Subgroups — C₂²×C₅₉×#350 #28445447

Order 15104.

1174 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×#131×C₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₉×#350	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#41×C₅₉	C₄	1	1	yes	characteristic
6	C₂×#131×C₅₉	—	2	3	no	—
7	C₂×#131×C₅₉	—	2	3	no	—
8	C₂×#131×C₅₉	C₄	1	6	yes	—
9	C₂×#131×C₅₉	C₂²	1	6	yes	—
10	C₂²×M₁₆×C₅₉	—	2	1	no	—
11	C₂²×M₁₆×C₅₉	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₅₉	—	2	1	no	—
13	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₅₉	—	2	3	no	—
15	C₂×M₃₂×C₅₉	C₂²	1	6	yes	complemented
16	C₂²×C₁₆×C₅₉	—	2	1	no	—
17	C₂²×C₁₆×C₅₉	—	2	3	no	—
18	C₂²×C₁₆×C₅₉	—	2	1	no	—
19	C₂²×C₁₆×C₅₉	C₂²	1	6	yes	complemented
20	C₅₉×#350	C₂²	1	16	yes	direct product factor
21	C₂×#41×C₅₉	—	2	3	no	—
22	C₂×#41×C₅₉	—	2	3	no	—
23	C₂×#41×C₅₉	C₈	1	6	yes	—
24	C₂×#41×C₅₉	C₂×C₄	1	6	yes	—
25	C₂²×Q₈×C₅₉	C₈	1	1	yes	characteristic
26	C₂²×D₄×C₅₉	—	2	1	no	—
27	C₂²×D₄×C₅₉	C₈	1	1	yes	characteristic
28	C₂³×C₄×C₅₉	—	2	1	no	—
29	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
30	#131×C₅₉	—	2	12	no	—
31	#131×C₅₉	—	2	6	no	—
32	#131×C₅₉	—	2	6	no	—
33	#131×C₅₉	C₂×C₄	1	12	yes	—
34	#131×C₅₉	C₂³	1	4	yes	—
35	C₂×M₁₆×C₅₉	—	2	3	no	—
36	C₂×M₁₆×C₅₉	—	2	6	no	—
37	C₂×M₁₆×C₅₉	—	2	6	no	—
38	C₂×M₁₆×C₅₉	C₂×C₄	1	6	yes	—
39	C₂²×C₈×C₅₉	—	2	1	no	—
40	C₂²×C₈×C₅₉	—	2	6	no	—
41	C₂²×C₈×C₅₉	—	2	6	no	—
42	C₂²×C₈×C₅₉	C₂×C₄	1	3	yes	—
43	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₉	D₄	1	3	yes	—
45	C₂²×C₈×C₅₉	D₄	1	3	yes	—
46	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₉	C₂³	1	3	yes	—
48	M₃₂×C₅₉	—	2	6	no	—
49	M₃₂×C₅₉	C₂³	1	4	yes	complemented
50	C₂×C₁₆×C₅₉	—	2	6	no	—
51	C₂×C₁₆×C₅₉	—	2	6	no	—
52	C₂×C₁₆×C₅₉	—	2	6	no	—
53	C₂×C₁₆×C₅₉	C₂³	1	4	yes	complemented
54	C₂⁴×C₅₉	—	2	1	no	—
55	C₂⁴×C₅₉	M₁₆	1	1	yes	characteristic
56	#41×C₅₉	—	2	12	no	—
57	#41×C₅₉	—	2	6	no	—
58	#41×C₅₉	—	2	6	no	—
59	#41×C₅₉	C₂×C₈	1	12	yes	—
60	#41×C₅₉	C₂²×C₄	1	4	yes	—
61	C₂×Q₈×C₅₉	—	2	3	no	—
62	C₂×Q₈×C₅₉	C₂×C₈	1	6	yes	—
63	C₂×D₄×C₅₉	—	2	6	no	—
64	C₂×D₄×C₅₉	—	2	6	no	—
65	C₂×D₄×C₅₉	—	2	3	no	—
66	C₂×D₄×C₅₉	C₂×C₈	1	6	yes	—
67	C₂²×C₄×C₅₉	—	2	1	no	—
68	C₂²×C₄×C₅₉	—	2	6	no	—
69	C₂²×C₄×C₅₉	—	2	6	no	—
70	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₉	C₂²⋊C₄	1	3	yes	—
72	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₅₉	C₂×C₈	1	3	yes	—
74	C₂²×C₄×C₅₉	M₁₆	1	3	yes	—
75	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
76	M₁₆×C₅₉	—	2	6	no	—
77	M₁₆×C₅₉	—	2	12	no	—
78	M₁₆×C₅₉	—	2	4	no	—
79	M₁₆×C₅₉	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₅₉	—	2	6	no	—
