Subgroups — #28482453

Order 15104.

382 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424248	C₂	1	1	yes	maximal, characteristic
2	#21424213	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837155	C₂	1	2	yes	maximal
4	C₂×#16837153	C₂	1	2	yes	maximal
5	C₅₉×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
10	#16837157	—	2	2	no	—
11	#16837155	—	2	4	no	—
12	#16837153	—	2	4	no	—
13	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
14	C₄³×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#336	C₄	1	2	yes	—
16	C₅₉×#336	C₄	1	2	yes	—
17	C₂²×C₅₉⋊C₈	—	2	1	no	—
18	C₂²×C₅₉⋊C₈	—	2	1	no	—
19	C₂²×C₅₉⋊C₈	—	2	2	no	—
20	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
21	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
22	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₅₉⋊C₈)	—	4	4	no	—
25	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₉	D₄	1	2	yes	—
29	C₂×C₄²×C₅₉	Q₈	1	2	yes	—
30	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₅₉	D₄	1	4	yes	—
32	C₄×C₈×C₅₉	Q₈	1	4	yes	—
33	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₅₉	—	2	4	no	—
39	C₂×C₈×C₅₉	—	2	4	no	—
40	C₂×C₈×C₅₉	C₄⋊C₄	1	8	yes	—
41	C₄²×C₅₉	—	2	4	no	—
42	C₄²×C₅₉	—	2	2	no	—
43	C₄²×C₅₉	—	2	2	no	—
44	C₄²×C₅₉	—	2	2	no	—
45	C₄²×C₅₉	—	2	2	no	—
46	C₄²×C₅₉	C₂×D₄	1	2	yes	—
47	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
48	C₂×C₅₉⋊C₈	—	4	2	no	—
49	C₂×C₅₉⋊C₈	—	4	1	no	—
50	C₂×C₅₉⋊C₈	—	4	1	no	—
51	C₂×C₅₉⋊C₈	—	4	4	no	—
52	C₂×C₅₉⋊C₈	—	4	2	no	—
53	C₂×C₅₉⋊C₈	—	4	2	no	—
54	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₉	—	2	2	no	—
56	C₂×C₄×C₅₉	—	2	2	no	—
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	—	2	2	no	—
59	C₂×C₄×C₅₉	—	2	2	no	—
60	C₂×C₄×C₅₉	—	2	2	no	—
61	C₂×C₄×C₅₉	#108	1	4	yes	—
62	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
63	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
65	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#131	1	2	yes	—
67	C₂×C₄×C₅₉	#131	1	2	yes	—
68	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×C₄×C₅₉	#137	1	2	yes	—
70	C₈×C₅₉	—	2	8	no	—
71	C₅₉⋊C₈	—	8	2	no	—
72	C₅₉⋊C₈	—	8	1	no	—
73	C₅₉⋊C₈	—	8	1	no	—
74	#8812	—	59	1	no	maximal, Sylow
75	C₂²×C₅₉	#384	1	1	yes	characteristic
76	C₂²×C₅₉	#424	1	1	yes	characteristic
77	C₂²×C₅₉	#424	1	1	yes	characteristic
78	C₂²×C₅₉	#428	1	2	yes	—
79	C₂²×C₅₉	C₂×#108	1	2	yes	—
80	C₄×C₅₉	—	2	4	no	—
81	C₄×C₅₉	—	2	2	no	—
82	C₄×C₅₉	—	4	2	no	—
83	C₄×C₅₉	#424	1	4	yes	—
84	C₄×C₅₉	#496	1	1	yes	characteristic
85	C₄×C₅₉	#497	1	1	yes	characteristic
86	C₄×C₅₉	#501	1	2	yes	—
87	#1842	—	59	1	no	—
88	#1831	—	59	1	no	—
89	#1744	D₅₉	1	1	yes	characteristic
90	C₂×#335	—	59	2	no	—
91	C₂×#334	—	59	2	no	—
92	C₂×C₅₉	#1573	1	2	yes	—
93	C₂×C₅₉	#1574	1	2	yes	—
94	C₂×C₅₉	#1843	1	1	yes	characteristic
95	C₂×C₅₉	#1847	1	1	yes	characteristic
96	C₂×C₅₉	#1848	1	1	yes	characteristic
97	C₂×C₄⋊C₈	—	59	1	no	—
98	C₂×C₄⋊C₈	—	59	2	no	—
