Subgroups — C₅₉×#8050 #28420159

Order 15104.

604 conjugacy classes of subgroups.

The subgroup lattice has 224 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	1	yes	maximal, characteristic
2	C₅₉×#2283	C₂	1	2	yes	maximal
3	C₅₉×#1297	C₂	1	4	yes	maximal
4	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
5	C₅₉×#432	—	2	2	no	—
6	C₅₉×#431	—	2	2	no	—
7	C₅₉×#423	C₄	1	2	yes	complemented
8	C₅₉×#407	C₂²	1	2	yes	—
9	C₄×M₁₆×C₅₉	C₄	1	2	yes	complemented
10	C₅₉×#383	C₄	1	4	yes	complemented
11	C₅₉×#382	C₄	1	4	yes	complemented
12	C₅₉×#381	—	2	2	no	—
13	C₅₉×#380	—	2	2	no	—
14	C₄×C₄⋊C₄×C₅₉	—	2	2	no	—
15	C₄×C₂²⋊C₄×C₅₉	C₂²	1	4	yes	—
16	C₅₉×#376	—	2	2	no	—
17	C₂³×C₄×C₅₉	D₄	1	2	yes	—
18	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₅₉	C₂×C₄	1	4	yes	—
20	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
21	C₂×C₂²⋊C₄×C₅₉	—	2	4	no	—
22	C₂×C₄²×C₅₉	—	2	4	no	—
23	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₉	D₄	1	2	yes	—
28	C₂×C₄²×C₅₉	D₄	1	2	yes	—
29	C₂×C₄²×C₅₉	Q₈	1	2	yes	—
30	C₄⋊C₈×C₅₉	—	2	4	no	—
31	C₂²⋊C₈×C₅₉	—	2	4	no	—
32	#97×C₅₉	—	2	2	no	—
33	C₄×C₈×C₅₉	—	2	2	no	—
34	#95×C₅₉	—	2	4	no	—
35	#95×C₅₉	—	2	4	no	—
36	#95×C₅₉	—	2	4	no	—
37	#95×C₅₉	C₂×C₄	1	8	yes	—
38	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	—	2	8	no	—
40	C₂²×C₄×C₅₉	—	4	2	no	—
41	C₂²×C₄×C₅₉	—	4	4	no	—
42	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
52	M₁₆×C₅₉	—	4	4	no	—
53	C₂×C₈×C₅₉	—	2	4	no	—
54	C₄⋊C₄×C₅₉	—	4	8	no	—
55	C₂²⋊C₄×C₅₉	—	4	8	no	—
56	C₄²×C₅₉	—	4	8	no	—
57	C₄²×C₅₉	C₄²	1	2	yes	—
58	C₄²×C₅₉	C₄²	1	1	yes	characteristic
59	C₄²×C₅₉	C₄²	1	1	yes	characteristic
60	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
64	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
65	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
66	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
67	C₂³×C₅₉	—	2	4	no	—
68	C₂³×C₅₉	—	4	1	no	—
69	C₂³×C₅₉	C₂×C₂²⋊C₄	1	2	yes	—
70	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₉	—	2	4	no	—
72	C₂×C₄×C₅₉	—	2	4	no	—
73	C₂×C₄×C₅₉	—	2	4	no	—
74	C₂×C₄×C₅₉	—	2	2	no	—
75	C₂×C₄×C₅₉	—	4	2	no	—
76	C₂×C₄×C₅₉	—	4	2	no	—
77	C₂×C₄×C₅₉	—	4	2	no	—
78	C₂×C₄×C₅₉	—	4	2	no	—
79	C₂×C₄×C₅₉	—	4	2	no	—
80	C₂×C₄×C₅₉	—	4	1	no	—
81	C₂×C₄×C₅₉	—	4	8	no	—
82	C₂×C₄×C₅₉	—	4	8	no	—
83	C₂×C₄×C₅₉	—	4	8	no	—
84	C₂×C₄×C₅₉	#95	1	2	yes	—
85	C₂×C₄×C₅₉	#95	1	2	yes	—
86	C₂×C₄×C₅₉	#95	1	2	yes	—
87	C₂×C₄×C₅₉	#95	1	2	yes	—
88	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
91	C₈×C₅₉	—	4	4	no	—
92	#8050	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₅₉	—	2	2	no	—
94	C₂²×C₅₉	—	2	2	no	—
95	C₂²×C₅₉	—	2	2	no	—
96	C₂²×C₅₉	—	2	1	no	—
97	C₂²×C₅₉	—	4	4	no	—
98	C₂²×C₅₉	C₂×#95	1	1	yes	characteristic
99	C₂²×C₅₉	#421	1	4	yes	—
100	C₄×C₅₉	—	2	2	no	—
101	C₄×C₅₉	—	2	2	no	—
102	C₄×C₅₉	—	2	2	no	—
103	C₄×C₅₉	—	2	2	no	—
104	C₄×C₅₉	—	2	1	no	—
105	C₄×C₅₉	—	2	2	no	—
106	C₄×C₅₉	—	8	8	no	—
107	C₄×C₅₉	#410	1	2	yes	—
108	#2920	C₂×C₅₉	1	1	yes	characteristic
109	#2283	C₂×C₅₉	1	2	yes	—
110	#1297	C₂×C₅₉	1	4	yes	—
111	C₂×C₅₉	—	2	4	no	—
112	C₂×C₅₉	—	4	1	no	—
113	C₂×C₅₉	#1733	1	2	yes	—
114	C₂×C₅₉	#1740	1	1	yes	characteristic
115	C₂²×C₄²	C₂²×C₅₉	1	1	yes	characteristic
116	#432	—	2	2	no	—
117	#431	—	2	2	no	—
118	#423	C₄×C₅₉	1	2	yes	complemented
119	#407	C₂²×C₅₉	1	2	yes	—
120	C₄×M₁₆	C₄×C₅₉	1	2	yes	complemented
121	#383	C₄×C₅₉	1	4	yes	complemented
122	#382	C₄×C₅₉	1	4	yes	complemented
123	#381	—	2	2	no	—
124	#380	—	2	2	no	—
