Subgroups — C₅₉×#8040 #28420149

Order 15104.

726 conjugacy classes of subgroups.

The subgroup lattice has 234 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	2	yes	maximal
2	C₂×C₅₉×#384	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1296	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄⋊C₄×C₅₉	—	2	2	no	—
5	C₂²×C₄⋊C₄×C₅₉	C₄	1	2	yes	—
6	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
7	C₅₉×#432	—	2	4	no	—
8	C₅₉×#432	C₄	1	8	yes	complemented
9	C₂×C₄⋊C₈×C₅₉	C₂²	1	4	yes	—
10	C₂×C₂²⋊C₈×C₅₉	—	2	2	no	—
11	C₂×#97×C₅₉	—	2	2	no	—
12	C₅₉×#384	—	2	2	no	—
13	C₅₉×#384	C₄	1	2	yes	—
14	C₅₉×#384	C₂²	1	2	yes	—
15	C₂³×C₄×C₅₉	—	2	2	no	—
16	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₅₉	D₄	1	2	yes	—
18	C₂²×C₈×C₅₉	—	2	4	no	—
19	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
20	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
21	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
22	C₂×C₄⋊C₄×C₅₉	—	4	4	no	—
23	C₂×C₄⋊C₄×C₅₉	—	4	8	no	—
24	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₉	C₂×C₄	1	4	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	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₉	—	2	4	no	—
32	C₄⋊C₈×C₅₉	C₂×C₄	1	8	yes	complemented
33	C₂²⋊C₈×C₅₉	—	2	8	no	—
34	#97×C₅₉	—	2	8	no	—
35	C₂⁴×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	—	2	2	no	—
37	C₂²×C₄×C₅₉	—	2	2	no	—
38	C₂²×C₄×C₅₉	—	2	4	no	—
39	C₂²×C₄×C₅₉	—	2	4	no	—
40	C₂²×C₄×C₅₉	—	2	4	no	—
41	C₂²×C₄×C₅₉	—	4	8	no	—
42	C₂²×C₄×C₅₉	—	4	4	no	—
43	C₂²×C₄×C₅₉	C₄²	1	2	yes	—
44	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	C₂²⋊C₄	1	4	yes	—
47	C₂²×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₅₉	—	2	8	no	—
52	C₂×C₈×C₅₉	—	4	4	no	—
53	C₂×C₈×C₅₉	—	4	4	no	—
54	C₄⋊C₄×C₅₉	—	4	4	no	—
55	C₄⋊C₄×C₅₉	—	4	8	no	—
56	C₄⋊C₄×C₅₉	—	4	4	no	—
57	C₄⋊C₄×C₅₉	—	4	8	no	—
58	C₄²×C₅₉	C₄²	1	2	yes	—
59	C₄²×C₅₉	C₂²⋊C₄	1	4	yes	—
60	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
61	C₄²×C₅₉	C₄⋊C₄	1	4	yes	—
62	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
63	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
65	C₂³×C₅₉	—	2	2	no	—
66	C₂³×C₅₉	—	2	2	no	—
67	C₂³×C₅₉	#95	1	2	yes	—
68	C₂³×C₅₉	#100	1	2	yes	—
69	C₂³×C₅₉	#101	1	2	yes	—
70	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₉	—	2	4	no	—
72	C₂×C₄×C₅₉	—	2	4	no	—
73	C₂×C₄×C₅₉	—	2	1	no	—
74	C₂×C₄×C₅₉	—	2	1	no	—
75	C₂×C₄×C₅₉	—	2	1	no	—
76	C₂×C₄×C₅₉	—	2	1	no	—
