Subgroups — #28505035

Order 15104.

366 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#21422173	C₂	1	4	yes	maximal
4	C₂×C₅₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
6	C₂×C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅₉	C₄	1	2	yes	—
8	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
9	C₄²×Dic₅₉	C₄	1	1	yes	characteristic
10	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
11	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×C₄⋊C₁₆	—	2	2	no	—
14	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₅₉	C₈	1	1	yes	characteristic
16	C₂×C₄×Dic₅₉	C₈	1	2	yes	—
17	C₂×C₄×Dic₅₉	C₈	1	1	yes	characteristic
18	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
19	C₂×C₅₉⋊C₁₆	—	2	2	no	—
20	C₂×C₅₉⋊C₁₆	—	2	4	no	—
21	C₄×C₅₉⋊C₈	—	2	2	no	—
22	C₈×Dic₅₉	—	2	2	no	—
23	C₈×Dic₅₉	—	2	2	no	—
24	C₈×Dic₅₉	—	2	4	no	—
25	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₉	—	2	2	no	—
29	C₂×C₁₆×C₅₉	—	2	4	no	—
30	C₄×C₈×C₅₉	D₄	1	2	yes	—
31	C₄×C₈×C₅₉	Q₈	1	2	yes	—
32	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₅₉	C₂×C₈	1	2	yes	—
36	C₂²×Dic₅₉	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₅₉	—	2	2	no	—
38	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
39	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₅₉	#41	1	2	yes	—
42	C₂×C₈×C₅₉	#41	1	2	yes	—
43	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
44	C₄²×C₅₉	M₁₆	1	2	yes	—
45	C₄×Dic₅₉	—	2	2	no	—
46	C₄×Dic₅₉	—	2	2	no	—
47	C₄×Dic₅₉	—	2	2	no	—
48	C₄×Dic₅₉	—	2	4	no	—
49	C₄×Dic₅₉	C₁₆	1	4	yes	complemented
50	C₂×C₅₉⋊C₈	—	2	2	no	—
51	C₂×C₅₉⋊C₈	—	2	2	no	—
52	C₂×C₅₉⋊C₈	M₁₆	1	4	yes	—
53	C₁₆×C₅₉	—	4	2	no	—
54	C₅₉⋊C₁₆	—	4	2	no	—
55	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₅₉	—	2	2	no	—
57	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
58	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₅₉	C₂×C₁₆	1	2	yes	—
60	C₂×C₄×C₅₉	M₃₂	1	2	yes	—
61	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#117	1	2	yes	—
63	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
64	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
65	C₂×Dic₅₉	—	2	2	no	—
66	C₂×Dic₅₉	—	2	2	no	—
67	C₂×Dic₅₉	—	2	2	no	—
68	C₂×Dic₅₉	—	2	2	no	—
69	C₂×Dic₅₉	C₂×C₁₆	1	4	yes	—
70	C₂×Dic₅₉	M₃₂	1	4	yes	—
71	C₈×C₅₉	—	2	2	no	—
72	C₈×C₅₉	#124	1	1	yes	characteristic
73	C₈×C₅₉	#124	1	1	yes	characteristic
74	C₈×C₅₉	#125	1	2	yes	—
75	C₅₉⋊C₈	—	2	4	no	—
76	#12879	—	59	1	no	maximal, Sylow
77	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
78	C₂²×C₅₉	#431	1	2	yes	—
79	C₂²×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
80	C₂²×C₅₉	C₂×M₃₂	1	2	yes	—
81	C₂²×C₅₉	C₂×M₃₂	1	1	yes	characteristic
82	C₄×C₅₉	—	2	2	no	—
83	C₄×C₅₉	C₄⋊C₁₆	1	4	yes	—
84	C₄×C₅₉	#382	1	1	yes	characteristic
85	C₄×C₅₉	#423	1	1	yes	characteristic
86	C₄×C₅₉	#432	1	2	yes	—
87	Dic₅₉	—	2	4	no	—
88	Dic₅₉	—	2	4	no	—
89	C₂×C₄⋊C₁₆	—	59	1	no	—
90	C₂×C₄⋊C₁₆	D₅₉	1	1	yes	characteristic
91	C₄²×C₈	—	59	1	no	—
92	#1374	—	59	4	no	—
93	C₂×C₅₉	#1831	1	1	yes	characteristic
94	C₂×C₅₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
95	C₂×C₅₉	#2150	1	1	yes	characteristic
96	C₂×C₅₉	#2162	1	2	yes	—
97	C₂×C₅₉	#2163	1	2	yes	—
