Subgroups — #28506725

Order 15104.

326 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837230	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#21424830	C₂	1	2	yes	maximal
4	#21422187	C₂	1	2	yes	maximal
5	C₂×C₅₉×#367	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14273661	C₂²	1	2	yes	—
8	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
9	#16837301	—	2	2	no	—
10	#16837230	—	2	2	no	—
11	C₁₆×Dic₅₉	—	2	2	no	—
12	#16837164	—	2	2	no	—
13	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×#107×C₅₉	C₂²	1	2	yes	—
15	C₅₉×#367	C₄	1	4	yes	complemented
16	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₅₉	—	2	2	no	—
18	C₂×C₄⋊Dic₅₉	—	4	2	no	—
19	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
20	C₂×C₅₉⋊C₁₆	—	2	1	no	—
21	C₂×C₅₉⋊C₁₆	—	2	1	no	—
22	C₂×C₅₉⋊C₁₆	Q₈	1	2	yes	—
23	#14273661	—	2	2	no	—
24	#14273661	—	2	2	no	—
25	C₈×Dic₅₉	—	2	1	no	—
26	C₈×Dic₅₉	—	2	1	no	—
27	C₅₉⋊#95	—	4	2	no	—
28	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
30	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
32	C₂×C₁₆×C₅₉	Q₈	1	2	yes	—
33	#107×C₅₉	—	2	2	no	—
34	#107×C₅₉	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₅₉	—	4	2	no	—
36	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₅₉	—	4	2	no	—
38	C₂²×Dic₅₉	D₈	1	2	yes	—
39	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
44	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
45	C₄⋊C₄×C₅₉	—	2	4	no	—
46	C₄⋊C₄×C₅₉	—	4	2	no	—
47	C₄⋊Dic₅₉	—	4	2	no	—
48	C₄⋊Dic₅₉	—	4	2	no	—
49	C₄⋊Dic₅₉	—	8	4	no	—
50	C₄×Dic₅₉	—	2	1	no	—
51	C₄×Dic₅₉	—	2	1	no	—
52	C₂×C₅₉⋊C₈	—	2	1	no	—
53	C₂×C₅₉⋊C₈	—	2	1	no	—
54	C₂×C₅₉⋊C₈	Q₁₆	1	2	yes	—
55	C₁₆×C₅₉	C₄⋊C₄	1	4	yes	complemented
56	C₅₉⋊C₁₆	—	2	2	no	—
57	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₅₉	—	4	4	no	—
59	C₂×C₄×C₅₉	—	8	2	no	—
60	C₂×C₄×C₅₉	—	8	2	no	—
61	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
67	C₂×Dic₅₉	—	2	2	no	—
68	C₂×Dic₅₉	—	2	2	no	—
69	C₂×Dic₅₉	—	8	2	no	—
70	C₂×Dic₅₉	—	8	2	no	—
71	C₂×Dic₅₉	—	8	2	no	—
72	C₂×Dic₅₉	SD₃₂	1	4	yes	—
73	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
75	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₅₉	#125	1	1	yes	characteristic
77	C₅₉⋊C₈	—	2	2	no	—
78	#13166	—	59	1	no	maximal, Sylow
79	C₂²×C₅₉	C₄×D₈	1	1	yes	characteristic
80	C₂²×C₅₉	#466	1	1	yes	characteristic
81	C₂²×C₅₉	#500	1	1	yes	characteristic
82	C₂²×C₅₉	C₂×SD₃₂	1	2	yes	—
83	C₂²×C₅₉	#508	1	2	yes	—
84	C₄×C₅₉	—	8	4	no	—
85	C₄×C₅₉	#389	1	1	yes	characteristic
86	C₄×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
87	C₄×C₅₉	#467	1	1	yes	characteristic
88	C₄×C₅₉	#499	1	1	yes	characteristic
89	Dic₅₉	—	2	4	no	—
90	Dic₅₉	—	16	2	no	—
91	C₂×#367	—	59	1	no	—
92	C₂×#367	D₅₉	1	1	yes	characteristic
93	C₂×C₄×C₁₆	—	59	1	no	—
94	#1942	—	59	2	no	—
95	#1380	—	59	2	no	—
96	C₂×C₅₉	#1942	1	1	yes	characteristic
97	C₂×C₅₉	C₄×SD₃₂	1	2	yes	—
98	C₂×C₅₉	#2217	1	2	yes	—
99	C₂×C₅₉	#2253	1	1	yes	characteristic
100	C₂×C₅₉	#2254	1	1	yes	characteristic
101	C₂²×C₁₆	—	59	1	no	—
102	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
103	C₂×#107	—	59	2	no	—
104	C₂×#107	C₂×D₅₉	1	2	yes	—
