Subgroups — C₅₉×#8048 #28420157

Order 15104.

788 conjugacy classes of subgroups.

The subgroup lattice has 230 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	C₄×#117×C₅₉	C₂	1	2	yes	maximal
3	C₅₉×#1297	C₂	1	4	yes	maximal
4	C₂×#117×C₅₉	—	2	2	no	—
5	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
6	C₅₉×#432	—	2	2	no	—
7	C₅₉×#431	—	2	2	no	—
8	C₅₉×#423	C₄	1	2	yes	complemented
9	C₅₉×#407	C₂²	1	2	yes	—
10	C₄×M₁₆×C₅₉	C₄	1	2	yes	complemented
11	C₄×C₄⋊C₄×C₅₉	—	2	2	no	—
12	C₄×C₄⋊C₄×C₅₉	C₄	1	4	yes	complemented
13	C₄×C₂²⋊C₄×C₅₉	—	2	2	no	—
14	C₄×C₂²⋊C₄×C₅₉	C₂²	1	4	yes	—
15	C₅₉×#376	C₄	1	4	yes	complemented
16	C₄³×C₅₉	—	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	#117×C₅₉	—	2	16	no	—
21	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
22	C₂×C₂²⋊C₄×C₅₉	—	2	4	no	—
23	C₂×C₄²×C₅₉	—	2	4	no	—
24	C₂×C₄²×C₅₉	—	2	4	no	—
25	C₂×C₄²×C₅₉	—	2	4	no	—
26	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₉	D₄	1	2	yes	—
31	C₂×C₄²×C₅₉	D₄	1	2	yes	—
32	C₂×C₄²×C₅₉	Q₈	1	2	yes	—
33	C₄⋊C₈×C₅₉	—	2	4	no	—
34	C₂²⋊C₈×C₅₉	—	2	4	no	—
35	#97×C₅₉	—	2	2	no	—
36	C₄×C₈×C₅₉	—	2	2	no	—
37	#95×C₅₉	—	2	4	no	—
38	#95×C₅₉	C₂×C₄	1	8	yes	—
39	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	—	2	8	no	—
41	C₂²×C₄×C₅₉	—	2	8	no	—
42	C₂²×C₄×C₅₉	—	4	2	no	—
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	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
53	M₁₆×C₅₉	—	4	4	no	—
54	C₂×C₈×C₅₉	—	2	4	no	—
55	C₄⋊C₄×C₅₉	—	2	16	no	—
56	C₂²⋊C₄×C₅₉	—	2	16	no	—
57	C₄²×C₅₉	—	2	16	no	—
58	C₄²×C₅₉	—	4	8	no	—
59	C₄²×C₅₉	—	4	8	no	—
60	C₄²×C₅₉	C₄²	1	1	yes	characteristic
61	C₄²×C₅₉	C₄²	1	2	yes	—
62	C₄²×C₅₉	C₄²	1	1	yes	characteristic
63	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
67	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
68	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
69	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
70	C₂³×C₅₉	—	2	4	no	—
71	C₂³×C₅₉	—	4	1	no	—
72	C₂³×C₅₉	C₂×C₂²⋊C₄	1	2	yes	—
73	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₉	—	2	4	no	—
75	C₂×C₄×C₅₉	—	2	4	no	—
76	C₂×C₄×C₅₉	—	2	4	no	—
77	C₂×C₄×C₅₉	—	2	4	no	—
78	C₂×C₄×C₅₉	—	2	4	no	—
79	C₂×C₄×C₅₉	—	2	4	no	—
80	C₂×C₄×C₅₉	—	2	2	no	—
81	C₂×C₄×C₅₉	—	2	16	no	—
82	C₂×C₄×C₅₉	—	4	2	no	—
83	C₂×C₄×C₅₉	—	4	1	no	—
84	C₂×C₄×C₅₉	—	4	8	no	—
85	C₂×C₄×C₅₉	—	4	8	no	—
86	C₂×C₄×C₅₉	#95	1	2	yes	—
87	C₂×C₄×C₅₉	#95	1	2	yes	—
88	C₂×C₄×C₅₉	#95	1	2	yes	—
89	C₂×C₄×C₅₉	#95	1	2	yes	—
90	C₂×C₄×C₅₉	#104	1	8	yes	—
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×C₄⋊C₄	1	2	yes	—
94	C₈×C₅₉	—	4	4	no	—
95	#8048	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂²×C₅₉	—	2	2	no	—
97	C₂²×C₅₉	—	2	2	no	—
98	C₂²×C₅₉	—	2	2	no	—
99	C₂²×C₅₉	—	2	1	no	—
100	C₂²×C₅₉	—	4	4	no	—
101	C₂²×C₅₉	C₂×#95	1	1	yes	characteristic
102	C₂²×C₅₉	C₂×#104	1	4	yes	—
103	C₄×C₅₉	—	2	2	no	—
104	C₄×C₅₉	—	2	2	no	—
105	C₄×C₅₉	—	2	1	no	—
106	C₄×C₅₉	—	2	2	no	—
107	C₄×C₅₉	—	4	16	no	—
108	C₄×C₅₉	#339	1	4	yes	—
109	C₄×C₅₉	#339	1	4	yes	—
110	C₄×C₅₉	#410	1	2	yes	—
111	#2920	C₂×C₅₉	1	1	yes	characteristic
112	C₄×#117	C₂×C₅₉	1	2	yes	—
113	#1297	C₂×C₅₉	1	4	yes	—
114	C₂×C₅₉	—	2	4	no	—
115	C₂×C₅₉	—	4	1	no	—
116	C₂×C₅₉	C₂×#339	1	2	yes	—
117	C₂×C₅₉	#1740	1	1	yes	characteristic
118	C₂×#117	—	2	2	no	—
119	C₂²×C₄²	C₂²×C₅₉	1	1	yes	characteristic
120	#432	—	2	2	no	—
121	#431	—	2	2	no	—
122	#423	C₄×C₅₉	1	2	yes	complemented
123	#407	C₂²×C₅₉	1	2	yes	—
124	C₄×M₁₆	C₄×C₅₉	1	2	yes	complemented
125	C₄×C₄⋊C₄	—	2	2	no	—
126	C₄×C₄⋊C₄	C₄×C₅₉	1	4	yes	complemented
127	C₄×C₂²⋊C₄	—	2	2	no	—
128	C₄×C₂²⋊C₄	C₂²×C₅₉	1	4	yes	—
