Subgroups — #28482459

Order 15104.

454 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	#21424255	C₂	1	2	yes	maximal
2	#21424252	C₂	1	2	yes	maximal
3	#21424224	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273660	—	2	4	no	—
7	C₂×#14273658	—	2	4	no	—
8	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	#16837272	C₂²	1	2	yes	—
11	#16837165	—	2	2	no	—
12	C₈×C₅₉⋊C₈	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#384	C₂²	1	2	yes	—
15	C₅₉×#340	—	2	2	no	—
16	C₂²×C₅₉⋊C₈	D₄	1	2	yes	—
17	C₂²×C₅₉⋊C₈	D₄	1	2	yes	—
18	C₂×C₄⋊Dic₅₉	—	2	2	no	—
19	C₂×C₄⋊Dic₅₉	—	2	2	no	—
20	C₂×C₄⋊Dic₅₉	—	2	2	no	—
21	#14273660	—	4	8	no	—
22	#14273658	—	4	8	no	—
23	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
25	C₄×C₅₉⋊C₈	Q₈	1	2	yes	—
26	C₄×C₅₉⋊C₈	Q₈	1	2	yes	—
27	C₂²×C₈×C₅₉	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
31	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₅₉	Q₈	1	2	yes	—
34	C₂²×C₄×C₅₉	—	4	2	no	—
35	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₅₉	—	4	2	no	—
39	C₂×C₈×C₅₉	—	2	4	no	—
40	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
41	C₄⋊C₄×C₅₉	—	4	4	no	—
42	C₄⋊C₄×C₅₉	—	4	4	no	—
43	C₄⋊C₄×C₅₉	—	4	4	no	—
44	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
48	C₄⋊Dic₅₉	—	4	4	no	—
49	C₄⋊Dic₅₉	—	4	4	no	—
50	C₄⋊Dic₅₉	—	4	4	no	—
51	C₂×C₅₉⋊C₈	C₄⋊C₄	1	4	yes	—
52	C₂×C₅₉⋊C₈	C₄⋊C₄	1	4	yes	—
53	C₂×C₅₉⋊C₈	D₈	1	4	yes	—
54	C₂×C₅₉⋊C₈	SD₁₆	1	8	yes	—
55	C₂×C₅₉⋊C₈	Q₁₆	1	4	yes	—
56	C₂³×C₅₉	#127	1	1	yes	characteristic
57	C₂×C₄×C₅₉	—	8	2	no	—
58	C₂×C₄×C₅₉	—	8	2	no	—
59	C₂×C₄×C₅₉	—	8	2	no	—
60	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
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	C₄⋊Q₈	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	C₂×D₈	1	2	yes	—
67	C₂×C₄×C₅₉	C₂×SD₁₆	1	4	yes	—
68	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
69	C₂×C₄×C₅₉	#135	1	4	yes	—
70	C₂×Dic₅₉	—	8	2	no	—
71	C₂×Dic₅₉	—	8	2	no	—
72	C₂×Dic₅₉	—	8	2	no	—
73	C₈×C₅₉	—	2	4	no	—
74	C₅₉⋊C₈	#106	1	8	yes	complemented
75	C₅₉⋊C₈	#107	1	8	yes	complemented
76	#8814	—	59	1	no	maximal, Sylow
77	C₂²×C₅₉	#384	1	1	yes	characteristic
78	C₂²×C₅₉	#492	1	2	yes	—
79	C₂²×C₅₉	#493	1	1	yes	characteristic
80	C₂²×C₅₉	#494	1	1	yes	characteristic
81	C₂²×C₅₉	#495	1	2	yes	—
82	C₄×C₅₉	—	16	2	no	—
83	C₄×C₅₉	C₂×#106	1	2	yes	—
84	C₄×C₅₉	C₂×#107	1	2	yes	—
85	C₄×C₅₉	#427	1	2	yes	—
86	C₄×C₅₉	#498	1	2	yes	—
87	C₄×C₅₉	#499	1	2	yes	—
88	C₄×C₅₉	#500	1	2	yes	—
89	Dic₅₉	—	16	2	no	—
90	#1845	—	59	2	no	—
91	#1844	—	59	2	no	—
92	#1834	—	59	1	no	—
93	#1834	D₅₉	1	1	yes	characteristic
94	C₂×C₈²	—	59	1	no	—
95	C₂×C₅₉	#1706	1	2	yes	—
96	C₂×C₅₉	#1707	1	2	yes	—
97	C₂×C₅₉	#1844	1	1	yes	characteristic
98	C₂×C₅₉	#1845	1	1	yes	characteristic
99	C₂×C₅₉	#1846	1	1	yes	characteristic
100	C₂×#107	—	118	4	no	—
101	C₂×#106	—	118	4	no	—
102	C₂×C₄×C₈	—	59	1	no	—
103	C₂×C₄×C₈	—	59	1	no	—
104	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
105	#384	—	59	2	no	—