105	C₂×C₄×C₈	—	59	1	no	—
106	#389	—	118	2	no	—
107	#367	—	118	2	no	—
108	#367	Dic₅₉	1	4	yes	complemented
109	C₄×C₁₆	—	118	2	no	—
110	#340	—	118	2	no	—
111	C₅₉	#13166	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₈	—	59	1	no	—
113	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	118	2	no	—
116	C₂×C₄⋊C₄	—	236	2	no	—
117	C₂×C₄²	—	59	1	no	—
118	C₂×C₁₆	—	59	2	no	—
119	C₂×C₁₆	—	118	1	no	—
120	C₂×C₁₆	—	118	1	no	—
121	C₂×C₁₆	Dic₁₁₈	1	2	yes	—
122	C₂×C₁₆	D₂₃₆	1	2	yes	—
123	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	118	2	no	—
126	#107	—	118	2	no	—
127	#107	C₂×Dic₅₉	1	4	yes	—
128	C₄×C₈	—	118	1	no	—
129	C₄×C₈	—	118	1	no	—
130	#95	—	236	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	59	2	no	—
133	C₂²×C₄	—	236	2	no	—
134	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
135	C₂×C₈	—	59	2	no	—
136	C₂×C₈	—	118	1	no	—
137	C₂×C₈	—	118	1	no	—
138	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
139	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
140	C₂×C₈	#11745687	1	1	yes	characteristic
141	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
142	C₂×C₈	#11745689	1	1	yes	characteristic
143	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	472	4	no	—
147	C₄⋊C₄	—	236	2	no	—
148	C₄⋊C₄	—	236	2	no	—
149	C₄²	—	118	1	no	—
150	C₄²	—	118	1	no	—
151	C₁₆	—	118	2	no	—
152	C₁₆	C₄⋊Dic₅₉	1	4	yes	complemented
153	C₂³	D₈×D₅₉	1	1	yes	characteristic
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	59	4	no	—
158	C₂×C₄	—	118	2	no	—
159	C₂×C₄	—	118	2	no	—
160	C₂×C₄	—	472	2	no	—
161	C₂×C₄	—	472	2	no	—
162	C₂×C₄	—	472	2	no	—
163	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
164	C₂×C₄	#14273702	1	1	yes	characteristic
165	C₂×C₄	#14273726	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
167	C₂×C₄	#14273746	1	1	yes	characteristic
168	C₂×C₄	#14273750	1	1	yes	characteristic
169	C₈	—	118	2	no	—
170	C₈	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
171	C₈	Q₈×Dic₅₉	1	1	yes	characteristic
172	C₈	#14273707	1	1	yes	characteristic
173	C₈	#14273718	1	1	yes	characteristic
174	C₂²	D₈×Dic₅₉	1	1	yes	characteristic
175	C₂²	#16837639	1	1	yes	characteristic
176	C₂²	#16837806	1	1	yes	characteristic
177	C₂²	SD₃₂×D₅₉	1	2	yes	—
178	C₂²	#16837833	1	2	yes	—
179	C₄	—	8	4	no	—
180	C₄	—	118	4	no	—
181	C₄	—	944	2	no	—
182	C₄	#16837301	1	1	yes	characteristic
183	C₄	Q₁₆×Dic₅₉	1	1	yes	characteristic
184	C₄	#16837644	1	1	yes	characteristic
185	C₄	#16837803	1	1	yes	characteristic
186	C₂	#21424830	1	1	yes	characteristic
187	C₂	SD₃₂×Dic₅₉	1	2	yes	—
188	C₂	#21426193	1	2	yes	—
189	C₂	#21426374	1	1	yes	characteristic
190	C₂	#21426377	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#367	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#367	D₅₉
hypocenter	C₅₉	#13166
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
15. C₅₉×#367	C₄	semidirect
55. C₁₆×C₅₉	C₄⋊C₄	semidirect
108. #367	Dic₅₉	semidirect
111. C₅₉	#13166	semidirect
152. C₁₆	C₄⋊Dic₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837230	2
C₂×C₁₆×Dic₅₉	2
#21424830	2
#21422187	2
C₂×C₅₉×#367	2
#13166	59