Subgroups — #28505124

Order 15104.

374 conjugacy classes of subgroups.

The subgroup lattice has 184 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×#16837229	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic
4	#21424254	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×#14273660	—	2	2	no	—
9	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
10	#16837271	—	2	2	no	—
11	#16837230	—	2	2	no	—
12	#16837229	—	2	4	no	—
13	C₁₆×Dic₅₉	C₄	1	4	yes	complemented
14	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×#107×C₅₉	C₂²	1	2	yes	—
16	C₅₉×#367	—	2	2	no	—
17	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₅₉	—	2	2	no	—
19	C₂×Dic₅₉⋊C₄	—	4	4	no	—
20	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
21	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₅₉⋊C₁₆	D₄	1	2	yes	—
23	C₂×C₅₉⋊C₁₆	Q₈	1	2	yes	—
24	#14273661	—	2	2	no	—
25	#14273661	—	2	2	no	—
26	#14273660	—	4	4	no	—
27	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
28	C₈×Dic₅₉	Q₈	1	2	yes	—
29	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
31	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
33	C₂×C₁₆×C₅₉	Q₈	1	2	yes	—
34	#107×C₅₉	—	2	2	no	—
35	#107×C₅₉	—	2	2	no	—
36	C₂²×C₄×C₅₉	—	4	2	no	—
37	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₅₉	—	4	2	no	—
39	C₂²×Dic₅₉	D₈	1	2	yes	—
40	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
46	C₄⋊C₄×C₅₉	—	4	2	no	—
47	C₄⋊C₄×C₅₉	—	4	2	no	—
48	C₄⋊Dic₅₉	—	4	2	no	—
49	C₄⋊Dic₅₉	—	4	2	no	—
50	Dic₅₉⋊C₄	—	8	8	no	—
51	C₄×Dic₅₉	C₄⋊C₄	1	2	yes	—
52	C₄×Dic₅₉	Q₁₆	1	2	yes	—
53	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
54	C₂×C₅₉⋊C₈	D₈	1	2	yes	—
55	C₂×C₅₉⋊C₈	Q₁₆	1	2	yes	—
56	C₁₆×C₅₉	C₄⋊C₄	1	4	yes	complemented
57	C₅₉⋊C₁₆	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₅₉	—	8	2	no	—
60	C₂×C₄×C₅₉	—	8	4	no	—
61	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×Dic₅₉	—	8	2	no	—
68	C₂×Dic₅₉	—	8	4	no	—
69	C₂×Dic₅₉	#107	1	4	yes	—
70	C₂×Dic₅₉	SD₃₂	1	8	yes	—
71	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
74	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
75	C₅₉⋊C₈	#107	1	4	yes	—
76	#12906	—	59	1	no	maximal, Sylow
77	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
78	C₂²×C₅₉	#493	1	1	yes	characteristic
79	C₂²×C₅₉	#500	1	1	yes	characteristic
80	C₂²×C₅₉	C₂×SD₃₂	1	4	yes	—
81	C₄×C₅₉	—	16	2	no	—
82	C₄×C₅₉	#384	1	1	yes	characteristic
83	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
84	C₄×C₅₉	#494	1	1	yes	characteristic
85	C₄×C₅₉	#500	1	1	yes	characteristic
86	Dic₅₉	—	16	2	no	—
87	Dic₅₉	#367	1	8	yes	complemented
88	C₂×#367	—	59	1	no	—
89	C₂×#367	—	59	2	no	—
90	C₂×#367	D₅₉	1	1	yes	characteristic
91	C₂×C₄×C₁₆	—	59	1	no	—
92	#1845	—	59	2	no	—
93	C₂×C₅₉	#1845	1	1	yes	characteristic
94	C₂×C₅₉	C₂×#367	1	2	yes	—
95	C₂×C₅₉	#2248	1	2	yes	—
96	C₂×C₅₉	#2254	1	2	yes	—
97	C₂²×C₁₆	—	59	1	no	—
98	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
99	C₂×#107	—	59	2	no	—
100	C₂×#107	—	118	2	no	—
