Subgroups — #28504980

Order 15104.

470 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#107×D₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₂×D₅₉)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#21424122	C₂	1	1	yes	maximal, characteristic
4	#21422188	C₂	1	1	yes	maximal, characteristic, complemented
5	#21422187	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422187	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₅₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×D₅₉	—	2	1	no	—
10	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×#14273661	C₂²	1	1	yes	characteristic
12	C₂×#14273661	C₂²	1	1	yes	characteristic
13	C₂×#14273660	—	2	1	no	—
14	#107×D₅₉	—	2	2	no	—
15	#107×D₅₉	C₄	1	4	yes	complemented
16	#16837259	—	2	1	no	—
17	(C₂×D₅₉)⋊C₁₆	—	2	2	no	—
18	#16837164	—	2	1	no	—
19	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
22	C₂²×C₄×D₅₉	—	4	1	no	—
23	C₂²×C₄×D₅₉	D₄	1	1	yes	characteristic
24	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
25	C₂×C₈×D₅₉	—	2	1	no	—
26	C₂×C₈×D₅₉	—	2	1	no	—
27	C₂×C₈×D₅₉	—	2	1	no	—
28	C₂×C₈×D₅₉	—	2	1	no	—
29	C₂×C₈×D₅₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₅₉	D₄	1	2	yes	—
31	C₂×C₄⋊Dic₅₉	—	2	1	no	—
32	C₂×C₄⋊Dic₅₉	—	2	1	no	—
33	C₂×Dic₅₉⋊C₄	—	4	1	no	—
34	C₄⋊C₄×D₅₉	—	2	4	no	—
35	C₄⋊C₄×D₅₉	—	4	2	no	—
36	C₂×(C₂×D₅₉)⋊C₄	—	4	1	no	—
37	C₂×C₅₉⋊C₁₆	—	2	1	no	—
38	C₂×C₅₉⋊C₁₆	—	2	1	no	—
39	C₂×C₅₉⋊C₁₆	—	2	1	no	—
40	#14273661	—	2	1	no	—
41	#14273661	—	2	1	no	—
42	#14273661	—	2	1	no	—
43	#14273661	C₂×C₄	1	2	yes	—
44	#14273660	—	2	4	no	—
45	C₅₉⋊#95	—	4	1	no	—
46	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
48	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
49	C₂×C₁₆×C₅₉	—	2	1	no	—
50	C₂×C₁₆×C₅₉	—	2	1	no	—
51	C₂×C₁₆×C₅₉	—	2	1	no	—
52	#107×C₅₉	—	2	1	no	—
53	#107×C₅₉	—	2	1	no	—
54	#107×C₅₉	—	2	1	no	—
55	#107×C₅₉	C₂×C₄	1	2	yes	—
56	C₂³×D₅₉	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₉	—	4	1	no	—
58	C₂²×C₄×C₅₉	—	4	1	no	—
59	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₅₉	—	4	1	no	—
61	C₂²×Dic₅₉	—	4	1	no	—
62	C₂²×Dic₅₉	D₈	1	1	yes	characteristic
63	C₂×C₄×D₅₉	—	2	1	no	—
64	C₂×C₄×D₅₉	—	2	1	no	—
65	C₂×C₄×D₅₉	—	2	1	no	—
66	C₂×C₄×D₅₉	—	2	1	no	—
67	C₂×C₄×D₅₉	—	4	4	no	—
68	C₂×C₄×D₅₉	—	8	2	no	—
69	C₂×C₄×D₅₉	—	8	2	no	—
70	C₂×C₄×D₅₉	C₂²⋊C₄	1	2	yes	—
71	C₂×C₄×D₅₉	SD₁₆	1	2	yes	—
72	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
75	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
76	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
77	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
78	C₄⋊C₄×C₅₉	—	2	2	no	—
79	C₄⋊C₄×C₅₉	—	4	1	no	—
80	C₄⋊C₄×C₅₉	—	4	1	no	—
