Subgroups — #28487553

Order 15104.

606 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424891	C₂	1	2	yes	maximal, complemented
2	C₂×#16837155	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊(C₅₉⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×D₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
8	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	#16837155	—	2	2	no	—
11	D₄⋊(C₅₉⋊C₈)	—	2	4	no	—
12	C₈×C₅₉⋊C₈	—	2	2	no	—
13	C₂²×D₈×C₅₉	C₄	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₉	C₂²	1	2	yes	—
15	C₄×D₈×C₅₉	—	2	2	no	—
16	C₄×D₈×C₅₉	—	2	2	no	—
17	C₂×#107×C₅₉	C₄	1	1	yes	characteristic
18	C₂×#102×C₅₉	C₄	1	2	yes	—
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	C₂²×C₅₉⋊C₈	D₄	1	2	yes	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₅₉⋊C₈)	—	4	8	no	—
26	C₄×C₅₉⋊C₈	—	2	1	no	—
27	C₄×C₅₉⋊C₈	—	2	1	no	—
28	C₄×C₅₉⋊C₈	—	2	1	no	—
29	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
30	C₂²×D₄×C₅₉	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₅₉	—	2	2	no	—
32	C₂×D₈×C₅₉	—	2	2	no	—
33	C₂×D₈×C₅₉	—	2	2	no	—
34	C₂×D₈×C₅₉	C₈	1	4	yes	complemented
35	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₄×D₄×C₅₉	—	2	2	no	—
38	C₄×D₄×C₅₉	—	2	2	no	—
39	C₄×D₄×C₅₉	—	2	2	no	—
40	C₄×D₄×C₅₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
43	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
44	#107×C₅₉	—	2	2	no	—
45	#102×C₅₉	—	2	4	no	—
46	C₄×C₈×C₅₉	D₄	1	2	yes	—
47	C₄×C₈×C₅₉	D₄	1	2	yes	—
48	C₂⁴×C₅₉	—	2	2	no	—
49	C₂×D₄×C₅₉	—	2	2	no	—
50	C₂×D₄×C₅₉	—	2	2	no	—
51	C₂×D₄×C₅₉	—	2	2	no	—
52	C₂×D₄×C₅₉	—	2	2	no	—
53	C₂×D₄×C₅₉	—	2	2	no	—
54	C₂×D₄×C₅₉	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₅₉	—	2	2	no	—
56	C₂²×C₄×C₅₉	—	4	2	no	—
57	C₂²×C₄×C₅₉	—	4	2	no	—
58	C₂²×C₄×C₅₉	—	4	4	no	—
59	C₂²×C₄×C₅₉	—	4	4	no	—
60	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
63	D₈×C₅₉	—	2	8	no	—
64	C₂×C₈×C₅₉	—	2	1	no	—
65	C₂×C₈×C₅₉	—	2	1	no	—
66	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
70	C₄⋊C₄×C₅₉	—	2	2	no	—
71	C₄⋊C₄×C₅₉	—	2	2	no	—
72	C₂²⋊C₄×C₅₉	—	4	2	no	—
73	C₂²⋊C₄×C₅₉	—	4	2	no	—
74	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₉	#41	1	1	yes	characteristic
78	C₂×C₅₉⋊C₈	—	2	4	no	—
79	C₂×C₅₉⋊C₈	—	2	2	no	—
80	C₂×C₅₉⋊C₈	—	2	2	no	—
81	C₂×C₅₉⋊C₈	—	2	2	no	—
82	C₂×C₅₉⋊C₈	—	4	2	no	—
83	C₂×C₅₉⋊C₈	—	4	4	no	—
84	C₂×C₅₉⋊C₈	D₈	1	4	yes	complemented
85	C₂³×C₅₉	—	2	4	no	—
86	C₂³×C₅₉	—	4	2	no	—
87	C₂³×C₅₉	—	4	2	no	—
88	C₂³×C₅₉	—	4	4	no	—
89	C₂³×C₅₉	—	4	4	no	—
90	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
91	D₄×C₅₉	—	2	4	no	—
92	D₄×C₅₉	—	2	4	no	—
93	D₄×C₅₉	—	4	2	no	—
94	D₄×C₅₉	—	4	2	no	—
95	C₂×C₄×C₅₉	—	2	2	no	—
96	C₂×C₄×C₅₉	—	4	2	no	—
97	C₂×C₄×C₅₉	—	4	4	no	—
98	C₂×C₄×C₅₉	—	8	4	no	—
