Subgroups — D₅₉×#2252 #28744163

Order 15104.

558 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₅₉	C₂	1	2	yes	maximal
2	#21426371	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426370	C₂	1	4	yes	maximal, complemented
4	#21426369	C₂	1	2	yes	maximal, complemented
5	#366×D₅₉	C₂	1	4	yes	maximal
6	C₄×C₁₆×D₅₉	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#2252	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄⋊Q₈×D₅₉	—	2	2	no	—
9	C₂×C₁₆×D₅₉	C₂²	1	2	yes	—
10	#16837807	C₂²	1	2	yes	—
11	#16837806	—	2	2	no	—
12	#16837806	C₂²	1	4	yes	—
13	#16837805	—	2	2	no	—
14	#107×D₅₉	—	2	2	no	—
15	#107×D₅₉	C₂²	1	4	yes	—
16	C₄×C₈×D₅₉	C₂²	1	1	yes	characteristic
17	#16837228	C₂²	1	4	yes	—
18	#16837227	C₂²	1	8	yes	—
19	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₁₆×Dic₅₉	C₂²	1	2	yes	—
21	C₅₉×#500	C₂²	1	2	yes	—
22	C₅₉×#366	C₂²	1	4	yes	—
23	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×Q₈×D₅₉	—	4	2	no	—
25	C₂×C₈×D₅₉	D₄	1	1	yes	characteristic
26	C₂×C₈×D₅₉	C₂³	1	1	yes	characteristic
27	#14273727	—	2	2	no	—
28	#14273726	—	2	4	no	—
29	#14273726	—	4	2	no	—
30	#14273725	—	4	2	no	—
31	C₄⋊C₄×D₅₉	—	2	4	no	—
32	C₄⋊C₄×D₅₉	—	4	2	no	—
33	C₄²×D₅₉	D₄	1	1	yes	characteristic
34	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
35	C₁₆×D₅₉	Q₈	1	8	yes	complemented
36	#14273661	—	2	2	no	—
37	#14273661	C₂³	1	4	yes	—
38	#14273660	—	2	4	no	—
39	#14273660	—	2	4	no	—
40	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
41	C₈×Dic₅₉	D₄	1	1	yes	characteristic
42	C₈×Dic₅₉	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₅₉	—	2	2	no	—
44	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
45	#107×C₅₉	—	2	2	no	—
46	#107×C₅₉	C₂³	1	4	yes	—
47	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₅₉	—	4	2	no	—
49	Q₈×D₅₉	—	8	8	no	—
50	C₂×C₄×D₅₉	—	4	4	no	—
51	C₂×C₄×D₅₉	D₈	1	1	yes	characteristic
52	C₂×C₄×D₅₉	D₈	1	1	yes	characteristic
53	C₂×C₄×D₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₁₁₈	—	4	2	no	—
55	C₂×Dic₁₁₈	—	4	4	no	—
56	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
58	C₄⋊C₄×C₅₉	—	2	4	no	—
59	C₄⋊C₄×C₅₉	—	4	2	no	—
60	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₄⋊Dic₅₉	—	2	4	no	—
62	C₄⋊Dic₅₉	—	4	2	no	—
63	Dic₅₉⋊C₄	—	4	4	no	—
64	Dic₅₉⋊C₄	—	4	4	no	—
65	C₄×Dic₅₉	D₈	1	1	yes	characteristic
66	C₄×Dic₅₉	D₈	1	1	yes	characteristic
67	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
70	C₈×D₅₉	Q₁₆	1	4	yes	—
71	C₈×D₅₉	C₂×Q₈	1	4	yes	—
72	C₁₆×C₅₉	C₂×Q₈	1	4	yes	complemented
73	C₅₉⋊C₁₆	C₂×Q₈	1	4	yes	complemented
