Subgroups — #28505162

Order 15104.

374 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21437198	C₂	1	1	yes	maximal, characteristic
2	#21425854	C₂	1	1	yes	maximal, characteristic, complemented
3	#21425709	C₂	1	1	yes	maximal, characteristic
4	#21423897	C₂	1	1	yes	maximal, characteristic
5	#21422182	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422181	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#2151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273733	C₂²	1	1	yes	characteristic
9	#16837820	—	2	2	no	—
10	#16837818	—	2	2	no	—
11	C₂×#14273667	C₂²	1	1	yes	characteristic
12	C₂×#14273667	C₂²	1	1	yes	characteristic
13	M₃₂×D₅₉	—	2	2	no	—
14	#16837384	C₄	1	1	yes	characteristic
15	#16837375	C₄	1	1	yes	characteristic
16	#16837357	C₂²	1	1	yes	characteristic
17	Dic₅₉⋊C₁₆	—	2	2	no	—
18	(C₂×D₅₉)⋊C₁₆	—	2	2	no	—
19	#16837174	C₄	1	1	yes	characteristic
20	#16837172	C₄	1	1	yes	characteristic
21	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
23	C₅₉×#405	C₂²	1	1	yes	characteristic
24	C₅₉×C₄⋊C₁₆	—	2	2	no	—
25	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
26	#14273733	—	2	1	no	—
27	#14273733	—	2	1	no	—
28	#14273733	—	2	1	no	—
29	#14273733	—	2	1	no	—
30	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
31	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
32	C₂×#11745659	Q₈	1	2	yes	—
33	C₂×C₈×D₅₉	D₄	1	2	yes	—
34	#14273686	C₂×C₄	1	1	yes	characteristic
35	#14273667	—	2	1	no	—
36	#14273667	—	2	1	no	—
37	#14273667	—	2	1	no	—
38	#14273667	—	2	1	no	—
39	#14273666	—	2	4	no	—
40	C₂×C₅₉⋊C₁₆	—	2	1	no	—
41	C₂×C₅₉⋊C₁₆	D₄	1	2	yes	—
42	C₁₆×D₅₉	—	2	4	no	—
43	#14273645	—	2	1	no	—
44	#14273643	—	2	1	no	—
45	#14273641	C₈	1	4	yes	—
46	#14273637	—	2	1	no	—
47	C₈×Dic₅₉	—	2	1	no	—
48	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
50	#117×C₅₉	C₂×C₄	1	1	yes	characteristic
51	M₃₂×C₅₉	—	2	1	no	—
52	M₃₂×C₅₉	—	2	1	no	—
53	C₂×C₁₆×C₅₉	—	2	1	no	—
54	C₂×C₁₆×C₅₉	—	2	1	no	—
55	C₂×C₁₆×C₅₉	—	2	1	no	—
56	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
57	#97×C₅₉	—	2	1	no	—
58	C₄×C₈×C₅₉	—	2	1	no	—
59	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₅₉	C₂×C₈	1	2	yes	—
61	#11745685	—	2	1	no	—
62	#11745685	—	2	1	no	—
63	#11745685	—	2	1	no	—
64	#11745685	—	2	1	no	—
65	C₂×D₂₃₆	M₁₆	1	1	yes	characteristic
66	C₂×C₄×D₅₉	C₄⋊C₄	1	2	yes	—
67	C₂×Dic₁₁₈	M₁₆	1	1	yes	characteristic
68	M₁₆×C₅₉	—	2	1	no	—
69	M₁₆×C₅₉	—	2	1	no	—
70	C₂×C₈×C₅₉	—	2	1	no	—
71	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
76	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
77	C₄⋊C₄×C₅₉	—	2	1	no	—
78	C₂²⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
79	C₄²×C₅₉	—	2	1	no	—
80	C₂²⋊Dic₅₉	C₂×C₈	1	2	yes	—
81	C₄⋊Dic₅₉	—	2	1	no	—
82	C₄×Dic₅₉	—	2	1	no	—
83	#11745664	—	2	1	no	—
84	#11745664	—	2	1	no	—
85	#11745664	—	2	1	no	—
86	#11745664	—	2	1	no	—
87	C₂×C₅₉⋊C₈	—	2	1	no	—
88	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
89	#11745659	—	2	2	no	—
90	#11745659	—	2	2	no	—
91	C₈×D₅₉	—	2	2	no	—
92	C₈×D₅₉	—	2	2	no	—
93	C₁₆×C₅₉	—	2	2	no	—
