Subgroups — #20291493

Order 6784.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811703	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811701	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811668	C₂	1	4	yes	maximal, complemented
4	#16811666	C₂	1	4	yes	maximal, complemented
5	D₅₃⋊#108	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811622	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811621	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811586	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133238	C₂	1	1	yes	maximal, characteristic
10	#13133305	—	2	2	no	—
11	#13133305	C₂²	1	4	yes	complemented
12	#13133304	—	2	2	no	—
13	#13133304	C₂²	1	4	yes	complemented
14	D₅₃⋊M₁₆	—	2	2	no	—
15	D₅₃⋊M₁₆	C₂²	1	1	yes	characteristic
16	D₅₃⋊M₁₆	C₂²	1	1	yes	characteristic
17	D₅₃⋊M₁₆	C₂²	1	1	yes	characteristic
18	C₂×#11719940	—	2	2	no	—
19	#13133296	C₂²	1	1	yes	characteristic
20	#13133288	C₂²	1	2	yes	complemented
21	#13133287	C₂²	1	2	yes	complemented
22	#13133282	C₂²	1	4	yes	complemented
23	#13133281	C₂²	1	4	yes	complemented
24	#13133271	C₂²	1	2	yes	complemented
25	C₈×#6157	C₂²	1	2	yes	complemented
26	C₂×#11719949	C₂²	1	1	yes	characteristic
27	C₂×#11719947	C₂²	1	1	yes	characteristic
28	#13133238	C₄	1	2	yes	complemented
29	#13133238	C₄	1	2	yes	complemented
30	#13133238	C₂²	1	2	yes	—
31	#13133238	C₂²	1	2	yes	—
32	C₂×#11719925	C₄	1	1	yes	characteristic, complemented
33	C₂×#11719913	C₄	1	1	yes	characteristic, complemented
34	C₂×#11719922	C₄	1	1	yes	characteristic, complemented
35	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
36	C₂×SD₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
37	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊D₅₃	—	2	1	no	—
40	C₂²×Dic₅₃	—	2	1	no	—
41	#11719949	—	2	1	no	—
42	#11719949	—	2	1	no	—
43	#11719949	C₂×C₄	1	2	yes	—
44	#11719949	C₂³	1	2	yes	—
45	#11719947	—	2	1	no	—
46	#11719947	—	2	1	no	—
47	#11719947	C₂×C₄	1	2	yes	—
48	#11719947	C₂³	1	2	yes	—
49	C₂×D₂₁₂	—	2	1	no	—
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	C₂×C₄	1	1	yes	characteristic
54	C₂²⋊#6157	—	2	1	no	—
55	#11719940	—	2	1	no	—
56	#11719940	—	2	1	no	—
57	#11719940	—	2	4	no	—
58	#11719940	D₄	1	2	yes	—
59	C₂×#88324	—	2	4	no	—
60	C₄⋊#6157	—	2	1	no	—
61	C₄×#6157	C₂³	1	2	yes	—
62	#11719936	—	2	4	no	—
63	#11719936	D₄	1	2	yes	—
64	#11719936	C₂³	1	2	yes	—
65	#11719936	C₂³	1	2	yes	—
66	D₅₃⋊C₈	—	2	1	no	—
67	D₅₃⋊C₈	—	2	1	no	—
68	D₅₃⋊C₈	—	2	4	no	—
69	D₅₃⋊C₈	C₂³	1	2	yes	—
70	SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
71	SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	#11719925	C₂×C₄	1	2	yes	—
74	#11719925	C₂×C₄	1	2	yes	—
75	#11719922	C₂×C₄	1	2	yes	—
76	#11719922	C₂×C₄	1	2	yes	—
77	C₂×#88322	C₂×C₄	1	1	yes	characteristic
78	#11719913	C₂×C₄	1	2	yes	—
79	#11719913	C₂×C₄	1	2	yes	—
80	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
