Subgroups — C₅₉×#33687 #28445796

Order 15104.

520 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3177	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#2950	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2237	C₂	1	2	yes	maximal
4	C₅₉×#2229	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2209	C₂	1	2	yes	maximal, complemented
6	C₅₉×#2188	C₂	1	2	yes	maximal
7	C₅₉×#2176	C₂	1	2	yes	maximal, complemented
8	C₄×Q₃₂×C₅₉	C₂	1	2	yes	maximal, complemented
9	C₅₉×#2164	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
11	C₅₉×#541	—	2	1	no	—
12	C₅₉×#519	—	2	1	no	—
13	C₂×Q₃₂×C₅₉	C₂²	1	2	yes	—
14	C₂×Q₃₂×C₅₉	C₂²	1	2	yes	—
15	C₅₉×#499	C₂²	1	1	yes	characteristic
16	C₅₉×#494	C₂²	1	1	yes	characteristic
17	C₅₉×#491	—	2	1	no	—
18	C₅₉×#470	—	2	1	no	—
19	C₅₉×#467	C₂²	1	2	yes	—
20	C₅₉×#441	—	2	1	no	—
21	C₅₉×#441	—	2	1	no	—
22	C₅₉×#441	—	2	1	no	—
23	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
26	C₅₉×#427	C₂²	1	1	yes	characteristic
27	C₅₉×#419	—	2	1	no	—
28	C₅₉×#405	C₂²	1	1	yes	characteristic
29	C₅₉×#367	C₂²	1	2	yes	—
30	C₅₉×#366	C₂²	1	2	yes	—
31	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
32	C₅₉×#358	C₂²	1	2	yes	—
33	C₅₉×#358	C₂²	1	2	yes	—
34	C₅₉×#358	C₂²	1	2	yes	—
35	C₅₉×#358	C₂²	1	2	yes	—
36	C₅₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
37	C₅₉×#346	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₅₉	—	2	1	no	—
40	C₂×Q₁₆×C₅₉	—	2	4	no	—
41	C₂×Q₁₆×C₅₉	—	2	1	no	—
42	C₂×Q₁₆×C₅₉	—	2	1	no	—
43	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
44	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
45	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₅₉	—	2	1	no	—
48	#126×C₅₉	—	4	2	no	—
49	#125×C₅₉	—	2	1	no	—
50	#125×C₅₉	—	4	2	no	—
51	C₂²⋊Q₈×C₅₉	—	2	2	no	—
52	C₄×Q₈×C₅₉	—	2	2	no	—
53	C₄×Q₈×C₅₉	—	2	2	no	—
54	C₄×Q₈×C₅₉	—	2	2	no	—
55	C₄×Q₈×C₅₉	—	4	2	no	—
56	#117×C₅₉	—	2	1	no	—
57	#117×C₅₉	—	4	2	no	—
58	#117×C₅₉	D₄	1	1	yes	characteristic
59	Q₃₂×C₅₉	—	2	4	no	—
60	Q₃₂×C₅₉	—	2	4	no	—
61	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
62	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
63	#107×C₅₉	C₂³	1	2	yes	—
64	#107×C₅₉	C₂³	1	2	yes	—
65	#106×C₅₉	—	2	1	no	—
66	#106×C₅₉	—	2	1	no	—
67	#103×C₅₉	—	2	2	no	—
68	#103×C₅₉	—	2	2	no	—
69	#103×C₅₉	—	2	2	no	—
70	#103×C₅₉	—	2	2	no	—
71	#97×C₅₉	D₄	1	1	yes	characteristic
72	#97×C₅₉	D₄	1	1	yes	characteristic
73	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₅₉	—	2	4	no	—
76	C₂×Q₈×C₅₉	—	2	2	no	—
77	C₂×Q₈×C₅₉	—	2	2	no	—
78	C₂×Q₈×C₅₉	—	4	2	no	—
79	C₂×Q₈×C₅₉	—	4	1	no	—
80	C₂²×C₄×C₅₉	—	4	1	no	—
81	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
82	Q₁₆×C₅₉	—	2	4	no	—
83	Q₁₆×C₅₉	—	2	2	no	—
84	Q₁₆×C₅₉	—	2	2	no	—
85	Q₁₆×C₅₉	—	2	2	no	—
86	Q₁₆×C₅₉	—	4	2	no	—
87	Q₁₆×C₅₉	#41	1	4	yes	—
88	M₁₆×C₅₉	—	4	1	no	—
89	C₂×C₈×C₅₉	—	2	2	no	—
90	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₅₉	—	2	2	no	—
95	C₄⋊C₄×C₅₉	—	2	2	no	—
