Subgroups — #28742236

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	#21426005	C₂	1	2	yes	maximal, complemented
2	#21425928	C₂	1	2	yes	maximal, complemented
3	#21425922	C₂	1	2	yes	maximal, complemented
4	#21425922	C₂	1	2	yes	maximal, complemented
5	#21425903	C₂	1	2	yes	maximal, complemented
6	#21425899	C₂	1	1	yes	maximal, characteristic, complemented
7	#21425693	C₂	1	2	yes	maximal, complemented
8	C₄×#14273667	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#2966	C₂	1	1	yes	maximal, characteristic
10	C₂×#14273667	C₂²	1	2	yes	—
11	C₂×#14273667	C₂²	1	2	yes	—
12	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
13	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
14	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
15	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
16	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
17	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
18	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
19	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
20	#16837178	C₂²	1	2	yes	—
21	#16837178	C₂²	1	2	yes	—
22	#16837178	C₂²	1	2	yes	—
23	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	—
24	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₅₉×#529	C₄	1	1	yes	characteristic
27	C₅₉×#446	C₄	1	1	yes	characteristic
28	C₈×Q₈×C₅₉	C₂²	1	1	yes	characteristic
29	C₅₉×#436	C₄	1	1	yes	characteristic
30	C₅₉×#435	C₄	1	2	yes	—
31	C₈×D₄×C₅₉	C₂²	1	1	yes	characteristic
32	C₈×D₄×C₅₉	C₂²	1	2	yes	—
33	C₅₉×#433	C₄	1	2	yes	—
34	C₅₉×#432	C₄	1	1	yes	characteristic
35	C₅₉×#405	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
37	#14273667	—	2	4	no	—
38	#14273667	—	2	4	no	—
39	C₂×C₅₉⋊C₁₆	—	2	4	no	—
40	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
41	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
42	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
43	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
44	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
48	#126×C₅₉	C₈	1	2	yes	—
49	#125×C₅₉	C₈	1	1	yes	characteristic
50	#124×C₅₉	C₈	1	1	yes	characteristic
51	#123×C₅₉	C₈	1	2	yes	—
52	C₂²⋊Q₈×C₅₉	C₈	1	1	yes	characteristic
53	#121×C₅₉	C₈	1	1	yes	characteristic
54	C₄×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
55	C₄×D₄×C₅₉	C₂×C₄	1	2	yes	—
56	C₄×D₄×C₅₉	C₂×C₄	1	1	yes	characteristic
57	#117×C₅₉	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
59	C₄⋊C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
62	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
63	C₂²⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
64	C₂²⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
65	C₂²⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
66	#97×C₅₉	C₂×C₄	1	2	yes	—
67	#97×C₅₉	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₅₉	C₂³	1	2	yes	—
72	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₂×Q₈×C₅₉	C₂×C₈	1	1	yes	characteristic
74	C₂×D₄×C₅₉	C₂×C₈	1	2	yes	—
75	C₂×D₄×C₅₉	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
80	M₁₆×C₅₉	—	2	4	no	—
81	C₂×C₈×C₅₉	—	2	2	no	—
82	C₂×C₈×C₅₉	—	2	2	no	—
83	C₂×C₈×C₅₉	—	2	2	no	—
84	C₂×C₈×C₅₉	—	2	2	no	—
85	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
92	C₄⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
93	C₄⋊C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
94	C₄⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
