Subgroups — #28478066

Order 15104.

582 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14273637	C₂	1	1	yes	maximal, characteristic
2	C₄×#14273637	C₂	1	2	yes	maximal
3	C₂×#16837124	C₂	1	2	yes	maximal
4	C₂×C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₉×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273637	C₄	1	8	yes	—
7	C₂×#14273637	C₄	1	4	yes	—
8	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
11	#16837124	C₄	1	8	yes	complemented
12	C₈×C₅₉⋊C₈	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
15	C₅₉×#322	C₄	1	4	yes	complemented
16	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
17	C₂²×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₄×Dic₅₉	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
20	#14273637	—	2	8	no	—
21	#14273637	—	2	16	no	—
22	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
23	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
24	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
28	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
33	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₅₉	C₄²	1	4	yes	—
35	C₂×C₈×C₅₉	—	2	8	no	—
36	C₂×C₈×C₅₉	C₄²	1	8	yes	—
37	C₂×C₈×C₅₉	C₄²	1	4	yes	—
38	C₂×C₈×C₅₉	M₁₆	1	8	yes	—
39	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
41	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
42	C₄×Dic₅₉	—	2	8	no	—
43	C₄×Dic₅₉	—	2	4	no	—
44	C₂×C₅₉⋊C₈	—	2	4	no	—
45	C₂×C₅₉⋊C₈	C₄²	1	4	yes	—
46	C₂×C₅₉⋊C₈	C₄²	1	8	yes	—
47	C₂×C₅₉⋊C₈	M₁₆	1	16	yes	—
48	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
50	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
53	C₂×C₄×C₅₉	C₂×M₁₆	1	8	yes	—
54	C₂×C₄×C₅₉	C₂×M₁₆	1	4	yes	—
55	C₂×Dic₅₉	—	2	4	no	—
56	C₂×Dic₅₉	—	2	8	no	—
57	C₈×C₅₉	—	2	8	no	—
58	C₈×C₅₉	#97	1	8	yes	complemented
59	C₅₉⋊C₈	—	2	4	no	—
60	C₅₉⋊C₈	#97	1	16	yes	complemented
61	#7846	—	59	1	no	maximal, Sylow
62	C₂²×C₅₉	C₄³	1	1	yes	characteristic
63	C₂²×C₅₉	C₄×M₁₆	1	2	yes	—
64	C₂²×C₅₉	C₄×M₁₆	1	4	yes	—
65	C₄×C₅₉	C₂×#97	1	4	yes	—
66	C₄×C₅₉	C₂×#97	1	2	yes	—
67	C₄×C₅₉	C₄×M₁₆	1	2	yes	—
68	C₄×C₅₉	C₄×M₁₆	1	4	yes	—
69	Dic₅₉	—	4	4	no	—
70	C₄×#97	—	59	2	no	—
71	C₄×#97	—	59	1	no	—
72	C₂×#322	—	59	2	no	—
73	C₂×#322	D₅₉	1	1	yes	characteristic
74	C₂×C₈²	—	59	1	no	—
75	C₂×C₅₉	#1455	1	4	yes	—
76	C₂×C₅₉	C₄×#97	1	2	yes	—
77	C₂×C₅₉	C₄×#97	1	1	yes	characteristic
78	C₂×#97	—	59	8	no	—
79	C₂×#97	—	59	4	no	—
80	C₂×C₄×C₈	—	59	2	no	—
81	C₂×C₄×C₈	—	59	1	no	—
82	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
84	C₄³	—	59	1	no	—
85	#322	—	59	8	no	—
86	#322	Dic₅₉	1	4	yes	complemented
87	C₈²	—	59	4	no	—
88	C₅₉	#7846	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	—	59	4	no	—
90	C₂²×C₈	—	59	2	no	—
91	C₂²×C₈	C₄×D₅₉	1	4	yes	—
92	C₂²×C₈	C₄×D₅₉	1	2	yes	—
93	C₂×C₄²	—	59	4	no	—
94	C₂×C₄²	—	59	2	no	—
95	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
96	#97	—	118	16	no	—
97	#97	—	118	8	no	—
98	C₄×C₈	—	59	4	no	—
99	C₄×C₈	—	59	4	no	—
100	C₄×C₈	—	59	4	no	—
101	C₄×C₈	C₄×D₅₉	1	2	yes	—
102	C₄×C₈	C₄×D₅₉	1	4	yes	—
103	C₄×C₈	C₂×Dic₅₉	1	4	yes	—
104	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
105	C₂²×C₄	—	59	4	no	—
106	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
107	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
108	C₂×C₈	—	2	8	no	—
109	C₂×C₈	—	59	8	no	—
110	C₂×C₈	—	59	16	no	—
111	C₂×C₈	—	59	4	no	—
112	C₂×C₈	—	118	4	no	—
113	C₂×C₈	#11745659	1	8	yes	—
114	C₂×C₈	C₄×Dic₅₉	1	8	yes	—
115	C₂×C₈	C₄×Dic₅₉	1	4	yes	—
116	C₄²	—	118	8	no	—
117	C₄²	—	118	4	no	—
118	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
119	C₄²	C₂×C₄×D₅₉	1	2	yes	—
120	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
121	C₂³	C₄²×D₅₉	1	1	yes	characteristic
122	C₂×C₄	—	118	4	no	—
123	C₂×C₄	—	118	8	no	—
124	C₂×C₄	C₄²×D₅₉	1	2	yes	—
125	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
126	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×Dic₅₉	1	2	yes	—
128	C₂×C₄	M₁₆×D₅₉	1	8	yes	—
129	C₂×C₄	C₂×#11745659	1	4	yes	—
130	C₈	—	2	8	no	—
131	C₈	—	59	16	no	—
132	C₈	—	118	4	no	—
133	C₈	#14273637	1	8	yes	complemented
134	C₂²	C₄²×Dic₅₉	1	1	yes	characteristic
135	C₂²	C₄×#11745659	1	2	yes	—
136	C₂²	#16837354	1	4	yes	—
137	C₄	—	236	4	no	—
138	C₄	#97×D₅₉	1	4	yes	—
139	C₄	C₂×#14273637	1	2	yes	—
140	C₄	C₄×#11745659	1	2	yes	—
141	C₄	M₁₆×Dic₅₉	1	4	yes	—
142	C₂	#21422400	1	4	yes	—
143	C₂	#97×Dic₅₉	1	2	yes	—
144	C₂	C₄×#14273637	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₉
commutator	C₂²×C₅₉	C₄³
Fitting	C₂×C₅₉×#322	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₄²
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#322	D₅₉
hypocenter	C₅₉	#7846
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
11. #16837124	C₄	semidirect
12. C₈×C₅₉⋊C₈	C₄	semidirect
15. C₅₉×#322	C₄	semidirect
58. C₈×C₅₉	#97	semidirect
60. C₅₉⋊C₈	#97	semidirect
86. #322	Dic₅₉	semidirect
88. C₅₉	#7846	semidirect
133. C₈	#14273637	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14273637	2
C₂×#16837124	2
C₂×C₈×C₅₉⋊C₈	2
C₂×C₅₉×#322	2
#7846	59