Subgroups — #28482338

Order 15104.

310 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21423782	C₂	1	1	yes	maximal, characteristic
2	#21422844	C₂	1	1	yes	maximal, characteristic
3	#21421940	C₂	1	2	yes	maximal
4	#21421924	C₂	1	2	yes	maximal
5	C₅₉×#1564	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	C₂×#14273638	C₂²	1	1	yes	characteristic
8	C₂×#14273637	C₄	1	1	yes	characteristic
9	C₂×#14273637	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₉	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₉	C₄	1	1	yes	characteristic
12	#16837248	C₂²	1	1	yes	characteristic
13	#16837155	—	2	1	no	—
14	#16837153	—	2	1	no	—
15	C₅₉×#431	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#335	—	2	1	no	—
18	C₅₉×#334	—	2	1	no	—
19	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
25	#14273638	—	2	1	no	—
26	#14273638	—	2	1	no	—
27	#14273637	—	2	2	no	—
28	#14273637	C₈	1	4	yes	complemented
29	C₄×C₅₉⋊C₈	—	2	2	no	—
30	C₈×Dic₅₉	—	2	2	no	—
31	C₈×Dic₅₉	—	2	2	no	—
32	C₅₉⋊#95	—	2	1	no	—
33	C₅₉⋊#95	—	2	1	no	—
34	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₅₉	—	2	2	no	—
38	C₂²⋊C₈×C₅₉	—	2	2	no	—
39	C₄×C₈×C₅₉	—	2	2	no	—
40	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
42	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
43	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
47	C₂²×Dic₅₉	—	2	1	no	—
48	C₂²×Dic₅₉	—	2	1	no	—
49	C₂×C₈×C₅₉	—	2	1	no	—
50	C₂×C₈×C₅₉	—	2	1	no	—
51	C₂×C₈×C₅₉	—	4	2	no	—
52	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
58	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₅₉	#41	1	1	yes	characteristic
61	C₄²×C₅₉	#41	1	1	yes	characteristic
62	C₄×Dic₅₉	—	2	1	no	—
63	C₄×Dic₅₉	—	2	1	no	—
64	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
65	C₄×Dic₅₉	M₁₆	1	2	yes	—
66	C₂×C₅₉⋊C₈	—	2	1	no	—
67	C₂×C₅₉⋊C₈	—	2	1	no	—
68	C₂×C₅₉⋊C₈	—	2	1	no	—
69	C₂×C₅₉⋊C₈	—	2	1	no	—
70	C₂×C₅₉⋊C₈	—	4	2	no	—
71	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
72	C₂×C₅₉⋊C₈	M₁₆	1	2	yes	—
73	C₂³×C₅₉	#117	1	1	yes	characteristic
74	C₂×C₄×C₅₉	—	4	1	no	—
75	C₂×C₄×C₅₉	—	4	1	no	—
76	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
78	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
79	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
80	C₂×C₄×C₅₉	#125	1	1	yes	characteristic
81	C₂×C₄×C₅₉	#125	1	1	yes	characteristic
82	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
86	C₂×Dic₅₉	—	2	2	no	—
87	C₂×Dic₅₉	—	2	2	no	—
88	C₂×Dic₅₉	—	4	1	no	—
89	C₂×Dic₅₉	—	4	1	no	—
90	C₂×Dic₅₉	—	4	1	no	—
91	C₂×Dic₅₉	—	4	1	no	—
92	C₈×C₅₉	—	2	1	no	—
93	C₈×C₅₉	—	2	1	no	—
94	C₈×C₅₉	—	8	2	no	—
95	C₈×C₅₉	C₄⋊C₈	1	2	yes	—
96	C₈×C₅₉	C₄⋊C₈	1	2	yes	—
97	C₅₉⋊C₈	—	2	2	no	—
98	C₅₉⋊C₈	—	2	2	no	—
99	C₅₉⋊C₈	—	8	2	no	—
100	#8781	—	59	1	no	maximal, Sylow
101	C₂²×C₅₉	—	2	1	no	—
102	C₂²×C₅₉	—	2	1	no	—
103	C₂²×C₅₉	#382	1	1	yes	characteristic
104	C₂²×C₅₉	#431	1	1	yes	characteristic
105	C₂²×C₅₉	#432	1	1	yes	characteristic
106	C₄×C₅₉	—	2	1	no	—
107	C₄×C₅₉	—	2	1	no	—
108	C₄×C₅₉	—	2	1	no	—
109	C₄×C₅₉	—	2	1	no	—
110	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
111	C₄×C₅₉	#423	1	1	yes	characteristic
112	C₄×C₅₉	#431	1	1	yes	characteristic
113	C₄×C₅₉	#432	1	1	yes	characteristic
114	Dic₅₉	—	4	2	no	—
115	Dic₅₉	—	4	2	no	—
116	#1726	—	59	1	no	—
117	#1564	—	59	1	no	—
118	#1564	D₅₉	1	1	yes	characteristic
119	#1281	—	59	2	no	—
120	#1274	—	59	2	no	—
121	C₂×C₅₉	—	2	1	no	—
122	C₂×C₅₉	—	2	1	no	—
123	C₂×C₅₉	#1831	1	1	yes	characteristic
124	C₂×C₅₉	#1833	1	1	yes	characteristic
125	C₂×C₅₉	#1833	1	1	yes	characteristic
126	#431	—	59	2	no	—
127	#431	C₂×D₅₉	1	2	yes	—
128	C₂×#97	—	59	1	no	—
129	C₂×#97	—	59	1	no	—
130	C₂×#97	—	59	1	no	—
131	C₂×C₄×C₈	—	59	1	no	—
132	C₂×C₄×C₈	—	59	1	no	—
133	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
134	#376	—	59	1	no	—