106	#384	C₂×D₅₉	1	2	yes	—
107	#340	—	2	2	no	—
108	#340	—	118	2	no	—
109	C₈²	—	59	4	no	—
110	C₅₉	#8814	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	—	59	2	no	—
112	C₂²×C₈	—	59	2	no	—
113	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	118	2	no	—
118	C₂×C₄⋊C₄	—	118	2	no	—
119	C₂×C₄⋊C₄	—	118	2	no	—
120	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
121	#107	—	236	8	no	—
122	#106	—	236	8	no	—
123	C₄×C₈	—	59	2	no	—
124	C₄×C₈	—	59	2	no	—
125	C₄×C₈	—	59	2	no	—
126	C₄×C₈	—	59	2	no	—
127	C₄×C₈	Dic₁₁₈	1	2	yes	—
128	C₄×C₈	C₄×D₅₉	1	2	yes	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	—	236	2	no	—
131	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
132	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	59	4	no	—
136	C₂×C₈	—	59	4	no	—
137	C₂×C₈	—	59	4	no	—
138	C₂×C₈	—	59	8	no	—
139	C₂×C₈	—	59	4	no	—
140	C₂×C₈	Dic₅₉⋊C₄	1	4	yes	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	236	4	no	—
145	C₄⋊C₄	—	236	4	no	—
146	C₄⋊C₄	—	236	4	no	—
147	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
148	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
149	C₄²	Q₈×D₅₉	1	1	yes	characteristic
150	C₄²	Q₈×D₅₉	1	1	yes	characteristic
151	C₂³	#14273723	1	1	yes	characteristic
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	472	2	no	—
156	C₂×C₄	—	472	2	no	—
157	C₂×C₄	—	472	2	no	—
158	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
159	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
160	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
161	C₂×C₄	#14273725	1	1	yes	characteristic
162	C₂×C₄	#14273726	1	1	yes	characteristic
163	C₂×C₄	#14273726	1	1	yes	characteristic
164	C₂×C₄	D₈×D₅₉	1	2	yes	—
165	C₂×C₄	SD₁₆×D₅₉	1	4	yes	—
166	C₂×C₄	Q₁₆×D₅₉	1	2	yes	—
167	C₂×C₄	#14273747	1	4	yes	—
168	C₈	—	2	4	no	—
169	C₈	—	59	8	no	—
170	C₈	—	59	8	no	—
171	C₂²	#16837271	1	1	yes	characteristic
172	C₂²	#16837773	1	2	yes	—
173	C₂²	#16837777	1	1	yes	characteristic
174	C₂²	#16837780	1	1	yes	characteristic
175	C₂²	#16837784	1	2	yes	—
176	C₄	—	16	2	no	—
177	C₄	—	944	2	no	—
178	C₄	#106×D₅₉	1	2	yes	—
179	C₄	#107×D₅₉	1	2	yes	—
180	C₄	#16837438	1	2	yes	—
181	C₄	#16837799	1	2	yes	—
182	C₄	#16837801	1	2	yes	—
183	C₄	#16837806	1	2	yes	—
184	C₂	#21423687	1	2	yes	—
185	C₂	#21423692	1	2	yes	—
186	C₂	#21424251	1	1	yes	characteristic
187	C₂	#21424254	1	1	yes	characteristic
188	C₂	#21424258	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273723
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1834	C₂
socle, prime radical	C₂³×C₅₉	#127
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1834	D₅₉
hypocenter	C₅₉	#8814
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
12. C₈×C₅₉⋊C₈	C₄	semidirect
74. C₅₉⋊C₈	#106	semidirect
75. C₅₉⋊C₈	#107	semidirect
110. C₅₉	#8814	semidirect

Maximal subgroups

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

subgroup	index
#21424255	2
#21424252	2
#21424224	2
C₂×C₈×C₅₉⋊C₈	2
C₅₉×#1834	2
#8814	59