Subgroups — #21436883

Order 7552.

384 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838134	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837763	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837761	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837747	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837693	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837690	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837685	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837619	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837595	C₂	1	1	yes	maximal, characteristic
11	#16837531	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837514	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837512	C₂	1	1	yes	maximal, characteristic, complemented
14	SD₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#463	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₁₁₈	—	2	1	no	—
17	C₂×#11745669	C₂²	1	1	yes	characteristic
18	#14273727	C₂²	1	1	yes	characteristic
19	#14273726	C₂²	1	1	yes	characteristic
20	#14273726	C₂²	1	1	yes	characteristic
21	#14273723	C₂²	1	1	yes	characteristic
22	#14273716	C₂²	1	1	yes	characteristic
23	#14273713	—	2	1	no	—
24	#14273711	—	2	1	no	—
25	#14273709	—	2	1	no	—
26	#14273706	—	2	1	no	—
27	#14273704	—	2	1	no	—
28	#14273704	—	2	1	no	—
29	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
31	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
32	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
33	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
34	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
35	(C₂²⋊D₅₉)⋊C₄	—	2	1	no	—
36	#14273659	C₂²	1	1	yes	characteristic
37	#14273659	C₂²	1	1	yes	characteristic
38	#14273658	C₂²	1	1	yes	characteristic
39	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
41	#14273651	C₂²	1	1	yes	characteristic
42	#14273651	C₂²	1	1	yes	characteristic
43	#14273649	C₂²	1	1	yes	characteristic
44	#14273649	C₂²	1	1	yes	characteristic
45	#14273648	C₂²	1	1	yes	characteristic
46	#14273648	C₂²	1	1	yes	characteristic
47	#14273648	C₂²	1	1	yes	characteristic
48	#14273647	C₂²	1	1	yes	characteristic
49	#14273638	C₂²	1	1	yes	characteristic
50	#14273637	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
54	#124×C₅₉	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	#102×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	—	2	1	no	—
63	C₂×C₂²⋊D₅₉	—	2	1	no	—
64	C₂×C₂²⋊D₅₉	—	2	1	no	—
65	C₂²×Dic₅₉	—	2	1	no	—
66	C₂²×Dic₅₉	—	2	1	no	—
67	C₂²×Dic₅₉	—	2	1	no	—
68	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₅₉	—	2	1	no	—
70	C₂×Dic₁₁₈	—	2	1	no	—
71	C₂×Dic₁₁₈	—	2	1	no	—
72	C₂×Dic₁₁₈	—	2	1	no	—
73	C₂×Dic₁₁₈	—	4	4	no	—
74	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
75	SD₁₆×C₅₉	—	2	2	no	—
76	D₈×C₅₉	—	2	2	no	—
77	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
80	C₂²⋊C₄×C₅₉	—	2	1	no	—
81	C₂²⋊C₄×C₅₉	—	2	1	no	—
82	C₂²⋊C₄×C₅₉	—	2	1	no	—
83	C₄²×C₅₉	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₅₉	—	2	1	no	—
85	C₂²⋊Dic₅₉	—	2	1	no	—
86	C₂²⋊Dic₅₉	—	2	1	no	—
87	#11745669	—	2	2	no	—
88	(C₂×D₅₉)⋊C₄	—	2	1	no	—
89	(C₂×D₅₉)⋊C₄	—	2	1	no	—
90	(C₂×D₅₉)⋊C₄	—	2	1	no	—
91	C₄⋊Dic₅₉	—	2	1	no	—
92	C₄⋊Dic₅₉	—	2	1	no	—
93	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
94	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
97	Dic₅₉⋊C₄	—	2	1	no	—
98	Dic₅₉⋊C₄	—	2	1	no	—
