Subgroups — #28742234

Order 15104.

508 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433071	C₂	1	1	yes	maximal, characteristic
2	#21426002	C₂	1	2	yes	maximal, complemented
3	#21425926	C₂	1	2	yes	maximal, complemented
4	#21425923	C₂	1	2	yes	maximal, complemented
5	#21425919	C₂	1	2	yes	maximal, complemented
6	#21425901	C₂	1	2	yes	maximal, complemented
7	#21425899	C₂	1	1	yes	maximal, characteristic, complemented
8	#21425691	C₂	1	2	yes	maximal, complemented
9	C₄×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837936	C₄	1	1	yes	characteristic
11	C₂×#14273666	C₂²	1	2	yes	—
12	C₂×C₁₆×D₅₉	C₂²	1	2	yes	—
13	#16837532	C₄	1	1	yes	characteristic
14	C₈×Dic₁₁₈	C₂²	1	1	yes	characteristic
15	#16837481	C₄	1	1	yes	characteristic
16	#16837473	C₄	1	2	yes	—
17	C₈×D₂₃₆	C₂²	1	1	yes	characteristic
18	C₈×C₂²⋊D₅₉	C₂²	1	2	yes	—
19	#16837461	C₄	1	2	yes	—
20	#16837459	C₄	1	1	yes	characteristic
21	#16837356	C₂²	1	2	yes	—
22	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
23	Dic₅₉⋊C₁₆	C₂²	1	2	yes	—
24	Dic₅₉⋊C₁₆	C₂²	1	2	yes	—
25	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
26	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
27	(C₂×D₅₉)⋊C₁₆	C₂²	1	2	yes	—
28	(C₂×D₅₉)⋊C₁₆	C₂²	1	2	yes	—
29	#16837178	C₂²	1	2	yes	—
30	#16837177	C₂²	1	2	yes	—
31	C₁₆×Dic₅₉	C₂²	1	2	yes	—
32	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
34	C₅₉×#346	C₂²	1	2	yes	—
35	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
36	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
37	C₂×#11745659	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₅₉	C₂³	1	2	yes	—
39	#14273721	C₈	1	2	yes	—
40	#14273719	C₈	1	1	yes	characteristic
41	#14273715	C₈	1	1	yes	characteristic
42	#14273710	C₈	1	2	yes	—
43	#14273706	C₈	1	1	yes	characteristic
44	#14273701	C₈	1	1	yes	characteristic
45	C₄×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
46	C₄×D₂₃₆	C₂×C₄	1	1	yes	characteristic
47	C₄×C₂²⋊D₅₉	C₂×C₄	1	2	yes	—
48	#14273684	C₂×C₄	1	2	yes	—
49	#14273666	—	2	4	no	—
50	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
51	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
52	C₁₆×D₅₉	—	2	4	no	—
53	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
54	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
55	Dic₅₉⋊C₈	C₂×C₄	1	2	yes	—
56	Dic₅₉⋊C₈	C₂×C₄	1	2	yes	—
57	C₂²⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
58	(C₂×D₅₉)⋊C₈	C₂×C₄	1	2	yes	—
59	(C₂×D₅₉)⋊C₈	C₂×C₄	1	2	yes	—
60	#14273638	C₂×C₄	1	2	yes	—
61	#14273637	C₂×C₄	1	2	yes	—
62	C₈×Dic₅₉	C₂³	1	2	yes	—
63	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
66	M₃₂×C₅₉	—	2	4	no	—
67	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
68	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
69	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₂²⋊D₅₉	C₂×C₈	1	2	yes	—
77	C₂×D₂₃₆	C₂×C₈	1	1	yes	characteristic
78	C₂×C₄×D₅₉	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
80	C₂×C₈×C₅₉	—	2	2	no	—
81	C₂×C₈×C₅₉	—	2	2	no	—
82	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
86	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
87	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
88	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
90	C₂²⋊Dic₅₉	C₂×C₈	1	2	yes	—
91	(C₂×D₅₉)⋊C₄	C₂×C₈	1	2	yes	—
92	(C₂×D₅₉)⋊C₄	C₂×C₈	1	2	yes	—
93	(C₂×D₅₉)⋊C₄	C₂×C₈	1	2	yes	—
94	(C₂×D₅₉)⋊C₄	C₂×C₈	1	2	yes	—
95	C₄⋊Dic₅₉	C₂×C₈	1	1	yes	characteristic
96	C₄⋊Dic₅₉	C₂×C₈	1	1	yes	characteristic
