Subgroups — #28505185

Order 15104.

376 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	#21437198	C₂	1	1	yes	maximal, characteristic
2	#21425862	C₂	1	1	yes	maximal, characteristic, complemented
3	#21425851	C₂	1	1	yes	maximal, characteristic, complemented
4	#21425723	C₂	1	1	yes	maximal, characteristic
5	#21423935	C₂	1	1	yes	maximal, characteristic
6	#21422177	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1378	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	C₂×#14273653	C₄	1	1	yes	characteristic
15	#16837396	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
17	#16837223	—	2	2	no	—
18	Dic₅₉⋊C₁₆	—	2	2	no	—
19	#16837185	—	2	2	no	—
20	#16837166	C₄	1	1	yes	characteristic
21	#16837161	C₄	1	1	yes	characteristic
22	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
24	C₅₉×#405	C₂²	1	1	yes	characteristic
25	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
26	#14273733	—	2	1	no	—
27	#14273733	—	2	1	no	—
28	#14273733	D₄	1	2	yes	—
29	#14273733	Q₈	1	2	yes	—
30	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
31	C₂×#11745659	—	2	1	no	—
32	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₅₉	—	2	1	no	—
34	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
35	#14273667	—	2	1	no	—
36	#14273667	—	2	1	no	—
37	#14273667	—	2	1	no	—
38	#14273667	D₄	1	2	yes	—
39	#14273666	—	2	4	no	—
40	C₂×C₅₉⋊C₁₆	—	2	1	no	—
41	C₂×C₅₉⋊C₁₆	—	2	1	no	—
42	C₁₆×D₅₉	—	2	4	no	—
43	#14273653	C₈	1	2	yes	—
44	#14273653	C₈	1	2	yes	—
45	#14273650	—	2	1	no	—
46	#14273647	—	2	1	no	—
47	C₈×Dic₅₉	—	2	1	no	—
48	C₈×Dic₅₉	—	2	1	no	—
49	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
51	#117×C₅₉	C₂×C₄	1	1	yes	characteristic
52	M₃₂×C₅₉	—	2	1	no	—
53	M₃₂×C₅₉	—	2	1	no	—
54	M₃₂×C₅₉	—	2	1	no	—
55	M₃₂×C₅₉	—	2	1	no	—
56	C₂×C₁₆×C₅₉	—	2	1	no	—
57	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
58	#97×C₅₉	—	2	1	no	—
59	C₄×C₈×C₅₉	—	2	1	no	—
60	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₅₉	—	2	1	no	—
62	C₂²×Dic₅₉	—	2	1	no	—
63	#11745685	—	2	1	no	—
64	#11745685	—	2	1	no	—
65	#11745685	C₄⋊C₄	1	2	yes	—
66	#11745685	C₂×C₈	1	2	yes	—
67	C₂×D₂₃₆	M₁₆	1	1	yes	characteristic
68	C₂×C₄×D₅₉	—	2	1	no	—
69	C₂×Dic₁₁₈	M₁₆	1	1	yes	characteristic
70	M₁₆×C₅₉	—	2	1	no	—
71	M₁₆×C₅₉	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₅₉	—	2	1	no	—
73	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
78	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
79	C₄⋊C₄×C₅₉	—	2	1	no	—
80	C₂²⋊C₄×C₅₉	—	2	1	no	—
81	C₄²×C₅₉	—	2	1	no	—
82	C₄×Dic₅₉	—	2	1	no	—
83	C₄×Dic₅₉	C₂×C₈	1	1	yes	characteristic
84	C₄×Dic₅₉	C₂×C₈	1	1	yes	characteristic
85	#11745664	—	2	1	no	—
86	#11745664	C₄⋊C₄	1	2	yes	—
87	C₂×C₅₉⋊C₈	—	2	1	no	—
88	C₂×C₅₉⋊C₈	—	2	1	no	—
89	C₂×C₅₉⋊C₈	—	2	1	no	—
90	C₂×C₅₉⋊C₈	—	2	1	no	—
91	#11745659	—	2	2	no	—
92	#11745659	—	2	2	no	—
93	C₈×D₅₉	—	2	2	no	—
94	C₈×D₅₉	—	2	2	no	—
95	C₁₆×C₅₉	—	2	2	no	—
96	C₁₆×C₅₉	—	4	1	no	—
97	C₅₉⋊C₁₆	—	2	2	no	—
98	C₅₉⋊C₁₆	—	4	1	no	—
99	C₂³×C₅₉	#131	1	1	yes	characteristic
100	C₂²×D₅₉	—	2	1	no	—
