Subgroups — #28493612

Order 15104.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 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	#21424814	C₂	1	1	yes	maximal, characteristic
3	#21424256	C₂	1	1	yes	maximal, characteristic
4	#21421980	C₂	1	2	yes	maximal
5	C₅₉×#1277	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×Dic₅₉⋊C₈	—	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	C₂²×C₅₉⋊C₈	—	2	1	no	—
18	C₂×C₄⋊Dic₅₉	—	2	2	no	—
19	C₂×C₄⋊Dic₅₉	—	2	2	no	—
20	C₂×C₄⋊Dic₅₉	—	2	2	no	—
21	C₂×C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅₉	—	2	1	no	—
24	#14273661	—	4	2	no	—
25	#14273661	—	4	2	no	—
26	#14273659	—	4	4	no	—
27	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
28	Dic₅₉⋊C₈	—	4	2	no	—
29	C₅₉⋊#95	—	2	1	no	—
30	C₂²×C₈×C₅₉	—	2	2	no	—
31	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₉	—	2	2	no	—
35	C₄⋊C₈×C₅₉	Q₈	1	4	yes	—
36	C₄×C₈×C₅₉	—	2	2	no	—
37	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
40	C₂²×Dic₅₉	—	2	1	no	—
41	C₂²×Dic₅₉	—	2	1	no	—
42	C₂²×Dic₅₉	—	4	2	no	—
43	C₂×C₈×C₅₉	—	2	1	no	—
44	C₂×C₈×C₅₉	—	2	1	no	—
45	C₂×C₈×C₅₉	—	2	1	no	—
46	C₂×C₈×C₅₉	—	2	1	no	—
47	C₂×C₈×C₅₉	—	2	4	no	—
48	C₂×C₈×C₅₉	—	4	2	no	—
49	C₂×C₈×C₅₉	—	4	2	no	—
50	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
51	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
52	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
53	C₄²×C₅₉	#41	1	2	yes	—
54	C₄⋊Dic₅₉	—	2	1	no	—
55	C₄⋊Dic₅₉	—	2	2	no	—
56	C₄⋊Dic₅₉	—	4	2	no	—
57	C₄⋊Dic₅₉	—	4	2	no	—
58	C₄⋊Dic₅₉	—	4	2	no	—
59	C₄⋊Dic₅₉	—	4	2	no	—
60	C₄⋊Dic₅₉	—	8	2	no	—
61	C₄⋊Dic₅₉	Q₁₆	1	2	yes	—
62	C₄×Dic₅₉	—	4	1	no	—
63	C₄×Dic₅₉	—	4	1	no	—
64	C₂×C₅₉⋊C₈	—	4	1	no	—
65	C₂×C₅₉⋊C₈	—	4	1	no	—
66	C₂×C₅₉⋊C₈	—	4	1	no	—
67	C₂³×C₅₉	#121	1	1	yes	characteristic
68	C₂×C₄×C₅₉	—	2	1	no	—
69	C₂×C₄×C₅₉	—	2	1	no	—
70	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
72	C₂×C₄×C₅₉	#125	1	2	yes	—
73	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
78	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
81	C₂×Dic₅₉	—	2	2	no	—
82	C₂×Dic₅₉	—	4	1	no	—
83	C₂×Dic₅₉	—	4	1	no	—
84	C₂×Dic₅₉	—	4	1	no	—
85	C₂×Dic₅₉	—	4	1	no	—
86	C₂×Dic₅₉	—	4	1	no	—
87	C₂×Dic₅₉	—	8	2	no	—
88	C₂×Dic₅₉	—	8	2	no	—
89	C₂×Dic₅₉	—	8	2	no	—
90	C₈×C₅₉	—	2	2	no	—
91	C₈×C₅₉	—	2	2	no	—
92	C₈×C₅₉	—	4	4	no	—
93	C₅₉⋊C₈	—	8	1	no	—
94	#10652	—	59	1	no	maximal, Sylow
95	C₂²×C₅₉	#398	1	1	yes	characteristic
96	C₂²×C₅₉	#460	1	1	yes	characteristic
97	C₂²×C₅₉	#461	1	1	yes	characteristic
98	C₂²×C₅₉	#467	1	1	yes	characteristic
99	C₂²×C₅₉	#469	1	1	yes	characteristic
100	C₂²×C₅₉	#472	1	1	yes	characteristic
101	C₂²×C₅₉	#473	1	1	yes	characteristic
102	C₄×C₅₉	—	2	2	no	—
103	C₄×C₅₉	#477	1	2	yes	—
104	C₄×C₅₉	#478	1	2	yes	—
105	C₄×C₅₉	#481	1	1	yes	characteristic
106	C₄×C₅₉	#483	1	1	yes	characteristic
107	C₄×C₅₉	#498	1	1	yes	characteristic
108	C₄×C₅₉	#501	1	1	yes	characteristic
109	Dic₅₉	—	4	2	no	—
110	Dic₅₉	—	8	1	no	—
111	Dic₅₉	—	16	2	no	—
112	#1948	—	59	2	no	—
113	#1939	—	59	1	no	—
114	#1845	—	59	1	no	—
115	#1296	—	59	2	no	—
116	#1277	D₅₉	1	1	yes	characteristic
117	C₂×C₅₉	#1670	1	1	yes	characteristic
118	C₂×C₅₉	#1675	1	1	yes	characteristic
119	C₂×C₅₉	#1688	1	1	yes	characteristic
120	C₂×C₅₉	#1693	1	1	yes	characteristic
121	C₂×C₅₉	#2084	1	1	yes	characteristic
122	C₂×C₅₉	#2087	1	1	yes	characteristic
123	C₂×C₅₉	#2089	1	1	yes	characteristic
124	C₂×#107	—	118	1	no	—
125	C₂×#107	—	118	1	no	—
126	C₂×#106	—	118	2	no	—
127	C₂×C₄⋊C₈	—	59	1	no	—
128	C₂×C₄⋊C₈	—	118	1	no	—
129	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
130	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
