Subgroups — C₅₉×#8034 #28420143

Order 15104.

706 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₉×#432	C₂	1	1	yes	maximal, characteristic
2	C₂²×#97×C₅₉	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2285	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#1295	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₅₉	—	2	1	no	—
6	C₂×#117×C₅₉	C₄	1	2	yes	—
7	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
8	C₅₉×#432	—	2	2	no	—
9	C₅₉×#432	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₈×C₅₉	—	2	1	no	—
11	C₂×C₂²⋊C₈×C₅₉	—	2	1	no	—
12	C₂×#97×C₅₉	—	2	1	no	—
13	C₂×#97×C₅₉	—	2	2	no	—
14	C₂×#97×C₅₉	C₄	1	4	yes	complemented
15	C₂×#97×C₅₉	C₂²	1	2	yes	—
16	C₂×#97×C₅₉	C₂²	1	2	yes	—
17	C₅₉×#384	C₄	1	2	yes	—
18	C₅₉×#382	C₂²	1	2	yes	—
19	C₅₉×#380	—	2	1	no	—
20	C₅₉×#380	—	2	1	no	—
21	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
22	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
23	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₉	—	2	8	no	—
27	C₂²×C₈×C₅₉	—	2	4	no	—
28	C₂²×C₈×C₅₉	—	2	2	no	—
29	C₂²×C₈×C₅₉	—	2	2	no	—
30	#117×C₅₉	—	4	4	no	—
31	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
35	C₂×C₄²×C₅₉	—	2	2	no	—
36	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₉	D₄	1	2	yes	—
42	C₂×C₄²×C₅₉	Q₈	1	2	yes	—
43	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₅₉	—	2	4	no	—
45	C₂²⋊C₈×C₅₉	—	2	4	no	—
46	#97×C₅₉	—	2	4	no	—
47	#97×C₅₉	—	2	2	no	—
48	#97×C₅₉	—	2	2	no	—
49	#97×C₅₉	C₂×C₄	1	4	yes	—
50	#97×C₅₉	C₂×C₄	1	4	yes	—
51	#95×C₅₉	—	2	2	no	—
52	#95×C₅₉	—	2	2	no	—
53	C₂⁴×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	—	2	2	no	—
55	C₂²×C₄×C₅₉	—	2	2	no	—
56	C₂²×C₄×C₅₉	—	2	4	no	—
57	C₂²×C₄×C₅₉	—	2	4	no	—
58	C₂²×C₄×C₅₉	—	4	2	no	—
59	C₂²×C₄×C₅₉	C₄²	1	2	yes	—
60	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₉	—	2	8	no	—
72	C₂×C₈×C₅₉	—	2	16	no	—
73	C₂×C₈×C₅₉	—	2	4	no	—
74	C₂×C₈×C₅₉	—	2	4	no	—
75	C₂×C₈×C₅₉	—	4	4	no	—
76	C₄⋊C₄×C₅₉	—	4	4	no	—
77	C₄⋊C₄×C₅₉	—	4	4	no	—
78	C₄⋊C₄×C₅₉	—	4	4	no	—
79	C₂²⋊C₄×C₅₉	—	4	4	no	—
80	C₄²×C₅₉	—	4	4	no	—
81	C₄²×C₅₉	C₄²	1	2	yes	—
82	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
85	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
86	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
87	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
91	C₂³×C₅₉	—	2	2	no	—
92	C₂³×C₅₉	—	2	2	no	—
93	C₂³×C₅₉	#95	1	2	yes	—
94	C₂³×C₅₉	#100	1	2	yes	—
95	C₂³×C₅₉	#101	1	2	yes	—
96	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₉	—	2	4	no	—
98	C₂×C₄×C₅₉	—	2	4	no	—
99	C₂×C₄×C₅₉	—	2	1	no	—
100	C₂×C₄×C₅₉	—	2	1	no	—
101	C₂×C₄×C₅₉	—	2	1	no	—
102	C₂×C₄×C₅₉	—	2	1	no	—
103	C₂×C₄×C₅₉	—	2	2	no	—
104	C₂×C₄×C₅₉	—	2	2	no	—
105	C₂×C₄×C₅₉	—	2	2	no	—
106	C₂×C₄×C₅₉	—	2	4	no	—
107	C₂×C₄×C₅₉	—	2	4	no	—
108	C₂×C₄×C₅₉	—	4	2	no	—
109	C₂×C₄×C₅₉	—	4	2	no	—
