Subgroups — C₅₉×#22530 #28434639

Order 15104.

1428 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	C₂×C₄×#41×C₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₅₉	C₂	1	2	yes	maximal, complemented
3	C₅₉×#1954	C₂	1	6	yes	maximal, complemented
4	C₅₉×#1797	C₂	1	3	yes	maximal, complemented
5	C₄×C₂²⋊C₈×C₅₉	C₂	1	3	yes	maximal, complemented
6	C₂²×#41×C₅₉	C₄	1	1	yes	characteristic
7	C₄×#41×C₅₉	—	2	8	no	—
8	C₂×C₄×Q₈×C₅₉	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₅₉	C₄	1	3	yes	—
11	C₂×C₄×D₄×C₅₉	C₂²	1	3	yes	—
12	C₂×#117×C₅₉	C₄	1	3	yes	—
13	C₂²×C₄²×C₅₉	C₂²	1	3	yes	—
14	C₈×Q₈×C₅₉	—	2	8	no	—
15	C₂×C₄⋊C₈×C₅₉	C₂²	1	6	yes	—
16	C₂×C₂²⋊C₈×C₅₉	C₂²	1	6	yes	—
17	C₂×C₂²⋊C₈×C₅₉	C₂²	1	6	yes	—
18	C₂×C₄×C₈×C₅₉	C₂²	1	6	yes	—
19	C₅₉×#336	C₂²	1	1	yes	characteristic
20	C₅₉×#336	C₂²	1	3	yes	—
21	C₂×#41×C₅₉	—	2	4	no	—
22	C₂×#41×C₅₉	—	2	4	no	—
23	C₂×#41×C₅₉	C₈	1	8	yes	complemented
24	C₂²×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×D₄×C₅₉	C₂×C₄	1	3	yes	—
26	C₂³×C₄×C₅₉	C₂×C₄	1	3	yes	—
27	C₂³×C₄×C₅₉	C₂×C₄	1	3	yes	—
28	C₂³×C₄×C₅₉	C₂³	1	3	yes	—
29	C₂²×C₈×C₅₉	C₂³	1	6	yes	—
30	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
31	C₄×Q₈×C₅₉	—	2	4	no	—
32	C₄×Q₈×C₅₉	D₄	1	8	yes	—
33	C₄×D₄×C₅₉	—	2	12	no	—
34	C₄×D₄×C₅₉	—	2	12	no	—
35	#117×C₅₉	—	2	12	no	—
36	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	3	yes	—
37	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	3	yes	—
38	C₂×C₂²⋊C₄×C₅₉	C₂×C₄	1	6	yes	—
39	C₂×C₄²×C₅₉	—	2	12	no	—
40	C₂×C₄²×C₅₉	C₂×C₄	1	3	yes	—
41	C₂×C₄²×C₅₉	C₂×C₄	1	3	yes	—
42	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅₉	C₂³	1	3	yes	—
44	C₄⋊C₈×C₅₉	—	2	12	no	—
45	C₂²⋊C₈×C₅₉	—	2	12	no	—
46	C₂²⋊C₈×C₅₉	—	4	6	no	—
47	C₄×C₈×C₅₉	—	2	12	no	—
48	C₂⁴×C₅₉	C₂²×C₄	1	3	yes	—
49	#41×C₅₉	—	2	32	no	—
50	C₂×Q₈×C₅₉	C₂²⋊C₄	1	4	yes	—
51	C₂×Q₈×C₅₉	C₂×C₈	1	4	yes	—
52	C₂×Q₈×C₅₉	M₁₆	1	4	yes	—
53	C₂×D₄×C₅₉	—	2	6	no	—
54	C₂×D₄×C₅₉	—	2	6	no	—
55	C₂×D₄×C₅₉	C₂×C₈	1	12	yes	—
56	C₂²×C₄×C₅₉	—	2	6	no	—
57	C₂²×C₄×C₅₉	—	2	6	no	—
58	C₂²×C₄×C₅₉	—	2	6	no	—
59	C₂²×C₄×C₅₉	—	2	6	no	—
60	C₂²×C₄×C₅₉	—	2	6	no	—
61	C₂²×C₄×C₅₉	—	2	6	no	—
62	C₂²×C₄×C₅₉	—	2	3	no	—
63	C₂²×C₄×C₅₉	—	2	3	no	—
64	C₂²×C₄×C₅₉	—	4	3	no	—
65	C₂²×C₄×C₅₉	C₂×C₈	1	12	yes	—
66	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
71	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₅₉	—	2	2	no	—
73	C₂×C₈×C₅₉	—	2	2	no	—
74	C₂×C₈×C₅₉	—	2	6	no	—
75	C₂×C₈×C₅₉	—	2	6	no	—
76	C₂×C₈×C₅₉	—	2	6	no	—
77	C₂×C₈×C₅₉	#41	1	4	yes	complemented
78	C₄⋊C₄×C₅₉	—	2	6	no	—
79	C₄⋊C₄×C₅₉	C₂²⋊C₄	1	12	yes	—
