Subgroups — #28505136

Order 15104.

314 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	C₂×#16837232	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837231	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic
4	#21424266	C₂	1	2	yes	maximal
5	C₂×C₅₉×#368	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14273663	C₂²	1	2	yes	—
8	C₂×#14273662	—	2	2	no	—
9	#16837425	—	2	2	no	—
10	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
11	#16837232	—	2	2	no	—
12	#16837231	—	2	4	no	—
13	C₁₆×Dic₅₉	C₄	1	2	yes	—
14	C₁₆×Dic₅₉	C₄	1	2	yes	—
15	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×#108×C₅₉	C₂²	1	2	yes	—
17	C₅₉×#368	—	2	2	no	—
18	C₂×#11745664	—	2	2	no	—
19	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
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	#14273663	—	2	2	no	—
25	#14273663	—	2	2	no	—
26	#14273662	—	4	4	no	—
27	Dic₅₉⋊C₈	—	4	4	no	—
28	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₅₉	Q₈	1	1	yes	characteristic
31	C₈×Dic₅₉	Q₈	1	1	yes	characteristic
32	C₂×M₁₆×C₅₉	—	2	2	no	—
33	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
34	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
35	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
36	C₂×C₁₆×C₅₉	Q₈	1	2	yes	—
37	#108×C₅₉	—	2	2	no	—
38	#108×C₅₉	—	2	2	no	—
39	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₅₉	Q₁₆	1	2	yes	—
41	M₁₆×C₅₉	—	4	2	no	—
42	M₁₆×C₅₉	—	4	2	no	—
43	C₂×C₈×C₅₉	—	4	2	no	—
44	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
50	C₄×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
51	C₄×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
52	C₄×Dic₅₉	D₈	1	2	yes	—
53	#11745664	—	4	2	no	—
54	#11745664	—	4	2	no	—
55	C₂×C₅₉⋊C₈	—	4	2	no	—
56	C₂×C₅₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₅₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₅₉⋊C₈	D₈	1	2	yes	—
59	C₂×C₅₉⋊C₈	Q₁₆	1	2	yes	—
60	C₁₆×C₅₉	C₄⋊C₄	1	4	yes	—
61	C₅₉⋊C₁₆	C₄⋊C₄	1	4	yes	—
62	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×Dic₅₉	—	2	4	no	—
70	C₂×Dic₅₉	#107	1	2	yes	—
71	C₂×Dic₅₉	#107	1	2	yes	—
72	C₈×C₅₉	—	8	2	no	—
73	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
76	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
77	C₅₉⋊C₈	—	8	2	no	—
78	C₅₉⋊C₈	#107	1	2	yes	—
79	C₅₉⋊C₈	#107	1	2	yes	—
80	#12909	—	59	1	no	maximal, Sylow
81	C₂²×C₅₉	—	2	2	no	—
82	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
83	C₂²×C₅₉	#494	1	1	yes	characteristic
84	C₂²×C₅₉	#500	1	1	yes	characteristic
85	C₄×C₅₉	#384	1	1	yes	characteristic
86	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
87	C₄×C₅₉	#493	1	1	yes	characteristic
88	C₄×C₅₉	#500	1	1	yes	characteristic
89	Dic₅₉	—	2	2	no	—
90	Dic₅₉	#368	1	4	yes	complemented
91	C₂×#368	—	59	1	no	—
92	C₂×#368	—	59	2	no	—
93	C₂×#368	D₅₉	1	1	yes	characteristic
94	C₂×C₄×C₁₆	—	59	1	no	—
95	#1848	—	59	2	no	—
96	C₂×C₅₉	—	2	1	no	—
97	C₂×C₅₉	—	2	1	no	—
98	C₂×C₅₉	#1845	1	1	yes	characteristic
99	C₂×C₅₉	C₂×#368	1	2	yes	—
100	C₂²×C₁₆	—	59	1	no	—
101	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
102	C₂×#108	—	59	2	no	—
103	C₂×#108	—	118	2	no	—
