Subgroups — #20291475

Order 6784.

492 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811691	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811690	C₂	1	1	yes	maximal, characteristic
3	#16811662	C₂	1	4	yes	maximal, complemented
4	SD₁₆×#6157	C₂	1	4	yes	maximal, complemented
5	#16811633	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811616	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811615	C₂	1	1	yes	maximal, characteristic
8	#16811586	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×SD₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
10	D₅₃⋊M₁₆	C₂²	1	1	yes	characteristic
11	Q₈×#6157	—	2	2	no	—
12	Q₈×#6157	C₂²	1	4	yes	complemented
13	D₄×#6157	—	2	2	no	—
14	D₄×#6157	C₂²	1	4	yes	—
15	#13133296	—	2	2	no	—
16	#13133296	C₂²	1	1	yes	characteristic
17	#13133296	C₂²	1	1	yes	characteristic
18	#13133296	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊#6157	—	2	2	no	—
20	#13133286	C₂²	1	2	yes	complemented
21	#13133285	C₂²	1	2	yes	complemented
22	#13133280	C₂²	1	4	yes	complemented
23	#13133279	C₂²	1	4	yes	—
24	#13133271	C₂²	1	2	yes	complemented
25	C₈×#6157	C₂²	1	2	yes	complemented
26	C₂×Q₈×D₅₃	C₂²	1	1	yes	characteristic
27	C₂×D₄×D₅₃	C₂²	1	1	yes	characteristic
28	C₂×#11719913	C₄	1	1	yes	characteristic, complemented
29	C₂×#11719923	C₄	1	1	yes	characteristic, complemented
30	C₂×#11719924	C₄	1	1	yes	characteristic, complemented
31	SD₁₆×D₅₃	C₄	1	2	yes	complemented
32	SD₁₆×D₅₃	C₄	1	2	yes	complemented
33	SD₁₆×D₅₃	C₂²	1	2	yes	—
34	SD₁₆×D₅₃	C₂²	1	2	yes	—
35	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
36	C₂×SD₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
37	C₂³×D₅₃	—	2	1	no	—
38	C₂²×#6157	—	2	4	no	—
39	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
40	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊D₅₃	—	2	1	no	—
42	Q₈×D₅₃	—	2	1	no	—
43	Q₈×D₅₃	—	2	1	no	—
44	Q₈×D₅₃	C₂×C₄	1	2	yes	complemented
45	Q₈×D₅₃	C₂³	1	2	yes	—
46	D₄×D₅₃	—	2	1	no	—
47	D₄×D₅₃	—	2	1	no	—
48	D₄×D₅₃	C₂×C₄	1	2	yes	—
49	D₄×D₅₃	C₂³	1	2	yes	—
50	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₅₃	—	2	1	no	—
52	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
53	C₂×Dic₁₀₆	—	2	1	no	—
54	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic, complemented
55	C₂²⋊#6157	—	2	1	no	—
56	C₂²⋊#6157	—	2	1	no	—
57	C₂²⋊#6157	—	2	4	no	—
58	C₂²⋊#6157	D₄	1	2	yes	—
59	#11719940	—	2	1	no	—
60	C₄⋊#6157	—	2	4	no	—
61	C₄⋊#6157	D₄	1	2	yes	—
62	C₄⋊#6157	C₂³	1	2	yes	—
63	C₄⋊#6157	C₂³	1	2	yes	—
64	C₄×#6157	—	2	4	no	—
65	C₄×#6157	—	2	1	no	—