80	C₂²⋊C₄×C₅₉	—	2	12	no	—
81	C₄²×C₅₉	—	2	6	no	—
82	C₄²×C₅₉	—	2	6	no	—
83	C₄²×C₅₉	—	2	2	no	—
84	C₄²×C₅₉	C₂×D₄	1	12	yes	—
85	C₂³×C₅₉	—	2	3	no	—
86	C₂³×C₅₉	—	2	3	no	—
87	C₂³×C₅₉	—	2	3	no	—
88	C₂³×C₅₉	—	2	3	no	—
89	C₂³×C₅₉	—	2	3	no	—
90	C₂³×C₅₉	—	2	3	no	—
91	C₂³×C₅₉	C₂²×C₈	1	6	yes	—
92	C₂³×C₅₉	C₂³×C₄	1	1	yes	characteristic
93	Q₈×C₅₉	C₂²⋊C₈	1	16	yes	complemented
94	D₄×C₅₉	—	2	24	no	—
95	C₂×C₄×C₅₉	—	2	1	no	—
96	C₂×C₄×C₅₉	—	2	1	no	—
97	C₂×C₄×C₅₉	—	2	1	no	—
98	C₂×C₄×C₅₉	—	2	3	no	—
99	C₂×C₄×C₅₉	—	2	1	no	—
100	C₂×C₄×C₅₉	—	2	3	no	—
101	C₂×C₄×C₅₉	—	2	24	no	—
102	C₂×C₄×C₅₉	—	2	3	no	—
103	C₂×C₄×C₅₉	—	2	3	no	—
104	C₂×C₄×C₅₉	—	2	3	no	—
105	C₂×C₄×C₅₉	—	4	12	no	—
106	C₂×C₄×C₅₉	—	4	6	no	—
107	C₂×C₄×C₅₉	—	4	6	no	—
108	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	6	yes	—
109	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	6	yes	—
110	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₅₉	C₂²×C₈	1	6	yes	—
112	C₂×C₄×C₅₉	C₂×M₁₆	1	6	yes	—
113	C₂×C₄×C₅₉	#131	1	2	yes	—
114	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
115	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
116	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
118	C₈×C₅₉	—	2	4	no	—
119	C₈×C₅₉	—	4	6	no	—
120	#22530	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₅₉	—	2	6	no	—
122	C₂²×C₅₉	—	2	6	no	—
123	C₂²×C₅₉	—	2	6	no	—
124	C₂²×C₅₉	—	4	3	no	—
125	C₂²×C₅₉	—	4	3	no	—
126	C₂²×C₅₉	—	4	3	no	—
127	C₂²×C₅₉	—	4	3	no	—
128	C₂²×C₅₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₅₉	C₄×#41	1	1	yes	characteristic
130	C₂²×C₅₉	#520	1	1	yes	characteristic
131	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
132	C₂²×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
133	C₂²×C₅₉	C₂×#131	1	1	yes	characteristic
134	C₂²×C₅₉	#568	1	1	yes	characteristic
135	C₄×C₅₉	—	2	2	no	—
136	C₄×C₅₉	—	2	2	no	—
137	C₄×C₅₉	—	2	6	no	—
138	C₄×C₅₉	—	4	3	no	—
139	C₄×C₅₉	C₂×C₂²⋊C₈	1	12	yes	—
140	C₄×C₅₉	#548	1	2	yes	—
141	C₄×C₅₉	#550	1	2	yes	—
142	C₂×C₄×#41	C₂×C₅₉	1	1	yes	characteristic
143	C₂×C₈×Q₈	C₂×C₅₉	1	2	yes	complemented
144	#1954	C₂×C₅₉	1	6	yes	complemented
145	#1797	C₂×C₅₉	1	3	yes	complemented
146	C₄×C₂²⋊C₈	C₂×C₅₉	1	3	yes	complemented
147	C₂×C₅₉	—	4	6	no	—
148	C₂×C₅₉	#2341	1	1	yes	characteristic
149	C₂×C₅₉	C₂²×C₂²⋊C₈	1	1	yes	characteristic
150	C₂×C₅₉	#2880	1	1	yes	characteristic
151	C₂×C₅₉	C₈×#41	1	1	yes	characteristic
152	C₂×C₅₉	#2971	1	1	yes	characteristic
153	C₂×C₅₉	#2988	1	1	yes	characteristic
154	C₂×C₅₉	#2991	1	1	yes	characteristic
155	C₂²×#41	C₄×C₅₉	1	1	yes	characteristic
156	C₄×#41	—	2	8	no	—
