Subgroups — C₂×C₅₉×#2250 #28445823

Order 15104.

930 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SD₃₂×C₅₉	C₂	1	2	yes	maximal, complemented
2	C₂²×D₁₆×C₅₉	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₉×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₉×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#2250	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅₉×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
8	C₂²×SD₁₆×C₅₉	—	2	1	no	—
9	C₂²×D₈×C₅₉	—	2	1	no	—
10	C₂²×D₈×C₅₉	C₂²	1	2	yes	—
11	C₂²×D₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×#127×C₅₉	—	2	1	no	—
13	C₂×#124×C₅₉	—	2	1	no	—
14	C₂×SD₃₂×C₅₉	—	2	8	no	—
15	C₂×SD₃₂×C₅₉	C₂²	1	8	yes	complemented
16	C₂×D₁₆×C₅₉	—	2	8	no	—
17	C₂×D₁₆×C₅₉	C₂²	1	8	yes	complemented
18	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
19	C₅₉×#495	—	2	2	no	—
20	C₅₉×#495	C₂²	1	4	yes	complemented
21	C₅₉×#493	—	2	2	no	—
22	C₅₉×#493	C₂²	1	4	yes	complemented
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₅₉×#346	C₂²	1	4	yes	complemented
25	C₂²×Q₈×C₅₉	—	2	1	no	—
26	C₂²×D₄×C₅₉	—	2	1	no	—
27	C₂²×D₄×C₅₉	—	2	2	no	—
28	C₂²×D₄×C₅₉	—	4	2	no	—
29	C₂×Q₁₆×C₅₉	—	2	4	no	—
30	C₂×Q₁₆×C₅₉	—	2	1	no	—
31	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
32	C₂×SD₁₆×C₅₉	—	2	4	no	—
33	C₂×SD₁₆×C₅₉	—	4	4	no	—
34	C₂×D₈×C₅₉	—	2	4	no	—
35	C₂×D₈×C₅₉	—	2	4	no	—
36	C₂×D₈×C₅₉	—	2	4	no	—
37	C₂×D₈×C₅₉	—	2	2	no	—
38	C₂×D₈×C₅₉	—	2	4	no	—
39	C₂×D₈×C₅₉	—	2	1	no	—
40	C₂×D₈×C₅₉	—	4	4	no	—
41	C₂×D₈×C₅₉	C₂³	1	4	yes	—
42	C₂×D₈×C₅₉	C₂³	1	2	yes	—
43	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
45	#127×C₅₉	—	2	4	no	—
46	#127×C₅₉	—	4	2	no	—
47	#124×C₅₉	—	2	4	no	—
48	#124×C₅₉	—	4	2	no	—
49	C₂×C₂²⋊C₄×C₅₉	—	4	2	no	—
50	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
51	SD₃₂×C₅₉	—	2	16	no	—
52	D₁₆×C₅₉	—	2	16	no	—
53	C₂×C₁₆×C₅₉	D₄	1	8	yes	complemented
54	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
55	C₄×C₈×C₅₉	D₄	1	2	yes	—
56	C₄×C₈×C₅₉	C₂³	1	2	yes	—
57	C₂⁴×C₅₉	—	4	1	no	—
58	C₂⁴×C₅₉	—	4	2	no	—
59	C₂×Q₈×C₅₉	—	2	2	no	—
60	C₂×Q₈×C₅₉	—	4	1	no	—
61	C₂×Q₈×C₅₉	—	4	2	no	—
62	C₂×Q₈×C₅₉	—	4	2	no	—
63	C₂×D₄×C₅₉	—	2	2	no	—
64	C₂×D₄×C₅₉	—	2	4	no	—
65	C₂×D₄×C₅₉	—	4	2	no	—
66	C₂×D₄×C₅₉	—	4	4	no	—
67	C₂×D₄×C₅₉	—	4	4	no	—
68	C₂×D₄×C₅₉	—	4	2	no	—
69	C₂×D₄×C₅₉	—	4	1	no	—
70	C₂×D₄×C₅₉	—	4	8	no	—
71	C₂×D₄×C₅₉	—	4	2	no	—
72	C₂×D₄×C₅₉	—	8	8	no	—
