Subgroups — #28504080

Order 15104.

318 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×#16837232	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#21423977	C₂	1	2	yes	maximal
4	#21422191	C₂	1	2	yes	maximal
5	C₅₉×#2158	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14273663	C₂²	1	2	yes	—
8	#16837358	—	2	2	no	—
9	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
10	#16837232	C₄	1	4	yes	complemented
11	C₁₆×Dic₅₉	C₄	1	4	yes	complemented
12	#16837161	—	2	2	no	—
13	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#427	C₂²	1	2	yes	—
15	C₅₉×#367	C₄	1	2	yes	complemented
16	C₅₉×#366	C₄	1	2	yes	complemented
17	C₂×#11745664	—	2	2	no	—
18	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
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	C₂×C₄	1	2	yes	—
23	#14273663	—	2	2	no	—
24	#14273663	C₂×C₄	1	4	yes	—
25	#14273638	—	4	2	no	—
26	C₄×C₅₉⋊C₈	—	4	2	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	#117×C₅₉	—	2	2	no	—
31	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
35	#107×C₅₉	C₂×C₄	1	4	yes	—
36	#106×C₅₉	—	2	2	no	—
37	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₅₉	—	2	1	no	—
39	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₂²×C₄	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	C₂²⋊C₄×C₅₉	—	4	2	no	—
47	C₄²×C₅₉	—	4	2	no	—
48	C₄×Dic₅₉	—	2	1	no	—
49	C₄×Dic₅₉	C₄⋊C₄	1	2	yes	—
50	#11745664	—	2	4	no	—
51	#11745664	—	4	2	no	—
52	C₂×C₅₉⋊C₈	—	4	2	no	—
53	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
54	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
55	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
56	C₂×C₅₉⋊C₈	Q₁₆	1	2	yes	—
57	C₁₆×C₅₉	C₄²	1	2	yes	—
58	C₁₆×C₅₉	C₄²	1	2	yes	—
59	C₅₉⋊C₁₆	—	2	2	no	—
60	C₂³×C₅₉	#135	1	1	yes	characteristic
61	C₂×C₄×C₅₉	—	4	4	no	—
62	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
68	C₂×Dic₅₉	—	2	4	no	—
69	C₂×Dic₅₉	—	2	2	no	—
70	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
71	C₈×C₅₉	#117	1	1	yes	characteristic
72	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
73	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
74	C₅₉⋊C₈	—	4	4	no	—
75	C₅₉⋊C₈	#107	1	2	yes	—
76	C₅₉⋊C₈	#107	1	2	yes	—
77	#12670	—	59	1	no	maximal, Sylow
78	C₂²×C₅₉	—	2	2	no	—
79	C₂²×C₅₉	#427	1	1	yes	characteristic
80	C₂²×C₅₉	C₄×D₈	1	1	yes	characteristic
81	C₂²×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
82	C₄×C₅₉	—	8	4	no	—
83	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
85	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
86	C₄×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
87	Dic₅₉	—	2	4	no	—
88	C₂×#368	—	59	1	no	—
89	#2158	D₅₉	1	1	yes	characteristic
90	C₂×C₄×C₁₆	—	59	1	no	—
91	#1772	—	59	2	no	—
92	#1382	—	59	2	no	—
93	C₂×C₅₉	—	2	1	no	—
94	C₂×C₅₉	—	2	1	no	—
95	C₂×C₅₉	C₄×#107	1	1	yes	characteristic
96	C₂×C₅₉	#2178	1	2	yes	—
97	C₂²×C₁₆	—	59	1	no	—
98	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
99	C₂×#108	—	59	2	no	—
100	#427	C₂×D₅₉	1	2	yes	—
101	#405	—	118	2	no	—
102	C₂×C₄×C₈	—	59	1	no	—
103	#368	—	59	4	no	—
