Subgroups — C₂³×C₂²⋊C₄×D₅₉ #29495239

Order 15104.

43678 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×D₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄×D₅₉	C₂	1	2	yes	maximal, complemented
3	C₂³×C₂²⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×(C₂×D₅₉)⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂²×C₂²⋊C₄×D₅₉	C₂	1	56	yes	maximal, direct product factor
6	C₂³×C₂²⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₅₉	—	2	32	no	—
8	C₂⁵×D₅₉	C₄	1	32	yes	complemented
9	C₂⁵×D₅₉	C₂²	1	28	yes	—
10	C₂⁵×D₅₉	C₂²	1	1	yes	characteristic
11	C₂⁵×D₅₉	C₂²	1	1	yes	characteristic
12	C₂⁴×Dic₅₉	C₂²	1	2	yes	complemented
13	C₂³×C₄×D₅₉	—	2	32	no	—
14	C₂³×C₄×D₅₉	C₂²	1	56	yes	complemented
15	C₂²×C₂²⋊Dic₅₉	C₂²	1	28	yes	complemented
16	C₂²×(C₂×D₅₉)⋊C₄	C₂²	1	56	yes	complemented
17	C₂×C₂²⋊C₄×D₅₉	C₂²	1	448	yes	direct product factor
18	C₂⁶×C₅₉	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₅₉	C₂²	1	2	yes	complemented
20	C₂²×C₂²⋊C₄×C₅₉	C₂²	1	28	yes	complemented
21	C₂⁴×D₅₉	—	2	16	no	—
22	C₂⁴×D₅₉	—	2	512	no	—
23	C₂⁴×D₅₉	—	2	448	no	—
24	C₂⁴×D₅₉	C₂×C₄	1	16	yes	—
25	C₂⁴×D₅₉	C₂×C₄	1	448	yes	complemented
26	C₂⁴×D₅₉	C₂×C₄	1	16	yes	complemented
27	C₂⁴×D₅₉	D₄	1	32	yes	—
28	C₂⁴×D₅₉	C₂³	1	112	yes	—
29	C₂⁴×D₅₉	C₂³	1	14	yes	—
30	C₂⁴×D₅₉	C₂³	1	14	yes	—
31	C₂³×Dic₅₉	—	2	16	no	—
32	C₂³×Dic₅₉	C₂³	1	28	yes	complemented
33	C₂²×C₄×D₅₉	—	2	448	no	—
34	C₂²×C₄×D₅₉	C₂³	1	224	yes	complemented
35	C₂×C₂²⋊Dic₅₉	C₂³	1	112	yes	complemented
36	C₂×(C₂×D₅₉)⋊C₄	C₂³	1	224	yes	complemented
37	C₂²⋊C₄×D₅₉	C₂³	1	512	yes	direct product factor
38	C₂⁵×C₅₉	—	2	16	no	—
39	C₂⁵×C₅₉	C₂×C₄	1	16	yes	complemented
40	C₂⁵×C₅₉	C₂³	1	14	yes	—
41	C₂⁵×C₅₉	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₅₉	—	2	16	no	—
43	C₂³×C₄×C₅₉	C₂³	1	28	yes	complemented
44	C₂×C₂²⋊C₄×C₅₉	C₂³	1	112	yes	complemented
45	C₂⁴×C₅₉	—	2	128	no	—
46	C₂⁴×C₅₉	—	2	112	no	—
47	C₂⁴×C₅₉	C₂²×C₄	1	8	yes	—
48	C₂⁴×C₅₉	C₂²×C₄	1	112	yes	complemented
49	C₂⁴×C₅₉	C₂×D₄	1	16	yes	—
50	C₂⁴×C₅₉	C₂⁴	1	28	yes	—
51	C₂⁴×C₅₉	C₂⁴	1	7	yes	—
