Subgroups — C₅₉×#13251 #28425360

Order 15104.

478 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₉×#359	C₂	1	4	yes	maximal, complemented
4	C₄×#108×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#135×C₅₉	—	2	1	no	—
6	C₂×#135×C₅₉	C₂²	1	2	yes	—
7	C₅₉×#532	—	2	1	no	—
8	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
9	C₅₉×#495	—	2	1	no	—
10	C₅₉×#495	C₄	1	2	yes	—
11	C₅₉×#494	C₄	1	1	yes	characteristic
12	C₅₉×#493	C₄	1	1	yes	characteristic
13	C₅₉×#492	—	2	1	no	—
14	C₂×#108×C₅₉	C₂²	1	2	yes	—
15	C₄×M₁₆×C₅₉	—	2	1	no	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×C₄⋊C₁₆	—	2	1	no	—
18	C₅₉×C₄⋊C₁₆	—	2	1	no	—
19	C₅₉×#359	—	2	8	no	—
20	C₅₉×#341	—	2	1	no	—
21	C₂×#41×C₅₉	—	2	2	no	—
22	#135×C₅₉	—	2	4	no	—
23	#135×C₅₉	—	2	4	no	—
24	#135×C₅₉	—	4	4	no	—
25	C₂×Q₁₆×C₅₉	—	2	1	no	—
26	C₂×Q₁₆×C₅₉	C₂×C₄	1	2	yes	—
27	C₂×SD₁₆×C₅₉	—	2	2	no	—
28	C₂×SD₁₆×C₅₉	—	2	2	no	—
29	C₂×D₈×C₅₉	—	2	1	no	—
30	C₂×D₈×C₅₉	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₅₉	—	2	2	no	—
32	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₅₉	—	2	1	no	—
35	#127×C₅₉	—	2	1	no	—
36	#124×C₅₉	—	2	2	no	—
37	#121×C₅₉	—	4	1	no	—
38	#121×C₅₉	—	4	1	no	—
39	C₄×D₄×C₅₉	—	4	1	no	—
40	C₄×D₄×C₅₉	—	4	1	no	—
41	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₅₉	—	2	4	no	—
43	C₂×C₁₆×C₅₉	—	2	2	no	—
44	#108×C₅₉	—	2	2	no	—
45	#108×C₅₉	D₄	1	4	yes	complemented
46	#97×C₅₉	—	4	1	no	—
47	C₄×C₈×C₅₉	—	4	1	no	—
48	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
52	#41×C₅₉	—	4	4	no	—
53	#41×C₅₉	—	4	4	no	—
54	C₂×Q₈×C₅₉	—	2	2	no	—
55	C₂×D₄×C₅₉	—	2	2	no	—
56	C₂×D₄×C₅₉	—	4	2	no	—
57	C₂×D₄×C₅₉	—	4	1	no	—
58	C₂×D₄×C₅₉	—	4	1	no	—
59	C₂²×C₄×C₅₉	—	4	2	no	—
60	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₅₉	—	2	4	no	—
64	Q₁₆×C₅₉	—	4	2	no	—
65	SD₁₆×C₅₉	—	4	4	no	—
66	SD₁₆×C₅₉	—	4	2	no	—
67	SD₁₆×C₅₉	—	4	2	no	—
68	D₈×C₅₉	—	2	4	no	—
69	D₈×C₅₉	—	4	2	no	—
70	M₁₆×C₅₉	—	2	4	no	—
71	M₁₆×C₅₉	—	4	2	no	—
72	C₂×C₈×C₅₉	—	2	2	no	—
73	C₂×C₈×C₅₉	—	4	2	no	—
74	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	#41	1	2	yes	—
80	C₄⋊C₄×C₅₉	—	4	1	no	—
81	C₄⋊C₄×C₅₉	—	4	1	no	—
82	C₂²⋊C₄×C₅₉	—	4	2	no	—
83	C₂²⋊C₄×C₅₉	—	4	2	no	—
84	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₅₉	D₈	1	1	yes	characteristic
87	C₄²×C₅₉	D₈	1	1	yes	characteristic
88	C₁₆×C₅₉	—	4	2	no	—
89	C₂³×C₅₉	—	4	2	no	—
90	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
91	Q₈×C₅₉	—	4	2	no	—
92	Q₈×C₅₉	—	4	2	no	—
93	D₄×C₅₉	—	4	2	no	—
94	D₄×C₅₉	—	4	2	no	—
95	D₄×C₅₉	—	8	2	no	—
96	D₄×C₅₉	—	8	4	no	—
97	D₄×C₅₉	—	8	2	no	—
98	C₂×C₄×C₅₉	—	2	2	no	—
99	C₂×C₄×C₅₉	—	4	1	no	—
100	C₂×C₄×C₅₉	—	4	2	no	—
101	C₂×C₄×C₅₉	—	8	4	no	—
102	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
103	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
104	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