81	C₄⋊C₄×C₅₉	—	4	1	no	—
82	(C₂×D₅₉)⋊C₄	—	8	2	no	—
83	C₄⋊Dic₅₉	—	2	2	no	—
84	C₄⋊Dic₅₉	—	4	1	no	—
85	C₄⋊Dic₅₉	—	4	1	no	—
86	C₄⋊Dic₅₉	—	4	1	no	—
87	Dic₅₉⋊C₄	—	4	4	no	—
88	C₂×C₅₉⋊C₈	—	2	1	no	—
89	C₂×C₅₉⋊C₈	—	2	1	no	—
90	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
91	C₈×D₅₉	—	2	2	no	—
92	C₈×D₅₉	—	2	2	no	—
93	C₈×D₅₉	—	2	2	no	—
94	C₈×D₅₉	—	2	2	no	—
95	C₁₆×C₅₉	—	4	1	no	—
96	C₅₉⋊C₁₆	—	4	1	no	—
97	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
98	C₂²×D₅₉	—	2	1	no	—
99	C₂²×D₅₉	—	2	1	no	—
100	C₂²×D₅₉	—	2	1	no	—
101	C₂²×D₅₉	—	2	1	no	—
102	C₂²×D₅₉	#102	1	2	yes	—
103	C₂²×D₅₉	SD₃₂	1	2	yes	—
104	C₂²×D₅₉	Q₃₂	1	2	yes	—
105	C₂×C₄×C₅₉	—	4	2	no	—
106	C₂×C₄×C₅₉	—	8	1	no	—
107	C₂×C₄×C₅₉	—	8	1	no	—
108	C₂×C₄×C₅₉	—	8	1	no	—
109	C₂×C₄×C₅₉	—	8	1	no	—
110	C₂×C₄×C₅₉	—	8	1	no	—
111	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
113	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
116	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
117	C₂×Dic₅₉	—	2	1	no	—
118	C₂×Dic₅₉	—	2	1	no	—
119	C₂×Dic₅₉	—	4	2	no	—
120	C₂×Dic₅₉	—	8	1	no	—
121	C₂×Dic₅₉	—	8	1	no	—
122	C₂×Dic₅₉	—	8	1	no	—
123	C₂×Dic₅₉	—	8	1	no	—
124	C₂×Dic₅₉	—	8	1	no	—
125	C₂×Dic₅₉	#102	1	2	yes	—
126	C₄×D₅₉	—	2	2	no	—
127	C₄×D₅₉	—	2	2	no	—
128	C₄×D₅₉	—	2	2	no	—
129	C₄×D₅₉	—	2	2	no	—
130	C₄×D₅₉	—	8	8	no	—
131	C₈×C₅₉	#117	1	1	yes	characteristic
132	C₈×C₅₉	#117	1	1	yes	characteristic
133	C₈×C₅₉	#123	1	1	yes	characteristic
134	C₈×C₅₉	#123	1	1	yes	characteristic
135	C₅₉⋊C₈	—	2	2	no	—
136	#12867	—	59	1	no	maximal, Sylow
137	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
138	C₂²×C₅₉	#480	1	1	yes	characteristic
139	C₂²×C₅₉	#480	1	1	yes	characteristic
140	C₂²×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
141	C₂²×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
142	C₂²×C₅₉	#509	1	1	yes	characteristic
143	C₂²×C₅₉	#509	1	1	yes	characteristic
144	C₂×D₅₉	—	2	2	no	—
145	C₂×D₅₉	—	2	2	no	—
146	C₂×D₅₉	—	2	2	no	—
147	C₂×D₅₉	—	2	2	no	—
148	C₂×D₅₉	—	2	2	no	—
149	C₂×D₅₉	—	2	2	no	—
150	C₂×D₅₉	#358	1	4	yes	—
151	C₄×C₅₉	—	8	2	no	—
152	C₄×C₅₉	—	16	1	no	—
153	C₄×C₅₉	#381	1	1	yes	characteristic
154	C₄×C₅₉	#419	1	1	yes	characteristic
155	C₄×C₅₉	#483	1	1	yes	characteristic
156	C₄×C₅₉	#483	1	1	yes	characteristic
157	Dic₅₉	—	2	2	no	—
158	Dic₅₉	—	8	2	no	—
159	Dic₅₉	—	16	1	no	—
160	C₂²×#107	—	59	1	no	—
161	C₂×C₂²⋊C₁₆	—	59	1	no	—
162	#1808	—	59	1	no	—
163	#1380	—	59	1	no	—
164	#1380	—	59	1	no	—
165	#1380	—	59	1	no	—
166	#1380	D₅₉	1	1	yes	characteristic, complemented
167	C₂×C₅₉	#1823	1	1	yes	characteristic
168	C₂×C₅₉	C₂×#358	1	1	yes	characteristic
169	C₂×C₅₉	#2140	1	1	yes	characteristic
170	C₂×C₅₉	#2233	1	1	yes	characteristic
