Subgroups — #28487991

Order 15104.

642 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425024	C₂	1	2	yes	maximal, complemented
2	#21423963	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16837153	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16837146	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₉×#493	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×#14273648	—	2	4	no	—
9	C₂×#14273637	—	2	1	no	—
10	C₄×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
11	#16837156	—	2	1	no	—
12	#16837153	—	2	2	no	—
13	#16837146	—	2	4	no	—
14	C₂²×D₈×C₅₉	—	2	1	no	—
15	C₂²×D₈×C₅₉	—	2	1	no	—
16	C₂×#127×C₅₉	C₂²	1	2	yes	—
17	C₅₉×#493	—	2	2	no	—
18	C₅₉×#493	C₄	1	4	yes	complemented
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₅₉⋊C₈	—	2	1	no	—
21	C₂²×C₅₉⋊C₈	—	2	2	no	—
22	C₂×C₄⋊Dic₅₉	D₄	1	2	yes	—
23	C₂×C₄×Dic₅₉	—	2	1	no	—
24	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
27	C₄⋊(C₅₉⋊C₈)	D₄	1	2	yes	—
28	#14273648	—	4	8	no	—
29	#14273637	—	4	2	no	—
30	C₅₉⋊#95	—	2	1	no	—
31	C₂²×D₄×C₅₉	—	2	2	no	—
32	C₂²×D₄×C₅₉	—	2	2	no	—
33	C₂²×D₄×C₅₉	—	2	2	no	—
34	C₂×D₈×C₅₉	—	2	4	no	—
35	C₂×D₈×C₅₉	—	2	4	no	—
36	C₂×D₈×C₅₉	—	4	2	no	—
37	C₂×D₈×C₅₉	—	4	2	no	—
38	C₂×D₈×C₅₉	—	4	2	no	—
39	C₂×D₈×C₅₉	—	4	2	no	—
40	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
42	#127×C₅₉	—	2	2	no	—
43	#127×C₅₉	—	2	2	no	—
44	#127×C₅₉	—	2	2	no	—
45	#127×C₅₉	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₅₉	D₄	1	2	yes	—
49	C₂⁴×C₅₉	—	4	2	no	—
50	C₂×D₄×C₅₉	—	2	4	no	—
51	C₂×D₄×C₅₉	—	2	4	no	—
52	C₂×D₄×C₅₉	—	2	4	no	—
53	C₂×D₄×C₅₉	—	4	2	no	—
54	C₂×D₄×C₅₉	—	4	2	no	—
55	C₂×D₄×C₅₉	—	4	2	no	—
56	C₂×D₄×C₅₉	—	4	2	no	—
57	C₂×D₄×C₅₉	—	4	2	no	—
58	C₂×D₄×C₅₉	—	4	2	no	—
59	C₂×D₄×C₅₉	—	4	2	no	—
60	C₂×D₄×C₅₉	—	4	4	no	—
61	C₂×D₄×C₅₉	—	4	4	no	—
62	C₂×D₄×C₅₉	—	8	4	no	—
63	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
66	C₂²×Dic₅₉	—	2	2	no	—
67	D₈×C₅₉	—	4	8	no	—
68	D₈×C₅₉	—	4	8	no	—
69	C₂×C₈×C₅₉	—	2	1	no	—
70	C₂×C₈×C₅₉	—	2	1	no	—
71	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₅₉	D₈	1	2	yes	—
74	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
75	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₉	#41	1	1	yes	characteristic
79	C₄⋊Dic₅₉	—	2	4	no	—
80	C₄×Dic₅₉	—	4	1	no	—
81	C₄×Dic₅₉	—	4	1	no	—
82	C₂×C₅₉⋊C₈	—	2	2	no	—
83	C₂×C₅₉⋊C₈	—	4	1	no	—
84	C₂×C₅₉⋊C₈	—	4	1	no	—
85	C₂×C₅₉⋊C₈	—	4	2	no	—
86	C₂×C₅₉⋊C₈	—	4	4	no	—
87	C₂³×C₅₉	—	4	4	no	—
88	C₂³×C₅₉	—	8	2	no	—
89	C₂³×C₅₉	—	8	2	no	—
90	C₂³×C₅₉	—	8	4	no	—
91	C₂³×C₅₉	—	8	4	no	—
92	C₂³×C₅₉	#121	1	1	yes	characteristic
93	D₄×C₅₉	—	4	4	no	—
94	D₄×C₅₉	—	4	4	no	—
95	D₄×C₅₉	—	4	8	no	—
96	D₄×C₅₉	—	8	4	no	—
97	D₄×C₅₉	—	8	4	no	—
98	D₄×C₅₉	—	8	2	no	—
99	D₄×C₅₉	—	8	2	no	—
100	D₄×C₅₉	—	8	2	no	—
101	D₄×C₅₉	—	8	2	no	—
102	C₂×C₄×C₅₉	—	2	2	no	—
103	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
107	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
108	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
112	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×Dic₅₉	—	4	2	no	—