81	C₂×C₈×C₅₉	—	2	12	no	—
82	C₂×C₈×C₅₉	—	2	4	no	—
83	C₂×C₈×C₅₉	C₂²⋊C₄	1	3	yes	—
84	C₂×C₈×C₅₉	C₂²⋊C₄	1	3	yes	—
85	C₂×C₈×C₅₉	C₂²⋊C₄	1	6	yes	—
86	C₂×C₈×C₅₉	C₂²×C₄	1	3	yes	—
87	C₂×C₈×C₅₉	C₂×D₄	1	3	yes	—
88	C₂×C₈×C₅₉	C₂×D₄	1	3	yes	—
89	C₂×C₈×C₅₉	C₂×D₄	1	3	yes	—
90	C₂×C₈×C₅₉	C₂×D₄	1	3	yes	—
91	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
92	C₁₆×C₅₉	—	2	4	no	—
93	C₁₆×C₅₉	—	2	4	no	—
94	C₂³×C₅₉	—	2	4	no	—
95	C₂³×C₅₉	—	2	6	no	—
96	C₂³×C₅₉	—	4	4	no	—
97	C₂³×C₅₉	C₂²⋊C₈	1	3	yes	—
98	C₂³×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
99	C₂³×C₅₉	C₂×M₁₆	1	3	yes	—
100	Q₈×C₅₉	—	2	6	no	—
101	Q₈×C₅₉	C₂²×C₈	1	4	yes	—
102	D₄×C₅₉	—	2	4	no	—
103	D₄×C₅₉	—	2	12	no	—
104	D₄×C₅₉	—	2	6	no	—
105	D₄×C₅₉	C₂²×C₈	1	4	yes	—
106	C₂×C₄×C₅₉	—	2	6	no	—
107	C₂×C₄×C₅₉	—	2	4	no	—
108	C₂×C₄×C₅₉	—	2	12	no	—
109	C₂×C₄×C₅₉	C₂²⋊C₈	1	3	yes	—
110	C₂×C₄×C₅₉	C₂²⋊C₈	1	3	yes	—
111	C₂×C₄×C₅₉	C₂²⋊C₈	1	6	yes	—
112	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	3	yes	—
113	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	3	yes	—
114	C₂×C₄×C₅₉	C₂²×C₈	1	3	yes	—
115	C₂×C₄×C₅₉	C₂²×C₈	1	3	yes	—
116	C₂×C₄×C₅₉	C₂×M₁₆	1	3	yes	—
117	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
118	C₈×C₅₉	—	2	4	no	—
119	C₈×C₅₉	C₂×C₂²⋊C₄	1	3	yes	—
120	C₈×C₅₉	C₂×C₂²⋊C₄	1	3	yes	—
121	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
122	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
123	C₂²×#350	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₅₉	—	2	6	no	—
125	C₂²×C₅₉	—	2	6	no	—
126	C₂²×C₅₉	—	2	4	no	—
127	C₂²×C₅₉	—	4	12	no	—
128	C₂²×C₅₉	#350	1	4	yes	direct product factor
129	C₂²×C₅₉	C₂×C₂²⋊C₈	1	3	yes	—
130	C₂²×C₅₉	C₂×C₂²⋊C₈	1	3	yes	—
131	C₂²×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
132	C₄×C₅₉	—	2	4	no	—
133	C₄×C₅₉	C₂×C₂²⋊C₈	1	3	yes	—
134	C₄×C₅₉	C₂×C₂²⋊C₈	1	3	yes	—
135	C₄×C₅₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₅₉	C₂³×C₈	1	1	yes	characteristic
137	C₂²×#131	C₂×C₅₉	1	1	yes	characteristic
138	C₂²×M₃₂	C₂×C₅₉	1	1	yes	characteristic, complemented
139	C₂³×C₁₆	C₂×C₅₉	1	1	yes	characteristic, complemented
140	C₂×#350	C₂×C₅₉	1	12	yes	direct product factor
141	C₂×C₅₉	—	2	1	no	—
142	C₂×C₅₉	—	2	3	no	—
143	C₂×C₅₉	—	4	4	no	—
144	C₂×C₅₉	C₂×#350	1	6	yes	direct product factor
145	C₂×C₅₉	C₂²×C₂²⋊C₈	1	1	yes	characteristic
146	C₂²×#41	C₄×C₅₉	1	1	yes	characteristic
147	C₂×#131	—	2	3	no	—
148	C₂×#131	—	2	3	no	—
149	C₂×#131	C₄×C₅₉	1	6	yes	—
150	C₂×#131	C₂²×C₅₉	1	6	yes	—
151	C₂²×M₁₆	—	2	1	no	—
152	C₂²×M₁₆	C₄×C₅₉	1	1	yes	characteristic
153	C₂³×C₈	—	2	1	no	—
154	C₂³×C₈	C₂²×C₅₉	1	1	yes	characteristic
155	C₂×M₃₂	—	2	3	no	—
