Subgroups — #28479350

Order 15104.

498 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21426570	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#21423841	C₂	1	1	yes	maximal, characteristic
4	C₂×#16837134	C₂	1	1	yes	maximal, characteristic
5	C₂×#16837132	C₂	1	1	yes	maximal, characteristic
6	#21421937	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1279	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₉	—	2	1	no	—
12	#16837307	C₄	1	2	yes	—
13	#16837298	C₄	1	2	yes	—
14	#16837284	—	2	2	no	—
15	C₄×Dic₅₉⋊C₄	—	2	1	no	—
16	#16837248	—	2	1	no	—
17	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
18	#16837156	—	2	1	no	—
19	#16837156	—	2	2	no	—
20	#16837134	C₄	1	4	yes	complemented
21	#16837132	C₄	1	4	yes	complemented
22	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
25	C₂²×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅₉⋊C₈	—	2	1	no	—
27	C₂²×C₅₉⋊C₈	—	2	1	no	—
28	C₄×Dic₁₁₈	—	2	1	no	—
29	C₄×Dic₁₁₈	—	2	1	no	—
30	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
31	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₅₉⋊C₄	—	2	1	no	—
34	C₂×Dic₅₉⋊C₄	—	2	2	no	—
35	C₂×C₄×Dic₅₉	—	2	1	no	—
36	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
38	C₂×C₄×Dic₅₉	D₄	1	2	yes	—
39	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
40	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
41	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
42	C₄×C₅₉⋊C₈	—	2	1	no	—
43	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
44	C₈×Dic₅₉	—	4	2	no	—
45	C₅₉⋊#95	—	2	2	no	—
46	C₅₉⋊#95	—	2	2	no	—
47	C₅₉⋊#95	—	2	2	no	—
48	C₂²×C₈×C₅₉	—	2	1	no	—
49	C₂²×C₈×C₅₉	—	2	1	no	—
50	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₅₉	—	2	1	no	—
52	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₅₉	—	2	1	no	—
54	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₅₉	—	2	1	no	—
56	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
57	#95×C₅₉	—	2	1	no	—
58	C₂²×C₄×C₅₉	—	2	2	no	—
59	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₅₉	—	2	1	no	—
63	C₂²×Dic₅₉	—	2	1	no	—
64	C₂²×Dic₅₉	C₂²⋊C₄	1	2	yes	—
65	C₂²×Dic₅₉	C₂²⋊C₄	1	2	yes	—
66	C₂×Dic₁₁₈	—	2	2	no	—
67	C₂×Dic₁₁₈	—	2	2	no	—
68	C₂×Dic₁₁₈	—	2	1	no	—
69	C₂×Dic₁₁₈	C₄²	1	2	yes	—
70	C₂×C₈×C₅₉	—	2	2	no	—
71	C₂×C₈×C₅₉	—	2	2	no	—
72	C₂×C₈×C₅₉	—	4	2	no	—
73	C₂×C₈×C₅₉	—	4	1	no	—
74	C₂×C₈×C₅₉	—	4	1	no	—
75	C₄⋊C₄×C₅₉	—	2	2	no	—
76	C₄⋊C₄×C₅₉	—	2	2	no	—
77	C₄²×C₅₉	—	4	1	no	—
78	C₄²×C₅₉	—	4	1	no	—
79	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₅₉	#41	1	1	yes	characteristic
82	C₄²×C₅₉	#41	1	1	yes	characteristic
83	C₄⋊Dic₅₉	—	2	1	no	—
84	C₄⋊Dic₅₉	C₄²	1	2	yes	—
85	Dic₅₉⋊C₄	—	2	2	no	—
86	Dic₅₉⋊C₄	—	2	2	no	—
87	Dic₅₉⋊C₄	—	4	4	no	—
88	C₄×Dic₅₉	—	2	2	no	—
89	C₄×Dic₅₉	—	2	2	no	—
90	C₄×Dic₅₉	—	2	2	no	—
91	C₄×Dic₅₉	—	2	2	no	—
92	C₄×Dic₅₉	—	2	1	no	—
93	C₄×Dic₅₉	—	2	1	no	—
94	C₄×Dic₅₉	—	4	2	no	—
95	C₄×Dic₅₉	SD₁₆	1	2	yes	—
96	C₄×Dic₅₉	Q₁₆	1	2	yes	—
97	C₂×C₅₉⋊C₈	—	2	2	no	—
98	C₂×C₅₉⋊C₈	—	2	2	no	—
99	C₂×C₅₉⋊C₈	—	4	2	no	—
100	C₂×C₅₉⋊C₈	—	4	1	no	—
101	C₂×C₅₉⋊C₈	—	4	1	no	—
102	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₉	—	2	2	no	—
104	C₂×C₄×C₅₉	—	4	2	no	—
105	C₂×C₄×C₅₉	—	4	2	no	—
106	C₂×C₄×C₅₉	—	4	2	no	—
107	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₅₉	#117	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	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₉	#131	1	2	yes	—
