Subgroups — #28560719

Order 15104.

564 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433198	C₂	1	2	yes	maximal
2	#21423005	C₂	1	2	yes	maximal, complemented
3	#21422947	C₂	1	4	yes	maximal, complemented
4	#21422915	C₂	1	2	yes	maximal, complemented
5	#21422844	C₂	1	1	yes	maximal, characteristic
6	#21422517	C₂	1	2	yes	maximal
7	#21422385	C₂	1	1	yes	maximal, characteristic
8	C₅₉×#3170	C₂	1	1	yes	maximal, characteristic
9	#16837533	—	2	2	no	—
10	#16837533	C₂²	1	4	yes	—
11	Q₈×C₅₉⋊C₈	—	2	2	no	—
12	Q₈×C₅₉⋊C₈	C₂²	1	4	yes	—
13	#16837463	—	2	4	no	—
14	#16837455	C₂²	1	2	yes	—
15	#16837455	C₂²	1	1	yes	characteristic
16	C₄×#11745664	C₂²	1	1	yes	characteristic
17	#16837155	C₂²	1	2	yes	—
18	#16837153	C₂²	1	1	yes	characteristic
19	#16837153	C₂²	1	1	yes	characteristic
20	Q₈⋊(C₅₉⋊C₈)	C₂²	1	4	yes	—
21	Q₈⋊(C₅₉⋊C₈)	C₂²	1	4	yes	—
22	#16837124	C₂²	1	2	yes	—
23	#16837123	C₂²	1	2	yes	—
24	C₅₉×#533	C₂²	1	2	yes	—
25	C₅₉×#500	C₄	1	1	yes	characteristic
26	C₅₉×#494	C₄	1	1	yes	characteristic
27	C₅₉×#487	C₄	1	2	yes	—
28	C₅₉×#467	C₄	1	2	yes	—
29	C₅₉×#467	C₄	1	2	yes	—
30	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
31	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
33	C₂×#11745664	—	2	1	no	—
34	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
35	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
36	C₄⋊(C₅₉⋊C₈)	D₄	1	2	yes	—
37	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
38	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
39	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
40	C₂²⋊(C₅₉⋊C₈)	D₄	1	2	yes	—
41	#14273638	—	2	1	no	—
42	#14273638	—	2	4	no	—
43	C₄×C₅₉⋊C₈	—	2	2	no	—
44	C₄×C₅₉⋊C₈	—	2	4	no	—
45	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
46	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
47	C₂×Q₁₆×C₅₉	C₂×C₄	1	2	yes	—
48	C₂×Q₁₆×C₅₉	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
52	C₄⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
53	#125×C₅₉	—	2	2	no	—
54	C₂²⋊Q₈×C₅₉	—	2	2	no	—
55	C₂²⋊Q₈×C₅₉	C₂×C₄	1	4	yes	—
56	C₄×Q₈×C₅₉	—	2	2	no	—
57	C₄×Q₈×C₅₉	C₂³	1	4	yes	—
58	#117×C₅₉	—	2	2	no	—
59	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
60	#107×C₅₉	C₂×C₄	1	2	yes	—
61	#107×C₅₉	C₂×C₄	1	2	yes	—
62	#103×C₅₉	C₂×C₄	1	4	yes	—
63	#103×C₅₉	C₂×C₄	1	4	yes	—
64	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
68	C₂×Q₈×C₅₉	—	2	2	no	—
69	C₂×Q₈×C₅₉	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
73	Q₁₆×C₅₉	—	2	4	no	—
74	Q₁₆×C₅₉	—	4	2	no	—
75	C₂×C₈×C₅₉	—	2	2	no	—
76	C₂×C₈×C₅₉	—	4	1	no	—
77	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₅₉	—	2	2	no	—
82	C₄⋊C₄×C₅₉	—	2	2	no	—
83	C₄⋊C₄×C₅₉	—	2	2	no	—
84	C₄⋊C₄×C₅₉	—	2	4	no	—
85	C₄⋊C₄×C₅₉	—	2	4	no	—
86	C₄⋊C₄×C₅₉	C₂²×C₄	1	4	yes	—
87	C₂²⋊C₄×C₅₉	—	2	4	no	—
88	C₄²×C₅₉	—	2	4	no	—