157	C₂×C₄×Q₈	C₄×C₅₉	1	1	yes	characteristic
158	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
159	C₂×C₄×D₄	C₄×C₅₉	1	3	yes	—
160	C₂×C₄×D₄	C₂²×C₅₉	1	3	yes	—
161	C₂×#117	C₄×C₅₉	1	3	yes	—
162	C₂²×C₄²	C₂²×C₅₉	1	3	yes	—
163	C₈×Q₈	—	2	8	no	—
164	C₂×C₄⋊C₈	C₂²×C₅₉	1	6	yes	—
165	C₂×C₂²⋊C₈	C₂²×C₅₉	1	6	yes	—
166	C₂×C₂²⋊C₈	C₂²×C₅₉	1	6	yes	—
167	C₂×C₄×C₈	C₂²×C₅₉	1	6	yes	—
168	#336	C₂²×C₅₉	1	3	yes	—
169	#336	C₂²×C₅₉	1	1	yes	characteristic
170	C₅₉	#22530	1	1	yes	characteristic, direct product factor, Sylow
171	C₂×#41	—	2	4	no	—
172	C₂×#41	—	2	4	no	—
173	C₂×#41	C₈×C₅₉	1	8	yes	complemented
174	C₂²×Q₈	C₂×C₄×C₅₉	1	1	yes	characteristic
175	C₂²×D₄	C₂×C₄×C₅₉	1	3	yes	—
176	C₂³×C₄	C₂×C₄×C₅₉	1	3	yes	—
177	C₂³×C₄	C₂×C₄×C₅₉	1	3	yes	—
178	C₂³×C₄	C₂³×C₅₉	1	3	yes	—
179	C₂²×C₈	C₂³×C₅₉	1	6	yes	—
180	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
181	C₄×Q₈	—	2	4	no	—
182	C₄×Q₈	D₄×C₅₉	1	8	yes	—
183	C₄×D₄	—	2	12	no	—
184	C₄×D₄	—	2	12	no	—
185	#117	—	2	12	no	—
186	C₂×C₄⋊C₄	C₂×C₄×C₅₉	1	3	yes	—
187	C₂×C₄⋊C₄	C₂×C₄×C₅₉	1	3	yes	—
188	C₂×C₂²⋊C₄	C₂×C₄×C₅₉	1	6	yes	—
189	C₂×C₄²	—	2	12	no	—
190	C₂×C₄²	C₂×C₄×C₅₉	1	3	yes	—
191	C₂×C₄²	C₂×C₄×C₅₉	1	3	yes	—
192	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₅₉	1	3	yes	—
194	C₄⋊C₈	—	2	12	no	—
195	C₂²⋊C₈	—	2	12	no	—
196	C₂²⋊C₈	—	4	6	no	—
197	C₄×C₈	—	2	12	no	—
198	C₂⁴	C₂²×C₄×C₅₉	1	3	yes	—
199	#41	—	2	32	no	—
200	C₂×Q₈	C₂²⋊C₄×C₅₉	1	4	yes	—
201	C₂×Q₈	C₂×C₈×C₅₉	1	4	yes	—
202	C₂×Q₈	M₁₆×C₅₉	1	4	yes	—
203	C₂×D₄	—	2	6	no	—
204	C₂×D₄	—	2	6	no	—
205	C₂×D₄	C₂×C₈×C₅₉	1	12	yes	—
206	C₂²×C₄	—	2	6	no	—
207	C₂²×C₄	—	2	6	no	—
208	C₂²×C₄	—	2	6	no	—
209	C₂²×C₄	—	2	6	no	—
210	C₂²×C₄	—	2	6	no	—
211	C₂²×C₄	—	2	6	no	—
212	C₂²×C₄	—	2	3	no	—
213	C₂²×C₄	—	2	3	no	—
214	C₂²×C₄	—	4	3	no	—
215	C₂²×C₄	C₂×C₈×C₅₉	1	12	yes	—
216	C₂²×C₄	C₂²×C₄×C₅₉	1	3	yes	—
217	C₂²×C₄	C₂²×C₄×C₅₉	1	3	yes	—
218	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₅₉	1	3	yes	—
221	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	6	no	—
225	C₂×C₈	—	2	6	no	—
226	C₂×C₈	—	2	6	no	—
227	C₂×C₈	#41×C₅₉	1	4	yes	complemented
228	C₄⋊C₄	—	2	6	no	—
229	C₄⋊C₄	C₂²⋊C₄×C₅₉	1	12	yes	—
230	C₂²⋊C₄	—	2	12	no	—
231	C₄²	—	2	6	no	—
232	C₄²	—	2	6	no	—
233	C₄²	—	2	2	no	—
234	C₄²	C₂×D₄×C₅₉	1	12	yes	—
235	C₂³	—	2	3	no	—
236	C₂³	—	2	3	no	—
237	C₂³	—	2	3	no	—
238	C₂³	—	2	3	no	—
239	C₂³	—	2	3	no	—
240	C₂³	—	2	3	no	—
241	C₂³	C₂²×C₈×C₅₉	1	6	yes	—
242	C₂³	C₂³×C₄×C₅₉	1	1	yes	characteristic
243	Q₈	C₂²⋊C₈×C₅₉	1	16	yes	complemented
244	D₄	—	2	24	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	3	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	3	no	—
