Subgroups — #28589140

Order 15104.

1070 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837444	C₂	1	2	yes	maximal
2	C₂×#16837440	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14273661	C₂	1	1	yes	maximal, characteristic
4	#21426719	C₂	1	2	yes	maximal
5	#21424264	C₂	1	4	yes	maximal, complemented
6	#21424260	C₂	1	2	yes	maximal, complemented
7	#21424256	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₉×#405	C₂	1	1	yes	maximal, characteristic
9	C₂×#14273686	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₅₉	C₂²	1	2	yes	—
11	#16837444	—	2	8	no	—
12	#16837440	—	2	4	no	—
13	C₂×#14273661	—	2	4	no	—
14	C₂×#14273661	C₂²	1	2	yes	—
15	C₂×#14273661	C₂²	1	2	yes	—
16	C₂×#14273661	C₂²	1	1	yes	characteristic
17	C₂×#14273661	C₂²	1	1	yes	characteristic
18	C₂×#14273661	C₂²	1	4	yes	—
19	C₂×#14273659	C₂²	1	2	yes	—
20	C₂×#14273659	C₂²	1	4	yes	—
21	#16837301	—	2	4	no	—
22	#16837277	—	2	4	no	—
23	#16837272	C₂²	1	4	yes	—
24	#16837268	C₂²	1	4	yes	—
25	C₂²×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
26	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#117×C₅₉	C₂²	1	1	yes	characteristic
28	C₅₉×#405	C₄	1	8	yes	complemented
29	C₂×#97×C₅₉	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
31	C₂³×Dic₅₉	—	2	2	no	—
32	#14273686	—	2	4	no	—
33	#14273686	—	2	4	no	—
34	C₂×C₄⋊Dic₅₉	—	2	4	no	—
35	C₂×C₄⋊Dic₅₉	—	2	4	no	—
36	C₂×C₄⋊Dic₅₉	—	2	2	no	—
37	C₂×C₄⋊Dic₅₉	—	2	8	no	—
38	C₂×C₄⋊Dic₅₉	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₅₉	C₂³	1	4	yes	—
40	C₂×C₄⋊Dic₅₉	C₂³	1	2	yes	—
41	C₂×C₂²⋊Dic₅₉	—	2	2	no	—
42	C₂×C₄×Dic₅₉	—	2	2	no	—
43	#14273661	—	2	4	no	—
44	#14273661	—	2	4	no	—
45	#14273661	—	2	2	no	—
46	#14273661	—	2	2	no	—
47	#14273661	—	2	8	no	—
48	#14273659	—	2	4	no	—
49	#14273659	—	2	8	no	—
50	C₅₉⋊#95	—	2	8	no	—
51	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₅₉	C₂×C₄	1	4	yes	—
53	C₂×M₁₆×C₅₉	D₄	1	4	yes	—
54	C₂×M₁₆×C₅₉	Q₈	1	4	yes	—
55	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
56	C₂²×C₈×C₅₉	D₄	1	4	yes	—
57	C₂²×C₈×C₅₉	Q₈	1	4	yes	—
58	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
59	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
61	#117×C₅₉	C₂×C₄	1	4	yes	—
62	#117×C₅₉	Q₈	1	4	yes	—
63	C₂×C₄⋊C₄×C₅₉	D₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₅₉	D₄	1	2	yes	—
65	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
66	#97×C₅₉	C₂×C₄	1	8	yes	complemented
67	C₄×C₈×C₅₉	C₂×C₄	1	8	yes	complemented
68	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₅₉	—	2	2	no	—
79	C₂²×Dic₅₉	—	2	4	no	—
80	C₂²×Dic₅₉	—	2	2	no	—
81	C₂²×Dic₅₉	—	4	4	no	—
82	C₂²×Dic₅₉	—	8	4	no	—
83	M₁₆×C₅₉	C₄⋊C₄	1	16	yes	complemented
84	C₂×C₈×C₅₉	C₄⋊C₄	1	16	yes	complemented
85	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