97	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
98	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
99	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
100	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
101	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
102	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
103	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
104	#11745659	—	2	4	no	—
105	C₈×D₅₉	—	2	2	no	—
106	C₈×D₅₉	—	2	2	no	—
107	C₁₆×C₅₉	—	2	2	no	—
108	C₁₆×C₅₉	#41	1	4	yes	—
109	C₅₉⋊C₁₆	—	2	2	no	—
110	C₅₉⋊C₁₆	—	2	2	no	—
111	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
112	C₂²×D₅₉	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₅₉	—	2	1	no	—
114	C₂×C₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	—	4	1	no	—
116	C₂×C₄×C₅₉	—	4	1	no	—
117	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
123	C₂²⋊D₅₉	—	4	2	no	—
124	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
125	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
126	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
127	D₂₃₆	—	4	1	no	—
128	C₄×D₅₉	—	2	2	no	—
129	C₄×D₅₉	—	2	2	no	—
130	Dic₁₁₈	—	4	1	no	—
131	C₈×C₅₉	—	2	1	no	—
132	C₈×C₅₉	—	2	1	no	—
133	C₈×C₅₉	#131	1	2	yes	—
134	C₈×C₅₉	C₂×#41	1	2	yes	—
135	C₅₉⋊C₈	—	2	2	no	—
136	C₅₉⋊C₈	—	2	2	no	—
137	#33481	—	59	1	no	maximal, Sylow
138	C₂²×C₅₉	—	2	1	no	—
139	C₂²×C₅₉	—	2	1	no	—
140	C₂²×C₅₉	—	2	1	no	—
141	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
142	C₂×D₅₉	—	2	2	no	—
143	C₂×D₅₉	—	2	2	no	—
144	C₂×D₅₉	—	2	2	no	—
145	C₄×C₅₉	—	2	1	no	—
146	C₄×C₅₉	—	2	1	no	—
147	C₄×C₅₉	—	2	1	no	—
148	C₄×C₅₉	—	2	1	no	—
149	C₄×C₅₉	—	2	1	no	—
150	C₄×C₅₉	C₄×#41	1	1	yes	characteristic
151	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
152	Dic₅₉	—	2	2	no	—
153	Dic₅₉	—	2	2	no	—
154	Dic₅₉	—	2	2	no	—
155	#2966	—	59	1	no	—
156	#2183	—	59	2	no	—
157	#2169	—	59	2	no	—
158	#2168	—	59	2	no	—
159	#2168	—	59	2	no	—
160	#2164	—	59	2	no	—
161	#2163	—	59	1	no	—
162	#2108	—	59	2	no	—
163	C₄×M₃₂	D₅₉	1	1	yes	characteristic, complemented
164	C₂×C₅₉	—	4	1	no	—
165	C₂×C₅₉	C₈×#41	1	1	yes	characteristic
166	C₂×C₅₉	#3407	1	1	yes	characteristic
167	C₂×C₅₉	#3407	1	1	yes	characteristic
168	D₅₉	—	4	2	no	—
169	#529	—	59	1	no	—
170	C₂×M₃₂	—	59	2	no	—
171	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
172	C₂²×C₁₆	—	59	2	no	—
173	#446	—	59	1	no	—
174	C₈×Q₈	—	59	1	no	—
175	#436	—	59	1	no	—
176	#435	—	59	2	no	—
177	C₈×D₄	—	59	2	no	—
178	C₈×D₄	—	59	1	no	—
179	#433	—	59	2	no	—
180	#432	—	59	1	no	—
181	#405	—	59	2	no	—
182	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
183	C₄⋊C₁₆	—	59	2	no	—
184	C₄⋊C₁₆	—	59	2	no	—
185	C₄⋊C₁₆	—	59	2	no	—
186	C₂²⋊C₁₆	—	59	1	no	—
187	C₂²⋊C₁₆	—	59	1	no	—
188	C₂²⋊C₁₆	—	59	2	no	—
189	C₂²⋊C₁₆	—	59	2	no	—
190	#346	—	59	2	no	—
191	#346	—	59	2	no	—
192	#346	C₂×D₅₉	1	2	yes	—
193	C₄×C₁₆	—	59	2	no	—
194	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
195	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
196	C₅₉	#33481	1	1	yes	characteristic, complemented, Sylow
197	C₂×M₁₆	—	59	2	no	—
198	C₂²×C₈	—	59	2	no	—
199	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
201	#126	—	59	2	no	—
202	#125	—	59	1	no	—
203	#124	—	59	1	no	—
204	#123	—	59	2	no	—
205	C₂²⋊Q₈	—	59	1	no	—
206	#121	—	59	1	no	—
207	C₄×Q₈	—	59	1	no	—
208	C₄×D₄	—	59	2	no	—
209	C₄×D₄	—	59	1	no	—
210	#117	—	59	2	no	—
211	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
212	M₃₂	—	2	4	no	—
213	M₃₂	—	118	4	no	—