73	C₂²×C₄×C₅₉	—	4	1	no	—
74	C₂²×C₄×C₅₉	D₈	1	2	yes	—
75	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
76	Q₁₆×C₅₉	—	2	4	no	—
77	Q₁₆×C₅₉	—	4	2	no	—
78	SD₁₆×C₅₉	—	4	8	no	—
79	D₈×C₅₉	—	2	4	no	—
80	D₈×C₅₉	—	2	4	no	—
81	D₈×C₅₉	—	2	4	no	—
82	D₈×C₅₉	—	4	2	no	—
83	D₈×C₅₉	—	4	4	no	—
84	D₈×C₅₉	—	4	8	no	—
85	C₂×C₈×C₅₉	D₈	1	4	yes	—
86	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
88	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
90	C₂²⋊C₄×C₅₉	—	4	8	no	—
91	C₄²×C₅₉	D₈	1	2	yes	—
92	C₄²×C₅₉	C₂×D₄	1	2	yes	—
93	C₁₆×C₅₉	C₂×D₄	1	8	yes	complemented
94	C₂³×C₅₉	—	4	2	no	—
95	C₂³×C₅₉	—	4	4	no	—
96	C₂³×C₅₉	—	8	4	no	—
97	C₂³×C₅₉	—	8	2	no	—
98	C₂³×C₅₉	—	8	8	no	—
99	C₂³×C₅₉	—	8	4	no	—
100	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
101	Q₈×C₅₉	—	4	2	no	—
102	Q₈×C₅₉	—	4	2	no	—
103	Q₈×C₅₉	—	8	2	no	—
104	D₄×C₅₉	—	4	2	no	—
105	D₄×C₅₉	—	4	4	no	—
106	D₄×C₅₉	—	4	4	no	—
107	D₄×C₅₉	—	4	2	no	—
108	D₄×C₅₉	—	8	16	no	—
109	D₄×C₅₉	—	8	4	no	—
110	D₄×C₅₉	—	8	2	no	—
111	C₂×C₄×C₅₉	—	2	4	no	—
112	C₂×C₄×C₅₉	—	4	2	no	—
113	C₂×C₄×C₅₉	—	8	2	no	—
114	C₂×C₄×C₅₉	#127	1	2	yes	—
115	C₂×C₄×C₅₉	C₂×D₈	1	4	yes	—
116	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
119	C₈×C₅₉	#127	1	2	yes	—
120	C₈×C₅₉	C₂×D₈	1	4	yes	—
121	C₈×C₅₉	C₂²×D₄	1	2	yes	—
122	C₂×#2250	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₅₉	—	8	4	no	—
124	C₂²×C₅₉	—	8	2	no	—
125	C₂²×C₅₉	—	8	8	no	—
126	C₂²×C₅₉	—	8	4	no	—
127	C₂²×C₅₉	—	16	8	no	—
128	C₂²×C₅₉	—	16	4	no	—
129	C₂²×C₅₉	#493	1	2	yes	—
130	C₂²×C₅₉	#509	1	4	yes	—
131	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₅₉	—	2	4	no	—
133	C₄×C₅₉	—	8	2	no	—
134	C₄×C₅₉	#493	1	2	yes	—
135	C₄×C₅₉	C₂×#127	1	1	yes	characteristic
136	C₄×C₅₉	C₂²×D₈	1	1	yes	characteristic
137	C₂²×SD₃₂	C₂×C₅₉	1	2	yes	complemented
138	C₂²×D₁₆	C₂×C₅₉	1	2	yes	complemented
139	C₂×#495	C₂×C₅₉	1	1	yes	characteristic, complemented
140	C₂×#493	C₂×C₅₉	1	1	yes	characteristic, complemented
141	#2250	C₂×C₅₉	1	8	yes	direct product factor
142	C₂×#346	C₂×C₅₉	1	1	yes	characteristic, complemented
143	C₂×C₅₉	—	16	4	no	—
144	C₂×C₅₉	—	16	2	no	—
145	C₂×C₅₉	#2250	1	4	yes	direct product factor
146	C₂×C₅₉	C₂×#493	1	1	yes	characteristic
147	C₂×C₅₉	C₂×#509	1	2	yes	—
148	C₂²×Q₁₆	C₂²×C₅₉	1	1	yes	characteristic
149	C₂²×SD₁₆	—	2	1	no	—
150	C₂²×D₈	—	2	1	no	—
151	C₂²×D₈	C₂²×C₅₉	1	2	yes	—
152	C₂²×D₈	C₂²×C₅₉	1	1	yes	characteristic
153	C₂×#127	—	2	1	no	—
