Subgroups — #28560507

Order 15104.

578 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434682	C₂	1	1	yes	maximal, characteristic
2	#21433043	C₂	1	2	yes	maximal
3	#21423006	C₂	1	2	yes	maximal, complemented
4	#21422969	C₂	1	4	yes	maximal, complemented
5	#21422916	C₂	1	2	yes	maximal, complemented
6	C₂×#16837153	C₂	1	1	yes	maximal, characteristic
7	#21422500	C₂	1	2	yes	maximal
8	C₅₉×#1451	C₂	1	1	yes	maximal, characteristic
9	#16837959	C₂²	1	2	yes	—
10	#16837807	C₄	1	1	yes	characteristic
11	#16837782	C₄	1	1	yes	characteristic
12	#16837753	C₄	1	2	yes	—
13	#16837643	C₄	1	2	yes	—
14	#16837643	C₄	1	2	yes	—
15	#16837532	C₂²	1	4	yes	—
16	#16837530	—	2	4	no	—
17	C₈×Dic₁₁₈	C₂²	1	4	yes	—
18	C₄×Dic₂₃₆	C₂²	1	2	yes	—
19	C₄×Dic₂₃₆	C₂²	1	2	yes	—
20	#16837430	—	2	2	no	—
21	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
22	#16837360	—	2	2	no	—
23	#16837153	C₂²	1	1	yes	characteristic
24	#16837153	C₂²	1	2	yes	—
25	#16837153	C₂²	1	1	yes	characteristic
26	#16837134	C₂²	1	4	yes	—
27	#16837134	C₂²	1	4	yes	—
28	C₅₉×#431	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
31	C₅₉×#322	C₂²	1	2	yes	—
32	C₅₉×#322	C₂²	1	2	yes	—
33	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
34	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
35	C₂²×C₅₉⋊C₈	—	2	2	no	—
36	#14273727	C₂×C₄	1	2	yes	—
37	#14273727	C₂×C₄	1	2	yes	—
38	#14273719	—	2	2	no	—
39	#14273706	—	2	2	no	—
40	#14273706	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₁₈	—	2	2	no	—
42	C₄×Dic₁₁₈	C₂³	1	4	yes	—
43	#14273686	—	2	2	no	—
44	#14273661	C₂×C₄	1	2	yes	—
45	#14273661	C₂×C₄	1	2	yes	—
46	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
47	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
48	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
49	Dic₅₉⋊C₈	—	2	4	no	—
50	Dic₅₉⋊C₈	—	2	4	no	—
51	#14273652	C₂×C₄	1	4	yes	—
52	#14273652	C₂×C₄	1	4	yes	—
53	#14273637	—	2	4	no	—
54	C₈×Dic₅₉	—	2	4	no	—
55	C₂×M₁₆×C₅₉	D₄	1	2	yes	—
56	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₅₉	—	2	1	no	—
60	C₂²⋊C₈×C₅₉	—	2	1	no	—
61	#97×C₅₉	D₄	1	2	yes	—
62	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₉	C₂³	1	2	yes	—
65	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₉	C₂³	1	2	yes	—
68	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
71	C₂×Dic₁₁₈	—	2	2	no	—
72	C₂×Dic₁₁₈	C₂²×C₄	1	4	yes	—
73	M₁₆×C₅₉	—	2	4	no	—
74	C₂×C₈×C₅₉	—	2	2	no	—
75	C₂×C₈×C₅₉	—	2	2	no	—
76	C₂×C₈×C₅₉	—	4	1	no	—
77	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
82	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₉	#41	1	1	yes	characteristic
85	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
86	C₂²⋊Dic₅₉	—	2	4	no	—
87	C₄⋊Dic₅₉	—	2	2	no	—
88	C₄⋊Dic₅₉	—	2	2	no	—
89	C₄⋊Dic₅₉	—	2	2	no	—
90	C₄⋊Dic₅₉	C₂²×C₄	1	4	yes	—
91	Dic₅₉⋊C₄	—	2	4	no	—
92	Dic₅₉⋊C₄	—	2	4	no	—