99	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
100	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
101	C₄×Dic₅₉	—	2	1	no	—
102	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
103	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
104	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
106	C₂³×C₅₉	—	2	1	no	—
107	C₂³×C₅₉	—	2	1	no	—
108	C₂²×D₅₉	—	2	1	no	—
109	Q₈×C₅₉	—	2	1	no	—
110	Q₈×C₅₉	—	2	1	no	—
111	D₄×C₅₉	—	2	1	no	—
112	D₄×C₅₉	—	2	1	no	—
113	D₄×C₅₉	—	2	1	no	—
114	D₄×C₅₉	#41	1	2	yes	—
115	C₂×C₄×C₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	—	2	1	no	—
117	C₂×C₄×C₅₉	—	4	2	no	—
118	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₅₉	—	4	2	no	—
122	C₂²⋊D₅₉	—	8	1	no	—
123	C₂×Dic₅₉	—	2	1	no	—
124	C₂×Dic₅₉	—	2	1	no	—
125	C₂×Dic₅₉	—	2	1	no	—
126	C₂×Dic₅₉	—	4	2	no	—
127	C₂×Dic₅₉	—	4	2	no	—
128	C₂×Dic₅₉	—	8	1	no	—
129	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
133	D₂₃₆	—	2	1	no	—
134	D₂₃₆	—	2	1	no	—
135	C₄×D₅₉	—	8	1	no	—
136	Dic₁₁₈	—	2	2	no	—
137	Dic₁₁₈	—	4	2	no	—
138	Dic₁₁₈	—	4	2	no	—
139	Dic₁₁₈	—	4	2	no	—
140	C₈×C₅₉	—	2	1	no	—
141	C₈×C₅₉	—	2	1	no	—
142	C₅₉⋊C₈	—	2	1	no	—
143	C₅₉⋊C₈	—	2	1	no	—
144	C₂²×C₅₉	—	2	2	no	—
145	C₂²×C₅₉	—	4	1	no	—
146	C₂²×C₅₉	—	4	1	no	—
147	C₂²×C₅₉	—	4	1	no	—
148	C₂²×C₅₉	—	4	1	no	—
149	C₂²×C₅₉	—	4	1	no	—
150	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₅₉	—	4	1	no	—
152	C₂×D₅₉	—	4	1	no	—
153	C₂×D₅₉	—	4	1	no	—
154	C₄×C₅₉	—	2	1	no	—
155	C₄×C₅₉	—	2	1	no	—
156	C₄×C₅₉	—	4	1	no	—
157	C₄×C₅₉	—	4	1	no	—
158	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
159	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₅₉	—	2	1	no	—
161	Dic₅₉	—	2	1	no	—
162	Dic₅₉	—	2	1	no	—
163	Dic₅₉	—	4	1	no	—
164	Dic₅₉	—	4	1	no	—
165	Dic₅₉	—	4	1	no	—
166	Dic₅₉	AGL(1,ℤ/8)	1	2	yes	—
167	#3373	—	59	1	no	maximal, Sylow
168	C₂×C₅₉	—	4	2	no	—
169	C₂×C₅₉	—	8	1	no	—
170	C₂×C₅₉	#548	1	1	yes	characteristic
171	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₂×C₅₉	#577	1	1	yes	characteristic
173	D₅₉	—	8	1	no	—
174	#559	—	59	1	no	—
175	D₄×Q₈	—	59	1	no	—
176	#489	—	59	1	no	—
177	#489	—	59	1	no	—
178	#486	—	59	1	no	—
179	#476	—	59	1	no	—
180	#476	—	59	1	no	—
181	#475	—	59	1	no	—
182	#463	—	59	1	no	—
183	#463	D₅₉	1	1	yes	characteristic, complemented
184	#459	—	59	1	no	—
185	#446	—	59	1	no	—
186	#442	—	59	1	no	—
187	#442	—	59	1	no	—
188	C₄×SD₁₆	—	59	1	no	—
189	C₅₉	#3373	1	1	yes	characteristic, complemented, Sylow
190	C₂²×Q₈	—	118	1	no	—
191	C₂×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
192	C₂×D₈	—	59	1	no	—
193	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
194	C₄⋊Q₈	—	59	1	no	—
195	C₄⋊Q₈	—	59	1	no	—
196	C₄⋊Q₈	—	59	1	no	—
197	#127	—	59	1	no	—
198	#124	—	59	1	no	—
199	#124	—	118	1	no	—
200	#124	C₂×D₅₉	1	1	yes	characteristic
201	#123	—	118	1	no	—
202	#123	—	118	1	no	—
203	C₂²⋊Q₈	—	118	1	no	—
204	C₂²⋊Q₈	—	118	1	no	—
205	C₂²⋊Q₈	—	118	1	no	—
206	C₄×Q₈	—	59	1	no	—
207	C₄×Q₈	—	59	1	no	—
208	C₄×D₄	—	59	1	no	—
209	C₄×D₄	—	59	1	no	—
210	C₄×D₄	—	59	1	no	—
211	C₄×D₄	—	59	1	no	—
212	C₄×D₄	—	118	1	no	—
213	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
214	#106	—	59	1	no	—
215	#106	—	59	1	no	—
216	#106	—	59	1	no	—
217	C₄⋊C₈	—	59	1	no	—
218	C₄⋊C₈	—	59	1	no	—
219	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
220	#103	—	59	1	no	—
221	#103	—	59	1	no	—
222	#103	C₂×D₅₉	1	1	yes	characteristic
223	#102	—	59	1	no	—
