Subgroups — C₅₉×#33247 #28445356

Order 15104.

922 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₉×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₉×#495	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₉×#489	C₂	1	2	yes	maximal, complemented
4	C₅₉×#2640	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1973	C₂	1	4	yes	maximal, complemented
6	C₅₉×#1970	C₂	1	2	yes	maximal
7	C₅₉×#1969	C₂	1	2	yes	maximal, complemented
8	C₅₉×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
10	C₂²×SD₁₆×C₅₉	C₂²	1	2	yes	—
11	C₂²×D₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
13	C₂×#124×C₅₉	C₂²	1	2	yes	—
14	C₅₉×#500	—	2	4	no	—
15	C₅₉×#495	—	2	4	no	—
16	C₅₉×#489	—	2	8	no	—
17	C₂×#107×C₅₉	C₂²	1	2	yes	—
18	C₂×#107×C₅₉	C₂²	1	2	yes	—
19	C₂×#103×C₅₉	C₂²	1	4	yes	—
20	C₂×#102×C₅₉	C₂²	1	4	yes	—
21	C₂×#97×C₅₉	C₂²	1	2	yes	—
22	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₅₉×#397	—	2	4	no	—
26	C₅₉×#393	—	2	4	no	—
27	C₅₉×#391	C₂²	1	4	yes	—
28	C₅₉×#390	C₂²	1	4	yes	—
29	C₅₉×#376	C₂²	1	1	yes	characteristic
30	C₂²×Q₈×C₅₉	—	2	2	no	—
31	C₂²×Q₈×C₅₉	C₂³	1	2	yes	—
32	C₂²×D₄×C₅₉	C₂³	1	2	yes	—
33	C₂×Q₁₆×C₅₉	—	2	2	no	—
34	C₂×Q₁₆×C₅₉	—	2	4	no	—
35	C₂×SD₁₆×C₅₉	—	2	4	no	—
36	C₂×SD₁₆×C₅₉	—	2	8	no	—
37	C₂×D₈×C₅₉	—	2	2	no	—
38	C₂×D₈×C₅₉	—	2	4	no	—
39	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈×C₅₉	—	2	4	no	—
43	C₄⋊Q₈×C₅₉	—	2	4	no	—
44	#124×C₅₉	—	2	4	no	—
45	#124×C₅₉	—	2	4	no	—
46	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
48	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₅₉	—	2	8	no	—
51	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
52	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
53	#107×C₅₉	—	2	4	no	—
54	#107×C₅₉	—	2	4	no	—
55	#103×C₅₉	—	2	8	no	—
56	#102×C₅₉	—	2	8	no	—
57	#97×C₅₉	—	2	4	no	—
58	#97×C₅₉	D₄	1	4	yes	—
59	C₄×C₈×C₅₉	—	2	4	no	—
60	C₄×C₈×C₅₉	D₄	1	4	yes	—
61	#95×C₅₉	—	2	8	no	—
62	#95×C₅₉	D₄	1	2	yes	—
63	#95×C₅₉	D₄	1	2	yes	—
64	C₂⁴×C₅₉	—	2	2	no	—
65	C₂×Q₈×C₅₉	—	2	2	no	—
66	C₂×Q₈×C₅₉	—	2	2	no	—
67	C₂×Q₈×C₅₉	—	2	4	no	—
68	C₂×Q₈×C₅₉	—	2	4	no	—
69	C₂×Q₈×C₅₉	—	4	4	no	—
70	C₂×Q₈×C₅₉	—	8	4	no	—
71	C₂×D₄×C₅₉	—	2	2	no	—
72	C₂×D₄×C₅₉	—	2	2	no	—
73	C₂×D₄×C₅₉	—	2	4	no	—
74	C₂×D₄×C₅₉	—	2	4	no	—
75	C₂²×C₄×C₅₉	—	2	2	no	—
76	C₂²×C₄×C₅₉	—	2	4	no	—
77	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₅₉	—	4	4	no	—
82	SD₁₆×C₅₉	—	4	8	no	—
83	D₈×C₅₉	—	4	4	no	—
84	C₂×C₈×C₅₉	—	2	2	no	—
85	C₂×C₈×C₅₉	—	2	2	no	—
86	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
87	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₅₉	#41	1	4	yes	—
90	C₂×C₈×C₅₉	#41	1	4	yes	—
91	C₄⋊C₄×C₅₉	—	2	4	no	—
92	C₄⋊C₄×C₅₉	—	2	4	no	—
93	C₄⋊C₄×C₅₉	—	4	4	no	—
94	C₂²⋊C₄×C₅₉	—	4	4	no	—
95	C₂²⋊C₄×C₅₉	—	4	4	no	—
96	C₂²⋊C₄×C₅₉	—	8	8	no	—
97	C₄²×C₅₉	—	2	2	no	—
98	C₄²×C₅₉	—	2	2	no	—
99	C₄²×C₅₉	C₂×D₄	1	2	yes	—
100	C₄²×C₅₉	C₂×D₄	1	2	yes	—
101	C₂³×C₅₉	—	4	2	no	—
102	C₂³×C₅₉	—	4	4	no	—
103	C₂³×C₅₉	—	4	4	no	—
104	C₂³×C₅₉	—	4	4	no	—
105	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
106	Q₈×C₅₉	—	4	2	no	—
107	Q₈×C₅₉	—	4	2	no	—
