Subgroups — C₅₉×#35679 #28447788

Order 15104.

2470 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#124×C₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₉×#519	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#2482	C₂	1	8	yes	maximal, complemented
4	C₅₉×#2448	C₂	1	16	yes	maximal, complemented
5	C₅₉×#2310	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₉×#2309	C₂	1	2	yes	maximal
7	C₅₉×#2296	C₂	1	2	yes	maximal, complemented
8	C₂³×Q₈×C₅₉	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₅₉	C₂²	1	1	yes	characteristic
10	C₂×#124×C₅₉	—	2	8	no	—
11	C₂×#124×C₅₉	C₂²	1	8	yes	—
12	C₅₉×#519	—	2	8	no	—
13	C₂×C₄×Q₈×C₅₉	C₂²	1	8	yes	complemented
14	C₂×#117×C₅₉	C₂²	1	4	yes	—
15	C₂×#117×C₅₉	C₂²	1	2	yes	—
16	C₂×#117×C₅₉	C₂²	1	2	yes	—
17	C₂²×C₂²⋊C₄×C₅₉	C₂²	1	4	yes	—
18	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
19	C₅₉×#399	C₂²	1	16	yes	complemented
20	C₅₉×#397	C₂²	1	16	yes	complemented
21	C₅₉×#391	C₂²	1	8	yes	complemented
22	C₅₉×#390	C₂²	1	8	yes	—
23	C₅₉×#390	C₂²	1	8	yes	—
24	C₅₉×#389	C₂²	1	8	yes	complemented
25	C₅₉×#388	C₂²	1	32	yes	complemented
26	C₅₉×#385	C₂²	1	8	yes	—
27	C₅₉×#381	C₂²	1	4	yes	—
28	C₅₉×#379	C₂²	1	8	yes	—
29	C₄×C₄⋊C₄×C₅₉	C₂²	1	8	yes	complemented
30	C₂⁵×C₅₉	C₂³	1	2	yes	—
31	C₂²×Q₈×C₅₉	—	2	8	no	—
32	C₂²×Q₈×C₅₉	D₄	1	8	yes	—
33	C₂²×Q₈×C₅₉	C₂³	1	4	yes	—
34	C₂²×D₄×C₅₉	—	2	4	no	—
35	C₂²×D₄×C₅₉	—	2	8	no	—
36	C₂²×D₄×C₅₉	C₂³	1	4	yes	—
37	C₂³×C₄×C₅₉	C₂³	1	2	yes	—
38	C₂³×C₄×C₅₉	C₂³	1	2	yes	—
39	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
40	#124×C₅₉	—	2	32	no	—
41	C₄×Q₈×C₅₉	—	2	32	no	—
42	#117×C₅₉	—	2	16	no	—
43	#117×C₅₉	—	2	8	no	—
44	#117×C₅₉	—	2	8	no	—
45	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₅₉	C₂³	1	8	yes	—
48	C₂×C₄⋊C₄×C₅₉	C₂³	1	16	yes	—
49	C₂×C₂²⋊C₄×C₅₉	—	2	16	no	—
50	C₂×C₂²⋊C₄×C₅₉	C₂³	1	4	yes	—
51	C₂×C₂²⋊C₄×C₅₉	C₂³	1	4	yes	—
52	C₂×C₂²⋊C₄×C₅₉	C₂³	1	16	yes	—
53	C₂×C₂²⋊C₄×C₅₉	C₂³	1	32	yes	—
54	C₂×C₄²×C₅₉	—	2	4	no	—
55	C₂×C₄²×C₅₉	C₂³	1	4	yes	—
56	C₂×C₄²×C₅₉	C₂³	1	4	yes	—
57	C₂×C₄²×C₅₉	C₂³	1	4	yes	—
58	C₂×C₄²×C₅₉	C₂³	1	8	yes	—
59	#95×C₅₉	C₂³	1	16	yes	—
60	#95×C₅₉	C₂³	1	16	yes	—
61	C₂⁴×C₅₉	—	2	4	no	—
62	C₂⁴×C₅₉	—	2	8	no	—
63	C₂⁴×C₅₉	—	4	8	no	—
64	C₂⁴×C₅₉	C₂⁴	1	4	yes	—
65	C₂⁴×C₅₉	C₂⁴	1	1	yes	characteristic
66	C₂×Q₈×C₅₉	—	2	32	no	—
67	C₂×Q₈×C₅₉	—	2	8	no	—
68	C₂×Q₈×C₅₉	—	2	8	no	—
69	C₂×Q₈×C₅₉	C₂×D₄	1	16	yes	—
70	C₂×D₄×C₅₉	—	2	8	no	—
71	C₂×D₄×C₅₉	—	2	8	no	—