154	C₂×#124	—	2	1	no	—
155	C₂×SD₃₂	—	2	8	no	—
156	C₂×SD₃₂	C₂²×C₅₉	1	8	yes	complemented
157	C₂×D₁₆	—	2	8	no	—
158	C₂×D₁₆	C₂²×C₅₉	1	8	yes	complemented
159	C₂²×C₁₆	C₂²×C₅₉	1	2	yes	—
160	#495	—	2	2	no	—
161	#495	C₂²×C₅₉	1	4	yes	complemented
162	#493	—	2	2	no	—
163	#493	C₂²×C₅₉	1	4	yes	complemented
164	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
165	#346	C₂²×C₅₉	1	4	yes	complemented
166	C₅₉	C₂×#2250	1	1	yes	characteristic, direct product factor, Sylow
167	C₂²×Q₈	—	2	1	no	—
168	C₂²×D₄	—	2	1	no	—
169	C₂²×D₄	—	2	2	no	—
170	C₂²×D₄	—	4	2	no	—
171	C₂×Q₁₆	—	2	4	no	—
172	C₂×Q₁₆	—	2	1	no	—
173	C₂×Q₁₆	C₂³×C₅₉	1	2	yes	—
174	C₂×SD₁₆	—	2	4	no	—
175	C₂×SD₁₆	—	4	4	no	—
176	C₂×D₈	—	2	4	no	—
177	C₂×D₈	—	2	4	no	—
178	C₂×D₈	—	2	4	no	—
179	C₂×D₈	—	2	4	no	—
180	C₂×D₈	—	2	2	no	—
181	C₂×D₈	—	2	1	no	—
182	C₂×D₈	—	4	4	no	—
183	C₂×D₈	C₂³×C₅₉	1	4	yes	—
184	C₂×D₈	C₂³×C₅₉	1	2	yes	—
185	C₂²×C₈	D₄×C₅₉	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
187	#127	—	2	4	no	—
188	#127	—	4	2	no	—
189	#124	—	2	4	no	—
190	#124	—	4	2	no	—
191	C₂×C₂²⋊C₄	—	4	2	no	—
192	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
193	SD₃₂	—	2	16	no	—
194	D₁₆	—	2	16	no	—
195	C₂×C₁₆	D₄×C₅₉	1	8	yes	complemented
196	C₂×C₁₆	C₂³×C₅₉	1	4	yes	—
197	C₄×C₈	D₄×C₅₉	1	2	yes	—
198	C₄×C₈	C₂³×C₅₉	1	2	yes	—
199	C₂⁴	—	4	1	no	—
200	C₂⁴	—	4	2	no	—
201	C₂×Q₈	—	2	2	no	—
202	C₂×Q₈	—	4	1	no	—
203	C₂×Q₈	—	4	2	no	—
204	C₂×Q₈	—	4	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	4	no	—
209	C₂×D₄	—	4	4	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	8	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂×D₄	—	8	8	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	D₈×C₅₉	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
218	Q₁₆	—	2	4	no	—
219	Q₁₆	—	4	2	no	—
220	SD₁₆	—	4	8	no	—
221	D₈	—	2	4	no	—
222	D₈	—	2	4	no	—
223	D₈	—	2	4	no	—
224	D₈	—	4	2	no	—
225	D₈	—	4	4	no	—
226	D₈	—	4	8	no	—
227	C₂×C₈	D₈×C₅₉	1	4	yes	—
228	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
229	C₂×C₈	C₂×D₄×C₅₉	1	4	yes	—
230	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
231	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
232	C₂²⋊C₄	—	4	8	no	—
233	C₄²	D₈×C₅₉	1	2	yes	—
234	C₄²	C₂×D₄×C₅₉	1	2	yes	—
235	C₁₆	C₂×D₄×C₅₉	1	8	yes	complemented
236	C₂³	—	4	2	no	—
237	C₂³	—	4	4	no	—
238	C₂³	—	8	4	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	8	no	—