108	Q₈×C₅₉	—	4	4	no	—
109	Q₈×C₅₉	—	8	8	no	—
110	D₄×C₅₉	—	4	2	no	—
111	D₄×C₅₉	—	4	2	no	—
112	D₄×C₅₉	—	4	4	no	—
113	C₂×C₄×C₅₉	—	2	2	no	—
114	C₂×C₄×C₅₉	—	2	2	no	—
115	C₂×C₄×C₅₉	—	2	4	no	—
116	C₂×C₄×C₅₉	—	2	4	no	—
117	C₂×C₄×C₅₉	—	2	4	no	—
118	C₂×C₄×C₅₉	—	4	2	no	—
119	C₂×C₄×C₅₉	—	4	4	no	—
120	C₂×C₄×C₅₉	—	4	4	no	—
121	C₂×C₄×C₅₉	—	4	8	no	—
122	C₂×C₄×C₅₉	#127	1	2	yes	—
123	C₂×C₄×C₅₉	#127	1	2	yes	—
124	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
127	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
128	C₈×C₅₉	—	2	4	no	—
129	C₈×C₅₉	#124	1	4	yes	—
130	C₈×C₅₉	#124	1	4	yes	—
131	#33247	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂²×C₅₉	—	8	2	no	—
133	C₂²×C₅₉	—	8	2	no	—
134	C₂²×C₅₉	—	8	2	no	—
135	C₂²×C₅₉	—	8	8	no	—
136	C₂²×C₅₉	C₂×#127	1	1	yes	characteristic
137	C₂²×C₅₉	#532	1	2	yes	—
138	C₂²×C₅₉	#576	1	2	yes	—
139	C₂²×C₅₉	#578	1	2	yes	—
140	C₄×C₅₉	—	2	2	no	—
141	C₄×C₅₉	—	2	2	no	—
142	C₄×C₅₉	—	4	4	no	—
143	C₄×C₅₉	—	8	2	no	—
144	C₄×C₅₉	—	8	4	no	—
145	C₄×C₅₉	C₂×#124	1	1	yes	characteristic
146	C₄×C₅₉	C₂×#124	1	1	yes	characteristic
147	C₄×C₅₉	#532	1	2	yes	—
148	C₂×#500	C₂×C₅₉	1	1	yes	characteristic, complemented
149	C₂×#495	C₂×C₅₉	1	1	yes	characteristic
150	C₂×#489	C₂×C₅₉	1	2	yes	complemented
151	#2640	C₂×C₅₉	1	2	yes	complemented
152	#1973	C₂×C₅₉	1	4	yes	complemented
153	#1970	C₂×C₅₉	1	2	yes	—
154	#1969	C₂×C₅₉	1	2	yes	complemented
155	#1726	C₂×C₅₉	1	1	yes	characteristic, complemented
156	C₂×C₅₉	—	16	2	no	—
157	C₂×C₅₉	#2858	1	1	yes	characteristic
158	C₂×C₅₉	#3155	1	1	yes	characteristic
159	C₂×C₅₉	#3156	1	1	yes	characteristic
160	C₂×C₅₉	#3175	1	4	yes	—
161	C₂²×Q₁₆	C₂²×C₅₉	1	1	yes	characteristic
162	C₂²×SD₁₆	C₂²×C₅₉	1	2	yes	—
163	C₂²×D₈	C₂²×C₅₉	1	1	yes	characteristic
164	C₂×C₄⋊Q₈	C₂²×C₅₉	1	2	yes	—
165	C₂×#124	C₂²×C₅₉	1	2	yes	—
166	#500	—	2	4	no	—
167	#495	—	2	4	no	—
168	#489	—	2	8	no	—
169	C₂×#107	C₂²×C₅₉	1	2	yes	—
170	C₂×#107	C₂²×C₅₉	1	2	yes	—
171	C₂×#103	C₂²×C₅₉	1	4	yes	—
172	C₂×#102	C₂²×C₅₉	1	4	yes	—
173	C₂×#97	C₂²×C₅₉	1	2	yes	—
174	C₂×#97	C₂²×C₅₉	1	1	yes	characteristic
175	C₂×C₄×C₈	C₂²×C₅₉	1	2	yes	—
176	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
177	#397	—	2	4	no	—
178	#393	—	2	4	no	—
179	#391	C₂²×C₅₉	1	4	yes	—
180	#390	C₂²×C₅₉	1	4	yes	—
181	#376	C₂²×C₅₉	1	1	yes	characteristic
182	C₅₉	#33247	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×Q₈	—	2	2	no	—
184	C₂²×Q₈	C₂³×C₅₉	1	2	yes	—
185	C₂²×D₄	C₂³×C₅₉	1	2	yes	—
186	C₂×Q₁₆	—	2	2	no	—
187	C₂×Q₁₆	—	2	4	no	—
188	C₂×SD₁₆	—	2	4	no	—
189	C₂×SD₁₆	—	2	8	no	—
190	C₂×D₈	—	2	2	no	—
191	C₂×D₈	—	2	4	no	—
192	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
193	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
195	C₄⋊Q₈	—	2	4	no	—
196	C₄⋊Q₈	—	2	4	no	—
197	#124	—	2	4	no	—
198	#124	—	2	4	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
201	C₂×C₂²⋊C₄	—	2	2	no	—
202	C₂×C₂²⋊C₄	—	2	2	no	—
203	C₂×C₂²⋊C₄	—	2	8	no	—
204	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
205	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
206	#107	—	2	4	no	—
