Subgroups — #20283282

Order 6784.

484 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133225	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133223	C₂	1	1	yes	maximal, characteristic
3	#16811091	C₂	1	1	yes	maximal, characteristic
4	#16810650	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810648	C₂	1	2	yes	maximal, complemented
6	#16810648	C₂	1	2	yes	maximal, complemented
7	#16810646	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810645	C₂	1	2	yes	maximal, complemented
9	C₂×#13133146	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₈×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₅₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	1	yes	characteristic
13	#13133225	—	2	4	no	—
14	#13133223	—	2	4	no	—
15	C₂×#11719917	C₂²	1	1	yes	characteristic
16	C₂×#11719912	C₂²	1	2	yes	—
17	C₂×#11719912	C₂²	1	2	yes	—
18	C₂²×#88322	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₅₃	C₂²	1	2	yes	—
20	C₂×C₈×D₅₃	C₂²	1	2	yes	—
21	#13133182	C₄	1	1	yes	characteristic
22	#13133182	C₄	1	1	yes	characteristic
23	#13133179	C₄	1	1	yes	characteristic
24	#13133179	C₄	1	1	yes	characteristic
25	#13133177	C₄	1	2	yes	—
26	#13133177	C₄	1	2	yes	—
27	#13133174	C₂²	1	2	yes	—
28	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
29	C₄²×D₅₃	C₂²	1	2	yes	—
30	C₄⋊#88322	C₂²	1	2	yes	—
31	C₄⋊#88322	C₂²	1	2	yes	—
32	#13133146	C₂²	1	1	yes	characteristic
33	#13133146	C₂²	1	2	yes	—
34	#13133146	C₂²	1	1	yes	characteristic
35	#13133146	C₂²	1	1	yes	characteristic
36	#13133146	C₂²	1	1	yes	characteristic
37	#13133146	C₂²	1	2	yes	—
38	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
40	#117×C₅₃	C₂²	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
42	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
45	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
47	#11719945	D₄	1	4	yes	—
48	#11719945	Q₈	1	4	yes	—
49	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₅₃	C₂³	1	2	yes	—
53	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
54	M₁₆×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂³	1	2	yes	—
59	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
62	C₄²×C₅₃	C₂³	1	2	yes	—
63	#11719921	C₂×C₄	1	2	yes	—
64	#11719921	C₂×C₄	1	2	yes	—
65	#11719919	C₂×C₄	1	2	yes	—
66	#11719919	C₂×C₄	1	2	yes	—
67	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
68	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
69	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
72	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
73	#11719917	—	2	2	no	—
74	C₂×#88322	—	2	2	no	—
75	C₂×#88322	C₂³	1	1	yes	characteristic
76	C₂×#88322	C₂³	1	2	yes	—
77	C₂×#88322	C₂³	1	1	yes	characteristic
78	#11719912	—	2	4	no	—
79	#11719912	—	2	4	no	—
80	C₈×D₅₃	—	2	4	no	—
81	C₈×D₅₃	—	2	4	no	—
82	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
89	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₂²⋊D₅₃	C₄⋊C₄	1	8	yes	—
91	C₂×Dic₅₃	—	2	2	no	—
92	C₂×Dic₅₃	—	4	1	no	—
93	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
97	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
98	D₂₁₂	C₄⋊C₄	1	4	yes	—
99	C₄×D₅₃	—	2	4	no	—
100	C₄×D₅₃	—	4	2	no	—
101	C₄×D₅₃	C₂×D₄	1	4	yes	—
102	C₄×D₅₃	C₂×Q₈	1	4	yes	—
103	Dic₁₀₆	C₄⋊C₄	1	4	yes	—
104	C₈×C₅₃	—	2	1	no	—
105	C₈×C₅₃	—	2	2	no	—
106	C₈×C₅₃	—	2	1	no	—
107	#88322	—	2	1	no	—
108	#88322	—	2	2	no	—
109	#88322	—	2	1	no	—
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×D₅₃	—	2	2	no	—
115	C₂×D₅₃	—	2	2	no	—
116	C₂×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
122	Dic₅₃	—	2	1	no	—
123	Dic₅₃	—	2	1	no	—
124	Dic₅₃	—	2	1	no	—
125	Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
126	Dic₅₃	#131	1	2	yes	complemented
127	#2906	—	53	1	no	maximal, Sylow
128	C₂×C₅₃	—	2	2	no	—
129	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
131	C₂×C₅₃	#568	1	1	yes	characteristic
132	D₅₃	—	2	4	no	—
133	C₂×#131	—	53	1	no	—
134	C₂×#131	—	53	1	no	—
135	C₄×#41	—	53	1	no	—
136	#424	—	53	1	no	—
137	#424	—	53	2	no	—
138	#424	—	53	2	no	—
