Subgroups — C₅₉×#8936 #28421045

Order 15104.

356 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#2986	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#1994	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1574	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#1574	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1544	C₂	1	1	yes	maximal, characteristic
6	C₅₉×#1276	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1276	C₂	1	1	yes	maximal, characteristic
8	C₅₉×#533	C₂²	1	1	yes	characteristic
9	C₅₉×#446	—	2	1	no	—
10	C₅₉×#446	—	2	1	no	—
11	C₈×Q₈×C₅₉	—	2	1	no	—
12	C₈×Q₈×C₅₉	—	2	1	no	—
13	C₅₉×#433	—	2	2	no	—
14	C₅₉×#431	C₂²	1	1	yes	characteristic
15	C₅₉×#431	C₂²	1	1	yes	characteristic
16	C₅₉×#431	C₂²	1	1	yes	characteristic
17	C₅₉×#431	C₂²	1	1	yes	characteristic
18	C₅₉×#423	C₂²	1	1	yes	characteristic
19	C₄³×C₅₉	C₂²	1	1	yes	characteristic
20	C₅₉×#339	—	2	2	no	—
21	C₅₉×#335	—	2	1	no	—
22	C₅₉×#335	—	2	1	no	—
23	C₅₉×#334	—	2	1	no	—
24	C₅₉×#334	—	2	1	no	—
25	C₅₉×#332	C₄	1	1	yes	characteristic, complemented
26	C₅₉×#332	C₄	1	1	yes	characteristic, complemented
27	C₅₉×#332	C₄	1	1	yes	characteristic, complemented
28	C₅₉×#332	C₄	1	1	yes	characteristic, complemented
29	C₂×M₁₆×C₅₉	—	2	1	no	—
30	C₄⋊Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
32	#125×C₅₉	Q₈	1	2	yes	—
33	C₂²⋊Q₈×C₅₉	—	2	1	no	—
34	C₂²⋊Q₈×C₅₉	—	2	1	no	—
35	C₄×Q₈×C₅₉	—	2	1	no	—
36	C₄×Q₈×C₅₉	—	2	1	no	—
37	#117×C₅₉	D₄	1	2	yes	—
38	C₂×C₄²×C₅₉	—	2	1	no	—
39	C₂×C₄²×C₅₉	—	2	1	no	—
40	C₂×C₄²×C₅₉	—	2	1	no	—
41	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₅₉	—	2	2	no	—
43	C₄⋊C₈×C₅₉	—	2	2	no	—
44	C₄⋊C₈×C₅₉	—	2	1	no	—
45	C₄⋊C₈×C₅₉	—	2	1	no	—
46	C₄⋊C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
47	C₄⋊C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
51	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
52	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
53	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
54	C₂²⋊C₈×C₅₉	—	2	1	no	—
55	C₂²⋊C₈×C₅₉	—	2	1	no	—
56	C₂²⋊C₈×C₅₉	—	2	1	no	—
57	C₂²⋊C₈×C₅₉	—	2	1	no	—
58	#97×C₅₉	—	2	2	no	—
59	C₄×C₈×C₅₉	—	2	2	no	—
60	C₄×C₈×C₅₉	—	2	1	no	—
61	C₄×C₈×C₅₉	—	2	1	no	—
62	C₄×C₈×C₅₉	—	2	1	no	—
63	C₄×C₈×C₅₉	—	2	1	no	—
64	C₂×Q₈×C₅₉	—	2	1	no	—
65	C₂×Q₈×C₅₉	—	2	1	no	—
66	C₂²×C₄×C₅₉	—	4	1	no	—
67	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
70	M₁₆×C₅₉	—	4	2	no	—
71	C₂×C₈×C₅₉	—	2	1	no	—
72	C₂×C₈×C₅₉	—	2	1	no	—
73	C₂×C₈×C₅₉	—	2	1	no	—
74	C₂×C₈×C₅₉	—	2	1	no	—
75	C₂×C₈×C₅₉	—	2	1	no	—
76	C₂×C₈×C₅₉	—	2	1	no	—
77	C₂×C₈×C₅₉	—	2	1	no	—
78	C₂×C₈×C₅₉	—	2	1	no	—
79	C₂×C₈×C₅₉	—	4	1	no	—
80	C₄⋊C₄×C₅₉	—	2	1	no	—
81	C₄⋊C₄×C₅₉	—	2	1	no	—
82	C₄⋊C₄×C₅₉	—	2	2	no	—
83	C₄⋊C₄×C₅₉	—	2	2	no	—
84	C₄⋊C₄×C₅₉	C₄⋊C₄	1	2	yes	—
85	C₄⋊C₄×C₅₉	C₄⋊C₄	1	2	yes	—
86	C₂²⋊C₄×C₅₉	—	2	2	no	—
87	C₄²×C₅₉	—	2	2	no	—
88	C₄²×C₅₉	—	2	2	no	—
89	C₄²×C₅₉	—	2	2	no	—
90	C₄²×C₅₉	—	2	2	no	—
91	C₄²×C₅₉	—	4	1	no	—
