Subgroups — #28482457

Order 15104.

454 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424256	C₂	1	1	yes	maximal, characteristic
2	#21424252	C₂	1	2	yes	maximal
3	#21424223	C₂	1	2	yes	maximal
4	C₂×C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273661	—	2	2	no	—
7	C₂×#14273658	—	2	4	no	—
8	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	#16837272	C₂²	1	2	yes	—
11	#16837164	—	2	4	no	—
12	C₈×C₅₉⋊C₈	C₄	1	4	yes	complemented
13	C₂×#107×C₅₉	—	2	2	no	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#384	C₂²	1	2	yes	—
16	C₂²×C₅₉⋊C₈	D₄	1	2	yes	—
17	C₂²×C₅₉⋊C₈	D₄	1	2	yes	—
18	C₂×C₄⋊Dic₅₉	—	2	2	no	—
19	C₂×C₄⋊Dic₅₉	—	2	2	no	—
20	C₂×C₄⋊Dic₅₉	—	2	2	no	—
21	#14273661	—	4	4	no	—
22	#14273658	—	4	8	no	—
23	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
25	C₄×C₅₉⋊C₈	Q₈	1	2	yes	—
26	C₄×C₅₉⋊C₈	Q₈	1	2	yes	—
27	C₂²×C₈×C₅₉	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
31	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
32	#107×C₅₉	—	4	4	no	—
33	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₉	Q₈	1	2	yes	—
35	C₂²×C₄×C₅₉	—	4	2	no	—
36	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₅₉	—	4	2	no	—
40	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
41	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
42	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
43	C₄⋊C₄×C₅₉	—	4	4	no	—
44	C₄⋊C₄×C₅₉	—	4	4	no	—
45	C₄⋊C₄×C₅₉	—	4	4	no	—
46	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
50	C₄⋊Dic₅₉	—	4	4	no	—
51	C₄⋊Dic₅₉	—	4	4	no	—
52	C₄⋊Dic₅₉	—	4	4	no	—
53	C₂×C₅₉⋊C₈	—	2	2	no	—
54	C₂×C₅₉⋊C₈	—	2	2	no	—
55	C₂×C₅₉⋊C₈	C₄⋊C₄	1	4	yes	—
56	C₂×C₅₉⋊C₈	C₄⋊C₄	1	4	yes	—
57	C₂×C₅₉⋊C₈	D₈	1	4	yes	—
58	C₂×C₅₉⋊C₈	Q₁₆	1	4	yes	—
59	C₂³×C₅₉	#127	1	1	yes	characteristic
60	C₂×C₄×C₅₉	—	8	2	no	—
61	C₂×C₄×C₅₉	—	8	2	no	—
62	C₂×C₄×C₅₉	—	8	2	no	—
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	C₂×C₄⋊C₄	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	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
71	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
72	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
73	C₂×C₄×C₅₉	#135	1	2	yes	—
74	C₂×C₄×C₅₉	#135	1	2	yes	—
75	C₂×Dic₅₉	—	8	2	no	—
76	C₂×Dic₅₉	—	8	2	no	—
77	C₂×Dic₅₉	—	8	2	no	—
78	C₈×C₅₉	#106	1	8	yes	complemented
79	C₅₉⋊C₈	—	2	4	no	—
80	C₅₉⋊C₈	#107	1	8	yes	complemented
81	#8814	—	59	1	no	maximal, Sylow
82	C₂²×C₅₉	#384	1	1	yes	characteristic
83	C₂²×C₅₉	#492	1	1	yes	characteristic
84	C₂²×C₅₉	#492	1	1	yes	characteristic
85	C₂²×C₅₉	#493	1	1	yes	characteristic
86	C₂²×C₅₉	#494	1	1	yes	characteristic
87	C₂²×C₅₉	#495	1	1	yes	characteristic
88	C₂²×C₅₉	#495	1	1	yes	characteristic
89	C₄×C₅₉	—	16	2	no	—
90	C₄×C₅₉	C₂×#106	1	2	yes	—
91	C₄×C₅₉	C₂×#107	1	2	yes	—
92	C₄×C₅₉	#427	1	2	yes	—
93	C₄×C₅₉	#498	1	2	yes	—
94	C₄×C₅₉	#499	1	2	yes	—
95	C₄×C₅₉	#500	1	2	yes	—
96	Dic₅₉	—	16	2	no	—
97	#1845	—	59	1	no	—
98	#1845	D₅₉	1	1	yes	characteristic
99	#1844	—	59	2	no	—
100	#1834	—	59	2	no	—
101	C₂×C₈²	—	59	1	no	—
102	C₂×C₅₉	#1706	1	1	yes	characteristic
103	C₂×C₅₉	#1706	1	1	yes	characteristic
104	C₂×C₅₉	#1707	1	1	yes	characteristic
105	C₂×C₅₉	#1707	1	1	yes	characteristic
106	C₂×C₅₉	#1844	1	1	yes	characteristic
107	C₂×C₅₉	#1845	1	1	yes	characteristic
108	C₂×C₅₉	#1846	1	1	yes	characteristic
109	C₂×#107	—	2	2	no	—
110	C₂×#107	—	118	2	no	—
111	C₂×#106	—	118	4	no	—
112	C₂×C₄×C₈	—	59	1	no	—
113	C₂×C₄×C₈	—	59	1	no	—
114	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