224	#102	—	59	1	no	—
225	#102	—	59	1	no	—
226	#102	—	59	1	no	—
227	#102	—	59	1	no	—
228	#102	—	59	1	no	—
229	#102	C₂×D₅₉	1	1	yes	characteristic
230	#97	—	59	1	no	—
231	#97	—	59	1	no	—
232	C₄×C₈	—	59	1	no	—
233	C₂×Q₈	—	59	1	no	—
234	C₂×Q₈	—	118	1	no	—
235	C₂×Q₈	—	118	1	no	—
236	C₂×Q₈	—	118	1	no	—
237	C₂×Q₈	—	236	4	no	—
238	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
239	C₂×D₄	—	59	1	no	—
240	C₂×D₄	—	118	1	no	—
241	C₂×D₄	—	118	1	no	—
242	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
243	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	118	1	no	—
246	C₂²×C₄	—	118	1	no	—
247	C₂²×C₄	—	118	1	no	—
248	C₂²×C₄	—	118	1	no	—
249	SD₁₆	—	2	2	no	—
250	D₈	—	2	2	no	—
251	D₈	—	118	2	no	—
252	C₂×C₈	—	59	1	no	—
253	C₂×C₈	—	59	1	no	—
254	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
256	C₄⋊C₄	—	59	1	no	—
257	C₄⋊C₄	—	59	1	no	—
258	C₄⋊C₄	—	59	1	no	—
259	C₄⋊C₄	—	59	1	no	—
260	C₄⋊C₄	—	59	1	no	—
261	C₄⋊C₄	—	59	1	no	—
262	C₄⋊C₄	—	118	1	no	—
263	C₄⋊C₄	—	118	1	no	—
264	C₄⋊C₄	—	118	1	no	—
265	C₄⋊C₄	—	118	1	no	—
266	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	118	1	no	—
271	C₂²⋊C₄	—	118	1	no	—
272	C₂²⋊C₄	—	118	1	no	—
273	C₂²⋊C₄	—	118	1	no	—
274	C₂²⋊C₄	—	118	1	no	—
275	C₂²⋊C₄	—	118	1	no	—
276	C₄²	—	59	1	no	—
277	C₄²	—	59	1	no	—
278	C₄²	—	118	1	no	—
279	C₄²	C₂²×D₅₉	1	1	yes	characteristic
280	C₂³	—	2	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	118	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	118	2	no	—
286	Q₈	—	236	2	no	—
287	Q₈	—	236	2	no	—
288	Q₈	—	236	2	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	118	1	no	—
293	D₄	—	118	1	no	—
294	D₄	—	472	1	no	—
295	D₄	—	236	2	no	—
296	D₄	#11745687	1	2	yes	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	59	1	no	—
301	C₂×C₄	—	59	1	no	—
302	C₂×C₄	—	59	1	no	—
303	C₂×C₄	—	59	1	no	—
304	C₂×C₄	—	118	1	no	—
305	C₂×C₄	—	118	1	no	—
306	C₂×C₄	—	118	1	no	—
307	C₂×C₄	—	472	1	no	—
308	C₂×C₄	—	472	1	no	—
309	C₂×C₄	—	236	2	no	—
310	C₂×C₄	—	236	2	no	—
311	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
312	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
313	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	118	1	no	—
317	C₈	—	118	1	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	236	1	no	—
325	C₂²	—	236	1	no	—
326	C₂²	—	236	1	no	—
327	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	59	2	no	—
333	C₄	—	118	1	no	—
334	C₄	—	118	1	no	—
335	C₄	—	118	1	no	—
336	C₄	—	236	1	no	—
337	C₄	—	236	1	no	—
338	C₄	—	236	1	no	—
339	C₄	C₂×#11745687	1	1	yes	characteristic
340	C₄	#14273774	1	1	yes	characteristic
341	C₂	—	4	2	no	—
342	C₂	—	8	1	no	—
343	C₂	—	472	1	no	—
344	C₂	#16838058	1	1	yes	characteristic
345	C₂	AGL(1,ℤ/8)×D₅₉	1	1	yes	characteristic
346	C₂	#16838209	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#463	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#463	D₅₉
hypocenter	C₅₉	#3373
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #16838134	C₂	semidirect
2. D₄×Dic₁₁₈	C₂	semidirect
3. #16837763	C₂	semidirect
4. #16837761	C₂	semidirect
5. #16837747	C₂	semidirect
6. #16837693	C₂	semidirect
7. #16837690	C₂	semidirect
8. #16837685	C₂	semidirect
9. #16837619	C₂	semidirect
11. #16837531	C₂	semidirect
12. #16837514	C₂	semidirect
13. #16837512	C₂	semidirect
14. SD₁₆×Dic₅₉	C₂	semidirect
15. C₅₉×#463	C₂	semidirect
183. #463	D₅₉	semidirect
189. C₅₉	#3373	semidirect