Subgroups — C₅₉×#35322 #28447431

Order 15104.

2074 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×C₅₉×#533	C₂	1	2	yes	maximal
2	C₅₉×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#2862	C₂	1	2	yes	maximal, complemented
4	C₅₉×#2862	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#2621	C₂	1	4	yes	maximal
6	C₅₉×#2565	C₂	1	8	yes	maximal, complemented
7	C₅₉×#2499	C₂	1	8	yes	maximal
8	C₅₉×#2327	C₂	1	4	yes	maximal, complemented
9	C₄×#117×C₅₉	C₂	1	1	yes	maximal, characteristic
10	C₅₉×#533	—	2	16	no	—
11	C₂×C₄⋊Q₈×C₅₉	C₂²	1	4	yes	—
12	C₂×C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
14	C₂×#125×C₅₉	C₂²	1	4	yes	—
15	C₂×C₂²⋊Q₈×C₅₉	C₂²	1	4	yes	—
16	C₂×C₂²⋊Q₈×C₅₉	C₂²	1	4	yes	—
17	C₂×C₄×Q₈×C₅₉	C₂²	1	4	yes	—
18	C₂×C₄×Q₈×C₅₉	C₂²	1	4	yes	—
19	C₂×#117×C₅₉	C₂²	1	4	yes	—
20	C₂×#117×C₅₉	C₂²	1	2	yes	—
21	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
22	C₅₉×#398	C₂²	1	16	yes	—
23	C₅₉×#397	C₂²	1	16	yes	—
24	C₅₉×#391	C₂²	1	8	yes	—
25	C₅₉×#391	C₂²	1	8	yes	—
26	C₅₉×#391	C₂²	1	8	yes	—
27	C₅₉×#387	C₂²	1	32	yes	—
28	C₅₉×#384	C₂²	1	8	yes	—
29	C₅₉×#382	C₂²	1	8	yes	—
30	C₅₉×#382	C₂²	1	4	yes	—
31	C₄×C₄⋊C₄×C₅₉	C₂²	1	4	yes	—
32	C₄×C₂²⋊C₄×C₅₉	C₂²	1	4	yes	—
33	C₅₉×#376	C₂²	1	2	yes	—
34	C₄³×C₅₉	C₂²	1	1	yes	characteristic
35	C₄³×C₅₉	C₂²	1	1	yes	characteristic
36	C₂²×Q₈×C₅₉	C₂³	1	4	yes	—
37	C₂²×Q₈×C₅₉	C₂³	1	4	yes	—
38	C₂³×C₄×C₅₉	C₂³	1	2	yes	—
39	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₅₉	—	2	8	no	—
41	C₄⋊Q₈×C₅₉	—	2	8	no	—
42	C₄⋊Q₈×C₅₉	—	2	16	no	—
43	#125×C₅₉	—	2	16	no	—
44	C₂²⋊Q₈×C₅₉	—	2	16	no	—
45	C₂²⋊Q₈×C₅₉	—	2	16	no	—
46	C₄×Q₈×C₅₉	—	2	16	no	—
47	C₄×Q₈×C₅₉	—	2	16	no	—
48	#117×C₅₉	—	2	16	no	—
49	#117×C₅₉	—	2	8	no	—
50	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
52	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
53	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
54	C₂×C₄⋊C₄×C₅₉	C₂³	1	16	yes	—
55	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
56	C₂×C₄⋊C₄×C₅₉	C₂³	1	8	yes	—
57	C₂×C₄⋊C₄×C₅₉	C₂³	1	8	yes	—
58	C₂×C₂²⋊C₄×C₅₉	C₂³	1	4	yes	—
59	C₂×C₂²⋊C₄×C₅₉	C₂³	1	8	yes	—
60	C₂×C₄²×C₅₉	D₄	1	8	yes	—
61	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
62	C₂×C₄²×C₅₉	C₂³	1	4	yes	—
63	C₂×C₄²×C₅₉	C₂³	1	4	yes	—
64	C₂×C₄²×C₅₉	C₂³	1	8	yes	—
65	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
68	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
69	#95×C₅₉	C₂³	1	8	yes	—
70	#95×C₅₉	C₂³	1	32	yes	—
71	#95×C₅₉	C₂³	1	16	yes	—
72	C₂⁴×C₅₉	C₂⁴	1	1	yes	characteristic
73	C₂×Q₈×C₅₉	—	2	8	no	—
74	C₂×Q₈×C₅₉	—	2	8	no	—