66	C₄×#6157	—	2	1	no	—
67	C₄×#6157	C₂³	1	2	yes	—
68	#11719936	—	2	1	no	—
69	D₅₃⋊C₈	C₂³	1	2	yes	—
70	SD₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
71	SD₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
72	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
73	#11719924	C₂×C₄	1	2	yes	complemented
74	#11719924	C₂×C₄	1	2	yes	complemented
75	#11719923	C₂×C₄	1	2	yes	complemented
76	#11719923	C₂×C₄	1	2	yes	complemented
77	C₂×#88322	C₂×C₄	1	1	yes	characteristic, complemented
78	#11719913	C₂×C₄	1	2	yes	complemented
79	#11719913	C₂×C₄	1	2	yes	complemented
80	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic, complemented
81	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic, complemented
82	C₈×D₅₃	C₂³	1	1	yes	characteristic
83	C₈×D₅₃	C₂³	1	1	yes	characteristic
84	C₂³×C₅₃	—	2	1	no	—
85	C₂²×D₅₃	—	2	1	no	—
86	C₂²×D₅₃	—	2	2	no	—
87	C₂²×D₅₃	—	2	2	no	—
88	C₂²×D₅₃	—	8	2	no	—
89	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×#6157	—	2	2	no	—
91	C₂×#6157	—	2	2	no	—
92	C₂×#6157	—	8	4	no	—
93	C₂×#6157	C₂×D₄	1	4	yes	—
94	Q₈×C₅₃	—	2	1	no	—
95	Q₈×C₅₃	C₂²×C₄	1	2	yes	—
96	D₄×C₅₃	—	2	1	no	—
97	D₄×C₅₃	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²⋊D₅₃	—	2	2	no	—
101	C₂²⋊D₅₃	—	2	2	no	—
102	C₂×Dic₅₃	—	2	1	no	—
103	C₂×Dic₅₃	#41	1	1	yes	characteristic
104	D₂₁₂	—	2	1	no	—
105	D₂₁₂	C₂²×C₄	1	2	yes	—
106	C₄×D₅₃	—	2	2	no	—
107	C₄×D₅₃	—	2	2	no	—
108	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
110	C₄×D₅₃	#41	1	1	yes	characteristic
111	C₄×D₅₃	C₂⁴	1	1	yes	characteristic
112	Dic₁₀₆	—	2	1	no	—
113	Dic₁₀₆	—	2	2	no	—
114	Dic₁₀₆	—	2	2	no	—
115	Dic₁₀₆	C₂²×C₄	1	2	yes	—
116	#88324	—	2	2	no	—
117	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	#88322	C₂²×C₄	1	1	yes	characteristic
120	#88322	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	—	8	1	no	—
123	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
124	C₂×D₅₃	—	2	2	no	—
125	C₂×D₅₃	—	2	2	no	—
126	C₂×D₅₃	—	4	4	no	—
127	C₂×D₅₃	—	8	2	no	—
128	C₂×D₅₃	—	8	1	no	—
129	C₂×D₅₃	C₄×D₄	1	1	yes	characteristic
130	C₂×D₅₃	C₂²×D₄	1	1	yes	characteristic
131	#6157	—	2	4	no	—
132	#6157	—	4	2	no	—
133	#6157	—	4	2	no	—
134	C₄×C₅₃	—	2	2	no	—
135	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
136	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
137	Dic₅₃	—	2	2	no	—
138	Dic₅₃	C₄×D₄	1	1	yes	characteristic
139	Dic₅₃	C₂×#41	1	1	yes	characteristic
140	#2949	—	53	1	no	maximal, Sylow
141	C₂×C₅₃	—	2	1	no	—
142	C₂×C₅₃	—	4	2	no	—
143	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
