Subgroups — #28479894

Order 15104.

414 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21423951	C₂	1	2	yes	maximal
2	C₄×#14273637	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837155	C₂	1	1	yes	maximal, characteristic
4	#21421930	C₂	1	2	yes	maximal
5	C₂×C₅₉×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14273637	—	2	2	no	—
8	C₂×#14273637	C₄	1	2	yes	—
9	C₂×#14273637	C₄	1	2	yes	—
10	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
12	#16837156	—	2	2	no	—
13	#16837155	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#334	C₄	1	4	yes	complemented
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)	C₂×C₄	1	4	yes	—
24	#14273637	—	2	4	no	—
25	#14273637	—	2	4	no	—
26	#14273637	—	4	4	no	—
27	C₄×C₅₉⋊C₈	—	2	1	no	—
28	C₄×C₅₉⋊C₈	—	2	1	no	—
29	C₂²×C₈×C₅₉	—	2	2	no	—
30	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₅₉	—	2	2	no	—
34	C₄⋊C₈×C₅₉	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
40	C₂²×Dic₅₉	C₄⋊C₄	1	4	yes	—
41	C₂×C₈×C₅₉	—	2	2	no	—
42	C₂×C₈×C₅₉	—	2	4	no	—
43	C₂×C₈×C₅₉	—	4	2	no	—
44	C₂×C₈×C₅₉	—	4	2	no	—
45	C₂×C₈×C₅₉	C₄²	1	2	yes	—
46	C₂×C₈×C₅₉	C₄²	1	2	yes	—
47	C₂×C₈×C₅₉	M₁₆	1	4	yes	—
48	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₅₉	#41	1	1	yes	characteristic
51	C₄²×C₅₉	#41	1	1	yes	characteristic
52	C₄×Dic₅₉	—	2	2	no	—
53	C₄×Dic₅₉	—	2	2	no	—
54	C₄×Dic₅₉	—	2	2	no	—
55	C₄×Dic₅₉	—	2	2	no	—
56	C₄×Dic₅₉	—	4	2	no	—
57	C₂×C₅₉⋊C₈	—	2	2	no	—
58	C₂×C₅₉⋊C₈	—	2	2	no	—
59	C₂×C₅₉⋊C₈	—	2	4	no	—
60	C₂×C₅₉⋊C₈	—	4	2	no	—
61	C₂×C₅₉⋊C₈	—	4	2	no	—
62	C₂×C₅₉⋊C₈	M₁₆	1	4	yes	—
63	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	—	2	2	no	—
65	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
72	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
73	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₅₉	#137	1	2	yes	—
75	C₂×Dic₅₉	—	2	4	no	—
76	C₂×Dic₅₉	—	2	4	no	—
77	C₂×Dic₅₉	—	2	4	no	—
78	C₈×C₅₉	—	2	2	no	—
79	C₈×C₅₉	—	4	4	no	—
80	C₈×C₅₉	#97	1	4	yes	complemented
81	C₅₉⋊C₈	—	2	4	no	—
82	C₅₉⋊C₈	—	4	4	no	—
83	#8276	—	59	1	no	maximal, Sylow
84	C₂²×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₅₉	#423	1	2	yes	—
86	C₂²×C₅₉	#428	1	2	yes	—
87	C₂²×C₅₉	#430	1	2	yes	—
88	C₄×C₅₉	—	2	1	no	—
89	C₄×C₅₉	—	2	1	no	—
90	C₄×C₅₉	C₂×#97	1	1	yes	characteristic
91	C₄×C₅₉	C₄×M₁₆	1	1	yes	characteristic
92	C₄×C₅₉	#432	1	2	yes	—
93	C₄×C₅₉	#440	1	2	yes	—
94	C₄×C₅₉	#441	1	1	yes	characteristic
95	C₄×C₅₉	#442	1	1	yes	characteristic
96	Dic₅₉	—	4	4	no	—
97	#1767	—	59	2	no	—
98	C₄×#97	—	59	1	no	—
99	C₂×#335	—	59	1	no	—
100	C₂×#334	D₅₉	1	1	yes	characteristic
101	#1277	—	59	2	no	—
102	C₂×C₅₉	#1569	1	4	yes	—
103	C₂×C₅₉	#1767	1	1	yes	characteristic
104	C₂×C₅₉	#1776	1	1	yes	characteristic
105	C₂×C₅₉	#1778	1	1	yes	characteristic
106	C₂×C₄⋊C₈	—	59	2	no	—
107	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
108	C₂×#97	—	59	2	no	—
109	C₂×#97	—	59	2	no	—
110	C₂×#97	—	118	2	no	—
111	C₂×C₄×C₈	—	59	1	no	—
112	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
113	C₄³	—	59	1	no	—
114	#336	—	118	2	no	—
115	#335	—	59	4	no	—
116	#334	Dic₅₉	1	4	yes	complemented
117	C₅₉	#8276	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	—	59	2	no	—
