Subgroups — #28507160

Order 15104.

494 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837782	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837205	C₂	1	4	yes	maximal
4	C₄×#107×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₃₆	—	2	1	no	—
6	C₂²×Dic₂₃₆	C₂²	1	2	yes	—
7	C₂×#14273727	—	2	1	no	—
8	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
9	#16837782	—	2	2	no	—
10	#16837782	C₄	1	4	yes	complemented
11	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
12	#16837205	—	2	8	no	—
13	C₂×#107×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
16	C₅₉×#340	—	2	1	no	—
17	C₂²×Dic₁₁₈	—	2	2	no	—
18	C₂×Dic₂₃₆	—	2	4	no	—
19	C₂×Dic₂₃₆	—	2	4	no	—
20	C₂×Dic₂₃₆	—	2	4	no	—
21	C₂×Dic₂₃₆	—	2	2	no	—
22	C₂×Dic₂₃₆	—	4	4	no	—
23	C₂×Dic₂₃₆	C₂×C₄	1	4	yes	—
24	#14273727	—	2	4	no	—
25	#14273727	—	4	2	no	—
26	C₂×C₄⋊Dic₅₉	—	4	1	no	—
27	C₂×C₄⋊Dic₅₉	—	4	1	no	—
28	C₂×C₅₉⋊C₁₆	—	2	4	no	—
29	C₂×C₅₉⋊C₁₆	—	2	2	no	—
30	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
33	C₂×C₄²×C₅₉	—	4	1	no	—
34	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
35	#107×C₅₉	—	2	2	no	—
36	#107×C₅₉	D₄	1	4	yes	—
37	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₅₉	D₄	1	2	yes	—
39	#95×C₅₉	—	4	1	no	—
40	C₂²×C₄×C₅₉	—	4	2	no	—
41	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₅₉	—	4	2	no	—
45	C₂×Dic₁₁₈	—	2	4	no	—
46	C₂×Dic₁₁₈	—	4	4	no	—
47	C₂×Dic₁₁₈	—	4	4	no	—
48	C₂×Dic₁₁₈	—	4	2	no	—
49	C₂×C₈×C₅₉	—	2	2	no	—
50	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₅₉	#41	1	2	yes	—
55	C₄⋊C₄×C₅₉	—	2	4	no	—
56	C₄⋊C₄×C₅₉	—	4	2	no	—
57	C₄²×C₅₉	—	8	2	no	—
58	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
59	C₄²×C₅₉	SD₁₆	1	2	yes	—
60	C₄⋊Dic₅₉	—	4	4	no	—
61	C₄⋊Dic₅₉	—	4	4	no	—
62	Dic₂₃₆	—	2	8	no	—
63	Dic₂₃₆	—	4	8	no	—
64	Dic₂₃₆	—	4	2	no	—
65	Dic₂₃₆	—	4	2	no	—
66	C₅₉⋊C₁₆	—	4	2	no	—
67	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₅₉	—	2	2	no	—
69	C₂×C₄×C₅₉	—	4	4	no	—
70	C₂×C₄×C₅₉	—	8	2	no	—
71	C₂×C₄×C₅₉	—	8	2	no	—
72	C₂×C₄×C₅₉	#102	1	2	yes	—
73	C₂×C₄×C₅₉	#102	1	2	yes	—
74	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
75	C₂×C₄×C₅₉	Q₃₂	1	2	yes	—
76	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₉	#121	1	2	yes	—
78	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×Dic₅₉	—	4	4	no	—
81	C₂×Dic₅₉	—	8	4	no	—
82	Dic₁₁₈	—	4	4	no	—
83	Dic₁₁₈	—	4	4	no	—
84	Dic₁₁₈	—	8	4	no	—
85	C₈×C₅₉	—	2	2	no	—
86	C₈×C₅₉	C₄×D₄	1	2	yes	—
87	C₈×C₅₉	#124	1	1	yes	characteristic
88	C₈×C₅₉	#127	1	1	yes	characteristic
89	#13253	—	59	1	no	maximal, Sylow
90	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
91	C₂²×C₅₉	#459	1	2	yes	—
92	C₂²×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
93	C₂²×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
94	C₂²×C₅₉	#510	1	2	yes	—
95	C₄×C₅₉	—	2	2	no	—
96	C₄×C₅₉	—	8	4	no	—
97	C₄×C₅₉	#358	1	4	yes	complemented
98	C₄×C₅₉	#390	1	1	yes	characteristic
99	C₄×C₅₉	#418	1	1	yes	characteristic
100	C₄×C₅₉	#460	1	2	yes	—
101	Dic₅₉	—	8	4	no	—
102	C₂×#494	—	59	1	no	—
103	C₂×C₄⋊C₁₆	—	59	1	no	—
104	C₂×#358	—	59	4	no	—
105	C₄×#107	D₅₉	1	1	yes	characteristic
106	C₂×C₅₉	#1982	1	1	yes	characteristic
107	C₂×C₅₉	C₂×#358	1	1	yes	characteristic
108	C₂×C₅₉	#2141	1	1	yes	characteristic
109	C₂×C₅₉	#2208	1	2	yes	—
110	C₂×C₅₉	#2209	1	2	yes	—
111	C₂²×Q₁₆	—	59	2	no	—
112	C₂²×Q₁₆	—	118	1	no	—
113	C₂×C₄⋊Q₈	—	118	1	no	—
114	C₂²×C₁₆	—	59	2	no	—
115	#494	—	59	4	no	—
