Subgroups — C₅₉×#13271 #28425380

Order 15104.

332 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	C₅₉×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₉×#364	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1772	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#1389	C₂	1	2	yes	maximal
5	C₅₉×#1384	C₂	1	2	yes	maximal
6	C₅₉×#533	—	2	1	no	—
7	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
8	C₅₉×#500	—	2	1	no	—
9	C₅₉×#499	—	2	1	no	—
10	C₅₉×#499	C₄	1	2	yes	complemented
11	C₅₉×#498	C₄	1	2	yes	complemented
12	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#427	—	2	1	no	—
14	C₅₉×#427	C₂²	1	2	yes	—
15	C₅₉×#427	C₂²	1	1	yes	characteristic
16	C₅₉×#405	—	2	1	no	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#364	—	2	1	no	—
19	C₅₉×#364	—	2	1	no	—
20	C₅₉×#339	—	2	1	no	—
21	C₂×M₁₆×C₅₉	—	2	1	no	—
22	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
24	C₄⋊Q₈×C₅₉	—	2	2	no	—
25	#125×C₅₉	—	2	2	no	—
26	C₂²⋊Q₈×C₅₉	—	4	2	no	—
27	C₄×Q₈×C₅₉	—	4	2	no	—
28	#117×C₅₉	—	2	1	no	—
29	#117×C₅₉	—	2	2	no	—
30	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
31	M₃₂×C₅₉	—	2	2	no	—
32	M₃₂×C₅₉	—	2	2	no	—
33	C₂×C₁₆×C₅₉	—	2	2	no	—
34	#108×C₅₉	—	2	1	no	—
35	#108×C₅₉	Q₈	1	2	yes	complemented
36	#107×C₅₉	—	2	2	no	—
37	#107×C₅₉	—	2	2	no	—
38	#107×C₅₉	—	2	1	no	—
39	#106×C₅₉	—	2	2	no	—
40	#106×C₅₉	C₂×C₄	1	4	yes	—
41	#106×C₅₉	Q₈	1	2	yes	complemented
42	#97×C₅₉	—	4	1	no	—
43	C₄×C₈×C₅₉	—	4	1	no	—
44	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
48	C₂×Q₈×C₅₉	—	4	2	no	—
49	C₂²×C₄×C₅₉	—	2	1	no	—
50	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	M₁₆×C₅₉	—	2	2	no	—
52	M₁₆×C₅₉	—	4	1	no	—
53	C₂×C₈×C₅₉	—	4	1	no	—
54	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
57	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
58	C₂×C₈×C₅₉	Q₁₆	1	1	yes	characteristic
59	C₂×C₈×C₅₉	Q₁₆	1	1	yes	characteristic
60	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
65	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
66	C₄⋊C₄×C₅₉	—	2	4	no	—
67	C₄⋊C₄×C₅₉	—	2	2	no	—
68	C₄⋊C₄×C₅₉	—	4	4	no	—
69	C₄⋊C₄×C₅₉	—	4	2	no	—
70	C₂²⋊C₄×C₅₉	—	4	1	no	—
71	C₂²⋊C₄×C₅₉	—	4	2	no	—
72	C₄²×C₅₉	—	2	1	no	—
73	C₄²×C₅₉	—	4	1	no	—
74	C₄²×C₅₉	—	4	2	no	—
75	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₁₆×C₅₉	—	4	2	no	—
78	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	Q₈×C₅₉	—	8	4	no	—
80	C₂×C₄×C₅₉	—	2	1	no	—
81	C₂×C₄×C₅₉	—	2	1	no	—
82	C₂×C₄×C₅₉	—	4	2	no	—
83	C₂×C₄×C₅₉	—	4	4	no	—
84	C₂×C₄×C₅₉	—	4	2	no	—
85	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₈×C₅₉	—	4	2	no	—
92	C₈×C₅₉	#103	1	1	yes	characteristic
93	C₈×C₅₉	#103	1	1	yes	characteristic
94	C₈×C₅₉	#103	1	1	yes	characteristic
95	C₈×C₅₉	#103	1	1	yes	characteristic
96	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
97	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
98	C₈×C₅₉	#124	1	1	yes	characteristic
99	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
100	#13271	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₅₉	—	4	1	no	—
102	C₂²×C₅₉	#418	1	1	yes	characteristic
103	C₂²×C₅₉	#476	1	1	yes	characteristic
104	C₂²×C₅₉	#477	1	1	yes	characteristic
105	C₄×C₅₉	—	2	2	no	—
106	C₄×C₅₉	—	4	1	no	—
107	C₄×C₅₉	—	8	2	no	—
108	C₄×C₅₉	—	8	4	no	—
109	C₄×C₅₉	#391	1	1	yes	characteristic
110	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
111	C₄×C₅₉	#476	1	1	yes	characteristic
112	C₄×C₅₉	#477	1	1	yes	characteristic
113	#3160	C₂×C₅₉	1	1	yes	characteristic
114	C₂×#364	C₂×C₅₉	1	1	yes	characteristic
115	#1772	C₂×C₅₉	1	1	yes	characteristic
116	#1389	C₂×C₅₉	1	2	yes	—
117	#1384	C₂×C₅₉	1	2	yes	—
118	C₂×C₅₉	—	2	1	no	—
119	C₂×C₅₉	—	2	1	no	—
120	C₂×C₅₉	—	2	1	no	—
121	C₂×C₅₉	#1987	1	1	yes	characteristic
122	#533	—	2	1	no	—
123	C₂×M₃₂	C₂²×C₅₉	1	2	yes	—
124	#500	—	2	1	no	—
125	#499	—	2	1	no	—
126	#499	C₄×C₅₉	1	2	yes	complemented
127	#498	C₄×C₅₉	1	2	yes	complemented
128	C₂×#108	C₂²×C₅₉	1	1	yes	characteristic
129	#427	—	2	1	no	—
130	#427	C₂²×C₅₉	1	2	yes	—
