Subgroups — #28505123

Order 15104.

374 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837228	C₂	1	4	yes	maximal
2	#21424256	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273661	—	2	2	no	—
5	C₂×#14273661	C₂²	1	4	yes	—
6	#16837272	—	2	2	no	—
7	#16837228	—	2	8	no	—
8	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₅₉	C₄	1	4	yes	complemented
11	C₂×C₄⋊Dic₅₉	—	2	4	no	—
12	C₂×C₄⋊Dic₅₉	—	4	4	no	—
13	#14273661	—	2	8	no	—
14	#14273661	—	4	4	no	—
15	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	—
19	C₂×C₁₆×C₅₉	D₄	1	4	yes	—
20	C₂×C₁₆×C₅₉	Q₈	1	4	yes	—
21	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₉	Q₈	1	2	yes	—
23	C₂²×C₄×C₅₉	D₈	1	2	yes	—
24	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
25	C₂²×Dic₅₉	—	4	4	no	—
26	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₅₉	D₈	1	2	yes	—
28	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
29	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
31	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
32	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
34	C₄²×C₅₉	Q₁₆	1	2	yes	—
35	C₄⋊Dic₅₉	—	4	4	no	—
36	C₄⋊Dic₅₉	—	4	4	no	—
37	C₄⋊Dic₅₉	—	8	8	no	—
38	C₁₆×C₅₉	C₄⋊C₄	1	8	yes	complemented
39	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
40	C₂×C₄×C₅₉	#107	1	4	yes	—
41	C₂×C₄×C₅₉	D₁₆	1	4	yes	—
42	C₂×C₄×C₅₉	Q₃₂	1	4	yes	—
43	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
49	C₂×Dic₅₉	—	8	4	no	—
50	C₂×Dic₅₉	—	8	8	no	—
51	C₈×C₅₉	#107	1	4	yes	—
52	C₈×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
53	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
54	#12905	—	59	1	no	maximal, Sylow
55	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
56	C₂²×C₅₉	#493	1	1	yes	characteristic
57	C₂²×C₅₉	#500	1	1	yes	characteristic
58	C₂²×C₅₉	C₂×D₁₆	1	2	yes	—
59	C₂²×C₅₉	C₂×Q₃₂	1	2	yes	—
60	C₄×C₅₉	#366	1	8	yes	complemented
61	C₄×C₅₉	#384	1	1	yes	characteristic
62	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
63	C₄×C₅₉	#494	1	1	yes	characteristic
64	C₄×C₅₉	#500	1	1	yes	characteristic
65	Dic₅₉	—	16	4	no	—
66	C₂×#366	—	59	4	no	—
67	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic
68	#1845	—	59	2	no	—
69	C₂×C₅₉	#1845	1	1	yes	characteristic
70	C₂×C₅₉	C₂×#366	1	2	yes	—
71	C₂×C₅₉	#2246	1	1	yes	characteristic
72	C₂×C₅₉	#2247	1	1	yes	characteristic
73	C₂×C₅₉	#2252	1	2	yes	—
74	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
75	C₂×#107	—	59	4	no	—
76	C₂×#107	—	118	2	no	—
77	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
78	#384	—	118	2	no	—
79	#366	—	118	8	no	—
80	C₄×C₁₆	Dic₅₉	1	4	yes	complemented
81	C₅₉	#12905	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
83	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
84	C₂×C₄⋊C₄	—	118	4	no	—
85	C₂×C₄⋊C₄	—	236	4	no	—
86	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
87	C₂×C₁₆	Dic₁₁₈	1	4	yes	—
88	C₂×C₁₆	D₂₃₆	1	4	yes	—
89	C₂×C₁₆	C₂×Dic₅₉	1	4	yes	—
90	#107	—	118	8	no	—
91	#107	—	236	4	no	—
92	C₄×C₈	Dic₁₁₈	1	2	yes	—
93	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
94	C₂²×C₄	—	236	4	no	—
95	C₂²×C₄	D₄₇₂	1	2	yes	—
96	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
97	C₂×C₈	D₄₇₂	1	2	yes	—
98	C₂×C₈	Dic₂₃₆	1	2	yes	—
99	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
100	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
101	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
102	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
103	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
104	C₄⋊C₄	—	472	8	no	—
105	C₄⋊C₄	—	236	4	no	—
106	C₄⋊C₄	—	236	4	no	—
107	C₄²	Dic₂₃₆	1	2	yes	—
108	C₄²	C₄⋊Dic₅₉	1	2	yes	—
109	C₁₆	C₄⋊Dic₅₉	1	8	yes	complemented
110	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
111	C₂×C₄	—	472	4	no	—
112	C₂×C₄	—	472	8	no	—
113	C₂×C₄	#14273661	1	4	yes	—
114	C₂×C₄	D₉₄₄	1	4	yes	—
115	C₂×C₄	Dic₄₇₂	1	4	yes	—
116	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
117	C₂×C₄	#14273724	1	1	yes	characteristic
118	C₂×C₄	#14273727	1	1	yes	characteristic
119	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
121	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
122	C₈	#14273661	1	4	yes	—
123	C₈	C₂×C₄⋊Dic₅₉	1	2	yes	—
124	C₈	#14273727	1	2	yes	—
125	C₂²	C₂×#14273661	1	1	yes	characteristic
126	C₂²	#16837779	1	1	yes	characteristic
127	C₂²	#16837807	1	1	yes	characteristic
128	C₂²	C₂×D₉₄₄	1	2	yes	—
129	C₂²	C₂×Dic₄₇₂	1	2	yes	—
130	C₄	—	944	4	no	—
131	C₄	#16837228	1	8	yes	complemented
132	C₄	#16837272	1	1	yes	characteristic
133	C₄	C₂×#14273661	1	1	yes	characteristic
134	C₄	#16837782	1	1	yes	characteristic
135	C₄	#16837807	1	1	yes	characteristic
136	C₂	#21424256	1	1	yes	characteristic
137	C₂	C₂×#16837228	1	2	yes	—
138	C₂	#21426346	1	1	yes	characteristic
139	C₂	#21426349	1	1	yes	characteristic
140	C₂	#21426371	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	#12905
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. C₄×C₁₆×C₅₉	C₄	semidirect
38. C₁₆×C₅₉	C₄⋊C₄	semidirect
60. C₄×C₅₉	#366	semidirect
80. C₄×C₁₆	Dic₅₉	semidirect
81. C₅₉	#12905	semidirect
109. C₁₆	C₄⋊Dic₅₉	semidirect
131. C₄	#16837228	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837228	2
#21424256	2
C₂×C₄×C₁₆×C₅₉	2
#12905	59