125	C₄×C₄⋊C₄	—	2	2	no	—
126	C₄×C₂²⋊C₄	C₂²×C₅₉	1	4	yes	—
127	#376	—	2	2	no	—
128	C₅₉	#8050	1	1	yes	characteristic, direct product factor, Sylow
129	C₂³×C₄	D₄×C₅₉	1	2	yes	—
130	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
131	C₂×M₁₆	C₂×C₄×C₅₉	1	4	yes	—
132	C₂×C₄⋊C₄	—	2	4	no	—
133	C₂×C₂²⋊C₄	—	2	4	no	—
134	C₂×C₄²	—	2	4	no	—
135	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
136	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
137	C₂×C₄²	C₂×C₄×C₅₉	1	2	yes	—
138	C₂×C₄²	C₂×C₄×C₅₉	1	2	yes	—
139	C₂×C₄²	D₄×C₅₉	1	2	yes	—
140	C₂×C₄²	D₄×C₅₉	1	2	yes	—
141	C₂×C₄²	Q₈×C₅₉	1	2	yes	—
142	C₄⋊C₈	—	2	4	no	—
143	C₂²⋊C₈	—	2	4	no	—
144	#97	—	2	2	no	—
145	C₄×C₈	—	2	2	no	—
146	#95	—	2	4	no	—
147	#95	—	2	4	no	—
148	#95	—	2	4	no	—
149	#95	C₂×C₄×C₅₉	1	8	yes	—
150	C₂⁴	C₂×D₄×C₅₉	1	1	yes	characteristic
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	—	4	4	no	—
154	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
155	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
156	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
157	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
158	C₂²×C₄	C₄⋊C₄×C₅₉	1	2	yes	—
159	C₂²×C₄	C₄⋊C₄×C₅₉	1	2	yes	—
160	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
161	C₂²×C₄	C₂²×C₄×C₅₉	1	2	yes	—
162	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
163	C₂²×C₄	C₂×Q₈×C₅₉	1	1	yes	characteristic
164	M₁₆	—	4	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₄⋊C₄	—	4	8	no	—
167	C₂²⋊C₄	—	4	8	no	—
168	C₄²	—	4	8	no	—
169	C₄²	C₄²×C₅₉	1	2	yes	—
170	C₄²	C₄²×C₅₉	1	1	yes	characteristic
171	C₄²	C₄²×C₅₉	1	1	yes	characteristic
172	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
173	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
174	C₄²	C₂²⋊C₄×C₅₉	1	2	yes	—
175	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
176	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
177	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
178	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
179	C₂³	—	2	4	no	—
180	C₂³	—	4	1	no	—
181	C₂³	C₂×C₂²⋊C₄×C₅₉	1	2	yes	—
182	C₂³	C₂×C₄⋊C₄×C₅₉	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	8	no	—
194	C₂×C₄	—	4	8	no	—
195	C₂×C₄	—	4	8	no	—
196	C₂×C₄	#95×C₅₉	1	2	yes	—
197	C₂×C₄	#95×C₅₉	1	2	yes	—
198	C₂×C₄	#95×C₅₉	1	2	yes	—
199	C₂×C₄	#95×C₅₉	1	2	yes	—
200	C₂×C₄	C₂×C₄²×C₅₉	1	1	yes	characteristic
201	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄⋊C₄×C₅₉	1	2	yes	—
203	C₈	—	4	4	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	4	4	no	—
209	C₂²	C₂×#95×C₅₉	1	1	yes	characteristic
210	C₂²	C₅₉×#421	1	4	yes	—
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	—	8	8	no	—
218	C₄	C₅₉×#410	1	2	yes	—
219	C₂	—	2	4	no	—
220	C₂	—	4	1	no	—
221	C₂	C₅₉×#1733	1	2	yes	—
222	C₂	C₅₉×#1740	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅₉
Frattini, cosocle	C₂³×C₄	C₂³×C₅₉
socle	C₂²×C₅₉	C₂×#95
prime radical	C₂⁴×C₅₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
7. C₅₉×#423	C₄	semidirect
9. C₄×M₁₆×C₅₉	C₄	semidirect
10. C₅₉×#383	C₄	semidirect
11. C₅₉×#382	C₄	semidirect
92. #8050	C₅₉	direct (N × H)
118. #423	C₄×C₅₉	semidirect
120. C₄×M₁₆	C₄×C₅₉	semidirect
121. #383	C₄×C₅₉	semidirect
122. #382	C₄×C₅₉	semidirect
128. C₅₉	#8050	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
C₅₉×#2283	2
C₅₉×#1297	2
#8050	59

Subgroups — C59×#8050 #28420159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#8050 #28420159