241	C₂³	—	8	4	no	—
242	C₂³	C₂×D₈×C₅₉	1	1	yes	characteristic
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	Q₈	—	8	2	no	—
246	D₄	—	4	2	no	—
247	D₄	—	4	4	no	—
248	D₄	—	4	4	no	—
249	D₄	—	4	2	no	—
250	D₄	—	8	16	no	—
251	D₄	—	8	4	no	—
252	D₄	—	8	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	#127×C₅₉	1	2	yes	—
257	C₂×C₄	C₂×D₈×C₅₉	1	4	yes	—
258	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
259	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
260	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
261	C₈	#127×C₅₉	1	2	yes	—
262	C₈	C₂×D₈×C₅₉	1	4	yes	—
263	C₈	C₂²×D₄×C₅₉	1	2	yes	—
264	C₂²	—	8	4	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	8	no	—
267	C₂²	—	8	4	no	—
268	C₂²	—	16	8	no	—
269	C₂²	—	16	4	no	—
270	C₂²	C₅₉×#493	1	2	yes	—
271	C₂²	C₅₉×#509	1	4	yes	—
272	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	8	2	no	—
275	C₄	C₅₉×#493	1	2	yes	—
276	C₄	C₂×#127×C₅₉	1	1	yes	characteristic
277	C₄	C₂²×D₈×C₅₉	1	1	yes	characteristic
278	C₂	—	16	4	no	—
279	C₂	—	16	2	no	—
280	C₂	C₅₉×#2250	1	4	yes	direct product factor
281	C₂	C₂×C₅₉×#493	1	1	yes	characteristic
282	C₂	C₂×C₅₉×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
1. C₂²×SD₃₂×C₅₉	C₂	semidirect
2. C₂²×D₁₆×C₅₉	C₂	semidirect
3. C₂×C₅₉×#495	C₂	semidirect
4. C₂×C₅₉×#493	C₂	semidirect
5. C₅₉×#2250	C₂	direct (N × H)
6. C₂×C₅₉×#346	C₂	semidirect
15. C₂×SD₃₂×C₅₉	C₂²	semidirect
17. C₂×D₁₆×C₅₉	C₂²	semidirect
20. C₅₉×#495	C₂²	semidirect
22. C₅₉×#493	C₂²	semidirect
24. C₅₉×#346	C₂²	semidirect
53. C₂×C₁₆×C₅₉	D₄	semidirect
93. C₁₆×C₅₉	C₂×D₄	semidirect
122. C₂×#2250	C₅₉	direct (N × H)
137. C₂²×SD₃₂	C₂×C₅₉	semidirect
138. C₂²×D₁₆	C₂×C₅₉	semidirect
139. C₂×#495	C₂×C₅₉	semidirect
140. C₂×#493	C₂×C₅₉	semidirect
141. #2250	C₂×C₅₉	direct (N × H)
142. C₂×#346	C₂×C₅₉	semidirect
145. C₂×C₅₉	#2250	direct (N × H)
156. C₂×SD₃₂	C₂²×C₅₉	semidirect
158. C₂×D₁₆	C₂²×C₅₉	semidirect
161. #495	C₂²×C₅₉	semidirect
163. #493	C₂²×C₅₉	semidirect
165. #346	C₂²×C₅₉	semidirect
166. C₅₉	C₂×#2250	direct (N × H)
195. C₂×C₁₆	D₄×C₅₉	semidirect
235. C₁₆	C₂×D₄×C₅₉	semidirect
280. C₂	C₅₉×#2250	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×SD₃₂×C₅₉	2
C₂²×D₁₆×C₅₉	2
C₂×C₅₉×#495	2
C₂×C₅₉×#493	2
C₅₉×#2250	2
C₂×C₅₉×#346	2
C₂×#2250	59

Subgroups — C2×C59×#2250 #28445823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₉×#2250 #28445823