110	C₂×C₄×C₅₉	—	4	4	no	—
111	C₂×C₄×C₅₉	—	4	4	no	—
112	C₂×C₄×C₅₉	—	4	4	no	—
113	C₂×C₄×C₅₉	#95	1	2	yes	—
114	C₂×C₄×C₅₉	#95	1	2	yes	—
115	C₂×C₄×C₅₉	#95	1	2	yes	—
116	C₂×C₄×C₅₉	#104	1	4	yes	—
117	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₈×C₅₉	—	2	8	no	—
124	C₈×C₅₉	—	4	4	no	—
125	#8034	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₅₉	—	2	2	no	—
127	C₂²×C₅₉	—	2	2	no	—
128	C₂²×C₅₉	—	2	4	no	—
129	C₂²×C₅₉	—	2	4	no	—
130	C₂²×C₅₉	#343	1	4	yes	—
131	C₂²×C₅₉	C₂×#95	1	1	yes	characteristic
132	C₂²×C₅₉	C₂×#100	1	1	yes	characteristic
133	C₂²×C₅₉	C₂×#101	1	1	yes	characteristic
134	C₂²×C₅₉	C₂×#104	1	2	yes	—
135	C₂²×C₅₉	#421	1	2	yes	—
136	C₄×C₅₉	—	2	1	no	—
137	C₄×C₅₉	—	2	1	no	—
138	C₄×C₅₉	—	2	1	no	—
139	C₄×C₅₉	—	2	2	no	—
140	C₄×C₅₉	—	2	1	no	—
141	C₄×C₅₉	—	2	2	no	—
142	C₄×C₅₉	—	2	1	no	—
143	C₄×C₅₉	—	2	1	no	—
144	C₄×C₅₉	—	8	4	no	—
145	C₄×C₅₉	#339	1	4	yes	—
146	C₂×#432	C₂×C₅₉	1	1	yes	characteristic
147	C₂²×#97	C₂×C₅₉	1	1	yes	characteristic
148	#2285	C₂×C₅₉	1	1	yes	characteristic
149	#1295	C₂×C₅₉	1	4	yes	complemented
150	C₂×C₅₉	—	2	4	no	—
151	C₂×C₅₉	#1530	1	2	yes	—
152	C₂×C₅₉	#1531	1	2	yes	—
153	C₂×C₅₉	C₂×#339	1	1	yes	characteristic
154	C₂×C₅₉	#1733	1	1	yes	characteristic
155	C₂×C₅₉	C₂×#343	1	1	yes	characteristic
156	C₂³×C₈	—	2	1	no	—
157	C₂×#117	C₄×C₅₉	1	2	yes	—
158	C₂²×C₄²	C₂²×C₅₉	1	1	yes	characteristic
159	#432	—	2	2	no	—
160	#432	C₄×C₅₉	1	4	yes	complemented
161	C₂×C₄⋊C₈	—	2	1	no	—
162	C₂×C₂²⋊C₈	—	2	1	no	—
163	C₂×#97	—	2	1	no	—
164	C₂×#97	—	2	2	no	—
165	C₂×#97	C₄×C₅₉	1	4	yes	complemented
166	C₂×#97	C₂²×C₅₉	1	2	yes	—
167	C₂×#97	C₂²×C₅₉	1	2	yes	—
168	#384	C₄×C₅₉	1	2	yes	—
169	#382	C₂²×C₅₉	1	2	yes	—
170	#380	—	2	1	no	—
171	#380	—	2	1	no	—
172	C₄×C₄⋊C₄	—	2	1	no	—
173	C₄×C₄⋊C₄	—	2	1	no	—
174	C₅₉	#8034	1	1	yes	characteristic, direct product factor, Sylow
175	C₂³×C₄	C₂×C₄×C₅₉	1	1	yes	characteristic
176	C₂³×C₄	D₄×C₅₉	1	1	yes	characteristic
177	C₂³×C₄	D₄×C₅₉	1	1	yes	characteristic
178	C₂²×C₈	—	2	8	no	—
179	C₂²×C₈	—	2	4	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	#117	—	4	4	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₅₉	1	4	yes	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
189	C₂×C₄²	C₂×C₄×C₅₉	1	2	yes	—
190	C₂×C₄²	C₂×C₄×C₅₉	1	2	yes	—
191	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
192	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
193	C₂×C₄²	D₄×C₅₉	1	2	yes	—
194	C₂×C₄²	Q₈×C₅₉	1	2	yes	—
195	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
196	C₄⋊C₈	—	2	4	no	—
197	C₂²⋊C₈	—	2	4	no	—
198	#97	—	2	4	no	—
199	#97	—	2	2	no	—
200	#97	—	2	2	no	—
201	#97	C₂×C₄×C₅₉	1	4	yes	—
202	#97	C₂×C₄×C₅₉	1	4	yes	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	C₂⁴	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	C₄²×C₅₉	1	2	yes	—
212	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
