Subgroups — #28560468

Order 15104.

586 conjugacy classes of subgroups.

The subgroup lattice has 294 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	#21422994	C₂	1	2	yes	maximal, complemented
4	#21422948	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₂₃₆	C₂	1	2	yes	maximal, complemented
6	C₂×#16837155	C₂	1	1	yes	maximal, characteristic
7	#21422500	C₂	1	2	yes	maximal
8	C₈×M₁₆×C₅₉	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	#16837155	C₂²	1	1	yes	characteristic
24	#16837155	C₂²	1	2	yes	—
25	#16837155	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	C₈²×C₅₉	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	1	yes	characteristic
56	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₅₉	—	2	1	no	—
61	C₂²⋊C₈×C₅₉	—	2	1	no	—
62	#97×C₅₉	D₄	1	2	yes	—
63	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₉	C₂³	1	2	yes	—
66	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₅₉	C₂³	1	2	yes	—
69	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
72	C₂×Dic₁₁₈	—	2	2	no	—
73	C₂×Dic₁₁₈	C₂²×C₄	1	4	yes	—
74	M₁₆×C₅₉	—	2	2	no	—
75	M₁₆×C₅₉	Q₁₆	1	4	yes	complemented
76	C₂×C₈×C₅₉	—	2	2	no	—
77	C₂×C₈×C₅₉	—	2	2	no	—
78	C₂×C₈×C₅₉	—	4	1	no	—
79	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
85	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₉	#41	1	1	yes	characteristic
88	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
89	C₂²⋊Dic₅₉	—	2	4	no	—
90	C₄⋊Dic₅₉	—	2	2	no	—
91	C₄⋊Dic₅₉	—	2	2	no	—
92	C₄⋊Dic₅₉	—	2	2	no	—
93	C₄⋊Dic₅₉	C₂²×C₄	1	4	yes	—
94	Dic₅₉⋊C₄	—	2	4	no	—
95	Dic₅₉⋊C₄	—	2	4	no	—
96	C₄×Dic₅₉	—	2	4	no	—
97	C₂×C₅₉⋊C₈	—	2	2	no	—
98	C₂×C₅₉⋊C₈	—	2	2	no	—
99	C₂×C₅₉⋊C₈	—	4	4	no	—
100	Dic₂₃₆	—	2	4	no	—
101	Dic₂₃₆	—	4	2	no	—
102	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₉	—	2	1	no	—
104	C₂×C₄×C₅₉	—	2	1	no	—
105	C₂×C₄×C₅₉	—	4	1	no	—
106	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₅₉	#135	1	2	yes	—
111	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
114	C₂×Dic₅₉	—	2	4	no	—
115	C₂×Dic₅₉	—	2	4	no	—
116	Dic₁₁₈	—	2	4	no	—
117	Dic₁₁₈	—	2	4	no	—
118	Dic₁₁₈	—	8	2	no	—
119	C₈×C₅₉	—	2	1	no	—
120	C₈×C₅₉	—	2	1	no	—
121	C₈×C₅₉	—	2	1	no	—
122	C₈×C₅₉	—	2	2	no	—
123	C₈×C₅₉	—	4	2	no	—
124	C₈×C₅₉	#131	1	2	yes	—
125	C₈×C₅₉	C₂×Q₁₆	1	4	yes	—
126	C₅₉⋊C₈	—	4	2	no	—
127	C₅₉⋊C₈	—	4	2	no	—
128	#18068	—	59	1	no	maximal, Sylow
129	C₂²×C₅₉	—	2	1	no	—
130	C₂²×C₅₉	—	2	1	no	—
131	C₂²×C₅₉	C₄×Q₁₆	1	2	yes	—
132	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
133	C₄×C₅₉	—	2	1	no	—
134	C₄×C₅₉	—	2	1	no	—
135	C₄×C₅₉	—	2	1	no	—
136	C₄×C₅₉	C₄×Q₁₆	1	2	yes	—
137	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
138	C₄×C₅₉	#568	1	1	yes	characteristic
139	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
140	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
141	Dic₅₉	—	4	4	no	—
142	Dic₅₉	—	4	4	no	—
143	#3170	—	59	1	no	—
144	#2962	—	59	2	no	—
145	#1591	—	59	2	no	—
146	#1584	—	59	4	no	—
147	C₈×Q₁₆	—	59	2	no	—
148	C₂×#335	—	59	1	no	—
149	#1481	—	59	2	no	—
150	C₈×M₁₆	D₅₉	1	1	yes	characteristic
151	C₂×C₅₉	—	2	2	no	—
152	C₂×C₅₉	#2935	1	1	yes	characteristic
153	C₂×C₅₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
154	C₂×C₅₉	#2957	1	1	yes	characteristic
155	#533	—	59	2	no	—
156	#500	—	59	1	no	—
157	#494	—	59	1	no	—
158	#487	—	59	2	no	—
159	#467	—	59	2	no	—
160	#467	—	59	2	no	—
161	#446	—	59	4	no	—
162	#446	—	118	4	no	—
163	C₈×Q₈	—	59	4	no	—
164	C₄×Q₁₆	—	59	2	no	—
165	C₄×Q₁₆	—	59	2	no	—
166	#431	C₂×D₅₉	1	1	yes	characteristic
167	#424	—	118	2	no	—
168	C₂×C₄⋊C₈	—	59	2	no	—
169	#405	—	118	2	no	—
170	C₄×M₁₆	C₂×D₅₉	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
172	#335	—	59	1	no	—
173	#335	—	59	2	no	—
174	#335	—	59	1	no	—
175	Q₈⋊C₈	—	59	4	no	—