116	#494	—	118	2	no	—
117	C₂×#107	C₂×D₅₉	1	2	yes	—
118	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
119	C₄×C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₁₆	—	118	2	no	—
121	#358	—	118	8	no	—
122	#340	—	2	1	no	—
123	C₅₉	#13253	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	—	118	2	no	—
125	C₂×Q₁₆	—	59	4	no	—
126	C₂×Q₁₆	—	118	4	no	—
127	C₂×Q₁₆	—	118	4	no	—
128	C₂×Q₁₆	—	118	2	no	—
129	C₂×Q₁₆	—	118	4	no	—
130	C₂×Q₁₆	—	236	4	no	—
131	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
133	C₄⋊Q₈	—	118	4	no	—
134	C₄⋊Q₈	—	236	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	236	1	no	—
137	C₂×C₄⋊C₄	—	236	1	no	—
138	C₂×C₄²	—	4	1	no	—
139	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
140	C₂×C₁₆	—	118	4	no	—
141	C₂×C₁₆	—	118	2	no	—
142	#107	—	2	2	no	—
143	#107	D₂₃₆	1	4	yes	—
144	C₄×C₈	C₄×D₅₉	1	2	yes	—
145	C₄×C₈	D₂₃₆	1	2	yes	—
146	#95	—	4	1	no	—
147	C₂×Q₈	—	118	4	no	—
148	C₂×Q₈	—	236	4	no	—
149	C₂×Q₈	—	236	4	no	—
150	C₂×Q₈	—	236	2	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	—	236	2	no	—
153	C₂²×C₄	#11745669	1	1	yes	characteristic
154	C₂²×C₄	#11745669	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
156	Q₁₆	—	118	8	no	—
157	Q₁₆	—	236	8	no	—
158	Q₁₆	—	236	2	no	—
159	Q₁₆	—	236	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
162	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
163	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
164	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
165	C₂×C₈	#11745685	1	2	yes	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	236	4	no	—
169	C₄⋊C₄	—	236	4	no	—
170	C₄²	—	8	2	no	—
171	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
172	C₄²	#11745671	1	2	yes	—
173	C₁₆	—	236	2	no	—
174	C₂³	C₂×#11745669	1	1	yes	characteristic
175	Q₈	—	472	4	no	—
176	Q₈	—	236	4	no	—
177	Q₈	—	236	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	472	4	no	—
183	C₂×C₄	—	236	4	no	—
184	C₂×C₄	#14273647	1	2	yes	—
185	C₂×C₄	#14273647	1	2	yes	—
186	C₂×C₄	#14273671	1	2	yes	—
187	C₂×C₄	#14273673	1	2	yes	—
188	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
189	C₂×C₄	#14273701	1	2	yes	—
190	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
191	C₂×C₄	#14273752	1	2	yes	—
192	C₈	—	2	2	no	—
193	C₈	C₄×D₂₃₆	1	2	yes	—
194	C₈	#14273715	1	1	yes	characteristic
195	C₈	#14273724	1	1	yes	characteristic
196	C₂²	C₂×#14273647	1	1	yes	characteristic
197	C₂²	#16837597	1	2	yes	—
198	C₂²	C₂×#14273671	1	1	yes	characteristic
199	C₂²	C₂×#14273673	1	1	yes	characteristic
200	C₂²	#16837842	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	8	4	no	—
203	C₄	—	472	4	no	—
204	C₄	#16837205	1	4	yes	complemented
205	C₄	#16837304	1	1	yes	characteristic
206	C₄	#16837406	1	1	yes	characteristic
207	C₄	#16837602	1	2	yes	—
208	C₂	#21425023	1	1	yes	characteristic
209	C₂	C₂×#16837205	1	1	yes	characteristic
210	C₂	#21425815	1	1	yes	characteristic
211	C₂	#21426141	1	2	yes	—
212	C₂	#21426146	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₄×#107×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄×#107	D₅₉
hypocenter	C₅₉	#13253
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
10. #16837782	C₄	semidirect
97. C₄×C₅₉	#358	semidirect
123. C₅₉	#13253	semidirect
204. C₄	#16837205	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837782	2
C₂×C₄⋊(C₅₉⋊C₁₆)	2
C₂×#16837205	2
C₄×#107×C₅₉	2
#13253	59