99	C₂×C₄×C₅₉	—	8	2	no	—
100	C₂×C₄×C₅₉	—	8	2	no	—
101	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
104	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
105	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
106	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
110	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
111	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₅₉	#135	1	2	yes	—
113	C₈×C₅₉	—	2	2	no	—
114	C₈×C₅₉	C₂²⋊C₈	1	4	yes	complemented
115	C₅₉⋊C₈	—	2	4	no	—
116	C₅₉⋊C₈	—	2	4	no	—
117	C₅₉⋊C₈	—	8	2	no	—
118	#9657	—	59	1	no	maximal, Sylow
119	C₂²×C₅₉	—	4	8	no	—
120	C₂²×C₅₉	—	4	4	no	—
121	C₂²×C₅₉	—	8	4	no	—
122	C₂²×C₅₉	—	8	2	no	—
123	C₂²×C₅₉	—	8	2	no	—
124	C₂²×C₅₉	#390	1	1	yes	characteristic
125	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
126	C₂²×C₅₉	#436	1	1	yes	characteristic
127	C₂²×C₅₉	C₄×D₈	1	2	yes	—
128	C₂²×C₅₉	#443	1	2	yes	—
129	C₄×C₅₉	—	2	1	no	—
130	C₄×C₅₉	—	2	1	no	—
131	C₄×C₅₉	—	8	2	no	—
132	C₄×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₄×C₅₉	#408	1	1	yes	characteristic
134	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
135	C₄×C₅₉	#436	1	1	yes	characteristic
136	C₄×C₅₉	#466	1	2	yes	—
137	C₄×C₅₉	#493	1	1	yes	characteristic
138	C₄×C₅₉	#495	1	1	yes	characteristic
139	C₂×C₄×D₈	D₅₉	1	1	yes	characteristic
140	#1954	—	59	2	no	—
141	C₂×#335	—	59	1	no	—
142	C₂×D₄⋊C₈	—	59	2	no	—
143	C₂×C₈²	—	59	1	no	—
144	C₂×C₅₉	—	8	4	no	—
145	C₂×C₅₉	C₈×D₈	1	2	yes	—
146	C₂×C₅₉	#1589	1	2	yes	—
147	C₂×C₅₉	#1954	1	1	yes	characteristic
148	C₂×C₅₉	#1970	1	1	yes	characteristic
149	C₂×C₅₉	#1974	1	1	yes	characteristic
150	C₂²×D₈	Dic₅₉	1	1	yes	characteristic
151	C₂×C₄×D₄	C₂×D₅₉	1	2	yes	—
152	C₄×D₈	—	2	2	no	—
153	C₄×D₈	—	2	2	no	—
154	C₂×#107	Dic₅₉	1	1	yes	characteristic
155	C₂×C₄⋊C₈	—	59	2	no	—
156	C₂×#102	Dic₅₉	1	2	yes	—
157	C₂×C₂²⋊C₈	—	118	4	no	—
158	C₂×C₄×C₈	—	59	1	no	—
159	C₂×C₄×C₈	—	59	1	no	—
160	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
161	#335	—	118	2	no	—
162	D₄⋊C₈	—	118	4	no	—
163	C₈²	—	118	2	no	—
164	C₅₉	#9657	1	1	yes	characteristic, complemented, Sylow
165	C₂²×D₄	C₂×Dic₅₉	1	2	yes	—
166	C₂³×C₄	—	2	2	no	—
167	C₂×D₈	—	2	2	no	—
168	C₂×D₈	—	2	2	no	—
169	C₂×D₈	C₅₉⋊C₈	1	4	yes	complemented
170	C₂²×C₈	—	59	2	no	—
171	C₂²×C₈	—	118	1	no	—
172	C₂²×C₈	—	118	2	no	—
173	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
174	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
175	C₄×D₄	—	2	2	no	—
176	C₄×D₄	—	2	2	no	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	—	2	2	no	—
179	C₂×C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
180	C₂×C₂²⋊C₄	—	2	2	no	—
181	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
182	#107	—	2	2	no	—
183	C₄⋊C₈	—	118	2	no	—