156	C₂×M₃₂	C₂²×C₅₉	1	6	yes	complemented
157	C₂²×C₁₆	—	2	1	no	—
158	C₂²×C₁₆	—	2	3	no	—
159	C₂²×C₁₆	—	2	1	no	—
160	C₂²×C₁₆	C₂²×C₅₉	1	6	yes	complemented
161	#350	C₂²×C₅₉	1	16	yes	direct product factor
162	C₅₉	C₂²×#350	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×#41	—	2	3	no	—
164	C₂×#41	—	2	3	no	—
165	C₂×#41	C₈×C₅₉	1	6	yes	—
166	C₂×#41	C₂×C₄×C₅₉	1	6	yes	—
167	C₂²×Q₈	C₈×C₅₉	1	1	yes	characteristic
168	C₂²×D₄	—	2	1	no	—
169	C₂²×D₄	C₈×C₅₉	1	1	yes	characteristic
170	C₂³×C₄	—	2	1	no	—
171	C₂³×C₄	C₂×C₄×C₅₉	1	1	yes	characteristic
172	#131	—	2	12	no	—
173	#131	—	2	6	no	—
174	#131	—	2	6	no	—
175	#131	C₂×C₄×C₅₉	1	12	yes	—
176	#131	C₂³×C₅₉	1	4	yes	—
177	C₂×M₁₆	—	2	3	no	—
178	C₂×M₁₆	—	2	6	no	—
179	C₂×M₁₆	—	2	6	no	—
180	C₂×M₁₆	C₂×C₄×C₅₉	1	6	yes	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	6	no	—
183	C₂²×C₈	—	2	6	no	—
184	C₂²×C₈	C₂×C₄×C₅₉	1	3	yes	—
185	C₂²×C₈	D₄×C₅₉	1	1	yes	characteristic
186	C₂²×C₈	D₄×C₅₉	1	1	yes	characteristic
187	C₂²×C₈	D₄×C₅₉	1	3	yes	—
188	C₂²×C₈	D₄×C₅₉	1	3	yes	—
189	C₂²×C₈	C₂³×C₅₉	1	3	yes	—
190	M₃₂	—	2	6	no	—
191	M₃₂	C₂³×C₅₉	1	4	yes	complemented
192	C₂×C₁₆	—	2	6	no	—
193	C₂×C₁₆	—	2	6	no	—
194	C₂×C₁₆	—	2	6	no	—
195	C₂×C₁₆	C₂³×C₅₉	1	4	yes	complemented
196	C₂⁴	—	2	1	no	—
197	C₂⁴	M₁₆×C₅₉	1	1	yes	characteristic
198	#41	—	2	12	no	—
199	#41	—	2	6	no	—
200	#41	—	2	6	no	—
201	#41	C₂×C₈×C₅₉	1	12	yes	—
202	#41	C₂²×C₄×C₅₉	1	4	yes	—
203	C₂×Q₈	—	2	3	no	—
204	C₂×Q₈	C₂×C₈×C₅₉	1	6	yes	—
205	C₂×D₄	—	2	6	no	—
206	C₂×D₄	—	2	6	no	—
207	C₂×D₄	—	2	3	no	—
208	C₂×D₄	C₂×C₈×C₅₉	1	6	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	6	no	—
211	C₂²×C₄	—	2	6	no	—
212	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
213	C₂²×C₄	C₂²⋊C₄×C₅₉	1	3	yes	—
214	C₂²×C₄	C₂×C₈×C₅₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₈×C₅₉	1	3	yes	—
216	C₂²×C₄	M₁₆×C₅₉	1	3	yes	—
217	C₂²×C₄	C₂²×C₄×C₅₉	1	3	yes	—
218	M₁₆	—	2	6	no	—
219	M₁₆	—	2	12	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	C₂²×C₄×C₅₉	1	4	yes	—
222	C₂×C₈	—	2	6	no	—
223	C₂×C₈	—	2	12	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	C₂²⋊C₄×C₅₉	1	3	yes	—
226	C₂×C₈	C₂²⋊C₄×C₅₉	1	3	yes	—
227	C₂×C₈	C₂²⋊C₄×C₅₉	1	6	yes	—
228	C₂×C₈	C₂²×C₄×C₅₉	1	3	yes	—
229	C₂×C₈	C₂×D₄×C₅₉	1	3	yes	—
230	C₂×C₈	C₂×D₄×C₅₉	1	3	yes	—
231	C₂×C₈	C₂×D₄×C₅₉	1	3	yes	—
232	C₂×C₈	C₂×D₄×C₅₉	1	3	yes	—
233	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
234	C₁₆	—	2	4	no	—
235	C₁₆	—	2	4	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	2	6	no	—
238	C₂³	—	4	4	no	—
239	C₂³	C₂²⋊C₈×C₅₉	1	3	yes	—
240	C₂³	C₂²⋊C₈×C₅₉	1	1	yes	characteristic
