Subgroups — #28481738

Order 15104.

454 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424253	C₂	1	1	yes	maximal, characteristic
2	#21424101	C₂	1	2	yes	maximal
3	C₂×#16837153	C₂	1	1	yes	maximal, characteristic
4	#21421980	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273659	—	2	1	no	—
7	C₂×#14273659	—	2	1	no	—
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×#14273651	—	2	2	no	—
10	#16837272	C₂²	1	2	yes	—
11	#16837164	—	2	2	no	—
12	#16837153	—	2	2	no	—
13	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₅₉	—	2	2	no	—
15	C₈×Q₈×C₅₉	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₈×C₅₉	—	2	1	no	—
17	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₉	—	2	1	no	—
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₅₉⋊C₈	—	2	2	no	—
21	C₂×C₄⋊Dic₅₉	—	2	2	no	—
22	C₂×C₄⋊Dic₅₉	—	2	2	no	—
23	C₂×C₄⋊Dic₅₉	—	2	2	no	—
24	#14273659	—	4	2	no	—
25	#14273659	—	4	2	no	—
26	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
27	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
28	#14273651	—	4	4	no	—
29	C₂²×Q₈×C₅₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₉	—	2	1	no	—
31	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₄×Q₈×C₅₉	—	2	1	no	—
34	C₄×Q₈×C₅₉	—	2	1	no	—
35	C₄×Q₈×C₅₉	C₂×C₄	1	2	yes	—
36	C₄×Q₈×C₅₉	Q₈	1	2	yes	—
37	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₉	—	2	1	no	—
40	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₉	—	2	4	no	—
42	C₄⋊C₈×C₅₉	—	2	1	no	—
43	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₉	—	2	4	no	—
45	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₅₉	D₄	1	2	yes	—
47	C₂×Q₈×C₅₉	—	2	1	no	—
48	C₂×Q₈×C₅₉	—	2	2	no	—
49	C₂×Q₈×C₅₉	C₄⋊C₄	1	2	yes	—
50	C₂×Q₈×C₅₉	SD₁₆	1	4	yes	—
51	C₂²×C₄×C₅₉	—	2	1	no	—
52	C₂²×C₄×C₅₉	—	2	1	no	—
53	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
56	C₂²×Dic₅₉	—	4	2	no	—
57	C₂×C₈×C₅₉	—	2	1	no	—
58	C₂×C₈×C₅₉	—	2	1	no	—
59	C₂×C₈×C₅₉	—	2	2	no	—
60	C₂×C₈×C₅₉	—	4	1	no	—
61	C₂×C₈×C₅₉	—	4	1	no	—
62	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
65	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
66	C₄⋊C₄×C₅₉	—	2	2	no	—
67	C₄⋊C₄×C₅₉	—	2	1	no	—
68	C₄⋊C₄×C₅₉	—	2	2	no	—
69	C₄⋊C₄×C₅₉	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅₉	—	2	2	no	—
71	C₄²×C₅₉	—	2	2	no	—
72	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
75	C₄²×C₅₉	#41	1	1	yes	characteristic
76	C₄⋊Dic₅₉	—	4	2	no	—
77	C₄⋊Dic₅₉	—	4	2	no	—
78	C₄⋊Dic₅₉	—	4	2	no	—
79	C₄⋊Dic₅₉	—	4	2	no	—
80	C₄⋊Dic₅₉	—	8	2	no	—
81	C₂×C₅₉⋊C₈	—	4	2	no	—
82	C₂×C₅₉⋊C₈	—	4	2	no	—
83	C₂×C₅₉⋊C₈	—	4	2	no	—
84	C₂³×C₅₉	#121	1	1	yes	characteristic
85	Q₈×C₅₉	—	2	2	no	—
86	Q₈×C₅₉	—	2	4	no	—
87	Q₈×C₅₉	#106	1	4	yes	complemented
88	C₂×C₄×C₅₉	—	2	2	no	—
89	C₂×C₄×C₅₉	—	2	4	no	—
90	C₂×C₄×C₅₉	—	2	2	no	—
91	C₂×C₄×C₅₉	—	2	2	no	—
92	C₂×C₄×C₅₉	—	4	2	no	—
93	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
96	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
97	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
101	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₉	#135	1	2	yes	—
103	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
105	C₂×Dic₅₉	—	8	2	no	—
106	C₂×Dic₅₉	—	8	2	no	—
107	C₂×Dic₅₉	—	8	2	no	—
108	C₈×C₅₉	—	2	2	no	—
109	C₈×C₅₉	—	4	2	no	—
110	C₈×C₅₉	#103	1	4	yes	complemented
111	C₅₉⋊C₈	—	8	2	no	—
112	#8643	—	59	1	no	maximal, Sylow
113	C₂²×C₅₉	#380	1	1	yes	characteristic
114	C₂²×C₅₉	#460	1	1	yes	characteristic
115	C₂²×C₅₉	#462	1	1	yes	characteristic
116	C₂²×C₅₉	#465	1	2	yes	—
117	C₂²×C₅₉	#471	1	2	yes	—
118	C₄×C₅₉	—	2	4	no	—
119	C₄×C₅₉	—	2	1	no	—
120	C₄×C₅₉	—	2	1	no	—
121	C₄×C₅₉	—	4	2	no	—
122	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
123	C₄×C₅₉	#417	1	1	yes	characteristic
124	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
125	C₄×C₅₉	#427	1	1	yes	characteristic
126	C₄×C₅₉	#465	1	2	yes	—
127	C₄×C₅₉	#475	1	1	yes	characteristic
128	C₄×C₅₉	#477	1	1	yes	characteristic
129	Dic₅₉	—	16	2	no	—
130	C₂×C₈×Q₈	D₅₉	1	1	yes	characteristic
131	#1844	—	59	1	no	—
132	#1803	—	59	2	no	—
133	C₂×#334	—	59	1	no	—
134	#1296	—	59	2	no	—
135	C₂×C₅₉	#1666	1	2	yes	—
136	C₂×C₅₉	#1672	1	2	yes	—
137	C₂×C₅₉	#1803	1	1	yes	characteristic
138	C₂×C₅₉	#1807	1	1	yes	characteristic
139	C₂×C₅₉	#1815	1	1	yes	characteristic
140	C₂×C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
141	C₈×Q₈	—	2	2	no	—
142	C₈×Q₈	Dic₅₉	1	4	yes	complemented
143	C₂×#106	—	118	1	no	—
