Subgroups — #28615321

Order 15104.

632 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424815	C₂	1	4	yes	maximal
2	#21424655	C₂	1	4	yes	maximal
3	#21424655	C₂	1	4	yes	maximal
4	#21424213	C₂	1	2	yes	maximal
5	C₄×#125×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	4	yes	—
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄×C₅₉⋊C₈	C₂²	1	4	yes	—
10	#16837157	C₂²	1	4	yes	—
11	#16837157	C₂²	1	4	yes	—
12	#16837157	C₂²	1	4	yes	—
13	#16837157	C₂²	1	4	yes	—
14	C₂×#125×C₅₉	C₄	1	1	yes	characteristic
15	C₅₉×#389	C₄	1	2	yes	—
16	C₅₉×#387	C₄	1	4	yes	—
17	C₅₉×#382	C₄	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
19	C₄×C₄⋊C₄×C₅₉	C₂²	1	4	yes	—
20	C₄³×C₅₉	C₂²	1	1	yes	characteristic
21	C₂²×C₅₉⋊C₈	C₂³	1	4	yes	—
22	C₂²×C₅₉⋊C₈	C₂³	1	4	yes	—
23	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
24	C₄⋊(C₅₉⋊C₈)	—	4	4	no	—
25	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
26	C₄×C₅₉⋊C₈	—	4	4	no	—
27	#125×C₅₉	C₈	1	8	yes	complemented
28	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
30	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
37	C₂×C₄²×C₅₉	C₂³	1	4	yes	—
38	#95×C₅₉	C₂×C₄	1	4	yes	—
39	#95×C₅₉	C₂×C₄	1	4	yes	—
40	#95×C₅₉	C₂×C₄	1	4	yes	—
41	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	C₂²×C₄	1	4	yes	—
43	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₅₉	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
48	C₄⋊C₄×C₅₉	—	4	2	no	—
49	C₄⋊C₄×C₅₉	—	4	4	no	—
50	C₄⋊C₄×C₅₉	C₂×C₈	1	16	yes	—
51	C₄⋊C₄×C₅₉	C₂×C₈	1	8	yes	—
52	C₄²×C₅₉	—	2	4	no	—
53	C₄²×C₅₉	—	2	2	no	—
54	C₄²×C₅₉	—	2	1	no	—
55	C₄²×C₅₉	—	2	1	no	—
56	C₄²×C₅₉	—	2	2	no	—
57	C₄²×C₅₉	—	2	4	no	—
58	C₄²×C₅₉	—	4	1	no	—
59	C₄²×C₅₉	—	4	2	no	—
60	C₄²×C₅₉	C₂×C₈	1	4	yes	—
61	C₂×C₅₉⋊C₈	—	2	4	no	—
62	C₂×C₅₉⋊C₈	—	2	4	no	—
63	C₂×C₅₉⋊C₈	—	2	4	no	—
64	C₂×C₅₉⋊C₈	—	2	4	no	—
65	C₂×C₅₉⋊C₈	—	2	4	no	—
66	C₂×C₅₉⋊C₈	—	2	4	no	—
67	C₂³×C₅₉	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₉	—	2	1	no	—
69	C₂×C₄×C₅₉	—	2	4	no	—
70	C₂×C₄×C₅₉	—	2	2	no	—
71	C₂×C₄×C₅₉	—	2	1	no	—
72	C₂×C₄×C₅₉	—	2	2	no	—
73	C₂×C₄×C₅₉	—	2	4	no	—
74	C₂×C₄×C₅₉	—	2	1	no	—
75	C₂×C₄×C₅₉	—	2	4	no	—
76	C₂×C₄×C₅₉	—	2	2	no	—
77	C₂×C₄×C₅₉	—	2	2	no	—
78	C₂×C₄×C₅₉	—	2	2	no	—
79	C₂×C₄×C₅₉	—	2	2	no	—
80	C₂×C₄×C₅₉	—	2	4	no	—
81	C₂×C₄×C₅₉	—	2	4	no	—
82	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₅₉	C₂²×C₈	1	4	yes	—
84	C₂×C₄×C₅₉	C₂²×C₈	1	8	yes	—
85	C₂×C₄×C₅₉	H(5,𝔽₂)	1	2	yes	—
86	C₂×C₄×C₅₉	2₋¹⁺⁴	1	2	yes	—
87	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
88	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
89	C₅₉⋊C₈	—	4	4	no	—
90	C₅₉⋊C₈	—	4	4	no	—
91	#23369	—	59	1	no	maximal, Sylow
92	C₂²×C₅₉	#519	1	1	yes	characteristic
93	C₂²×C₅₉	#520	1	2	yes	—
94	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
95	C₂²×C₅₉	#568	1	1	yes	characteristic
96	C₂²×C₅₉	#568	1	2	yes	—
97	C₄×C₅₉	—	2	2	no	—
98	C₄×C₅₉	—	2	4	no	—
99	C₄×C₅₉	—	2	8	no	—
100	C₄×C₅₉	—	4	1	no	—
101	C₄×C₅₉	—	4	2	no	—
102	C₄×C₅₉	—	4	4	no	—
103	C₄×C₅₉	#562	1	1	yes	characteristic
104	C₄×C₅₉	#563	1	2	yes	—
105	C₄×C₅₉	#564	1	1	yes	characteristic
106	C₄×#125	D₅₉	1	1	yes	characteristic
107	#1939	—	59	4	no	—
108	#1916	—	59	4	no	—
109	#1916	—	59	4	no	—
110	#1831	—	59	2	no	—
111	C₂×C₅₉	#2382	1	1	yes	characteristic
112	C₂×C₅₉	#2982	1	1	yes	characteristic
113	C₂×C₅₉	#2987	1	1	yes	characteristic
114	C₂×C₅₉	#2988	1	2	yes	—
115	C₂×C₅₉	#2995	1	2	yes	—
116	C₂×#125	Dic₅₉	1	1	yes	characteristic
117	C₂×C₄⋊C₈	—	59	2	no	—
118	C₂×C₄⋊C₈	—	59	2	no	—