115	#384	—	59	2	no	—
116	#384	C₂×D₅₉	1	2	yes	—
117	#340	—	118	4	no	—
118	C₈²	—	59	4	no	—
119	C₅₉	#8814	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₈	—	59	2	no	—
121	C₂²×C₈	—	59	2	no	—
122	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	118	2	no	—
127	C₂×C₄⋊C₄	—	118	2	no	—
128	C₂×C₄⋊C₄	—	118	2	no	—
129	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
130	#107	—	4	4	no	—
131	#107	—	236	4	no	—
132	#106	—	236	8	no	—
133	C₄×C₈	—	59	2	no	—
134	C₄×C₈	—	59	2	no	—
135	C₄×C₈	—	59	2	no	—
136	C₄×C₈	—	59	2	no	—
137	C₄×C₈	Dic₁₁₈	1	2	yes	—
138	C₄×C₈	C₄×D₅₉	1	2	yes	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	—	236	2	no	—
141	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
142	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
144	C₂×C₈	—	59	4	no	—
145	C₂×C₈	—	59	4	no	—
146	C₂×C₈	—	59	4	no	—
147	C₂×C₈	—	59	4	no	—
148	C₂×C₈	—	118	2	no	—
149	C₂×C₈	—	118	2	no	—
150	C₂×C₈	Dic₅₉⋊C₄	1	4	yes	—
151	C₂×C₈	#11745670	1	4	yes	—
152	C₂×C₈	#11745671	1	4	yes	—
153	C₄⋊C₄	—	4	4	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	236	4	no	—
157	C₄⋊C₄	—	236	4	no	—
158	C₄⋊C₄	—	236	4	no	—
159	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
160	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
161	C₄²	Q₈×D₅₉	1	1	yes	characteristic
162	C₄²	Q₈×D₅₉	1	1	yes	characteristic
163	C₂³	#14273723	1	1	yes	characteristic
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	472	2	no	—
168	C₂×C₄	—	472	2	no	—
169	C₂×C₄	—	472	2	no	—
170	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
171	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
172	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
173	C₂×C₄	#14273725	1	1	yes	characteristic
174	C₂×C₄	#14273726	1	1	yes	characteristic
175	C₂×C₄	#14273726	1	1	yes	characteristic
176	C₂×C₄	D₈×D₅₉	1	2	yes	—
177	C₂×C₄	C₂×#11745671	1	2	yes	—
178	C₂×C₄	C₂×#11745670	1	2	yes	—
179	C₂×C₄	Q₁₆×D₅₉	1	2	yes	—
180	C₂×C₄	#14273746	1	2	yes	—
181	C₂×C₄	#14273750	1	2	yes	—
182	C₈	—	59	8	no	—
183	C₈	—	118	4	no	—
184	C₈	#14273658	1	8	yes	complemented
185	C₂²	#16837271	1	1	yes	characteristic
186	C₂²	#16837774	1	1	yes	characteristic
187	C₂²	#16837776	1	1	yes	characteristic
188	C₂²	#16837777	1	1	yes	characteristic
189	C₂²	#16837780	1	1	yes	characteristic
190	C₂²	#16837785	1	1	yes	characteristic
191	C₂²	#16837787	1	1	yes	characteristic
192	C₄	—	16	2	no	—
193	C₄	—	944	2	no	—
194	C₄	C₂×#14273658	1	2	yes	—
195	C₄	#107×D₅₉	1	2	yes	—
196	C₄	#16837439	1	2	yes	—
197	C₄	#16837798	1	2	yes	—
198	C₄	#16837803	1	2	yes	—
199	C₄	#16837806	1	2	yes	—
200	C₂	#21423685	1	1	yes	characteristic
201	C₂	#21423689	1	1	yes	characteristic
202	C₂	#21423690	1	1	yes	characteristic
203	C₂	#21423694	1	1	yes	characteristic
204	C₂	#21424252	1	1	yes	characteristic
205	C₂	#21424254	1	1	yes	characteristic
206	C₂	#21424259	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273723
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1845	C₂
socle, prime radical	C₂³×C₅₉	#127
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1845	D₅₉
hypocenter	C₅₉	#8814
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
12. C₈×C₅₉⋊C₈	C₄	semidirect
78. C₈×C₅₉	#106	semidirect
80. C₅₉⋊C₈	#107	semidirect
119. C₅₉	#8814	semidirect
184. C₈	#14273658	semidirect

Maximal subgroups

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

subgroup	index
#21424256	2
#21424252	2
#21424223	2
C₂×C₈×C₅₉⋊C₈	2
C₅₉×#1845	2
#8814	59