120	C₂²×C₈	—	118	2	no	—
121	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
122	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
123	C₂×C₄²	—	59	2	no	—
124	C₂×C₄²	—	59	2	no	—
125	C₂×C₄²	—	59	2	no	—
126	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	59	4	no	—
129	C₄⋊C₈	—	118	2	no	—
130	C₄⋊C₈	C₂×Dic₅₉	1	4	yes	—
131	#97	—	118	4	no	—
132	#97	—	118	4	no	—
133	#97	—	236	4	no	—
134	C₄×C₈	—	118	1	no	—
135	C₄×C₈	—	118	1	no	—
136	C₄×C₈	C₄×D₅₉	1	2	yes	—
137	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
138	C₂²×C₄	—	59	4	no	—
139	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
140	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
141	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	59	4	no	—
147	C₂×C₈	—	118	2	no	—
148	C₂×C₈	—	118	2	no	—
149	C₂×C₈	—	118	4	no	—
150	C₂×C₈	—	236	2	no	—
151	C₂×C₈	—	236	2	no	—
152	C₂×C₈	#11745659	1	4	yes	—
153	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
154	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
155	C₄²	—	118	2	no	—
156	C₄²	—	118	2	no	—
157	C₄²	—	118	2	no	—
158	C₄²	—	118	2	no	—
159	C₄²	—	236	2	no	—
160	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
161	C₄²	#11745687	1	1	yes	characteristic
162	C₄²	#11745689	1	1	yes	characteristic
163	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
164	C₂³	C₄⋊C₄×D₅₉	1	1	yes	characteristic
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	118	4	no	—
167	C₂×C₄	—	118	4	no	—
168	C₂×C₄	—	118	4	no	—
169	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
170	C₂×C₄	#14273685	1	1	yes	characteristic
171	C₂×C₄	#14273689	1	1	yes	characteristic
172	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
173	C₂×C₄	#14273692	1	1	yes	characteristic
174	C₂×C₄	Q₈×Dic₅₉	1	1	yes	characteristic
175	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
176	C₂×C₄	C₂×#11745659	1	2	yes	—
177	C₂×C₄	#14273754	1	2	yes	—
178	C₂×C₄	#14273756	1	2	yes	—
179	C₈	—	2	2	no	—
180	C₈	—	4	4	no	—
181	C₈	—	118	4	no	—
182	C₈	—	236	4	no	—
183	C₈	#14273637	1	4	yes	complemented
184	C₂²	C₄⋊C₄×Dic₅₉	1	1	yes	characteristic
185	C₂²	#16837424	1	2	yes	—
186	C₂²	#16837441	1	2	yes	—
187	C₂²	#16837449	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	236	4	no	—
191	C₄	C₂×#14273637	1	1	yes	characteristic
192	C₄	M₁₆×Dic₅₉	1	1	yes	characteristic
193	C₄	#16837458	1	2	yes	—
194	C₄	#16837503	1	2	yes	—
195	C₄	#16837508	1	1	yes	characteristic
196	C₄	#16837513	1	1	yes	characteristic
197	C₂	#21422860	1	4	yes	—
198	C₂	#21423951	1	1	yes	characteristic
199	C₂	#21423996	1	1	yes	characteristic
200	C₂	#21424004	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#334	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#334	D₅₉
hypocenter	C₅₉	#8276
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
13. #16837155	C₄	semidirect
16. C₅₉×#334	C₄	semidirect
80. C₈×C₅₉	#97	semidirect
116. #334	Dic₅₉	semidirect
117. C₅₉	#8276	semidirect
183. C₈	#14273637	semidirect

Maximal subgroups

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

subgroup	index
#21423951	2
C₄×#14273637	2
C₂×#16837155	2
#21421930	2
C₂×C₅₉×#334	2
#8276	59