Subgroups — C₂³×#14273686 #29495249

Order 15104.

14334 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273686	C₂	1	56	yes	maximal, direct product factor
2	C₂³×C₄⋊Dic₅₉	C₂	1	2	yes	maximal, complemented
3	C₂³×C₂²⋊Dic₅₉	C₂	1	2	yes	maximal
4	C₂³×C₄×Dic₅₉	C₂	1	2	yes	maximal, complemented
5	C₂⁵×C₄×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×Dic₅₉	C₂²	1	4	yes	—
7	C₂×#14273686	C₂²	1	448	yes	direct product factor
8	C₂²×C₄⋊Dic₅₉	C₂²	1	56	yes	complemented
9	C₂²×C₂²⋊Dic₅₉	C₂²	1	56	yes	—
10	C₂²×C₄×Dic₅₉	C₂²	1	56	yes	complemented
11	C₂⁶×C₅₉	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₅₉	C₄	1	32	yes	complemented
13	C₂⁴×C₄×C₅₉	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₅₉	C₂²	1	28	yes	—
15	C₂⁴×C₄×C₅₉	C₂²	1	1	yes	characteristic
16	C₂³×Dic₅₉	—	2	32	no	—
17	C₂³×Dic₅₉	C₂³	1	56	yes	—
18	#14273686	C₂³	1	512	yes	direct product factor
19	C₂×C₄⋊Dic₅₉	C₂³	1	224	yes	complemented
20	C₂×C₂²⋊Dic₅₉	C₂³	1	224	yes	—
21	C₂×C₄×Dic₅₉	C₂³	1	224	yes	complemented
22	C₂⁵×C₅₉	—	2	16	no	—
23	C₂⁵×C₅₉	C₂×C₄	1	16	yes	—
24	C₂⁵×C₅₉	C₂³	1	14	yes	—
25	C₂⁵×C₅₉	C₂³	1	1	yes	characteristic
26	C₂³×C₄×C₅₉	C₂×C₄	1	16	yes	—
27	C₂³×C₄×C₅₉	C₂×C₄	1	16	yes	—
28	C₂³×C₄×C₅₉	C₂×C₄	1	448	yes	complemented
29	C₂³×C₄×C₅₉	C₂³	1	14	yes	—
30	C₂³×C₄×C₅₉	C₂³	1	14	yes	—
31	C₂³×C₄×C₅₉	C₂³	1	112	yes	—
32	C₂⁴×C₅₉	—	2	128	no	—
33	C₂⁴×C₅₉	—	2	112	no	—
34	C₂⁴×C₅₉	C₂²×C₄	1	112	yes	—
35	C₂⁴×C₅₉	C₂²×C₄	1	8	yes	—
36	C₂⁴×C₅₉	#41	1	16	yes	—
37	C₂⁴×C₅₉	C₂⁴	1	28	yes	—
38	C₂⁴×C₅₉	C₂⁴	1	7	yes	—
39	C₂²×C₄×C₅₉	C₂²×C₄	1	112	yes	—
40	C₂²×C₄×C₅₉	C₂²×C₄	1	112	yes	—
41	C₂²×C₄×C₅₉	C₂²×C₄	1	896	yes	complemented
42	C₂²×C₄×C₅₉	C₂⁴	1	28	yes	—
43	C₂²×C₄×C₅₉	C₂⁴	1	28	yes	—
44	C₂²×C₄×C₅₉	C₂⁴	1	64	yes	—
45	C₂²×Dic₅₉	—	2	224	no	—
46	C₂²×Dic₅₉	C₂⁴	1	112	yes	—
47	C₂²⋊Dic₅₉	C₂⁴	1	128	yes	—
48	C₄⋊Dic₅₉	C₂⁴	1	128	yes	complemented
49	C₄×Dic₅₉	C₂⁴	1	128	yes	complemented
50	C₂³×C₅₉	—	2	448	no	—
51	C₂³×C₅₉	—	2	112	no	—
52	C₂³×C₅₉	#117	1	64	yes	complemented
53	C₂³×C₅₉	C₂³×C₄	1	112	yes	—
54	C₂³×C₅₉	C₂³×C₄	1	28	yes	—
55	C₂³×C₅₉	C₂×#41	1	56	yes	—
56	C₂³×C₅₉	C₂⁵	1	8	yes	—
57	C₂³×C₅₉	C₂⁵	1	7	yes	—
58	C₂×C₄×C₅₉	C₂³×C₄	1	112	yes	—
59	C₂×C₄×C₅₉	C₂³×C₄	1	112	yes	—
60	C₂×C₄×C₅₉	C₂³×C₄	1	256	yes	complemented
