Subgroups — #28505126

Order 15104.

374 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837230	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	#16837230	—	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	SD₃₂	1	8	yes	—
42	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
44	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
48	C₂×Dic₅₉	—	8	4	no	—
49	C₂×Dic₅₉	—	8	8	no	—
50	C₈×C₅₉	#107	1	4	yes	—
51	C₈×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
52	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
53	#12906	—	59	1	no	maximal, Sylow
54	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
55	C₂²×C₅₉	#493	1	1	yes	characteristic
56	C₂²×C₅₉	#500	1	1	yes	characteristic
57	C₂²×C₅₉	C₂×SD₃₂	1	4	yes	—
58	C₄×C₅₉	#367	1	8	yes	complemented
59	C₄×C₅₉	#384	1	1	yes	characteristic
60	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
61	C₄×C₅₉	#494	1	1	yes	characteristic
62	C₄×C₅₉	#500	1	1	yes	characteristic
63	Dic₅₉	—	16	4	no	—
64	C₂×#367	—	59	4	no	—
65	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic
66	#1845	—	59	2	no	—
67	C₂×C₅₉	#1845	1	1	yes	characteristic
68	C₂×C₅₉	C₂×#367	1	2	yes	—
69	C₂×C₅₉	#2248	1	2	yes	—
70	C₂×C₅₉	#2254	1	2	yes	—
71	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
72	C₂×#107	—	59	4	no	—
73	C₂×#107	—	118	2	no	—
74	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
75	#384	—	118	2	no	—
76	#367	—	118	8	no	—
77	C₄×C₁₆	Dic₅₉	1	4	yes	complemented
78	C₅₉	#12906	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
80	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
81	C₂×C₄⋊C₄	—	118	4	no	—
82	C₂×C₄⋊C₄	—	236	4	no	—
83	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
84	C₂×C₁₆	Dic₁₁₈	1	4	yes	—
85	C₂×C₁₆	D₂₃₆	1	4	yes	—
86	C₂×C₁₆	C₂×Dic₅₉	1	4	yes	—
87	#107	—	118	8	no	—
88	#107	—	236	4	no	—
89	C₄×C₈	Dic₁₁₈	1	2	yes	—
90	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
91	C₂²×C₄	—	236	4	no	—
92	C₂²×C₄	D₄₇₂	1	2	yes	—
93	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
94	C₂×C₈	D₄₇₂	1	2	yes	—
95	C₂×C₈	Dic₂₃₆	1	2	yes	—
96	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
97	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
98	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
99	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
100	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
101	C₄⋊C₄	—	472	8	no	—
102	C₄⋊C₄	—	236	4	no	—
103	C₄⋊C₄	—	236	4	no	—
104	C₄²	Dic₂₃₆	1	2	yes	—
105	C₄²	C₄⋊Dic₅₉	1	2	yes	—
106	C₁₆	C₄⋊Dic₅₉	1	8	yes	complemented
107	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
108	C₂×C₄	—	472	4	no	—
109	C₂×C₄	—	472	8	no	—
110	C₂×C₄	#14273661	1	4	yes	—
111	C₂×C₄	#14273672	1	8	yes	—
112	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
113	C₂×C₄	#14273724	1	1	yes	characteristic
114	C₂×C₄	#14273727	1	1	yes	characteristic
115	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
116	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
117	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
118	C₈	#14273661	1	4	yes	—
119	C₈	C₂×C₄⋊Dic₅₉	1	2	yes	—
120	C₈	#14273727	1	2	yes	—
121	C₂²	C₂×#14273661	1	1	yes	characteristic
122	C₂²	#16837779	1	1	yes	characteristic
123	C₂²	#16837807	1	1	yes	characteristic
124	C₂²	C₂×#14273672	1	4	yes	—
125	C₄	—	944	4	no	—
126	C₄	#16837230	1	8	yes	complemented
127	C₄	#16837272	1	1	yes	characteristic
128	C₄	C₂×#14273661	1	1	yes	characteristic
129	C₄	#16837782	1	1	yes	characteristic
130	C₄	#16837807	1	1	yes	characteristic
131	C₂	#21424256	1	1	yes	characteristic
132	C₂	C₂×#16837230	1	2	yes	—
133	C₂	#21426352	1	2	yes	—
134	C₂	#21426378	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	#12906
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
58. C₄×C₅₉	#367	semidirect
77. C₄×C₁₆	Dic₅₉	semidirect
78. C₅₉	#12906	semidirect
106. C₁₆	C₄⋊Dic₅₉	semidirect
126. C₄	#16837230	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837230	2
#21424256	2
C₂×C₄×C₁₆×C₅₉	2
#12906	59