251	C₂×C₄	—	2	24	no	—
252	C₂×C₄	—	2	3	no	—
253	C₂×C₄	—	2	3	no	—
254	C₂×C₄	—	2	3	no	—
255	C₂×C₄	—	4	12	no	—
256	C₂×C₄	—	4	6	no	—
257	C₂×C₄	—	4	6	no	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	6	yes	—
259	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	6	yes	—
260	C₂×C₄	C₂²×C₈×C₅₉	1	2	yes	—
261	C₂×C₄	C₂²×C₈×C₅₉	1	6	yes	—
262	C₂×C₄	C₂×M₁₆×C₅₉	1	6	yes	—
263	C₂×C₄	#131×C₅₉	1	2	yes	—
264	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
265	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
266	C₂×C₄	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
267	C₂×C₄	2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	—	4	6	no	—
270	C₂²	—	2	6	no	—
271	C₂²	—	2	6	no	—
272	C₂²	—	2	6	no	—
273	C₂²	—	4	3	no	—
274	C₂²	—	4	3	no	—
275	C₂²	—	4	3	no	—
276	C₂²	—	4	3	no	—
277	C₂²	C₂²×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
278	C₂²	C₄×#41×C₅₉	1	1	yes	characteristic
279	C₂²	C₅₉×#520	1	1	yes	characteristic
280	C₂²	C₂³×C₈×C₅₉	1	1	yes	characteristic
281	C₂²	C₂²×M₁₆×C₅₉	1	1	yes	characteristic
282	C₂²	C₂×#131×C₅₉	1	1	yes	characteristic
283	C₂²	C₅₉×#568	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	6	no	—
287	C₄	—	4	3	no	—
288	C₄	C₂×C₂²⋊C₈×C₅₉	1	12	yes	—
289	C₄	C₅₉×#548	1	2	yes	—
290	C₄	C₅₉×#550	1	2	yes	—
291	C₂	—	4	6	no	—
292	C₂	C₅₉×#2341	1	1	yes	characteristic
293	C₂	C₂²×C₂²⋊C₈×C₅₉	1	1	yes	characteristic
294	C₂	C₅₉×#2880	1	1	yes	characteristic
295	C₂	C₈×#41×C₅₉	1	1	yes	characteristic
296	C₂	C₅₉×#2971	1	1	yes	characteristic
297	C₂	C₅₉×#2988	1	1	yes	characteristic
298	C₂	C₅₉×#2991	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₉	C₂⁴
commutator	C₂²	C₂³×C₈×C₅₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
socle	C₂³×C₅₉	C₂³×C₄
prime radical	C₂²×#41×C₅₉	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Q₈×C₅₉	C₂	semidirect
3. C₅₉×#1954	C₂	semidirect
4. C₅₉×#1797	C₂	semidirect
5. C₄×C₂²⋊C₈×C₅₉	C₂	semidirect
23. C₂×#41×C₅₉	C₈	semidirect
77. C₂×C₈×C₅₉	#41	semidirect
93. Q₈×C₅₉	C₂²⋊C₈	semidirect
120. #22530	C₅₉	direct (N × H)
143. C₂×C₈×Q₈	C₂×C₅₉	semidirect
144. #1954	C₂×C₅₉	semidirect
145. #1797	C₂×C₅₉	semidirect
146. C₄×C₂²⋊C₈	C₂×C₅₉	semidirect
170. C₅₉	#22530	direct (N × H)
173. C₂×#41	C₈×C₅₉	semidirect
227. C₂×C₈	#41×C₅₉	semidirect
243. Q₈	C₂²⋊C₈×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₅₉	2
C₂×C₈×Q₈×C₅₉	2
C₅₉×#1954	2
C₅₉×#1797	2
C₄×C₂²⋊C₈×C₅₉	2
#22530	59

Subgroups — C59×#22530 #28434639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#22530 #28434639