131	#384	—	59	2	no	—
132	C₄×C₄⋊C₄	—	59	1	no	—
133	#340	—	118	2	no	—
134	#336	—	118	1	no	—
135	C₅₉	#10652	1	1	yes	characteristic, complemented, Sylow
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	118	1	no	—
138	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
139	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	59	1	no	—
141	C₂×C₄⋊C₄	—	59	1	no	—
142	C₂×C₄⋊C₄	—	118	2	no	—
143	C₂×C₄⋊C₄	—	118	2	no	—
144	C₂×C₄⋊C₄	—	118	2	no	—
145	C₂×C₄²	—	118	1	no	—
146	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
147	#107	—	236	2	no	—
148	#107	—	236	2	no	—
149	#106	—	236	4	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	118	2	no	—
152	C₄⋊C₈	—	236	2	no	—
153	C₄⋊C₈	Dic₁₁₈	1	4	yes	—
154	C₄×C₈	—	2	2	no	—
155	#95	—	118	1	no	—
156	C₂²×C₄	—	118	1	no	—
157	C₂²×C₄	—	118	1	no	—
158	C₂²×C₄	—	236	2	no	—
159	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
160	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
161	C₂²×C₄	#11745689	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	236	1	no	—
170	C₂×C₈	—	236	1	no	—
171	C₂×C₈	—	236	1	no	—
172	C₂×C₈	#11745660	1	2	yes	—
173	C₂×C₈	#11745660	1	2	yes	—
174	C₄⋊C₄	—	59	2	no	—
175	C₄⋊C₄	—	118	1	no	—
176	C₄⋊C₄	—	118	2	no	—
177	C₄⋊C₄	—	472	2	no	—
178	C₄⋊C₄	—	236	2	no	—
179	C₄⋊C₄	—	236	2	no	—
180	C₄⋊C₄	—	236	2	no	—
181	C₄⋊C₄	—	236	2	no	—
182	C₄²	—	236	1	no	—
183	C₄²	—	236	1	no	—
184	C₄²	C₂×Dic₁₁₈	1	2	yes	—
185	C₄²	#11745687	1	2	yes	—
186	C₂³	#14273702	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	118	2	no	—
190	C₂×C₄	—	472	2	no	—
191	C₂×C₄	—	472	2	no	—
192	C₂×C₄	—	472	2	no	—
193	C₂×C₄	—	236	1	no	—
194	C₂×C₄	—	236	1	no	—
195	C₂×C₄	—	236	1	no	—
196	C₂×C₄	—	236	1	no	—
197	C₂×C₄	—	236	1	no	—
198	C₂×C₄	#14273704	1	2	yes	—
199	C₂×C₄	#14273711	1	1	yes	characteristic
200	C₂×C₄	#14273718	1	2	yes	—
201	C₂×C₄	#14273727	1	1	yes	characteristic
202	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
203	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
204	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
205	C₂×C₄	#14273746	1	1	yes	characteristic
206	C₂×C₄	#14273754	1	2	yes	—
207	C₂×C₄	#14273755	1	1	yes	characteristic
208	C₂×C₄	#14273760	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	4	4	no	—
212	C₈	—	472	1	no	—
213	C₂²	#16837334	1	1	yes	characteristic
214	C₂²	#16837603	1	1	yes	characteristic
215	C₂²	#16837608	1	1	yes	characteristic
216	C₂²	#16837644	1	1	yes	characteristic
217	C₂²	#16837656	1	1	yes	characteristic
218	C₂²	#16837673	1	1	yes	characteristic
219	C₂²	#16837678	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	944	2	no	—
222	C₄	—	472	1	no	—
223	C₄	—	236	2	no	—
224	C₄	#16837695	1	2	yes	—
225	C₄	#16837700	1	2	yes	—
226	C₄	#16837721	1	1	yes	characteristic
227	C₄	#16837733	1	1	yes	characteristic
228	C₄	#16837800	1	1	yes	characteristic
229	C₄	#16837811	1	1	yes	characteristic
230	C₂	#21423467	1	1	yes	characteristic
231	C₂	#21423496	1	1	yes	characteristic
232	C₂	#21423581	1	1	yes	characteristic
233	C₂	#21423610	1	1	yes	characteristic
234	C₂	#21425574	1	1	yes	characteristic
235	C₂	#21425591	1	1	yes	characteristic
236	C₂	#21425603	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273702
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1277	C₂
socle, prime radical	C₂³×C₅₉	#121
hypercenter	#1277	D₅₉
hypocenter	C₅₉	#10652
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
135. C₅₉	#10652	semidirect

Maximal subgroups

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

subgroup	index
#21424860	2
#21424814	2
#21424256	2
#21421980	2
C₅₉×#1277	2
#10652	59