Subgroups — #20283773

Order 6784.

402 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	#16811130	C₂	1	1	yes	maximal, characteristic
2	#16810699	C₂	1	2	yes	maximal, complemented
3	#16810698	C₂	1	2	yes	maximal, complemented
4	#16810694	C₂	1	2	yes	maximal, complemented
5	#16810691	C₂	1	2	yes	maximal, complemented
6	#16810681	C₂	1	2	yes	maximal, complemented
7	#16810590	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810585	C₂	1	2	yes	maximal, complemented
9	C₅₃×#408	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
11	C₂×#11719917	C₂²	1	1	yes	characteristic, complemented
12	C₂×#11719912	C₂²	1	2	yes	—
13	C₂×#11719912	C₂²	1	2	yes	—
14	C₂²×#88322	C₂²	1	1	yes	characteristic, complemented
15	#13133198	C₄	1	1	yes	characteristic
16	#13133198	C₄	1	1	yes	characteristic
17	#13133193	C₄	1	2	yes	—
18	#13133188	C₄	1	2	yes	—
19	#13133185	C₄	1	1	yes	characteristic
20	#13133185	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
22	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
23	#13133176	C₂²	1	1	yes	characteristic
24	#13133146	C₂²	1	2	yes	complemented
25	#13133146	C₂²	1	2	yes	complemented
26	C₂²⋊#88322	C₂²	1	2	yes	—
27	C₂²⋊#88322	C₂²	1	2	yes	complemented
28	#13133129	C₂²	1	2	yes	—
29	#13133129	C₂²	1	2	yes	—
30	#13133127	C₂²	1	2	yes	complemented
31	C₈×Dic₅₃	C₂²	1	2	yes	complemented
32	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
34	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
35	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
36	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	complemented
37	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
38	#41×C₅₃	—	4	1	no	—
39	#41×C₅₃	—	4	1	no	—
40	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
49	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₅₃	—	4	1	no	—
51	C₂×C₄×D₅₃	—	4	1	no	—
52	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₅₃	C₂³	1	2	yes	—
54	M₁₆×C₅₃	—	4	1	no	—
55	C₂×C₈×C₅₃	—	4	1	no	—
56	C₂×C₈×C₅₃	C₂³	1	2	yes	—
57	C₂×C₈×C₅₃	C₂³	1	2	yes	—
58	C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
62	#11719921	C₂×C₄	1	2	yes	—
63	#11719921	C₂×C₄	1	2	yes	—
64	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
65	#11719919	C₂×C₄	1	2	yes	—
66	#11719919	C₂×C₄	1	2	yes	—
67	C₄×Dic₅₃	C₂³	1	2	yes	—
68	#11719917	—	4	1	no	—
69	C₂×#88322	—	4	1	no	—
70	C₂×#88322	C₂³	1	2	yes	—
71	C₂×#88322	C₂³	1	2	yes	—
72	#11719912	—	4	2	no	—
73	#11719912	—	4	2	no	—
74	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₅₃	—	4	1	no	—
78	C₂²×D₅₃	—	4	2	no	—
79	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
80	Q₈×C₅₃	—	4	1	no	—
81	D₄×C₅₃	—	4	1	no	—
82	D₄×C₅₃	—	4	1	no	—
83	D₄×C₅₃	—	4	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₅₃	—	4	2	no	—
95	C₂²⋊D₅₃	—	4	2	no	—
96	C₂×Dic₅₃	—	4	1	no	—
97	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
98	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
99	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
100	C₄×D₅₃	—	2	2	no	—
101	C₄×D₅₃	—	2	2	no	—
102	C₄×D₅₃	—	4	2	no	—
103	C₈×C₅₃	—	2	2	no	—
104	C₈×C₅₃	—	2	2	no	—
105	#88322	—	2	2	no	—
106	#88322	—	2	2	no	—
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	—	2	1	no	—
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×D₅₃	—	2	2	no	—
115	C₂×D₅₃	—	2	2	no	—
116	C₂×D₅₃	—	2	2	no	—
117	C₂×D₅₃	—	4	4	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	2	1	no	—
121	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
122	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
123	Dic₅₃	—	2	2	no	—
124	Dic₅₃	—	2	2	no	—
125	Dic₅₃	—	2	2	no	—
126	#2978	—	53	1	no	maximal, Sylow
127	C₂×C₅₃	—	4	1	no	—
128	C₂×C₅₃	—	4	1	no	—
129	C₂×C₅₃	—	4	1	no	—
130	C₂×C₅₃	#518	1	1	yes	characteristic
131	C₂×C₅₃	#568	1	2	yes	—
132	D₅₃	—	4	2	no	—
133	#525	—	53	1	no	—
134	#436	—	53	2	no	—
135	#436	—	53	2	no	—
136	#435	—	53	2	no	—
137	#435	—	53	2	no	—
138	#433	—	53	2	no	—