213	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
214	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
215	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
216	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
217	C₂²×C₄	C₄⋊C₄×C₅₉	1	2	yes	—
218	C₂²×C₄	C₄⋊C₄×C₅₉	1	2	yes	—
219	C₂²×C₄	C₄⋊C₄×C₅₉	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
223	C₂×C₈	—	2	8	no	—
224	C₂×C₈	—	2	16	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	4	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₂²⋊C₄	—	4	4	no	—
232	C₄²	—	4	4	no	—
233	C₄²	C₄²×C₅₉	1	2	yes	—
234	C₄²	C₂²⋊C₄×C₅₉	1	2	yes	—
235	C₄²	C₂²⋊C₄×C₅₉	1	2	yes	—
236	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
237	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
238	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
239	C₄²	C₂²×C₄×C₅₉	1	1	yes	characteristic
240	C₄²	C₂²×C₄×C₅₉	1	1	yes	characteristic
241	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
242	C₄²	C₂×Q₈×C₅₉	1	1	yes	characteristic
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	#95×C₅₉	1	2	yes	—
246	C₂³	#100×C₅₉	1	2	yes	—
247	C₂³	#101×C₅₉	1	2	yes	—
248	C₂³	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	#95×C₅₉	1	2	yes	—
266	C₂×C₄	#95×C₅₉	1	2	yes	—
267	C₂×C₄	#95×C₅₉	1	2	yes	—
268	C₂×C₄	#104×C₅₉	1	4	yes	—
269	C₂×C₄	C₂×C₄²×C₅₉	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
271	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄⋊C₄×C₅₉	1	1	yes	characteristic
273	C₂×C₄	C₂×C₄⋊C₄×C₅₉	1	1	yes	characteristic
274	C₂×C₄	C₂×C₄⋊C₄×C₅₉	1	1	yes	characteristic
275	C₈	—	2	8	no	—
276	C₈	—	4	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	C₅₉×#343	1	4	yes	—
282	C₂²	C₂×#95×C₅₉	1	1	yes	characteristic
283	C₂²	C₂×#100×C₅₉	1	1	yes	characteristic
284	C₂²	C₂×#101×C₅₉	1	1	yes	characteristic
285	C₂²	C₂×#104×C₅₉	1	2	yes	—
286	C₂²	C₅₉×#421	1	2	yes	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	8	4	no	—
296	C₄	C₅₉×#339	1	4	yes	—
297	C₂	—	2	4	no	—
298	C₂	C₅₉×#1530	1	2	yes	—
299	C₂	C₅₉×#1531	1	2	yes	—
300	C₂	C₂×C₅₉×#339	1	1	yes	characteristic
301	C₂	C₅₉×#1733	1	1	yes	characteristic
302	C₂	C₂×C₅₉×#343	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅₉
Frattini, cosocle	C₂×C₄²	C₂³×C₅₉
prime radical	C₂⁴×C₅₉	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₂²×C₄²×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₉×#1295	C₂	semidirect
9. C₅₉×#432	C₄	semidirect
14. C₂×#97×C₅₉	C₄	semidirect
125. #8034	C₅₉	direct (N × H)
149. #1295	C₂×C₅₉	semidirect
160. #432	C₄×C₅₉	semidirect
165. C₂×#97	C₄×C₅₉	semidirect
174. C₅₉	#8034	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×#432	2
C₂²×#97×C₅₉	2
C₅₉×#2285	2
C₅₉×#1295	2
#8034	59

Subgroups — C59×#8034 #28420143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#8034 #28420143