94	C₁₆×C₅₉	—	4	1	no	—
95	C₅₉⋊C₁₆	—	2	2	no	—
96	C₅₉⋊C₁₆	—	4	1	no	—
97	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
98	C₂²×D₅₉	C₄⋊C₈	1	2	yes	—
99	C₂×C₄×C₅₉	—	2	2	no	—
100	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
105	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
106	C₂²⋊D₅₉	—	2	2	no	—
107	C₂²⋊D₅₉	—	2	2	no	—
108	C₂×Dic₅₉	—	2	2	no	—
109	C₂×Dic₅₉	C₄⋊C₈	1	2	yes	—
110	D₂₃₆	—	2	1	no	—
111	D₂₃₆	—	2	1	no	—
112	C₄×D₅₉	—	2	2	no	—
113	C₄×D₅₉	—	2	2	no	—
114	Dic₁₁₈	—	2	1	no	—
115	Dic₁₁₈	—	2	1	no	—
116	C₈×C₅₉	—	4	1	no	—
117	C₈×C₅₉	#120	1	1	yes	characteristic
118	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₈×C₅₉	#123	1	1	yes	characteristic
121	C₅₉⋊C₈	—	2	2	no	—
122	C₅₉⋊C₈	—	4	1	no	—
123	#12915	—	59	1	no	maximal, Sylow
124	C₂²×C₅₉	—	4	1	no	—
125	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
126	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
127	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
128	C₂×D₅₉	—	2	2	no	—
129	C₂×D₅₉	—	4	2	no	—
130	C₄×C₅₉	—	4	2	no	—
131	C₄×C₅₉	#385	1	1	yes	characteristic
132	C₄×C₅₉	#424	1	1	yes	characteristic
133	C₄×C₅₉	#435	1	1	yes	characteristic
134	C₄×C₅₉	#435	1	1	yes	characteristic
135	Dic₅₉	—	2	2	no	—
136	Dic₅₉	—	4	2	no	—
137	#3407	—	59	1	no	—
138	#2151	—	59	1	no	—
139	#2151	D₅₉	1	1	yes	characteristic, complemented
140	#2113	—	59	1	no	—
141	#1757	—	59	1	no	—
142	#1378	—	59	1	no	—
143	#1378	—	59	1	no	—
144	C₂×C₅₉	—	2	1	no	—
145	C₂×C₅₉	—	2	1	no	—
146	C₂×C₅₉	—	2	1	no	—
147	C₂×C₅₉	#1850	1	1	yes	characteristic
148	D₅₉	—	4	2	no	—
149	C₂×#131	—	59	1	no	—
150	#504	—	118	2	no	—
151	#504	—	118	2	no	—
152	C₂×M₃₂	—	59	1	no	—
153	C₂×M₃₂	—	59	1	no	—
154	C₂×M₃₂	—	118	2	no	—
155	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
156	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
157	#413	—	59	1	no	—
158	#410	—	59	1	no	—
159	#405	—	59	1	no	—
160	#405	C₂×D₅₉	1	1	yes	characteristic
161	C₄⋊C₁₆	—	2	2	no	—
162	C₄⋊C₁₆	—	118	2	no	—
163	C₂²⋊C₁₆	—	118	2	no	—
164	#344	—	59	1	no	—
165	#344	—	59	1	no	—
166	C₅₉	#12915	1	1	yes	characteristic, complemented, Sylow
167	C₂×#41	—	59	1	no	—
168	#131	—	118	1	no	—
169	#131	—	118	1	no	—
170	#131	—	118	1	no	—
171	#131	—	118	1	no	—
172	C₂×M₁₆	—	59	1	no	—
173	C₂×M₁₆	—	59	2	no	—
174	C₂×M₁₆	—	59	1	no	—
175	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
176	C₂²×C₈	—	59	2	no	—
177	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
178	#117	—	59	1	no	—
179	#117	C₄×D₅₉	1	1	yes	characteristic
180	M₃₂	—	2	1	no	—
181	M₃₂	—	2	1	no	—
182	M₃₂	—	118	4	no	—
183	M₃₂	—	118	1	no	—
184	M₃₂	—	118	1	no	—
185	M₃₂	—	118	1	no	—
186	M₃₂	—	118	1	no	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	59	2	no	—
191	C₂×C₁₆	—	118	4	no	—
192	C₂×C₁₆	—	118	1	no	—
193	C₂×C₁₆	C₂²⋊D₅₉	1	2	yes	—
194	#101	—	118	1	no	—
195	#100	—	118	1	no	—
196	#99	—	59	4	no	—
197	#97	—	2	1	no	—
198	#97	—	118	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	118	1	no	—
201	#41	—	118	1	no	—
202	#41	—	118	1	no	—
203	#41	—	118	1	no	—
204	#41	—	118	1	no	—
205	C₂×Q₈	—	59	1	no	—
