Subgroups — #28487388

Order 15104.

298 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424871	C₂	1	2	yes	maximal
2	#21422852	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837153	C₂	1	1	yes	maximal, characteristic
4	#21421992	C₂	1	2	yes	maximal
5	C₈×M₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273662	—	2	2	no	—
7	#16837426	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	#16837166	—	2	2	no	—
10	#16837153	—	2	1	no	—
11	#16837153	—	2	1	no	—
12	#16837153	C₄	1	2	yes	—
13	#16837153	C₄	1	2	yes	—
14	C₅₉×#431	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#322	—	2	1	no	—
18	C₈²×C₅₉	—	2	1	no	—
19	C₂×#11745664	—	2	2	no	—
20	C₂²×C₅₉⋊C₈	—	2	2	no	—
21	#14273662	—	4	4	no	—
22	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
26	C₂×M₁₆×C₅₉	D₄	1	2	yes	—
27	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
28	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₉	—	2	1	no	—
31	C₂²⋊C₈×C₅₉	Q₈	1	2	yes	—
32	#97×C₅₉	—	2	1	no	—
33	C₄×C₈×C₅₉	—	2	1	no	—
34	C₄×C₈×C₅₉	—	2	1	no	—
35	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
38	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
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	—	2	2	no	—
43	M₁₆×C₅₉	SD₁₆	1	4	yes	—
44	C₂×C₈×C₅₉	—	2	2	no	—
45	C₂×C₈×C₅₉	—	2	2	no	—
46	C₂×C₈×C₅₉	—	2	2	no	—
47	C₂×C₈×C₅₉	—	4	1	no	—
48	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₅₉	#41	1	1	yes	characteristic
55	C₄²×C₅₉	#41	1	1	yes	characteristic
56	#11745664	—	4	2	no	—
57	#11745664	—	4	2	no	—
58	C₂×C₅₉⋊C₈	—	2	4	no	—
59	C₂×C₅₉⋊C₈	—	4	2	no	—
60	C₂×C₅₉⋊C₈	—	4	2	no	—
61	C₂×C₅₉⋊C₈	—	4	2	no	—
62	C₂³×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅₉	—	2	1	no	—
64	C₂×C₄×C₅₉	—	2	1	no	—
65	C₂×C₄×C₅₉	—	4	1	no	—
66	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₉	#125	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₅₉	#135	1	2	yes	—
74	C₈×C₅₉	—	2	1	no	—
75	C₈×C₅₉	—	2	1	no	—
76	C₈×C₅₉	—	2	1	no	—
77	C₈×C₅₉	—	2	4	no	—
78	C₈×C₅₉	—	4	2	no	—
79	C₈×C₅₉	#108	1	2	yes	complemented
80	C₅₉⋊C₈	—	4	4	no	—
81	C₅₉⋊C₈	—	8	2	no	—
82	#9623	—	59	1	no	maximal, Sylow
83	C₂²×C₅₉	—	2	1	no	—
84	C₂²×C₅₉	—	2	1	no	—
85	C₂²×C₅₉	#389	1	1	yes	characteristic
86	C₂²×C₅₉	#498	1	1	yes	characteristic
87	C₂²×C₅₉	#499	1	1	yes	characteristic
88	C₄×C₅₉	—	2	1	no	—
89	C₄×C₅₉	—	2	1	no	—
90	C₄×C₅₉	—	2	1	no	—
91	C₄×C₅₉	C₂×#108	1	1	yes	characteristic
92	C₄×C₅₉	#430	1	1	yes	characteristic
93	C₄×C₅₉	C₄×SD₁₆	1	2	yes	—
94	C₄×C₅₉	#465	1	2	yes	—
95	#1950	—	59	2	no	—
96	#1566	—	59	1	no	—
97	C₂×#334	—	59	1	no	—
98	C₈×M₁₆	D₅₉	1	1	yes	characteristic
99	#1301	—	59	2	no	—
100	C₂×C₅₉	—	2	1	no	—
101	C₂×C₅₉	—	2	1	no	—
102	C₂×C₅₉	#1941	1	1	yes	characteristic
103	C₂×C₅₉	#1950	1	1	yes	characteristic
104	C₂×C₅₉	#1953	1	1	yes	characteristic
105	#431	C₂×D₅₉	1	1	yes	characteristic
106	C₂×#108	—	118	2	no	—
107	#423	—	59	2	no	—
108	C₂×C₄⋊C₈	—	59	2	no	—
109	C₄×M₁₆	C₂×D₅₉	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
111	#341	—	118	2	no	—
112	#334	—	59	2	no	—
113	#334	—	59	2	no	—
114	#334	—	118	1	no	—
115	#334	—	118	1	no	—
116	#322	—	2	1	no	—
117	C₈²	—	2	1	no	—
118	C₅₉	#9623	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₁₆	—	118	2	no	—
120	C₂×M₁₆	D₂₃₆	1	2	yes	—
121	C₂²×C₈	—	118	2	no	—
122	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
123	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
124	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
125	#108	—	236	4	no	—
126	C₄⋊C₈	—	2	1	no	—
127	C₄⋊C₈	—	59	4	no	—
128	C₄⋊C₈	—	118	2	no	—
129	C₄⋊C₈	—	118	2	no	—
130	C₄⋊C₈	—	118	2	no	—
131	C₂²⋊C₈	Dic₁₁₈	1	2	yes	—
132	#97	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₄×C₈	—	2	1	no	—
135	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
136	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
139	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
142	M₁₆	—	2	2	no	—
143	M₁₆	—	236	2	no	—
144	M₁₆	—	236	2	no	—
145	M₁₆	#11745660	1	4	yes	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	118	4	no	—
151	C₂×C₈	—	236	2	no	—
152	C₂×C₈	—	236	2	no	—
153	C₂×C₈	—	236	2	no	—
154	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
155	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
156	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
157	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
158	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
159	C₄²	#11745685	1	1	yes	characteristic
160	C₄²	#11745685	1	1	yes	characteristic
161	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
162	C₂³	#14273727	1	1	yes	characteristic
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
167	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₁₁₈	1	1	yes	characteristic
169	C₂×C₄	#14273701	1	1	yes	characteristic
170	C₂×C₄	#14273706	1	1	yes	characteristic
171	C₂×C₄	#14273719	1	1	yes	characteristic
172	C₂×C₄	C₂×#11745660	1	2	yes	—
173	C₂×C₄	#14273749	1	2	yes	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	4	no	—
178	C₈	—	4	2	no	—
179	C₈	—	472	2	no	—
180	C₈	—	236	4	no	—
181	C₈	#14273662	1	2	yes	complemented
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	#16837300	1	1	yes	characteristic
185	C₂²	#16837800	1	1	yes	characteristic
186	C₂²	#16837804	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	C₂×#14273662	1	1	yes	characteristic
191	C₄	#16837450	1	1	yes	characteristic
192	C₄	C₄×#11745660	1	2	yes	—
193	C₄	#16837631	1	2	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#21424824	1	1	yes	characteristic
197	C₂	#21424871	1	1	yes	characteristic
198	C₂	#21424886	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273701
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₈×M₁₆×C₅₉	C₂
socle	C₂²×C₅₉	#389
prime radical	C₂³×C₅₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₈×M₁₆	D₅₉
hypocenter	C₅₉	#9623
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
79. C₈×C₅₉	#108	semidirect
118. C₅₉	#9623	semidirect
181. C₈	#14273662	semidirect

Maximal subgroups

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

subgroup	index
#21424871	2
#21422852	2
C₂×#16837153	2
#21421992	2
C₈×M₁₆×C₅₉	2
#9623	59