Subgroups — #28746070

Order 15104.

470 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21436467	C₂	1	2	yes	maximal
2	#21426373	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426303	C₂	1	2	yes	maximal, complemented
4	#21426291	C₂	1	2	yes	maximal, complemented
5	#21426262	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	#21425877	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16837228	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#2216	C₂	1	1	yes	maximal, characteristic, complemented
12	#16838094	—	2	2	no	—
13	C₂×C₁₆×D₅₉	C₂²	1	1	yes	characteristic
14	#16837803	C₂²	1	2	yes	—
15	#16837742	—	2	2	no	—
16	#16837730	—	2	2	no	—
17	#16837711	—	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	#16837439	C₂²	1	2	yes	—
22	C₂×#14273661	C₂²	1	2	yes	—
23	#16837229	C₂²	1	2	yes	—
24	#16837229	C₂²	1	2	yes	—
25	#16837228	C₂²	1	1	yes	characteristic
26	#16837228	C₂²	1	1	yes	characteristic
27	#16837228	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	#14273722	—	4	2	no	—
41	#14273718	—	2	2	no	—
42	#14273718	—	4	2	no	—
43	#14273711	—	4	2	no	—
44	#14273703	—	2	2	no	—
45	Q₈×Dic₅₉	—	2	2	no	—
46	C₄×C₂²⋊D₅₉	D₄	1	1	yes	characteristic
47	#14273685	—	2	2	no	—
48	#14273683	—	4	2	no	—
49	C₂×C₄⋊Dic₅₉	—	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	#14273658	—	2	2	no	—
57	#14273658	—	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	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₅₉	—	2	2	no	—
79	Q₁₆×C₅₉	—	2	2	no	—
80	C₂×C₈×C₅₉	—	2	1	no	—
81	C₂×C₈×C₅₉	—	2	1	no	—
82	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₅₉	—	2	2	no	—
85	C₄⋊C₄×C₅₉	—	4	2	no	—
86	C₂²⋊C₄×C₅₉	—	4	2	no	—
87	C₂²⋊Dic₅₉	D₈	1	1	yes	characteristic
88	(C₂×D₅₉)⋊C₄	—	4	2	no	—
89	(C₂×D₅₉)⋊C₄	—	4	2	no	—
90	(C₂×D₅₉)⋊C₄	D₈	1	1	yes	characteristic
91	C₄⋊Dic₅₉	—	2	2	no	—
92	C₄⋊Dic₅₉	—	2	2	no	—
93	C₄⋊Dic₅₉	—	4	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	D₈	1	1	yes	characteristic
99	C₄×Dic₅₉	—	4	2	no	—
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	—	2	2	no	—
121	C₂×Dic₅₉	—	4	2	no	—
122	C₂×Dic₅₉	—	4	2	no	—
123	C₂×Dic₅₉	—	8	4	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	C₈×C₅₉	—	2	1	no	—
130	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
131	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
132	C₅₉⋊C₈	—	2	1	no	—
133	C₅₉⋊C₈	—	2	1	no	—
134	#33903	—	59	1	no	maximal, Sylow
135	C₂²×C₅₉	—	2	1	no	—
136	C₂²×C₅₉	—	2	1	no	—
137	C₂²×C₅₉	#508	1	2	yes	—
138	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
139	C₂×D₅₉	—	2	1	no	—
140	C₂×D₅₉	—	2	1	no	—
141	C₂×D₅₉	—	2	1	no	—
142	C₄×C₅₉	—	2	1	no	—
143	C₄×C₅₉	—	8	2	no	—
144	C₄×C₅₉	—	8	2	no	—
145	C₄×C₅₉	#547	1	1	yes	characteristic
146	C₄×C₅₉	#578	1	1	yes	characteristic
147	Dic₅₉	—	2	1	no	—
148	Dic₅₉	—	2	1	no	—
149	Dic₅₉	—	2	1	no	—
150	Dic₅₉	—	8	2	no	—
151	Dic₅₉	—	8	2	no	—
152	#3336	—	59	2	no	—
153	#2253	—	59	1	no	—
154	#2237	—	59	2	no	—
155	#2235	—	59	2	no	—
156	#2229	—	59	2	no	—
157	#2216	—	59	1	no	—
158	#2216	D₅₉	1	1	yes	characteristic, complemented
159	C₄×Q₃₂	—	59	1	no	—
160	D₄×C₁₆	—	59	1	no	—
161	#2158	—	59	1	no	—
162	C₂×#366	—	59	1	no	—
163	C₂×C₅₉	—	2	2	no	—
164	C₂×C₅₉	#3334	1	1	yes	characteristic
165	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
166	C₂×C₅₉	#3417	1	1	yes	characteristic
167	D₅₉	—	4	1	no	—
168	#552	—	118	2	no	—
169	C₂×Q₃₂	C₂×D₅₉	1	1	yes	characteristic
170	C₂²×C₁₆	—	59	1	no	—
171	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
172	#499	—	59	2	no	—
173	#485	—	118	2	no	—
174	#483	—	118	2	no	—
175	#479	—	118	2	no	—
176	#467	—	59	2	no	—
177	#467	C₂×D₅₉	1	2	yes	—
178	C₄×Q₁₆	—	59	2	no	—
179	C₈×D₄	—	59	1	no	—
180	#427	—	59	2	no	—
181	C₂×#107	—	59	2	no	—
182	#367	—	59	2	no	—
183	#367	—	59	2	no	—
184	#366	—	59	1	no	—
185	#366	—	59	1	no	—
186	#366	—	59	2	no	—
187	#366	C₂×D₅₉	1	1	yes	characteristic
188	C₄⋊C₁₆	—	59	1	no	—
