Subgroups — #28487343

Order 15104.

382 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424860	C₂	1	2	yes	maximal
2	#21424256	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
4	#21421980	C₂	1	2	yes	maximal
5	C₂×C₅₉×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273661	—	2	1	no	—
7	C₂×#14273661	—	2	1	no	—
8	C₂×#14273659	—	2	2	no	—
9	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₅₉	—	2	1	no	—
11	#16837272	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
13	#16837165	—	2	2	no	—
14	#16837156	—	2	1	no	—
15	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#335	C₄	1	4	yes	complemented
18	C₂²×C₅₉⋊C₈	—	2	1	no	—
19	C₂×C₄⋊Dic₅₉	—	2	2	no	—
20	C₂×C₄⋊Dic₅₉	—	2	2	no	—
21	C₂×C₄⋊Dic₅₉	—	2	2	no	—
22	C₂×C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₅₉	—	2	1	no	—
25	#14273661	—	4	2	no	—
26	#14273661	—	4	2	no	—
27	#14273659	—	4	4	no	—
28	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
29	C₄⋊(C₅₉⋊C₈)	Q₈	1	2	yes	—
30	C₈×Dic₅₉	—	4	2	no	—
31	C₅₉⋊#95	—	2	1	no	—
32	C₂²×C₈×C₅₉	—	2	2	no	—
33	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₅₉	—	2	2	no	—
37	C₄⋊C₈×C₅₉	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₉	Q₈	1	2	yes	—
40	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
43	C₂²×Dic₅₉	—	2	1	no	—
44	C₂²×Dic₅₉	—	2	1	no	—
45	C₂²×Dic₅₉	—	4	2	no	—
46	C₂×C₈×C₅₉	—	2	1	no	—
47	C₂×C₈×C₅₉	—	2	1	no	—
48	C₂×C₈×C₅₉	—	2	4	no	—
49	C₂×C₈×C₅₉	—	4	2	no	—
50	C₂×C₈×C₅₉	—	4	2	no	—
51	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
54	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
55	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₅₉	#41	1	1	yes	characteristic
59	C₄⋊Dic₅₉	—	2	1	no	—
60	C₄⋊Dic₅₉	—	2	1	no	—
61	C₄⋊Dic₅₉	—	4	2	no	—
62	C₄⋊Dic₅₉	—	4	2	no	—
63	C₄⋊Dic₅₉	—	4	2	no	—
64	C₄⋊Dic₅₉	—	4	2	no	—
65	C₄⋊Dic₅₉	—	8	2	no	—
66	C₄⋊Dic₅₉	D₈	1	2	yes	—
67	C₄⋊Dic₅₉	Q₁₆	1	2	yes	—
68	C₄×Dic₅₉	—	4	1	no	—
69	C₄×Dic₅₉	—	4	1	no	—
70	C₂×C₅₉⋊C₈	—	2	2	no	—
71	C₂×C₅₉⋊C₈	—	4	1	no	—
72	C₂×C₅₉⋊C₈	—	4	1	no	—
73	C₂³×C₅₉	#121	1	1	yes	characteristic
74	C₂×C₄×C₅₉	—	2	1	no	—
75	C₂×C₄×C₅₉	—	2	1	no	—
76	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₅₉	#125	1	1	yes	characteristic
81	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
87	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
88	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
90	C₂×Dic₅₉	—	2	2	no	—
91	C₂×Dic₅₉	—	2	2	no	—
92	C₂×Dic₅₉	—	4	1	no	—
93	C₂×Dic₅₉	—	4	1	no	—
94	C₂×Dic₅₉	—	4	1	no	—
95	C₂×Dic₅₉	—	4	1	no	—
96	C₂×Dic₅₉	—	8	2	no	—
97	C₂×Dic₅₉	—	8	2	no	—
98	C₂×Dic₅₉	—	8	2	no	—
99	C₈×C₅₉	—	2	2	no	—
100	C₈×C₅₉	—	4	4	no	—
101	C₈×C₅₉	#106	1	4	yes	complemented
102	C₅₉⋊C₈	—	4	2	no	—
103	#9615	—	59	1	no	maximal, Sylow
104	C₂²×C₅₉	#389	1	1	yes	characteristic
105	C₂²×C₅₉	#460	1	1	yes	characteristic
106	C₂²×C₅₉	#461	1	1	yes	characteristic
107	C₂²×C₅₉	#466	1	1	yes	characteristic
108	C₂²×C₅₉	#467	1	1	yes	characteristic
