Subgroups — #28746088

Order 15104.

484 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	#21436485	C₂	1	2	yes	maximal
2	#21426370	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426297	C₂	1	2	yes	maximal, complemented
4	#21426296	C₂	1	2	yes	maximal, complemented
5	#21426246	C₂	1	2	yes	maximal, complemented
6	#21426184	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₃₂×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₆×C₂²⋊D₅₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#16837228	C₂	1	1	yes	maximal, characteristic, complemented
10	#366×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#2216	C₂	1	1	yes	maximal, characteristic, complemented
12	#16838110	—	2	2	no	—
13	C₂×C₁₆×D₅₉	C₂²	1	1	yes	characteristic
14	#16837806	C₂²	1	2	yes	—
15	#16837736	—	2	2	no	—
16	#16837735	—	2	2	no	—
17	#16837695	—	2	2	no	—
18	#16837640	C₂²	1	2	yes	—
19	Q₁₆×Dic₅₉	C₂²	1	2	yes	—
20	C₈×C₂²⋊D₅₉	C₂²	1	1	yes	characteristic
21	C₂×#14273661	C₂²	1	2	yes	—
22	#107×D₅₉	C₂²	1	2	yes	—
23	#16837228	C₂²	1	1	yes	characteristic
24	#16837228	C₂²	1	1	yes	characteristic
25	#16837228	C₂²	1	2	yes	—
26	#16837227	C₂²	1	2	yes	—
27	#16837227	C₂²	1	2	yes	—
28	Dic₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
29	#16837206	C₂²	1	2	yes	—
30	#16837206	C₂²	1	2	yes	—
31	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
32	(C₂×D₅₉)⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
34	C₂×Q₃₂×C₅₉	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
36	C₅₉×#467	C₂²	1	2	yes	—
37	C₅₉×#366	C₂²	1	1	yes	characteristic
38	C₅₉×#358	C₂²	1	2	yes	—
39	C₂×C₈×D₅₉	C₂³	1	1	yes	characteristic
40	#14273726	—	2	2	no	—
41	#14273717	—	4	2	no	—
42	#14273707	—	4	2	no	—
43	#14273704	—	4	2	no	—
44	#14273703	—	2	2	no	—
45	Q₈×Dic₅₉	—	2	2	no	—
46	C₄×C₂²⋊D₅₉	D₄	1	1	yes	characteristic
47	(C₂²⋊D₅₉)⋊C₄	—	4	2	no	—
48	C₂×C₄⋊Dic₅₉	—	2	2	no	—
49	C₄⋊C₄×D₅₉	—	2	2	no	—
50	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
52	C₁₆×D₅₉	—	2	2	no	—
53	#14273661	—	2	2	no	—
54	#14273661	C₂³	1	2	yes	—
55	#14273661	C₂³	1	2	yes	—
56	#14273660	—	2	2	no	—
57	#14273660	—	2	2	no	—
58	Dic₅₉⋊C₈	D₄	1	1	yes	characteristic
59	#14273651	—	2	2	no	—
60	#14273651	—	2	2	no	—
61	C₂²⋊(C₅₉⋊C₈)	D₄	1	1	yes	characteristic
62	(C₂×D₅₉)⋊C₈	D₄	1	1	yes	characteristic
63	C₈×Dic₅₉	C₂³	1	1	yes	characteristic
64	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
65	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
66	C₂²⋊Q₈×C₅₉	—	2	2	no	—
67	Q₃₂×C₅₉	—	2	2	no	—
68	C₂×C₁₆×C₅₉	—	2	2	no	—
