Subgroups — #28505178

Order 15104.

384 conjugacy classes of subgroups.

The subgroup lattice has 316 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×#16837222	C₂	1	1	yes	maximal, characteristic, complemented
3	#21425855	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16837185	C₂	1	1	yes	maximal, characteristic
5	#21423935	C₂	1	1	yes	maximal, characteristic
6	#21422173	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	—	2	2	no	—
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	#16837222	—	2	2	no	—
19	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
20	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
21	#16837185	—	2	2	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	D₄	1	2	yes	—
31	#14273733	Q₈	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	—	2	1	no	—
39	#14273666	—	2	4	no	—
40	C₂×C₅₉⋊C₁₆	—	2	1	no	—
41	C₂×C₅₉⋊C₁₆	—	2	1	no	—
42	C₂×C₅₉⋊C₁₆	—	2	1	no	—
43	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
44	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
45	C₁₆×D₅₉	—	2	4	no	—
46	#14273653	—	2	2	no	—
47	#14273650	C₈	1	1	yes	characteristic
48	#14273650	C₈	1	1	yes	characteristic
49	#14273647	C₈	1	1	yes	characteristic
50	#14273647	C₈	1	1	yes	characteristic
51	C₈×Dic₅₉	—	2	1	no	—
52	C₈×Dic₅₉	—	2	1	no	—
53	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
55	#117×C₅₉	C₂×C₄	1	1	yes	characteristic
56	M₃₂×C₅₉	—	2	1	no	—
57	M₃₂×C₅₉	D₄	1	2	yes	—
58	C₂×C₁₆×C₅₉	—	2	1	no	—
59	C₂×C₁₆×C₅₉	—	2	1	no	—
60	C₂×C₁₆×C₅₉	—	2	1	no	—
61	C₂×C₁₆×C₅₉	—	2	1	no	—
62	#97×C₅₉	—	2	1	no	—
63	C₄×C₈×C₅₉	—	2	1	no	—
64	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₂²⋊D₅₉	—	2	1	no	—
66	C₂²×Dic₅₉	—	2	1	no	—
67	#11745685	—	2	1	no	—
68	#11745685	—	2	1	no	—
69	#11745685	C₄⋊C₄	1	2	yes	—
70	#11745685	M₁₆	1	2	yes	—
71	C₂×D₂₃₆	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₅₉	—	2	1	no	—
73	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
74	M₁₆×C₅₉	—	2	1	no	—
75	M₁₆×C₅₉	—	2	1	no	—
76	C₂×C₈×C₅₉	—	2	1	no	—
77	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
82	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
83	C₄⋊C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
84	C₄⋊C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
85	C₂²⋊C₄×C₅₉	—	2	1	no	—
86	C₄²×C₅₉	—	2	1	no	—
87	C₄×Dic₅₉	—	2	1	no	—
88	C₄×Dic₅₉	—	2	1	no	—
89	#11745664	—	2	1	no	—
90	#11745664	C₄⋊C₄	1	2	yes	—
91	C₂×C₅₉⋊C₈	—	2	1	no	—
92	C₂×C₅₉⋊C₈	—	2	1	no	—
93	C₂×C₅₉⋊C₈	—	2	1	no	—
94	C₂×C₅₉⋊C₈	C₂×C₈	1	1	yes	characteristic
95	C₂×C₅₉⋊C₈	C₂×C₈	1	1	yes	characteristic
96	#11745659	—	2	2	no	—
97	#11745659	—	2	2	no	—
98	C₈×D₅₉	—	2	2	no	—
99	C₈×D₅₉	—	2	2	no	—
100	C₁₆×C₅₉	—	2	2	no	—
101	C₁₆×C₅₉	—	4	1	no	—
102	C₅₉⋊C₁₆	—	2	1	no	—
103	C₅₉⋊C₁₆	—	2	1	no	—
104	C₅₉⋊C₁₆	—	4	1	no	—
105	C₂³×C₅₉	#131	1	1	yes	characteristic
106	C₂²×D₅₉	—	2	1	no	—
107	C₂×C₄×C₅₉	—	2	1	no	—
108	C₂×C₄×C₅₉	—	2	1	no	—
109	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
115	C₂²⋊D₅₉	—	2	2	no	—
116	C₂²⋊D₅₉	—	2	2	no	—
117	C₂×Dic₅₉	—	2	1	no	—
