Subgroups — #20284497

Order 6784.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11719945	C₂	1	1	yes	maximal, characteristic
2	#16810927	C₂	1	2	yes	maximal, complemented
3	#16810920	C₂	1	2	yes	maximal, complemented
4	#16810915	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810910	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810905	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810900	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810659	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
10	#16810617	C₂	1	2	yes	maximal
11	C₅₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	1	yes	characteristic
13	C₂²×#88322	C₂²	1	2	yes	—
14	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
16	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
17	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
18	C₄×C₂²⋊D₅₃	—	2	1	no	—
19	C₄×C₂²⋊D₅₃	—	2	1	no	—
20	#13133176	C₂²	1	1	yes	characteristic
21	#13133174	—	2	1	no	—
22	C₄²×D₅₃	—	2	1	no	—
23	#13133150	C₂²	1	2	yes	—
24	#13133150	C₂²	1	2	yes	—
25	#13133148	C₂²	1	2	yes	—
26	#13133148	C₂²	1	2	yes	—
27	C₄⋊#88322	C₂²	1	1	yes	characteristic
28	C₄⋊#88322	C₂²	1	1	yes	characteristic
29	C₄⋊#88322	C₂²	1	2	yes	—
30	#13133140	C₂²	1	2	yes	—
31	#13133140	C₂²	1	2	yes	—
32	#13133137	C₂²	1	2	yes	—
33	#13133137	C₂²	1	2	yes	—
34	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
36	#125×C₅₃	C₂²	1	2	yes	—
37	C₂²⋊Q₈×C₅₃	—	2	1	no	—
38	C₂²⋊Q₈×C₅₃	—	2	1	no	—
39	C₄×Q₈×C₅₃	—	2	1	no	—
40	C₄×Q₈×C₅₃	—	2	1	no	—
41	#117×C₅₃	C₂²	1	2	yes	—
42	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×Q₈×C₅₃	—	2	1	no	—
44	C₂×Q₈×C₅₃	—	2	1	no	—
45	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₅₃	—	2	1	no	—
49	#11719945	—	2	2	no	—
50	#11719945	Q₈	1	4	yes	complemented
51	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₅₃	—	2	1	no	—
53	C₂×C₄×D₅₃	—	2	1	no	—
54	C₂×C₄×D₅₃	—	2	1	no	—
55	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₄⋊C₄×C₅₃	—	2	1	no	—
59	C₄⋊C₄×C₅₃	—	2	1	no	—
60	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
61	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
62	C₂²⋊C₄×C₅₃	—	2	2	no	—
63	C₄²×C₅₃	—	2	2	no	—
64	C₄²×C₅₃	D₄	1	1	yes	characteristic
65	C₄²×C₅₃	D₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₂²⋊Dic₅₃	—	2	1	no	—
69	#11719921	—	2	1	no	—
70	#11719921	—	2	1	no	—
71	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
73	#11719919	—	2	1	no	—
74	#11719919	—	2	1	no	—
75	C₄×Dic₅₃	—	2	1	no	—
76	C₄×Dic₅₃	—	2	1	no	—
77	C₄×Dic₅₃	—	2	1	no	—
78	C₂×#88322	—	2	4	no	—
79	C₂×#88322	C₂³	1	2	yes	—
80	C₂×#88322	C₂³	1	2	yes	—
81	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₅₃	—	2	1	no	—
83	Q₈×C₅₃	—	4	2	no	—
84	Q₈×C₅₃	—	4	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	4	1	no	—
89	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
95	C₂×C₄×C₅₃	#41	1	2	yes	—
96	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₅₃	—	2	4	no	—
98	C₂×Dic₅₃	—	2	1	no	—
99	C₂×Dic₅₃	—	2	1	no	—
100	C₂×Dic₅₃	—	2	1	no	—
101	D₂₁₂	—	2	1	no	—
102	D₂₁₂	C₂×Q₈	1	2	yes	—
103	C₄×D₅₃	—	2	4	no	—
104	C₄×D₅₃	—	4	2	no	—
105	C₄×D₅₃	—	4	2	no	—
106	Dic₁₀₆	—	2	1	no	—
107	Dic₁₀₆	C₂×Q₈	1	2	yes	—
108	#88322	—	2	2	no	—
109	#88322	—	2	2	no	—
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
113	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×D₅₃	—	2	2	no	—
115	C₂×D₅₃	—	4	1	no	—
116	C₂×D₅₃	—	4	1	no	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	4	2	no	—
121	C₄×C₅₃	—	4	2	no	—
122	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
123	C₄×C₅₃	#135	1	2	yes	—
124	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
125	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
126	Dic₅₃	—	2	2	no	—
127	Dic₅₃	—	4	1	no	—
128	Dic₅₃	—	4	1	no	—
129	#3076	—	53	1	no	maximal, Sylow
130	C₂×C₅₃	—	2	2	no	—
131	C₂×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
