Subgroups — #28479117

Order 15104.

446 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21423841	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	C₅₉⋊#1273	C₂	1	1	yes	maximal, characteristic
4	C₅₉⋊#1273	C₂	1	2	yes	maximal
5	C₅₉×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
8	C₂×C₈×Dic₅₉	C₄	1	2	yes	—
9	C₂×C₈×Dic₅₉	C₄	1	2	yes	—
10	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
11	#16837156	C₄	1	4	yes	—
12	#16837156	C₄	1	4	yes	—
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
15	C₂²×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
16	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
21	C₄×C₅₉⋊C₈	—	2	1	no	—
22	C₄×C₅₉⋊C₈	—	2	2	no	—
23	C₄×C₅₉⋊C₈	—	2	1	no	—
24	C₄×C₅₉⋊C₈	—	2	2	no	—
25	C₈×Dic₅₉	—	2	4	no	—
26	C₈×Dic₅₉	C₈	1	8	yes	complemented
27	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₅₉	—	2	2	no	—
31	C₄×C₈×C₅₉	—	2	2	no	—
32	C₄×C₈×C₅₉	—	2	2	no	—
33	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
35	C₂²×Dic₅₉	C₄²	1	1	yes	characteristic
36	C₂²×Dic₅₉	C₄²	1	1	yes	characteristic
37	C₂²×Dic₅₉	C₄²	1	2	yes	—
38	C₂×C₈×C₅₉	—	2	2	no	—
39	C₂×C₈×C₅₉	—	2	2	no	—
40	C₂×C₈×C₅₉	—	2	4	no	—
41	C₂×C₈×C₅₉	—	2	4	no	—
42	C₂×C₈×C₅₉	—	2	4	no	—
43	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
44	C₄²×C₅₉	#41	1	2	yes	—
45	C₄²×C₅₉	#41	1	1	yes	characteristic
46	C₄²×C₅₉	#41	1	1	yes	characteristic
47	C₄×Dic₅₉	—	2	4	no	—
48	C₄×Dic₅₉	—	2	4	no	—
49	C₄×Dic₅₉	C₂×C₈	1	4	yes	—
50	C₄×Dic₅₉	M₁₆	1	4	yes	—
51	C₂×C₅₉⋊C₈	—	2	2	no	—
52	C₂×C₅₉⋊C₈	—	2	2	no	—
53	C₂×C₅₉⋊C₈	—	2	4	no	—
54	C₂×C₅₉⋊C₈	—	2	4	no	—
55	C₂×C₅₉⋊C₈	—	2	4	no	—
56	C₂×C₅₉⋊C₈	C₂×C₈	1	4	yes	—
57	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
59	C₂×C₄×C₅₉	#117	1	2	yes	—
60	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#117	1	2	yes	—
62	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
63	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
64	C₂×C₄×C₅₉	#131	1	4	yes	—
65	C₂×C₄×C₅₉	#131	1	4	yes	—
66	C₂×Dic₅₉	—	2	2	no	—
67	C₂×Dic₅₉	—	2	2	no	—
68	C₂×Dic₅₉	—	2	2	no	—
69	C₂×Dic₅₉	—	2	2	no	—
70	C₂×Dic₅₉	C₄×C₈	1	2	yes	—
71	C₂×Dic₅₉	C₄×C₈	1	2	yes	—
72	C₂×Dic₅₉	#97	1	4	yes	—
73	C₈×C₅₉	—	2	4	no	—
74	C₈×C₅₉	—	4	4	no	—
75	C₅₉⋊C₈	—	2	4	no	—
76	C₅₉⋊C₈	—	4	4	no	—
77	#8133	—	59	1	no	maximal, Sylow
78	C₂²×C₅₉	#376	1	1	yes	characteristic
79	C₂²×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
80	C₂²×C₅₉	C₂×#97	1	1	yes	characteristic
81	C₂²×C₅₉	#405	1	2	yes	—
82	C₂²×C₅₉	#405	1	2	yes	—
83	C₄×C₅₉	#431	1	2	yes	—
84	C₄×C₅₉	#431	1	2	yes	—
85	C₄×C₅₉	#433	1	4	yes	—
86	C₄×C₅₉	#433	1	4	yes	—
87	Dic₅₉	—	2	2	no	—
88	Dic₅₉	—	2	2	no	—
89	Dic₅₉	—	2	4	no	—
90	#1744	—	59	2	no	—
91	C₄²×C₈	—	59	1	no	—
92	#1273	—	59	2	no	—
93	#1273	—	59	1	no	—
94	#1273	D₅₉	1	1	yes	characteristic
95	C₂×C₅₉	#1450	1	1	yes	characteristic
96	C₂×C₅₉	#1450	1	1	yes	characteristic
97	C₂×C₅₉	#1452	1	2	yes	—