115	C₂×Dic₅₉	—	4	2	no	—
116	C₂×Dic₅₉	—	4	2	no	—
117	C₈×C₅₉	—	2	2	no	—
118	C₈×C₅₉	#102	1	4	yes	complemented
119	C₅₉⋊C₈	—	4	2	no	—
120	C₅₉⋊C₈	—	8	2	no	—
121	#9743	—	59	1	no	maximal, Sylow
122	C₂²×C₅₉	—	8	4	no	—
123	C₂²×C₅₉	—	8	4	no	—
124	C₂²×C₅₉	—	8	4	no	—
125	C₂²×C₅₉	—	16	4	no	—
126	C₂²×C₅₉	—	16	4	no	—
127	C₂²×C₅₉	#390	1	1	yes	characteristic
128	C₂²×C₅₉	#459	1	1	yes	characteristic
129	C₂²×C₅₉	#460	1	1	yes	characteristic
130	C₂²×C₅₉	#469	1	2	yes	—
131	C₂²×C₅₉	#472	1	2	yes	—
132	C₄×C₅₉	—	2	1	no	—
133	C₄×C₅₉	—	2	1	no	—
134	C₄×C₅₉	C₂×#102	1	1	yes	characteristic
135	C₄×C₅₉	#418	1	1	yes	characteristic
136	C₄×C₅₉	#442	1	2	yes	—
137	C₄×C₅₉	#459	1	1	yes	characteristic
138	C₄×C₅₉	#460	1	1	yes	characteristic
139	C₄×C₅₉	#496	1	2	yes	—
140	Dic₅₉	—	8	2	no	—
141	C₂×#493	D₅₉	1	1	yes	characteristic
142	#1982	—	59	2	no	—
143	#1770	—	59	1	no	—
144	C₂×#334	—	59	1	no	—
145	C₂×#331	—	59	2	no	—
146	C₂×C₅₉	—	16	4	no	—
147	C₂×C₅₉	#1687	1	2	yes	—
148	C₂×C₅₉	#1688	1	2	yes	—
149	C₂×C₅₉	#1972	1	1	yes	characteristic
150	C₂×C₅₉	#1978	1	1	yes	characteristic
151	C₂×C₅₉	#1982	1	1	yes	characteristic
152	C₂²×D₈	—	2	1	no	—
153	C₂²×D₈	—	2	1	no	—
154	C₂×#127	C₂×D₅₉	1	2	yes	—
155	#493	—	2	2	no	—
156	#493	Dic₅₉	1	4	yes	complemented
157	C₂×C₄⋊C₈	—	59	1	no	—
158	C₂×C₄⋊C₈	—	59	2	no	—
159	C₂×#102	—	118	4	no	—
160	C₂×#97	—	118	1	no	—
161	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
162	C₄×C₄⋊C₄	—	59	1	no	—
163	#336	—	118	1	no	—
164	#334	—	118	2	no	—
165	#331	—	118	4	no	—
166	C₅₉	#9743	1	1	yes	characteristic, complemented, Sylow
167	C₂²×D₄	—	2	2	no	—
168	C₂²×D₄	—	2	2	no	—
169	C₂²×D₄	—	2	2	no	—
170	C₂×D₈	—	2	4	no	—
171	C₂×D₈	—	2	4	no	—
172	C₂×D₈	—	4	2	no	—
173	C₂×D₈	—	4	2	no	—
174	C₂×D₈	—	4	2	no	—
175	C₂×D₈	—	4	2	no	—
176	C₂²×C₈	—	118	1	no	—
177	C₂²×C₈	—	118	2	no	—
178	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
179	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
180	#127	—	2	2	no	—
181	#127	—	2	2	no	—
182	#127	—	2	2	no	—
183	#127	C₂×Dic₅₉	1	4	yes	—
184	C₂×C₄⋊C₄	—	59	2	no	—
185	C₂×C₄²	—	118	1	no	—
186	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
187	C₄⋊C₈	—	59	2	no	—
188	C₄⋊C₈	—	118	2	no	—
189	C₄⋊C₈	—	118	1	no	—
190	C₄⋊C₈	—	118	2	no	—
191	#102	—	236	8	no	—
192	#97	—	236	2	no	—
193	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
194	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
195	#95	—	118	1	no	—
196	C₂⁴	—	4	2	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	4	2	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	4	no	—
206	C₂×D₄	—	4	4	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	8	4	no	—
210	C₂²×C₄	—	118	2	no	—
211	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
212	C₂²×C₄	#11745687	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
214	D₈	—	4	8	no	—
215	D₈	—	4	8	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	118	2	no	—
219	C₂×C₈	—	236	2	no	—
220	C₂×C₈	—	236	4	no	—
221	C₂×C₈	—	236	1	no	—
222	C₂×C₈	—	236	1	no	—