96	C₄⋊C₄×C₅₉	—	4	2	no	—
97	C₄⋊C₄×C₅₉	—	4	2	no	—
98	C₄⋊C₄×C₅₉	—	4	2	no	—
99	C₄⋊C₄×C₅₉	—	4	2	no	—
100	C₄⋊C₄×C₅₉	—	4	2	no	—
101	C₄⋊C₄×C₅₉	—	4	2	no	—
102	C₄⋊C₄×C₅₉	D₈	1	2	yes	—
103	C₂²⋊C₄×C₅₉	—	4	1	no	—
104	C₂²⋊C₄×C₅₉	—	4	2	no	—
105	C₂²⋊C₄×C₅₉	D₈	1	2	yes	—
106	C₄²×C₅₉	—	4	1	no	—
107	C₄²×C₅₉	—	4	2	no	—
108	C₄²×C₅₉	—	4	2	no	—
109	C₄²×C₅₉	—	4	2	no	—
110	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₅₉	—	2	2	no	—
113	C₁₆×C₅₉	—	2	2	no	—
114	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
115	Q₈×C₅₉	—	2	4	no	—
116	Q₈×C₅₉	—	4	4	no	—
117	Q₈×C₅₉	—	4	2	no	—
118	Q₈×C₅₉	—	4	2	no	—
119	Q₈×C₅₉	—	4	2	no	—
120	C₂×C₄×C₅₉	—	2	1	no	—
121	C₂×C₄×C₅₉	—	2	1	no	—
122	C₂×C₄×C₅₉	—	4	4	no	—
123	C₂×C₄×C₅₉	—	4	2	no	—
124	C₂×C₄×C₅₉	—	4	2	no	—
125	C₂×C₄×C₅₉	—	4	2	no	—
126	C₂×C₄×C₅₉	—	4	2	no	—
127	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
131	C₈×C₅₉	—	2	1	no	—
132	C₈×C₅₉	—	2	1	no	—
133	C₈×C₅₉	—	2	1	no	—
134	C₈×C₅₉	C₂×#41	1	2	yes	—
135	#33687	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₅₉	—	2	1	no	—
137	C₂²×C₅₉	—	2	2	no	—
138	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
139	C₄×C₅₉	—	2	1	no	—
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	2	no	—
145	C₄×C₅₉	—	8	2	no	—
146	C₄×C₅₉	#525	1	1	yes	characteristic
147	C₄×C₅₉	#575	1	1	yes	characteristic
148	#3177	C₂×C₅₉	1	1	yes	characteristic
149	#2950	C₂×C₅₉	1	1	yes	characteristic
150	#2237	C₂×C₅₉	1	2	yes	—
151	#2229	C₂×C₅₉	1	2	yes	complemented
152	#2209	C₂×C₅₉	1	2	yes	complemented
153	#2188	C₂×C₅₉	1	2	yes	—
154	#2176	C₂×C₅₉	1	2	yes	complemented
155	C₄×Q₃₂	C₂×C₅₉	1	2	yes	complemented
156	#2164	C₂×C₅₉	1	1	yes	characteristic
157	C₂×C₅₉	—	4	1	no	—
158	C₂×C₅₉	#3100	1	1	yes	characteristic
159	C₂×C₅₉	#3417	1	2	yes	—
160	C₂²×Q₁₆	C₂²×C₅₉	1	1	yes	characteristic
161	#541	—	2	1	no	—
162	#519	—	2	1	no	—
163	C₂×Q₃₂	C₂²×C₅₉	1	2	yes	—
164	C₂×Q₃₂	C₂²×C₅₉	1	2	yes	—
165	#499	C₂²×C₅₉	1	1	yes	characteristic
166	#494	C₂²×C₅₉	1	1	yes	characteristic
167	#491	—	2	1	no	—
168	#470	—	2	1	no	—
169	#467	C₂²×C₅₉	1	2	yes	—
170	#441	—	2	1	no	—
171	#441	—	2	1	no	—
172	#441	—	2	1	no	—
173	C₄×Q₁₆	C₂²×C₅₉	1	2	yes	—
174	C₄×Q₁₆	C₂²×C₅₉	1	2	yes	—
175	C₄×Q₁₆	C₂²×C₅₉	1	2	yes	—
176	#427	C₂²×C₅₉	1	1	yes	characteristic
177	#419	—	2	1	no	—
178	#405	C₂²×C₅₉	1	1	yes	characteristic
179	#367	C₂²×C₅₉	1	2	yes	—
180	#366	C₂²×C₅₉	1	2	yes	—
181	C₄⋊C₁₆	C₂²×C₅₉	1	2	yes	—
182	#358	C₂²×C₅₉	1	2	yes	—
183	#358	C₂²×C₅₉	1	2	yes	—
184	#358	C₂²×C₅₉	1	2	yes	—
185	#358	C₂²×C₅₉	1	2	yes	—
186	C₂²⋊C₁₆	C₂²×C₅₉	1	2	yes	—
187	#346	C₂²×C₅₉	1	1	yes	characteristic
188	C₄×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
189	C₅₉	#33687	1	1	yes	characteristic, direct product factor, Sylow
190	C₂²×Q₈	—	2	1	no	—
191	C₂×Q₁₆	—	2	4	no	—
192	C₂×Q₁₆	—	2	1	no	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	C₂³×C₅₉	1	2	yes	—
195	C₂×Q₁₆	C₂³×C₅₉	1	2	yes	—
196	C₂×M₁₆	D₄×C₅₉	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
