Subgroups — #28508900

Order 15104.

290 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837236	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16837236	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422238	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₉×#504	C₂	1	1	yes	maximal, characteristic
5	#16837236	—	2	1	no	—
6	#16837236	—	2	1	no	—
7	#16837236	—	2	1	no	—
8	#16837236	C₂²	1	2	yes	complemented
9	C₂×C₅₉⋊C₃₂	—	2	1	no	—
10	C₂×C₅₉⋊C₃₂	C₂²	1	2	yes	complemented
11	C₂×#131×C₅₉	C₄	1	1	yes	characteristic
12	C₅₉×#504	—	2	1	no	—
13	C₅₉×#504	—	2	1	no	—
14	C₅₉×#504	C₄	1	2	yes	—
15	C₅₉×#504	C₂²	1	2	yes	—
16	C₂×M₃₂×C₅₉	—	2	1	no	—
17	C₂×M₃₂×C₅₉	C₄	1	1	yes	characteristic
18	C₂²×C₁₆×C₅₉	—	2	1	no	—
19	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₅₉⋊C₃₂	—	2	2	no	—
21	C₅₉⋊C₃₂	—	2	2	no	—
22	C₂×#41×C₅₉	C₈	1	1	yes	characteristic
23	#131×C₅₉	—	2	1	no	—
24	#131×C₅₉	—	2	1	no	—
25	#131×C₅₉	C₈	1	2	yes	—
26	#131×C₅₉	C₂×C₄	1	2	yes	—
27	C₂×M₁₆×C₅₉	—	2	1	no	—
28	C₂×M₁₆×C₅₉	C₈	1	1	yes	characteristic
29	C₂²×C₈×C₅₉	—	2	1	no	—
30	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	M₃₂×C₅₉	—	2	1	no	—
32	M₃₂×C₅₉	—	2	2	no	—
33	M₃₂×C₅₉	—	2	2	no	—
34	M₃₂×C₅₉	C₂×C₄	1	2	yes	—
35	C₂×C₁₆×C₅₉	—	2	1	no	—
36	C₂×C₁₆×C₅₉	—	2	2	no	—
37	C₂×C₁₆×C₅₉	—	2	2	no	—
38	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
43	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
44	#41×C₅₉	—	2	1	no	—
45	#41×C₅₉	—	2	1	no	—
46	#41×C₅₉	C₁₆	1	2	yes	—
47	#41×C₅₉	C₂×C₈	1	2	yes	—
48	C₂×Q₈×C₅₉	C₁₆	1	1	yes	characteristic
49	C₂×D₄×C₅₉	—	2	1	no	—
50	C₂×D₄×C₅₉	C₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	—	2	1	no	—
52	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
53	M₁₆×C₅₉	—	2	1	no	—
54	M₁₆×C₅₉	—	2	2	no	—
55	M₁₆×C₅₉	—	2	2	no	—
56	M₁₆×C₅₉	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₅₉	—	2	1	no	—
58	C₂×C₈×C₅₉	—	2	2	no	—
59	C₂×C₈×C₅₉	—	2	2	no	—
60	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
63	C₂×C₈×C₅₉	M₁₆	1	1	yes	characteristic
64	C₂×C₈×C₅₉	M₁₆	1	1	yes	characteristic
65	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
66	C₁₆×C₅₉	—	2	2	no	—
67	C₁₆×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₁₆×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₁₆×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₁₆×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂³×C₅₉	—	2	1	no	—
72	C₂³×C₅₉	M₃₂	1	1	yes	characteristic
73	Q₈×C₅₉	—	2	1	no	—
74	Q₈×C₅₉	C₂×C₁₆	1	2	yes	—
75	D₄×C₅₉	—	2	2	no	—
76	D₄×C₅₉	—	2	2	no	—
77	D₄×C₅₉	—	2	1	no	—
78	D₄×C₅₉	C₂×C₁₆	1	2	yes	—
79	C₂×C₄×C₅₉	—	2	1	no	—
80	C₂×C₄×C₅₉	—	2	2	no	—
81	C₂×C₄×C₅₉	—	2	2	no	—
82	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
85	C₂×C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
86	C₂×C₄×C₅₉	M₃₂	1	1	yes	characteristic
87	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
88	C₈×C₅₉	—	2	2	no	—
89	C₈×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
90	C₈×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
91	C₈×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
93	#13558	—	59	1	no	maximal, Sylow
94	C₂²×C₅₉	—	2	2	no	—
95	C₂²×C₅₉	—	4	1	no	—
96	C₂²×C₅₉	—	4	2	no	—
97	C₂²×C₅₉	C₂²⋊C₁₆	1	1	yes	characteristic
98	C₂²×C₅₉	C₂²⋊C₁₆	1	1	yes	characteristic
99	C₂²×C₅₉	C₂×M₃₂	1	1	yes	characteristic
100	C₄×C₅₉	—	2	2	no	—
101	C₄×C₅₉	C₂²⋊C₁₆	1	1	yes	characteristic
102	C₄×C₅₉	C₂²⋊C₁₆	1	1	yes	characteristic
103	C₄×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₄×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
105	C₂×#504	D₅₉	1	1	yes	characteristic
106	C₂×M₆₄	—	59	1	no	—
107	C₂×M₆₄	—	59	1	no	—
108	#1401	—	59	4	no	—
109	C₂×C₅₉	—	2	1	no	—
110	C₂×C₅₉	—	2	1	no	—
111	C₂×C₅₉	—	2	1	no	—
112	C₂×C₅₉	—	4	2	no	—
113	C₂×C₅₉	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
114	C₂×#131	Dic₅₉	1	1	yes	characteristic
115	#504	—	2	1	no	—