93	C₄×Dic₅₉	—	2	4	no	—
94	C₂×C₅₉⋊C₈	—	2	2	no	—
95	C₂×C₅₉⋊C₈	—	2	2	no	—
96	C₂×C₅₉⋊C₈	—	4	4	no	—
97	Dic₂₃₆	—	2	4	no	—
98	Dic₂₃₆	—	4	2	no	—
99	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₉	—	2	1	no	—
101	C₂×C₄×C₅₉	—	2	1	no	—
102	C₂×C₄×C₅₉	—	4	1	no	—
103	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₉	#137	1	4	yes	—
107	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
110	C₂×Dic₅₉	—	2	4	no	—
111	C₂×Dic₅₉	—	2	4	no	—
112	Dic₁₁₈	—	2	4	no	—
113	Dic₁₁₈	—	2	4	no	—
114	Dic₁₁₈	—	8	2	no	—
115	C₈×C₅₉	—	2	1	no	—
116	C₈×C₅₉	—	2	1	no	—
117	C₈×C₅₉	—	2	1	no	—
118	C₈×C₅₉	—	2	4	no	—
119	C₈×C₅₉	—	4	2	no	—
120	C₈×C₅₉	#131	1	2	yes	—
121	C₅₉⋊C₈	—	4	2	no	—
122	C₅₉⋊C₈	—	4	2	no	—
123	#18072	—	59	1	no	maximal, Sylow
124	C₂²×C₅₉	—	2	1	no	—
125	C₂²×C₅₉	—	2	1	no	—
126	C₂²×C₅₉	#441	1	2	yes	—
127	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
128	C₄×C₅₉	—	2	1	no	—
129	C₄×C₅₉	—	2	1	no	—
130	C₄×C₅₉	—	2	1	no	—
131	C₄×C₅₉	#441	1	2	yes	—
132	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
133	C₄×C₅₉	#568	1	1	yes	characteristic
134	C₄×C₅₉	C₂×#137	1	2	yes	—
135	Dic₅₉	—	4	4	no	—
136	Dic₅₉	—	4	4	no	—
137	#3170	—	59	1	no	—
138	#2962	—	59	2	no	—
139	#1593	—	59	2	no	—
140	#1587	—	59	4	no	—
141	#1579	—	59	2	no	—
142	C₂×#334	—	59	1	no	—
143	#1481	—	59	2	no	—
144	#1451	D₅₉	1	1	yes	characteristic
145	C₂×C₅₉	—	2	2	no	—
146	C₂×C₅₉	#2935	1	1	yes	characteristic
147	C₂×C₅₉	C₂×#441	1	1	yes	characteristic
148	C₂×C₅₉	#2957	1	1	yes	characteristic
149	#533	—	59	2	no	—
150	#500	—	59	1	no	—
151	#494	—	59	1	no	—
152	#487	—	59	2	no	—
153	#467	—	59	2	no	—
154	#467	—	59	2	no	—
155	#446	—	59	4	no	—
156	#446	—	118	4	no	—
157	C₈×Q₈	—	59	4	no	—
158	C₄×Q₁₆	—	59	2	no	—
159	C₄×Q₁₆	—	59	2	no	—
160	#431	C₂×D₅₉	1	1	yes	characteristic
161	#424	—	118	2	no	—
162	C₂×C₄⋊C₈	—	59	2	no	—
163	#405	—	118	2	no	—
164	C₄×M₁₆	C₂×D₅₉	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
166	#334	—	59	1	no	—
167	#334	—	59	2	no	—
168	#334	—	59	1	no	—
169	Q₈⋊C₈	—	59	4	no	—
170	Q₈⋊C₈	—	59	4	no	—
171	#322	C₂×D₅₉	1	2	yes	—
172	#322	C₂×D₅₉	1	2	yes	—
173	C₅₉	#18072	1	1	yes	characteristic, complemented, Sylow
174	C₂×Q₁₆	—	59	2	no	—
175	C₂×Q₁₆	—	59	2	no	—
176	C₂×M₁₆	D₂₃₆	1	2	yes	—
177	C₂²×C₈	—	118	2	no	—
178	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
179	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
180	C₄⋊Q₈	—	59	2	no	—
181	C₄⋊Q₈	—	59	2	no	—
182	#125	—	118	2	no	—
183	C₂²⋊Q₈	—	59	4	no	—
184	C₂²⋊Q₈	—	118	2	no	—
185	C₄×Q₈	—	59	4	no	—
186	C₄×Q₈	—	118	2	no	—
187	#117	—	118	2	no	—
188	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
189	#107	—	59	2	no	—
190	#107	—	59	2	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	59	2	no	—
193	C₄⋊C₈	—	59	2	no	—
194	C₄⋊C₈	—	118	4	no	—
195	C₄⋊C₈	—	118	4	no	—
196	C₄⋊C₈	—	118	2	no	—
197	#103	—	59	4	no	—
