Subgroups — #28479928

Order 15104.

414 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14273659	C₂	1	1	yes	maximal, characteristic
2	#21423963	C₂	1	2	yes	maximal
3	C₂×#16837153	C₂	1	1	yes	maximal, characteristic
4	#21421936	C₂	1	2	yes	maximal
5	C₂×C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273659	C₄	1	2	yes	—
7	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×#14273637	—	2	2	no	—
9	C₄×C₄⋊Dic₅₉	C₂²	1	2	yes	—
10	#16837165	C₄	1	2	yes	—
11	#16837156	—	2	2	no	—
12	#16837153	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₈²×C₅₉	C₄	1	4	yes	complemented
17	C₂²×C₅₉⋊C₈	—	2	2	no	—
18	C₂×C₄⋊Dic₅₉	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₅₉	—	2	2	no	—
20	#14273659	—	2	4	no	—
21	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
22	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
23	#14273637	—	4	4	no	—
24	C₅₉⋊#95	—	2	2	no	—
25	C₂²×C₈×C₅₉	—	2	1	no	—
26	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₅₉	—	2	1	no	—
32	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₅₉	Q₈	1	2	yes	—
37	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
40	C₂²×Dic₅₉	—	2	4	no	—
41	C₂×C₈×C₅₉	—	2	4	no	—
42	C₂×C₈×C₅₉	—	2	2	no	—
43	C₂×C₈×C₅₉	—	2	2	no	—
44	C₂×C₈×C₅₉	—	2	2	no	—
45	C₂×C₈×C₅₉	C₄²	1	2	yes	—
46	C₂×C₈×C₅₉	C₄²	1	2	yes	—
47	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₅₉	M₁₆	1	4	yes	—
50	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
51	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₅₉	#41	1	1	yes	characteristic
55	C₄⋊Dic₅₉	—	2	4	no	—
56	C₄⋊Dic₅₉	—	2	4	no	—
57	C₄×Dic₅₉	—	4	2	no	—
58	C₄×Dic₅₉	—	4	2	no	—
59	C₂×C₅₉⋊C₈	—	2	4	no	—
60	C₂×C₅₉⋊C₈	—	4	2	no	—
61	C₂×C₅₉⋊C₈	—	4	2	no	—
62	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	—	2	2	no	—
64	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#117	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	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₅₉	#131	1	2	yes	—
72	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₅₉	#135	1	2	yes	—
74	C₂×Dic₅₉	—	4	4	no	—
75	C₂×Dic₅₉	—	4	4	no	—
76	C₂×Dic₅₉	—	4	4	no	—
77	C₈×C₅₉	—	2	4	no	—
78	C₈×C₅₉	—	2	2	no	—
79	C₈×C₅₉	—	2	2	no	—
80	C₈×C₅₉	#97	1	4	yes	complemented
81	C₈×C₅₉	#106	1	4	yes	complemented
82	C₅₉⋊C₈	—	4	4	no	—
83	#8283	—	59	1	no	maximal, Sylow
84	C₂²×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₅₉	#436	1	2	yes	—
86	C₂²×C₅₉	C₄×SD₁₆	1	2	yes	—
87	C₂²×C₅₉	#443	1	2	yes	—
88	C₄×C₅₉	—	2	1	no	—
89	C₄×C₅₉	—	2	1	no	—
90	C₄×C₅₉	C₂×#97	1	1	yes	characteristic
91	C₄×C₅₉	#405	1	1	yes	characteristic
92	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
93	C₄×C₅₉	#427	1	1	yes	characteristic
94	C₄×C₅₉	C₄×SD₁₆	1	2	yes	—
95	C₄×C₅₉	#446	1	2	yes	—
96	Dic₅₉	—	8	4	no	—
97	C₄×#106	—	59	1	no	—
98	#1770	—	59	2	no	—
99	C₂×#334	—	59	1	no	—
100	C₂×C₈²	D₅₉	1	1	yes	characteristic
101	#1279	—	59	2	no	—
102	C₂×C₅₉	#1590	1	4	yes	—
103	C₂×C₅₉	#1770	1	1	yes	characteristic
104	C₂×C₅₉	#1772	1	1	yes	characteristic
105	C₂×C₅₉	C₄×#106	1	1	yes	characteristic
106	C₂×#106	—	59	2	no	—
107	C₂×C₄⋊C₈	—	59	2	no	—
108	C₂×#97	—	118	2	no	—
109	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
112	C₄×C₄⋊C₄	—	59	2	no	—
113	#340	—	59	2	no	—
114	#336	—	118	2	no	—
115	#334	—	59	4	no	—
116	C₈²	Dic₅₉	1	4	yes	complemented
117	C₅₉	#8283	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₈	—	2	1	no	—
119	C₂²×C₈	—	118	2	no	—