116	#504	—	2	1	no	—
117	#504	Dic₅₉	1	2	yes	—
118	#504	C₂×D₅₉	1	2	yes	—
119	C₂×M₃₂	—	2	1	no	—
120	C₂×M₃₂	Dic₅₉	1	1	yes	characteristic
121	C₂²×C₁₆	—	2	1	no	—
122	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
123	M₆₄	—	59	2	no	—
124	M₆₄	—	118	1	no	—
125	M₆₄	—	118	1	no	—
126	M₆₄	—	118	1	no	—
127	C₂×C₃₂	—	59	2	no	—
128	C₂×C₃₂	—	118	1	no	—
129	C₅₉	#13558	1	1	yes	characteristic, complemented, Sylow
130	C₂×#41	C₅₉⋊C₈	1	1	yes	characteristic
131	#131	—	2	1	no	—
132	#131	—	2	1	no	—
133	#131	C₅₉⋊C₈	1	2	yes	—
134	#131	C₂×Dic₅₉	1	2	yes	—
135	C₂×M₁₆	—	2	1	no	—
136	C₂×M₁₆	C₅₉⋊C₈	1	1	yes	characteristic
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
139	M₃₂	—	2	1	no	—
140	M₃₂	—	2	2	no	—
141	M₃₂	—	2	2	no	—
142	M₃₂	C₂×Dic₅₉	1	2	yes	—
143	C₂×C₁₆	—	2	1	no	—
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
147	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
148	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
149	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
150	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
151	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
152	C₃₂	—	118	2	no	—
153	C₃₂	—	118	2	no	—
154	#41	—	2	1	no	—
155	#41	—	2	1	no	—
156	#41	C₅₉⋊C₁₆	1	2	yes	—
157	#41	C₂×C₅₉⋊C₈	1	2	yes	—
158	C₂×Q₈	C₅₉⋊C₁₆	1	1	yes	characteristic
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	C₅₉⋊C₁₆	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
163	M₁₆	—	2	1	no	—
164	M₁₆	—	2	2	no	—
165	M₁₆	—	2	2	no	—
166	M₁₆	C₂×C₅₉⋊C₈	1	2	yes	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
171	C₂×C₈	#11745664	1	1	yes	characteristic
172	C₂×C₈	#11745664	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
174	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
175	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
176	C₁₆	—	2	2	no	—
177	C₁₆	C₂²⋊Dic₅₉	1	1	yes	characteristic
178	C₁₆	C₂²⋊Dic₅₉	1	1	yes	characteristic
179	C₁₆	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
180	C₁₆	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
181	C₂³	—	2	1	no	—
182	C₂³	#14273667	1	1	yes	characteristic
183	Q₈	—	2	1	no	—
184	Q₈	C₂×C₅₉⋊C₁₆	1	2	yes	—
185	D₄	—	2	2	no	—
186	D₄	—	2	2	no	—
187	D₄	—	2	1	no	—
188	D₄	C₂×C₅₉⋊C₁₆	1	2	yes	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
193	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
194	C₂×C₄	C₂×C₅₉⋊C₁₆	1	1	yes	characteristic
195	C₂×C₄	C₂×C₅₉⋊C₁₆	1	1	yes	characteristic
196	C₂×C₄	#14273667	1	1	yes	characteristic
197	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
200	C₈	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
201	C₈	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
202	C₈	C₂×#11745664	1	1	yes	characteristic
203	C₂²	—	2	2	no	—
204	C₂²	—	4	1	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²⋊(C₅₉⋊C₁₆)	1	1	yes	characteristic
207	C₂²	C₂²⋊(C₅₉⋊C₁₆)	1	1	yes	characteristic
208	C₂²	C₂×#14273667	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	C₂²⋊(C₅₉⋊C₁₆)	1	1	yes	characteristic
211	C₄	C₂²⋊(C₅₉⋊C₁₆)	1	1	yes	characteristic
212	C₄	C₂×C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
213	C₄	C₂²×C₅₉⋊C₁₆	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	—	4	2	no	—
218	C₂	C₂×C₂²⋊(C₅₉⋊C₁₆)	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂×C₂²⋊D₅₉
commutator	C₄×C₅₉	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₅₉
Fitting	C₂×C₅₉×#504	C₂
socle	C₂×C₅₉	C₂×C₂²⋊C₁₆
prime radical	C₂×#41×C₅₉	C₈
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	C₂×#504	D₅₉
hypocenter	C₅₉	#13558
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#16837236	C₂	semidirect
2. C₂×#16837236	C₂	semidirect
3. #21422238	C₂	semidirect
8. #16837236	C₂²	semidirect
10. C₂×C₅₉⋊C₃₂	C₂²	semidirect
129. C₅₉	#13558	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837236	2
#21422238	2
C₂×C₅₉×#504	2
#13558	59