104	#367	Dic₅₉	1	2	yes	complemented
105	#366	Dic₅₉	1	2	yes	complemented
106	C₄×C₁₆	—	59	4	no	—
107	#339	—	118	2	no	—
108	C₅₉	#12670	1	1	yes	characteristic, complemented, Sylow
109	C₂×M₁₆	—	118	2	no	—
110	C₂²×C₈	—	59	1	no	—
111	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
112	#117	—	2	2	no	—
113	C₂×C₄²	—	59	1	no	—
114	C₂×C₁₆	—	59	2	no	—
115	C₂×C₁₆	—	118	1	no	—
116	C₂×C₁₆	—	118	1	no	—
117	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
118	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
119	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
120	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
121	#108	—	59	4	no	—
122	#108	—	118	2	no	—
123	#107	C₂×Dic₅₉	1	4	yes	—
124	#106	—	2	2	no	—
125	#97	—	236	2	no	—
126	C₄×C₈	—	59	2	no	—
127	C₄×C₈	—	59	2	no	—
128	C₄×C₈	—	236	2	no	—
129	C₂²×C₄	—	118	1	no	—
130	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
131	M₁₆	—	118	4	no	—
132	M₁₆	—	236	2	no	—
133	C₂×C₈	—	59	2	no	—
134	C₂×C₈	—	59	1	no	—
135	C₂×C₈	—	59	1	no	—
136	C₂×C₈	—	59	2	no	—
137	C₂×C₈	—	236	2	no	—
138	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
139	C₂×C₈	C₂×C₄×D₅₉	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	C₂²⋊C₄	—	4	2	no	—
146	C₄²	—	4	2	no	—
147	C₄²	—	59	2	no	—
148	C₄²	—	118	1	no	—
149	C₁₆	—	118	2	no	—
150	C₁₆	C₄×Dic₅₉	1	2	yes	—
151	C₁₆	C₄×Dic₅₉	1	2	yes	—
152	C₂³	#14273746	1	1	yes	characteristic
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	118	4	no	—
155	C₂×C₄	—	118	2	no	—
156	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
157	C₂×C₄	#14273689	1	1	yes	characteristic
158	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
159	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
160	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
161	C₂×C₄	#14273750	1	1	yes	characteristic
162	C₈	—	59	2	no	—
163	C₈	—	59	2	no	—
164	C₈	—	236	4	no	—
165	C₈	C₂×C₄×Dic₅₉	1	1	yes	characteristic
166	C₈	#14273685	1	1	yes	characteristic
167	C₈	#14273692	1	1	yes	characteristic
168	C₈	Q₈×Dic₅₉	1	1	yes	characteristic
169	C₂²	—	2	2	no	—
170	C₂²	#16837439	1	1	yes	characteristic
171	C₂²	D₈×Dic₅₉	1	1	yes	characteristic
172	C₂²	#16837496	1	1	yes	characteristic
173	C₄	—	8	4	no	—
174	C₄	—	118	4	no	—
175	C₄	C₄⋊C₄×Dic₅₉	1	1	yes	characteristic
176	C₄	#107×D₅₉	1	1	yes	characteristic
177	C₄	#16837484	1	1	yes	characteristic
178	C₄	Q₁₆×Dic₅₉	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	#107×Dic₅₉	1	1	yes	characteristic
182	C₂	#21425977	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₉
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2158	C₂
socle	C₂²×C₅₉	C₄×D₈
prime radical	C₂³×C₅₉	#135
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2158	D₅₉
hypocenter	C₅₉	#12670
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
10. #16837232	C₄	semidirect
11. C₁₆×Dic₅₉	C₄	semidirect
15. C₅₉×#367	C₄	semidirect
16. C₅₉×#366	C₄	semidirect
104. #367	Dic₅₉	semidirect
105. #366	Dic₅₉	semidirect
108. C₅₉	#12670	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×C₁₆×Dic₅₉	2
#21423977	2
#21422191	2
C₅₉×#2158	2
#12670	59