69	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
71	#107×C₅₉	C₂³	1	2	yes	—
72	#103×C₅₉	—	2	2	no	—
73	C₂×Q₈×C₅₉	—	2	2	no	—
74	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₂²⋊D₅₉	D₈	1	1	yes	characteristic
76	C₂²×Dic₅₉	—	4	2	no	—
77	C₂×C₄×D₅₉	—	4	2	no	—
78	C₂×C₄×D₅₉	C₂×D₄	1	1	yes	characteristic
79	C₂×Dic₁₁₈	—	4	2	no	—
80	Q₁₆×C₅₉	—	2	2	no	—
81	Q₁₆×C₅₉	—	2	2	no	—
82	C₂×C₈×C₅₉	—	2	1	no	—
83	C₂×C₈×C₅₉	—	2	1	no	—
84	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₅₉	—	2	2	no	—
87	C₄⋊C₄×C₅₉	—	4	2	no	—
88	C₂²⋊C₄×C₅₉	—	4	2	no	—
89	C₂²⋊Dic₅₉	D₈	1	1	yes	characteristic
90	(C₂×D₅₉)⋊C₄	—	4	2	no	—
91	(C₂×D₅₉)⋊C₄	D₈	1	1	yes	characteristic
92	C₄⋊Dic₅₉	—	2	2	no	—
93	C₄⋊Dic₅₉	—	2	2	no	—
94	C₄⋊Dic₅₉	—	4	2	no	—
95	C₄⋊Dic₅₉	—	4	2	no	—
96	Dic₅₉⋊C₄	—	4	2	no	—
97	Dic₅₉⋊C₄	—	4	2	no	—
98	Dic₅₉⋊C₄	—	4	2	no	—
99	Dic₅₉⋊C₄	D₈	1	1	yes	characteristic
100	C₄×Dic₅₉	—	4	2	no	—
101	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
104	C₈×D₅₉	—	2	1	no	—
105	C₈×D₅₉	—	2	1	no	—
106	C₁₆×C₅₉	—	2	1	no	—
107	C₁₆×C₅₉	#41	1	2	yes	—
108	C₅₉⋊C₁₆	—	2	1	no	—
109	C₅₉⋊C₁₆	—	2	1	no	—
110	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
111	C₂²×D₅₉	C₂×D₈	1	1	yes	characteristic
112	Q₈×C₅₉	—	4	2	no	—
113	Q₈×C₅₉	—	4	2	no	—
114	C₂×C₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	—	4	2	no	—
117	C₂×C₄×C₅₉	—	4	2	no	—
118	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
120	C₂²⋊D₅₉	Q₃₂	1	4	yes	complemented
121	C₂×Dic₅₉	—	4	2	no	—
122	C₂×Dic₅₉	—	4	2	no	—
123	C₂×Dic₅₉	—	16	2	no	—
124	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
125	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
126	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
127	C₄×D₅₉	—	2	1	no	—
128	C₄×D₅₉	—	2	1	no	—
129	C₄×D₅₉	—	16	2	no	—
130	Dic₁₁₈	—	16	2	no	—
131	C₈×C₅₉	—	2	1	no	—
132	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
133	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
134	C₅₉⋊C₈	—	2	1	no	—
135	C₅₉⋊C₈	—	2	1	no	—
136	#33905	—	59	1	no	maximal, Sylow
137	C₂²×C₅₉	—	2	1	no	—
138	C₂²×C₅₉	—	2	1	no	—
139	C₂²×C₅₉	C₂×Q₃₂	1	2	yes	—
140	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
141	C₂×D₅₉	—	2	1	no	—
142	C₂×D₅₉	—	2	1	no	—
143	C₂×D₅₉	C₂×Q₃₂	1	2	yes	—
144	C₄×C₅₉	—	2	1	no	—
145	C₄×C₅₉	—	8	2	no	—
146	C₄×C₅₉	—	8	2	no	—
147	C₄×C₅₉	#547	1	1	yes	characteristic
148	C₄×C₅₉	#578	1	1	yes	characteristic
149	Dic₅₉	—	2	1	no	—
150	Dic₅₉	—	2	1	no	—
151	Dic₅₉	—	8	2	no	—
152	Dic₅₉	—	8	2	no	—
153	Dic₅₉	C₂×Q₃₂	1	2	yes	—
154	#3338	—	59	2	no	—
155	#2252	—	59	1	no	—
