Subgroups — #21437014

Order 7552.

360 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838129	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838067	C₂	1	1	yes	maximal, characteristic
3	#16837809	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837806	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837701	C₂	1	1	yes	maximal, characteristic
6	#16837699	C₂	1	1	yes	maximal, characteristic
7	#16837696	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837694	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837686	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837681	C₂	1	1	yes	maximal, characteristic
11	#16837530	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837504	C₂	1	1	yes	maximal, characteristic
13	#16837501	C₂	1	1	yes	maximal, characteristic
14	#16837489	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#501	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₅₉	—	2	1	no	—
17	C₂×#11745660	C₂²	1	1	yes	characteristic
18	#14273726	—	2	1	no	—
19	#14273726	C₂²	1	1	yes	characteristic
20	#14273726	C₂²	1	1	yes	characteristic
21	#14273726	C₂²	1	1	yes	characteristic
22	#14273718	C₂²	1	1	yes	characteristic
23	#14273717	—	2	1	no	—
24	#14273717	—	2	1	no	—
25	#14273705	—	2	1	no	—
26	#14273705	—	2	1	no	—
27	#14273703	—	2	1	no	—
28	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
30	#14273692	—	2	1	no	—
31	#14273692	C₂²	1	1	yes	characteristic
32	#14273692	C₂²	1	1	yes	characteristic
33	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
34	#14273689	C₂²	1	1	yes	characteristic
35	#14273689	C₂²	1	1	yes	characteristic
36	#14273661	C₂²	1	1	yes	characteristic
37	#14273660	C₂²	1	1	yes	characteristic
38	#14273660	C₂²	1	1	yes	characteristic
39	#14273660	C₂²	1	1	yes	characteristic
40	#14273658	C₂²	1	1	yes	characteristic
41	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
42	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
44	#14273652	C₂²	1	1	yes	characteristic
45	#14273650	C₂²	1	1	yes	characteristic
46	#14273650	C₂²	1	1	yes	characteristic
47	#14273649	C₂²	1	1	yes	characteristic
48	#14273647	C₂²	1	1	yes	characteristic
49	#14273647	C₂²	1	1	yes	characteristic
50	#14273637	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
53	#125×C₅₉	C₂²	1	1	yes	characteristic
54	#107×C₅₉	C₂²	1	1	yes	characteristic
55	#107×C₅₉	C₂²	1	1	yes	characteristic
56	#106×C₅₉	C₂²	1	1	yes	characteristic
57	#106×C₅₉	C₂²	1	1	yes	characteristic
58	#97×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	—	2	1	no	—
60	Q₈×D₅₉	—	4	4	no	—
61	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₅₉	—	2	1	no	—
63	C₂×C₄×D₅₉	—	2	1	no	—
64	C₂×C₄×D₅₉	—	2	1	no	—
65	C₂×Dic₁₁₈	—	2	1	no	—
66	C₂×Dic₁₁₈	—	2	1	no	—
67	C₂×Dic₁₁₈	—	2	1	no	—
68	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
69	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₅₉	—	2	1	no	—
73	C₄⋊C₄×C₅₉	—	2	1	no	—
74	C₄⋊C₄×C₅₉	—	2	1	no	—
75	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
79	C₄²×C₅₉	C₂³	1	1	yes	characteristic
80	(C₂×D₅₉)⋊C₄	—	2	1	no	—
81	(C₂×D₅₉)⋊C₄	—	2	1	no	—
82	(C₂×D₅₉)⋊C₄	—	2	1	no	—
83	C₄⋊Dic₅₉	—	2	1	no	—
84	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