109	C₂²×C₅₉	#472	1	1	yes	characteristic
110	C₂²×C₅₉	#473	1	1	yes	characteristic
111	C₄×C₅₉	—	2	1	no	—
112	C₄×C₅₉	—	2	1	no	—
113	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
114	C₄×C₅₉	#428	1	1	yes	characteristic
115	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
116	C₄×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
117	C₄×C₅₉	#475	1	1	yes	characteristic
118	C₄×C₅₉	#476	1	1	yes	characteristic
119	C₄×C₅₉	#499	1	1	yes	characteristic
120	C₄×C₅₉	#500	1	1	yes	characteristic
121	Dic₅₉	—	4	2	no	—
122	Dic₅₉	—	4	2	no	—
123	Dic₅₉	—	16	2	no	—
124	#1948	—	59	2	no	—
125	#1845	—	59	1	no	—
126	C₈×C₄⋊C₄	—	59	1	no	—
127	C₂×#335	D₅₉	1	1	yes	characteristic
128	#1296	—	59	2	no	—
129	C₂×C₅₉	#1668	1	1	yes	characteristic
130	C₂×C₅₉	#1670	1	1	yes	characteristic
131	C₂×C₅₉	#1673	1	1	yes	characteristic
132	C₂×C₅₉	#1675	1	1	yes	characteristic
133	C₂×C₅₉	#1942	1	1	yes	characteristic
134	C₂×C₅₉	#1948	1	1	yes	characteristic
135	C₂×C₅₉	#1952	1	1	yes	characteristic
136	C₂×#107	—	118	1	no	—
137	C₂×#107	—	118	1	no	—
138	C₂×#106	—	118	2	no	—
139	C₂×C₄⋊C₈	—	59	1	no	—
140	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
141	C₂×C₄×C₈	—	118	1	no	—
142	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
143	#384	—	59	2	no	—
144	C₄×C₄⋊C₄	—	59	1	no	—
145	#340	—	118	2	no	—
146	#336	—	118	1	no	—
147	#335	Dic₅₉	1	4	yes	complemented
148	C₅₉	#9615	1	1	yes	characteristic, complemented, Sylow
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	118	1	no	—
151	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
152	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	59	1	no	—
154	C₂×C₄⋊C₄	—	59	1	no	—
155	C₂×C₄⋊C₄	—	118	2	no	—
156	C₂×C₄⋊C₄	—	118	2	no	—
157	C₂×C₄⋊C₄	—	118	2	no	—
158	C₂×C₄²	—	118	1	no	—
159	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
160	#107	—	236	2	no	—
161	#107	—	236	2	no	—
162	#106	—	236	4	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	59	2	no	—
165	C₄⋊C₈	—	118	1	no	—
166	C₄⋊C₈	C₂×Dic₅₉	1	4	yes	—
167	C₄×C₈	—	236	2	no	—
168	C₄×C₈	Dic₁₁₈	1	2	yes	—
169	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
170	#95	—	118	1	no	—
171	C₂²×C₄	—	118	1	no	—
172	C₂²×C₄	—	118	1	no	—
173	C₂²×C₄	—	236	2	no	—
174	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
175	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
176	C₂²×C₄	#11745689	1	1	yes	characteristic
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	118	2	no	—
183	C₂×C₈	—	236	1	no	—
184	C₂×C₈	—	236	1	no	—
185	C₂×C₈	#11745660	1	2	yes	—
186	C₂×C₈	#11745660	1	2	yes	—
187	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
188	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
189	C₄⋊C₄	—	59	2	no	—
190	C₄⋊C₄	—	59	2	no	—
191	C₄⋊C₄	—	118	1	no	—
192	C₄⋊C₄	—	118	1	no	—
193	C₄⋊C₄	—	472	2	no	—
194	C₄⋊C₄	—	236	2	no	—
195	C₄⋊C₄	—	236	2	no	—
196	C₄⋊C₄	—	236	2	no	—
197	C₄⋊C₄	—	236	2	no	—
198	C₄²	—	236	1	no	—