139	#424	D₅₃	1	1	yes	characteristic, complemented
140	#423	—	53	1	no	—
141	#423	—	53	2	no	—
142	C₂×C₄⋊C₈	—	53	1	no	—
143	C₂×C₄⋊C₈	—	53	2	no	—
144	C₅₃	#2906	1	1	yes	characteristic, complemented, Sylow
145	C₂×#41	—	53	1	no	—
146	#131	—	106	4	no	—
147	#131	—	106	4	no	—
148	C₂×M₁₆	—	53	1	no	—
149	C₂×M₁₆	—	53	2	no	—
150	C₂×M₁₆	—	53	2	no	—
151	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
152	C₂²×C₈	—	53	1	no	—
153	C₂²×C₈	—	53	2	no	—
154	C₂²×C₈	—	53	2	no	—
155	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
156	C₄×Q₈	—	53	1	no	—
157	C₄×Q₈	—	53	1	no	—
158	C₄×D₄	—	53	1	no	—
159	C₄×D₄	—	53	1	no	—
160	C₄×D₄	—	53	2	no	—
161	C₄×D₄	—	53	2	no	—
162	#117	—	53	2	no	—
163	#117	C₂×D₅₃	1	1	yes	characteristic
164	C₂×C₄²	—	53	1	no	—
165	C₂×C₄²	—	53	2	no	—
166	C₄⋊C₈	—	53	1	no	—
167	C₄⋊C₈	—	53	2	no	—
168	C₄⋊C₈	—	53	1	no	—
169	C₄⋊C₈	—	53	1	no	—
170	C₄⋊C₈	—	53	1	no	—
171	C₄⋊C₈	—	53	2	no	—
172	C₄⋊C₈	—	53	2	no	—
173	C₄⋊C₈	—	53	2	no	—
174	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
175	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
176	#41	—	53	4	no	—
177	#41	—	53	4	no	—
178	C₂×Q₈	—	53	1	no	—
179	C₂×D₄	—	53	2	no	—
180	C₂×D₄	—	53	1	no	—
181	C₂²×C₄	—	53	2	no	—
182	C₂²×C₄	—	53	1	no	—
183	C₂²×C₄	—	53	1	no	—
184	C₂²×C₄	—	53	2	no	—
185	C₂²×C₄	—	53	2	no	—
186	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
187	M₁₆	—	2	2	no	—
188	M₁₆	—	106	4	no	—
189	M₁₆	—	106	4	no	—
190	M₁₆	—	106	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	53	1	no	—
193	C₂×C₈	—	53	1	no	—
194	C₂×C₈	—	53	2	no	—
195	C₂×C₈	—	106	4	no	—
196	C₂×C₈	—	106	4	no	—
197	C₂×C₈	—	106	2	no	—
198	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
199	C₂×C₈	C₂²×D₅₃	1	2	yes	—
200	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
201	C₄⋊C₄	—	53	2	no	—
202	C₄⋊C₄	—	53	2	no	—
203	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
204	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
205	C₂²⋊C₄	—	53	2	no	—
206	C₂²⋊C₄	—	53	2	no	—
207	C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
208	C₄²	—	53	2	no	—
209	C₄²	—	53	1	no	—
210	C₄²	—	53	2	no	—
211	C₄²	—	53	1	no	—
212	C₄²	—	53	1	no	—
213	C₄²	—	53	1	no	—
214	C₄²	C₂²×D₅₃	1	2	yes	—
215	C₂³	—	53	2	no	—
216	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
217	Q₈	—	53	4	no	—
218	D₄	—	53	8	no	—
219	D₄	—	53	4	no	—
220	C₂×C₄	—	53	1	no	—
221	C₂×C₄	—	53	2	no	—
222	C₂×C₄	—	53	4	no	—
223	C₂×C₄	—	53	4	no	—
224	C₂×C₄	—	53	1	no	—
225	C₂×C₄	—	53	1	no	—
226	C₂×C₄	—	53	1	no	—
227	C₂×C₄	—	106	4	no	—
228	C₂×C₄	—	106	2	no	—
229	C₂×C₄	—	212	2	no	—
230	C₂×C₄	—	212	1	no	—
231	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
232	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
233	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
234	C₂×C₄	D₄×D₅₃	1	2	yes	—
235	C₂×C₄	Q₈×D₅₃	1	2	yes	—
236	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	2	no	—
240	C₈	—	106	1	no	—
241	C₈	—	106	2	no	—
242	C₈	—	106	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	53	4	no	—
246	C₂²	—	106	2	no	—
247	C₂²	—	106	2	no	—
248	C₂²	C₄⋊C₄×D₅₃	1	2	yes	—
249	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	53	4	no	—
253	C₄	—	53	2	no	—
254	C₄	—	106	1	no	—
255	C₄	—	106	1	no	—
256	C₄	—	106	1	no	—
257	C₄	C₄⋊C₄×D₅₃	1	2	yes	—
258	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
259	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
260	C₂	—	2	2	no	—
261	C₂	—	106	4	no	—
262	C₂	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
263	C₂	#131×D₅₃	1	1	yes	characteristic
264	C₂	#16811388	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#424	C₂
hypercenter	#424	D₅₃
hypocenter	C₅₃	#2906
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
4. #16810650	C₂	semidirect
5. #16810648	C₂	semidirect
6. #16810648	C₂	semidirect
7. #16810646	C₂	semidirect
8. #16810645	C₂	semidirect
9. C₂×#13133146	C₂	semidirect
10. C₄⋊C₈×D₅₃	C₂	semidirect
11. C₅₃×#424	C₂	semidirect
126. Dic₅₃	#131	semidirect
139. #424	D₅₃	semidirect
144. C₅₃	#2906	semidirect