72	C₂×D₄×C₅₉	—	2	8	no	—
73	C₂×D₄×C₅₉	—	4	16	no	—
74	C₂²×C₄×C₅₉	—	2	8	no	—
75	C₂²×C₄×C₅₉	—	2	4	no	—
76	C₂²×C₄×C₅₉	—	4	4	no	—
77	C₂²×C₄×C₅₉	C₂×D₄	1	8	yes	—
78	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₅₉	#41	1	8	yes	—
80	C₂²×C₄×C₅₉	C₂⁴	1	8	yes	—
81	C₂²×C₄×C₅₉	C₂⁴	1	4	yes	—
82	C₂²×C₄×C₅₉	C₂⁴	1	8	yes	—
83	C₂²×C₄×C₅₉	C₂⁴	1	4	yes	—
84	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
85	C₄⋊C₄×C₅₉	—	2	16	no	—
86	C₄⋊C₄×C₅₉	—	2	8	no	—
87	C₄⋊C₄×C₅₉	—	2	8	no	—
88	C₄⋊C₄×C₅₉	—	4	16	no	—
89	C₂²⋊C₄×C₅₉	—	2	8	no	—
90	C₂²⋊C₄×C₅₉	—	2	8	no	—
91	C₂²⋊C₄×C₅₉	—	2	32	no	—
92	C₂²⋊C₄×C₅₉	—	4	32	no	—
93	C₄²×C₅₉	—	2	8	no	—
94	C₄²×C₅₉	—	2	16	no	—
95	C₄²×C₅₉	—	2	8	no	—
96	C₄²×C₅₉	—	2	8	no	—
97	C₂³×C₅₉	—	2	4	no	—
98	C₂³×C₅₉	—	2	4	no	—
99	C₂³×C₅₉	—	2	8	no	—
100	C₂³×C₅₉	—	2	16	no	—
101	C₂³×C₅₉	—	4	16	no	—
102	C₂³×C₅₉	—	4	8	no	—
103	C₂³×C₅₉	—	8	16	no	—
104	C₂³×C₅₉	C₂²×D₄	1	2	yes	—
105	C₂³×C₅₉	C₂×#41	1	4	yes	—
106	C₂³×C₅₉	C₂⁵	1	1	yes	characteristic
107	Q₈×C₅₉	—	2	32	no	—
108	D₄×C₅₉	—	4	16	no	—
109	C₂×C₄×C₅₉	—	2	16	no	—
110	C₂×C₄×C₅₉	—	2	8	no	—
111	C₂×C₄×C₅₉	—	2	8	no	—
112	C₂×C₄×C₅₉	—	2	4	no	—
113	C₂×C₄×C₅₉	—	2	8	no	—
114	C₂×C₄×C₅₉	—	2	16	no	—
115	C₂×C₄×C₅₉	—	2	16	no	—
116	C₂×C₄×C₅₉	—	2	8	no	—
117	C₂×C₄×C₅₉	—	4	8	no	—
118	C₂×C₄×C₅₉	#123	1	16	yes	—
119	C₂×C₄×C₅₉	C₂²×D₄	1	8	yes	—
120	C₂×C₄×C₅₉	C₂²×D₄	1	4	yes	—
121	C₂×C₄×C₅₉	C₂×#41	1	4	yes	—
122	C₂×C₄×C₅₉	C₂×#41	1	4	yes	—
123	#35679	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₅₉	—	2	4	no	—
125	C₂²×C₅₉	—	2	8	no	—
126	C₂²×C₅₉	—	4	4	no	—
127	C₂²×C₅₉	—	4	8	no	—
128	C₂²×C₅₉	—	4	16	no	—
129	C₂²×C₅₉	—	8	16	no	—
130	C₂²×C₅₉	C₂×#123	1	4	yes	—
131	C₂²×C₅₉	C₂³×D₄	1	1	yes	characteristic
132	C₂²×C₅₉	C₂²×#41	1	2	yes	—
133	C₂²×C₅₉	#585	1	4	yes	—
134	C₄×C₅₉	—	2	8	no	—
135	C₄×C₅₉	—	4	8	no	—
136	C₄×C₅₉	—	4	4	no	—
137	C₄×C₅₉	—	4	8	no	—
138	C₄×C₅₉	C₂×#123	1	8	yes	—
139	C₂²×#124	C₂×C₅₉	1	1	yes	characteristic
140	C₂×#519	C₂×C₅₉	1	1	yes	characteristic, complemented
141	#2482	C₂×C₅₉	1	8	yes	complemented
142	#2448	C₂×C₅₉	1	16	yes	complemented
143	#2310	C₂×C₅₉	1	1	yes	characteristic, complemented
144	#2309	C₂×C₅₉	1	2	yes	—
145	#2296	C₂×C₅₉	1	2	yes	complemented
146	C₂×C₅₉	—	2	4	no	—
147	C₂×C₅₉	—	8	4	no	—
148	C₂×C₅₉	C₂²×#123	1	1	yes	characteristic
149	C₂×C₅₉	#3487	1	2	yes	—
150	C₂×C₅₉	#3510	1	4	yes	—
151	C₂³×Q₈	C₂²×C₅₉	1	1	yes	characteristic
152	C₂³×D₄	C₂²×C₅₉	1	1	yes	characteristic
153	C₂×#124	—	2	8	no	—