81	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
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	#41	1	1	yes	characteristic
87	C₂×#6157	—	2	2	no	—
88	Q₈×C₅₃	—	2	1	no	—
89	Q₈×C₅₃	C₂²×C₄	1	2	yes	—
90	D₄×C₅₃	—	2	1	no	—
91	D₄×C₅₃	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²⋊D₅₃	—	2	2	no	—
95	C₂²⋊D₅₃	—	2	2	no	—
96	C₂×Dic₅₃	—	2	1	no	—
97	C₂×Dic₅₃	—	2	2	no	—
98	C₂×Dic₅₃	—	2	2	no	—
99	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
100	D₂₁₂	—	2	2	no	—
101	D₂₁₂	—	2	2	no	—
102	D₂₁₂	—	2	1	no	—
103	D₂₁₂	C₂²×C₄	1	2	yes	—
104	C₄×D₅₃	—	2	2	no	—
105	C₄×D₅₃	—	2	2	no	—
106	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
108	C₄×D₅₃	#41	1	1	yes	characteristic
109	C₄×D₅₃	C₂⁴	1	1	yes	characteristic
110	Dic₁₀₆	—	2	1	no	—
111	Dic₁₀₆	C₂²×C₄	1	2	yes	—
112	#88324	—	2	2	no	—
113	#88324	—	2	2	no	—
114	#88324	C₂×D₄	1	4	yes	—
115	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	#88322	C₂²×C₄	1	1	yes	characteristic
118	#88322	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	8	1	no	—
121	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×D₅₃	—	2	2	no	—
123	C₂×D₅₃	—	4	1	no	—
124	C₂×D₅₃	—	8	1	no	—
125	C₂×D₅₃	C₄×D₄	1	1	yes	characteristic
126	C₂×D₅₃	C₂×#41	1	1	yes	characteristic
127	#6157	—	4	2	no	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
130	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
131	Dic₅₃	—	2	2	no	—
132	Dic₅₃	C₄×D₄	1	1	yes	characteristic
133	Dic₅₃	C₂²×D₄	1	1	yes	characteristic
134	#2957	—	53	1	no	maximal, Sylow
135	C₂×C₅₃	—	2	1	no	—
136	C₂×C₅₃	—	4	2	no	—
137	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
138	D₅₃	—	2	1	no	—
139	D₅₃	—	2	1	no	—
140	D₅₃	—	4	2	no	—
141	C₂×#135	—	53	1	no	—
142	#568	—	53	1	no	—
143	#568	—	53	1	no	—
144	#444	—	53	4	no	—
145	#443	—	53	4	no	—
146	C₂×#108	—	53	1	no	—
147	#421	—	53	1	no	—
148	#421	—	53	1	no	—
149	#405	—	53	1	no	—
150	C₅₃	#2957	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	—	53	1	no	—
152	C₂×#41	—	53	1	no	—
153	#135	—	53	2	no	—
154	#135	—	53	2	no	—
155	#135	—	53	2	no	—
156	#135	—	53	2	no	—
157	C₂×Q₁₆	—	53	1	no	—
158	C₂×SD₁₆	—	53	1	no	—
159	C₂×SD₁₆	#6157	1	1	yes	characteristic, complemented
160	C₂×D₈	—	53	1	no	—
161	#131	—	53	4	no	—
162	#131	—	53	4	no	—
163	#131	—	106	2	no	—
164	#131	—	106	2	no	—
165	C₂×M₁₆	—	53	1	no	—
166	C₂×M₁₆	—	53	1	no	—
167	C₂×M₁₆	—	53	1	no	—
168	C₂×M₁₆	—	106	2	no	—
169	C₂×M₁₆	—	106	2	no	—
170	C₂²×C₈	—	53	1	no	—
171	#117	—	53	1	no	—
172	#108	—	53	2	no	—
173	#108	—	53	2	no	—
174	#104	—	53	4	no	—
175	#104	—	53	4	no	—
176	#97	—	53	2	no	—
177	C₄×C₈	—	53	2	no	—
178	#41	—	53	2	no	—
179	#41	—	53	2	no	—
180	#41	—	53	2	no	—
181	#41	—	53	2	no	—
182	#41	—	106	1	no	—
183	#41	—	106	1	no	—
184	#41	—	106	1	no	—
185	#41	—	106	1	no	—
186	C₂×Q₈	—	53	1	no	—
187	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
188	C₂×D₄	—	53	1	no	—
189	C₂×D₄	—	106	1	no	—
190	C₂×D₄	—	106	1	no	—