98	C₂×C₅₉	#1744	1	1	yes	characteristic
99	C₂×C₅₉	#1746	1	2	yes	—
100	C₂×C₄×C₈	—	59	2	no	—
101	C₂×C₄×C₈	—	59	2	no	—
102	C₂×C₄×C₈	—	59	1	no	—
103	C₂×C₄×C₈	—	59	2	no	—
104	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
106	C₄³	—	59	1	no	—
107	#336	—	59	4	no	—
108	#336	—	59	4	no	—
109	C₅₉	#8133	1	1	yes	characteristic, complemented, Sylow
110	C₂²×C₈	—	59	2	no	—
111	C₂²×C₈	—	59	4	no	—
112	C₂²×C₈	C₄×D₅₉	1	4	yes	—
113	C₂²×C₈	C₄×D₅₉	1	2	yes	—
114	C₂×C₄²	—	59	1	no	—
115	C₂×C₄²	—	59	1	no	—
116	C₂×C₄²	—	59	2	no	—
117	C₂×C₄²	—	59	2	no	—
118	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	59	8	no	—
123	C₄×C₈	—	118	4	no	—
124	C₄×C₈	—	118	1	no	—
125	C₄×C₈	—	118	2	no	—
126	C₄×C₈	—	118	1	no	—
127	C₄×C₈	—	118	2	no	—
128	C₂²×C₄	—	59	1	no	—
129	C₂²×C₄	—	59	1	no	—
130	C₂²×C₄	—	59	2	no	—
131	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	59	4	no	—
139	C₂×C₈	—	118	2	no	—
140	C₂×C₈	—	118	2	no	—
141	C₂×C₈	—	118	4	no	—
142	C₂×C₈	—	118	4	no	—
143	C₂×C₈	—	118	4	no	—
144	C₂×C₈	C₈×D₅₉	1	4	yes	—
145	C₄²	—	59	4	no	—
146	C₄²	—	59	4	no	—
147	C₄²	—	118	4	no	—
148	C₄²	—	118	4	no	—
149	C₄²	#11745687	1	2	yes	—
150	C₄²	#11745687	1	1	yes	characteristic
151	C₄²	#11745689	1	1	yes	characteristic
152	C₂³	C₄²×D₅₉	1	1	yes	characteristic
153	C₂×C₄	—	59	2	no	—
154	C₂×C₄	—	59	2	no	—
155	C₂×C₄	—	59	4	no	—
156	C₂×C₄	—	118	2	no	—
157	C₂×C₄	—	118	2	no	—
158	C₂×C₄	—	118	2	no	—
159	C₂×C₄	—	118	2	no	—
160	C₂×C₄	#14273683	1	1	yes	characteristic
161	C₂×C₄	#14273683	1	2	yes	—
162	C₂×C₄	#14273685	1	1	yes	characteristic
163	C₂×C₄	#14273685	1	2	yes	—
164	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
165	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
166	C₂×C₄	#14273733	1	4	yes	—
167	C₂×C₄	#14273735	1	4	yes	—
168	C₈	—	2	4	no	—
169	C₈	—	4	4	no	—
170	C₈	—	118	4	no	—
171	C₈	—	236	4	no	—
172	C₂²	#16837247	1	1	yes	characteristic
173	C₂²	C₄×C₈×D₅₉	1	1	yes	characteristic
174	C₂²	#97×D₅₉	1	1	yes	characteristic
175	C₂²	#16837356	1	2	yes	—
176	C₂²	#16837359	1	2	yes	—
177	C₄	—	118	2	no	—
178	C₄	—	118	2	no	—
179	C₄	—	118	4	no	—
180	C₄	#16837452	1	2	yes	—
181	C₄	#16837454	1	2	yes	—
182	C₄	#16837460	1	4	yes	—
183	C₄	#16837462	1	4	yes	—
184	C₂	#21422378	1	1	yes	characteristic
185	C₂	#21422381	1	1	yes	characteristic
186	C₂	#21422387	1	2	yes	—
187	C₂	#21423840	1	1	yes	characteristic
188	C₂	#21423846	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂×C₄×C₈
Fitting	C₅₉×#1273	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₄²
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1273	D₅₉
hypocenter	C₅₉	#8133
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
26. C₈×Dic₅₉	C₈	semidirect
109. C₅₉	#8133	semidirect

Maximal subgroups

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

subgroup	index
#21423841	2
C₄×C₈×Dic₅₉	2
C₅₉⋊#1273	2
C₅₉×#1273	2
#8133	59