154	C₂×#124	C₂²×C₅₉	1	8	yes	—
155	#519	—	2	8	no	—
156	C₂×C₄×Q₈	C₂²×C₅₉	1	8	yes	complemented
157	C₂×#117	C₂²×C₅₉	1	2	yes	—
158	C₂×#117	C₂²×C₅₉	1	2	yes	—
159	C₂×#117	C₂²×C₅₉	1	4	yes	—
160	C₂²×C₂²⋊C₄	C₂²×C₅₉	1	4	yes	—
161	C₂²×C₄²	C₂²×C₅₉	1	1	yes	characteristic
162	#399	C₂²×C₅₉	1	16	yes	complemented
163	#397	C₂²×C₅₉	1	16	yes	complemented
164	#391	C₂²×C₅₉	1	8	yes	complemented
165	#390	C₂²×C₅₉	1	8	yes	—
166	#390	C₂²×C₅₉	1	8	yes	—
167	#389	C₂²×C₅₉	1	8	yes	complemented
168	#388	C₂²×C₅₉	1	32	yes	complemented
169	#385	C₂²×C₅₉	1	8	yes	—
170	#381	C₂²×C₅₉	1	4	yes	—
171	#379	C₂²×C₅₉	1	8	yes	—
172	C₄×C₄⋊C₄	C₂²×C₅₉	1	8	yes	complemented
173	C₅₉	#35679	1	1	yes	characteristic, direct product factor, Sylow
174	C₂⁵	C₂³×C₅₉	1	2	yes	—
175	C₂²×Q₈	—	2	8	no	—
176	C₂²×Q₈	D₄×C₅₉	1	8	yes	—
177	C₂²×Q₈	C₂³×C₅₉	1	4	yes	—
178	C₂²×D₄	—	2	4	no	—
179	C₂²×D₄	—	2	8	no	—
180	C₂²×D₄	C₂³×C₅₉	1	4	yes	—
181	C₂³×C₄	C₂³×C₅₉	1	2	yes	—
182	C₂³×C₄	C₂³×C₅₉	1	2	yes	—
183	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
184	#124	—	2	32	no	—
185	C₄×Q₈	—	2	32	no	—
186	#117	—	2	16	no	—
187	#117	—	2	8	no	—
188	#117	—	2	8	no	—
189	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
190	C₂×C₄⋊C₄	C₂³×C₅₉	1	8	yes	—
191	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
192	C₂×C₄⋊C₄	C₂³×C₅₉	1	16	yes	—
193	C₂×C₂²⋊C₄	—	2	16	no	—
194	C₂×C₂²⋊C₄	C₂³×C₅₉	1	16	yes	—
195	C₂×C₂²⋊C₄	C₂³×C₅₉	1	4	yes	—
196	C₂×C₂²⋊C₄	C₂³×C₅₉	1	4	yes	—
197	C₂×C₂²⋊C₄	C₂³×C₅₉	1	32	yes	—
198	C₂×C₄²	—	2	4	no	—
199	C₂×C₄²	C₂³×C₅₉	1	4	yes	—
200	C₂×C₄²	C₂³×C₅₉	1	8	yes	—
201	C₂×C₄²	C₂³×C₅₉	1	4	yes	—
202	C₂×C₄²	C₂³×C₅₉	1	4	yes	—
203	#95	C₂³×C₅₉	1	16	yes	—
204	#95	C₂³×C₅₉	1	16	yes	—
205	C₂⁴	—	2	4	no	—
206	C₂⁴	—	2	8	no	—
207	C₂⁴	—	4	8	no	—
208	C₂⁴	C₂⁴×C₅₉	1	4	yes	—
209	C₂⁴	C₂⁴×C₅₉	1	1	yes	characteristic
210	C₂×Q₈	—	2	32	no	—
211	C₂×Q₈	—	2	8	no	—
212	C₂×Q₈	—	2	8	no	—
213	C₂×Q₈	C₂×D₄×C₅₉	1	16	yes	—
214	C₂×D₄	—	2	8	no	—
215	C₂×D₄	—	2	8	no	—
216	C₂×D₄	—	2	8	no	—
217	C₂×D₄	—	4	16	no	—
218	C₂²×C₄	—	2	8	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	C₂×D₄×C₅₉	1	8	yes	—
222	C₂²×C₄	C₂×D₄×C₅₉	1	4	yes	—
223	C₂²×C₄	#41×C₅₉	1	8	yes	—
224	C₂²×C₄	C₂⁴×C₅₉	1	8	yes	—
225	C₂²×C₄	C₂⁴×C₅₉	1	4	yes	—
226	C₂²×C₄	C₂⁴×C₅₉	1	8	yes	—
227	C₂²×C₄	C₂⁴×C₅₉	1	4	yes	—
228	C₂²×C₄	C₂⁴×C₅₉	1	2	yes	—
229	C₄⋊C₄	—	2	16	no	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	4	16	no	—
233	C₂²⋊C₄	—	2	8	no	—
234	C₂²⋊C₄	—	2	8	no	—