198	#103	—	59	4	no	—
199	C₂²⋊C₈	—	2	1	no	—
200	#97	—	118	4	no	—
201	#97	D₂₃₆	1	2	yes	—
202	C₄×C₈	—	118	4	no	—
203	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
204	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
205	C₄×C₈	C₂²×D₅₉	1	2	yes	—
206	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₅₉	1	2	yes	—
209	C₂×Q₈	—	59	4	no	—
210	C₂×Q₈	—	118	2	no	—
211	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
213	C₂²×C₄	#11745685	1	1	yes	characteristic
214	Q₁₆	—	118	4	no	—
215	Q₁₆	—	236	2	no	—
216	M₁₆	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	118	2	no	—
221	C₂×C₈	—	118	2	no	—
222	C₂×C₈	—	236	4	no	—
223	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
227	C₂×C₈	C₂×D₂₃₆	1	4	yes	—
228	C₄⋊C₄	—	59	4	no	—
229	C₄⋊C₄	—	118	2	no	—
230	C₄⋊C₄	—	118	2	no	—
231	C₄⋊C₄	—	118	4	no	—
232	C₄⋊C₄	—	118	2	no	—
233	C₄⋊C₄	—	118	4	no	—
234	C₂²⋊C₄	—	118	4	no	—
235	C₄²	—	118	4	no	—
236	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
237	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
238	C₄²	#11745685	1	1	yes	characteristic
239	C₄²	C₂³×D₅₉	1	1	yes	characteristic
240	C₂³	C₄×D₂₃₆	1	1	yes	characteristic
241	Q₈	—	118	4	no	—
242	Q₈	—	118	4	no	—
243	Q₈	—	472	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	118	4	no	—
248	C₂×C₄	—	118	4	no	—
249	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
250	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
251	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
252	C₂×C₄	#14273760	1	4	yes	—
253	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
255	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	2	4	no	—
260	C₈	—	4	2	no	—
261	C₈	—	236	2	no	—
262	C₈	—	236	2	no	—
263	C₈	#14273735	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	#16837510	1	2	yes	—
267	C₂²	C₂×C₄×D₂₃₆	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	236	4	no	—
272	C₄	—	236	4	no	—
273	C₄	#16837510	1	2	yes	—
274	C₄	C₂×#14273735	1	1	yes	characteristic
275	C₄	#16838168	1	1	yes	characteristic
276	C₄	C₂×#14273760	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	#21432842	1	1	yes	characteristic
279	C₂	C₂×#16837510	1	1	yes	characteristic
280	C₂	#21433014	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₄²	C₂³×D₅₉
Fitting	C₅₉×#1451	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	#1451	D₅₉
hypocenter	C₅₉	#18072
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

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

N	H ≅ G/N	type
3. #21423006	C₂	semidirect
4. #21422969	C₂	semidirect
5. #21422916	C₂	semidirect
173. C₅₉	#18072	semidirect

Maximal subgroups

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

subgroup	index
#21434682	2
#21433043	2
#21423006	2
#21422969	2
#21422916	2
C₂×#16837153	2
#21422500	2
C₅₉×#1451	2
#18072	59