156	#2236	—	59	2	no	—
157	#2236	—	59	2	no	—
158	#2227	—	59	2	no	—
159	#2216	—	59	1	no	—
160	#2216	D₅₉	1	1	yes	characteristic, complemented
161	C₄×Q₃₂	—	59	1	no	—
162	D₄×C₁₆	—	59	1	no	—
163	C₂×#366	—	59	1	no	—
164	C₂×#366	—	59	1	no	—
165	C₂×C₅₉	—	2	2	no	—
166	C₂×C₅₉	#3334	1	1	yes	characteristic
167	C₂×C₅₉	C₂²×Q₃₂	1	1	yes	characteristic
168	C₂×C₅₉	#3415	1	1	yes	characteristic
169	D₅₉	—	2	2	no	—
170	#554	—	118	2	no	—
171	C₂×Q₃₂	C₂×D₅₉	1	1	yes	characteristic
172	C₂²×C₁₆	—	59	1	no	—
173	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
174	#500	—	59	2	no	—
175	#484	—	118	2	no	—
176	#484	—	118	2	no	—
177	#477	—	118	2	no	—
178	#467	—	59	2	no	—
179	#467	C₂×D₅₉	1	2	yes	—
180	C₄×Q₁₆	—	59	2	no	—
181	C₈×D₄	—	59	1	no	—
182	C₂×#107	—	59	2	no	—
183	C₂×#107	—	59	2	no	—
184	#366	—	59	1	no	—
185	#366	—	59	1	no	—
186	#366	—	59	2	no	—
187	#366	—	59	2	no	—
188	#366	—	59	2	no	—
189	#366	C₂×D₅₉	1	1	yes	characteristic
190	C₄⋊C₁₆	—	59	1	no	—
191	#358	—	59	2	no	—
192	#358	—	59	2	no	—
193	#358	C₂×D₅₉	1	2	yes	—
194	C₂²⋊C₁₆	—	59	1	no	—
195	C₂²⋊C₁₆	—	59	1	no	—
196	C₄×C₁₆	—	59	1	no	—
197	C₅₉	#33905	1	1	yes	characteristic, complemented, Sylow
198	C₂×Q₁₆	C₂²×D₅₉	1	2	yes	—
199	C₂²×C₈	—	59	1	no	—
200	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
201	C₄⋊Q₈	—	118	2	no	—
202	#125	—	236	2	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₂²⋊Q₈	—	118	2	no	—
205	C₂²⋊Q₈	—	236	2	no	—
206	C₂²⋊Q₈	—	236	2	no	—
207	C₄×Q₈	—	118	2	no	—
208	C₄×D₄	—	59	1	no	—
209	C₄×D₄	—	236	2	no	—
210	C₂×C₄⋊C₄	—	118	2	no	—
211	C₂×C₄⋊C₄	—	118	2	no	—
212	Q₃₂	—	2	2	no	—
213	C₂×C₁₆	—	2	2	no	—
214	C₂×C₁₆	—	59	1	no	—
215	C₂×C₁₆	—	59	1	no	—
216	C₂×C₁₆	—	118	2	no	—
217	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
218	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
219	#107	—	59	2	no	—
220	#107	—	59	2	no	—
221	#107	—	118	2	no	—
222	#107	—	118	2	no	—
223	#107	—	118	2	no	—
224	#107	C₂²×D₅₉	1	2	yes	—
225	C₄⋊C₈	—	59	1	no	—
226	#103	—	2	2	no	—
227	#103	—	118	2	no	—
228	#103	—	118	2	no	—
229	C₂²⋊C₈	—	59	1	no	—
230	C₂²⋊C₈	—	59	1	no	—
231	C₄×C₈	—	59	1	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	236	2	no	—
234	C₂×D₄	—	59	1	no	—
235	C₂²×C₄	—	59	1	no	—
236	C₂²×C₄	—	236	2	no	—
237	C₂²×C₄	—	236	2	no	—
238	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
239	Q₁₆	—	2	2	no	—
240	Q₁₆	—	2	2	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	59	1	no	—
244	C₂×C₈	—	59	1	no	—
245	C₂×C₈	—	118	1	no	—
246	C₂×C₈	—	118	1	no	—
247	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