171	C₂×C₅₉	#2233	1	1	yes	characteristic
172	C₂×C₅₉	#2236	1	1	yes	characteristic
173	C₂×C₅₉	#2236	1	1	yes	characteristic
174	D₅₉	—	2	4	no	—
175	C₂³×C₈	—	59	1	no	—
176	C₂²×C₄⋊C₄	—	118	1	no	—
177	C₂²×C₁₆	—	59	1	no	—
178	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
179	C₂×#107	—	59	1	no	—
180	C₂×#107	—	59	4	no	—
181	C₂×#107	—	59	1	no	—
182	C₂×#107	—	118	2	no	—
183	C₂×#107	—	118	1	no	—
184	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
185	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
186	#380	—	118	1	no	—
187	C₂²⋊C₁₆	—	118	2	no	—
188	#340	—	118	1	no	—
189	C₅₉	#12867	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₄	—	59	1	no	—
191	C₂³×C₄	—	236	1	no	—
192	C₂²×C₈	—	59	1	no	—
193	C₂²×C₈	—	59	2	no	—
194	C₂²×C₈	—	59	2	no	—
195	C₂²×C₈	—	118	1	no	—
196	C₂²×C₈	—	118	1	no	—
197	C₂²×C₈	—	118	1	no	—
198	C₂²×C₈	—	118	1	no	—
199	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	118	4	no	—
203	C₂×C₄⋊C₄	—	118	1	no	—
204	C₂×C₄⋊C₄	—	118	1	no	—
205	C₂×C₄⋊C₄	—	236	2	no	—
206	C₂×C₄⋊C₄	—	236	1	no	—
207	C₂×C₂²⋊C₄	—	236	1	no	—
208	C₂×C₁₆	—	2	1	no	—
209	C₂×C₁₆	—	2	1	no	—
210	C₂×C₁₆	—	2	1	no	—
211	C₂×C₁₆	—	118	1	no	—
212	C₂×C₁₆	—	118	1	no	—
213	C₂×C₁₆	—	118	1	no	—
214	#107	—	2	1	no	—
215	#107	—	2	1	no	—
216	#107	—	2	1	no	—
217	#107	—	59	2	no	—
218	#107	—	118	4	no	—
219	#107	—	118	1	no	—
220	#107	—	118	1	no	—
221	#107	—	118	1	no	—
222	#107	C₄×D₅₉	1	2	yes	—
223	#95	—	236	1	no	—
224	C₂⁴	—	59	1	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	—	59	1	no	—
228	C₂²×C₄	—	59	2	no	—
229	C₂²×C₄	—	59	2	no	—
230	C₂²×C₄	—	118	1	no	—
231	C₂²×C₄	—	118	1	no	—
232	C₂²×C₄	—	118	1	no	—
233	C₂²×C₄	—	118	1	no	—
234	C₂²×C₄	—	472	2	no	—
235	C₂²×C₄	—	472	2	no	—
236	C₂²×C₄	—	236	4	no	—
237	C₂²×C₄	—	236	1	no	—
238	C₂²×C₄	—	236	1	no	—
239	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
240	C₂×C₈	—	59	2	no	—
241	C₂×C₈	—	118	1	no	—
242	C₂×C₈	—	118	1	no	—
243	C₂×C₈	—	118	2	no	—
244	C₂×C₈	—	118	2	no	—
245	C₂×C₈	—	118	2	no	—
246	C₂×C₈	—	118	2	no	—
247	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
248	C₂×C₈	#11745685	1	1	yes	characteristic
249	C₂×C₈	#11745685	1	1	yes	characteristic
250	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
251	C₂×C₈	#11745687	1	1	yes	characteristic
252	C₂×C₈	#11745689	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	118	2	no	—
258	C₄⋊C₄	—	236	4	no	—
259	C₄⋊C₄	—	236	1	no	—
260	C₄⋊C₄	—	236	1	no	—
261	C₄⋊C₄	—	236	1	no	—
262	C₂²⋊C₄	—	472	2	no	—
263	C₁₆	—	4	1	no	—
264	C₁₆	—	236	1	no	—
265	C₂³	—	59	2	no	—
266	C₂³	—	59	2	no	—
267	C₂³	—	59	2	no	—
268	C₂³	—	118	1	no	—
269	C₂³	—	118	1	no	—
270	C₂³	—	118	1	no	—