85	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
87	Dic₅₉⋊C₄	—	2	1	no	—
88	Dic₅₉⋊C₄	—	2	1	no	—
89	Dic₅₉⋊C₄	—	2	1	no	—
90	Dic₅₉⋊C₄	—	2	1	no	—
91	Dic₅₉⋊C₄	—	2	1	no	—
92	Dic₅₉⋊C₄	—	2	1	no	—
93	Dic₅₉⋊C₄	—	2	1	no	—
94	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
95	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
96	C₄×Dic₅₉	—	2	1	no	—
97	C₄×Dic₅₉	—	2	1	no	—
98	C₄×Dic₅₉	—	2	1	no	—
99	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
100	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
101	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
103	#11745660	Q₈	1	4	yes	complemented
104	C₂²×D₅₉	—	2	1	no	—
105	Q₈×C₅₉	—	4	2	no	—
106	C₂×C₄×C₅₉	—	2	1	no	—
107	C₂×C₄×C₅₉	—	2	1	no	—
108	C₂×C₄×C₅₉	—	2	1	no	—
109	C₂×C₄×C₅₉	—	2	1	no	—
110	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
113	C₂×Dic₅₉	—	2	1	no	—
114	C₂×Dic₅₉	—	2	1	no	—
115	C₂×Dic₅₉	—	2	1	no	—
116	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
120	D₂₃₆	—	2	1	no	—
121	D₂₃₆	C₂×Q₈	1	2	yes	—
122	C₄×D₅₉	—	4	2	no	—
123	C₄×D₅₉	—	4	2	no	—
124	C₄×D₅₉	—	4	2	no	—
125	Dic₁₁₈	—	2	1	no	—
126	Dic₁₁₈	—	2	2	no	—
127	Dic₁₁₈	—	4	2	no	—
128	Dic₁₁₈	—	4	2	no	—
129	Dic₁₁₈	—	4	2	no	—
130	Dic₁₁₈	C₂×Q₈	1	2	yes	complemented
131	C₈×C₅₉	—	2	1	no	—
132	C₈×C₅₉	C₂×Q₈	1	2	yes	complemented
133	C₅₉⋊C₈	—	2	1	no	—
134	C₅₉⋊C₈	—	2	1	no	—
135	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×D₅₉	—	2	2	no	—
137	C₂×D₅₉	—	4	1	no	—
138	C₂×D₅₉	—	4	1	no	—
139	C₄×C₅₉	—	2	1	no	—
140	C₄×C₅₉	—	2	1	no	—
141	C₄×C₅₉	—	4	1	no	—
142	C₄×C₅₉	—	4	1	no	—
143	C₄×C₅₉	—	4	1	no	—
144	C₄×C₅₉	—	4	1	no	—
145	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
146	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
147	Dic₅₉	—	2	2	no	—
148	Dic₅₉	—	2	1	no	—
149	Dic₅₉	—	2	1	no	—
150	Dic₅₉	—	2	1	no	—
151	Dic₅₉	—	4	1	no	—
152	Dic₅₉	—	4	1	no	—
153	Dic₅₉	AGL(1,ℤ/8)	1	2	yes	—
154	#3386	—	59	1	no	maximal, Sylow
155	C₂×C₅₉	D₄×Q₈	1	1	yes	characteristic
156	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₅₉	#578	1	1	yes	characteristic
158	D₅₉	—	4	2	no	—
159	#557	—	59	1	no	—
160	D₄×Q₈	—	59	1	no	—
161	#501	—	59	1	no	—
162	#501	D₅₉	1	1	yes	characteristic, complemented
163	#500	—	59	1	no	—
164	#478	—	59	1	no	—
165	#478	—	59	1	no	—
166	#477	—	59	1	no	—
167	#477	—	59	1	no	—
168	#475	—	59	1	no	—
169	#474	—	59	1	no	—
170	#446	—	59	1	no	—
171	#440	—	59	1	no	—
172	#440	—	59	1	no	—
173	C₄×SD₁₆	—	59	1	no	—
174	C₅₉	#3386	1	1	yes	characteristic, complemented, Sylow
175	C₂²×Q₈	—	118	1	no	—
176	C₂×SD₁₆	—	59	1	no	—
177	C₄⋊Q₈	—	59	1	no	—
178	C₄⋊Q₈	—	59	1	no	—
179	C₄⋊Q₈	—	59	1	no	—
180	C₄⋊Q₈	—	118	1	no	—
181	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
182	#125	—	59	1	no	—
183	#125	—	118	1	no	—
184	#125	—	118	1	no	—
185	#125	C₂×D₅₉	1	1	yes	characteristic
186	C₂²⋊Q₈	—	118	1	no	—
187	C₂²⋊Q₈	—	118	1	no	—
188	C₂²⋊Q₈	—	118	1	no	—
189	C₄×Q₈	—	59	1	no	—
190	C₄×Q₈	—	59	1	no	—
191	C₄×Q₈	—	59	1	no	—
192	C₄×Q₈	—	59	1	no	—
193	C₄×Q₈	—	118	1	no	—
194	C₄×D₄	—	59	1	no	—
195	C₄×D₄	—	59	1	no	—
196	C₄×D₄	—	59	1	no	—
197	#107	—	59	1	no	—
198	#107	—	59	1	no	—
199	#107	—	59	1	no	—
200	#107	—	59	1	no	—
201	#107	C₂×D₅₉	1	1	yes	characteristic
202	#107	C₂×D₅₉	1	1	yes	characteristic
203	#106	—	59	1	no	—