89	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₉	#41	1	1	yes	characteristic
92	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
93	#11745664	—	4	2	no	—
94	C₂×C₅₉⋊C₈	—	2	2	no	—
95	C₂×C₅₉⋊C₈	—	2	2	no	—
96	C₂×C₅₉⋊C₈	—	2	2	no	—
97	C₂×C₅₉⋊C₈	C₂×D₄	1	4	yes	—
98	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
99	Q₈×C₅₉	—	2	4	no	—
100	Q₈×C₅₉	—	2	4	no	—
101	Q₈×C₅₉	—	4	4	no	—
102	C₂×C₄×C₅₉	—	2	1	no	—
103	C₂×C₄×C₅₉	—	2	1	no	—
104	C₂×C₄×C₅₉	—	2	2	no	—
105	C₂×C₄×C₅₉	—	2	2	no	—
106	C₂×C₄×C₅₉	—	2	4	no	—
107	C₂×C₄×C₅₉	—	2	4	no	—
108	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
114	C₈×C₅₉	—	2	1	no	—
115	C₈×C₅₉	—	2	1	no	—
116	C₈×C₅₉	—	2	1	no	—
117	C₈×C₅₉	#131	1	2	yes	—
118	C₅₉⋊C₈	—	2	4	no	—
119	C₅₉⋊C₈	—	4	2	no	—
120	C₅₉⋊C₈	—	4	2	no	—
121	C₅₉⋊C₈	—	4	2	no	—
122	#18090	—	59	1	no	maximal, Sylow
123	C₂²×C₅₉	—	2	1	no	—
124	C₂²×C₅₉	—	2	1	no	—
125	C₂²×C₅₉	—	2	1	no	—
126	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄×C₅₉	—	2	1	no	—
128	C₄×C₅₉	—	2	1	no	—
129	C₄×C₅₉	—	2	1	no	—
130	C₄×C₅₉	—	4	4	no	—
131	C₄×C₅₉	—	4	4	no	—
132	C₄×C₅₉	#444	1	2	yes	—
133	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
134	C₄×C₅₉	#568	1	1	yes	characteristic
135	C₄×C₅₉	#578	1	2	yes	—
136	#3170	D₅₉	1	1	yes	characteristic
137	#2986	—	59	2	no	—
138	#1593	—	59	2	no	—
139	#1584	—	59	4	no	—
140	#1579	—	59	2	no	—
141	#1564	—	59	1	no	—
142	#1485	—	59	2	no	—
143	#1451	—	59	1	no	—
144	C₂×C₅₉	—	4	1	no	—
145	C₂×C₅₉	#2935	1	1	yes	characteristic
146	C₂×C₅₉	#2950	1	1	yes	characteristic
147	C₂×C₅₉	C₂×#444	1	1	yes	characteristic
148	#533	C₂×D₅₉	1	2	yes	—
149	#500	Dic₅₉	1	1	yes	characteristic
150	#494	Dic₅₉	1	1	yes	characteristic
151	#487	Dic₅₉	1	2	yes	—
152	#467	Dic₅₉	1	2	yes	—
153	#467	Dic₅₉	1	2	yes	—
154	#446	—	59	4	no	—
155	#446	—	118	2	no	—
156	C₈×Q₈	—	59	4	no	—
157	C₈×Q₈	—	118	2	no	—
158	C₄×Q₁₆	C₂×D₅₉	1	2	yes	—
159	C₄×Q₁₆	C₂×D₅₉	1	2	yes	—
160	#433	—	118	4	no	—
161	#431	—	59	1	no	—
162	#431	—	59	2	no	—
163	C₄×M₁₆	—	59	1	no	—
164	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
165	#335	—	59	2	no	—
166	#334	—	59	1	no	—
167	#334	—	59	1	no	—
168	Q₈⋊C₈	—	59	4	no	—
169	Q₈⋊C₈	—	59	4	no	—
170	#322	—	59	2	no	—
171	#322	—	59	2	no	—
172	C₅₉	#18090	1	1	yes	characteristic, complemented, Sylow
173	C₂×Q₁₆	C₂×Dic₅₉	1	2	yes	—
174	C₂×Q₁₆	C₂×Dic₅₉	1	2	yes	—
175	C₂×M₁₆	—	118	1	no	—
176	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
177	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
178	C₄⋊Q₈	C₂×Dic₅₉	1	2	yes	—
179	C₄⋊Q₈	C₂×Dic₅₉	1	2	yes	—
180	#125	—	2	2	no	—
181	C₂²⋊Q₈	—	2	2	no	—
182	C₂²⋊Q₈	C₂×Dic₅₉	1	4	yes	—
183	C₄×Q₈	—	2	2	no	—
184	C₄×Q₈	C₂²×D₅₉	1	4	yes	—
185	#117	—	2	2	no	—
186	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
187	#107	C₂×Dic₅₉	1	2	yes	—
188	#107	C₂×Dic₅₉	1	2	yes	—
189	C₄⋊C₈	—	59	2	no	—
