Subgroups — C₅₉×#8000 #28420109

Order 15104.

542 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#2920	C₂	1	2	yes	maximal
2	C₂×C₅₉×#431	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1292	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
5	C₅₉×#432	—	2	2	no	—
6	C₅₉×#432	C₄	1	4	yes	—
7	C₅₉×#431	—	2	2	no	—
8	C₅₉×#431	—	2	2	no	—
9	C₅₉×#431	—	2	2	no	—
10	C₅₉×#431	C₄	1	4	yes	—
11	C₅₉×#423	C₂²	1	4	yes	—
12	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
13	C₅₉×#407	—	2	2	no	—
14	C₂×C₂²⋊C₈×C₅₉	C₄	1	2	yes	—
15	C₄×M₁₆×C₅₉	—	2	2	no	—
16	C₂×C₄×C₈×C₅₉	C₄	1	2	yes	—
17	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₅₉	Q₈	1	1	yes	characteristic
20	C₂×M₁₆×C₅₉	—	2	4	no	—
21	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₉	D₄	1	2	yes	—
27	C₂×C₄²×C₅₉	D₄	1	2	yes	—
28	C₂×C₄²×C₅₉	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₉	—	2	4	no	—
31	C₄⋊C₈×C₅₉	—	2	4	no	—
32	C₄⋊C₈×C₅₉	C₂×C₄	1	8	yes	—
33	C₂²⋊C₈×C₅₉	—	2	4	no	—
34	C₂²⋊C₈×C₅₉	—	2	4	no	—
35	C₂²⋊C₈×C₅₉	—	2	4	no	—
36	#97×C₅₉	—	2	4	no	—
37	C₄×C₈×C₅₉	—	2	2	no	—
38	C₄×C₈×C₅₉	—	2	2	no	—
39	C₄×C₈×C₅₉	—	2	4	no	—
40	C₂⁴×C₅₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	—	2	4	no	—
42	C₂²×C₄×C₅₉	—	4	2	no	—
43	C₂²×C₄×C₅₉	—	4	2	no	—
44	C₂²×C₄×C₅₉	C₄²	1	2	yes	—
45	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
56	M₁₆×C₅₉	—	4	8	no	—
57	C₂×C₈×C₅₉	—	2	4	no	—
58	C₂×C₈×C₅₉	—	2	8	no	—
59	C₂×C₈×C₅₉	—	4	4	no	—
60	C₄²×C₅₉	C₄²	1	1	yes	characteristic
61	C₄²×C₅₉	C₄²	1	1	yes	characteristic
62	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
67	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
68	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
69	C₄²×C₅₉	C₂×D₄	1	2	yes	—
70	C₂³×C₅₉	—	2	2	no	—
71	C₂³×C₅₉	—	4	1	no	—
72	C₂³×C₅₉	—	4	1	no	—
73	C₂³×C₅₉	#95	1	2	yes	—
74	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₉	—	2	2	no	—
76	C₂×C₄×C₅₉	—	2	2	no	—
77	C₂×C₄×C₅₉	—	2	4	no	—
78	C₂×C₄×C₅₉	—	2	2	no	—
79	C₂×C₄×C₅₉	—	2	4	no	—
80	C₂×C₄×C₅₉	—	2	4	no	—
81	C₂×C₄×C₅₉	—	2	4	no	—
82	C₂×C₄×C₅₉	—	4	2	no	—
83	C₂×C₄×C₅₉	—	4	2	no	—
84	C₂×C₄×C₅₉	—	4	1	no	—
85	C₂×C₄×C₅₉	—	4	1	no	—
86	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
87	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
88	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
89	C₂×C₄×C₅₉	#95	1	2	yes	—
90	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
91	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₅₉	#135	1	4	yes	—
96	C₈×C₅₉	—	4	4	no	—
97	C₈×C₅₉	—	4	8	no	—
98	#8000	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₅₉	—	2	1	no	—
100	C₂²×C₅₉	—	2	1	no	—
101	C₂²×C₅₉	—	2	2	no	—
102	C₂²×C₅₉	—	2	2	no	—
103	C₂²×C₅₉	—	2	2	no	—
104	C₂²×C₅₉	—	4	4	no	—
105	C₂²×C₅₉	C₂×#95	1	1	yes	characteristic
106	C₂²×C₅₉	#427	1	2	yes	—
107	C₄×C₅₉	—	2	1	no	—
108	C₄×C₅₉	—	2	2	no	—
109	C₄×C₅₉	—	2	1	no	—
110	C₄×C₅₉	—	2	1	no	—
111	C₄×C₅₉	—	2	2	no	—
112	C₄×C₅₉	—	2	1	no	—
113	C₄×C₅₉	#338	1	2	yes	—
114	C₄×C₅₉	#413	1	2	yes	—
115	C₄×C₅₉	#416	1	4	yes	—
116	#2920	C₂×C₅₉	1	2	yes	—
117	C₂×#431	C₂×C₅₉	1	1	yes	characteristic
118	#1292	C₂×C₅₉	1	4	yes	complemented
119	C₂×C₅₉	—	2	2	no	—
120	C₂×C₅₉	—	4	2	no	—
121	C₂×C₅₉	C₂×#338	1	1	yes	characteristic
122	C₂×C₅₉	#1735	1	1	yes	characteristic
123	C₂×C₅₉	#1742	1	1	yes	characteristic
124	C₂²×C₄²	C₂²×C₅₉	1	1	yes	characteristic
125	#432	—	2	2	no	—
126	#432	C₄×C₅₉	1	4	yes	—
127	#431	—	2	2	no	—
128	#431	—	2	2	no	—
129	#431	—	2	2	no	—
130	#431	C₄×C₅₉	1	4	yes	—
131	#423	C₂²×C₅₉	1	4	yes	—
132	C₂×C₄⋊C₈	C₂²×C₅₉	1	2	yes	—
133	#407	—	2	2	no	—
134	C₂×C₂²⋊C₈	C₄×C₅₉	1	2	yes	—
135	C₄×M₁₆	—	2	2	no	—
136	C₂×C₄×C₈	C₄×C₅₉	1	2	yes	—