61	C₂×C₄×C₅₉	C₂⁵	1	8	yes	—
62	C₂×C₄×C₅₉	C₂⁵	1	8	yes	—
63	C₂×Dic₅₉	—	2	224	no	—
64	C₂×Dic₅₉	C₂⁵	1	32	yes	—
65	C₂³×#117	—	59	1	no	maximal, Sylow
66	C₂²×C₅₉	—	2	16	no	—
67	C₂²×C₅₉	—	2	224	no	—
68	C₂²×C₅₉	C₂×#117	1	112	yes	complemented
69	C₂²×C₅₉	C₂⁴×C₄	1	16	yes	—
70	C₂²×C₅₉	C₂⁴×C₄	1	14	yes	—
71	C₂²×C₅₉	C₂²×#41	1	28	yes	—
72	C₂²×C₅₉	C₂⁶	1	1	yes	characteristic
73	C₄×C₅₉	C₂⁴×C₄	1	16	yes	—
74	C₄×C₅₉	C₂⁴×C₄	1	16	yes	—
75	Dic₅₉	—	2	32	no	—
76	C₂⁵×C₄	D₅₉	1	1	yes	characteristic
77	C₂²×#117	—	59	56	no	—
78	C₂³×C₄⋊C₄	—	59	2	no	—
79	C₂³×C₂²⋊C₄	—	59	2	no	—
80	C₂³×C₄²	—	59	2	no	—
81	C₂×C₅₉	—	2	16	no	—
82	C₂×C₅₉	C₂²×#117	1	28	yes	complemented
83	C₂×C₅₉	C₂⁵×C₄	1	1	yes	characteristic
84	C₂×C₅₉	C₂³×#41	1	2	yes	—
85	C₂⁶	C₂×D₅₉	1	1	yes	characteristic
86	C₂⁴×C₄	—	59	4	no	—
87	C₂⁴×C₄	Dic₅₉	1	32	yes	complemented
88	C₂⁴×C₄	C₂×D₅₉	1	1	yes	characteristic
89	C₂⁴×C₄	C₂×D₅₉	1	28	yes	—
90	C₂⁴×C₄	C₂×D₅₉	1	1	yes	characteristic
91	C₂×#117	—	59	448	no	—
92	C₂²×C₄⋊C₄	—	59	56	no	—
93	C₂²×C₂²⋊C₄	—	59	56	no	—
94	C₂²×C₄²	—	59	56	no	—
95	C₅₉	C₂³×#117	1	1	yes	characteristic, complemented, Sylow
96	C₂⁵	—	2	16	no	—
97	C₂⁵	C₂×Dic₅₉	1	16	yes	—
98	C₂⁵	C₂²×D₅₉	1	14	yes	—
99	C₂⁵	C₂²×D₅₉	1	1	yes	characteristic
100	C₂³×C₄	—	59	56	no	—
101	C₂³×C₄	—	118	32	no	—
102	C₂³×C₄	C₂×Dic₅₉	1	16	yes	—
103	C₂³×C₄	C₂×Dic₅₉	1	16	yes	—
104	C₂³×C₄	C₂×Dic₅₉	1	448	yes	complemented
105	C₂³×C₄	C₂²×D₅₉	1	14	yes	—
106	C₂³×C₄	C₂²×D₅₉	1	14	yes	—
107	C₂³×C₄	C₂²×D₅₉	1	112	yes	—
108	#117	—	59	512	no	—
109	C₂×C₄⋊C₄	—	59	224	no	—
110	C₂×C₂²⋊C₄	—	59	224	no	—
111	C₂×C₄²	—	59	224	no	—
112	C₂⁴	—	2	128	no	—
113	C₂⁴	—	2	112	no	—
114	C₂⁴	#11745685	1	16	yes	—
115	C₂⁴	C₂²×Dic₅₉	1	112	yes	—
116	C₂⁴	C₂²×Dic₅₉	1	8	yes	—
117	C₂⁴	C₂³×D₅₉	1	28	yes	—
118	C₂⁴	C₂³×D₅₉	1	7	yes	—
119	C₂²×C₄	—	59	112	no	—
120	C₂²×C₄	—	118	224	no	—
121	C₂²×C₄	C₂²×Dic₅₉	1	112	yes	—
122	C₂²×C₄	C₂²×Dic₅₉	1	112	yes	—
123	C₂²×C₄	C₂²×Dic₅₉	1	896	yes	complemented
124	C₂²×C₄	C₂³×D₅₉	1	28	yes	—
125	C₂²×C₄	C₂³×D₅₉	1	28	yes	—
126	C₂²×C₄	C₂³×D₅₉	1	64	yes	—
127	C₄⋊C₄	—	59	128	no	—
128	C₂²⋊C₄	—	59	128	no	—
129	C₄²	—	59	128	no	—
130	C₂³	—	2	112	no	—
131	C₂³	—	2	448	no	—
132	C₂³	#14273686	1	64	yes	direct product factor