199	C₄²	—	236	1	no	—
200	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
201	C₄²	#11745687	1	1	yes	characteristic
202	C₄²	Q₈×D₅₉	1	1	yes	characteristic
203	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
204	C₂³	#14273702	1	1	yes	characteristic
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	118	2	no	—
208	C₂×C₄	—	118	2	no	—
209	C₂×C₄	—	472	2	no	—
210	C₂×C₄	—	472	2	no	—
211	C₂×C₄	—	472	2	no	—
212	C₂×C₄	—	236	1	no	—
213	C₂×C₄	—	236	1	no	—
214	C₂×C₄	—	236	1	no	—
215	C₂×C₄	—	236	1	no	—
216	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
217	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
218	C₂×C₄	Q₈×Dic₅₉	1	1	yes	characteristic
219	C₂×C₄	#14273707	1	1	yes	characteristic
220	C₂×C₄	#14273718	1	1	yes	characteristic
221	C₂×C₄	#14273726	1	1	yes	characteristic
222	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
223	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
224	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
225	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
226	C₂×C₄	#14273746	1	1	yes	characteristic
227	C₂×C₄	#14273750	1	1	yes	characteristic
228	C₂×C₄	#14273755	1	1	yes	characteristic
229	C₂×C₄	#14273760	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	4	4	no	—
232	C₈	—	236	2	no	—
233	C₈	#14273659	1	4	yes	complemented
234	C₂²	#16837301	1	1	yes	characteristic
235	C₂²	#16837603	1	1	yes	characteristic
236	C₂²	#16837608	1	1	yes	characteristic
237	C₂²	#16837639	1	1	yes	characteristic
238	C₂²	#16837644	1	1	yes	characteristic
239	C₂²	#16837673	1	1	yes	characteristic
240	C₂²	#16837678	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	944	2	no	—
244	C₄	—	236	2	no	—
245	C₄	—	236	2	no	—
246	C₄	C₂×#14273659	1	1	yes	characteristic
247	C₄	#16837444	1	1	yes	characteristic
248	C₄	D₈×Dic₅₉	1	1	yes	characteristic
249	C₄	Q₁₆×Dic₅₉	1	1	yes	characteristic
250	C₄	#16837688	1	1	yes	characteristic
251	C₄	#16837693	1	1	yes	characteristic
252	C₄	#16837803	1	1	yes	characteristic
253	C₄	#16837806	1	1	yes	characteristic
254	C₂	#21423457	1	1	yes	characteristic
255	C₂	#21423467	1	1	yes	characteristic
256	C₂	#21423486	1	1	yes	characteristic
257	C₂	#21423496	1	1	yes	characteristic
258	C₂	#21424830	1	1	yes	characteristic
259	C₂	#21424860	1	1	yes	characteristic
260	C₂	#21424882	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273702
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#335	C₂
socle, prime radical	C₂³×C₅₉	#121
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#335	D₅₉
hypocenter	C₅₉	#9615
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
17. C₅₉×#335	C₄	semidirect
101. C₈×C₅₉	#106	semidirect
147. #335	Dic₅₉	semidirect
148. C₅₉	#9615	semidirect
233. C₈	#14273659	semidirect

Maximal subgroups

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

subgroup	index
#21424860	2
#21424256	2
C₈×C₄⋊Dic₅₉	2
#21421980	2
C₂×C₅₉×#335	2
#9615	59