101	C₂×C₄×C₅₉	—	4	1	no	—
102	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
108	C₂²⋊D₅₉	—	2	2	no	—
109	C₂²⋊D₅₉	—	2	2	no	—
110	C₂×Dic₅₉	—	2	1	no	—
111	C₂×Dic₅₉	—	2	1	no	—
112	C₂×Dic₅₉	—	2	1	no	—
113	C₂×Dic₅₉	—	4	2	no	—
114	D₂₃₆	—	2	1	no	—
115	D₂₃₆	C₄⋊C₈	1	2	yes	—
116	C₄×D₅₉	—	2	2	no	—
117	C₄×D₅₉	—	2	2	no	—
118	Dic₁₁₈	—	2	1	no	—
119	Dic₁₁₈	C₄⋊C₈	1	2	yes	—
120	C₈×C₅₉	—	2	2	no	—
121	C₈×C₅₉	#120	1	1	yes	characteristic
122	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
123	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
124	C₈×C₅₉	#123	1	1	yes	characteristic
125	C₅₉⋊C₈	—	2	2	no	—
126	C₅₉⋊C₈	—	4	1	no	—
127	#12918	—	59	1	no	maximal, Sylow
128	C₂²×C₅₉	—	4	1	no	—
129	C₂²×C₅₉	#424	1	1	yes	characteristic
130	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
131	C₂²×C₅₉	#435	1	1	yes	characteristic
132	C₂×D₅₉	—	2	2	no	—
133	C₂×D₅₉	—	8	1	no	—
134	C₄×C₅₉	—	8	1	no	—
135	C₄×C₅₉	#385	1	1	yes	characteristic
136	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
137	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
138	C₄×C₅₉	#435	1	1	yes	characteristic
139	Dic₅₉	—	2	2	no	—
140	Dic₅₉	—	2	2	no	—
141	Dic₅₉	—	4	1	no	—
142	#3407	—	59	1	no	—
143	#2153	—	59	1	no	—
144	#2150	—	59	1	no	—
145	#2117	—	59	1	no	—
146	#1764	—	59	1	no	—
147	#1378	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	—	2	1	no	—
152	C₂×C₅₉	#1850	1	1	yes	characteristic
153	D₅₉	—	4	2	no	—
154	C₂×#131	—	59	1	no	—
155	#504	—	118	2	no	—
156	#504	—	118	2	no	—
157	C₂×M₃₂	—	59	1	no	—
158	C₂×M₃₂	—	59	1	no	—
159	C₂×M₃₂	—	118	2	no	—
160	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
161	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
162	C₂×#104	—	59	1	no	—
163	#416	—	59	1	no	—
164	#405	C₂×D₅₉	1	1	yes	characteristic
165	C₂×C₄×C₈	—	59	1	no	—
166	#364	—	118	2	no	—
167	C₄⋊C₁₆	—	118	2	no	—
168	#350	—	118	2	no	—
169	#341	—	59	1	no	—
170	#339	—	59	1	no	—
171	C₅₉	#12918	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	59	1	no	—
173	#131	—	59	2	no	—
174	#131	—	59	2	no	—
175	#131	—	118	1	no	—
176	#131	—	118	1	no	—
177	C₂×M₁₆	—	59	1	no	—
178	C₂×M₁₆	—	118	1	no	—
179	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
180	C₂²×C₈	—	59	1	no	—
181	C₂²×C₈	—	118	1	no	—
182	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
183	#117	C₄×D₅₉	1	1	yes	characteristic
184	C₂×C₄²	—	59	1	no	—
185	M₃₂	—	2	1	no	—
186	M₃₂	—	2	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	M₃₂	—	59	2	no	—
190	M₃₂	—	118	4	no	—
191	M₃₂	—	118	1	no	—
192	M₃₂	—	118	1	no	—
193	M₃₂	—	118	1	no	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	—	118	4	no	—
196	C₂×C₁₆	—	118	1	no	—
197	C₂×C₁₆	—	118	1	no	—
198	C₂×C₁₆	D₂₃₆	1	2	yes	—
199	#104	—	59	2	no	—
200	#104	—	59	2	no	—
201	#103	—	118	1	no	—
202	#102	—	118	1	no	—
203	#97	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	118	1	no	—
206	C₄×C₈	—	118	1	no	—
207	#41	—	59	2	no	—
208	#41	—	59	2	no	—
209	#41	—	118	1	no	—
210	#41	—	118	1	no	—
211	C₂×Q₈	—	59	1	no	—
212	C₂×D₄	—	59	1	no	—
