Subgroups — #28508869

Order 15104.

238 conjugacy classes of subgroups.

The subgroup lattice has 130 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837236	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	#21422230	C₂	1	2	yes	maximal, complemented
4	C₄×C₅₉⋊C₃₂	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2108	C₂	1	1	yes	maximal, characteristic
6	#16837236	C₄	1	4	yes	complemented
7	C₂×C₅₉⋊C₃₂	C₄	1	4	yes	complemented
8	C₂×C₅₉⋊C₃₂	C₂²	1	1	yes	characteristic
9	C₂×C₅₉⋊C₃₂	C₂²	1	2	yes	—
10	C₂×C₅₉⋊C₃₂	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₅₉	C₄	1	1	yes	characteristic
12	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#405	C₄	1	1	yes	characteristic
14	C₅₉×#346	C₄	1	2	yes	—
15	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
16	C₅₉⋊C₃₂	C₂×C₄	1	2	yes	—
17	C₅₉⋊C₃₂	C₂×C₄	1	2	yes	—
18	C₅₉⋊C₃₂	C₂×C₄	1	4	yes	—
19	C₂×M₁₆×C₅₉	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
21	#117×C₅₉	C₈	1	1	yes	characteristic
22	M₃₂×C₅₉	C₈	1	4	yes	—
23	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
28	#97×C₅₉	C₈	1	2	yes	—
29	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
31	M₁₆×C₅₉	C₁₆	1	4	yes	—
32	C₂×C₈×C₅₉	C₄²	1	2	yes	—
33	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
36	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₅₉	C₁₆	1	2	yes	—
38	C₂²⋊C₄×C₅₉	C₁₆	1	2	yes	—
39	C₄²×C₅₉	C₂×C₈	1	2	yes	—
40	C₁₆×C₅₉	C₄²	1	2	yes	—
41	C₁₆×C₅₉	C₂×C₈	1	4	yes	—
42	C₁₆×C₅₉	C₂²×C₄	1	1	yes	characteristic
43	C₁₆×C₅₉	C₂²×C₄	1	1	yes	characteristic
44	C₂³×C₅₉	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₅₉	C₂×C₁₆	1	4	yes	—
47	C₂×C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂×C₁₆	1	2	yes	—
49	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₅₉	C₄×C₈	1	2	yes	—
51	C₈×C₅₉	C₂×C₁₆	1	4	yes	—
52	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
53	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
54	#13549	—	59	1	no	maximal, Sylow
55	C₂²×C₅₉	—	2	2	no	—
56	C₂²×C₅₉	C₄×C₁₆	1	2	yes	—
57	C₂²×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
58	C₄×C₅₉	—	2	4	no	—
59	C₄×C₅₉	C₄×C₁₆	1	2	yes	—
60	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
61	C₄×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
62	C₂×M₆₄	—	59	1	no	—
63	C₂²×C₃₂	—	59	1	no	—
64	#2108	D₅₉	1	1	yes	characteristic
65	#1397	—	59	2	no	—
66	C₄×C₃₂	—	59	2	no	—
67	C₂×C₅₉	—	2	2	no	—
68	C₂×C₅₉	C₂×C₄×C₁₆	1	1	yes	characteristic
69	C₂×C₅₉	#2258	1	2	yes	—
70	C₂×M₃₂	Dic₅₉	1	1	yes	characteristic
71	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
72	#405	Dic₅₉	1	1	yes	characteristic
73	M₆₄	—	59	4	no	—
74	C₂×C₃₂	—	59	4	no	—
75	C₂×C₃₂	—	59	1	no	—
76	C₂×C₃₂	—	59	2	no	—
77	C₂×C₃₂	—	59	1	no	—
78	#346	Dic₅₉	1	2	yes	—
79	C₄×C₁₆	C₂×D₅₉	1	2	yes	—
80	C₅₉	#13549	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₁₆	C₅₉⋊C₈	1	1	yes	characteristic
82	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
83	#117	C₅₉⋊C₈	1	1	yes	characteristic
84	M₃₂	C₅₉⋊C₈	1	4	yes	—
85	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
86	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
87	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
88	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
89	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
90	#97	C₅₉⋊C₈	1	2	yes	—
91	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
92	C₃₂	—	59	2	no	—
93	C₃₂	—	59	2	no	—
94	C₃₂	—	59	4	no	—
95	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
96	M₁₆	C₅₉⋊C₁₆	1	4	yes	—
97	C₂×C₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
98	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
99	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
100	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
101	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
102	C₄⋊C₄	C₅₉⋊C₁₆	1	2	yes	—
103	C₂²⋊C₄	C₅₉⋊C₁₆	1	2	yes	—
104	C₄²	C₂×C₅₉⋊C₈	1	2	yes	—
105	C₁₆	C₂×C₅₉⋊C₈	1	4	yes	—
106	C₁₆	C₄×Dic₅₉	1	2	yes	—
107	C₁₆	C₂×C₄×D₅₉	1	1	yes	characteristic
108	C₁₆	C₂×C₄×D₅₉	1	1	yes	characteristic
109	C₂³	C₂×C₅₉⋊C₁₆	1	1	yes	characteristic
110	C₂×C₄	C₄×C₅₉⋊C₈	1	2	yes	—
111	C₂×C₄	C₂×C₅₉⋊C₁₆	1	4	yes	—
112	C₂×C₄	C₂×C₅₉⋊C₁₆	1	1	yes	characteristic
113	C₂×C₄	C₂×C₅₉⋊C₁₆	1	2	yes	—
114	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
115	C₈	C₄×C₅₉⋊C₈	1	2	yes	—
116	C₈	C₂×C₅₉⋊C₁₆	1	4	yes	—
117	C₈	C₂×C₄×Dic₅₉	1	1	yes	characteristic
118	C₈	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
119	C₂²	—	2	2	no	—
120	C₂²	C₄×C₅₉⋊C₁₆	1	2	yes	—
121	C₂²	C₂²×C₅₉⋊C₁₆	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	C₄×C₅₉⋊C₁₆	1	2	yes	—
124	C₄	C₂×C₄×C₅₉⋊C₈	1	1	yes	characteristic
125	C₄	C₂²×C₅₉⋊C₁₆	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	C₂×C₄×C₅₉⋊C₁₆	1	1	yes	characteristic
128	C₂	#21426390	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₉
commutator	C₂×C₅₉	C₂×C₄×C₁₆
Fitting	C₅₉×#2108	C₂
socle	C₂²×C₅₉	C₂²×C₁₆
prime radical	C₂³×C₅₉	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	#2108	D₅₉
hypocenter	C₅₉	#13549
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
3. #21422230	C₂	semidirect
4. C₄×C₅₉⋊C₃₂	C₂	semidirect
6. #16837236	C₄	semidirect
7. C₂×C₅₉⋊C₃₂	C₄	semidirect
80. C₅₉	#13549	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837236	2
C₂²×C₅₉⋊C₃₂	2
#21422230	2
C₄×C₅₉⋊C₃₂	2
C₅₉×#2108	2
#13549	59