198	C₄⋊Q₈	—	2	1	no	—
199	#126	—	4	2	no	—
200	#125	—	2	1	no	—
201	#125	—	4	2	no	—
202	C₂²⋊Q₈	—	2	2	no	—
203	C₄×Q₈	—	2	2	no	—
204	C₄×Q₈	—	2	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	—	4	2	no	—
207	#117	—	2	1	no	—
208	#117	—	4	2	no	—
209	#117	D₄×C₅₉	1	1	yes	characteristic
210	Q₃₂	—	2	4	no	—
211	Q₃₂	—	2	4	no	—
212	C₂×C₁₆	C₂³×C₅₉	1	2	yes	—
213	C₂×C₁₆	C₂³×C₅₉	1	2	yes	—
214	#107	C₂³×C₅₉	1	2	yes	—
215	#107	C₂³×C₅₉	1	2	yes	—
216	#106	—	2	1	no	—
217	#106	—	2	1	no	—
218	#103	—	2	2	no	—
219	#103	—	2	2	no	—
220	#103	—	2	2	no	—
221	#103	—	2	2	no	—
222	#97	D₄×C₅₉	1	1	yes	characteristic
223	#97	D₄×C₅₉	1	1	yes	characteristic
224	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
225	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
226	C₂×Q₈	—	2	4	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
233	Q₁₆	—	2	4	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	2	2	no	—
237	Q₁₆	—	4	2	no	—
238	Q₁₆	#41×C₅₉	1	4	yes	—
239	M₁₆	—	4	1	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
242	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
243	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
244	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	D₈×C₅₉	1	2	yes	—
254	C₂²⋊C₄	—	4	1	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	D₈×C₅₉	1	2	yes	—
257	C₄²	—	4	1	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	—	4	2	no	—
261	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
262	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
263	C₁₆	—	2	2	no	—
264	C₁₆	—	2	2	no	—
265	C₂³	C₂×D₈×C₅₉	1	1	yes	characteristic
266	Q₈	—	2	4	no	—
267	Q₈	—	4	4	no	—
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
279	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
280	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	C₂×#41×C₅₉	1	2	yes	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	2	no	—
288	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	4	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	C₅₉×#525	1	1	yes	characteristic
297	C₄	C₅₉×#575	1	1	yes	characteristic
298	C₂	—	4	1	no	—
299	C₂	C₅₉×#3100	1	1	yes	characteristic
300	C₂	C₅₉×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
prime radical	C₂³×C₅₉	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ #117×C₅₉ ◁ C₅₉×#405 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₉×#2229	C₂	semidirect
5. C₅₉×#2209	C₂	semidirect
7. C₅₉×#2176	C₂	semidirect
8. C₄×Q₃₂×C₅₉	C₂	semidirect
135. #33687	C₅₉	direct (N × H)
151. #2229	C₂×C₅₉	semidirect
152. #2209	C₂×C₅₉	semidirect
154. #2176	C₂×C₅₉	semidirect
155. C₄×Q₃₂	C₂×C₅₉	semidirect
189. C₅₉	#33687	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₉×#3177	2
C₅₉×#2950	2
C₅₉×#2237	2
C₅₉×#2229	2
C₅₉×#2209	2
C₅₉×#2188	2
C₅₉×#2176	2
C₄×Q₃₂×C₅₉	2
C₅₉×#2164	2
#33687	59

Subgroups — C59×#33687 #28445796

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33687 #28445796