Subgroups — #28500247

Order 15104.

242 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21422844	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837153	C₂	1	1	yes	maximal, characteristic
3	#21422162	C₂	1	4	yes	maximal, complemented
4	C₄×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
5	#16837455	—	2	1	no	—
6	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
7	#16837155	—	2	1	no	—
8	#16837153	—	2	1	no	—
9	#16837153	C₂²	1	2	yes	—
10	#16837153	C₂²	1	2	yes	—
11	C₂×M₃₂×C₅₉	C₄	1	1	yes	characteristic
12	C₂×M₃₂×C₅₉	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#346	C₂²	1	2	yes	—
15	C₄×C₁₆×C₅₉	C₄	1	2	yes	—
16	C₂²×C₅₉⋊C₈	—	4	1	no	—
17	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
18	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
19	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
20	C₂²⋊(C₅₉⋊C₈)	—	4	1	no	—
21	C₄×C₅₉⋊C₈	—	4	1	no	—
22	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
25	M₃₂×C₅₉	—	2	2	no	—
26	M₃₂×C₅₉	C₈	1	4	yes	complemented
27	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₅₉	—	2	2	no	—
37	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
41	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
45	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
46	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₅₉⋊C₈	—	4	2	no	—
48	C₂×C₅₉⋊C₈	—	4	2	no	—
49	C₂×C₅₉⋊C₈	—	8	2	no	—
50	C₁₆×C₅₉	—	2	2	no	—
51	C₁₆×C₅₉	C₂×C₈	1	4	yes	complemented
52	C₂³×C₅₉	#106	1	1	yes	characteristic
53	C₂×C₄×C₅₉	—	2	2	no	—
54	C₂×C₄×C₅₉	—	4	1	no	—
55	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
57	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
58	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
59	C₂×C₄×C₅₉	#108	1	2	yes	—
60	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₅₉	—	2	1	no	—
64	C₈×C₅₉	—	2	1	no	—
65	C₈×C₅₉	C₄⋊C₈	1	2	yes	—
66	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
67	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
68	C₅₉⋊C₈	—	8	2	no	—
69	C₅₉⋊C₈	—	8	2	no	—
70	#12000	—	59	1	no	maximal, Sylow
71	C₂²×C₅₉	—	2	1	no	—
72	C₂²×C₅₉	—	2	1	no	—
73	C₂²×C₅₉	#334	1	2	yes	—
74	C₂²×C₅₉	C₂×#106	1	1	yes	characteristic
75	C₄×C₅₉	—	2	1	no	—
76	C₄×C₅₉	—	2	1	no	—
77	C₄×C₅₉	—	2	1	no	—
78	C₄×C₅₉	#334	1	2	yes	—
79	C₄×C₅₉	#365	1	2	yes	—
80	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
81	C₄×C₅₉	C₂×#108	1	1	yes	characteristic
82	C₄×M₃₂	D₅₉	1	1	yes	characteristic
83	#1564	—	59	1	no	—
84	C₂×#334	—	59	1	no	—
85	#1368	—	59	4	no	—
86	C₂×C₅₉	—	2	2	no	—
87	C₂×C₅₉	C₂×#334	1	1	yes	characteristic
88	C₂×C₅₉	C₂×#365	1	1	yes	characteristic
89	C₂×C₅₉	#2155	1	1	yes	characteristic
90	C₂×M₃₂	Dic₅₉	1	1	yes	characteristic
91	C₂×M₃₂	Dic₅₉	1	1	yes	characteristic
92	#431	—	118	1	no	—
93	C₂×C₄⋊C₈	—	118	1	no	—
94	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
95	#346	C₂×D₅₉	1	2	yes	—
96	C₄×C₁₆	Dic₅₉	1	2	yes	—
97	#335	—	118	1	no	—
98	#334	—	59	2	no	—
99	#334	—	59	2	no	—
100	#334	—	118	1	no	—
101	C₅₉	#12000	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₈	—	236	1	no	—
103	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
104	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
105	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	C₅₉⋊C₈	1	4	yes	complemented
108	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
109	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
110	C₄⋊C₈	—	118	2	no	—
111	C₄⋊C₈	—	118	2	no	—
112	C₄⋊C₈	—	236	1	no	—
113	C₂²⋊C₈	—	236	1	no	—
114	C₄×C₈	—	236	1	no	—
115	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
116	C₄×C₈	D₂₃₆	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
118	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
119	C₂²×C₄	#11745660	1	1	yes	characteristic
120	C₂²×C₄	#11745660	1	1	yes	characteristic
121	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	472	2	no	—
124	C₂×C₈	—	236	2	no	—
125	C₂×C₈	—	236	2	no	—
126	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
127	C₂×C₈	#11745664	1	2	yes	—
128	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
129	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
131	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
132	C₄²	#11745660	1	1	yes	characteristic
133	C₄²	#11745660	1	1	yes	characteristic
134	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
135	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
136	C₁₆	—	2	2	no	—
137	C₁₆	C₂×C₅₉⋊C₈	1	4	yes	complemented
138	C₂³	#14273659	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
142	C₂×C₄	#14273659	1	1	yes	characteristic
143	C₂×C₄	#14273659	1	1	yes	characteristic
144	C₂×C₄	#14273659	1	1	yes	characteristic
145	C₂×C₄	#14273663	1	2	yes	—
146	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
147	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
148	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	472	2	no	—
152	C₈	—	472	2	no	—
153	C₈	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
154	C₈	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
155	C₈	C₂×#11745664	1	1	yes	characteristic
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	#16837153	1	2	yes	—
159	C₂²	C₂×#14273659	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	#16837153	1	2	yes	—
164	C₄	#16837225	1	2	yes	—
165	C₄	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
166	C₄	C₂×#14273663	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	C₂×#16837153	1	1	yes	characteristic
169	C₂	C₂×#16837225	1	1	yes	characteristic
170	C₂	#21425869	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273659
commutator	C₈×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₄×M₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂×#106
prime radical	C₂³×C₅₉	#106
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₄×M₃₂	D₅₉
hypocenter	C₅₉	#12000
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #21422162	C₂	semidirect
26. M₃₂×C₅₉	C₈	semidirect
51. C₁₆×C₅₉	C₂×C₈	semidirect
101. C₅₉	#12000	semidirect
107. M₃₂	C₅₉⋊C₈	semidirect
137. C₁₆	C₂×C₅₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#21422844	2
C₂×#16837153	2
#21422162	2
C₄×M₃₂×C₅₉	2
#12000	59