235	C₂²⋊C₄	—	2	32	no	—
236	C₂²⋊C₄	—	4	32	no	—
237	C₄²	—	2	8	no	—
238	C₄²	—	2	16	no	—
239	C₄²	—	2	8	no	—
240	C₄²	—	2	8	no	—
241	C₂³	—	2	4	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	2	8	no	—
244	C₂³	—	2	16	no	—
245	C₂³	—	4	16	no	—
246	C₂³	—	4	8	no	—
247	C₂³	—	8	16	no	—
248	C₂³	C₂²×D₄×C₅₉	1	2	yes	—
249	C₂³	C₂×#41×C₅₉	1	4	yes	—
250	C₂³	C₂⁵×C₅₉	1	1	yes	characteristic
251	Q₈	—	2	32	no	—
252	D₄	—	4	16	no	—
253	C₂×C₄	—	2	16	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	16	no	—
259	C₂×C₄	—	2	16	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	4	8	no	—
262	C₂×C₄	#123×C₅₉	1	16	yes	—
263	C₂×C₄	C₂²×D₄×C₅₉	1	8	yes	—
264	C₂×C₄	C₂²×D₄×C₅₉	1	4	yes	—
265	C₂×C₄	C₂×#41×C₅₉	1	4	yes	—
266	C₂×C₄	C₂×#41×C₅₉	1	4	yes	—
267	C₂²	—	2	8	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	4	16	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	4	8	no	—
272	C₂²	—	8	16	no	—
273	C₂²	C₂×#123×C₅₉	1	4	yes	—
274	C₂²	C₂³×D₄×C₅₉	1	1	yes	characteristic
275	C₂²	C₂²×#41×C₅₉	1	2	yes	—
276	C₂²	C₅₉×#585	1	4	yes	—
277	C₄	—	2	8	no	—
278	C₄	—	4	8	no	—
279	C₄	—	4	4	no	—
280	C₄	—	4	8	no	—
281	C₄	C₂×#123×C₅₉	1	8	yes	—
282	C₂	—	2	4	no	—
283	C₂	—	8	4	no	—
284	C₂	C₂²×#123×C₅₉	1	1	yes	characteristic
285	C₂	C₅₉×#3487	1	2	yes	—
286	C₂	C₅₉×#3510	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₉
prime radical	C₂³×D₄×C₅₉	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₉×#519	C₂	semidirect
3. C₅₉×#2482	C₂	semidirect
4. C₅₉×#2448	C₂	semidirect
5. C₅₉×#2310	C₂	semidirect
7. C₅₉×#2296	C₂	semidirect
13. C₂×C₄×Q₈×C₅₉	C₂²	semidirect
19. C₅₉×#399	C₂²	semidirect
20. C₅₉×#397	C₂²	semidirect
21. C₅₉×#391	C₂²	semidirect
24. C₅₉×#389	C₂²	semidirect
25. C₅₉×#388	C₂²	semidirect
29. C₄×C₄⋊C₄×C₅₉	C₂²	semidirect
123. #35679	C₅₉	direct (N × H)
140. C₂×#519	C₂×C₅₉	semidirect
141. #2482	C₂×C₅₉	semidirect
142. #2448	C₂×C₅₉	semidirect
143. #2310	C₂×C₅₉	semidirect
145. #2296	C₂×C₅₉	semidirect
156. C₂×C₄×Q₈	C₂²×C₅₉	semidirect
162. #399	C₂²×C₅₉	semidirect
163. #397	C₂²×C₅₉	semidirect
164. #391	C₂²×C₅₉	semidirect
167. #389	C₂²×C₅₉	semidirect
168. #388	C₂²×C₅₉	semidirect
172. C₄×C₄⋊C₄	C₂²×C₅₉	semidirect
173. C₅₉	#35679	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#124×C₅₉	2
C₂×C₅₉×#519	2
C₅₉×#2482	2
C₅₉×#2448	2
C₅₉×#2310	2
C₅₉×#2309	2
C₅₉×#2296	2
#35679	59

Subgroups — C59×#35679 #28447788

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#35679 #28447788