207	#107	—	2	4	no	—
208	#103	—	2	8	no	—
209	#102	—	2	8	no	—
210	#97	—	2	4	no	—
211	#97	D₄×C₅₉	1	4	yes	—
212	C₄×C₈	—	2	4	no	—
213	C₄×C₈	D₄×C₅₉	1	4	yes	—
214	#95	—	2	8	no	—
215	#95	D₄×C₅₉	1	2	yes	—
216	#95	D₄×C₅₉	1	2	yes	—
217	C₂⁴	—	2	2	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	4	4	no	—
223	C₂×Q₈	—	8	4	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	C₂×D₄×C₅₉	1	4	yes	—
231	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
233	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
234	Q₁₆	—	4	4	no	—
235	SD₁₆	—	4	8	no	—
236	D₈	—	4	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	C₂×D₄×C₅₉	1	4	yes	—
240	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
241	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
242	C₂×C₈	#41×C₅₉	1	4	yes	—
243	C₂×C₈	#41×C₅₉	1	4	yes	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₂²⋊C₄	—	4	4	no	—
248	C₂²⋊C₄	—	4	4	no	—
249	C₂²⋊C₄	—	8	8	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	C₂×D₄×C₅₉	1	2	yes	—
253	C₄²	C₂×D₄×C₅₉	1	2	yes	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	4	no	—
256	C₂³	—	4	4	no	—
257	C₂³	—	4	4	no	—
258	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
259	Q₈	—	4	2	no	—
260	Q₈	—	4	2	no	—
261	Q₈	—	4	4	no	—
262	Q₈	—	8	8	no	—
263	D₄	—	4	2	no	—
264	D₄	—	4	2	no	—
265	D₄	—	4	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	8	no	—
275	C₂×C₄	#127×C₅₉	1	2	yes	—
276	C₂×C₄	#127×C₅₉	1	2	yes	—
277	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
280	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	#124×C₅₉	1	4	yes	—
283	C₈	#124×C₅₉	1	4	yes	—
284	C₂²	—	8	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	—	8	8	no	—
288	C₂²	C₂×#127×C₅₉	1	1	yes	characteristic
289	C₂²	C₅₉×#532	1	2	yes	—
290	C₂²	C₅₉×#576	1	2	yes	—
291	C₂²	C₅₉×#578	1	2	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	4	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	4	no	—
297	C₄	C₂×#124×C₅₉	1	1	yes	characteristic
298	C₄	C₂×#124×C₅₉	1	1	yes	characteristic
299	C₄	C₅₉×#532	1	2	yes	—
300	C₂	—	16	2	no	—
301	C₂	C₅₉×#2858	1	1	yes	characteristic
302	C₂	C₅₉×#3155	1	1	yes	characteristic
303	C₂	C₅₉×#3156	1	1	yes	characteristic
304	C₂	C₅₉×#3175	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
prime radical	C₂²×D₈×C₅₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₅₉×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₉×#500	C₂	semidirect
3. C₂×C₅₉×#489	C₂	semidirect
4. C₅₉×#2640	C₂	semidirect
5. C₅₉×#1973	C₂	semidirect
7. C₅₉×#1969	C₂	semidirect
8. C₅₉×#1726	C₂	semidirect
131. #33247	C₅₉	direct (N × H)
148. C₂×#500	C₂×C₅₉	semidirect
150. C₂×#489	C₂×C₅₉	semidirect
151. #2640	C₂×C₅₉	semidirect
152. #1973	C₂×C₅₉	semidirect
154. #1969	C₂×C₅₉	semidirect
155. #1726	C₂×C₅₉	semidirect
182. C₅₉	#33247	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₉×#500	2
C₂×C₅₉×#495	2
C₂×C₅₉×#489	2
C₅₉×#2640	2
C₅₉×#1973	2
C₅₉×#1970	2
C₅₉×#1969	2
C₅₉×#1726	2
#33247	59

Subgroups — C59×#33247 #28445356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33247 #28445356