133	C₂×C₅₃	#575	1	1	yes	characteristic
134	D₅₃	—	4	2	no	—
135	#533	D₅₃	1	1	yes	characteristic, complemented
136	C₄×#41	—	53	1	no	—
137	#479	—	53	2	no	—
138	#478	—	53	2	no	—
139	#477	—	53	1	no	—
140	#476	—	53	1	no	—
141	#475	—	53	1	no	—
142	#474	—	53	1	no	—
143	#427	—	53	2	no	—
144	C₂×C₄⋊C₈	—	53	1	no	—
145	#416	—	53	2	no	—
146	C₅₃	#3076	1	1	yes	characteristic, complemented, Sylow
147	C₂×#41	—	53	1	no	—
148	C₂²×C₈	—	53	2	no	—
149	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
150	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
151	#125	C₂×D₅₃	1	2	yes	—
152	C₂²⋊Q₈	—	2	1	no	—
153	C₂²⋊Q₈	—	2	1	no	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	—	2	1	no	—
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	—	106	1	no	—
161	C₄×D₄	—	106	1	no	—
162	#117	—	53	1	no	—
163	#117	—	106	1	no	—
164	#117	C₂×D₅₃	1	2	yes	—
165	C₂×C₄²	—	106	1	no	—
166	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
167	#107	—	53	2	no	—
168	#107	—	53	2	no	—
169	#106	—	53	2	no	—
170	#106	—	53	2	no	—
171	C₄⋊C₈	—	53	1	no	—
172	C₄⋊C₈	—	53	1	no	—
173	C₄⋊C₈	—	53	2	no	—
174	#103	—	53	2	no	—
175	#103	—	53	2	no	—
176	#102	—	53	2	no	—
177	#102	—	53	2	no	—
178	#41	—	53	4	no	—
179	#41	—	106	2	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	—	53	1	no	—
183	C₂×D₄	—	53	1	no	—
184	C₂×D₄	—	106	1	no	—
185	C₂²×C₄	—	106	1	no	—
186	C₂²×C₄	—	106	1	no	—
187	C₂²×C₄	—	106	1	no	—
188	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
191	C₂×C₈	—	53	2	no	—
192	C₂×C₈	—	53	2	no	—
193	C₂×C₈	—	106	4	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	53	1	no	—
199	C₄⋊C₄	—	53	1	no	—
200	C₄⋊C₄	—	106	1	no	—
201	C₄⋊C₄	—	106	1	no	—
202	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
203	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	106	1	no	—
206	C₂²⋊C₄	—	106	1	no	—
207	C₂²⋊C₄	—	106	1	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	106	1	no	—
210	C₄²	—	106	1	no	—
211	C₄²	—	106	1	no	—
212	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
213	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
214	C₄²	C₂²×D₅₃	1	1	yes	characteristic
215	C₄²	C₂²×D₅₃	1	1	yes	characteristic
216	C₂³	—	106	1	no	—
217	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
218	Q₈	—	4	2	no	—
219	Q₈	—	4	2	no	—
220	Q₈	—	53	2	no	—
221	Q₈	—	106	1	no	—
222	D₄	—	53	2	no	—
223	D₄	—	106	4	no	—
224	D₄	—	106	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	106	1	no	—
230	C₂×C₄	—	106	4	no	—
231	C₂×C₄	—	106	1	no	—
232	C₂×C₄	—	106	1	no	—
233	C₂×C₄	—	212	2	no	—
234	C₂×C₄	—	212	2	no	—
235	C₂×C₄	Q₈×D₅₃	1	2	yes	—
236	C₂×C₄	#11719949	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
241	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
242	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
243	C₈	—	106	2	no	—
244	C₈	—	106	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	106	2	no	—
248	C₂²	—	212	1	no	—
249	C₂²	—	212	1	no	—
250	C₂²	#13133193	1	2	yes	—
251	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	106	2	no	—
258	C₄	—	212	1	no	—
259	C₄	—	212	1	no	—
260	C₄	#13133193	1	2	yes	—
261	C₄	#13133237	1	2	yes	—
262	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
263	C₄	C₂×#11719949	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	212	2	no	—
266	C₂	C₂×#13133193	1	1	yes	characteristic
267	C₂	C₂×#13133237	1	1	yes	characteristic
268	C₂	#16811421	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#533	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#533	D₅₃
hypocenter	C₅₃	#3076
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
2. #16810927	C₂	semidirect
3. #16810920	C₂	semidirect
4. #16810915	C₂	semidirect
5. #16810910	C₂	semidirect
6. #16810905	C₂	semidirect
7. #16810900	C₂	semidirect
8. #16810659	C₂	semidirect
9. C₂×C₄⋊#88322	C₂	semidirect
11. C₅₃×#533	C₂	semidirect
50. #11719945	Q₈	semidirect
135. #533	D₅₃	semidirect
146. C₅₃	#3076	semidirect