95	C₄⋊C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
96	C₄⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
97	C₂²⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
98	C₂²⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
99	C₂²⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
100	C₂²⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
101	C₂²⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
102	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
103	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
104	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
107	C₅₉⋊C₁₆	—	2	2	no	—
108	C₅₉⋊C₁₆	—	2	2	no	—
109	C₅₉⋊C₁₆	—	2	2	no	—
110	C₅₉⋊C₁₆	#41	1	4	yes	—
111	C₂³×C₅₉	C₂²×C₈	1	2	yes	—
112	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
113	Q₈×C₅₉	—	4	1	no	—
114	D₄×C₅₉	—	4	1	no	—
115	D₄×C₅₉	—	4	2	no	—
116	C₂×C₄×C₅₉	—	2	2	no	—
117	C₂×C₄×C₅₉	—	2	1	no	—
118	C₂×C₄×C₅₉	—	2	1	no	—
119	C₂×C₄×C₅₉	—	2	2	no	—
120	C₂×C₄×C₅₉	—	4	1	no	—
121	C₂×C₄×C₅₉	—	4	1	no	—
122	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
131	C₈×C₅₉	—	2	2	no	—
132	C₈×C₅₉	—	2	1	no	—
133	C₈×C₅₉	—	2	1	no	—
134	C₈×C₅₉	—	2	2	no	—
135	C₈×C₅₉	#131	1	2	yes	—
136	C₈×C₅₉	C₂×#41	1	2	yes	—
137	#33481	—	59	1	no	maximal, Sylow
138	C₂²×C₅₉	—	2	1	no	—
139	C₂²×C₅₉	—	2	1	no	—
140	C₂²×C₅₉	—	2	2	no	—
141	C₂²×C₅₉	—	2	2	no	—
142	C₂²×C₅₉	—	2	1	no	—
143	C₂²×C₅₉	—	2	2	no	—
144	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
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	2	no	—
150	C₄×C₅₉	—	2	2	no	—
151	C₄×C₅₉	—	2	1	no	—
152	C₄×C₅₉	—	2	2	no	—
153	C₄×C₅₉	C₄×#41	1	1	yes	characteristic
154	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
155	#2966	D₅₉	1	1	yes	characteristic
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	—	59	1	no	—
164	C₂×C₅₉	—	4	1	no	—
165	C₂×C₅₉	—	4	2	no	—
166	C₂×C₅₉	C₈×#41	1	1	yes	characteristic
167	C₂×C₅₉	#3407	1	1	yes	characteristic
168	C₂×C₅₉	#3407	1	1	yes	characteristic
169	#529	Dic₅₉	1	1	yes	characteristic
170	C₂×M₃₂	—	59	2	no	—
171	C₂×M₃₂	—	59	2	no	—
172	C₂²×C₁₆	—	59	2	no	—
173	#446	Dic₅₉	1	1	yes	characteristic
174	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
175	#436	Dic₅₉	1	1	yes	characteristic
176	#435	Dic₅₉	1	2	yes	—
177	C₈×D₄	C₂×D₅₉	1	1	yes	characteristic
178	C₈×D₄	C₂×D₅₉	1	2	yes	—
179	#433	Dic₅₉	1	2	yes	—
180	#432	Dic₅₉	1	1	yes	characteristic
181	#405	C₂×D₅₉	1	2	yes	—
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	—	59	2	no	—
193	C₄×C₁₆	—	59	1	no	—
194	C₄×C₁₆	—	59	1	no	—
195	C₄×C₁₆	—	59	2	no	—
196	C₅₉	#33481	1	1	yes	characteristic, complemented, Sylow
197	C₂×M₁₆	C₂×Dic₅₉	1	2	yes	—
198	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
201	#126	C₅₉⋊C₈	1	2	yes	—
202	#125	C₅₉⋊C₈	1	1	yes	characteristic
203	#124	C₅₉⋊C₈	1	1	yes	characteristic
204	#123	C₅₉⋊C₈	1	2	yes	—
205	C₂²⋊Q₈	C₅₉⋊C₈	1	1	yes	characteristic
206	#121	C₅₉⋊C₈	1	1	yes	characteristic
207	C₄×Q₈	C₂×Dic₅₉	1	1	yes	characteristic
208	C₄×D₄	C₂×Dic₅₉	1	2	yes	—
209	C₄×D₄	C₂×Dic₅₉	1	1	yes	characteristic
210	#117	C₂×Dic₅₉	1	2	yes	—
211	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
212	M₃₂	—	118	4	no	—
213	M₃₂	—	118	4	no	—
214	C₂×C₁₆	—	59	2	no	—
215	C₂×C₁₆	—	59	2	no	—
216	C₂×C₁₆	—	59	2	no	—
217	C₂×C₁₆	—	59	2	no	—
218	C₂×C₁₆	—	118	4	no	—
219	C₄⋊C₈	C₂×Dic₅₉	1	1	yes	characteristic
220	C₄⋊C₈	C₂×Dic₅₉	1	1	yes	characteristic