98	C₂²×C₁₆	—	59	2	no	—
99	C₂²×C₁₆	C₂×D₅₉	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	C₂×D₅₉	1	1	yes	characteristic
104	C₄³	—	59	1	no	—
105	C₄⋊C₁₆	—	2	2	no	—
106	C₄⋊C₁₆	—	118	2	no	—
107	C₅₉	#12879	1	1	yes	characteristic, complemented, Sylow
108	C₂²×C₈	—	59	2	no	—
109	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
111	C₂×C₄²	—	59	1	no	—
112	C₂×C₄²	—	59	2	no	—
113	C₂×C₄²	—	59	1	no	—
114	C₂×C₄²	—	59	2	no	—
115	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
116	C₂×C₁₆	—	2	4	no	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	—	118	4	no	—
119	C₂×C₁₆	—	118	2	no	—
120	C₄×C₈	—	118	4	no	—
121	C₄×C₈	—	118	2	no	—
122	C₄×C₈	—	118	2	no	—
123	C₄×C₈	—	118	2	no	—
124	C₄×C₈	Dic₁₁₈	1	2	yes	—
125	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
126	C₂²×C₄	—	59	2	no	—
127	C₂²×C₄	—	59	2	no	—
128	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
129	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	59	4	no	—
133	C₂×C₈	—	118	2	no	—
134	C₂×C₈	—	118	2	no	—
135	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
136	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
137	C₂×C₈	#11745687	1	2	yes	—
138	C₂×C₈	#11745689	1	2	yes	—
139	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
140	C₄²	—	59	4	no	—
141	C₄²	—	118	4	no	—
142	C₄²	—	118	2	no	—
143	C₄²	—	118	2	no	—
144	C₄²	—	118	2	no	—
145	C₄²	#11745659	1	2	yes	—
146	C₄²	Dic₅₉⋊C₄	1	2	yes	—
147	C₁₆	—	4	2	no	—
148	C₁₆	—	236	2	no	—
149	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	59	4	no	—
152	C₂×C₄	—	59	4	no	—
153	C₂×C₄	—	118	2	no	—
154	C₂×C₄	—	118	2	no	—
155	C₂×C₄	—	118	2	no	—
156	C₂×C₄	—	118	2	no	—
157	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
158	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
159	C₂×C₄	C₁₆×D₅₉	1	2	yes	—
160	C₂×C₄	#14273666	1	2	yes	—
161	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
162	C₂×C₄	#14273685	1	2	yes	—
163	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
164	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	118	4	no	—
167	C₈	#14273714	1	1	yes	characteristic
168	C₈	#14273716	1	1	yes	characteristic
169	C₈	#14273718	1	2	yes	—
170	C₂²	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
171	C₂²	#16837454	1	2	yes	—
172	C₂²	C₂×C₁₆×D₅₉	1	1	yes	characteristic
173	C₂²	M₃₂×D₅₉	1	2	yes	—
174	C₂²	C₂×#14273666	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	118	4	no	—
177	C₄	—	118	4	no	—
178	C₄	Dic₅₉⋊C₁₆	1	4	yes	—
179	C₄	#16837267	1	1	yes	characteristic
180	C₄	#16837425	1	1	yes	characteristic
181	C₄	#16837458	1	2	yes	—
182	C₂	#21424212	1	1	yes	characteristic
183	C₂	C₂×Dic₅₉⋊C₁₆	1	1	yes	characteristic
184	C₂	#21425851	1	1	yes	characteristic
185	C₂	#21425894	1	2	yes	—
186	C₂	#21425898	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂²×C₁₆
Fitting	C₂×C₅₉×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅₉	C₂²×C₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₉
hypocenter	C₅₉	#12879
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
49. C₄×Dic₅₉	C₁₆	semidirect
107. C₅₉	#12879	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₅₉⋊C₁₆)	2
C₄×C₈×Dic₅₉	2
#21422173	2
C₂×C₅₉×C₄⋊C₁₆	2
#12879	59