104	C₂×#108	C₂×D₅₉	1	2	yes	—
105	#423	—	118	2	no	—
106	C₂×C₄×C₈	—	59	1	no	—
107	#368	—	2	2	no	—
108	#368	—	118	4	no	—
109	#368	—	118	2	no	—
110	C₄×C₁₆	—	59	2	no	—
111	C₄×C₁₆	—	59	2	no	—
112	C₅₉	#12909	1	1	yes	characteristic, complemented, Sylow
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	118	2	no	—
115	C₂²×C₈	—	59	1	no	—
116	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
117	C₂×C₄²	—	59	1	no	—
118	C₂×C₁₆	—	59	2	no	—
119	C₂×C₁₆	—	59	2	no	—
120	C₂×C₁₆	—	59	2	no	—
121	C₂×C₁₆	Dic₁₁₈	1	2	yes	—
122	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
123	C₂×C₁₆	C₂²⋊D₅₉	1	2	yes	—
124	#108	—	2	2	no	—
125	#108	—	2	2	no	—
126	#108	—	118	2	no	—
127	#108	—	118	2	no	—
128	#108	—	236	4	no	—
129	C₄⋊C₈	—	236	4	no	—
130	C₄×C₈	—	59	1	no	—
131	C₄×C₈	—	59	1	no	—
132	C₄×C₈	—	59	1	no	—
133	C₄×C₈	—	59	1	no	—
134	C₂²×C₄	—	59	2	no	—
135	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
136	M₁₆	—	4	2	no	—
137	M₁₆	—	4	2	no	—
138	M₁₆	—	236	2	no	—
139	M₁₆	—	236	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	59	1	no	—
142	C₂×C₈	—	59	1	no	—
143	C₂×C₈	—	59	2	no	—
144	C₂×C₈	—	59	2	no	—
145	C₂×C₈	—	236	2	no	—
146	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
147	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
148	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
149	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
150	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
151	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
152	C₄²	—	59	1	no	—
153	C₄²	—	59	1	no	—
154	C₄²	—	59	2	no	—
155	C₁₆	—	59	4	no	—
156	C₁₆	Dic₅₉⋊C₄	1	4	yes	—
157	C₂³	Q₁₆×D₅₉	1	1	yes	characteristic
158	C₂×C₄	—	59	2	no	—
159	C₂×C₄	—	59	2	no	—
160	C₂×C₄	—	118	4	no	—
161	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
162	C₂×C₄	#14273723	1	1	yes	characteristic
163	C₂×C₄	#14273726	1	1	yes	characteristic
164	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
165	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
167	C₈	—	8	2	no	—
168	C₈	—	59	2	no	—
169	C₈	—	59	2	no	—
170	C₈	—	472	2	no	—
171	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
172	C₈	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
173	C₈	#14273725	1	1	yes	characteristic
174	C₈	#14273726	1	1	yes	characteristic
175	C₂²	—	2	2	no	—
176	C₂²	#107×D₅₉	1	1	yes	characteristic
177	C₂²	#16837780	1	1	yes	characteristic
178	C₂²	#16837806	1	1	yes	characteristic
179	C₄	—	59	4	no	—
180	C₄	—	118	2	no	—
181	C₄	#16837271	1	1	yes	characteristic
182	C₄	#107×D₅₉	1	1	yes	characteristic
183	C₄	#16837777	1	1	yes	characteristic
184	C₄	#16837806	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	#21424254	1	1	yes	characteristic
188	C₂	#368×D₅₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#368	C₂
socle	C₂²×C₅₉	C₂×#107
prime radical	C₂³×C₅₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#368	D₅₉
hypocenter	C₅₉	#12909
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
90. Dic₅₉	#368	semidirect
112. C₅₉	#12909	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837232	2
C₂×#16837231	2
C₂×C₁₆×Dic₅₉	2
#21424266	2
C₂×C₅₉×#368	2
#12909	59