Subgroups — #21432997

Order 7552.

456 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838200	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273733	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837522	C₂	1	4	yes	maximal, complemented
4	#16837517	C₂	1	4	yes	maximal, complemented
5	#16837450	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837419	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837418	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837357	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11745685	C₂²	1	1	yes	characteristic
11	#14273758	C₄	1	2	yes	complemented
12	#14273757	C₄	1	2	yes	complemented
13	#14273753	C₄	1	2	yes	complemented
14	#14273752	C₄	1	2	yes	complemented
15	#14273747	C₂²	1	4	yes	—
16	#14273733	—	2	1	no	—
17	#14273733	—	2	1	no	—
18	#14273733	C₂²	1	2	yes	complemented
19	#14273733	C₂²	1	2	yes	complemented
20	C₂×#11745664	C₂²	1	1	yes	characteristic
21	C₂×#11745664	C₂²	1	1	yes	characteristic
22	C₂×#11745659	—	2	1	no	—
23	C₂×C₈×D₅₉	—	2	1	no	—
24	#14273686	C₂²	1	1	yes	characteristic
25	#14273662	C₂²	1	4	yes	complemented
26	#14273654	C₂²	1	4	yes	complemented
27	#14273653	C₂²	1	4	yes	complemented
28	#14273637	C₂²	1	2	yes	complemented
29	C₈×Dic₅₉	C₂²	1	2	yes	complemented
30	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
31	#131×C₅₉	—	2	1	no	—
32	#131×C₅₉	—	2	1	no	—
33	#131×C₅₉	C₂²	1	2	yes	complemented
34	#131×C₅₉	C₂²	1	2	yes	complemented
35	C₂×M₁₆×C₅₉	—	2	1	no	—
36	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₅₉	—	2	1	no	—
38	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
39	#41×C₅₉	—	2	1	no	—
40	#41×C₅₉	—	2	1	no	—
41	#41×C₅₉	C₂×C₄	1	2	yes	—
42	#41×C₅₉	C₂³	1	2	yes	—
43	C₂×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
44	C₂×D₄×C₅₉	—	2	1	no	—
45	C₂×D₄×C₅₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	—	2	1	no	—
47	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₅₉	—	2	1	no	—
49	#11745685	—	2	1	no	—
50	#11745685	—	2	1	no	—
51	#11745685	C₂×C₄	1	2	yes	—
52	#11745685	C₂³	1	2	yes	—
53	C₂×D₂₃₆	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₅₉	—	2	1	no	—
55	C₂×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
56	M₁₆×C₅₉	—	2	1	no	—
57	M₁₆×C₅₉	—	2	2	no	—
58	M₁₆×C₅₉	—	2	2	no	—
59	M₁₆×C₅₉	C₂³	1	2	yes	—
60	C₂×C₈×C₅₉	—	2	1	no	—
61	C₂×C₈×C₅₉	—	2	2	no	—
62	C₂×C₈×C₅₉	—	2	2	no	—
63	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
64	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
67	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
69	C₂²⋊Dic₅₉	—	2	1	no	—
70	#11745672	C₂×C₄	1	4	yes	—
71	#11745671	C₂×C₄	1	4	yes	—
72	#11745670	C₂×C₄	1	4	yes	—
73	#11745669	C₂×C₄	1	4	yes	—
74	C₄⋊Dic₅₉	—	2	1	no	—
75	C₄×Dic₅₉	C₂³	1	2	yes	—
76	#11745664	—	2	1	no	—
77	#11745664	C₂×C₄	1	2	yes	—
78	#11745664	C₂×C₄	1	2	yes	—
79	#11745664	C₂³	1	2	yes	—
80	C₂×C₅₉⋊C₈	—	2	1	no	—
81	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
82	#11745659	—	2	2	no	—
83	#11745659	—	2	2	no	—
84	C₈×D₅₉	—	2	2	no	—
85	C₈×D₅₉	—	2	2	no	—
86	C₂³×C₅₉	—	2	1	no	—
87	C₂³×C₅₉	#41	1	1	yes	characteristic
88	C₂²×D₅₉	—	2	1	no	—
89	Q₈×C₅₉	—	2	1	no	—
90	Q₈×C₅₉	C₂²×C₄	1	2	yes	—
91	D₄×C₅₉	—	2	2	no	—
92	D₄×C₅₉	—	2	2	no	—
93	D₄×C₅₉	—	2	1	no	—
94	D₄×C₅₉	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₅₉	—	2	1	no	—