131	#427	C₂²×C₅₉	1	1	yes	characteristic
132	#405	—	2	1	no	—
133	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
134	#364	—	2	1	no	—
135	#364	—	2	1	no	—
136	#339	—	2	1	no	—
137	C₅₉	#13271	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×M₁₆	—	2	1	no	—
139	C₂²×C₈	D₄×C₅₉	1	1	yes	characteristic
140	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
141	C₄⋊Q₈	—	2	2	no	—
142	#125	—	2	2	no	—
143	C₂²⋊Q₈	—	4	2	no	—
144	C₄×Q₈	—	4	2	no	—
145	#117	—	2	2	no	—
146	#117	—	2	1	no	—
147	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
148	M₃₂	—	2	2	no	—
149	M₃₂	—	2	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	#108	—	2	1	no	—
152	#108	Q₈×C₅₉	1	2	yes	complemented
153	#107	—	2	2	no	—
154	#107	—	2	2	no	—
155	#107	—	2	1	no	—
156	#106	—	2	2	no	—
157	#106	C₂×C₄×C₅₉	1	4	yes	—
158	#106	Q₈×C₅₉	1	2	yes	complemented
159	#97	—	4	1	no	—
160	C₄×C₈	—	4	1	no	—
161	C₄×C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
162	C₄×C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
163	C₄×C₈	D₄×C₅₉	1	1	yes	characteristic
164	C₄×C₈	D₄×C₅₉	1	1	yes	characteristic
165	C₂×Q₈	—	4	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
168	M₁₆	—	2	2	no	—
169	M₁₆	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
173	C₂×C₈	SD₁₆×C₅₉	1	1	yes	characteristic
174	C₂×C₈	SD₁₆×C₅₉	1	1	yes	characteristic
175	C₂×C₈	Q₁₆×C₅₉	1	1	yes	characteristic
176	C₂×C₈	Q₁₆×C₅₉	1	1	yes	characteristic
177	C₂×C₈	C₂²×C₄×C₅₉	1	1	yes	characteristic
178	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
179	C₂×C₈	C₂×Q₈×C₅₉	1	1	yes	characteristic
180	C₂×C₈	C₂×Q₈×C₅₉	1	1	yes	characteristic
181	C₂×C₈	#41×C₅₉	1	1	yes	characteristic
182	C₂×C₈	#41×C₅₉	1	1	yes	characteristic
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₂²⋊C₄	—	4	2	no	—
188	C₂²⋊C₄	—	4	1	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	4	1	no	—
191	C₄²	—	4	2	no	—
192	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
193	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
194	C₁₆	—	4	2	no	—
195	C₂³	AGL(1,ℤ/8)×C₅₉	1	1	yes	characteristic
196	Q₈	—	8	4	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
203	C₂×C₄	C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
204	C₂×C₄	C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
205	C₂×C₄	C₂×SD₁₆×C₅₉	1	1	yes	characteristic
206	C₂×C₄	C₂×Q₁₆×C₅₉	1	1	yes	characteristic
207	C₂×C₄	AGL(1,ℤ/8)×C₅₉	1	1	yes	characteristic
208	C₈	—	4	2	no	—
209	C₈	#103×C₅₉	1	1	yes	characteristic
210	C₈	#103×C₅₉	1	1	yes	characteristic
211	C₈	#103×C₅₉	1	1	yes	characteristic
212	C₈	#103×C₅₉	1	1	yes	characteristic
213	C₈	C₄×Q₈×C₅₉	1	1	yes	characteristic
214	C₈	C₄×Q₈×C₅₉	1	1	yes	characteristic
215	C₈	#124×C₅₉	1	1	yes	characteristic
216	C₈	C₄⋊Q₈×C₅₉	1	1	yes	characteristic
217	C₂²	—	4	1	no	—
218	C₂²	C₅₉×#418	1	1	yes	characteristic
219	C₂²	C₅₉×#476	1	1	yes	characteristic
220	C₂²	C₅₉×#477	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	8	4	no	—
224	C₄	—	8	2	no	—
225	C₄	C₅₉×#391	1	1	yes	characteristic
226	C₄	C₂×#103×C₅₉	1	1	yes	characteristic
227	C₄	C₅₉×#476	1	1	yes	characteristic
228	C₄	C₅₉×#477	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	C₅₉×#1987	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₉	#476
commutator	C₂×C₈	C₂²×C₄×C₅₉
Frattini, cosocle	C₂²×C₈	C₂³×C₅₉
socle	C₂×C₅₉	#1987
prime radical	C₂³×C₅₉	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₉ ◁ C₂²×C₄×C₅₉ ◁ C₂×C₄×C₈×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₅₉×#499	C₄	semidirect
11. C₅₉×#498	C₄	semidirect
35. #108×C₅₉	Q₈	semidirect
41. #106×C₅₉	Q₈	semidirect
100. #13271	C₅₉	direct (N × H)
126. #499	C₄×C₅₉	semidirect
127. #498	C₄×C₅₉	semidirect
137. C₅₉	#13271	direct (N × H)
152. #108	Q₈×C₅₉	semidirect
158. #106	Q₈×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#3160	2
C₂×C₅₉×#364	2
C₅₉×#1772	2
C₅₉×#1389	2
C₅₉×#1384	2
#13271	59

Subgroups — C59×#13271 #28425380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#13271 #28425380