144	D₅₃	—	2	1	no	—
145	D₅₃	—	4	2	no	—
146	D₅₃	#577	1	2	yes	—
147	C₂²×SD₁₆	—	53	1	no	—
148	#519	—	53	1	no	—
149	#518	—	53	1	no	—
150	#440	—	53	4	no	—
151	C₄×SD₁₆	—	53	4	no	—
152	#427	—	53	1	no	—
153	#419	—	53	1	no	—
154	#418	—	53	1	no	—
155	#405	—	53	1	no	—
156	C₅₃	#2949	1	1	yes	characteristic, complemented, Sylow
157	C₂²×Q₈	—	53	1	no	—
158	C₂²×D₄	—	53	1	no	—
159	C₂×SD₁₆	—	53	1	no	—
160	C₂×SD₁₆	—	53	1	no	—
161	C₂×SD₁₆	—	53	2	no	—
162	C₂×SD₁₆	—	53	2	no	—
163	C₂×SD₁₆	—	53	2	no	—
164	C₂×SD₁₆	—	53	2	no	—
165	C₂×SD₁₆	—	53	1	no	—
166	C₂×SD₁₆	#6157	1	1	yes	characteristic, complemented
167	C₂×M₁₆	—	53	1	no	—
168	C₂²×C₈	—	53	1	no	—
169	C₄×Q₈	—	53	4	no	—
170	C₄×Q₈	—	106	2	no	—
171	C₄×D₄	—	53	4	no	—
172	C₄×D₄	—	106	2	no	—
173	#117	—	53	1	no	—
174	#117	—	53	1	no	—
175	#117	—	53	1	no	—
176	#117	—	106	2	no	—
177	C₂×C₂²⋊C₄	—	106	2	no	—
178	#107	—	53	2	no	—
179	#106	—	53	2	no	—
180	#103	—	53	4	no	—
181	#102	—	53	4	no	—
182	#97	—	53	2	no	—
183	C₄×C₈	—	53	2	no	—
184	C₂⁴	—	106	1	no	—
185	C₂×Q₈	—	53	2	no	—
186	C₂×Q₈	—	53	2	no	—
187	C₂×Q₈	—	53	1	no	—
188	C₂×Q₈	—	106	1	no	—
189	C₂×Q₈	—	106	1	no	—
190	C₂×Q₈	—	106	1	no	—
191	C₂×Q₈	C₂×#6157	1	1	yes	characteristic, complemented
192	C₂×D₄	—	53	2	no	—
193	C₂×D₄	—	53	2	no	—
194	C₂×D₄	—	53	1	no	—
195	C₂×D₄	—	106	1	no	—
196	C₂×D₄	—	106	1	no	—
197	C₂×D₄	—	106	1	no	—
198	C₂×D₄	C₂×#6157	1	1	yes	characteristic
199	C₂²×C₄	—	53	1	no	—
200	C₂²×C₄	—	106	1	no	—
201	C₂²×C₄	—	106	4	no	—
202	SD₁₆	—	53	2	no	—
203	SD₁₆	—	53	2	no	—
204	SD₁₆	—	53	2	no	—
205	SD₁₆	—	53	2	no	—
206	SD₁₆	—	53	2	no	—
207	SD₁₆	—	53	2	no	—
208	SD₁₆	C₂×#6157	1	2	yes	complemented
209	SD₁₆	C₂×#6157	1	2	yes	complemented
210	M₁₆	—	106	1	no	—
211	M₁₆	—	106	1	no	—
212	C₂×C₈	—	53	1	no	—
213	C₂×C₈	—	53	1	no	—
214	C₂×C₈	—	53	1	no	—
215	C₂×C₈	—	53	1	no	—
216	C₂×C₈	—	53	1	no	—
217	C₂×C₈	—	53	2	no	—
218	C₂×C₈	C₂×#6157	1	1	yes	characteristic, complemented
219	C₄⋊C₄	—	53	2	no	—
220	C₄⋊C₄	—	53	2	no	—
221	C₄⋊C₄	—	53	2	no	—
222	C₄⋊C₄	—	106	4	no	—
223	C₂²⋊C₄	—	53	2	no	—
224	C₂²⋊C₄	—	106	4	no	—
225	C₂²⋊C₄	—	106	1	no	—
226	C₂²⋊C₄	—	106	1	no	—
227	C₄²	—	53	2	no	—
228	C₄²	—	106	1	no	—
229	C₄²	—	106	1	no	—
230	C₄²	—	106	4	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	53	1	no	—
233	C₂³	—	106	2	no	—
234	C₂³	—	106	2	no	—
235	C₂³	—	106	1	no	—
236	C₂³	—	424	2	no	—
237	Q₈	—	2	1	no	—
238	Q₈	—	53	2	no	—
239	Q₈	—	106	2	no	—
240	Q₈	—	106	2	no	—
241	Q₈	—	106	1	no	—
242	Q₈	C₂²×#6157	1	2	yes	—
243	D₄	—	2	1	no	—
244	D₄	—	53	2	no	—
245	D₄	—	106	2	no	—
246	D₄	—	106	2	no	—
247	D₄	—	106	1	no	—
