—	2	1	no	—
99	#88322	—	2	2	no	—
100	#88322	—	2	1	no	—
101	#88322	—	2	1	no	—
102	C₂²×C₅₃	—	2	2	no	—
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×D₅₃	—	2	1	no	—
107	C₂×D₅₃	—	2	1	no	—
108	C₂×D₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
112	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
114	Dic₅₃	—	2	1	no	—
115	Dic₅₃	—	2	1	no	—
116	Dic₅₃	—	2	2	no	—
117	Dic₅₃	C₂×M₁₆	1	4	yes	—
118	Dic₅₃	#131	1	2	yes	—
119	#2991	—	53	1	no	maximal, Sylow
120	C₂×C₅₃	—	4	2	no	—
121	C₂×C₅₃	#520	1	1	yes	characteristic
122	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
123	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
124	D₅₃	—	4	1	no	—
125	C₄×#41	—	53	1	no	—
126	C₈×Q₈	—	53	1	no	—
127	C₈×Q₈	—	53	1	no	—
128	#436	—	53	2	no	—
129	#436	—	53	1	no	—
130	#436	—	53	2	no	—
131	#436	D₅₃	1	1	yes	characteristic, complemented
132	#431	—	53	2	no	—
133	#431	—	53	1	no	—
134	#423	—	53	2	no	—
135	#423	—	53	1	no	—
136	C₅₃	#2991	1	1	yes	characteristic, complemented, Sylow
137	C₂×#41	—	53	1	no	—
138	C₂×M₁₆	—	53	2	no	—
139	C₂×M₁₆	—	53	2	no	—
140	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
141	C₄×Q₈	—	53	1	no	—
142	C₄×Q₈	—	53	1	no	—
143	C₄×D₄	—	53	1	no	—
144	C₄×D₄	—	53	2	no	—
145	C₄×D₄	—	53	2	no	—
146	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
147	#117	—	53	1	no	—
148	#117	—	53	2	no	—
149	C₂×C₄²	—	53	2	no	—
150	C₂×C₄²	—	53	1	no	—
151	C₄⋊C₈	—	53	2	no	—
152	C₄⋊C₈	—	53	1	no	—
153	C₄⋊C₈	—	53	2	no	—
154	C₄⋊C₈	—	53	2	no	—
155	C₄⋊C₈	—	53	1	no	—
156	C₄⋊C₈	—	53	1	no	—
157	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
158	C₂²⋊C₈	—	53	2	no	—
159	C₂²⋊C₈	—	53	1	no	—
160	C₂²⋊C₈	—	53	1	no	—
161	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
162	C₄×C₈	—	53	2	no	—
163	C₄×C₈	—	53	2	no	—
164	C₄×C₈	—	53	1	no	—
165	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
166	#41	—	106	4	no	—
167	C₂×Q₈	—	53	1	no	—
168	C₂×D₄	—	53	2	no	—
169	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
170	C₂²×C₄	—	53	2	no	—
171	C₂²×C₄	—	53	2	no	—
172	C₂²×C₄	—	53	1	no	—
173	C₂²×C₄	—	53	1	no	—
174	C₂²×C₄	—	53	1	no	—
175	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
176	M₁₆	—	4	2	no	—
177	M₁₆	—	212	2	no	—
178	M₁₆	—	212	2	no	—
179	C₂×C₈	—	53	1	no	—
180	C₂×C₈	—	53	2	no	—
181	C₂×C₈	—	53	1	no	—
182	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₅₃	1	2	yes	—
184	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
185	C₄⋊C₄	—	53	2	no	—
186	C₄⋊C₄	—	53	2	no	—
187	C₄⋊C₄	—	53	1	no	—
188	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
189	C₂²⋊C₄	—	53	2	no	—
190	C₂²⋊C₄	—	53	2	no	—
191	C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
192	C₄²	—	53	2	no	—
193	C₄²	—	53	1	no	—
194	C₄²	—	53	2	no	—
195	C₄²	—	53	1	no	—
196	C₄²	—	53	2	no	—
197	C₄²	—	53	1	no	—
198	C₄²	C₂²×D₅₃	1	1	yes	characteristic
199	C₂³	—	53	1	no	—
200	C₂³	C₂×C₄×D₅₃	1	2	yes	—
201	Q₈	—	53	4	no	—
202	D₄	—	2	2	no	—
203	D₄	—	106	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	53	1	no	—
207	C₂×C₄	—	53	2	no	—
208	C₂×C₄	—	53	1	no	—
209	C₂×C₄	—	53	1	no	—
210	C₂×C₄	—	53	2	no	—
211	C₂×C₄	—	106	2	no	—
212	C₂×C₄	—	106	4	no	—
213	C₂×C₄	—	106	4	no	—
214	C₂×C₄	—	106	2	no	—
215	C₂×C₄	—	106	1	no	—
216	C₂×C₄	—	106	1	no	—
217	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	—	106	2	no	—
225	C₈	—	106	1	no	—
226	C₈	—	106	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	106	1	no	—
231	C₂²	—	106	1	no	—
232	C₂²	—	106	1	no	—
233	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	53	4	no	—
237	C₄	—	53	2	no	—
238	C₄	—	106	1	no	—
239	C₄	—	106	1	no	—
240	C₄	—	106	2	no	—
241	C₄	M₁₆×D₅₃	1	2	yes	—
242	C₄	#13133262	1	1	yes	characteristic
243	C₄	#13133264	1	1	yes	characteristic
244	C₂	—	4	2	no	—
245	C₂	—	212	1	no	—
246	C₂	#16811102	1	1	yes	characteristic
247	C₂	C₂×M₁₆×D₅₃	1	1	yes	characteristic
248	C₂	#131×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#436	C₂
hypercenter	#436	D₅₃
hypocenter	C₅₃	#2991
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₁₀₆	C₂	semidirect
3. #16810747	C₂	semidirect
4. #16810701	C₂	semidirect
5. #16810699	C₂	semidirect
6. #16810698	C₂	semidirect
7. #16810674	C₂	semidirect
8. #16810673	C₂	semidirect
9. #16810646	C₂	semidirect
10. #16810645	C₂	semidirect
11. C₅₃×#436	C₂	semidirect
95. Dic₁₀₆	M₁₆	semidirect
131. #436	D₅₃	semidirect
136. C₅₃	#2991	semidirect