77	C₂×C₄×C₅₉	—	2	2	no	—
78	C₂×C₄×C₅₉	—	2	2	no	—
79	C₂×C₄×C₅₉	—	2	8	no	—
80	C₂×C₄×C₅₉	—	2	8	no	—
81	C₂×C₄×C₅₉	—	4	4	no	—
82	C₂×C₄×C₅₉	—	4	8	no	—
83	C₂×C₄×C₅₉	—	8	8	no	—
84	C₂×C₄×C₅₉	#95	1	4	yes	—
85	C₂×C₄×C₅₉	#95	1	2	yes	—
86	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	2	yes	—
88	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
91	C₂×C₄×C₅₉	#137	1	4	yes	—
92	C₈×C₅₉	—	4	8	no	—
93	#8040	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂²×C₅₉	—	2	2	no	—
95	C₂²×C₅₉	—	2	4	no	—
96	C₂²×C₅₉	—	2	2	no	—
97	C₂²×C₅₉	—	2	4	no	—
98	C₂²×C₅₉	#343	1	4	yes	—
99	C₂²×C₅₉	C₂×#95	1	1	yes	characteristic
100	C₂²×C₅₉	C₂×#100	1	1	yes	characteristic
101	C₂²×C₅₉	C₂×#101	1	1	yes	characteristic
102	C₂²×C₅₉	#418	1	2	yes	—
103	C₂²×C₅₉	#419	1	2	yes	—
104	C₄×C₅₉	—	2	4	no	—
105	C₄×C₅₉	—	2	2	no	—
106	C₄×C₅₉	—	2	1	no	—
107	C₄×C₅₉	—	2	1	no	—
108	C₄×C₅₉	—	8	4	no	—
109	C₄×C₅₉	#417	1	4	yes	—
110	C₄×C₅₉	#428	1	4	yes	—
111	C₂×#432	C₂×C₅₉	1	2	yes	—
112	C₂×#384	C₂×C₅₉	1	1	yes	characteristic
113	#1296	C₂×C₅₉	1	4	yes	complemented
114	C₂×C₅₉	—	2	4	no	—
115	C₂×C₅₉	#1555	1	1	yes	characteristic
116	C₂×C₅₉	#1556	1	2	yes	—
117	C₂×C₅₉	#1558	1	1	yes	characteristic
118	C₂×C₅₉	#1736	1	2	yes	—
119	C₂×C₅₉	C₂×#343	1	1	yes	characteristic
120	C₂²×C₄⋊C₄	—	2	2	no	—
121	C₂²×C₄⋊C₄	C₄×C₅₉	1	2	yes	—
122	C₂²×C₄²	C₂²×C₅₉	1	1	yes	characteristic
123	#432	—	2	4	no	—
124	#432	C₄×C₅₉	1	8	yes	complemented
125	C₂×C₄⋊C₈	C₂²×C₅₉	1	4	yes	—
126	C₂×C₂²⋊C₈	—	2	2	no	—
127	C₂×#97	—	2	2	no	—
128	#384	—	2	2	no	—
129	#384	C₄×C₅₉	1	2	yes	—
130	#384	C₂²×C₅₉	1	2	yes	—
131	C₅₉	#8040	1	1	yes	characteristic, direct product factor, Sylow
132	C₂³×C₄	—	2	2	no	—
133	C₂³×C₄	C₂×C₄×C₅₉	1	1	yes	characteristic
134	C₂³×C₄	D₄×C₅₉	1	2	yes	—
135	C₂²×C₈	—	2	4	no	—
136	C₂×C₄⋊C₄	—	2	4	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	4	no	—
139	C₂×C₄⋊C₄	—	4	4	no	—
140	C₂×C₄⋊C₄	—	4	8	no	—
141	C₂×C₄⋊C₄	C₂×C₄×C₅₉	1	4	yes	—
142	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
143	C₂×C₄²	C₂×C₄×C₅₉	1	4	yes	—
144	C₂×C₄²	D₄×C₅₉	1	2	yes	—
145	C₂×C₄²	D₄×C₅₉	1	2	yes	—
146	C₂×C₄²	Q₈×C₅₉	1	2	yes	—
147	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
148	C₄⋊C₈	—	2	4	no	—
149	C₄⋊C₈	C₂×C₄×C₅₉	1	8	yes	complemented
150	C₂²⋊C₈	—	2	8	no	—
151	#97	—	2	8	no	—
152	C₂⁴	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	4	8	no	—
159	C₂²×C₄	—	4	4	no	—
160	C₂²×C₄	C₄²×C₅₉	1	2	yes	—