248	D₄	C₂²×#6157	1	2	yes	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	53	1	no	—
251	C₂×C₄	—	53	1	no	—
252	C₂×C₄	—	53	1	no	—
253	C₂×C₄	—	53	1	no	—
254	C₂×C₄	—	53	1	no	—
255	C₂×C₄	—	53	4	no	—
256	C₂×C₄	—	106	2	no	—
257	C₂×C₄	—	106	2	no	—
258	C₂×C₄	—	106	1	no	—
259	C₂×C₄	—	106	2	no	—
260	C₂×C₄	—	106	2	no	—
261	C₂×C₄	—	424	4	no	—
262	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
263	C₈	—	53	1	no	—
264	C₈	—	53	1	no	—
265	C₈	—	106	2	no	—
266	C₈	C₂²×#6157	1	1	yes	characteristic
267	C₈	C₂²×#6157	1	1	yes	characteristic
268	C₂²	—	2	2	no	—
269	C₂²	—	8	1	no	—
270	C₂²	—	53	1	no	—
271	C₂²	—	53	1	no	—
272	C₂²	—	106	2	no	—
273	C₂²	—	106	2	no	—
274	C₂²	—	424	1	no	—
275	C₂²	—	424	2	no	—
276	C₂²	—	212	4	no	—
277	C₂²	D₄×#6157	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	53	1	no	—
280	C₄	—	53	1	no	—
281	C₄	—	106	4	no	—
282	C₄	—	106	2	no	—
283	C₄	—	212	2	no	—
284	C₄	—	212	2	no	—
285	C₄	D₄×#6157	1	1	yes	characteristic
286	C₄	C₂³×#6157	1	1	yes	characteristic
287	C₂	—	2	1	no	—
288	C₂	—	4	2	no	—
289	C₂	—	53	2	no	—
290	C₂	—	106	1	no	—
291	C₂	—	212	2	no	—
292	C₂	C₂×D₄×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#6157
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₄	C₂³×#6157
Fitting	C₂×SD₁₆×C₅₃	C₄
socle	C₂×C₅₃	C₂×C₄×D₄
hypercenter	C₂×SD₁₆	#6157
hypocenter	C₅₃	#2949
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #16811691	C₂	semidirect
3. #16811662	C₂	semidirect
4. SD₁₆×#6157	C₂	semidirect
5. #16811633	C₂	semidirect
6. #16811616	C₂	semidirect
8. #16811586	C₂	semidirect
12. Q₈×#6157	C₂²	semidirect
20. #13133286	C₂²	semidirect
21. #13133285	C₂²	semidirect
22. #13133280	C₂²	semidirect
24. #13133271	C₂²	semidirect
25. C₈×#6157	C₂²	semidirect
28. C₂×#11719913	C₄	semidirect
29. C₂×#11719923	C₄	semidirect
30. C₂×#11719924	C₄	semidirect
31. SD₁₆×D₅₃	C₄	semidirect
32. SD₁₆×D₅₃	C₄	semidirect
36. C₂×SD₁₆×C₅₃	C₄	semidirect
39. C₂×Q₈×C₅₃	C₂×C₄	semidirect
44. Q₈×D₅₃	C₂×C₄	semidirect
54. C₂×Dic₁₀₆	C₂×C₄	semidirect
70. SD₁₆×C₅₃	C₂×C₄	semidirect
71. SD₁₆×C₅₃	C₂×C₄	semidirect
72. C₂×C₈×C₅₃	C₂×C₄	semidirect
73. #11719924	C₂×C₄	semidirect
74. #11719924	C₂×C₄	semidirect
75. #11719923	C₂×C₄	semidirect
76. #11719923	C₂×C₄	semidirect
77. C₂×#88322	C₂×C₄	semidirect
78. #11719913	C₂×C₄	semidirect
79. #11719913	C₂×C₄	semidirect
80. C₈×D₅₃	C₂×C₄	semidirect
81. C₈×D₅₃	C₂×C₄	semidirect
156. C₅₃	#2949	semidirect
166. C₂×SD₁₆	#6157	semidirect
191. C₂×Q₈	C₂×#6157	semidirect
208. SD₁₆	C₂×#6157	semidirect
209. SD₁₆	C₂×#6157	semidirect
218. C₂×C₈	C₂×#6157	semidirect