92	C₄²×C₅₉	—	4	1	no	—
93	C₄²×C₅₉	—	4	1	no	—
94	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
98	C₂³×C₅₉	#123	1	1	yes	characteristic
99	Q₈×C₅₉	—	4	1	no	—
100	Q₈×C₅₉	—	4	1	no	—
101	Q₈×C₅₉	—	4	1	no	—
102	Q₈×C₅₉	—	4	1	no	—
103	C₂×C₄×C₅₉	—	2	1	no	—
104	C₂×C₄×C₅₉	—	2	1	no	—
105	C₂×C₄×C₅₉	—	2	1	no	—
106	C₂×C₄×C₅₉	—	2	1	no	—
107	C₂×C₄×C₅₉	—	2	1	no	—
108	C₂×C₄×C₅₉	—	2	1	no	—
109	C₂×C₄×C₅₉	—	2	2	no	—
110	C₂×C₄×C₅₉	—	2	2	no	—
111	C₂×C₄×C₅₉	—	4	2	no	—
112	C₂×C₄×C₅₉	—	4	2	no	—
113	C₂×C₄×C₅₉	—	4	2	no	—
114	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₈×C₅₉	—	4	1	no	—
121	C₈×C₅₉	—	4	1	no	—
122	C₈×C₅₉	—	4	1	no	—
123	C₈×C₅₉	—	4	1	no	—
124	C₈×C₅₉	—	4	1	no	—
125	C₈×C₅₉	—	4	1	no	—
126	C₈×C₅₉	—	4	1	no	—
127	C₈×C₅₉	—	4	1	no	—
128	C₈×C₅₉	—	8	1	no	—
129	#8936	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₅₉	—	2	1	no	—
131	C₂²×C₅₉	—	2	1	no	—
132	C₂²×C₅₉	—	2	1	no	—
133	C₂²×C₅₉	#385	1	1	yes	characteristic
134	C₄×C₅₉	—	2	1	no	—
135	C₄×C₅₉	—	2	1	no	—
136	C₄×C₅₉	—	2	1	no	—
137	C₄×C₅₉	—	4	4	no	—
138	C₄×C₅₉	—	4	2	no	—
139	C₄×C₅₉	—	4	2	no	—
140	C₄×C₅₉	#443	1	1	yes	characteristic
141	C₄×C₅₉	#443	1	1	yes	characteristic
142	C₄×C₅₉	#444	1	1	yes	characteristic
143	C₄×C₅₉	#444	1	1	yes	characteristic
144	C₄×C₅₉	#454	1	1	yes	characteristic
145	C₄×C₅₉	#456	1	1	yes	characteristic
146	#2986	C₂×C₅₉	1	1	yes	characteristic
147	#1994	C₂×C₅₉	1	1	yes	characteristic
148	#1574	C₂×C₅₉	1	1	yes	characteristic
149	#1574	C₂×C₅₉	1	1	yes	characteristic
150	#1544	C₂×C₅₉	1	1	yes	characteristic
151	#1276	C₂×C₅₉	1	1	yes	characteristic
152	#1276	C₂×C₅₉	1	1	yes	characteristic
153	C₂×C₅₉	—	4	1	no	—
154	C₂×C₅₉	#1857	1	1	yes	characteristic
155	C₂×C₅₉	#1866	1	1	yes	characteristic
156	C₂×C₅₉	#1866	1	1	yes	characteristic
157	#533	C₂²×C₅₉	1	1	yes	characteristic
158	#446	—	2	1	no	—
159	#446	—	2	1	no	—
160	C₈×Q₈	—	2	1	no	—
161	C₈×Q₈	—	2	1	no	—
162	#433	—	2	2	no	—
163	#431	C₂²×C₅₉	1	1	yes	characteristic
164	#431	C₂²×C₅₉	1	1	yes	characteristic
165	#431	C₂²×C₅₉	1	1	yes	characteristic
166	#431	C₂²×C₅₉	1	1	yes	characteristic
167	#423	C₂²×C₅₉	1	1	yes	characteristic
168	C₄³	C₂²×C₅₉	1	1	yes	characteristic
169	#339	—	2	2	no	—
170	#335	—	2	1	no	—
171	#335	—	2	1	no	—
172	#334	—	2	1	no	—
173	#334	—	2	1	no	—
174	#332	C₄×C₅₉	1	1	yes	characteristic, complemented
175	#332	C₄×C₅₉	1	1	yes	characteristic, complemented
176	#332	C₄×C₅₉	1	1	yes	characteristic, complemented
177	#332	C₄×C₅₉	1	1	yes	characteristic, complemented
178	C₅₉	#8936	1	1	yes	characteristic, direct product factor, Sylow
179	C₂×M₁₆	—	2	1	no	—
180	C₄⋊Q₈	C₂×C₄×C₅₉	1	1	yes	characteristic
181	C₄⋊Q₈	C₂×C₄×C₅₉	1	1	yes	characteristic
182	#125	Q₈×C₅₉	1	2	yes	—
183	C₂²⋊Q₈	—	2	1	no	—
184	C₂²⋊Q₈	—	2	1	no	—
185	C₄×Q₈	—	2	1	no	—
186	C₄×Q₈	—	2	1	no	—
187	#117	D₄×C₅₉	1	2	yes	—
188	C₂×C₄²	—	2	1	no	—
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
197	C₄⋊C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