96	C₂×C₄×C₅₉	—	2	2	no	—
97	C₂×C₄×C₅₉	—	2	2	no	—
98	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
103	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₅₉	—	2	2	no	—
105	C₂²⋊D₅₉	—	2	2	no	—
106	C₂×Dic₅₉	—	2	1	no	—
107	C₂×Dic₅₉	—	2	2	no	—
108	D₂₃₆	—	2	1	no	—
109	D₂₃₆	C₂²×C₄	1	2	yes	—
110	C₄×D₅₉	—	2	2	no	—
111	C₄×D₅₉	—	2	2	no	—
112	Dic₁₁₈	—	2	1	no	—
113	Dic₁₁₈	C₂²×C₄	1	2	yes	—
114	C₈×C₅₉	—	2	2	no	—
115	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
116	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
117	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
118	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
119	C₅₉⋊C₈	—	2	2	no	—
120	C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
121	C₂²×C₅₉	—	2	2	no	—
122	C₂²×C₅₉	—	4	1	no	—
123	C₂²×C₅₉	—	4	2	no	—
124	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
125	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
126	C₂²×C₅₉	C₂×#41	1	1	yes	characteristic
127	C₂×D₅₉	—	2	2	no	—
128	C₂×D₅₉	—	4	2	no	—
129	C₄×C₅₉	—	2	2	no	—
130	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
131	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
132	C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
133	C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
134	Dic₅₉	—	2	2	no	—
135	Dic₅₉	—	4	2	no	—
136	#2955	—	59	1	no	maximal, Sylow
137	C₂×C₅₉	—	2	1	no	—
138	C₂×C₅₉	—	2	1	no	—
139	C₂×C₅₉	—	2	1	no	—
140	C₂×C₅₉	—	4	2	no	—
141	C₂×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
142	D₅₉	—	4	2	no	—
143	#575	—	59	1	no	—
144	C₂×#131	—	59	1	no	—
145	C₂×#131	D₅₉	1	1	yes	characteristic, complemented
146	#444	—	59	4	no	—
147	#443	—	59	4	no	—
148	#430	—	59	1	no	—
149	#421	—	59	1	no	—
150	#421	—	59	1	no	—
151	#405	—	59	1	no	—
152	C₅₉	#2955	1	1	yes	characteristic, complemented, Sylow
153	C₂×#41	—	59	1	no	—
154	C₂×#41	C₂×D₅₉	1	1	yes	characteristic
155	#137	—	59	2	no	—
156	#137	—	59	2	no	—
157	AGL(1,ℤ/8)	—	59	2	no	—
158	AGL(1,ℤ/8)	—	59	2	no	—
159	#135	—	59	4	no	—
160	#131	—	2	1	no	—
161	#131	—	2	1	no	—
162	#131	—	59	2	no	—
163	#131	—	59	2	no	—
164	#131	—	118	1	no	—
165	#131	—	118	1	no	—
166	#131	C₂×D₅₉	1	2	yes	complemented
167	#131	C₂×D₅₉	1	2	yes	complemented
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	59	1	no	—
170	C₂×M₁₆	—	59	1	no	—
171	C₂×M₁₆	—	118	1	no	—
172	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	—	118	1	no	—
175	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
176	#117	—	59	1	no	—
177	#108	—	59	4	no	—
178	#104	—	59	4	no	—
179	#104	—	59	4	no	—
180	#97	—	59	2	no	—
181	C₄×C₈	—	59	2	no	—
182	#41	—	2	1	no	—
183	#41	—	2	1	no	—
184	#41	—	59	2	no	—
185	#41	—	59	2	no	—
186	#41	—	118	1	no	—
187	#41	—	118	1	no	—
188	#41	C₄×D₅₉	1	2	yes	—
189	#41	C₂²×D₅₉	1	2	yes	—
190	C₂×Q₈	—	59	1	no	—
191	C₂×Q₈	C₄×D₅₉	1	1	yes	characteristic
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	59	1	no	—
194	C₂×D₄	—	118	1	no	—
195	C₂×D₄	C₄×D₅₉	1	1	yes	characteristic
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	118	1	no	—
198	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
199	Q₁₆	—	59	4	no	—
200	SD₁₆	—	59	4	no	—
201	SD₁₆	—	59	4	no	—
202	D₈	—	59	4	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	59	2	no	—