99	C₂×C₄⋊C₈	—	59	1	no	—
100	C₂×C₄⋊C₈	—	118	2	no	—
101	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
102	C₄³	C₂×D₅₉	1	1	yes	characteristic
103	#336	—	118	2	no	—
104	#336	Dic₅₉	1	2	yes	—
105	#336	Dic₅₉	1	2	yes	—
106	#335	—	118	4	no	—
107	#334	—	118	4	no	—
108	C₅₉	#8812	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	118	1	no	—
110	C₂²×C₈	—	118	1	no	—
111	C₂²×C₈	—	118	2	no	—
112	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
113	C₂×C₄²	Dic₁₁₈	1	2	yes	—
114	C₂×C₄²	D₂₃₆	1	2	yes	—
115	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
116	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
117	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
118	C₄⋊C₈	—	118	2	no	—
119	C₄⋊C₈	—	118	2	no	—
120	C₄⋊C₈	—	118	2	no	—
121	C₄⋊C₈	—	118	2	no	—
122	C₄⋊C₈	—	236	4	no	—
123	C₄×C₈	Dic₁₁₈	1	4	yes	—
124	C₄×C₈	D₂₃₆	1	4	yes	—
125	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
126	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
127	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
128	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
129	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	—	236	2	no	—
133	C₂×C₈	—	236	1	no	—
134	C₂×C₈	—	236	1	no	—
135	C₂×C₈	—	236	4	no	—
136	C₂×C₈	—	236	2	no	—
137	C₂×C₈	—	236	2	no	—
138	C₂×C₈	C₄⋊Dic₅₉	1	8	yes	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	C₂×Dic₁₁₈	1	2	yes	—
145	C₄²	C₂×D₂₃₆	1	2	yes	—
146	C₂³	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	#14273663	1	4	yes	—
154	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
155	C₂×C₄	#14273724	1	1	yes	characteristic
156	C₂×C₄	#14273727	1	2	yes	—
157	C₂×C₄	#14273727	1	1	yes	characteristic
158	C₂×C₄	#14273734	1	2	yes	—
159	C₂×C₄	#14273734	1	2	yes	—
160	C₂×C₄	#14273755	1	2	yes	—
161	C₂×C₄	#14273760	1	2	yes	—
162	C₈	—	2	8	no	—
163	C₈	—	472	2	no	—
164	C₈	—	472	1	no	—
165	C₈	—	472	1	no	—
166	C₂²	#16837272	1	1	yes	characteristic
167	C₂²	#16837429	1	1	yes	characteristic
168	C₂²	#16837429	1	1	yes	characteristic
169	C₂²	#16837444	1	2	yes	—
170	C₂²	C₂×#14273663	1	2	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	2	no	—
173	C₄	—	4	2	no	—
174	C₄	#16837429	1	4	yes	—
175	C₄	#16837791	1	1	yes	characteristic
176	C₄	#16837796	1	1	yes	characteristic
177	C₄	#16837811	1	2	yes	—
178	C₂	#21422881	1	2	yes	—
179	C₂	#21422885	1	2	yes	—
180	C₂	#21424250	1	1	yes	characteristic
181	C₂	#21424264	1	1	yes	characteristic
182	C₂	#21424268	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃₆
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1744	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1744	D₅₉
hypocenter	C₅₉	#8812
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
108. C₅₉	#8812	semidirect

Maximal subgroups

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

subgroup	index
#21424248	2
#21424213	2
C₂×#16837155	2
C₂×#16837153	2
C₅₉×#1744	2
#8812	59