176	Q₈⋊C₈	—	59	4	no	—
177	#322	C₂×D₅₉	1	2	yes	—
178	C₈²	C₂×D₅₉	1	2	yes	—
179	C₅₉	#18068	1	1	yes	characteristic, complemented, Sylow
180	C₂×Q₁₆	—	59	2	no	—
181	C₂×Q₁₆	—	59	2	no	—
182	C₂×M₁₆	D₂₃₆	1	1	yes	characteristic
183	C₂×M₁₆	D₂₃₆	1	1	yes	characteristic
184	C₂²×C₈	—	118	2	no	—
185	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
187	C₄⋊Q₈	—	59	2	no	—
188	C₄⋊Q₈	—	59	2	no	—
189	#125	—	118	2	no	—
190	C₂²⋊Q₈	—	59	4	no	—
191	C₂²⋊Q₈	—	118	2	no	—
192	C₄×Q₈	—	59	4	no	—
193	C₄×Q₈	—	118	2	no	—
194	#117	—	118	2	no	—
195	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
196	#107	—	59	2	no	—
197	#107	—	59	2	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	59	2	no	—
200	C₄⋊C₈	—	59	2	no	—
201	C₄⋊C₈	—	118	4	no	—
202	C₄⋊C₈	—	118	4	no	—
203	C₄⋊C₈	—	118	2	no	—
204	#103	—	59	4	no	—
205	#103	—	59	4	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	#97	—	118	4	no	—
208	#97	D₂₃₆	1	2	yes	—
209	C₄×C₈	—	118	4	no	—
210	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
211	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₅₉	1	2	yes	—
213	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₅₉	1	2	yes	—
216	C₂×Q₈	—	59	4	no	—
217	C₂×Q₈	—	118	2	no	—
218	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
220	C₂²×C₄	#11745685	1	1	yes	characteristic
221	Q₁₆	—	118	4	no	—
222	Q₁₆	—	236	2	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	Dic₂₃₆	1	4	yes	complemented
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	118	2	no	—
229	C₂×C₈	—	118	2	no	—
230	C₂×C₈	—	236	4	no	—
231	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
235	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
236	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
237	C₄⋊C₄	—	59	4	no	—
238	C₄⋊C₄	—	118	2	no	—
239	C₄⋊C₄	—	118	2	no	—
240	C₄⋊C₄	—	118	4	no	—
241	C₄⋊C₄	—	118	2	no	—
242	C₄⋊C₄	—	118	4	no	—
243	C₂²⋊C₄	—	118	4	no	—
244	C₄²	—	118	4	no	—
245	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
246	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
247	C₄²	#11745685	1	1	yes	characteristic
248	C₄²	C₂³×D₅₉	1	1	yes	characteristic
249	C₂³	C₄×D₂₃₆	1	1	yes	characteristic
250	Q₈	—	118	4	no	—
251	Q₈	—	118	4	no	—
252	Q₈	—	472	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	118	4	no	—
257	C₂×C₄	—	118	4	no	—
258	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
259	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
260	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
261	C₂×C₄	C₂×Dic₂₃₆	1	2	yes	—
262	C₂×C₄	#14273749	1	2	yes	—
263	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
265	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	236	2	no	—
272	C₈	—	236	2	no	—
273	C₈	#14273735	1	2	yes	—
274	C₈	C₂×Dic₂₃₆	1	4	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	C₄×Dic₂₃₆	1	2	yes	—
278	C₂²	C₂×C₄×D₂₃₆	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	236	4	no	—
283	C₄	—	236	4	no	—
284	C₄	C₄×Dic₂₃₆	1	2	yes	—
285	C₄	C₂×#14273735	1	1	yes	characteristic
286	C₄	#16838168	1	1	yes	characteristic
287	C₄	C₂²×Dic₂₃₆	1	1	yes	characteristic
288	C₄	C₂×#14273749	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	#21432842	1	1	yes	characteristic
291	C₂	C₂×C₄×Dic₂₃₆	1	1	yes	characteristic
292	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	C₈×M₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	C₈×M₁₆	D₅₉
hypocenter	C₅₉	#18068
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #21422994	C₂	semidirect
4. #21422948	C₂	semidirect
5. C₈×Dic₂₃₆	C₂	semidirect
75. M₁₆×C₅₉	Q₁₆	semidirect
179. C₅₉	#18068	semidirect
224. M₁₆	Dic₂₃₆	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
#21422994	2
#21422948	2
C₈×Dic₂₃₆	2
C₂×#16837155	2
#21422500	2
C₈×M₁₆×C₅₉	2
#18068	59