75	C₂×Q₈×C₅₉	—	2	16	no	—
76	C₂×Q₈×C₅₉	—	2	16	no	—
77	C₂²×C₄×C₅₉	—	2	4	no	—
78	C₂²×C₄×C₅₉	—	2	8	no	—
79	C₂²×C₄×C₅₉	C₂×D₄	1	8	yes	—
80	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₅₉	C₂⁴	1	4	yes	—
82	C₂²×C₄×C₅₉	C₂⁴	1	8	yes	—
83	C₂²×C₄×C₅₉	C₂⁴	1	8	yes	—
84	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
85	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
86	C₂²×C₄×C₅₉	C₂⁴	1	4	yes	—
87	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
88	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
89	C₄⋊C₄×C₅₉	—	2	8	no	—
90	C₄⋊C₄×C₅₉	—	2	8	no	—
91	C₄⋊C₄×C₅₉	—	2	16	no	—
92	C₄⋊C₄×C₅₉	—	2	16	no	—
93	C₄⋊C₄×C₅₉	—	2	8	no	—
94	C₄⋊C₄×C₅₉	—	2	8	no	—
95	C₄⋊C₄×C₅₉	—	2	8	no	—
96	C₄⋊C₄×C₅₉	—	2	32	no	—
97	C₂²⋊C₄×C₅₉	—	2	16	no	—
98	C₂²⋊C₄×C₅₉	—	2	8	no	—
99	C₄²×C₅₉	—	2	16	no	—
100	C₄²×C₅₉	—	2	4	no	—
101	C₄²×C₅₉	—	2	8	no	—
102	C₄²×C₅₉	—	2	8	no	—
103	C₄²×C₅₉	C₂×D₄	1	4	yes	—
104	C₄²×C₅₉	C₂×D₄	1	8	yes	—
105	C₄²×C₅₉	C₂×D₄	1	4	yes	—
106	C₄²×C₅₉	#41	1	8	yes	—
107	C₂³×C₅₉	—	2	2	no	—
108	C₂³×C₅₉	—	2	4	no	—
109	C₂³×C₅₉	C₂²×D₄	1	2	yes	—
110	C₂³×C₅₉	C₂⁵	1	1	yes	characteristic
111	Q₈×C₅₉	—	4	16	no	—
112	Q₈×C₅₉	—	4	16	no	—
113	C₂×C₄×C₅₉	—	2	2	no	—
114	C₂×C₄×C₅₉	—	2	8	no	—
115	C₂×C₄×C₅₉	—	2	4	no	—
116	C₂×C₄×C₅₉	—	2	8	no	—
117	C₂×C₄×C₅₉	—	2	16	no	—
118	C₂×C₄×C₅₉	—	2	8	no	—
119	C₂×C₄×C₅₉	—	2	8	no	—
120	C₂×C₄×C₅₉	—	2	4	no	—
121	C₂×C₄×C₅₉	—	2	16	no	—
122	C₂×C₄×C₅₉	—	2	8	no	—
123	C₂×C₄×C₅₉	—	2	16	no	—
124	C₂×C₄×C₅₉	C₂²×D₄	1	4	yes	—
125	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
126	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
127	C₂×C₄×C₅₉	C₂²×D₄	1	4	yes	—
128	C₂×C₄×C₅₉	C₂×#41	1	8	yes	—
129	C₂×C₄×C₅₉	C₂×#41	1	4	yes	—
130	C₂×C₄×C₅₉	2₋¹⁺⁴	1	8	yes	—
131	#35322	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂²×C₅₉	—	2	2	no	—
133	C₂²×C₅₉	—	2	4	no	—
134	C₂²×C₅₉	—	4	4	no	—
135	C₂²×C₅₉	C₂³×D₄	1	1	yes	characteristic
136	C₂²×C₅₉	C₂²×#41	1	2	yes	—
137	C₂²×C₅₉	C₂×2₋¹⁺⁴	1	4	yes	—
138	C₄×C₅₉	—	2	2	no	—
139	C₄×C₅₉	—	2	4	no	—
140	C₄×C₅₉	—	2	8	no	—
141	C₄×C₅₉	—	4	8	no	—
142	C₄×C₅₉	—	4	8	no	—
143	C₄×C₅₉	—	4	4	no	—
144	C₄×C₅₉	#532	1	4	yes	—
145	C₄×C₅₉	#536	1	8	yes	—
146	C₂×#533	C₂×C₅₉	1	2	yes	—
147	#2868	C₂×C₅₉	1	1	yes	characteristic, complemented
148	#2862	C₂×C₅₉	1	1	yes	characteristic, complemented
149	#2862	C₂×C₅₉	1	2	yes	complemented
150	#2621	C₂×C₅₉	1	4	yes	—
151	#2565	C₂×C₅₉	1	8	yes	complemented
152	#2499	C₂×C₅₉	1	8	yes	—
153	#2327	C₂×C₅₉	1	4	yes	complemented
154	C₄×#117	C₂×C₅₉	1	1	yes	characteristic
155	C₂×C₅₉	—	4	2	no	—