52	C₂³×D₅₉	—	2	128	no	—
53	C₂³×D₅₉	—	2	112	no	—
54	C₂³×D₅₉	—	2	3584	no	—
55	C₂³×D₅₉	—	2	896	no	—
56	C₂³×D₅₉	C₂²⋊C₄	1	256	yes	direct product factor
57	C₂³×D₅₉	C₂²×C₄	1	112	yes	—
58	C₂³×D₅₉	C₂²×C₄	1	896	yes	complemented
59	C₂³×D₅₉	C₂²×C₄	1	112	yes	complemented
60	C₂³×D₅₉	C₂×D₄	1	224	yes	—
61	C₂³×D₅₉	C₂⁴	1	64	yes	—
62	C₂³×D₅₉	C₂⁴	1	28	yes	—
63	C₂³×D₅₉	C₂⁴	1	28	yes	—
64	C₂²×C₄×C₅₉	—	2	112	no	—
65	C₂²×C₄×C₅₉	C₂⁴	1	56	yes	complemented
66	C₂²×Dic₅₉	—	2	112	no	—
67	C₂²×Dic₅₉	C₂⁴	1	56	yes	complemented
68	C₂×C₄×D₅₉	—	2	896	no	—
69	C₂×C₄×D₅₉	C₂⁴	1	128	yes	complemented
70	C₂²⋊C₄×C₅₉	C₂⁴	1	64	yes	complemented
71	C₂²⋊Dic₅₉	C₂⁴	1	64	yes	complemented
72	(C₂×D₅₉)⋊C₄	C₂⁴	1	128	yes	complemented
73	C₂³×C₅₉	—	2	448	no	—
74	C₂³×C₅₉	—	2	112	no	—
75	C₂³×C₅₉	C₂×C₂²⋊C₄	1	64	yes	complemented
76	C₂³×C₅₉	C₂³×C₄	1	28	yes	—
77	C₂³×C₅₉	C₂³×C₄	1	112	yes	complemented
78	C₂³×C₅₉	C₂²×D₄	1	56	yes	—
79	C₂³×C₅₉	C₂⁵	1	8	yes	—
80	C₂³×C₅₉	C₂⁵	1	7	yes	—
81	C₂²×D₅₉	—	2	448	no	—
82	C₂²×D₅₉	—	2	112	no	—
83	C₂²×D₅₉	—	2	3584	no	—
84	C₂²×D₅₉	—	2	256	no	—
85	C₂²×D₅₉	C₂×C₂²⋊C₄	1	896	yes	direct product factor
86	C₂²×D₅₉	C₂³×C₄	1	112	yes	—
87	C₂²×D₅₉	C₂³×C₄	1	256	yes	complemented
88	C₂²×D₅₉	C₂³×C₄	1	112	yes	complemented
89	C₂²×D₅₉	C₂²×D₄	1	224	yes	—
90	C₂²×D₅₉	C₂⁵	1	8	yes	—
91	C₂²×D₅₉	C₂⁵	1	8	yes	—
92	C₂×C₄×C₅₉	—	2	112	no	—
93	C₂×C₄×C₅₉	C₂⁵	1	16	yes	complemented
94	C₂×Dic₅₉	—	2	112	no	—
95	C₂×Dic₅₉	C₂⁵	1	16	yes	complemented
96	C₄×D₅₉	—	2	256	no	—
97	C₂⁴×C₂²⋊C₄	—	59	1	no	maximal, Sylow
98	C₂²×C₅₉	—	2	16	no	—
99	C₂²×C₅₉	—	2	224	no	—
100	C₂²×C₅₉	C₂²×C₂²⋊C₄	1	112	yes	complemented
101	C₂²×C₅₉	C₂⁴×C₄	1	14	yes	—
102	C₂²×C₅₉	C₂⁴×C₄	1	16	yes	complemented
103	C₂²×C₅₉	C₂³×D₄	1	28	yes	—
104	C₂²×C₅₉	C₂⁶	1	1	yes	characteristic
105	C₂×D₅₉	—	2	16	no	—
106	C₂×D₅₉	—	2	224	no	—
107	C₂×D₅₉	—	2	512	no	—
108	C₂×D₅₉	C₂²×C₂²⋊C₄	1	448	yes	direct product factor
109	C₂×D₅₉	C₂⁴×C₄	1	16	yes	—
110	C₂×D₅₉	C₂⁴×C₄	1	16	yes	complemented
111	C₂×D₅₉	C₂³×D₄	1	32	yes	—
112	C₄×C₅₉	—	2	16	no	—
113	Dic₅₉	—	2	16	no	—
114	C₂⁷	—	59	1	no	—
