Subgroups — #20283789

Order 6784.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811170	C₂	1	1	yes	maximal, characteristic
2	#16810701	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810700	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810697	C₂	1	2	yes	maximal, complemented
5	#16810694	C₂	1	2	yes	maximal, complemented
6	C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810679	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810674	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810589	C₂	1	2	yes	maximal, complemented
10	#16810588	C₂	1	2	yes	maximal, complemented
11	C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	2	yes	—
13	C₂×#11719917	C₂²	1	2	yes	—
14	C₂×#11719912	C₂²	1	2	yes	—
15	C₂×C₈×D₅₃	C₂²	1	2	yes	—
16	#13133216	C₄	1	1	yes	characteristic
17	#13133213	C₄	1	1	yes	characteristic
18	#13133203	C₄	1	2	yes	—
19	#13133192	C₄	1	1	yes	characteristic
20	#13133190	C₄	1	2	yes	—
21	#13133188	C₄	1	1	yes	characteristic
22	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
24	C₄²×D₅₃	C₂²	1	1	yes	characteristic
25	C₄⋊#88322	C₂²	1	1	yes	characteristic
26	#13133146	C₂²	1	1	yes	characteristic
27	#13133146	C₂²	1	1	yes	characteristic
28	C₂²⋊#88322	C₂²	1	2	yes	—
29	#13133129	C₂²	1	1	yes	characteristic
30	#13133129	C₂²	1	2	yes	—
31	#13133129	C₂²	1	1	yes	characteristic
32	#13133129	C₂²	1	2	yes	—
33	#13133129	C₂²	1	2	yes	—
34	#13133127	C₂²	1	2	yes	—
35	C₄×#88322	C₂²	1	1	yes	characteristic
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×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
46	#11719945	—	4	2	no	—
47	#11719945	—	4	2	no	—
48	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
49	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₅₃	C₂³	1	2	yes	—
54	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
55	C₂×C₈×C₅₃	—	4	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	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
61	C₄²×C₅₃	C₂³	1	1	yes	characteristic
62	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
63	#11719921	C₂×C₄	1	2	yes	—
64	#11719921	C₂×C₄	1	2	yes	—
65	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
66	#11719919	C₂×C₄	1	2	yes	—
67	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
70	#11719917	—	4	2	no	—
71	C₂×#88322	C₂³	1	1	yes	characteristic
72	C₂×#88322	C₂³	1	1	yes	characteristic
73	C₂×#88322	C₂³	1	2	yes	—
74	#11719912	—	4	2	no	—
75	C₈×D₅₃	—	4	2	no	—
76	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
79	D₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₂²⋊D₅₃	—	4	2	no	—
87	C₂²⋊D₅₃	—	4	2	no	—
88	C₂²⋊D₅₃	—	4	2	no	—
89	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
91	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
93	D₂₁₂	—	2	2	no	—
94	D₂₁₂	—	4	2	no	—
95	C₄×D₅₃	—	2	2	no	—
96	C₄×D₅₃	—	2	1	no	—
97	C₄×D₅₃	—	2	1	no	—
98	C₄×D₅₃	—	2	2	no	—
99	C₄×D₅₃	—	2	2	no	—
100	C₄×D₅₃	—	4	1	no	—
101	Dic₁₀₆	—	4	2	no	—
102	C₈×C₅₃	—	2	1	no	—
103	C₈×C₅₃	—	2	2	no	—
104	C₈×C₅₃	—	2	1	no	—
105	#88322	—	2	2	no	—
106	#88322	—	2	1	no	—
107	#88322	—	2	1	no	—
108	C₂²×C₅₃	—	2	2	no	—
109	C₂²×C₅₃	—	2	2	no	—
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×D₅₃	—	2	1	no	—
113	C₂×D₅₃	—	2	2	no	—
114	C₂×D₅₃	—	2	1	no	—
115	C₂×D₅₃	—	2	2	no	—
116	C₂×D₅₃	—	2	1	no	—
117	C₂×D₅₃	—	2	2	no	—
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	#131	1	2	yes	—
121	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
122	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
123	Dic₅₃	—	2	1	no	—
124	Dic₅₃	—	2	2	no	—
125	Dic₅₃	—	2	1	no	—
126	Dic₅₃	—	2	1	no	—
127	#2981	—	53	1	no	maximal, Sylow
128	C₂×C₅₃	—	4	2	no	—
129	C₂×C₅₃	#518	1	1	yes	characteristic
130	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
131	C₂×C₅₃	#568	1	1	yes	characteristic
132	D₅₃	—	4	1	no	—
133	D₅₃	—	4	2	no	—
134	#532	—	53	1	no	—
135	#436	—	53	1	no	—