156	C₂×C₅₉	C₂×#532	1	1	yes	characteristic
157	C₂×C₅₉	C₂×#536	1	2	yes	—
158	C₂×C₅₉	#3509	1	4	yes	—
159	#533	—	2	16	no	—
160	C₂×C₄⋊Q₈	C₂²×C₅₉	1	4	yes	—
161	C₂×C₄⋊Q₈	C₂²×C₅₉	1	2	yes	—
162	C₂×C₄⋊Q₈	C₂²×C₅₉	1	2	yes	—
163	C₂×#125	C₂²×C₅₉	1	4	yes	—
164	C₂×C₂²⋊Q₈	C₂²×C₅₉	1	4	yes	—
165	C₂×C₂²⋊Q₈	C₂²×C₅₉	1	4	yes	—
166	C₂×C₄×Q₈	C₂²×C₅₉	1	4	yes	—
167	C₂×C₄×Q₈	C₂²×C₅₉	1	4	yes	—
168	C₂×#117	C₂²×C₅₉	1	2	yes	—
169	C₂×#117	C₂²×C₅₉	1	4	yes	—
170	C₂²×C₄²	C₂²×C₅₉	1	1	yes	characteristic
171	#398	C₂²×C₅₉	1	16	yes	—
172	#397	C₂²×C₅₉	1	16	yes	—
173	#391	C₂²×C₅₉	1	8	yes	—
174	#391	C₂²×C₅₉	1	8	yes	—
175	#391	C₂²×C₅₉	1	8	yes	—
176	#387	C₂²×C₅₉	1	32	yes	—
177	#384	C₂²×C₅₉	1	8	yes	—
178	#382	C₂²×C₅₉	1	8	yes	—
179	#382	C₂²×C₅₉	1	4	yes	—
180	C₄×C₄⋊C₄	C₂²×C₅₉	1	4	yes	—
181	C₄×C₂²⋊C₄	C₂²×C₅₉	1	4	yes	—
182	#376	C₂²×C₅₉	1	2	yes	—
183	C₄³	C₂²×C₅₉	1	1	yes	characteristic
184	C₄³	C₂²×C₅₉	1	1	yes	characteristic
185	C₅₉	#35322	1	1	yes	characteristic, direct product factor, Sylow
186	C₂²×Q₈	C₂³×C₅₉	1	4	yes	—
187	C₂²×Q₈	C₂³×C₅₉	1	4	yes	—
188	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
189	C₂³×C₄	C₂³×C₅₉	1	2	yes	—
190	C₄⋊Q₈	—	2	8	no	—
191	C₄⋊Q₈	—	2	8	no	—
192	C₄⋊Q₈	—	2	16	no	—
193	#125	—	2	16	no	—
194	C₂²⋊Q₈	—	2	16	no	—
195	C₂²⋊Q₈	—	2	16	no	—
196	C₄×Q₈	—	2	16	no	—
197	C₄×Q₈	—	2	16	no	—
198	#117	—	2	16	no	—
199	#117	—	2	8	no	—
200	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
201	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
202	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
203	C₂×C₄⋊C₄	C₂³×C₅₉	1	8	yes	—
204	C₂×C₄⋊C₄	C₂³×C₅₉	1	8	yes	—
205	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
206	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
207	C₂×C₄⋊C₄	C₂³×C₅₉	1	16	yes	—
208	C₂×C₂²⋊C₄	C₂³×C₅₉	1	8	yes	—
209	C₂×C₂²⋊C₄	C₂³×C₅₉	1	4	yes	—
210	C₂×C₄²	D₄×C₅₉	1	8	yes	—
211	C₂×C₄²	C₂³×C₅₉	1	8	yes	—
212	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
213	C₂×C₄²	C₂³×C₅₉	1	4	yes	—
214	C₂×C₄²	C₂³×C₅₉	1	4	yes	—
215	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
217	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
218	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
219	#95	C₂³×C₅₉	1	8	yes	—
220	#95	C₂³×C₅₉	1	32	yes	—
221	#95	C₂³×C₅₉	1	16	yes	—
222	C₂⁴	C₂⁴×C₅₉	1	1	yes	characteristic
223	C₂×Q₈	—	2	8	no	—
224	C₂×Q₈	—	2	8	no	—
225	C₂×Q₈	—	2	16	no	—
226	C₂×Q₈	—	2	16	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	8	no	—
229	C₂²×C₄	C₂×D₄×C₅₉	1	8	yes	—
230	C₂²×C₄	C₂×D₄×C₅₉	1	4	yes	—
231	C₂²×C₄	C₂⁴×C₅₉	1	4	yes	—
232	C₂²×C₄	C₂⁴×C₅₉	1	8	yes	—