207	M₁₆	—	59	2	no	—
208	M₁₆	—	59	2	no	—
209	M₁₆	—	118	1	no	—
210	M₁₆	—	118	2	no	—
211	M₁₆	—	118	2	no	—
212	M₁₆	C₂²×D₅₉	1	2	yes	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	59	2	no	—
217	C₂×C₈	—	118	1	no	—
218	C₂×C₈	—	118	2	no	—
219	C₂×C₈	—	118	2	no	—
220	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
221	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
222	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
223	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
224	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
225	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
226	C₄⋊C₄	—	118	1	no	—
227	C₂²⋊C₄	—	118	1	no	—
228	C₄²	—	59	2	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	118	1	no	—
231	C₂³	#11745685	1	1	yes	characteristic
232	Q₈	—	2	1	no	—
233	Q₈	—	59	2	no	—
234	Q₈	—	118	1	no	—
235	Q₈	C₂×C₄×D₅₉	1	2	yes	—
236	D₄	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	2	1	no	—
239	D₄	—	59	2	no	—
240	D₄	—	118	2	no	—
241	D₄	—	118	2	no	—
242	D₄	—	118	1	no	—
243	D₄	C₂×C₄×D₅₉	1	2	yes	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	118	1	no	—
248	C₂×C₄	—	118	2	no	—
249	C₂×C₄	—	118	2	no	—
250	C₂×C₄	—	118	2	no	—
251	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
253	C₂×C₄	#11745685	1	1	yes	characteristic
254	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
255	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
256	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	59	4	no	—
259	C₈	—	118	2	no	—
260	C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
261	C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
262	C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
263	C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	118	2	no	—
268	C₂²	—	236	2	no	—
269	C₂²	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
270	C₂²	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
271	C₂²	C₂×#11745685	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	118	2	no	—
274	C₄	—	236	2	no	—
275	C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
276	C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
277	C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
278	C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	4	2	no	—
283	C₂	—	236	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×C₂²⋊D₅₉
commutator	C₄×C₅₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₂×#131×C₅₉	C₂
socle	C₂×C₅₉	C₂×C₄×D₄
hypercenter	C₂×#131	D₅₉
hypocenter	C₅₉	#2955
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#14273733	C₂	semidirect
3. #16837522	C₂	semidirect
4. #16837517	C₂	semidirect
5. #16837450	C₂	semidirect
6. #16837419	C₂	semidirect
7. #16837418	C₂	semidirect
8. #16837357	C₂	semidirect
9. C₂×#131×C₅₉	C₂	semidirect
11. #14273758	C₄	semidirect
12. #14273757	C₄	semidirect
13. #14273753	C₄	semidirect
14. #14273752	C₄	semidirect
18. #14273733	C₂²	semidirect
19. #14273733	C₂²	semidirect
25. #14273662	C₂²	semidirect
26. #14273654	C₂²	semidirect
27. #14273653	C₂²	semidirect
28. #14273637	C₂²	semidirect
29. C₈×Dic₅₉	C₂²	semidirect
33. #131×C₅₉	C₂²	semidirect
34. #131×C₅₉	C₂²	semidirect
145. C₂×#131	D₅₉	semidirect
152. C₅₉	#2955	semidirect
166. #131	C₂×D₅₉	semidirect
167. #131	C₂×D₅₉	semidirect