101	C₂×#107	C₂×D₅₉	1	2	yes	—
102	C₂×C₄×C₈	—	59	1	no	—
103	#384	—	118	2	no	—
104	#367	—	2	2	no	—
105	#367	—	118	4	no	—
106	#367	—	118	2	no	—
107	C₄×C₁₆	—	59	4	no	—
108	C₅₉	#12906	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	59	1	no	—
110	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	118	2	no	—
113	C₂×C₄⋊C₄	—	236	4	no	—
114	C₂×C₄²	—	59	1	no	—
115	C₂×C₁₆	—	59	2	no	—
116	C₂×C₁₆	—	59	2	no	—
117	C₂×C₁₆	—	59	2	no	—
118	C₂×C₁₆	Dic₁₁₈	1	2	yes	—
119	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
120	C₂×C₁₆	C₂²⋊D₅₉	1	2	yes	—
121	#107	—	2	2	no	—
122	#107	—	2	2	no	—
123	#107	—	118	2	no	—
124	#107	—	118	2	no	—
125	#107	—	236	4	no	—
126	C₄×C₈	—	59	2	no	—
127	C₄×C₈	—	59	2	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	—	59	2	no	—
130	C₂²×C₄	—	236	2	no	—
131	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
132	C₂×C₈	—	59	2	no	—
133	C₂×C₈	—	59	2	no	—
134	C₂×C₈	—	59	2	no	—
135	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
137	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
139	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
140	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	472	8	no	—
144	C₄⋊C₄	—	236	2	no	—
145	C₄⋊C₄	—	236	2	no	—
146	C₄²	—	59	2	no	—
147	C₄²	—	59	2	no	—
148	C₁₆	—	59	4	no	—
149	C₁₆	Dic₅₉⋊C₄	1	4	yes	complemented
150	C₂³	D₈×D₅₉	1	1	yes	characteristic
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	—	59	4	no	—
154	C₂×C₄	—	59	8	no	—
155	C₂×C₄	—	472	2	no	—
156	C₂×C₄	—	472	4	no	—
157	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
158	C₂×C₄	#14273723	1	1	yes	characteristic
159	C₂×C₄	#14273726	1	1	yes	characteristic
160	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
161	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
162	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
163	C₈	—	59	4	no	—
164	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
165	C₈	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
166	C₈	#14273725	1	1	yes	characteristic
167	C₈	#14273726	1	1	yes	characteristic
168	C₂²	#107×D₅₉	1	1	yes	characteristic
169	C₂²	#16837777	1	1	yes	characteristic
170	C₂²	#16837806	1	1	yes	characteristic
171	C₂²	SD₃₂×D₅₉	1	4	yes	—
172	C₄	—	16	2	no	—
173	C₄	—	59	8	no	—
174	C₄	—	944	2	no	—
175	C₄	#16837271	1	1	yes	characteristic
176	C₄	#107×D₅₉	1	1	yes	characteristic
177	C₄	#16837780	1	1	yes	characteristic
178	C₄	#16837806	1	1	yes	characteristic
179	C₂	#21424254	1	1	yes	characteristic
180	C₂	#367×D₅₉	1	2	yes	—
181	C₂	#21426350	1	2	yes	—
182	C₂	#21426377	1	2	yes	—

	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	#12906
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
13. C₁₆×Dic₅₉	C₄	semidirect
56. C₁₆×C₅₉	C₄⋊C₄	semidirect
57. C₅₉⋊C₁₆	C₄⋊C₄	semidirect
87. Dic₅₉	#367	semidirect
108. C₅₉	#12906	semidirect
149. C₁₆	Dic₅₉⋊C₄	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×#16837229	2
C₂×C₁₆×Dic₅₉	2
#21424254	2
C₂×C₅₉×#367	2
#12906	59