223	C₂×C₈	#11745669	1	2	yes	—
224	C₂×C₈	#11745670	1	2	yes	—
225	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
226	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
227	C₄⋊C₄	—	118	4	no	—
228	C₄²	—	236	1	no	—
229	C₄²	—	236	1	no	—
230	C₄²	D₄×D₅₉	1	1	yes	characteristic
231	C₄²	#11745687	1	1	yes	characteristic
232	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
233	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
234	C₂³	—	4	4	no	—
235	C₂³	—	8	4	no	—
236	C₂³	—	8	4	no	—
237	C₂³	—	8	2	no	—
238	C₂³	—	8	2	no	—
239	C₂³	#14273698	1	1	yes	characteristic
240	D₄	—	4	4	no	—
241	D₄	—	4	4	no	—
242	D₄	—	4	8	no	—
243	D₄	—	8	4	no	—
244	D₄	—	8	4	no	—
245	D₄	—	8	2	no	—
246	D₄	—	8	2	no	—
247	D₄	—	8	2	no	—
248	D₄	—	8	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	236	2	no	—
251	C₂×C₄	—	236	2	no	—
252	C₂×C₄	—	236	2	no	—
253	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
254	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
255	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
256	C₂×C₄	#14273698	1	1	yes	characteristic
257	C₂×C₄	#14273714	1	1	yes	characteristic
258	C₂×C₄	#14273723	1	1	yes	characteristic
259	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
260	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
261	C₂×C₄	#14273751	1	4	yes	—
262	C₂×C₄	#14273753	1	1	yes	characteristic
263	C₂×C₄	#14273753	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	472	2	no	—
266	C₈	—	236	2	no	—
267	C₈	#14273648	1	4	yes	complemented
268	C₂²	—	8	4	no	—
269	C₂²	—	8	4	no	—
270	C₂²	—	8	4	no	—
271	C₂²	—	16	4	no	—
272	C₂²	—	16	4	no	—
273	C₂²	#16837305	1	1	yes	characteristic
274	C₂²	#16837594	1	1	yes	characteristic
275	C₂²	#16837599	1	1	yes	characteristic
276	C₂²	#16837652	1	2	yes	—
277	C₂²	#16837669	1	2	yes	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	472	2	no	—
281	C₄	C₂×#14273648	1	1	yes	characteristic
282	C₄	#16837407	1	1	yes	characteristic
283	C₄	#16837514	1	2	yes	—
284	C₄	#16837594	1	1	yes	characteristic
285	C₄	#16837599	1	1	yes	characteristic
286	C₄	#16837790	1	2	yes	—
287	C₂	—	16	4	no	—
288	C₂	#21423572	1	2	yes	—
289	C₂	#21423577	1	2	yes	—
290	C₂	#21424973	1	1	yes	characteristic
291	C₂	#21425007	1	1	yes	characteristic
292	C₂	#21425024	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273698
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting, prime radical	C₂×C₅₉×#493	C₂
socle	C₂³×C₅₉	#121
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#493	D₅₉
hypocenter	C₅₉	#9743
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #21425024	C₂	semidirect
2. #21423963	C₂	semidirect
3. C₂×#16837153	C₂	semidirect
4. C₂×#16837146	C₂	semidirect
18. C₅₉×#493	C₄	semidirect
118. C₈×C₅₉	#102	semidirect
156. #493	Dic₅₉	semidirect
166. C₅₉	#9743	semidirect
267. C₈	#14273648	semidirect

Maximal subgroups

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

subgroup	index
#21425024	2
#21423963	2
C₂×#16837153	2
C₂×#16837146	2
C₂×C₅₉×#493	2
#9743	59