248	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	59	1	no	—
252	C₄⋊C₄	—	118	2	no	—
253	C₄⋊C₄	—	118	2	no	—
254	C₄⋊C₄	—	236	2	no	—
255	C₄⋊C₄	—	236	2	no	—
256	C₄⋊C₄	—	236	2	no	—
257	C₄⋊C₄	—	236	2	no	—
258	C₄⋊C₄	—	236	2	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₂²⋊C₄	—	59	1	no	—
261	C₂²⋊C₄	—	59	1	no	—
262	C₂²⋊C₄	—	236	2	no	—
263	C₄²	—	59	1	no	—
264	C₄²	—	236	2	no	—
265	C₁₆	—	2	1	no	—
266	C₁₆	—	118	1	no	—
267	C₁₆	—	118	1	no	—
268	C₁₆	#11745687	1	2	yes	—
269	C₂³	—	59	1	no	—
270	C₂³	D₈×D₅₉	1	1	yes	characteristic
271	Q₈	—	4	2	no	—
272	Q₈	—	4	2	no	—
273	Q₈	—	944	2	no	—
274	D₄	—	59	4	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	59	1	no	—
280	C₂×C₄	—	59	1	no	—
281	C₂×C₄	—	59	1	no	—
282	C₂×C₄	—	118	1	no	—
283	C₂×C₄	—	118	1	no	—
284	C₂×C₄	—	944	2	no	—
285	C₂×C₄	—	944	2	no	—
286	C₂×C₄	—	236	2	no	—
287	C₂×C₄	—	236	2	no	—
288	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
289	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	118	1	no	—
292	C₈	—	118	1	no	—
293	C₈	C₂×#11745687	1	1	yes	characteristic
294	C₈	#14273774	1	1	yes	characteristic
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	59	2	no	—
298	C₂²	—	118	1	no	—
299	C₂²	—	118	1	no	—
300	C₂²	Q₃₂×D₅₉	1	2	yes	—
301	C₂²	C₂×D₈×D₅₉	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	59	2	no	—
306	C₄	—	118	1	no	—
307	C₄	—	118	1	no	—
308	C₄	—	472	2	no	—
309	C₄	—	472	2	no	—
310	C₄	#16838052	1	1	yes	characteristic
311	C₄	#16838216	1	1	yes	characteristic
312	C₂	—	2	2	no	—
313	C₂	—	118	2	no	—
314	C₂	#21436440	1	1	yes	characteristic
315	C₂	C₂×Q₃₂×D₅₉	1	1	yes	characteristic
316	C₂	#21437234	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₉
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2216	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂×C₂²⋊D₅₉	D₈
hypercenter	#2216	D₅₉
hypocenter	C₅₉	#33905
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

Complements

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

N	H ≅ G/N	type
2. #21426370	C₂	semidirect
3. #21426297	C₂	semidirect
4. #21426296	C₂	semidirect
5. #21426246	C₂	semidirect
6. #21426184	C₂	semidirect
7. Q₃₂×Dic₅₉	C₂	semidirect
9. C₂×#16837228	C₂	semidirect
10. #366×D₅₉	C₂	semidirect
11. C₅₉×#2216	C₂	semidirect
120. C₂²⋊D₅₉	Q₃₂	semidirect
160. #2216	D₅₉	semidirect
197. C₅₉	#33905	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21436485	2
#21426370	2
#21426297	2
#21426296	2
#21426246	2
#21426184	2
Q₃₂×Dic₅₉	2
C₁₆×C₂²⋊D₅₉	2
C₂×#16837228	2
#366×D₅₉	2
C₅₉×#2216	2
#33905	59