Subgroups — #28503798

Order 15104.

418 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837820	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837185	C₂	1	1	yes	maximal, characteristic
3	#21425693	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16837177	C₂	1	1	yes	maximal, characteristic, complemented
5	#21423935	C₂	1	1	yes	maximal, characteristic
6	#21422177	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273733	C₂²	1	1	yes	characteristic
9	#16837820	—	2	2	no	—
10	#16837820	C₄	1	4	yes	complemented
11	C₂×#14273667	C₂²	1	1	yes	characteristic
12	C₂×#14273666	—	2	1	no	—
13	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₅₉	—	2	1	no	—
15	C₂×#14273653	C₄	1	1	yes	characteristic
16	#16837396	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
18	#16837185	C₄	1	4	yes	complemented
19	#16837178	—	2	1	no	—
20	#16837177	—	2	2	no	—
21	C₄×C₅₉⋊C₁₆	—	2	1	no	—
22	#16837166	C₄	1	1	yes	characteristic
23	#16837161	C₄	1	1	yes	characteristic
24	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
26	C₅₉×#405	C₂²	1	1	yes	characteristic
27	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
28	#14273733	—	2	1	no	—
29	#14273733	—	2	1	no	—
30	#14273733	C₂×C₄	1	2	yes	—
31	#14273733	C₂×C₄	1	2	yes	—
32	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
33	C₂×#11745659	—	2	1	no	—
34	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₅₉	—	2	1	no	—
36	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
37	#14273667	—	2	1	no	—
38	#14273667	C₂×C₄	1	2	yes	—
39	#14273666	—	2	4	no	—
40	C₂×C₅₉⋊C₁₆	—	2	1	no	—
41	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
42	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
43	C₂×C₅₉⋊C₁₆	D₄	1	2	yes	—
44	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
45	C₁₆×D₅₉	—	2	4	no	—
46	#14273653	C₈	1	4	yes	—
47	#14273650	C₈	1	2	yes	—
48	#14273647	C₈	1	2	yes	—
49	C₈×Dic₅₉	—	2	1	no	—
50	C₈×Dic₅₉	—	2	1	no	—
51	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
53	#117×C₅₉	C₂×C₄	1	1	yes	characteristic
54	M₃₂×C₅₉	—	2	1	no	—
55	M₃₂×C₅₉	C₂×C₄	1	2	yes	—
56	C₂×C₁₆×C₅₉	—	2	1	no	—
57	C₂×C₁₆×C₅₉	—	2	1	no	—
58	C₂×C₁₆×C₅₉	—	2	1	no	—
59	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
60	#97×C₅₉	—	2	1	no	—
61	C₄×C₈×C₅₉	—	2	1	no	—
62	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₅₉	—	2	1	no	—
64	C₂²×Dic₅₉	—	2	1	no	—
65	#11745685	—	2	1	no	—
66	#11745685	—	2	1	no	—
67	#11745685	C₄²	1	2	yes	—
68	#11745685	C₂×C₈	1	2	yes	—
69	C₂×D₂₃₆	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×D₅₉	—	2	1	no	—
71	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
72	M₁₆×C₅₉	—	2	1	no	—
73	M₁₆×C₅₉	C₂×C₈	1	2	yes	—
74	C₂×C₈×C₅₉	—	2	1	no	—
75	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
80	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
81	C₄⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
82	C₂²⋊C₄×C₅₉	—	2	1	no	—
83	C₄²×C₅₉	—	2	1	no	—
84	C₄×Dic₅₉	—	2	1	no	—
85	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
86	#11745664	—	2	1	no	—
87	#11745664	C₄²	1	2	yes	—
88	C₂×C₅₉⋊C₈	—	2	1	no	—
89	C₂×C₅₉⋊C₈	—	2	1	no	—
90	C₂×C₅₉⋊C₈	—	2	1	no	—
91	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
92	#11745659	—	2	2	no	—
93	#11745659	—	2	2	no	—