74	C₂²×D₅₉	C₂×D₈	1	1	yes	characteristic
75	Q₈×C₅₉	—	8	4	no	—
76	C₂×C₄×C₅₉	—	4	4	no	—
77	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
80	C₂×Dic₅₉	—	4	4	no	—
81	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
82	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
83	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
84	C₄×D₅₉	—	8	8	no	—
85	C₄×D₅₉	D₁₆	1	4	yes	—
86	C₄×D₅₉	Q₃₂	1	4	yes	—
87	C₄×D₅₉	C₄⋊Q₈	1	2	yes	—
88	C₄×D₅₉	C₂×Q₁₆	1	2	yes	—
89	Dic₁₁₈	—	8	4	no	—
90	Dic₁₁₈	—	8	8	no	—
91	C₈×C₅₉	C₂×Q₁₆	1	2	yes	—
92	C₈×C₅₉	C₂²×Q₈	1	2	yes	—
93	C₅₉⋊C₈	C₄⋊Q₈	1	2	yes	—
94	C₅₉⋊C₈	C₂×Q₁₆	1	2	yes	—
95	C₂×#2252	—	59	1	no	maximal, Sylow
96	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
97	C₂×D₅₉	#500	1	2	yes	—
98	C₂×D₅₉	C₂×D₁₆	1	2	yes	—
99	C₂×D₅₉	C₂×Q₃₂	1	2	yes	—
100	C₄×C₅₉	—	8	4	no	—
101	C₄×C₅₉	C₂×D₁₆	1	2	yes	—
102	C₄×C₅₉	C₂×Q₃₂	1	2	yes	—
103	C₄×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
104	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
105	Dic₅₉	—	8	4	no	—
106	Dic₅₉	#500	1	2	yes	—
107	Dic₅₉	C₂×D₁₆	1	2	yes	—
108	Dic₅₉	C₂×Q₃₂	1	2	yes	—
109	C₂×#500	—	59	2	no	—
110	#2252	—	59	2	no	—
111	#2252	—	59	4	no	—
112	#2252	—	59	1	no	—
113	#2252	D₅₉	1	1	yes	characteristic, direct product factor
114	C₂×#366	—	59	4	no	—
115	C₂×C₄×C₁₆	—	59	1	no	—
116	C₂×C₅₉	C₂×#500	1	1	yes	characteristic
117	C₂×C₅₉	C₂²×D₁₆	1	1	yes	characteristic
118	C₂×C₅₉	C₂²×Q₃₂	1	1	yes	characteristic
119	D₅₉	#2252	1	4	yes	direct product factor
120	C₂×C₄⋊Q₈	—	118	2	no	—
121	C₂²×C₁₆	—	59	2	no	—
122	#500	—	59	4	no	—
123	#500	—	59	2	no	—
124	#500	—	118	2	no	—
125	#500	—	118	2	no	—
126	#500	C₂×D₅₉	1	2	yes	—
127	C₂×#107	—	59	4	no	—
128	C₂×#107	—	118	2	no	—
129	C₂×C₄×C₈	—	59	1	no	—
130	#366	—	59	4	no	—
131	#366	—	59	8	no	—
132	#366	C₂×D₅₉	1	4	yes	—
133	C₄×C₁₆	—	59	2	no	—
134	C₄×C₁₆	—	59	1	no	—
135	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
136	C₅₉	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₈	—	236	2	no	—
138	C₂²×C₈	—	59	1	no	—
139	C₂²×C₈	—	59	1	no	—
140	C₄⋊Q₈	—	2	2	no	—
141	C₄⋊Q₈	—	118	4	no	—
142	C₄⋊Q₈	—	118	2	no	—
143	C₄⋊Q₈	—	236	2	no	—
144	C₄⋊Q₈	—	236	2	no	—
145	C₂×C₄⋊C₄	—	118	4	no	—
146	C₂×C₄⋊C₄	—	236	2	no	—
147	C₂×C₄²	—	59	1	no	—
148	C₂×C₁₆	—	59	2	no	—
149	C₂×C₁₆	—	59	8	no	—
150	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
151	#107	—	2	2	no	—
152	#107	—	59	4	no	—
153	#107	—	118	4	no	—
154	#107	—	118	4	no	—
155	#107	—	118	2	no	—
156	#107	C₂²×D₅₉	1	4	yes	—
157	C₄×C₈	—	59	1	no	—
158	C₄×C₈	—	59	1	no	—
159	C₄×C₈	—	59	1	no	—