139	#408	—	53	1	no	—
140	#408	D₅₃	1	1	yes	characteristic, complemented
141	#407	—	53	2	no	—
142	C₅₃	#2978	1	1	yes	characteristic, complemented, Sylow
143	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
144	C₂³×C₄	—	53	1	no	—
145	C₂×M₁₆	—	53	1	no	—
146	C₂×M₁₆	—	53	2	no	—
147	C₂×M₁₆	—	53	2	no	—
148	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
149	C₂²×C₈	—	53	1	no	—
150	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
151	#123	—	53	1	no	—
152	#123	—	53	1	no	—
153	C₂²⋊Q₈	—	53	2	no	—
154	#121	—	53	2	no	—
155	#120	—	53	1	no	—
156	#120	—	53	1	no	—
157	C₄×D₄	—	53	2	no	—
158	C₄×D₄	—	53	2	no	—
159	#117	—	53	1	no	—
160	C₄⋊C₈	—	53	2	no	—
161	C₄⋊C₈	—	53	2	no	—
162	C₂²⋊C₈	—	53	2	no	—
163	C₂²⋊C₈	—	53	2	no	—
164	C₂²⋊C₈	—	53	2	no	—
165	C₂²⋊C₈	—	53	2	no	—
166	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
167	C₂²⋊C₈	C₂×D₅₃	1	2	yes	complemented
168	#97	—	53	2	no	—
169	C₄×C₈	—	53	2	no	—
170	C₂⁴	—	53	1	no	—
171	#41	—	4	1	no	—
172	#41	—	4	1	no	—
173	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
174	C₂×D₄	—	53	2	no	—
175	C₂×D₄	—	53	2	no	—
176	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
177	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
178	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
179	C₂²×C₄	—	53	1	no	—
180	C₂²×C₄	—	53	2	no	—
181	C₂²×C₄	—	53	2	no	—
182	C₂²×C₄	—	212	1	no	—
183	C₂²×C₄	—	212	1	no	—
184	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
187	M₁₆	—	4	1	no	—
188	M₁₆	—	212	2	no	—
189	M₁₆	—	212	2	no	—
190	M₁₆	—	212	1	no	—
191	C₂×C₈	—	4	1	no	—
192	C₂×C₈	—	53	2	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	—	212	1	no	—
195	C₂×C₈	C₂²×D₅₃	1	2	yes	—
196	C₂×C₈	C₂²×D₅₃	1	2	yes	—
197	C₄⋊C₄	—	53	2	no	—
198	C₄⋊C₄	—	53	2	no	—
199	C₄⋊C₄	—	53	2	no	—
200	C₂²⋊C₄	—	53	1	no	—
201	C₂²⋊C₄	—	53	1	no	—
202	C₂²⋊C₄	—	53	2	no	—
203	C₂²⋊C₄	—	53	2	no	—
204	C₂²⋊C₄	—	53	2	no	—
205	C₂²⋊C₄	—	53	2	no	—
206	C₄²	—	53	2	no	—
207	C₂³	—	53	2	no	—
208	C₂³	—	212	1	no	—
209	C₂³	—	212	2	no	—
210	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
211	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
212	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
213	Q₈	—	4	1	no	—
214	D₄	—	4	1	no	—
215	D₄	—	4	1	no	—
216	D₄	—	4	1	no	—
217	D₄	—	212	2	no	—
218	D₄	—	212	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	53	2	no	—
226	C₂×C₄	—	53	2	no	—
227	C₂×C₄	—	53	2	no	—
228	C₂×C₄	—	106	2	no	—
229	C₂×C₄	—	106	2	no	—
230	C₂×C₄	—	212	2	no	—
231	C₂×C₄	—	212	1	no	—
232	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	2	no	—
238	C₈	—	106	2	no	—
239	C₈	—	106	2	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	106	2	no	—
247	C₂²	—	106	2	no	—
248	C₂²	—	106	2	no	—
249	C₂²	—	212	4	no	—
250	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	106	2	no	—
255	C₄	—	106	2	no	—
256	C₄	—	106	2	no	—
257	C₄	#13133262	1	1	yes	characteristic
258	C₄	#13133262	1	1	yes	characteristic
259	C₂	—	4	1	no	—
260	C₂	—	4	1	no	—
261	C₂	—	4	1	no	—
262	C₂	—	212	2	no	—
263	C₂	#16811096	1	1	yes	characteristic
264	C₂	#16811388	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#408	C₂
hypercenter	#408	D₅₃
hypocenter	C₅₃	#2978
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
2. #16810699	C₂	semidirect
3. #16810698	C₂	semidirect
4. #16810694	C₂	semidirect
5. #16810691	C₂	semidirect
6. #16810681	C₂	semidirect
7. #16810590	C₂	semidirect
8. #16810585	C₂	semidirect
9. C₅₃×#408	C₂	semidirect
11. C₂×#11719917	C₂²	semidirect
14. C₂²×#88322	C₂²	semidirect
24. #13133146	C₂²	semidirect
25. #13133146	C₂²	semidirect
27. C₂²⋊#88322	C₂²	semidirect
30. #13133127	C₂²	semidirect
31. C₈×Dic₅₃	C₂²	semidirect
33. C₂×M₁₆×C₅₃	C₂²	semidirect
34. C₂²×C₈×C₅₃	C₂²	semidirect
36. C₂²⋊C₈×C₅₃	C₂²	semidirect
140. #408	D₅₃	semidirect
142. C₅₃	#2978	semidirect
148. C₂×M₁₆	C₂×D₅₃	semidirect
150. C₂²×C₈	C₂×D₅₃	semidirect
167. C₂²⋊C₈	C₂×D₅₃	semidirect