94	C₈×D₅₉	—	2	2	no	—
95	C₈×D₅₉	—	2	2	no	—
96	C₁₆×C₅₉	—	2	2	no	—
97	C₁₆×C₅₉	—	2	2	no	—
98	C₅₉⋊C₁₆	—	2	2	no	—
99	C₅₉⋊C₁₆	C₂²⋊C₄	1	2	yes	—
100	C₅₉⋊C₁₆	C₂²⋊C₄	1	2	yes	—
101	C₂³×C₅₉	#131	1	1	yes	characteristic
102	C₂²×D₅₉	—	2	1	no	—
103	C₂×C₄×C₅₉	—	2	2	no	—
104	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
110	C₂²⋊D₅₉	—	2	2	no	—
111	C₂²⋊D₅₉	—	2	2	no	—
112	C₂×Dic₅₉	—	2	1	no	—
113	C₂×Dic₅₉	—	2	2	no	—
114	C₂×Dic₅₉	—	4	2	no	—
115	D₂₃₆	—	2	1	no	—
116	D₂₃₆	C₄×C₈	1	2	yes	—
117	C₄×D₅₉	—	2	2	no	—
118	C₄×D₅₉	—	2	2	no	—
119	Dic₁₁₈	—	2	1	no	—
120	Dic₁₁₈	C₄×C₈	1	2	yes	—
121	C₈×C₅₉	—	2	2	no	—
122	C₈×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₈×C₅₉	#117	1	1	yes	characteristic
124	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
125	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
126	C₅₉⋊C₈	—	2	2	no	—
127	C₅₉⋊C₈	—	2	2	no	—
128	#12615	—	59	1	no	maximal, Sylow
129	C₂²×C₅₉	—	2	2	no	—
130	C₂²×C₅₉	#405	1	1	yes	characteristic
131	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
132	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
133	C₂×D₅₉	—	2	2	no	—
134	C₂×D₅₉	—	4	2	no	—
135	C₄×C₅₉	—	4	2	no	—
136	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
138	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
139	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
140	Dic₅₉	—	2	2	no	—
141	Dic₅₉	—	4	2	no	—
142	C₂×#504	—	59	1	no	—
143	C₂×#350	—	59	1	no	—
144	#2108	—	59	1	no	—
145	C₂×#346	—	59	1	no	—
146	#1764	—	59	1	no	—
147	#1379	D₅₉	1	1	yes	characteristic, complemented
148	#1376	—	59	1	no	—
149	C₂×C₅₉	—	2	1	no	—
150	C₂×C₅₉	—	2	1	no	—
151	C₂×C₅₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₅₉	#2171	1	2	yes	—
153	D₅₉	—	4	2	no	—
154	C₂×#131	—	59	1	no	—
155	#504	—	59	4	no	—
156	#504	—	118	2	no	—
157	C₂×M₃₂	—	59	1	no	—
158	C₂×M₃₂	—	118	1	no	—
159	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
160	C₂²×C₁₆	—	59	1	no	—
161	C₂²×C₁₆	—	118	1	no	—
162	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
163	C₂×#104	—	59	1	no	—
164	#416	—	59	1	no	—
165	#405	C₂×D₅₉	1	1	yes	characteristic
166	C₂×C₄×C₈	—	59	1	no	—
167	#350	—	59	4	no	—
168	#346	—	118	1	no	—
169	#346	—	118	2	no	—
170	C₄×C₁₆	—	118	1	no	—
171	#341	—	59	1	no	—
172	#339	—	59	1	no	—
173	C₅₉	#12615	1	1	yes	characteristic, complemented, Sylow
174	C₂×#41	—	59	1	no	—
175	#131	—	59	2	no	—
176	#131	—	59	2	no	—
177	#131	—	118	1	no	—
178	#131	—	118	1	no	—
179	C₂×M₁₆	—	59	1	no	—
180	C₂×M₁₆	—	118	1	no	—
181	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
182	C₂²×C₈	—	59	1	no	—
183	C₂²×C₈	—	118	1	no	—
184	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
185	#117	C₄×D₅₉	1	1	yes	characteristic
186	C₂×C₄²	—	59	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	59	2	no	—
189	M₃₂	—	118	4	no	—
190	M₃₂	—	118	1	no	—
191	M₃₂	C₄×D₅₉	1	2	yes	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	—	59	2	no	—
196	C₂×C₁₆	—	59	2	no	—
197	C₂×C₁₆	—	59	1	no	—
198	C₂×C₁₆	—	59	1	no	—
199	C₂×C₁₆	—	118	4	no	—
200	C₂×C₁₆	—	118	1	no	—
201	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
202	#104	—	59	4	no	—
203	#103	—	59	2	no	—
204	#102	—	59	2	no	—
205	#97	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	118	1	no	—