191	C₂×D₄	C₂×#6157	1	1	yes	characteristic
192	C₂²×C₄	—	53	1	no	—
193	C₂²×C₄	—	106	1	no	—
194	C₂²×C₄	—	106	1	no	—
195	Q₁₆	—	53	2	no	—
196	Q₁₆	—	53	2	no	—
197	SD₁₆	—	53	2	no	—
198	SD₁₆	—	53	2	no	—
199	SD₁₆	C₂×#6157	1	2	yes	—
200	SD₁₆	C₂×#6157	1	2	yes	—
201	D₈	—	53	2	no	—
202	D₈	—	53	2	no	—
203	M₁₆	—	53	2	no	—
204	M₁₆	—	53	2	no	—
205	M₁₆	—	53	2	no	—
206	M₁₆	—	53	2	no	—
207	M₁₆	—	106	4	no	—
208	M₁₆	—	106	4	no	—
209	M₁₆	—	106	1	no	—
210	M₁₆	—	106	1	no	—
211	C₂×C₈	—	53	2	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	—	106	4	no	—
218	C₂×C₈	—	106	4	no	—
219	C₂×C₈	—	106	1	no	—
220	C₂×C₈	—	106	1	no	—
221	C₂×C₈	C₂×#6157	1	1	yes	characteristic
222	C₄⋊C₄	—	106	1	no	—
223	C₂²⋊C₄	—	106	1	no	—
224	C₄²	—	53	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	53	1	no	—
227	C₂³	—	106	1	no	—
228	Q₈	—	2	1	no	—
229	Q₈	—	53	2	no	—
230	Q₈	—	106	1	no	—
231	Q₈	C₂²×#6157	1	2	yes	—
232	D₄	—	2	1	no	—
233	D₄	—	53	2	no	—
234	D₄	—	106	2	no	—
235	D₄	—	106	2	no	—
236	D₄	—	106	2	no	—
237	D₄	—	106	2	no	—
238	D₄	—	106	1	no	—
239	D₄	C₂²×#6157	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	53	1	no	—
242	C₂×C₄	—	53	1	no	—
243	C₂×C₄	—	53	1	no	—
244	C₂×C₄	—	53	1	no	—
245	C₂×C₄	—	53	1	no	—
246	C₂×C₄	—	106	2	no	—
247	C₂×C₄	—	106	2	no	—
248	C₂×C₄	—	106	2	no	—
249	C₂×C₄	—	106	2	no	—
250	C₂×C₄	—	106	1	no	—
251	C₂×C₄	—	106	2	no	—
252	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
253	C₈	—	53	4	no	—
254	C₈	—	53	1	no	—
255	C₈	—	53	1	no	—
256	C₈	—	106	2	no	—
257	C₈	—	106	2	no	—
258	C₈	C₂²×#6157	1	1	yes	characteristic
259	C₈	C₂²×#6157	1	1	yes	characteristic
260	C₂²	—	2	2	no	—
261	C₂²	—	8	1	no	—
262	C₂²	—	53	1	no	—
263	C₂²	—	53	1	no	—
264	C₂²	—	106	2	no	—
265	C₂²	—	424	1	no	—
266	C₂²	—	212	1	no	—
267	C₂²	D₄×#6157	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	53	1	no	—
270	C₄	—	53	1	no	—
271	C₄	—	106	2	no	—
272	C₄	—	212	2	no	—
273	C₄	D₄×#6157	1	1	yes	characteristic
274	C₄	C₂³×#6157	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	4	2	no	—
277	C₂	—	106	1	no	—
278	C₂	—	106	1	no	—
279	C₂	—	212	2	no	—
280	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	#2957
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. #16811703	C₂	semidirect
2. #16811701	C₂	semidirect
3. #16811668	C₂	semidirect
4. #16811666	C₂	semidirect
5. D₅₃⋊#108	C₂	semidirect
6. #16811622	C₂	semidirect
7. #16811621	C₂	semidirect
8. #16811586	C₂	semidirect
11. #13133305	C₂²	semidirect
13. #13133304	C₂²	semidirect
20. #13133288	C₂²	semidirect
21. #13133287	C₂²	semidirect
22. #13133282	C₂²	semidirect
23. #13133281	C₂²	semidirect
24. #13133271	C₂²	semidirect
25. C₈×#6157	C₂²	semidirect
28. #13133238	C₄	semidirect
29. #13133238	C₄	semidirect
32. C₂×#11719925	C₄	semidirect
33. C₂×#11719913	C₄	semidirect
34. C₂×#11719922	C₄	semidirect
36. C₂×SD₁₆×C₅₃	C₄	semidirect
150. C₅₃	#2957	semidirect
159. C₂×SD₁₆	#6157	semidirect