204	#106	C₂×D₅₉	1	1	yes	characteristic
205	#106	C₂×D₅₉	1	1	yes	characteristic
206	C₄⋊C₈	—	59	1	no	—
207	C₄⋊C₈	—	59	1	no	—
208	C₄⋊C₈	—	59	1	no	—
209	#103	—	59	1	no	—
210	#103	—	59	1	no	—
211	#103	—	59	1	no	—
212	#102	—	59	1	no	—
213	#102	—	59	1	no	—
214	#102	—	59	1	no	—
215	#97	—	59	1	no	—
216	#97	C₂×D₅₉	1	1	yes	characteristic
217	C₄×C₈	—	59	1	no	—
218	C₂×Q₈	—	2	1	no	—
219	C₂×Q₈	—	59	1	no	—
220	C₂×Q₈	—	59	1	no	—
221	C₂×Q₈	—	118	1	no	—
222	C₂×Q₈	—	118	1	no	—
223	C₂×Q₈	—	118	1	no	—
224	C₂×Q₈	—	236	4	no	—
225	C₂×D₄	—	59	1	no	—
226	C₂²×C₄	—	118	1	no	—
227	C₂²×C₄	—	118	1	no	—
228	C₂²×C₄	—	118	1	no	—
229	SD₁₆	—	59	4	no	—
230	C₂×C₈	—	59	1	no	—
231	C₂×C₈	—	59	1	no	—
232	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
233	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	59	1	no	—
238	C₄⋊C₄	—	59	1	no	—
239	C₄⋊C₄	—	59	1	no	—
240	C₄⋊C₄	—	59	1	no	—
241	C₄⋊C₄	—	59	1	no	—
242	C₄⋊C₄	—	118	1	no	—
243	C₄⋊C₄	—	118	1	no	—
244	C₄⋊C₄	—	118	1	no	—
245	C₄⋊C₄	—	118	1	no	—
246	C₄⋊C₄	—	118	1	no	—
247	C₄⋊C₄	—	118	1	no	—
248	C₄⋊C₄	—	118	1	no	—
249	C₄⋊C₄	—	118	1	no	—
250	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
251	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
253	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
254	C₂²⋊C₄	—	118	1	no	—
255	C₂²⋊C₄	—	118	1	no	—
256	C₂²⋊C₄	—	118	1	no	—
257	C₄²	—	59	1	no	—
258	C₄²	—	59	1	no	—
259	C₄²	—	118	1	no	—
260	C₄²	—	118	1	no	—
261	C₄²	—	118	1	no	—
262	C₄²	C₂²×D₅₉	1	1	yes	characteristic
263	C₂³	—	118	1	no	—
264	Q₈	—	4	2	no	—
265	Q₈	—	59	2	no	—
266	Q₈	—	118	2	no	—
267	Q₈	—	118	1	no	—
268	Q₈	—	236	2	no	—
269	Q₈	—	236	2	no	—
270	Q₈	—	236	2	no	—
271	D₄	—	59	2	no	—
272	D₄	—	118	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	59	1	no	—
278	C₂×C₄	—	59	1	no	—
279	C₂×C₄	—	59	1	no	—
280	C₂×C₄	—	59	1	no	—
281	C₂×C₄	—	118	1	no	—
282	C₂×C₄	—	118	1	no	—
283	C₂×C₄	—	118	1	no	—
284	C₂×C₄	—	236	2	no	—
285	C₂×C₄	—	236	2	no	—
286	C₂×C₄	—	236	2	no	—
287	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
288	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	118	1	no	—
292	C₈	—	118	1	no	—
293	C₈	Q₈×D₅₉	1	2	yes	complemented
294	C₂²	—	118	2	no	—
295	C₂²	—	236	1	no	—
296	C₂²	—	236	1	no	—
297	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	59	2	no	—
305	C₄	—	118	1	no	—
306	C₄	—	118	1	no	—
307	C₄	—	118	1	no	—
308	C₄	—	118	2	no	—
309	C₄	—	236	1	no	—
310	C₄	—	236	1	no	—
311	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
312	C₄	#14273775	1	1	yes	characteristic
313	C₂	—	236	2	no	—
314	C₂	#16838067	1	1	yes	characteristic
315	C₂	AGL(1,ℤ/8)×D₅₉	1	1	yes	characteristic
316	C₂	#16838218	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#501	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#501	D₅₉
hypocenter	C₅₉	#3386
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
1. #16838129	C₂	semidirect
3. #16837809	C₂	semidirect
4. #16837806	C₂	semidirect
7. #16837696	C₂	semidirect
8. #16837694	C₂	semidirect
9. #16837686	C₂	semidirect
11. #16837530	C₂	semidirect
15. C₅₉×#501	C₂	semidirect
103. #11745660	Q₈	semidirect
130. Dic₁₁₈	C₂×Q₈	semidirect
132. C₈×C₅₉	C₂×Q₈	semidirect
162. #501	D₅₉	semidirect
174. C₅₉	#3386	semidirect
293. C₈	Q₈×D₅₉	semidirect