105	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₉	#121	1	2	yes	—
108	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
110	C₈×C₅₉	—	2	1	no	—
111	C₈×C₅₉	—	2	1	no	—
112	C₈×C₅₉	—	4	4	no	—
113	C₈×C₅₉	C₄×D₄	1	2	yes	—
114	C₈×C₅₉	#124	1	1	yes	characteristic
115	C₈×C₅₉	#127	1	1	yes	characteristic
116	#13251	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂²×C₅₉	—	2	1	no	—
118	C₂²×C₅₉	—	2	1	no	—
119	C₂²×C₅₉	—	8	2	no	—
120	C₂²×C₅₉	—	8	2	no	—
121	C₂²×C₅₉	—	8	2	no	—
122	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
123	C₂²×C₅₉	#460	1	2	yes	—
124	C₄×C₅₉	—	2	1	no	—
125	C₄×C₅₉	—	2	1	no	—
126	C₄×C₅₉	—	2	1	no	—
127	C₄×C₅₉	—	8	2	no	—
128	C₄×C₅₉	#359	1	2	yes	complemented
129	C₄×C₅₉	#390	1	1	yes	characteristic
130	C₄×C₅₉	#418	1	1	yes	characteristic
131	C₄×C₅₉	#459	1	2	yes	—
132	#3147	C₂×C₅₉	1	1	yes	characteristic
133	C₂×C₄⋊C₁₆	C₂×C₅₉	1	1	yes	characteristic, complemented
134	C₂×#359	C₂×C₅₉	1	4	yes	complemented
135	C₄×#108	C₂×C₅₉	1	1	yes	characteristic, complemented
136	C₂×C₅₉	—	2	2	no	—
137	C₂×C₅₉	—	16	2	no	—
138	C₂×C₅₉	#1982	1	1	yes	characteristic
139	C₂×C₅₉	C₂×#359	1	1	yes	characteristic
140	C₂×C₅₉	#2143	1	1	yes	characteristic
141	C₂×#135	—	2	1	no	—
142	C₂×#135	C₂²×C₅₉	1	2	yes	—
143	#532	—	2	1	no	—
144	C₂²×C₁₆	C₂²×C₅₉	1	2	yes	—
145	#495	—	2	1	no	—
146	#495	C₄×C₅₉	1	2	yes	—
147	#494	C₄×C₅₉	1	1	yes	characteristic
148	#493	C₄×C₅₉	1	1	yes	characteristic
149	#492	—	2	1	no	—
150	C₂×#108	C₂²×C₅₉	1	2	yes	—
151	C₄×M₁₆	—	2	1	no	—
152	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
153	C₄⋊C₁₆	—	2	1	no	—
154	C₄⋊C₁₆	—	2	1	no	—
155	#359	—	2	8	no	—
156	#341	—	2	1	no	—
157	C₅₉	#13251	1	1	yes	characteristic, direct product factor, Sylow
158	C₂×#41	—	2	2	no	—
159	#135	—	2	4	no	—
160	#135	—	2	4	no	—
161	#135	—	4	4	no	—
162	C₂×Q₁₆	—	2	1	no	—
163	C₂×Q₁₆	C₂×C₄×C₅₉	1	2	yes	—
164	C₂×SD₁₆	—	2	2	no	—
165	C₂×SD₁₆	—	2	2	no	—
166	C₂×D₈	—	2	1	no	—
167	C₂×D₈	C₂×C₄×C₅₉	1	2	yes	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂²×C₈	D₄×C₅₉	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
171	C₄⋊Q₈	—	2	1	no	—
172	#127	—	2	1	no	—
173	#124	—	2	2	no	—
174	#121	—	4	1	no	—
175	#121	—	4	1	no	—
176	C₄×D₄	—	4	1	no	—
177	C₄×D₄	—	4	1	no	—
178	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
179	C₂×C₁₆	—	2	4	no	—
180	C₂×C₁₆	—	2	2	no	—
181	#108	—	2	2	no	—
182	#108	D₄×C₅₉	1	4	yes	complemented
183	#97	—	4	1	no	—
184	C₄×C₈	—	4	1	no	—
185	C₄×C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
186	C₄×C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
187	C₄×C₈	D₄×C₅₉	1	1	yes	characteristic
188	C₄×C₈	D₄×C₅₉	1	1	yes	characteristic
189	#41	—	4	4	no	—
190	#41	—	4	4	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	SD₁₆×C₅₉	1	1	yes	characteristic
198	C₂²×C₄	SD₁₆×C₅₉	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