91	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
94	C₄⋊C₄×C₅₉	C₄⋊C₄	1	8	yes	—
95	C₂²⋊C₄×C₅₉	C₄⋊C₄	1	8	yes	—
96	C₄²×C₅₉	C₂²×C₄	1	4	yes	—
97	C₄²×C₅₉	C₂×Q₈	1	4	yes	—
98	C₂²⋊Dic₅₉	—	4	4	no	—
99	C₄⋊Dic₅₉	—	2	2	no	—
100	C₄⋊Dic₅₉	—	2	4	no	—
101	C₄⋊Dic₅₉	—	2	2	no	—
102	C₄⋊Dic₅₉	—	2	2	no	—
103	C₄⋊Dic₅₉	—	2	4	no	—
104	C₄⋊Dic₅₉	—	2	2	no	—
105	C₄⋊Dic₅₉	—	4	4	no	—
106	C₄⋊Dic₅₉	—	8	8	no	—
107	C₄×Dic₅₉	—	4	4	no	—
108	C₂³×C₅₉	—	2	2	no	—
109	C₂³×C₅₉	—	2	2	no	—
110	C₂³×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
111	C₂³×C₅₉	#127	1	2	yes	—
112	C₂³×C₅₉	C₄⋊Q₈	1	2	yes	—
113	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	—	2	4	no	—
115	C₂×C₄×C₅₉	—	2	4	no	—
116	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	8	yes	—
117	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
118	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₅₉	#127	1	2	yes	—
121	C₂×C₄×C₅₉	C₄⋊Q₈	1	4	yes	—
122	C₂×C₄×C₅₉	C₄⋊Q₈	1	4	yes	—
123	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
130	C₂×Dic₅₉	—	4	2	no	—
131	C₂×Dic₅₉	—	4	2	no	—
132	C₂×Dic₅₉	—	4	4	no	—
133	C₂×Dic₅₉	—	4	4	no	—
134	C₂×Dic₅₉	—	4	8	no	—
135	C₂×Dic₅₉	—	4	4	no	—
136	C₂×Dic₅₉	—	8	8	no	—
137	C₈×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
138	C₈×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
139	C₈×C₅₉	C₂×C₄⋊C₄	1	8	yes	—
140	#20803	—	59	1	no	maximal, Sylow
141	C₂²×C₅₉	—	2	2	no	—
142	C₂²×C₅₉	—	2	2	no	—
143	C₂²×C₅₉	—	2	4	no	—
144	C₂²×C₅₉	—	2	4	no	—
145	C₂²×C₅₉	#384	1	4	yes	—
146	C₂²×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₅₉	C₂×#127	1	1	yes	characteristic
148	C₂²×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
149	C₂²×C₅₉	#576	1	2	yes	—
150	C₂²×C₅₉	#578	1	2	yes	—
151	C₄×C₅₉	—	2	8	no	—
152	C₄×C₅₉	#384	1	4	yes	—
153	C₄×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
156	C₄×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
157	Dic₅₉	—	8	2	no	—
158	Dic₅₉	—	8	4	no	—
159	Dic₅₉	—	8	2	no	—
160	C₂×#428	—	59	2	no	—
161	C₂×#427	—	59	1	no	—
162	C₂²×#107	—	59	1	no	—
163	C₂×#405	D₅₉	1	1	yes	characteristic
164	#2317	—	59	2	no	—
165	#1847	—	59	4	no	—
166	#1846	—	59	2	no	—
167	#1845	—	59	2	no	—
168	C₂×C₅₉	—	2	4	no	—
169	C₂×C₅₉	C₂×#384	1	1	yes	characteristic
170	C₂×C₅₉	#2913	1	2	yes	—
171	C₂×C₅₉	#3155	1	1	yes	characteristic
172	C₂×C₅₉	#3156	1	1	yes	characteristic
173	C₂×C₅₉	#3164	1	2	yes	—
174	C₂²×M₁₆	C₂×D₅₉	1	1	yes	characteristic
175	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
176	C₂×#117	—	59	2	no	—
177	C₂×#117	C₂×D₅₉	1	1	yes	characteristic
178	C₂²×C₄⋊C₄	—	59	2	no	—
179	#428	—	118	8	no	—
180	#427	—	118	4	no	—
181	C₂×#107	—	59	1	no	—
182	C₂×#107	—	59	2	no	—
183	C₂×#107	—	59	4	no	—
184	C₂×#107	—	59	1	no	—
185	C₂×#107	—	59	2	no	—
186	C₂×#107	—	118	4	no	—
187	C₂×#106	—	59	4	no	—