118	C₂×Dic₅₉	—	2	2	no	—
119	C₂×Dic₅₉	—	4	1	no	—
120	C₂×Dic₅₉	—	4	1	no	—
121	D₂₃₆	—	2	1	no	—
122	D₂₃₆	C₄⋊C₈	1	2	yes	—
123	C₄×D₅₉	—	2	2	no	—
124	C₄×D₅₉	—	2	2	no	—
125	Dic₁₁₈	—	2	1	no	—
126	Dic₁₁₈	C₄⋊C₈	1	2	yes	—
127	C₈×C₅₉	—	4	1	no	—
128	C₈×C₅₉	#120	1	1	yes	characteristic
129	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
130	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
131	C₈×C₅₉	#123	1	1	yes	characteristic
132	C₅₉⋊C₈	—	2	2	no	—
133	C₅₉⋊C₈	—	2	1	no	—
134	C₅₉⋊C₈	—	2	1	no	—
135	#12917	—	59	1	no	maximal, Sylow
136	C₂²×C₅₉	—	2	1	no	—
137	C₂²×C₅₉	—	2	1	no	—
138	C₂²×C₅₉	#424	1	1	yes	characteristic
139	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
140	C₂²×C₅₉	#435	1	1	yes	characteristic
141	C₂×D₅₉	—	2	2	no	—
142	C₂×D₅₉	—	4	1	no	—
143	C₂×D₅₉	—	4	1	no	—
144	C₄×C₅₉	—	4	1	no	—
145	C₄×C₅₉	—	4	1	no	—
146	C₄×C₅₉	#385	1	1	yes	characteristic
147	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
148	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
149	C₄×C₅₉	#435	1	1	yes	characteristic
150	Dic₅₉	—	2	2	no	—
151	Dic₅₉	—	4	2	no	—
152	C₂×#504	—	59	1	no	—
153	C₂×#364	—	59	1	no	—
154	#2151	—	59	1	no	—
155	C₂×#350	—	59	1	no	—
156	#1764	—	59	1	no	—
157	#1379	D₅₉	1	1	yes	characteristic, complemented
158	#1374	—	59	1	no	—
159	C₂×C₅₉	—	2	1	no	—
160	C₂×C₅₉	—	2	1	no	—
161	C₂×C₅₉	#1850	1	1	yes	characteristic
162	C₂×C₅₉	#2170	1	1	yes	characteristic
163	C₂×C₅₉	#2171	1	1	yes	characteristic
164	D₅₉	—	4	2	no	—
165	C₂×#131	—	59	1	no	—
166	#504	—	118	2	no	—
167	#504	—	118	2	no	—
168	C₂×M₃₂	—	59	1	no	—
169	C₂×M₃₂	—	118	1	no	—
170	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
171	C₂²×C₁₆	—	59	1	no	—
172	C₂²×C₁₆	—	118	1	no	—
173	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
174	C₂×#104	—	59	1	no	—
175	#416	—	59	1	no	—
176	#405	C₂×D₅₉	1	1	yes	characteristic
177	C₂×C₄×C₈	—	59	1	no	—
178	#364	—	118	2	no	—
179	C₄⋊C₁₆	—	118	1	no	—
180	C₄⋊C₁₆	—	118	1	no	—
181	#350	—	118	2	no	—
182	#341	—	59	1	no	—
183	#339	—	59	1	no	—
184	C₅₉	#12917	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	59	1	no	—
186	#131	—	59	2	no	—
187	#131	—	59	2	no	—
188	#131	—	118	1	no	—
189	#131	—	118	1	no	—
190	C₂×M₁₆	—	59	1	no	—
191	C₂×M₁₆	—	118	1	no	—
192	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
193	C₂²×C₈	—	59	1	no	—
194	C₂²×C₈	—	118	1	no	—
195	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
196	#117	C₄×D₅₉	1	1	yes	characteristic
197	C₂×C₄²	—	59	1	no	—
198	M₃₂	—	2	1	no	—
199	M₃₂	—	118	4	no	—
200	M₃₂	—	118	1	no	—
201	M₃₂	—	118	1	no	—
202	M₃₂	D₂₃₆	1	2	yes	—
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	59	1	no	—
208	C₂×C₁₆	—	59	1	no	—
209	C₂×C₁₆	—	118	4	no	—
210	C₂×C₁₆	—	118	1	no	—
211	C₂×C₁₆	—	118	1	no	—
212	C₂×C₁₆	—	118	1	no	—
213	#104	—	118	2	no	—
214	#103	—	59	1	no	—
215	#103	—	59	1	no	—
216	#102	—	59	1	no	—
217	#102	—	59	1	no	—
218	#97	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	118	1	no	—
221	C₄×C₈	—	118	1	no	—
222	#41	—	59	2	no	—
223	#41	—	59	2	no	—
224	#41	—	118	1	no	—
225	#41	—	118	1	no	—