208	C₄×C₈	—	118	1	no	—
209	#41	—	59	2	no	—
210	#41	—	59	2	no	—
211	#41	—	118	1	no	—
212	#41	—	118	1	no	—
213	C₂×Q₈	—	59	1	no	—
214	C₂×D₄	—	59	1	no	—
215	C₂×D₄	—	118	1	no	—
216	C₂²×C₄	—	118	1	no	—
217	C₂²×C₄	—	118	1	no	—
218	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
219	M₁₆	—	2	1	no	—
220	M₁₆	—	59	2	no	—
221	M₁₆	—	118	1	no	—
222	M₁₆	—	118	2	no	—
223	M₁₆	—	118	2	no	—
224	M₁₆	C₈×D₅₉	1	2	yes	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	59	2	no	—
227	C₂×C₈	—	118	1	no	—
228	C₂×C₈	—	118	1	no	—
229	C₂×C₈	—	118	1	no	—
230	C₂×C₈	—	118	2	no	—
231	C₂×C₈	—	118	2	no	—
232	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
234	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
235	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
236	C₂×C₈	#11745687	1	1	yes	characteristic
237	C₂×C₈	#11745689	1	1	yes	characteristic
238	C₄⋊C₄	C₈×D₅₉	1	2	yes	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	59	2	no	—
242	C₄²	—	118	1	no	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	—	59	2	no	—
246	C₁₆	—	59	2	no	—
247	C₁₆	—	118	2	no	—
248	C₂³	—	118	1	no	—
249	C₂³	#14273735	1	1	yes	characteristic
250	Q₈	—	59	2	no	—
251	Q₈	—	118	1	no	—
252	D₄	—	59	2	no	—
253	D₄	—	118	2	no	—
254	D₄	—	118	2	no	—
255	D₄	—	118	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	118	1	no	—
258	C₂×C₄	—	118	2	no	—
259	C₂×C₄	—	118	2	no	—
260	C₂×C₄	—	118	2	no	—
261	C₂×C₄	—	236	2	no	—
262	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
263	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
264	C₂×C₄	#14273689	1	1	yes	characteristic
265	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
266	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
267	C₂×C₄	#14273735	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	118	2	no	—
270	C₈	—	118	2	no	—
271	C₈	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
272	C₈	#14273685	1	1	yes	characteristic
273	C₈	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
274	C₈	#14273689	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	118	2	no	—
277	C₂²	—	236	2	no	—
278	C₂²	#16837359	1	1	yes	characteristic
279	C₂²	(C₂²⋊D₅₉)⋊C₈	1	1	yes	characteristic
280	C₂²	#16837467	1	1	yes	characteristic
281	C₄	—	4	2	no	—
282	C₄	—	118	2	no	—
283	C₄	—	236	2	no	—
284	C₄	#16837249	1	1	yes	characteristic
285	C₄	C₄×C₈×D₅₉	1	1	yes	characteristic
286	C₄	(C₂²⋊D₅₉)⋊C₈	1	1	yes	characteristic
287	C₄	#16837467	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	236	2	no	—
291	C₂	#21423866	1	1	yes	characteristic
292	C₂	#21425935	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₉
commutator	C₄×C₅₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1379	C₂
socle	C₂²×C₅₉	C₈×D₄
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1379	D₅₉
hypocenter	C₅₉	#12615
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
3. #21425693	C₂	semidirect
4. C₂×#16837177	C₂	semidirect
6. #21422177	C₂	semidirect
7. C₅₉×#1379	C₂	semidirect
10. #16837820	C₄	semidirect
18. #16837185	C₄	semidirect
147. #1379	D₅₉	semidirect
173. C₅₉	#12615	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837820	2
C₂×#16837185	2
#21425693	2
C₂×#16837177	2
#21423935	2
#21422177	2
C₅₉×#1379	2
#12615	59