233	C₂²×C₄	C₂⁴×C₅₉	1	8	yes	—
234	C₂²×C₄	C₂⁴×C₅₉	1	2	yes	—
235	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
236	C₂²×C₄	C₂⁴×C₅₉	1	4	yes	—
237	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
238	C₂²×C₄	C₂⁴×C₅₉	1	2	yes	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	2	8	no	—
241	C₄⋊C₄	—	2	16	no	—
242	C₄⋊C₄	—	2	16	no	—
243	C₄⋊C₄	—	2	8	no	—
244	C₄⋊C₄	—	2	8	no	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	2	32	no	—
247	C₂²⋊C₄	—	2	16	no	—
248	C₂²⋊C₄	—	2	8	no	—
249	C₄²	—	2	16	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	8	no	—
252	C₄²	—	2	8	no	—
253	C₄²	C₂×D₄×C₅₉	1	4	yes	—
254	C₄²	C₂×D₄×C₅₉	1	8	yes	—
255	C₄²	C₂×D₄×C₅₉	1	4	yes	—
256	C₄²	#41×C₅₉	1	8	yes	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	4	no	—
259	C₂³	C₂²×D₄×C₅₉	1	2	yes	—
260	C₂³	C₂⁵×C₅₉	1	1	yes	characteristic
261	Q₈	—	4	16	no	—
262	Q₈	—	4	16	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	16	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	16	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	16	no	—
274	C₂×C₄	C₂²×D₄×C₅₉	1	4	yes	—
275	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
276	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
277	C₂×C₄	C₂²×D₄×C₅₉	1	4	yes	—
278	C₂×C₄	C₂×#41×C₅₉	1	8	yes	—
279	C₂×C₄	C₂×#41×C₅₉	1	4	yes	—
280	C₂×C₄	2₋¹⁺⁴×C₅₉	1	8	yes	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	4	4	no	—
284	C₂²	C₂³×D₄×C₅₉	1	1	yes	characteristic
285	C₂²	C₂²×#41×C₅₉	1	2	yes	—
286	C₂²	C₂×2₋¹⁺⁴×C₅₉	1	4	yes	—
287	C₄	—	2	8	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	4	no	—
290	C₄	—	4	4	no	—
291	C₄	—	4	8	no	—
292	C₄	—	4	8	no	—
293	C₄	C₅₉×#532	1	4	yes	—
294	C₄	C₅₉×#536	1	8	yes	—
295	C₂	—	4	2	no	—
296	C₂	C₂×C₅₉×#532	1	1	yes	characteristic
297	C₂	C₂×C₅₉×#536	1	2	yes	—
298	C₂	C₅₉×#3509	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₉
prime radical	C₂⁴×C₅₉	C₂⁴
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. C₅₉×#2868	C₂	semidirect
3. C₅₉×#2862	C₂	semidirect
4. C₅₉×#2862	C₂	semidirect
6. C₅₉×#2565	C₂	semidirect
8. C₅₉×#2327	C₂	semidirect
131. #35322	C₅₉	direct (N × H)
147. #2868	C₂×C₅₉	semidirect
148. #2862	C₂×C₅₉	semidirect
149. #2862	C₂×C₅₉	semidirect
151. #2565	C₂×C₅₉	semidirect
153. #2327	C₂×C₅₉	semidirect
185. C₅₉	#35322	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×#533	2
C₅₉×#2868	2
C₅₉×#2862	2
C₅₉×#2621	2
C₅₉×#2565	2
C₅₉×#2499	2
C₅₉×#2327	2
C₄×#117×C₅₉	2
#35322	59

Subgroups — C59×#35322 #28447431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#35322 #28447431