214	C₂×C₁₆	—	59	2	no	—
215	C₂×C₁₆	—	59	2	no	—
216	C₂×C₁₆	—	118	4	no	—
217	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
218	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
219	C₄⋊C₈	—	59	2	no	—
220	C₄⋊C₈	—	59	2	no	—
221	C₄⋊C₈	—	59	1	no	—
222	C₄⋊C₈	—	59	1	no	—
223	C₂²⋊C₈	—	59	2	no	—
224	C₂²⋊C₈	—	59	2	no	—
225	C₂²⋊C₈	—	59	2	no	—
226	#97	—	59	2	no	—
227	#97	—	59	2	no	—
228	C₄×C₈	—	59	2	no	—
229	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
230	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
233	C₂×Q₈	—	59	1	no	—
234	C₂×D₄	—	59	2	no	—
235	C₂×D₄	—	59	1	no	—
236	C₂²×C₄	—	59	2	no	—
237	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
238	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
240	M₁₆	—	118	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	59	2	no	—
244	C₂×C₈	—	59	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	C₂×C₄×D₅₉	1	1	yes	characteristic
249	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
250	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
251	C₄⋊C₄	—	59	2	no	—
252	C₄⋊C₄	—	59	2	no	—
253	C₄⋊C₄	—	59	1	no	—
254	C₄⋊C₄	—	59	2	no	—
255	C₄⋊C₄	—	59	1	no	—
256	C₄⋊C₄	—	59	2	no	—
257	C₂²⋊C₄	—	59	2	no	—
258	C₂²⋊C₄	—	59	2	no	—
259	C₂²⋊C₄	—	59	2	no	—
260	C₂²⋊C₄	—	59	2	no	—
261	C₂²⋊C₄	—	59	2	no	—
262	C₄²	—	59	2	no	—
263	C₄²	C₈×D₅₉	1	1	yes	characteristic
264	C₄²	C₈×D₅₉	1	1	yes	characteristic
265	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
266	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
267	C₁₆	—	2	2	no	—
268	C₁₆	—	118	2	no	—
269	C₁₆	—	118	2	no	—
270	C₁₆	#11745685	1	4	yes	—
271	C₂³	—	59	2	no	—
272	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
273	Q₈	—	236	1	no	—
274	D₄	—	236	1	no	—
275	D₄	—	236	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	59	2	no	—
281	C₂×C₄	—	59	2	no	—
282	C₂×C₄	—	59	2	no	—
283	C₂×C₄	—	118	2	no	—
284	C₂×C₄	—	118	2	no	—
285	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
286	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
287	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
288	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
289	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	118	2	no	—
294	C₈	—	118	2	no	—
295	C₈	#14273733	1	2	yes	—
296	C₈	C₂×#11745685	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	118	2	no	—
301	C₂²	—	118	2	no	—
302	C₂²	—	118	2	no	—
303	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	118	2	no	—
310	C₄	—	118	2	no	—
311	C₄	—	118	2	no	—
312	C₄	C₄×#11745685	1	1	yes	characteristic
313	C₄	C₂×#14273733	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	236	2	no	—
316	C₂	C₈×#11745685	1	1	yes	characteristic
317	C₂	#21437198	1	1	yes	characteristic
318	C₂	#21437198	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₈
Fitting	C₄×M₃₂×C₅₉	C₂
prime radical	#14273701	C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₄×M₃₂	D₅₉
hypocenter	C₅₉	#33481
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #21426002	C₂	semidirect
3. #21425926	C₂	semidirect
4. #21425923	C₂	semidirect
5. #21425919	C₂	semidirect
6. #21425901	C₂	semidirect
7. #21425899	C₂	semidirect
8. #21425691	C₂	semidirect
9. C₄×M₃₂×C₅₉	C₂	semidirect
163. C₄×M₃₂	D₅₉	semidirect
196. C₅₉	#33481	semidirect

Maximal subgroups

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

subgroup	index
#21433071	2
#21426002	2
#21425926	2
#21425923	2
#21425919	2
#21425901	2
#21425899	2
#21425691	2
C₄×M₃₂×C₅₉	2
#33481	59