190	C₄⋊C₈	—	59	2	no	—
191	C₄⋊C₈	—	59	2	no	—
192	C₄⋊C₈	—	118	4	no	—
193	C₄⋊C₈	—	118	4	no	—
194	#103	C₂×Dic₅₉	1	4	yes	—
195	#103	C₂×Dic₅₉	1	4	yes	—
196	C₂²⋊C₈	—	59	2	no	—
197	C₂²⋊C₈	—	118	2	no	—
198	#97	—	118	4	no	—
199	#97	—	118	1	no	—
200	C₄×C₈	—	59	2	no	—
201	C₄×C₈	—	59	2	no	—
202	C₄×C₈	—	118	4	no	—
203	C₄×C₈	—	118	2	no	—
204	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
205	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	C₂²×Dic₅₉	1	4	yes	—
210	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
211	C₂²×C₄	#11745687	1	1	yes	characteristic
212	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
213	Q₁₆	—	2	4	no	—
214	Q₁₆	—	4	2	no	—
215	M₁₆	—	236	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	59	4	no	—
219	C₂×C₈	—	118	2	no	—
220	C₂×C₈	—	118	2	no	—
221	C₂×C₈	—	118	2	no	—
222	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
223	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
224	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
225	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	C₂²×Dic₅₉	1	4	yes	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	D₄×D₅₉	1	1	yes	characteristic
235	C₄²	#11745687	1	1	yes	characteristic
236	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
237	C₄²	C₂³×D₅₉	1	1	yes	characteristic
238	C₂³	D₄×Dic₅₉	1	1	yes	characteristic
239	Q₈	—	2	4	no	—
240	Q₈	—	2	4	no	—
241	Q₈	—	4	4	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
249	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
250	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
251	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
252	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
253	C₂×C₄	C₂×#11745687	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	118	4	no	—
258	C₈	—	236	2	no	—
259	C₈	—	236	2	no	—
260	C₈	—	236	2	no	—
261	C₈	#14273734	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	C₂×D₄×Dic₅₉	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	4	4	no	—
270	C₄	—	4	4	no	—
271	C₄	#16837524	1	2	yes	—
272	C₄	C₂×#14273734	1	1	yes	characteristic
273	C₄	#16838169	1	1	yes	characteristic
274	C₄	#16838215	1	2	yes	—
275	C₂	—	4	1	no	—
276	C₂	#21432840	1	1	yes	characteristic
277	C₂	#21432964	1	1	yes	characteristic
278	C₂	C₂×#16837524	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₄²	C₂³×D₅₉
Fitting	C₅₉×#3170	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	#3170	D₅₉
hypocenter	C₅₉	#18090
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #21423005	C₂	semidirect
3. #21422947	C₂	semidirect
4. #21422915	C₂	semidirect
172. C₅₉	#18090	semidirect

Maximal subgroups

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

subgroup	index
#21433198	2
#21423005	2
#21422947	2
#21422915	2
#21422844	2
#21422517	2
#21422385	2
C₅₉×#3170	2
#18090	59