161	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
162	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
163	C₂²×C₄	C₂²⋊C₄×C₅₉	1	4	yes	—
164	C₂²×C₄	C₄⋊C₄×C₅₉	1	4	yes	—
165	C₂²×C₄	C₄⋊C₄×C₅₉	1	2	yes	—
166	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
168	C₂×C₈	—	2	8	no	—
169	C₂×C₈	—	4	4	no	—
170	C₂×C₈	—	4	4	no	—
171	C₄⋊C₄	—	4	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	8	no	—
174	C₄⋊C₄	—	4	8	no	—
175	C₄²	C₄²×C₅₉	1	2	yes	—
176	C₄²	C₂²⋊C₄×C₅₉	1	4	yes	—
177	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
178	C₄²	C₄⋊C₄×C₅₉	1	4	yes	—
179	C₄²	C₂²×C₄×C₅₉	1	2	yes	—
180	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
181	C₄²	C₂×Q₈×C₅₉	1	1	yes	characteristic
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	#95×C₅₉	1	2	yes	—
185	C₂³	#100×C₅₉	1	2	yes	—
186	C₂³	#101×C₅₉	1	2	yes	—
187	C₂³	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	8	no	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	8	no	—
200	C₂×C₄	—	8	8	no	—
201	C₂×C₄	#95×C₅₉	1	4	yes	—
202	C₂×C₄	#95×C₅₉	1	2	yes	—
203	C₂×C₄	C₂×C₄²×C₅₉	1	1	yes	characteristic
204	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	2	yes	—
205	C₂×C₄	C₂×C₄⋊C₄×C₅₉	1	1	yes	characteristic
206	C₂×C₄	C₂×C₄⋊C₄×C₅₉	1	2	yes	—
207	C₂×C₄	AGL(1,ℤ/8)×C₅₉	1	4	yes	—
208	C₂×C₄	#137×C₅₉	1	4	yes	—
209	C₈	—	4	8	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	2	2	no	—
214	C₂²	C₅₉×#343	1	4	yes	—
215	C₂²	C₂×#95×C₅₉	1	1	yes	characteristic
216	C₂²	C₂×#100×C₅₉	1	1	yes	characteristic
217	C₂²	C₂×#101×C₅₉	1	1	yes	characteristic
218	C₂²	C₅₉×#418	1	2	yes	—
219	C₂²	C₅₉×#419	1	2	yes	—
220	C₄	—	2	4	no	—
221	C₄	—	2	2	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	8	4	no	—
225	C₄	C₅₉×#417	1	4	yes	—
226	C₄	C₅₉×#428	1	4	yes	—
227	C₂	—	2	4	no	—
228	C₂	C₅₉×#1555	1	1	yes	characteristic
229	C₂	C₅₉×#1556	1	2	yes	—
230	C₂	C₅₉×#1558	1	1	yes	characteristic
231	C₂	C₅₉×#1736	1	2	yes	—
232	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
3. C₅₉×#1296	C₂	semidirect
8. C₅₉×#432	C₄	semidirect
32. C₄⋊C₈×C₅₉	C₂×C₄	semidirect
93. #8040	C₅₉	direct (N × H)
113. #1296	C₂×C₅₉	semidirect
124. #432	C₄×C₅₉	semidirect
131. C₅₉	#8040	direct (N × H)
149. C₄⋊C₈	C₂×C₄×C₅₉	semidirect

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
C₂×C₅₉×#384	2
C₅₉×#1296	2
#8040	59

Subgroups — C59×#8040 #28420149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#8040 #28420149