226	C₂×Q₈	—	59	1	no	—
227	C₂×D₄	—	59	1	no	—
228	C₂×D₄	—	118	1	no	—
229	C₂²×C₄	—	118	1	no	—
230	C₂²×C₄	—	118	1	no	—
231	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
232	M₁₆	—	2	1	no	—
233	M₁₆	—	2	1	no	—
234	M₁₆	—	59	2	no	—
235	M₁₆	—	118	1	no	—
236	M₁₆	—	118	2	no	—
237	M₁₆	—	118	2	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	—	118	1	no	—
244	C₂×C₈	—	118	2	no	—
245	C₂×C₈	—	118	2	no	—
246	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
247	C₂×C₈	#11745685	1	1	yes	characteristic
248	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
249	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
250	C₂×C₈	#11745687	1	1	yes	characteristic
251	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
252	C₄⋊C₄	C₈×D₅₉	1	1	yes	characteristic
253	C₄⋊C₄	C₈×D₅₉	1	1	yes	characteristic
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	118	1	no	—
257	C₄²	—	118	1	no	—
258	C₁₆	—	2	2	no	—
259	C₁₆	—	4	1	no	—
260	C₁₆	—	118	1	no	—
261	C₁₆	—	118	1	no	—
262	C₁₆	—	236	1	no	—
263	C₂³	—	118	1	no	—
264	C₂³	#14273735	1	1	yes	characteristic
265	Q₈	—	59	2	no	—
266	Q₈	—	118	1	no	—
267	D₄	—	59	2	no	—
268	D₄	—	118	2	no	—
269	D₄	—	118	2	no	—
270	D₄	—	118	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	118	1	no	—
274	C₂×C₄	—	118	2	no	—
275	C₂×C₄	—	118	2	no	—
276	C₂×C₄	—	118	2	no	—
277	C₂×C₄	—	236	1	no	—
278	C₂×C₄	—	236	1	no	—
279	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
280	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
282	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
283	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
284	C₂×C₄	#14273733	1	1	yes	characteristic
285	C₈	—	4	1	no	—
286	C₈	—	118	1	no	—
287	C₈	—	118	1	no	—
288	C₈	—	118	2	no	—
289	C₈	#14273697	1	1	yes	characteristic
290	C₈	#14273705	1	1	yes	characteristic
291	C₈	#14273707	1	1	yes	characteristic
292	C₈	#14273709	1	1	yes	characteristic
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	118	2	no	—
296	C₂²	—	236	1	no	—
297	C₂²	—	236	1	no	—
298	C₂²	#16837427	1	1	yes	characteristic
299	C₂²	(C₂²⋊D₅₉)⋊C₈	1	1	yes	characteristic
300	C₂²	#16837475	1	1	yes	characteristic
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	118	2	no	—
304	C₄	—	236	2	no	—
305	C₄	#16837276	1	1	yes	characteristic
306	C₄	C₄⋊C₈×D₅₉	1	1	yes	characteristic
307	C₄	C₈×D₂₃₆	1	1	yes	characteristic
308	C₄	#16837476	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	236	2	no	—
312	C₂	#21424275	1	1	yes	characteristic
313	C₂	#21425930	1	1	yes	characteristic
314	C₂	#21425935	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×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	#12917
LCS	G ▷ C₂×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
2. C₂×#16837222	C₂	semidirect
3. #21425855	C₂	semidirect
6. #21422173	C₂	semidirect
7. C₅₉×#1379	C₂	semidirect
157. #1379	D₅₉	semidirect
184. C₅₉	#12917	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×#16837222	2
#21425855	2
C₂×#16837185	2
#21423935	2
#21422173	2
C₅₉×#1379	2
#12917	59