129	#376	C₄×C₅₉	1	4	yes	complemented
130	C₄³	—	2	2	no	—
131	C₅₉	#8048	1	1	yes	characteristic, direct product factor, Sylow
132	C₂³×C₄	D₄×C₅₉	1	2	yes	—
133	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
134	C₂×M₁₆	C₂×C₄×C₅₉	1	4	yes	—
135	#117	—	2	16	no	—
136	C₂×C₄⋊C₄	—	2	4	no	—
137	C₂×C₂²⋊C₄	—	2	4	no	—
138	C₂×C₄²	—	2	4	no	—
139	C₂×C₄²	—	2	4	no	—
140	C₂×C₄²	—	2	4	no	—
141	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
142	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
143	C₂×C₄²	C₂×C₄×C₅₉	1	2	yes	—
144	C₂×C₄²	C₂×C₄×C₅₉	1	2	yes	—
145	C₂×C₄²	D₄×C₅₉	1	2	yes	—
146	C₂×C₄²	D₄×C₅₉	1	2	yes	—
147	C₂×C₄²	Q₈×C₅₉	1	2	yes	—
148	C₄⋊C₈	—	2	4	no	—
149	C₂²⋊C₈	—	2	4	no	—
150	#97	—	2	2	no	—
151	C₄×C₈	—	2	2	no	—
152	#95	—	2	4	no	—
153	#95	C₂×C₄×C₅₉	1	8	yes	—
154	C₂⁴	C₂×D₄×C₅₉	1	1	yes	characteristic
155	C₂²×C₄	—	2	8	no	—
156	C₂²×C₄	—	2	8	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
159	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
160	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
161	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
162	C₂²×C₄	C₄⋊C₄×C₅₉	1	2	yes	—
163	C₂²×C₄	C₄⋊C₄×C₅₉	1	2	yes	—
164	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
165	C₂²×C₄	C₂²×C₄×C₅₉	1	2	yes	—
166	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
167	C₂²×C₄	C₂×Q₈×C₅₉	1	1	yes	characteristic
168	M₁₆	—	4	4	no	—
169	C₂×C₈	—	2	4	no	—
170	C₄⋊C₄	—	2	16	no	—
171	C₂²⋊C₄	—	2	16	no	—
172	C₄²	—	2	16	no	—
173	C₄²	—	4	8	no	—
174	C₄²	—	4	8	no	—
175	C₄²	C₄²×C₅₉	1	1	yes	characteristic
176	C₄²	C₄²×C₅₉	1	2	yes	—
177	C₄²	C₄²×C₅₉	1	1	yes	characteristic
178	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
179	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
180	C₄²	C₂²⋊C₄×C₅₉	1	2	yes	—
181	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
182	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
183	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
184	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
185	C₂³	—	2	4	no	—
186	C₂³	—	4	1	no	—
187	C₂³	C₂×C₂²⋊C₄×C₅₉	1	2	yes	—
188	C₂³	C₂×C₄⋊C₄×C₅₉	1	1	yes	characteristic
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	16	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	8	no	—
200	C₂×C₄	—	4	8	no	—
201	C₂×C₄	#95×C₅₉	1	2	yes	—
202	C₂×C₄	#95×C₅₉	1	2	yes	—
203	C₂×C₄	#95×C₅₉	1	2	yes	—
204	C₂×C₄	#95×C₅₉	1	2	yes	—
205	C₂×C₄	#104×C₅₉	1	8	yes	—
206	C₂×C₄	C₂×C₄²×C₅₉	1	1	yes	characteristic
207	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
208	C₂×C₄	C₂×C₄⋊C₄×C₅₉	1	2	yes	—
209	C₈	—	4	4	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	4	4	no	—
215	C₂²	C₂×#95×C₅₉	1	1	yes	characteristic
216	C₂²	C₂×#104×C₅₉	1	4	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	4	16	no	—
222	C₄	C₅₉×#339	1	4	yes	—
223	C₄	C₅₉×#339	1	4	yes	—
224	C₄	C₅₉×#410	1	2	yes	—
225	C₂	—	2	4	no	—
226	C₂	—	4	1	no	—
227	C₂	C₂×C₅₉×#339	1	2	yes	—
228	C₂	C₅₉×#1740	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₅₉	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 ◁ C₄²×C₅₉ ◁ C₂²×C₄²×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅₉×#423	C₄	semidirect
10. C₄×M₁₆×C₅₉	C₄	semidirect
12. C₄×C₄⋊C₄×C₅₉	C₄	semidirect
15. C₅₉×#376	C₄	semidirect
95. #8048	C₅₉	direct (N × H)
122. #423	C₄×C₅₉	semidirect
124. C₄×M₁₆	C₄×C₅₉	semidirect
126. C₄×C₄⋊C₄	C₄×C₅₉	semidirect
129. #376	C₄×C₅₉	semidirect
131. C₅₉	#8048	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
C₄×#117×C₅₉	2
C₅₉×#1297	2
#8048	59

Subgroups — C59×#8048 #28420157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#8048 #28420157