144	C₂×#106	—	118	1	no	—
145	C₂×C₄⋊C₈	—	2	1	no	—
146	C₂×C₄⋊C₈	—	59	2	no	—
147	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
148	C₂×#103	—	118	2	no	—
149	C₂×C₄×C₈	—	2	1	no	—
150	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
151	#384	—	59	2	no	—
152	#340	—	118	2	no	—
153	#334	—	118	2	no	—
154	C₅₉	#8643	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	D₂₃₆	1	1	yes	characteristic
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	118	2	no	—
158	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	—	2	1	no	—
162	C₄×Q₈	Dic₁₁₈	1	2	yes	—
163	C₄×Q₈	C₂×Dic₅₉	1	2	yes	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	118	2	no	—
166	C₂×C₄⋊C₄	—	118	2	no	—
167	C₂×C₄⋊C₄	—	118	2	no	—
168	C₂×C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
169	C₂×C₄²	—	2	1	no	—
170	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
171	#106	—	236	2	no	—
172	#106	—	236	2	no	—
173	C₄⋊C₈	—	2	4	no	—
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	118	2	no	—
176	C₄⋊C₈	—	118	2	no	—
177	C₄⋊C₈	C₂×Dic₅₉	1	2	yes	—
178	#103	—	236	4	no	—
179	C₄×C₈	—	2	4	no	—
180	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
181	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
182	C₂×Q₈	—	2	1	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	#11745660	1	4	yes	—
185	C₂×Q₈	C₄⋊Dic₅₉	1	2	yes	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	236	2	no	—
189	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
190	C₂²×C₄	#11745685	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	236	2	no	—
198	C₂×C₈	—	236	2	no	—
199	C₂×C₈	—	236	2	no	—
200	C₂×C₈	#11745671	1	2	yes	—
201	C₂×C₈	#11745672	1	2	yes	—
202	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
203	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	472	2	no	—
208	C₄⋊C₄	—	236	2	no	—
209	C₄⋊C₄	—	236	2	no	—
210	C₄⋊C₄	—	236	2	no	—
211	C₄⋊C₄	—	236	2	no	—
212	C₄⋊C₄	C₄⋊Dic₅₉	1	2	yes	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
216	C₄²	#11745685	1	1	yes	characteristic
217	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
218	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
219	C₂³	#14273701	1	1	yes	characteristic
220	Q₈	—	2	2	no	—
221	Q₈	—	2	4	no	—
222	Q₈	#14273659	1	4	yes	complemented
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	472	2	no	—
229	C₂×C₄	—	472	2	no	—
230	C₂×C₄	—	472	2	no	—
231	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
233	C₂×C₄	#14273686	1	1	yes	characteristic
234	C₂×C₄	#14273701	1	1	yes	characteristic
235	C₂×C₄	#14273706	1	1	yes	characteristic
236	C₂×C₄	#14273706	1	1	yes	characteristic
237	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
238	C₂×C₄	C₂×#11745660	1	2	yes	—
239	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
240	C₂×C₄	#14273749	1	2	yes	—
241	C₂×C₄	#14273752	1	1	yes	characteristic
242	C₂×C₄	#14273757	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	4	2	no	—
245	C₈	—	472	2	no	—
246	C₈	#14273651	1	4	yes	complemented
247	C₂²	#16837262	1	1	yes	characteristic
248	C₂²	#16837602	1	1	yes	characteristic
249	C₂²	#16837612	1	1	yes	characteristic
250	C₂²	#16837631	1	2	yes	—
251	C₂²	#16837665	1	2	yes	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	4	no	—
255	C₄	—	4	2	no	—
256	C₄	—	944	2	no	—
257	C₄	C₂×#14273651	1	1	yes	characteristic
258	C₄	#16837403	1	1	yes	characteristic
259	C₄	C₂×#14273659	1	1	yes	characteristic
260	C₄	#16837440	1	1	yes	characteristic
261	C₄	#16837631	1	2	yes	—
262	C₄	#16837687	1	1	yes	characteristic
263	C₄	#16837697	1	1	yes	characteristic
264	C₂	#21423444	1	2	yes	—
265	C₂	#21423478	1	2	yes	—
266	C₂	#21424101	1	1	yes	characteristic
267	C₂	#21424121	1	1	yes	characteristic
268	C₂	#21424155	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273701
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₈×Q₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	#121
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×C₈×Q₈	D₅₉
hypocenter	C₅₉	#8643
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₅₉	C₄	semidirect
87. Q₈×C₅₉	#106	semidirect
110. C₈×C₅₉	#103	semidirect
142. C₈×Q₈	Dic₅₉	semidirect
154. C₅₉	#8643	semidirect
222. Q₈	#14273659	semidirect
246. C₈	#14273651	semidirect

Maximal subgroups

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

subgroup	index
#21424253	2
#21424101	2
C₂×#16837153	2
#21421980	2
C₂×C₈×Q₈×C₅₉	2
#8643	59