188	C₂×#106	—	59	2	no	—
189	#405	Dic₅₉	1	8	yes	complemented
190	C₂×#97	C₂×D₅₉	1	2	yes	—
191	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
192	#389	—	118	4	no	—
193	#385	—	118	4	no	—
194	#384	—	59	4	no	—
195	#382	—	59	4	no	—
196	C₅₉	#20803	1	1	yes	characteristic, complemented, Sylow
197	C₂³×C₄	—	118	2	no	—
198	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
199	C₂×M₁₆	Dic₁₁₈	1	4	yes	—
200	C₂×M₁₆	D₂₃₆	1	4	yes	—
201	C₂×M₁₆	C₂×Dic₅₉	1	4	yes	—
202	C₂²×C₈	Dic₁₁₈	1	4	yes	—
203	C₂²×C₈	D₂₃₆	1	4	yes	—
204	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
205	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
208	#117	—	118	4	no	—
209	#117	—	118	4	no	—
210	#117	Dic₁₁₈	1	4	yes	—
211	#117	C₂×Dic₅₉	1	4	yes	—
212	C₂×C₄⋊C₄	—	59	4	no	—
213	C₂×C₄⋊C₄	—	59	2	no	—
214	C₂×C₄⋊C₄	—	59	2	no	—
215	C₂×C₄⋊C₄	—	118	4	no	—
216	C₂×C₄⋊C₄	—	118	4	no	—
217	C₂×C₄⋊C₄	—	118	8	no	—
218	C₂×C₄⋊C₄	—	118	2	no	—
219	C₂×C₄⋊C₄	D₂₃₆	1	2	yes	—
220	C₂×C₂²⋊C₄	—	118	2	no	—
221	C₂×C₂²⋊C₄	D₂₃₆	1	2	yes	—
222	C₂×C₄²	—	118	2	no	—
223	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
224	#107	—	118	4	no	—
225	#107	—	118	4	no	—
226	#107	—	118	2	no	—
227	#107	—	118	2	no	—
228	#107	—	118	8	no	—
229	#106	—	118	4	no	—
230	#106	—	118	8	no	—
231	#97	C₂×Dic₅₉	1	8	yes	complemented
232	C₄×C₈	C₂×Dic₅₉	1	8	yes	complemented
233	#95	—	118	8	no	—
234	C₂⁴	C₂×D₂₃₆	1	1	yes	characteristic
235	C₂²×C₄	—	118	2	no	—
236	C₂²×C₄	—	118	4	no	—
237	C₂²×C₄	—	118	2	no	—
238	C₂²×C₄	—	472	4	no	—
239	C₂²×C₄	—	236	4	no	—
240	C₂²×C₄	C₂×Dic₁₁₈	1	2	yes	—
241	C₂²×C₄	C₂×Dic₁₁₈	1	2	yes	—
242	C₂²×C₄	C₂×Dic₁₁₈	1	2	yes	—
243	C₂²×C₄	C₂×D₂₃₆	1	4	yes	—
244	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
246	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
247	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
248	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
249	M₁₆	C₄⋊Dic₅₉	1	16	yes	complemented
250	C₂×C₈	C₄⋊Dic₅₉	1	16	yes	complemented
251	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
252	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
253	C₂×C₈	C₂×Dic₁₁₈	1	4	yes	—
254	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
255	C₂×C₈	C₂×D₂₃₆	1	4	yes	—
256	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
257	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
258	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
259	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
260	C₄⋊C₄	—	118	2	no	—
261	C₄⋊C₄	—	118	4	no	—
262	C₄⋊C₄	—	118	2	no	—
263	C₄⋊C₄	—	118	2	no	—
264	C₄⋊C₄	—	118	4	no	—
265	C₄⋊C₄	—	118	2	no	—
266	C₄⋊C₄	—	472	8	no	—
267	C₄⋊C₄	—	236	4	no	—
268	C₄⋊C₄	C₄⋊Dic₅₉	1	8	yes	—
269	C₂²⋊C₄	—	236	4	no	—
270	C₂²⋊C₄	C₄⋊Dic₅₉	1	8	yes	—
271	C₄²	—	236	4	no	—