115	C₂⁵×C₄	—	59	2	no	—
116	C₂³×C₂²⋊C₄	—	59	1	no	—
117	C₂³×C₂²⋊C₄	—	59	2	no	—
118	C₂³×C₂²⋊C₄	—	59	56	no	—
119	C₂³×C₂²⋊C₄	D₅₉	1	1	yes	characteristic, direct product factor
120	C₂×C₅₉	—	2	16	no	—
121	C₂×C₅₉	C₂³×C₂²⋊C₄	1	28	yes	complemented
122	C₂×C₅₉	C₂⁵×C₄	1	1	yes	characteristic
123	C₂×C₅₉	C₂⁴×D₄	1	2	yes	—
124	D₅₉	—	2	16	no	—
125	D₅₉	C₂³×C₂²⋊C₄	1	32	yes	direct product factor
126	C₂⁶	—	59	1	no	—
127	C₂⁶	—	59	32	no	—
128	C₂⁶	—	59	28	no	—
129	C₂⁶	—	59	1	no	—
130	C₂⁶	—	118	32	no	—
131	C₂⁶	C₂×D₅₉	1	1	yes	characteristic
132	C₂⁴×C₄	—	59	2	no	—
133	C₂⁴×C₄	—	59	56	no	—
134	C₂⁴×C₄	—	118	32	no	—
135	C₂⁴×C₄	C₂×D₅₉	1	2	yes	complemented
136	C₂²×C₂²⋊C₄	—	59	28	no	—
137	C₂²×C₂²⋊C₄	—	59	56	no	—
138	C₂²×C₂²⋊C₄	—	59	448	no	—
139	C₂²×C₂²⋊C₄	C₂×D₅₉	1	28	yes	direct product factor
140	C₅₉	C₂⁴×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
141	C₂⁵	—	2	16	no	—
142	C₂⁵	—	59	16	no	—
143	C₂⁵	—	59	32	no	—
144	C₂⁵	—	59	14	no	—
145	C₂⁵	—	59	16	no	—
146	C₂⁵	—	59	448	no	—
147	C₂⁵	—	59	112	no	—
148	C₂⁵	—	59	14	no	—
149	C₂⁵	—	118	512	no	—
150	C₂⁵	—	118	448	no	—
151	C₂⁵	—	118	16	no	—
152	C₂⁵	C₄×D₅₉	1	16	yes	complemented
153	C₂⁵	C₂²×D₅₉	1	14	yes	—
154	C₂⁵	C₂²×D₅₉	1	1	yes	characteristic
155	C₂³×C₄	—	2	16	no	—
156	C₂³×C₄	—	59	28	no	—
157	C₂³×C₄	—	59	224	no	—
158	C₂³×C₄	—	118	448	no	—
159	C₂³×C₄	—	118	16	no	—
160	C₂³×C₄	C₂²×D₅₉	1	28	yes	complemented
161	C₂×C₂²⋊C₄	—	59	112	no	—
162	C₂×C₂²⋊C₄	—	59	224	no	—
163	C₂×C₂²⋊C₄	—	59	512	no	—
164	C₂×C₂²⋊C₄	C₂²×D₅₉	1	112	yes	direct product factor
165	C₂⁴	—	2	128	no	—
166	C₂⁴	—	2	112	no	—
167	C₂⁴	—	59	256	no	—
168	C₂⁴	—	59	112	no	—
169	C₂⁴	—	59	224	no	—
170	C₂⁴	—	59	28	no	—
171	C₂⁴	—	59	112	no	—
172	C₂⁴	—	59	896	no	—
173	C₂⁴	—	59	64	no	—
174	C₂⁴	—	59	28	no	—
175	C₂⁴	—	118	896	no	—
176	C₂⁴	—	118	3584	no	—
177	C₂⁴	—	118	128	no	—
178	C₂⁴	—	118	112	no	—
179	C₂⁴	C₂×C₄×D₅₉	1	8	yes	—
180	C₂⁴	C₂×C₄×D₅₉	1	112	yes	complemented
181	C₂⁴	D₄×D₅₉	1	16	yes	—
182	C₂⁴	C₂³×D₅₉	1	28	yes	—
183	C₂⁴	C₂³×D₅₉	1	7	yes	—
184	C₂²×C₄	—	2	112	no	—
185	C₂²×C₄	—	59	56	no	—