136	#436	—	53	1	no	—
137	#435	—	53	2	no	—
138	#435	—	53	2	no	—
139	C₈×D₄	—	53	1	no	—
140	C₈×D₄	D₅₃	1	1	yes	characteristic, complemented
141	#432	—	53	1	no	—
142	#431	—	53	1	no	—
143	#408	—	53	2	no	—
144	#408	—	53	2	no	—
145	C₅₃	#2981	1	1	yes	characteristic, complemented, Sylow
146	C₂×#41	—	53	2	no	—
147	C₂×M₁₆	—	53	2	no	—
148	C₂×M₁₆	—	53	2	no	—
149	C₂²×C₈	—	53	2	no	—
150	C₂²×C₈	C₂×D₅₃	1	2	yes	—
151	C₄⋊Q₈	—	53	1	no	—
152	#127	—	53	1	no	—
153	#124	—	53	2	no	—
154	#121	—	53	1	no	—
155	#121	—	53	1	no	—
156	#121	—	53	2	no	—
157	C₄×D₄	—	53	2	no	—
158	C₄×D₄	—	53	1	no	—
159	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
160	C₂×C₄²	—	53	1	no	—
161	C₄⋊C₈	—	53	1	no	—
162	C₄⋊C₈	—	53	1	no	—
163	C₄⋊C₈	—	53	1	no	—
164	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
165	C₂²⋊C₈	—	53	2	no	—
166	C₂²⋊C₈	—	53	1	no	—
167	C₂²⋊C₈	—	53	1	no	—
168	C₂²⋊C₈	—	53	2	no	—
169	C₂²⋊C₈	—	53	2	no	—
170	C₂²⋊C₈	—	53	2	no	—
171	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
172	#97	—	53	2	no	—
173	C₄×C₈	—	53	1	no	—
174	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
175	#41	—	212	2	no	—
176	#41	—	212	2	no	—
177	C₂×Q₈	—	53	2	no	—
178	C₂×D₄	—	53	2	no	—
179	C₂×D₄	—	53	2	no	—
180	C₂×D₄	—	53	2	no	—
181	C₂×D₄	—	53	1	no	—
182	C₂×D₄	—	53	2	no	—
183	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
184	C₂²×C₄	—	53	1	no	—
185	C₂²×C₄	—	53	1	no	—
186	C₂²×C₄	—	53	2	no	—
187	C₂²×C₄	—	53	1	no	—
188	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
189	M₁₆	—	212	2	no	—
190	M₁₆	—	212	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	53	1	no	—
193	C₂×C₈	—	53	1	no	—
194	C₂×C₈	—	53	2	no	—
195	C₂×C₈	—	212	2	no	—
196	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
197	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
198	C₂×C₈	C₂²×D₅₃	1	2	yes	—
199	C₄⋊C₄	—	53	2	no	—
200	C₄⋊C₄	—	53	1	no	—
201	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
202	C₂²⋊C₄	—	53	2	no	—
203	C₂²⋊C₄	—	53	2	no	—
204	C₂²⋊C₄	—	53	2	no	—
205	C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
206	C₄²	—	53	1	no	—
207	C₄²	—	53	1	no	—
208	C₄²	—	53	1	no	—
209	C₄²	C₂²×D₅₃	1	1	yes	characteristic
210	C₂³	—	53	2	no	—
211	C₂³	—	53	1	no	—
212	C₂³	C₂×C₄×D₅₃	1	2	yes	—
213	Q₈	—	212	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	106	2	no	—
216	D₄	—	212	2	no	—
217	D₄	—	212	2	no	—
218	D₄	—	212	2	no	—
219	D₄	—	212	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	53	1	no	—
223	C₂×C₄	—	53	1	no	—
224	C₂×C₄	—	53	1	no	—
225	C₂×C₄	—	53	2	no	—
226	C₂×C₄	—	106	2	no	—
227	C₂×C₄	—	106	2	no	—
228	C₂×C₄	—	106	1	no	—
229	C₂×C₄	—	106	2	no	—
230	C₂×C₄	—	106	1	no	—
231	C₂×C₄	—	212	1	no	—
232	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
234	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	2	1	no	—
239	C₈	—	106	1	no	—
240	C₈	—	106	2	no	—
241	C₈	—	106	1	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	106	1	no	—
246	C₂²	—	106	2	no	—
247	C₂²	—	106	1	no	—
248	C₂²	—	106	2	no	—
249	C₂²	—	106	1	no	—
250	C₂²	—	106	2	no	—
251	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	106	1	no	—
255	C₄	—	106	2	no	—
256	C₄	—	106	1	no	—
257	C₄	—	106	1	no	—
258	C₄	#13133223	1	2	yes	—
259	C₄	#13133262	1	1	yes	characteristic
260	C₄	#13133263	1	1	yes	characteristic
261	C₂	—	4	2	no	—
262	C₂	—	212	1	no	—
263	C₂	—	212	2	no	—
264	C₂	#16811095	1	1	yes	characteristic
265	C₂	C₂×#13133223	1	1	yes	characteristic
266	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	#2981
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. #16810701	C₂	semidirect
3. #16810700	C₂	semidirect
4. #16810697	C₂	semidirect
5. #16810694	C₂	semidirect
6. C₈×D₂₁₂	C₂	semidirect
7. #16810679	C₂	semidirect
8. #16810674	C₂	semidirect
9. #16810589	C₂	semidirect
10. #16810588	C₂	semidirect
11. C₈×D₄×C₅₃	C₂	semidirect
140. C₈×D₄	D₅₃	semidirect
145. C₅₃	#2981	semidirect