160	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	—	472	8	no	—
163	C₂×Q₈	—	236	4	no	—
164	C₂×Q₈	—	236	2	no	—
165	C₂²×C₄	—	59	1	no	—
166	C₂²×C₄	—	59	1	no	—
167	C₂²×C₄	—	59	1	no	—
168	C₂²×C₄	—	236	4	no	—
169	C₂×C₈	—	59	1	no	—
170	C₂×C₈	—	59	1	no	—
171	C₂×C₈	—	59	4	no	—
172	C₂×C₈	—	59	4	no	—
173	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
174	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	118	4	no	—
178	C₄⋊C₄	—	236	4	no	—
179	C₄⋊C₄	—	236	4	no	—
180	C₄⋊C₄	—	236	2	no	—
181	C₄²	—	59	1	no	—
182	C₄²	—	59	1	no	—
183	C₄²	—	59	1	no	—
184	C₄²	D₄×D₅₉	1	1	yes	characteristic
185	C₁₆	—	59	4	no	—
186	C₁₆	Q₈×D₅₉	1	4	yes	complemented
187	C₂³	—	59	1	no	—
188	Q₈	—	8	4	no	—
189	Q₈	—	472	8	no	—
190	Q₈	—	472	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	59	1	no	—
193	C₂×C₄	—	59	1	no	—
194	C₂×C₄	—	59	1	no	—
195	C₂×C₄	—	59	2	no	—
196	C₂×C₄	—	59	2	no	—
197	C₂×C₄	—	59	4	no	—
198	C₂×C₄	—	59	4	no	—
199	C₂×C₄	—	472	8	no	—
200	C₂×C₄	—	236	4	no	—
201	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
202	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
203	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
204	C₈	—	59	2	no	—
205	C₈	—	59	2	no	—
206	C₈	Q₁₆×D₅₉	1	2	yes	—
207	C₈	C₂×Q₈×D₅₉	1	2	yes	—
208	C₂²	—	59	2	no	—
209	C₂²	—	59	2	no	—
210	C₂²	—	59	2	no	—
211	C₂²	C₂×D₈×D₅₉	1	1	yes	characteristic
212	C₄	—	8	4	no	—
213	C₄	—	59	2	no	—
214	C₄	—	59	2	no	—
215	C₄	—	59	2	no	—
216	C₄	—	472	4	no	—
217	C₄	D₁₆×D₅₉	1	2	yes	—
218	C₄	Q₃₂×D₅₉	1	2	yes	—
219	C₄	C₄⋊Q₈×D₅₉	1	1	yes	characteristic
220	C₄	C₂×Q₁₆×D₅₉	1	1	yes	characteristic
221	C₂	—	59	4	no	—
222	C₂	#500×D₅₉	1	1	yes	characteristic
223	C₂	C₂×D₁₆×D₅₉	1	1	yes	characteristic
224	C₂	C₂×Q₃₂×D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₉
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2252	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂²×D₅₉	C₂×D₈
hypercenter	#2252	D₅₉
hypocenter	C₅₉	C₂×#2252
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
2. #21426371	C₂	semidirect
3. #21426370	C₂	semidirect
4. #21426369	C₂	semidirect
7. C₅₉×#2252	C₂	semidirect
35. C₁₆×D₅₉	Q₈	semidirect
72. C₁₆×C₅₉	C₂×Q₈	semidirect
73. C₅₉⋊C₁₆	C₂×Q₈	semidirect
113. #2252	D₅₉	direct (N × H)
119. D₅₉	#2252	direct (N × H)
136. C₅₉	C₂×#2252	semidirect
186. C₁₆	Q₈×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
#500×D₅₉	2
#21426371	2
#21426370	2
#21426369	2
#366×D₅₉	2
C₄×C₁₆×D₅₉	2
C₅₉×#2252	2
C₂×#2252	59

Subgroups — D59×#2252 #28744163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₉×#2252 #28744163