189	#358	—	59	2	no	—
190	#358	—	59	2	no	—
191	#358	C₂×D₅₉	1	2	yes	—
192	C₂²⋊C₁₆	—	59	1	no	—
193	C₂²⋊C₁₆	—	59	1	no	—
194	C₄×C₁₆	—	59	1	no	—
195	C₅₉	#33903	1	1	yes	characteristic, complemented, Sylow
196	C₂×Q₁₆	C₂²×D₅₉	1	2	yes	—
197	C₂²×C₈	—	59	1	no	—
198	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
199	#126	—	236	2	no	—
200	#125	—	118	2	no	—
201	#125	—	236	2	no	—
202	#123	—	236	2	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₂²⋊Q₈	—	118	2	no	—
205	C₄×Q₈	—	118	2	no	—
206	C₄×D₄	—	59	1	no	—
207	#117	—	118	2	no	—
208	#117	—	236	2	no	—
209	C₂×C₄⋊C₄	—	118	2	no	—
210	Q₃₂	—	2	2	no	—
211	C₂×C₁₆	—	2	2	no	—
212	C₂×C₁₆	—	59	1	no	—
213	C₂×C₁₆	—	59	1	no	—
214	C₂×C₁₆	—	118	2	no	—
215	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
216	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
217	#107	—	59	2	no	—
218	#107	—	59	2	no	—
219	#107	—	118	2	no	—
220	#107	C₂²×D₅₉	1	2	yes	—
221	#106	—	118	2	no	—
222	#106	—	118	2	no	—
223	C₄⋊C₈	—	59	1	no	—
224	#103	—	2	2	no	—
225	#103	—	118	2	no	—
226	#103	—	118	2	no	—
227	C₂²⋊C₈	—	59	1	no	—
228	C₂²⋊C₈	—	59	1	no	—
229	C₄×C₈	—	59	1	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×D₄	—	59	1	no	—
232	C₂²×C₄	—	59	1	no	—
233	C₂²×C₄	—	236	2	no	—
234	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	2	2	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	59	1	no	—
240	C₂×C₈	—	59	1	no	—
241	C₂×C₈	—	118	1	no	—
242	C₂×C₈	—	118	1	no	—
243	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
244	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	59	1	no	—
248	C₄⋊C₄	—	118	2	no	—
249	C₄⋊C₄	—	118	2	no	—
250	C₄⋊C₄	—	236	2	no	—
251	C₄⋊C₄	—	236	2	no	—
252	C₄⋊C₄	—	236	2	no	—
253	C₄⋊C₄	—	236	2	no	—
254	C₄⋊C₄	—	236	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	59	1	no	—
257	C₂²⋊C₄	—	59	1	no	—
258	C₂²⋊C₄	—	236	2	no	—
259	C₂²⋊C₄	—	236	2	no	—
260	C₄²	—	59	1	no	—
261	C₄²	—	236	2	no	—
262	C₄²	—	236	2	no	—
263	C₁₆	—	2	1	no	—
264	C₁₆	—	118	1	no	—
265	C₁₆	—	118	1	no	—
266	C₁₆	#11745687	1	2	yes	—
267	C₂³	—	59	1	no	—
268	C₂³	D₈×D₅₉	1	1	yes	characteristic
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	D₄	—	118	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	59	1	no	—
277	C₂×C₄	—	59	1	no	—
278	C₂×C₄	—	59	1	no	—
279	C₂×C₄	—	118	1	no	—
280	C₂×C₄	—	118	1	no	—
281	C₂×C₄	—	472	4	no	—
282	C₂×C₄	—	236	2	no	—
283	C₂×C₄	—	236	2	no	—
284	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
285	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
286	C₈	—	2	1	no	—
287	C₈	—	118	1	no	—
288	C₈	—	118	1	no	—
289	C₈	C₂×#11745687	1	1	yes	characteristic
290	C₈	#14273774	1	1	yes	characteristic
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	118	1	no	—
294	C₂²	—	118	1	no	—
295	C₂²	—	118	1	no	—
296	C₂²	#16837835	1	2	yes	—
297	C₂²	C₂×D₈×D₅₉	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	2	no	—
301	C₄	—	118	1	no	—
302	C₄	—	118	1	no	—
303	C₄	—	118	1	no	—
304	C₄	—	472	2	no	—
305	C₄	—	472	2	no	—
306	C₄	#16838052	1	1	yes	characteristic
307	C₄	#16838216	1	1	yes	characteristic
308	C₂	—	2	2	no	—
309	C₂	—	236	1	no	—
310	C₂	#21436440	1	1	yes	characteristic
311	C₂	C₂×#16837835	1	1	yes	characteristic
312	C₂	#21437246	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	#33903
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. #21426373	C₂	semidirect
3. #21426303	C₂	semidirect
4. #21426291	C₂	semidirect
5. #21426262	C₂	semidirect
6. #21426184	C₂	semidirect
7. Q₃₂×Dic₅₉	C₂	semidirect
9. #21425877	C₂	semidirect
10. C₂×#16837228	C₂	semidirect
11. C₅₉×#2216	C₂	semidirect
158. #2216	D₅₉	semidirect
195. C₅₉	#33903	semidirect

Maximal subgroups

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

subgroup	index
#21436467	2
#21426373	2
#21426303	2
#21426291	2
#21426262	2
#21426184	2
Q₃₂×Dic₅₉	2
C₁₆×C₂²⋊D₅₉	2
#21425877	2
C₂×#16837228	2
C₅₉×#2216	2
#33903	59