Subgroups — #28482443

Order 15104.

402 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424247	C₂	1	2	yes	maximal
2	#21423841	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837155	C₂	1	1	yes	maximal, characteristic
4	C₂×#16837154	C₂	1	2	yes	maximal
5	C₂×C₅₉×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×Dic₅₉⋊C₈	—	2	4	no	—
8	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
10	#16837156	C₄	1	2	yes	—
11	#16837156	C₄	1	2	yes	—
12	#16837155	—	2	2	no	—
13	#16837154	—	2	4	no	—
14	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#335	—	2	2	no	—
17	C₂²×C₅₉⋊C₈	—	2	2	no	—
18	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₅₉	D₄	1	2	yes	—
21	C₂×C₄×Dic₅₉	Q₈	1	2	yes	—
22	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	Dic₅₉⋊C₈	—	4	8	no	—
25	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
26	C₄×C₅₉⋊C₈	Q₈	1	2	yes	—
27	C₂²×C₈×C₅₉	—	2	2	no	—
28	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₉	—	2	2	no	—
32	C₄⋊C₈×C₅₉	—	2	2	no	—
33	C₄×C₈×C₅₉	D₄	1	2	yes	—
34	C₄×C₈×C₅₉	Q₈	1	2	yes	—
35	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
38	C₂²×Dic₅₉	C₄⋊C₄	1	4	yes	—
39	C₂×C₈×C₅₉	—	2	2	no	—
40	C₂×C₈×C₅₉	—	2	2	no	—
41	C₂×C₈×C₅₉	—	4	2	no	—
42	C₂×C₈×C₅₉	—	4	4	no	—
43	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
45	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
49	C₄×Dic₅₉	—	2	2	no	—
50	C₄×Dic₅₉	—	2	2	no	—
51	C₄×Dic₅₉	—	4	2	no	—
52	C₄×Dic₅₉	D₈	1	4	yes	—
53	C₄×Dic₅₉	Q₁₆	1	4	yes	—
54	C₂×C₅₉⋊C₈	—	2	2	no	—
55	C₂×C₅₉⋊C₈	—	2	2	no	—
56	C₂×C₅₉⋊C₈	—	4	2	no	—
57	C₂×C₅₉⋊C₈	—	4	4	no	—
58	C₂×C₅₉⋊C₈	C₄⋊C₄	1	4	yes	—
59	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₉	—	2	2	no	—
61	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#131	1	2	yes	—
68	C₂×C₄×C₅₉	#131	1	2	yes	—
69	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
71	C₂×Dic₅₉	—	2	4	no	—
72	C₂×Dic₅₉	—	2	4	no	—
73	C₂×Dic₅₉	#107	1	8	yes	—
74	C₈×C₅₉	—	2	2	no	—
75	C₈×C₅₉	—	2	2	no	—
76	C₈×C₅₉	—	8	2	no	—
77	C₅₉⋊C₈	—	2	4	no	—
78	C₅₉⋊C₈	—	8	2	no	—
79	#8810	—	59	1	no	maximal, Sylow
80	C₂²×C₅₉	#384	1	1	yes	characteristic
81	C₂²×C₅₉	#424	1	2	yes	—
82	C₂²×C₅₉	C₂×#107	1	2	yes	—
83	C₂²×C₅₉	#430	1	2	yes	—
84	C₄×C₅₉	—	2	1	no	—
85	C₄×C₅₉	—	2	1	no	—
86	C₄×C₅₉	#424	1	2	yes	—
87	C₄×C₅₉	#424	1	1	yes	characteristic
88	C₄×C₅₉	#424	1	1	yes	characteristic
89	C₄×C₅₉	#493	1	1	yes	characteristic
90	C₄×C₅₉	#494	1	1	yes	characteristic
91	C₄×C₅₉	#500	1	2	yes	—
92	Dic₅₉	—	2	8	no	—
93	#1842	—	59	2	no	—
94	#1744	—	59	1	no	—
95	C₂×#335	—	59	2	no	—
96	C₂×#335	—	59	1	no	—
97	C₂×#335	D₅₉	1	1	yes	characteristic
98	C₂×C₅₉	#1571	1	4	yes	—
99	C₂×C₅₉	#1843	1	1	yes	characteristic
100	C₂×C₅₉	#1845	1	1	yes	characteristic
101	C₂×C₅₉	#1849	1	1	yes	characteristic
102	C₂×C₄⋊C₈	—	59	2	no	—
103	C₂×C₄⋊C₈	—	118	4	no	—
104	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
105	C₂×C₄×C₈	—	59	1	no	—
106	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
107	C₄³	—	59	1	no	—
108	#336	—	59	2	no	—
109	#336	—	59	2	no	—