137	C₅₉	#8000	1	1	yes	characteristic, direct product factor, Sylow
138	C₂³×C₄	C₂×C₄×C₅₉	1	1	yes	characteristic
139	C₂³×C₄	D₄×C₅₉	1	1	yes	characteristic
140	C₂³×C₄	Q₈×C₅₉	1	1	yes	characteristic
141	C₂×M₁₆	—	2	4	no	—
142	C₂²×C₈	C₂×C₄×C₅₉	1	4	yes	—
143	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
144	C₂×C₄²	C₂×C₄×C₅₉	1	2	yes	—
145	C₂×C₄²	C₂×C₄×C₅₉	1	2	yes	—
146	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
147	C₂×C₄²	D₄×C₅₉	1	2	yes	—
148	C₂×C₄²	D₄×C₅₉	1	2	yes	—
149	C₂×C₄²	Q₈×C₅₉	1	1	yes	characteristic
150	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
151	C₄⋊C₈	—	2	4	no	—
152	C₄⋊C₈	—	2	4	no	—
153	C₄⋊C₈	C₂×C₄×C₅₉	1	8	yes	—
154	C₂²⋊C₈	—	2	4	no	—
155	C₂²⋊C₈	—	2	4	no	—
156	C₂²⋊C₈	—	2	4	no	—
157	#97	—	2	4	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	2	4	no	—
161	C₂⁴	C₄⋊C₄×C₅₉	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	C₄²×C₅₉	1	2	yes	—
166	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
167	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
168	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
169	C₂²×C₄	C₄⋊C₄×C₅₉	1	1	yes	characteristic
170	C₂²×C₄	C₄⋊C₄×C₅₉	1	1	yes	characteristic
171	C₂²×C₄	C₄⋊C₄×C₅₉	1	2	yes	—
172	C₂²×C₄	C₄⋊C₄×C₅₉	1	1	yes	characteristic
173	C₂²×C₄	C₄⋊C₄×C₅₉	1	2	yes	—
174	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
175	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
176	C₂²×C₄	C₂×Q₈×C₅₉	1	1	yes	characteristic
177	M₁₆	—	4	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	8	no	—
180	C₂×C₈	—	4	4	no	—
181	C₄²	C₄²×C₅₉	1	1	yes	characteristic
182	C₄²	C₄²×C₅₉	1	1	yes	characteristic
183	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
184	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
185	C₄²	C₂²⋊C₄×C₅₉	1	2	yes	—
186	C₄²	C₂²⋊C₄×C₅₉	1	2	yes	—
187	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
188	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
189	C₄²	C₂²×C₄×C₅₉	1	2	yes	—
190	C₄²	C₂×D₄×C₅₉	1	2	yes	—
191	C₂³	—	2	2	no	—
192	C₂³	—	4	1	no	—
193	C₂³	—	4	1	no	—
194	C₂³	#95×C₅₉	1	2	yes	—
195	C₂³	C₂×C₄⋊C₄×C₅₉	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	#95×C₅₉	1	1	yes	characteristic
208	C₂×C₄	#95×C₅₉	1	1	yes	characteristic
209	C₂×C₄	#95×C₅₉	1	1	yes	characteristic
210	C₂×C₄	#95×C₅₉	1	2	yes	—
211	C₂×C₄	#95×C₅₉	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄²×C₅₉	1	1	yes	characteristic
213	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
214	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	2	yes	—
215	C₂×C₄	C₂×C₄⋊C₄×C₅₉	1	2	yes	—
216	C₂×C₄	#135×C₅₉	1	4	yes	—
217	C₈	—	4	4	no	—
218	C₈	—	4	8	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	4	no	—
225	C₂²	C₂×#95×C₅₉	1	1	yes	characteristic
226	C₂²	C₅₉×#427	1	2	yes	—
227	C₄	—	2	1	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	C₅₉×#338	1	2	yes	—
234	C₄	C₅₉×#413	1	2	yes	—
235	C₄	C₅₉×#416	1	4	yes	—
236	C₂	—	2	2	no	—
237	C₂	—	4	2	no	—
238	C₂	C₂×C₅₉×#338	1	1	yes	characteristic
239	C₂	C₅₉×#1735	1	1	yes	characteristic
240	C₂	C₅₉×#1742	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₉	#95
commutator	C₂×C₄	C₂×C₄²×C₅₉
Frattini, cosocle	C₂×C₄²	C₂³×C₅₉
socle	C₂²×C₅₉	C₂×#95
prime radical	C₂⁴×C₅₉	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₉ ◁ C₂²×C₄²×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₉×#1292	C₂	semidirect
98. #8000	C₅₉	direct (N × H)
118. #1292	C₂×C₅₉	semidirect
137. C₅₉	#8000	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×#2920	2
C₂×C₅₉×#431	2
C₅₉×#1292	2
#8000	59

Subgroups — C59×#8000 #28420109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#8000 #28420109