213	C₂×D₄	—	118	1	no	—
214	C₂²×C₄	—	118	1	no	—
215	C₂²×C₄	—	118	1	no	—
216	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
217	M₁₆	—	2	1	no	—
218	M₁₆	—	59	2	no	—
219	M₁₆	—	118	1	no	—
220	M₁₆	—	118	2	no	—
221	M₁₆	—	118	2	no	—
222	M₁₆	C₈×D₅₉	1	2	yes	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	118	1	no	—
225	C₂×C₈	—	118	2	no	—
226	C₂×C₈	—	118	2	no	—
227	C₂×C₈	—	118	1	no	—
228	C₂×C₈	—	118	1	no	—
229	C₂×C₈	—	118	1	no	—
230	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
231	C₂×C₈	#11745685	1	1	yes	characteristic
232	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
233	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
234	C₂×C₈	#11745687	1	1	yes	characteristic
235	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	59	1	no	—
240	C₄²	—	59	1	no	—
241	C₄²	—	118	1	no	—
242	C₁₆	—	2	2	no	—
243	C₁₆	—	4	1	no	—
244	C₁₆	—	118	2	no	—
245	C₁₆	—	236	1	no	—
246	C₂³	—	118	1	no	—
247	C₂³	#14273733	1	1	yes	characteristic
248	Q₈	—	59	2	no	—
249	Q₈	—	118	1	no	—
250	D₄	—	59	2	no	—
251	D₄	—	118	2	no	—
252	D₄	—	118	2	no	—
253	D₄	—	118	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	118	1	no	—
256	C₂×C₄	—	118	1	no	—
257	C₂×C₄	—	118	2	no	—
258	C₂×C₄	—	118	2	no	—
259	C₂×C₄	—	118	1	no	—
260	C₂×C₄	—	236	2	no	—
261	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
262	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
264	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
265	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
266	C₂×C₄	#14273735	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	118	2	no	—
269	C₈	—	236	1	no	—
270	C₈	#14273697	1	1	yes	characteristic
271	C₈	#14273705	1	1	yes	characteristic
272	C₈	#14273707	1	1	yes	characteristic
273	C₈	#14273709	1	1	yes	characteristic
274	C₂²	—	4	1	no	—
275	C₂²	—	118	2	no	—
276	C₂²	—	472	1	no	—
277	C₂²	#16837427	1	1	yes	characteristic
278	C₂²	C₈×D₂₃₆	1	1	yes	characteristic
279	C₂²	#16837476	1	1	yes	characteristic
280	C₄	—	8	1	no	—
281	C₄	—	118	2	no	—
282	C₄	—	118	2	no	—
283	C₄	—	236	1	no	—
284	C₄	#16837276	1	1	yes	characteristic
285	C₄	C₄⋊C₈×D₅₉	1	1	yes	characteristic
286	C₄	(C₂²⋊D₅₉)⋊C₈	1	1	yes	characteristic
287	C₄	#16837475	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	236	2	no	—
292	C₂	#21424275	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14273697
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1378	C₂
socle	C₂×C₅₉	#1850
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1378	D₅₉
hypocenter	C₅₉	#12918
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1378

Complements

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

N	H ≅ G/N	type
2. #21425862	C₂	semidirect
3. #21425851	C₂	semidirect
6. #21422177	C₂	semidirect
7. C₅₉×#1378	C₂	semidirect
147. #1378	D₅₉	semidirect
171. C₅₉	#12918	semidirect

Maximal subgroups

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

subgroup	index
#21437198	2
#21425862	2
#21425851	2
#21425723	2
#21423935	2
#21422177	2
C₅₉×#1378	2
#12918	59