186	C₂²×C₄	—	59	128	no	—
187	C₂²×C₄	—	118	896	no	—
188	C₂²×C₄	—	118	112	no	—
189	C₂²×C₄	C₂³×D₅₉	1	56	yes	complemented
190	C₂²⋊C₄	—	59	64	no	—
191	C₂²⋊C₄	—	59	128	no	—
192	C₂²⋊C₄	C₂³×D₅₉	1	64	yes	direct product factor
193	C₂³	—	2	112	no	—
194	C₂³	—	2	448	no	—
195	C₂³	—	59	112	no	—
196	C₂³	—	59	224	no	—
197	C₂³	—	59	8	no	—
198	C₂³	—	59	896	no	—
199	C₂³	—	59	112	no	—
200	C₂³	—	59	256	no	—
201	C₂³	—	59	8	no	—
202	C₂³	—	118	3584	no	—
203	C₂³	—	118	256	no	—
204	C₂³	—	118	112	no	—
205	C₂³	—	118	448	no	—
206	C₂³	C₂²⋊C₄×D₅₉	1	64	yes	direct product factor
207	C₂³	C₂²×C₄×D₅₉	1	28	yes	—
208	C₂³	C₂²×C₄×D₅₉	1	112	yes	complemented
209	C₂³	C₂×D₄×D₅₉	1	56	yes	—
210	C₂³	C₂⁴×D₅₉	1	8	yes	—
211	C₂³	C₂⁴×D₅₉	1	7	yes	—
212	C₂×C₄	—	2	112	no	—
213	C₂×C₄	—	59	16	no	—
214	C₂×C₄	—	118	256	no	—
215	C₂×C₄	—	118	112	no	—
216	C₂×C₄	C₂⁴×D₅₉	1	16	yes	complemented
217	C₂²	—	2	224	no	—
218	C₂²	—	2	16	no	—
219	C₂²	—	59	448	no	—
220	C₂²	—	59	16	no	—
221	C₂²	—	59	32	no	—
222	C₂²	—	59	16	no	—
223	C₂²	—	118	512	no	—
224	C₂²	—	118	224	no	—
225	C₂²	—	118	16	no	—
226	C₂²	C₂×C₂²⋊C₄×D₅₉	1	112	yes	direct product factor
227	C₂²	C₂³×C₄×D₅₉	1	14	yes	—
228	C₂²	C₂³×C₄×D₅₉	1	16	yes	complemented
229	C₂²	C₂²×D₄×D₅₉	1	28	yes	—
230	C₂²	C₂⁵×D₅₉	1	1	yes	characteristic
231	C₄	—	2	16	no	—
232	C₄	—	118	16	no	—
233	C₂	—	2	16	no	—
234	C₂	—	59	32	no	—
235	C₂	—	118	16	no	—
236	C₂	C₂²×C₂²⋊C₄×D₅₉	1	28	yes	direct product factor
237	C₂	C₂⁴×C₄×D₅₉	1	1	yes	characteristic
238	C₂	C₂³×D₄×D₅₉	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₅₉
commutator	C₂×C₅₉	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₉
Fitting	C₂³×C₂²⋊C₄×C₅₉	C₂
socle	C₂⁵×C₅₉	C₂³
prime radical	C₂⁶×D₅₉	C₂
cosocle	C₂²×C₅₉	C₂⁶
hypercenter	C₂³×C₂²⋊C₄	D₅₉
hypocenter	C₅₉	C₂⁴×C₂²⋊C₄
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×D₅₉	C₂	semidirect
3. C₂³×C₂²⋊Dic₅₉	C₂	semidirect
4. C₂³×(C₂×D₅₉)⋊C₄	C₂	semidirect
5. C₂²×C₂²⋊C₄×D₅₉	C₂	direct (N × H)
6. C₂³×C₂²⋊C₄×C₅₉	C₂	semidirect
8. C₂⁵×D₅₉	C₄	semidirect
12. C₂⁴×Dic₅₉	C₂²	semidirect
14. C₂³×C₄×D₅₉	C₂²	semidirect