119	C₂×C₄⋊C₈	—	59	4	no	—
120	C₂×C₄×C₈	—	59	4	no	—
121	#389	Dic₅₉	1	2	yes	—
122	#387	Dic₅₉	1	4	yes	—
123	#382	Dic₅₉	1	1	yes	characteristic
124	C₄×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
125	C₄×C₄⋊C₄	C₂×D₅₉	1	4	yes	—
126	C₄³	C₂×D₅₉	1	1	yes	characteristic
127	#336	—	59	4	no	—
128	#336	—	59	4	no	—
129	#336	—	59	4	no	—
130	#336	—	59	4	no	—
131	C₅₉	#23369	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₈	—	59	4	no	—
133	C₂²×C₈	—	59	4	no	—
134	#125	C₅₉⋊C₈	1	8	yes	complemented
135	C₂×C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
136	C₂×C₄⋊C₄	C₂×Dic₅₉	1	4	yes	—
137	C₂×C₄⋊C₄	C₂×Dic₅₉	1	4	yes	—
138	C₂×C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
139	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
140	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
141	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
142	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
143	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
144	C₂×C₄²	C₂²×D₅₉	1	4	yes	—
145	C₄⋊C₈	—	236	4	no	—
146	C₄⋊C₈	—	236	2	no	—
147	C₄⋊C₈	—	236	2	no	—
148	C₄×C₈	—	236	4	no	—
149	#95	C₂×Dic₅₉	1	4	yes	—
150	#95	C₂×Dic₅₉	1	4	yes	—
151	#95	C₂×Dic₅₉	1	4	yes	—
152	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
153	C₂²×C₄	C₂²×Dic₅₉	1	4	yes	—
154	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
155	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
156	C₂²×C₄	C₂²×Dic₅₉	1	4	yes	—
157	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
158	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
159	C₂×C₈	—	118	4	no	—
160	C₂×C₈	—	118	4	no	—
161	C₂×C₈	—	118	4	no	—
162	C₂×C₈	—	118	4	no	—
163	C₂×C₈	—	118	4	no	—
164	C₂×C₈	—	118	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	C₂×C₅₉⋊C₈	1	16	yes	—
168	C₄⋊C₄	C₂×C₅₉⋊C₈	1	8	yes	—
169	C₄²	—	2	4	no	—
170	C₄²	—	2	2	no	—
171	C₄²	—	2	2	no	—
172	C₄²	—	2	4	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	4	1	no	—
176	C₄²	—	4	2	no	—
177	C₄²	C₂×C₅₉⋊C₈	1	4	yes	—
178	C₂³	C₂³×Dic₅₉	1	1	yes	characteristic
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	C₂²×C₅₉⋊C₈	1	2	yes	—
194	C₂×C₄	C₂²×C₅₉⋊C₈	1	4	yes	—
195	C₂×C₄	C₂²×C₅₉⋊C₈	1	8	yes	—
196	C₂×C₄	#14273773	1	2	yes	—
197	C₂×C₄	#14273774	1	2	yes	—
198	C₂×C₄	#14273775	1	1	yes	characteristic
199	C₂×C₄	#14273775	1	1	yes	characteristic
200	C₈	—	236	4	no	—
201	C₈	—	236	4	no	—
202	C₂²	#16837880	1	1	yes	characteristic
203	C₂²	#16837885	1	2	yes	—
204	C₂²	C₂³×C₅₉⋊C₈	1	1	yes	characteristic
205	C₂²	#16838169	1	1	yes	characteristic
206	C₂²	#16838169	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	2	4	no	—
209	C₄	—	2	8	no	—
210	C₄	—	4	1	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	4	no	—
213	C₄	#16838150	1	1	yes	characteristic
214	C₄	#16838156	1	2	yes	—
215	C₄	C₅₉⋊#564	1	1	yes	characteristic
216	C₂	#21427282	1	1	yes	characteristic
217	C₂	#21433177	1	1	yes	characteristic
218	C₂	#21433202	1	1	yes	characteristic
219	C₂	#21433207	1	2	yes	—
220	C₂	#21433266	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₉
commutator	C₂²×C₅₉	C₂³×C₈
Fitting	C₄×#125×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂³×C₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₄×#125	D₅₉
hypocenter	C₅₉	#23369
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
27. #125×C₅₉	C₈	semidirect
131. C₅₉	#23369	semidirect
134. #125	C₅₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#21424815	2
#21424655	2
#21424213	2
C₄×#125×C₅₉	2
#23369	59