110	#335	—	2	2	no	—
111	#335	—	118	4	no	—
112	#335	—	118	2	no	—
113	C₅₉	#8810	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	—	59	2	no	—
116	C₂²×C₈	—	118	2	no	—
117	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
118	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
119	C₂×C₄²	—	59	2	no	—
120	C₂×C₄²	—	59	2	no	—
121	C₂×C₄²	—	59	2	no	—
122	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	118	2	no	—
126	C₄⋊C₈	—	118	2	no	—
127	C₄⋊C₈	—	236	8	no	—
128	C₄×C₈	—	59	2	no	—
129	C₄×C₈	—	59	2	no	—
130	C₄×C₈	Dic₁₁₈	1	2	yes	—
131	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
132	C₂²×C₄	—	59	4	no	—
133	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
134	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
135	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	4	no	—
140	C₂×C₈	—	59	4	no	—
141	C₂×C₈	—	118	2	no	—
142	C₂×C₈	—	118	2	no	—
143	C₂×C₈	—	236	2	no	—
144	C₂×C₈	—	236	4	no	—
145	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
146	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
147	C₄²	—	59	4	no	—
148	C₄²	—	59	4	no	—
149	C₄²	—	118	2	no	—
150	C₄²	—	118	2	no	—
151	C₄²	—	236	2	no	—
152	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
153	C₄²	D₄×D₅₉	1	1	yes	characteristic
154	C₄²	Q₈×D₅₉	1	1	yes	characteristic
155	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
156	C₂³	C₄⋊C₄×D₅₉	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	59	8	no	—
159	C₂×C₄	—	118	4	no	—
160	C₂×C₄	—	118	4	no	—
161	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
162	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
163	C₂×C₄	#14273723	1	1	yes	characteristic
164	C₂×C₄	#14273725	1	1	yes	characteristic
165	C₂×C₄	#14273726	1	1	yes	characteristic
166	C₂×C₄	#14273726	1	1	yes	characteristic
167	C₂×C₄	#14273733	1	2	yes	—
168	C₂×C₄	#14273735	1	2	yes	—
169	C₂×C₄	D₈×D₅₉	1	2	yes	—
170	C₂×C₄	Q₁₆×D₅₉	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	2	2	no	—
173	C₈	—	8	2	no	—
174	C₈	—	118	4	no	—
175	C₈	—	472	2	no	—
176	C₂²	#16837271	1	1	yes	characteristic
177	C₂²	#16837427	1	2	yes	—
178	C₂²	#107×D₅₉	1	2	yes	—
179	C₂²	#16837449	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	118	8	no	—
183	C₄	#16837427	1	2	yes	—
184	C₄	#16837428	1	1	yes	characteristic
185	C₄	#16837430	1	1	yes	characteristic
186	C₄	#16837777	1	1	yes	characteristic
187	C₄	#16837780	1	1	yes	characteristic
188	C₄	#16837806	1	2	yes	—
189	C₂	#21422869	1	4	yes	—
190	C₂	#21424249	1	1	yes	characteristic
191	C₂	#21424254	1	1	yes	characteristic
192	C₂	#21424269	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₉
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#335	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#335	D₅₉
hypocenter	C₅₉	#8810
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
113. C₅₉	#8810	semidirect

Maximal subgroups

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

subgroup	index
#21424247	2
#21423841	2
C₂×#16837155	2
C₂×#16837154	2
C₂×C₅₉×#335	2
#8810	59