271	C₂³	—	118	1	no	—
272	C₂³	D₈×D₅₉	1	1	yes	characteristic
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	—	8	1	no	—
277	C₂×C₄	—	8	1	no	—
278	C₂×C₄	—	8	1	no	—
279	C₂×C₄	—	59	2	no	—
280	C₂×C₄	—	118	1	no	—
281	C₂×C₄	—	118	1	no	—
282	C₂×C₄	—	118	2	no	—
283	C₂×C₄	—	118	2	no	—
284	C₂×C₄	—	118	2	no	—
285	C₂×C₄	—	118	2	no	—
286	C₂×C₄	—	472	8	no	—
287	C₂×C₄	—	472	1	no	—
288	C₂×C₄	—	472	1	no	—
289	C₂×C₄	—	472	1	no	—
290	C₂×C₄	—	472	1	no	—
291	C₂×C₄	—	472	1	no	—
292	C₂×C₄	—	236	2	no	—
293	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
294	C₂×C₄	#14273709	1	1	yes	characteristic
295	C₂×C₄	#14273712	1	1	yes	characteristic
296	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
297	C₂×C₄	#14273756	1	1	yes	characteristic
298	C₂×C₄	#14273759	1	1	yes	characteristic
299	C₈	—	118	2	no	—
300	C₈	#14273684	1	1	yes	characteristic
301	C₈	#14273685	1	1	yes	characteristic
302	C₈	#14273709	1	1	yes	characteristic
303	C₈	#14273712	1	1	yes	characteristic
304	C₂²	—	59	4	no	—
305	C₂²	—	118	2	no	—
306	C₂²	—	118	2	no	—
307	C₂²	—	118	2	no	—
308	C₂²	—	118	2	no	—
309	C₂²	—	118	2	no	—
310	C₂²	—	118	2	no	—
311	C₂²	#102×D₅₉	1	1	yes	characteristic
312	C₂²	#16837714	1	1	yes	characteristic
313	C₂²	#16837717	1	1	yes	characteristic
314	C₂²	SD₃₂×D₅₉	1	1	yes	characteristic
315	C₂²	Q₃₂×D₅₉	1	1	yes	characteristic
316	C₂²	#16837836	1	1	yes	characteristic
317	C₂²	#16837839	1	1	yes	characteristic
318	C₄	—	8	2	no	—
319	C₄	—	16	1	no	—
320	C₄	—	118	2	no	—
321	C₄	—	944	1	no	—
322	C₄	—	472	2	no	—
323	C₄	#16837264	1	1	yes	characteristic
324	C₄	#16837409	1	1	yes	characteristic
325	C₄	#16837731	1	1	yes	characteristic
326	C₄	#16837734	1	1	yes	characteristic
327	C₂	—	118	4	no	—
328	C₂	#21424181	1	1	yes	characteristic
329	C₂	#358×D₅₉	1	1	yes	characteristic
330	C₂	#21425810	1	1	yes	characteristic
331	C₂	#21426280	1	1	yes	characteristic
332	C₂	#21426283	1	1	yes	characteristic
333	C₂	#21426297	1	1	yes	characteristic
334	C₂	#21426300	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1380	C₂
socle	C₂³×C₅₉	C₂×D₈
prime radical	C₂³×D₅₉	D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1380	D₅₉
hypocenter	C₅₉	#12867
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

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

N	H ≅ G/N	type
4. #21422188	C₂	semidirect
5. #21422187	C₂	semidirect
6. #21422187	C₂	semidirect
7. C₅₉×#1380	C₂	semidirect
15. #107×D₅₉	C₄	semidirect
166. #1380	D₅₉	semidirect
189. C₅₉	#12867	semidirect

Maximal subgroups

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

subgroup	index
C₂×#107×D₅₉	2
C₂×(C₂×D₅₉)⋊C₁₆	2
#21424122	2
#21422188	2
#21422187	2
C₅₉×#1380	2
#12867	59