206	C₂×D₄	—	59	2	no	—
207	C₂×D₄	—	59	1	no	—
208	C₂²×C₄	—	59	2	no	—
209	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
210	M₁₆	—	2	1	no	—
211	M₁₆	—	2	1	no	—
212	M₁₆	—	118	1	no	—
213	M₁₆	—	118	1	no	—
214	M₁₆	—	118	2	no	—
215	M₁₆	—	118	2	no	—
216	M₁₆	—	118	1	no	—
217	M₁₆	—	118	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	59	2	no	—
220	C₂×C₈	—	118	2	no	—
221	C₂×C₈	—	118	2	no	—
222	C₂×C₈	—	118	1	no	—
223	C₂×C₈	#11745685	1	1	yes	characteristic
224	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
225	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
226	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
227	C₂×C₈	#11745689	1	1	yes	characteristic
228	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	118	1	no	—
231	C₂²⋊C₄	—	59	2	no	—
232	C₂²⋊C₄	C₈×D₅₉	1	2	yes	—
233	C₄²	—	2	1	no	—
234	C₄²	—	118	1	no	—
235	C₁₆	—	2	2	no	—
236	C₁₆	—	4	1	no	—
237	C₁₆	—	118	2	no	—
238	C₁₆	—	236	1	no	—
239	C₂³	—	59	2	no	—
240	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
241	Q₈	—	118	1	no	—
242	Q₈	—	118	1	no	—
243	D₄	—	118	2	no	—
244	D₄	—	118	2	no	—
245	D₄	—	118	1	no	—
246	D₄	—	118	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	59	2	no	—
249	C₂×C₄	—	118	2	no	—
250	C₂×C₄	—	118	2	no	—
251	C₂×C₄	—	118	2	no	—
252	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
253	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
254	C₂×C₄	#14273689	1	1	yes	characteristic
255	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
256	C₂×C₄	#14273733	1	1	yes	characteristic
257	C₂×C₄	#14273735	1	1	yes	characteristic
258	C₈	—	4	1	no	—
259	C₈	—	118	2	no	—
260	C₈	—	236	1	no	—
261	C₈	#14273695	1	1	yes	characteristic
262	C₈	#14273703	1	1	yes	characteristic
263	C₈	#14273705	1	1	yes	characteristic
264	C₈	#14273712	1	1	yes	characteristic
265	C₂²	—	4	1	no	—
266	C₂²	—	118	2	no	—
267	C₂²	—	236	2	no	—
268	C₂²	C₄⋊C₈×D₅₉	1	1	yes	characteristic
269	C₂²	C₈×C₂²⋊D₅₉	1	1	yes	characteristic
270	C₂²	#16837467	1	1	yes	characteristic
271	C₄	—	4	2	no	—
272	C₄	—	118	2	no	—
273	C₄	—	236	2	no	—
274	C₄	#16837273	1	1	yes	characteristic
275	C₄	#16837427	1	1	yes	characteristic
276	C₄	#16837471	1	1	yes	characteristic
277	C₄	#16837472	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	236	2	no	—
282	C₂	#21424272	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14273695
commutator	C₂³×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2151	C₂
socle	C₂×C₅₉	#1850
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2151	D₅₉
hypocenter	C₅₉	#12915
LCS	G ▷ C₂³×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

Complements

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

N	H ≅ G/N	type
2. #21425854	C₂	semidirect
5. #21422182	C₂	semidirect
6. #21422181	C₂	semidirect
7. C₅₉×#2151	C₂	semidirect
139. #2151	D₅₉	semidirect
166. C₅₉	#12915	semidirect

Maximal subgroups

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

subgroup	index
#21437198	2
#21425854	2
#21425709	2
#21423897	2
#21422182	2
#21422181	2
C₅₉×#2151	2
#12915	59