15. C₂²×C₂²⋊Dic₅₉	C₂²	semidirect
16. C₂²×(C₂×D₅₉)⋊C₄	C₂²	semidirect
17. C₂×C₂²⋊C₄×D₅₉	C₂²	direct (N × H)
19. C₂⁴×C₄×C₅₉	C₂²	semidirect
20. C₂²×C₂²⋊C₄×C₅₉	C₂²	semidirect
25. C₂⁴×D₅₉	C₂×C₄	semidirect
26. C₂⁴×D₅₉	C₂×C₄	semidirect
32. C₂³×Dic₅₉	C₂³	semidirect
34. C₂²×C₄×D₅₉	C₂³	semidirect
35. C₂×C₂²⋊Dic₅₉	C₂³	semidirect
36. C₂×(C₂×D₅₉)⋊C₄	C₂³	semidirect
37. C₂²⋊C₄×D₅₉	C₂³	direct (N × H)
39. C₂⁵×C₅₉	C₂×C₄	semidirect
43. C₂³×C₄×C₅₉	C₂³	semidirect
44. C₂×C₂²⋊C₄×C₅₉	C₂³	semidirect
48. C₂⁴×C₅₉	C₂²×C₄	semidirect
56. C₂³×D₅₉	C₂²⋊C₄	direct (N × H)
58. C₂³×D₅₉	C₂²×C₄	semidirect
59. C₂³×D₅₉	C₂²×C₄	semidirect
65. C₂²×C₄×C₅₉	C₂⁴	semidirect
67. C₂²×Dic₅₉	C₂⁴	semidirect
69. C₂×C₄×D₅₉	C₂⁴	semidirect
70. C₂²⋊C₄×C₅₉	C₂⁴	semidirect
71. C₂²⋊Dic₅₉	C₂⁴	semidirect
72. (C₂×D₅₉)⋊C₄	C₂⁴	semidirect
75. C₂³×C₅₉	C₂×C₂²⋊C₄	semidirect
77. C₂³×C₅₉	C₂³×C₄	semidirect
85. C₂²×D₅₉	C₂×C₂²⋊C₄	direct (N × H)
87. C₂²×D₅₉	C₂³×C₄	semidirect
88. C₂²×D₅₉	C₂³×C₄	semidirect
93. C₂×C₄×C₅₉	C₂⁵	semidirect
95. C₂×Dic₅₉	C₂⁵	semidirect
100. C₂²×C₅₉	C₂²×C₂²⋊C₄	semidirect
102. C₂²×C₅₉	C₂⁴×C₄	semidirect
108. C₂×D₅₉	C₂²×C₂²⋊C₄	direct (N × H)
110. C₂×D₅₉	C₂⁴×C₄	semidirect
119. C₂³×C₂²⋊C₄	D₅₉	direct (N × H)
121. C₂×C₅₉	C₂³×C₂²⋊C₄	semidirect
125. D₅₉	C₂³×C₂²⋊C₄	direct (N × H)
135. C₂⁴×C₄	C₂×D₅₉	semidirect
139. C₂²×C₂²⋊C₄	C₂×D₅₉	direct (N × H)
140. C₅₉	C₂⁴×C₂²⋊C₄	semidirect
152. C₂⁵	C₄×D₅₉	semidirect
160. C₂³×C₄	C₂²×D₅₉	semidirect
164. C₂×C₂²⋊C₄	C₂²×D₅₉	direct (N × H)
180. C₂⁴	C₂×C₄×D₅₉	semidirect
189. C₂²×C₄	C₂³×D₅₉	semidirect
192. C₂²⋊C₄	C₂³×D₅₉	direct (N × H)
206. C₂³	C₂²⋊C₄×D₅₉	direct (N × H)
208. C₂³	C₂²×C₄×D₅₉	semidirect
216. C₂×C₄	C₂⁴×D₅₉	semidirect
226. C₂²	C₂×C₂²⋊C₄×D₅₉	direct (N × H)
228. C₂²	C₂³×C₄×D₅₉	semidirect
236. C₂	C₂²×C₂²⋊C₄×D₅₉	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁶×D₅₉	2
C₂⁴×C₄×D₅₉	2
C₂³×C₂²⋊Dic₅₉	2
C₂³×(C₂×D₅₉)⋊C₄	2
C₂²×C₂²⋊C₄×D₅₉	2
C₂³×C₂²⋊C₄×C₅₉	2
C₂⁴×C₂²⋊C₄	59

Subgroups — C23×C22⋊C4×D59 #29495239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂²⋊C₄×D₅₉ #29495239