241	C₂³	C₂×M₁₆×C₅₉	1	3	yes	—
242	Q₈	—	2	6	no	—
243	Q₈	C₂²×C₈×C₅₉	1	4	yes	—
244	D₄	—	2	12	no	—
245	D₄	—	2	4	no	—
246	D₄	—	2	6	no	—
247	D₄	C₂²×C₈×C₅₉	1	4	yes	—
248	C₂×C₄	—	2	6	no	—
249	C₂×C₄	—	2	12	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	C₂²⋊C₈×C₅₉	1	3	yes	—
252	C₂×C₄	C₂²⋊C₈×C₅₉	1	3	yes	—
253	C₂×C₄	C₂²⋊C₈×C₅₉	1	6	yes	—
254	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	3	yes	—
255	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	3	yes	—
256	C₂×C₄	C₂²×C₈×C₅₉	1	3	yes	—
257	C₂×C₄	C₂²×C₈×C₅₉	1	3	yes	—
258	C₂×C₄	C₂×M₁₆×C₅₉	1	3	yes	—
259	C₂×C₄	C₂³×C₄×C₅₉	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₈	C₂×C₂²⋊C₄×C₅₉	1	3	yes	—
262	C₈	C₂×C₂²⋊C₄×C₅₉	1	3	yes	—
263	C₈	C₂²×D₄×C₅₉	1	1	yes	characteristic
264	C₈	C₂²×D₄×C₅₉	1	1	yes	characteristic
265	C₂²	—	2	6	no	—
266	C₂²	—	2	6	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	4	12	no	—
269	C₂²	C₅₉×#350	1	4	yes	direct product factor
270	C₂²	C₂×C₂²⋊C₈×C₅₉	1	3	yes	—
271	C₂²	C₂×C₂²⋊C₈×C₅₉	1	3	yes	—
272	C₂²	C₂²×M₁₆×C₅₉	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	C₂×C₂²⋊C₈×C₅₉	1	3	yes	—
275	C₄	C₂×C₂²⋊C₈×C₅₉	1	3	yes	—
276	C₄	C₂²×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
277	C₄	C₂³×C₈×C₅₉	1	1	yes	characteristic
278	C₂	—	2	3	no	—
279	C₂	—	2	1	no	—
280	C₂	—	4	4	no	—
281	C₂	C₂×C₅₉×#350	1	6	yes	direct product factor
282	C₂	C₂²×C₂²⋊C₈×C₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₅₉	D₄
commutator	C₄	C₂³×C₈×C₅₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
socle	C₂³×C₅₉	C₂²⋊C₈
prime radical	C₂²×#41×C₅₉	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₅₉ ◁ C₂³×C₈×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×M₃₂×C₅₉	C₂	semidirect
3. C₂³×C₁₆×C₅₉	C₂	semidirect
4. C₂×C₅₉×#350	C₂	direct (N × H)
15. C₂×M₃₂×C₅₉	C₂²	semidirect
19. C₂²×C₁₆×C₅₉	C₂²	semidirect
20. C₅₉×#350	C₂²	direct (N × H)
49. M₃₂×C₅₉	C₂³	semidirect
53. C₂×C₁₆×C₅₉	C₂³	semidirect
123. C₂²×#350	C₅₉	direct (N × H)
128. C₂²×C₅₉	#350	direct (N × H)
138. C₂²×M₃₂	C₂×C₅₉	semidirect
139. C₂³×C₁₆	C₂×C₅₉	semidirect
140. C₂×#350	C₂×C₅₉	direct (N × H)
144. C₂×C₅₉	C₂×#350	direct (N × H)
156. C₂×M₃₂	C₂²×C₅₉	semidirect
160. C₂²×C₁₆	C₂²×C₅₉	semidirect
161. #350	C₂²×C₅₉	direct (N × H)
162. C₅₉	C₂²×#350	direct (N × H)
191. M₃₂	C₂³×C₅₉	semidirect
195. C₂×C₁₆	C₂³×C₅₉	semidirect
269. C₂²	C₅₉×#350	direct (N × H)
281. C₂	C₂×C₅₉×#350	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×#131×C₅₉	2
C₂²×M₃₂×C₅₉	2
C₂³×C₁₆×C₅₉	2
C₂×C₅₉×#350	2
C₂²×#350	59

Subgroups — C22×C59×#350 #28445447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₉×#350 #28445447