114	C₂×C₄×C₅₉	#131	1	2	yes	—
115	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
117	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
118	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
119	C₂×Dic₅₉	—	2	2	no	—
120	C₂×Dic₅₉	—	2	2	no	—
121	C₂×Dic₅₉	—	2	2	no	—
122	C₂×Dic₅₉	—	2	2	no	—
123	C₂×Dic₅₉	—	2	2	no	—
124	C₂×Dic₅₉	—	2	2	no	—
125	C₂×Dic₅₉	—	2	2	no	—
126	C₂×Dic₅₉	—	4	2	no	—
127	C₂×Dic₅₉	—	4	2	no	—
128	C₂×Dic₅₉	#103	1	4	yes	—
129	Dic₁₁₈	—	2	2	no	—
130	Dic₁₁₈	—	2	2	no	—
131	Dic₁₁₈	—	4	2	no	—
132	C₈×C₅₉	—	4	2	no	—
133	C₈×C₅₉	—	4	2	no	—
134	C₅₉⋊C₈	—	4	2	no	—
135	C₅₉⋊C₈	—	4	2	no	—
136	#8182	—	59	1	no	maximal, Sylow
137	C₂²×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₅₉	#408	1	2	yes	—
139	C₂²×C₅₉	C₂×#103	1	1	yes	characteristic
140	C₂²×C₅₉	#416	1	1	yes	characteristic
141	C₂²×C₅₉	#421	1	2	yes	—
142	C₄×C₅₉	—	2	1	no	—
143	C₄×C₅₉	—	2	1	no	—
144	C₄×C₅₉	—	8	2	no	—
145	C₄×C₅₉	#405	1	1	yes	characteristic
146	C₄×C₅₉	#405	1	1	yes	characteristic
147	C₄×C₅₉	#433	1	2	yes	—
148	C₄×C₅₉	C₄×SD₁₆	1	1	yes	characteristic
149	C₄×C₅₉	C₄×SD₁₆	1	1	yes	characteristic
150	C₄×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
151	C₄×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
152	Dic₅₉	—	2	4	no	—
153	Dic₅₉	—	4	2	no	—
154	Dic₅₉	—	4	2	no	—
155	Dic₅₉	—	4	2	no	—
156	#2295	—	59	1	no	—
157	C₄×C₄⋊C₈	—	59	1	no	—
158	#1744	—	59	1	no	—
159	C₂×Q₈⋊C₈	—	59	1	no	—
160	C₂×Q₈⋊C₈	—	59	1	no	—
161	#1279	—	59	1	no	—
162	#1279	D₅₉	1	1	yes	characteristic
163	C₂×C₅₉	#1481	1	2	yes	—
164	C₂×C₅₉	#1482	1	2	yes	—
165	C₂×C₅₉	#1750	1	1	yes	characteristic
166	C₂×C₅₉	#1759	1	1	yes	characteristic
167	C₂×C₅₉	C₄×#103	1	1	yes	characteristic
168	C₂×C₄×Q₈	—	59	1	no	—
169	C₂×C₄⋊C₈	—	59	1	no	—
170	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
171	C₂×C₄×C₈	—	59	1	no	—
172	C₂×C₄×C₈	—	118	1	no	—
173	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
174	#391	—	59	2	no	—
175	#389	—	59	2	no	—
176	#387	—	118	2	no	—
177	C₄×C₄⋊C₄	—	118	1	no	—
178	C₄×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
179	#376	—	118	1	no	—
180	C₄³	—	59	1	no	—
181	#336	—	118	2	no	—
182	#336	—	118	1	no	—
183	Q₈⋊C₈	—	59	4	no	—
184	Q₈⋊C₈	—	59	4	no	—
185	C₅₉	#8182	1	1	yes	characteristic, complemented, Sylow
186	C₂²×Q₈	—	59	1	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	118	1	no	—
190	C₂²×C₈	—	118	1	no	—
191	C₄×Q₈	—	59	2	no	—
192	C₄×Q₈	—	59	2	no	—
193	C₄×Q₈	—	118	1	no	—
194	C₄×Q₈	—	118	1	no	—
195	C₂×C₄⋊C₄	—	59	1	no	—
196	C₂×C₄⋊C₄	—	118	1	no	—
197	C₂×C₄⋊C₄	—	118	2	no	—
198	C₂×C₄⋊C₄	C₄×D₅₉	1	2	yes	—
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	—	59	1	no	—
201	C₂×C₄²	—	59	2	no	—
202	C₂×C₄²	—	59	1	no	—
203	C₂×C₄²	—	59	2	no	—
204	C₂×C₄²	—	118	1	no	—
205	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	59	2	no	—
208	C₄⋊C₈	—	118	1	no	—
209	C₄⋊C₈	C₄×D₅₉	1	2	yes	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	59	2	no	—
212	C₄×C₈	—	118	1	no	—
213	C₄×C₈	—	236	2	no	—
214	C₄×C₈	C₄×D₅₉	1	2	yes	—
215	#95	—	2	1	no	—
216	#95	—	118	2	no	—
217	#95	—	118	2	no	—
218	#95	—	118	2	no	—
219	C₂×Q₈	—	59	2	no	—
220	C₂×Q₈	—	118	1	no	—
221	C₂×Q₈	—	118	2	no	—
222	C₂×Q₈	—	118	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	59	2	no	—
