Subgroups — #28482489

Order 15104.

354 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	#21424267	C₂	1	2	yes	maximal
2	#21422852	C₂	1	1	yes	maximal, characteristic
3	#21422851	C₂	1	2	yes	maximal
4	C₂×C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273662	—	2	4	no	—
7	#16837426	C₂²	1	2	yes	—
8	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	#16837153	—	2	2	no	—
11	#16837152	—	2	4	no	—
12	C₈×C₅₉⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₅₉⋊C₈	C₄	1	1	yes	characteristic
14	C₈×C₅₉⋊C₈	C₄	1	1	yes	characteristic
15	C₈×C₅₉⋊C₈	C₄	1	1	yes	characteristic
16	C₅₉×#423	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#334	—	2	2	no	—
19	C₂×#11745664	—	2	2	no	—
20	C₂²×C₅₉⋊C₈	D₄	1	2	yes	—
21	C₂²×C₅₉⋊C₈	Q₈	1	2	yes	—
22	#14273662	—	4	8	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
25	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
30	C₄×C₅₉⋊C₈	Q₈	1	2	yes	—
31	C₂×M₁₆×C₅₉	—	2	2	no	—
32	C₂²×C₈×C₅₉	Q₈	1	2	yes	—
33	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₉	—	2	2	no	—
35	C₄⋊C₈×C₅₉	—	2	2	no	—
36	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₅₉	D₄	1	2	yes	—
39	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
42	M₁₆×C₅₉	—	4	4	no	—
43	C₂×C₈×C₅₉	—	2	4	no	—
44	C₂×C₈×C₅₉	—	4	2	no	—
45	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
47	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
51	#11745664	—	4	4	no	—
52	C₂×C₅₉⋊C₈	—	2	4	no	—
53	C₂×C₅₉⋊C₈	—	4	2	no	—
54	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
55	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
56	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
57	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
58	C₂×C₅₉⋊C₈	SD₁₆	1	8	yes	—
59	C₂³×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₅₉	—	2	2	no	—
61	C₂×C₄×C₅₉	—	2	2	no	—
62	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂×SD₁₆	1	4	yes	—
69	C₈×C₅₉	—	2	2	no	—
70	C₈×C₅₉	—	8	2	no	—
71	C₈×C₅₉	#108	1	4	yes	complemented
72	C₅₉⋊C₈	—	2	2	no	—
73	C₅₉⋊C₈	—	8	2	no	—
74	C₅₉⋊C₈	#106	1	4	yes	—
75	C₅₉⋊C₈	#106	1	4	yes	—
76	C₅₉⋊C₈	#108	1	4	yes	complemented
77	#8822	—	59	1	no	maximal, Sylow
78	C₂²×C₅₉	—	2	1	no	—
79	C₂²×C₅₉	—	2	1	no	—
80	C₂²×C₅₉	#384	1	1	yes	characteristic
81	C₂²×C₅₉	#498	1	2	yes	—
82	C₄×C₅₉	—	2	1	no	—
83	C₄×C₅₉	—	2	1	no	—
84	C₄×C₅₉	C₂×#106	1	2	yes	—
85	C₄×C₅₉	C₂×#108	1	2	yes	—
86	C₄×C₅₉	C₂×#108	1	2	yes	—
87	C₄×C₅₉	#492	1	2	yes	—
88	#1848	—	59	2	no	—
89	#1566	—	59	2	no	—
90	#1566	—	59	1	no	—
91	#1566	D₅₉	1	1	yes	characteristic
92	C₂×C₈²	—	59	1	no	—
93	C₂×C₅₉	—	2	2	no	—
94	C₂×C₅₉	#1844	1	1	yes	characteristic
95	C₂×C₅₉	#1848	1	1	yes	characteristic
96	C₂×C₅₉	#1848	1	1	yes	characteristic
97	C₂×#108	—	118	4	no	—
98	#423	—	59	2	no	—
99	#423	C₂×D₅₉	1	2	yes	—
100	C₂×C₄×C₈	—	59	1	no	—
101	C₂×C₄×C₈	—	59	1	no	—
102	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
103	#334	—	2	2	no	—
104	#334	—	118	4	no	—
105	#334	—	118	2	no	—
106	C₈²	—	59	1	no	—
107	C₈²	—	59	1	no	—
108	C₈²	—	59	1	no	—
109	C₈²	—	59	1	no	—
110	C₅₉	#8822	1	1	yes	characteristic, complemented, Sylow
111	C₂×M₁₆	—	2	2	no	—
112	C₂×M₁₆	—	118	2	no	—
113	C₂²×C₈	—	59	2	no	—
114	C₂²×C₈	—	59	2	no	—
115	C₂²×C₈	Dic₁₁₈	1	2	yes	—
116	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
117	#108	—	236	8	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	118	2	no	—
121	C₄⋊C₈	—	118	2	no	—
122	C₄×C₈	—	59	1	no	—
123	C₄×C₈	—	59	1	no	—
124	C₄×C₈	—	59	1	no	—
125	C₄×C₈	—	59	1	no	—
126	C₄×C₈	—	59	2	no	—
127	C₄×C₈	—	59	2	no	—
128	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
129	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
130	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
131	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
132	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
133	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
134	M₁₆	—	4	4	no	—
135	M₁₆	—	236	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	59	2	no	—
139	C₂×C₈	—	59	2	no	—
140	C₂×C₈	—	59	2	no	—
141	C₂×C₈	—	59	8	no	—
142	C₂×C₈	—	59	2	no	—
143	C₂×C₈	—	118	4	no	—
144	C₂×C₈	—	236	2	no	—
145	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
146	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
147	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
148	C₄²	D₄×D₅₉	1	1	yes	characteristic
149	C₄²	Q₈×D₅₉	1	1	yes	characteristic
150	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
151	C₂³	#14273726	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
155	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
156	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
157	C₂×C₄	#14273723	1	1	yes	characteristic
158	C₂×C₄	#14273725	1	1	yes	characteristic
159	C₂×C₄	#14273726	1	1	yes	characteristic
160	C₂×C₄	SD₁₆×D₅₉	1	4	yes	—
161	C₈	—	2	2	no	—
162	C₈	—	8	2	no	—
163	C₈	—	59	4	no	—
164	C₈	—	59	4	no	—
165	C₈	—	59	4	no	—
166	C₈	—	118	2	no	—
167	C₈	—	472	2	no	—
168	C₈	#14273662	1	4	yes	complemented
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#16837271	1	1	yes	characteristic
172	C₂²	#16837799	1	2	yes	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	#106×D₅₉	1	2	yes	—
176	C₄	#108×D₅₉	1	2	yes	—
177	C₄	C₂×#14273662	1	2	yes	—
178	C₄	#16837773	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	#21424251	1	1	yes	characteristic
181	C₂	#21424266	1	1	yes	characteristic
182	C₂	#21424267	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273723
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1566	C₂
socle	C₂²×C₅₉	#384
prime radical	C₂³×C₅₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1566	D₅₉
hypocenter	C₅₉	#8822
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
71. C₈×C₅₉	#108	semidirect
76. C₅₉⋊C₈	#108	semidirect
110. C₅₉	#8822	semidirect
168. C₈	#14273662	semidirect

Maximal subgroups

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

subgroup	index
#21424267	2
#21422852	2
#21422851	2
C₂×C₈×C₅₉⋊C₈	2
C₅₉×#1566	2
#8822	59