135	#335	—	2	1	no	—
136	#335	—	118	1	no	—
137	#334	—	2	1	no	—
138	#334	—	118	1	no	—
139	C₅₉	#8781	1	1	yes	characteristic, complemented, Sylow
140	C₂²×C₈	—	59	1	no	—
141	C₂²×C₈	—	59	1	no	—
142	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
143	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
144	C₂×C₄²	—	59	1	no	—
145	C₂×C₄²	—	59	1	no	—
146	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	118	2	no	—
149	C₂²⋊C₈	—	2	2	no	—
150	C₂²⋊C₈	—	118	2	no	—
151	#97	—	59	4	no	—
152	#97	—	118	2	no	—
153	#97	—	118	1	no	—
154	#97	—	118	1	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	118	2	no	—
157	C₄×C₈	—	118	2	no	—
158	C₄×C₈	—	118	2	no	—
159	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
160	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
163	#95	—	118	1	no	—
164	#95	—	118	1	no	—
165	C₂²×C₄	—	118	1	no	—
166	C₂²×C₄	—	118	1	no	—
167	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
168	C₂²×C₄	#11745687	1	1	yes	characteristic
169	C₂²×C₄	#11745689	1	1	yes	characteristic
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	59	2	no	—
174	C₂×C₈	—	59	2	no	—
175	C₂×C₈	—	118	1	no	—
176	C₂×C₈	—	118	1	no	—
177	C₂×C₈	—	118	1	no	—
178	C₂×C₈	—	118	1	no	—
179	C₂×C₈	—	236	2	no	—
180	C₂×C₈	C₈×D₅₉	1	2	yes	—
181	C₂×C₈	#11745659	1	2	yes	—
182	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
183	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
184	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
185	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
186	C₄²	—	59	2	no	—
187	C₄²	—	59	2	no	—
188	C₄²	—	118	1	no	—
189	C₄²	—	118	1	no	—
190	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
191	C₄²	#11745687	1	1	yes	characteristic
192	C₄²	#11745689	1	1	yes	characteristic
193	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
194	C₂³	#14273685	1	1	yes	characteristic
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	118	2	no	—
198	C₂×C₄	—	118	2	no	—
199	C₂×C₄	—	236	1	no	—
200	C₂×C₄	—	236	1	no	—
201	C₂×C₄	—	236	1	no	—
202	C₂×C₄	—	236	1	no	—
203	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
204	C₂×C₄	#14273685	1	1	yes	characteristic
205	C₂×C₄	#14273714	1	1	yes	characteristic
206	C₂×C₄	#14273716	1	1	yes	characteristic
207	C₂×C₄	#14273718	1	1	yes	characteristic
208	C₂×C₄	#14273718	1	1	yes	characteristic
209	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
210	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
211	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
212	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	—	8	2	no	—
216	C₈	—	118	2	no	—
217	C₈	—	118	2	no	—
218	C₈	—	472	2	no	—
219	C₈	Dic₅₉⋊C₈	1	2	yes	—
220	C₈	Dic₅₉⋊C₈	1	2	yes	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	#16837267	1	1	yes	characteristic
224	C₂²	#16837454	1	1	yes	characteristic
225	C₂²	#16837458	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	236	2	no	—
231	C₄	—	236	2	no	—
232	C₄	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
233	C₄	#16837425	1	1	yes	characteristic
234	C₄	#16837454	1	1	yes	characteristic
235	C₄	#16837458	1	1	yes	characteristic
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	#21424212	1	1	yes	characteristic
239	C₂	#21424219	1	1	yes	characteristic
240	C₂	#21424221	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273685
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1564	C₂
socle	C₂²×C₅₉	#382
prime radical	C₂³×C₅₉	#117
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1564	D₅₉
hypocenter	C₅₉	#8781
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
28. #14273637	C₈	semidirect
139. C₅₉	#8781	semidirect

Maximal subgroups

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

subgroup	index
#21423782	2
#21422844	2
#21421940	2
#21421924	2
C₅₉×#1564	2
#8781	59