120	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
121	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
122	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
123	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
124	C₂×C₄⋊C₄	—	59	4	no	—
125	C₂×C₄²	—	118	2	no	—
126	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
127	#106	—	118	4	no	—
128	C₄⋊C₈	—	59	4	no	—
129	C₄⋊C₈	—	118	2	no	—
130	#97	—	236	4	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	Dic₁₁₈	1	2	yes	—
133	C₄×C₈	C₄×D₅₉	1	2	yes	—
134	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
135	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
136	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
137	#95	—	118	2	no	—
138	C₂²×C₄	—	118	4	no	—
139	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
141	C₂²×C₄	#11745685	1	1	yes	characteristic
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	118	4	no	—
147	C₂×C₈	—	236	2	no	—
148	C₂×C₈	—	236	2	no	—
149	C₂×C₈	#11745659	1	4	yes	—
150	C₂×C₈	#11745660	1	4	yes	—
151	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
152	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
153	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
154	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
155	C₄⋊C₄	—	118	4	no	—
156	C₄⋊C₄	—	118	4	no	—
157	C₄²	—	236	2	no	—
158	C₄²	—	236	2	no	—
159	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
160	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
161	C₄²	#11745685	1	1	yes	characteristic
162	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
163	C₂³	C₄×D₂₃₆	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	236	4	no	—
166	C₂×C₄	—	236	4	no	—
167	C₂×C₄	—	236	4	no	—
168	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
169	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
170	C₂×C₄	#14273686	1	1	yes	characteristic
171	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
172	C₂×C₄	C₄×Dic₁₁₈	1	1	yes	characteristic
173	C₂×C₄	C₄×Dic₁₁₈	1	1	yes	characteristic
174	C₂×C₄	C₂×#11745659	1	2	yes	—
175	C₂×C₄	#14273733	1	2	yes	—
176	C₂×C₄	C₂×#11745660	1	2	yes	—
177	C₂×C₄	#14273749	1	2	yes	—
178	C₈	—	2	4	no	—
179	C₈	—	2	2	no	—
180	C₈	—	2	2	no	—
181	C₈	—	236	4	no	—
182	C₈	#14273637	1	4	yes	complemented
183	C₈	#14273659	1	4	yes	complemented
184	C₂²	C₄×C₄⋊Dic₅₉	1	1	yes	characteristic
185	C₂²	#16837481	1	2	yes	—
186	C₂²	C₄×#11745660	1	2	yes	—
187	C₂²	#16837520	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	472	4	no	—
191	C₄	C₂×#14273637	1	1	yes	characteristic
192	C₄	#16837357	1	1	yes	characteristic
193	C₄	C₂×#14273659	1	1	yes	characteristic
194	C₄	#16837440	1	1	yes	characteristic
195	C₄	C₄×#11745660	1	2	yes	—
196	C₄	#16837532	1	2	yes	—
197	C₂	#21422988	1	4	yes	—
198	C₂	#21423963	1	1	yes	characteristic
199	C₂	#21423975	1	1	yes	characteristic
200	C₂	C₄×#14273659	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₈²×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×C₈²	D₅₉
hypocenter	C₅₉	#8283
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
12. #16837153	C₄	semidirect
16. C₈²×C₅₉	C₄	semidirect
80. C₈×C₅₉	#97	semidirect
81. C₈×C₅₉	#106	semidirect
116. C₈²	Dic₅₉	semidirect
117. C₅₉	#8283	semidirect
182. C₈	#14273637	semidirect
183. C₈	#14273659	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14273659	2
#21423963	2
C₂×#16837153	2
#21421936	2
C₂×C₈²×C₅₉	2
#8283	59