198	C₄⋊C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
199	C₄⋊C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
200	C₄⋊C₈	D₄×C₅₉	1	1	yes	characteristic
201	C₄⋊C₈	D₄×C₅₉	1	1	yes	characteristic
202	C₄⋊C₈	D₄×C₅₉	1	1	yes	characteristic
203	C₄⋊C₈	D₄×C₅₉	1	1	yes	characteristic
204	C₂²⋊C₈	—	2	1	no	—
205	C₂²⋊C₈	—	2	1	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	C₂²⋊C₈	—	2	1	no	—
208	#97	—	2	2	no	—
209	C₄×C₈	—	2	2	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	2	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
218	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
219	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
220	M₁₆	—	4	2	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	C₄⋊C₄×C₅₉	1	2	yes	—
235	C₄⋊C₄	C₄⋊C₄×C₅₉	1	2	yes	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	C₂²×C₄×C₅₉	1	1	yes	characteristic
245	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
246	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
247	C₄²	C₂×Q₈×C₅₉	1	1	yes	characteristic
248	C₂³	#123×C₅₉	1	1	yes	characteristic
249	Q₈	—	4	1	no	—
250	Q₈	—	4	1	no	—
251	Q₈	—	4	1	no	—
252	Q₈	—	4	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	C₂×C₄⋊C₄×C₅₉	1	1	yes	characteristic
265	C₂×C₄	C₄×D₄×C₅₉	1	1	yes	characteristic
266	C₂×C₄	C₄×D₄×C₅₉	1	1	yes	characteristic
267	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
268	C₂×C₄	C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
269	C₂×C₄	C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
270	C₈	—	4	1	no	—
271	C₈	—	4	1	no	—
272	C₈	—	4	1	no	—
273	C₈	—	4	1	no	—
274	C₈	—	4	1	no	—
275	C₈	—	4	1	no	—
276	C₈	—	4	1	no	—
277	C₈	—	4	1	no	—
278	C₈	—	8	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	C₅₉×#385	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	C₅₉×#443	1	1	yes	characteristic
290	C₄	C₅₉×#443	1	1	yes	characteristic
291	C₄	C₅₉×#444	1	1	yes	characteristic
292	C₄	C₅₉×#444	1	1	yes	characteristic
293	C₄	C₅₉×#454	1	1	yes	characteristic
294	C₄	C₅₉×#456	1	1	yes	characteristic
295	C₂	—	4	1	no	—
296	C₂	C₅₉×#1857	1	1	yes	characteristic
297	C₂	C₅₉×#1866	1	1	yes	characteristic
298	C₂	C₅₉×#1866	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₉	#120
commutator	C₄²	C₂²×C₄×C₅₉
Frattini, cosocle	C₂×C₄²	C₂³×C₅₉
socle	C₂²×C₅₉	#385
prime radical	C₂³×C₅₉	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
25. C₅₉×#332	C₄	semidirect
26. C₅₉×#332	C₄	semidirect
27. C₅₉×#332	C₄	semidirect
28. C₅₉×#332	C₄	semidirect
129. #8936	C₅₉	direct (N × H)
174. #332	C₄×C₅₉	semidirect
175. #332	C₄×C₅₉	semidirect
176. #332	C₄×C₅₉	semidirect
177. #332	C₄×C₅₉	semidirect
178. C₅₉	#8936	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×#2986	2
C₅₉×#1994	2
C₅₉×#1574	2
C₅₉×#1544	2
C₅₉×#1276	2
#8936	59

Subgroups — C59×#8936 #28421045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#8936 #28421045