225	C₂²×C₄	—	59	2	no	—
226	C₂²×C₄	—	118	1	no	—
227	C₂²×C₄	—	118	1	no	—
228	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
229	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
230	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	118	2	no	—
237	C₂×C₈	—	118	2	no	—
238	C₂×C₈	—	236	2	no	—
239	C₂×C₈	—	236	1	no	—
240	C₂×C₈	—	236	1	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	59	2	no	—
244	C₄⋊C₄	—	118	1	no	—
245	C₄⋊C₄	—	118	2	no	—
246	C₄⋊C₄	—	118	2	no	—
247	C₄⋊C₄	—	236	4	no	—
248	C₄²	—	4	1	no	—
249	C₄²	—	4	1	no	—
250	C₄²	—	59	2	no	—
251	C₄²	—	59	2	no	—
252	C₄²	—	118	2	no	—
253	C₄²	—	118	2	no	—
254	C₄²	—	118	2	no	—
255	C₄²	—	118	2	no	—
256	C₄²	—	118	1	no	—
257	C₄²	—	118	1	no	—
258	C₄²	—	236	2	no	—
259	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
260	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
261	C₄²	#11745687	1	1	yes	characteristic
262	C₄²	#11745689	1	1	yes	characteristic
263	C₂³	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
264	Q₈	—	118	2	no	—
265	Q₈	—	118	2	no	—
266	Q₈	—	236	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	59	4	no	—
272	C₂×C₄	—	118	2	no	—
273	C₂×C₄	—	118	2	no	—
274	C₂×C₄	—	118	2	no	—
275	C₂×C₄	—	118	2	no	—
276	C₂×C₄	—	118	2	no	—
277	C₂×C₄	—	118	2	no	—
278	C₂×C₄	—	118	2	no	—
279	C₂×C₄	—	236	2	no	—
280	C₂×C₄	—	236	2	no	—
281	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
282	C₂×C₄	#14273685	1	1	yes	characteristic
283	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
285	C₂×C₄	#14273689	1	1	yes	characteristic
286	C₂×C₄	#14273689	1	1	yes	characteristic
287	C₂×C₄	#14273733	1	2	yes	—
288	C₂×C₄	#14273735	1	2	yes	—
289	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
290	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
291	C₂×C₄	#14273746	1	1	yes	characteristic
292	C₂×C₄	#14273748	1	1	yes	characteristic
293	C₈	—	4	2	no	—
294	C₈	—	4	2	no	—
295	C₈	—	236	2	no	—
296	C₈	—	236	2	no	—
297	C₂²	#16837249	1	1	yes	characteristic
298	C₂²	#16837367	1	2	yes	—
299	C₂²	#103×D₅₉	1	1	yes	characteristic
300	C₂²	#16837395	1	1	yes	characteristic
301	C₂²	#16837417	1	2	yes	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	8	2	no	—
305	C₄	—	118	4	no	—
306	C₄	—	236	2	no	—
307	C₄	—	236	2	no	—
308	C₄	—	236	2	no	—
309	C₄	#16837356	1	1	yes	characteristic
310	C₄	#16837359	1	1	yes	characteristic
311	C₄	#16837462	1	2	yes	—
312	C₄	#16837489	1	1	yes	characteristic
313	C₄	#16837493	1	1	yes	characteristic
314	C₄	#16837494	1	1	yes	characteristic
315	C₄	#16837496	1	1	yes	characteristic
316	C₂	#21422497	1	2	yes	—
317	C₂	#21422501	1	2	yes	—
318	C₂	#21423861	1	1	yes	characteristic
319	C₂	#21423905	1	1	yes	characteristic
320	C₂	#21423915	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1279	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1279	D₅₉
hypocenter	C₅₉	#8182
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1279

Complements

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

N	H ≅ G/N	type
20. #16837134	C₄	semidirect
21. #16837132	C₄	semidirect
185. C₅₉	#8182	semidirect

Maximal subgroups

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

subgroup	index
#21426570	2
C₄⋊C₈×Dic₅₉	2
#21423841	2
C₂×#16837134	2
C₂×#16837132	2
#21421937	2
C₅₉×#1279	2
#8182	59