133	C₂³	C₂³×Dic₅₉	1	112	yes	—
134	C₂³	C₂³×Dic₅₉	1	28	yes	—
135	C₂³	C₂×#11745685	1	56	yes	—
136	C₂³	C₂⁴×D₅₉	1	8	yes	—
137	C₂³	C₂⁴×D₅₉	1	7	yes	—
138	C₂×C₄	—	59	32	no	—
139	C₂×C₄	—	118	224	no	—
140	C₂×C₄	C₂³×Dic₅₉	1	112	yes	—
141	C₂×C₄	C₂³×Dic₅₉	1	112	yes	—
142	C₂×C₄	C₂³×Dic₅₉	1	256	yes	complemented
143	C₂×C₄	C₂⁴×D₅₉	1	8	yes	—
144	C₂×C₄	C₂⁴×D₅₉	1	8	yes	—
145	C₂²	—	2	224	no	—
146	C₂²	—	2	16	no	—
147	C₂²	C₂×#14273686	1	112	yes	direct product factor
148	C₂²	C₂⁴×Dic₅₉	1	16	yes	—
149	C₂²	C₂⁴×Dic₅₉	1	14	yes	—
150	C₂²	C₂²×#11745685	1	28	yes	—
151	C₂²	C₂⁵×D₅₉	1	1	yes	characteristic
152	C₄	—	118	32	no	—
153	C₄	C₂⁴×Dic₅₉	1	16	yes	—
154	C₄	C₂⁴×Dic₅₉	1	16	yes	—
155	C₂	—	2	16	no	—
156	C₂	C₂²×#14273686	1	28	yes	direct product factor
157	C₂	C₂⁵×Dic₅₉	1	1	yes	characteristic
158	C₂	C₂³×#11745685	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₅₉
commutator	C₂×C₅₉	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₉
Fitting	C₂⁵×C₄×C₅₉	C₂
socle	C₂⁵×C₅₉	C₂³
prime radical	C₂⁶×C₅₉	C₂²
cosocle	C₂²×C₅₉	C₂⁶
hypercenter	C₂⁵×C₄	D₅₉
hypocenter	C₅₉	C₂³×#117
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14273686	C₂	direct (N × H)
2. C₂³×C₄⋊Dic₅₉	C₂	semidirect
4. C₂³×C₄×Dic₅₉	C₂	semidirect
7. C₂×#14273686	C₂²	direct (N × H)
8. C₂²×C₄⋊Dic₅₉	C₂²	semidirect
10. C₂²×C₄×Dic₅₉	C₂²	semidirect
12. C₂⁴×C₄×C₅₉	C₄	semidirect
18. #14273686	C₂³	direct (N × H)
19. C₂×C₄⋊Dic₅₉	C₂³	semidirect
21. C₂×C₄×Dic₅₉	C₂³	semidirect
28. C₂³×C₄×C₅₉	C₂×C₄	semidirect
41. C₂²×C₄×C₅₉	C₂²×C₄	semidirect
48. C₄⋊Dic₅₉	C₂⁴	semidirect
49. C₄×Dic₅₉	C₂⁴	semidirect
52. C₂³×C₅₉	#117	semidirect
60. C₂×C₄×C₅₉	C₂³×C₄	semidirect
68. C₂²×C₅₉	C₂×#117	semidirect
82. C₂×C₅₉	C₂²×#117	semidirect
87. C₂⁴×C₄	Dic₅₉	semidirect
95. C₅₉	C₂³×#117	semidirect
104. C₂³×C₄	C₂×Dic₅₉	semidirect
123. C₂²×C₄	C₂²×Dic₅₉	semidirect
132. C₂³	#14273686	direct (N × H)
142. C₂×C₄	C₂³×Dic₅₉	semidirect
147. C₂²	C₂×#14273686	direct (N × H)
156. C₂	C₂²×#14273686	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14273686	2
C₂³×C₄⋊Dic₅₉	2
C₂³×C₂²⋊Dic₅₉	2
C₂³×C₄×Dic₅₉	2
C₂⁵×C₄×C₅₉	2
C₂³×#117	59

Subgroups — C23×#14273686 #29495249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14273686 #29495249