221	C₄⋊C₈	C₂×Dic₅₉	1	2	yes	—
222	C₄⋊C₈	C₂×Dic₅₉	1	2	yes	—
223	C₂²⋊C₈	C₂×Dic₅₉	1	2	yes	—
224	C₂²⋊C₈	C₂×Dic₅₉	1	2	yes	—
225	C₂²⋊C₈	C₂×Dic₅₉	1	2	yes	—
226	#97	C₂×Dic₅₉	1	2	yes	—
227	#97	C₂×Dic₅₉	1	2	yes	—
228	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
229	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₅₉	1	2	yes	—
232	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
233	C₂×Q₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
234	C₂×D₄	C₂×C₅₉⋊C₈	1	2	yes	—
235	C₂×D₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
238	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
239	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
240	M₁₆	—	2	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
246	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
248	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
249	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
250	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
251	C₄⋊C₄	C₂×C₅₉⋊C₈	1	2	yes	—
252	C₄⋊C₄	C₂×C₅₉⋊C₈	1	2	yes	—
253	C₄⋊C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
254	C₄⋊C₄	C₂×C₅₉⋊C₈	1	2	yes	—
255	C₄⋊C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
256	C₄⋊C₄	C₂×C₅₉⋊C₈	1	2	yes	—
257	C₂²⋊C₄	C₂×C₅₉⋊C₈	1	2	yes	—
258	C₂²⋊C₄	C₂×C₅₉⋊C₈	1	2	yes	—
259	C₂²⋊C₄	C₂×C₅₉⋊C₈	1	2	yes	—
260	C₂²⋊C₄	C₂×C₅₉⋊C₈	1	2	yes	—
261	C₂²⋊C₄	C₂×C₅₉⋊C₈	1	2	yes	—
262	C₄²	C₂×C₅₉⋊C₈	1	1	yes	characteristic
263	C₄²	C₂×C₅₉⋊C₈	1	1	yes	characteristic
264	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
265	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
266	C₄²	C₂²×Dic₅₉	1	2	yes	—
267	C₁₆	—	59	4	no	—
268	C₁₆	—	118	2	no	—
269	C₁₆	—	118	2	no	—
270	C₁₆	—	118	2	no	—
271	C₂³	C₂²×C₅₉⋊C₈	1	2	yes	—
272	C₂³	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
273	Q₈	—	4	1	no	—
274	D₄	—	4	1	no	—
275	D₄	—	4	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
284	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
286	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
287	C₂×C₄	C₂²×C₅₉⋊C₈	1	2	yes	—
288	C₂×C₄	C₂²×C₅₉⋊C₈	1	2	yes	—
289	C₂×C₄	C₂²×C₅₉⋊C₈	1	2	yes	—
290	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	2	no	—
293	C₈	—	2	2	no	—
294	C₈	—	2	1	no	—
295	C₈	#14273734	1	2	yes	—
296	C₈	#41×D₅₉	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	2	no	—
303	C₂²	C₂³×C₅₉⋊C₈	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	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	#41×Dic₅₉	1	1	yes	characteristic
313	C₄	C₂×#14273734	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	4	2	no	—
316	C₂	#41×C₅₉⋊C₈	1	1	yes	characteristic
317	C₂	#21437199	1	1	yes	characteristic
318	C₂	#21437199	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₈
Fitting	C₅₉×#2966	C₂
prime radical	#121×C₅₉	C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	#2966	D₅₉
hypocenter	C₅₉	#33481
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ #2966

Complements

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

N	H ≅ G/N	type
1. #21426005	C₂	semidirect
2. #21425928	C₂	semidirect
3. #21425922	C₂	semidirect
4. #21425922	C₂	semidirect
5. #21425903	C₂	semidirect
6. #21425899	C₂	semidirect
7. #21425693	C₂	semidirect
8. C₄×#14273667	C₂	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
#21426005	2
#21425928	2
#21425922	2
#21425903	2
#21425899	2
#21425693	2
C₄×#14273667	2
C₅₉×#2966	2
#33481	59