200	Q₁₆	—	2	4	no	—
201	Q₁₆	—	4	2	no	—
202	SD₁₆	—	4	4	no	—
203	SD₁₆	—	4	2	no	—
204	SD₁₆	—	4	2	no	—
205	D₈	—	2	4	no	—
206	D₈	—	4	2	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	4	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
213	C₂×C₈	C₂²×C₄×C₅₉	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
215	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
216	C₂×C₈	#41×C₅₉	1	2	yes	—
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₂²⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
222	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
223	C₄²	D₈×C₅₉	1	1	yes	characteristic
224	C₄²	D₈×C₅₉	1	1	yes	characteristic
225	C₁₆	—	4	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	C₂×SD₁₆×C₅₉	1	1	yes	characteristic
228	Q₈	—	4	2	no	—
229	Q₈	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	8	2	no	—
233	D₄	—	8	4	no	—
234	D₄	—	8	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	8	4	no	—
239	C₂×C₄	#102×C₅₉	1	1	yes	characteristic
240	C₂×C₄	#102×C₅₉	1	1	yes	characteristic
241	C₂×C₄	#102×C₅₉	1	1	yes	characteristic
242	C₂×C₄	#102×C₅₉	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
244	C₂×C₄	#121×C₅₉	1	2	yes	—
245	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
246	C₂×C₄	AGL(1,ℤ/8)×C₅₉	1	2	yes	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	4	4	no	—
250	C₈	C₄×D₄×C₅₉	1	2	yes	—
251	C₈	#124×C₅₉	1	1	yes	characteristic
252	C₈	#127×C₅₉	1	1	yes	characteristic
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	8	2	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	C₂×#102×C₅₉	1	1	yes	characteristic
259	C₂²	C₅₉×#460	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	8	2	no	—
264	C₄	C₅₉×#359	1	2	yes	complemented
265	C₄	C₅₉×#390	1	1	yes	characteristic
266	C₄	C₅₉×#418	1	1	yes	characteristic
267	C₄	C₅₉×#459	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	16	2	no	—
270	C₂	C₅₉×#1982	1	1	yes	characteristic
271	C₂	C₂×C₅₉×#359	1	1	yes	characteristic
272	C₂	C₅₉×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅₉
Frattini, cosocle	C₂²×C₈	C₂³×C₅₉
socle	C₂²×C₅₉	C₂×#102
prime radical	C₅₉×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ C₂×C₄×C₈×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₉×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₅₉×#359	C₂	semidirect
4. C₄×#108×C₅₉	C₂	semidirect
45. #108×C₅₉	D₄	semidirect
116. #13251	C₅₉	direct (N × H)
128. C₄×C₅₉	#359	semidirect
133. C₂×C₄⋊C₁₆	C₂×C₅₉	semidirect
134. C₂×#359	C₂×C₅₉	semidirect
135. C₄×#108	C₂×C₅₉	semidirect
157. C₅₉	#13251	direct (N × H)
182. #108	D₄×C₅₉	semidirect
264. C₄	C₅₉×#359	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#3147	2
C₂×C₅₉×C₄⋊C₁₆	2
C₂×C₅₉×#359	2
C₄×#108×C₅₉	2
#13251	59

Subgroups — C59×#13251 #28425360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#13251 #28425360