272	C₄²	C₂×Dic₁₁₈	1	4	yes	—
273	C₄²	C₂²×Dic₅₉	1	4	yes	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	C₂×C₄⋊Dic₅₉	1	2	yes	—
277	C₂³	#14273724	1	2	yes	—
278	C₂³	#14273727	1	2	yes	—
279	C₂³	C₂²×D₂₃₆	1	1	yes	characteristic
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	472	8	no	—
283	C₂×C₄	—	236	2	no	—
284	C₂×C₄	—	236	2	no	—
285	C₂×C₄	—	236	4	no	—
286	C₂×C₄	—	236	4	no	—
287	C₂×C₄	—	236	8	no	—
288	C₂×C₄	—	236	4	no	—
289	C₂×C₄	C₂×C₄⋊Dic₅₉	1	8	yes	—
290	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
291	C₂×C₄	C₂×C₄⋊Dic₅₉	1	4	yes	—
292	C₂×C₄	C₂×C₄⋊Dic₅₉	1	4	yes	—
293	C₂×C₄	#14273724	1	2	yes	—
294	C₂×C₄	#14273727	1	4	yes	—
295	C₂×C₄	#14273727	1	4	yes	—
296	C₂×C₄	#14273727	1	2	yes	—
297	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
298	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
300	C₂×C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
301	C₂×C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
302	C₂×C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
303	C₈	C₂×C₄⋊Dic₅₉	1	4	yes	—
304	C₈	C₂×C₄⋊Dic₅₉	1	4	yes	—
305	C₈	C₂×C₄⋊Dic₅₉	1	8	yes	—
306	C₂²	—	2	4	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	#16837272	1	4	yes	—
311	C₂²	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic
312	C₂²	C₂×#14273724	1	1	yes	characteristic
313	C₂²	C₂×#14273727	1	1	yes	characteristic
314	C₂²	#16838207	1	2	yes	—
315	C₂²	#16838219	1	2	yes	—
316	C₄	—	2	8	no	—
317	C₄	—	472	4	no	—
318	C₄	—	472	2	no	—
319	C₄	—	472	2	no	—
320	C₄	#16837272	1	4	yes	—
321	C₄	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic
322	C₄	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic
323	C₄	C₂×#14273727	1	1	yes	characteristic
324	C₄	C₂×#14273727	1	1	yes	characteristic
325	C₂	—	2	4	no	—
326	C₂	C₂×#16837272	1	1	yes	characteristic
327	C₂	#21432696	1	2	yes	—
328	C₂	#21434582	1	1	yes	characteristic
329	C₂	#21434590	1	1	yes	characteristic
330	C₂	#21434635	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₂×C₅₉×#405	C₂
socle	C₂³×C₅₉	C₂²×D₄
prime radical	C₂⁴×C₅₉	C₂×D₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₂×#405	D₅₉
hypocenter	C₅₉	#20803
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
5. #21424264	C₂	semidirect
6. #21424260	C₂	semidirect
7. #21424256	C₂	semidirect
28. C₅₉×#405	C₄	semidirect
66. #97×C₅₉	C₂×C₄	semidirect
67. C₄×C₈×C₅₉	C₂×C₄	semidirect
83. M₁₆×C₅₉	C₄⋊C₄	semidirect
84. C₂×C₈×C₅₉	C₄⋊C₄	semidirect
189. #405	Dic₅₉	semidirect
196. C₅₉	#20803	semidirect
231. #97	C₂×Dic₅₉	semidirect
232. C₄×C₈	C₂×Dic₅₉	semidirect
249. M₁₆	C₄⋊Dic₅₉	semidirect
250. C₂×C₈	C₄⋊Dic₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837444	2
C₂×#16837440	2
C₂²×#14273661	2
#21426719	2
#21424264	2
#21424260	2
#21424256	2
C₂×C₅₉×#405	2
#20803	59