184	C₄⋊C₈	—	118	2	no	—
185	#102	—	2	4	no	—
186	C₂²⋊C₈	—	236	8	no	—
187	C₄×C₈	—	59	2	no	—
188	C₄×C₈	—	118	1	no	—
189	C₄×C₈	—	118	1	no	—
190	C₄×C₈	—	118	1	no	—
191	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
192	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
193	C₂⁴	—	2	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	C₂×C₅₉⋊C₈	1	4	yes	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
206	C₂²×C₄	#11745687	1	1	yes	characteristic
207	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
208	D₈	—	2	8	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	59	4	no	—
212	C₂×C₈	—	118	4	no	—
213	C₂×C₈	—	118	2	no	—
214	C₂×C₈	—	118	2	no	—
215	C₂×C₈	—	118	2	no	—
216	C₂×C₈	—	236	2	no	—
217	C₂×C₈	—	236	4	no	—
218	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
219	C₂×C₈	#11745664	1	2	yes	—
220	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
221	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₄²	D₄×D₅₉	1	1	yes	characteristic
227	C₄²	#11745687	1	1	yes	characteristic
228	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
229	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
230	C₂³	—	2	4	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	4	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	2	no	—
235	C₂³	D₄×Dic₅₉	1	1	yes	characteristic
236	D₄	—	2	4	no	—
237	D₄	—	2	4	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
247	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
248	C₂×C₄	#14273698	1	1	yes	characteristic
249	C₂×C₄	#14273698	1	1	yes	characteristic
250	C₂×C₄	#14273714	1	1	yes	characteristic
251	C₂×C₄	#14273723	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
253	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
254	C₂×C₄	#14273734	1	1	yes	characteristic
255	C₂×C₄	#14273734	1	1	yes	characteristic
256	C₂×C₄	D₈×D₅₉	1	2	yes	—
257	C₂×C₄	#14273746	1	2	yes	—
258	C₈	—	2	2	no	—
259	C₈	—	118	4	no	—
260	C₈	—	118	4	no	—
261	C₈	—	472	2	no	—
262	C₈	C₂²⋊(C₅₉⋊C₈)	1	4	yes	complemented
263	C₂²	—	4	8	no	—
264	C₂²	—	4	4	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	8	4	no	—
268	C₂²	#16837305	1	1	yes	characteristic
269	C₂²	D₄×C₅₉⋊C₈	1	1	yes	characteristic
270	C₂²	#16837480	1	1	yes	characteristic
271	C₂²	D₈×Dic₅₉	1	2	yes	—
272	C₂²	#16837519	1	2	yes	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	8	2	no	—
276	C₄	C₂×C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
277	C₄	#16837369	1	1	yes	characteristic
278	C₄	D₄×C₅₉⋊C₈	1	1	yes	characteristic
279	C₄	#16837480	1	1	yes	characteristic
280	C₄	#16837635	1	2	yes	—
281	C₄	#16837777	1	1	yes	characteristic
282	C₄	#16837785	1	1	yes	characteristic
283	C₂	—	8	4	no	—
284	C₂	D₈×C₅₉⋊C₈	1	2	yes	—
285	C₂	#21422981	1	2	yes	—
286	C₂	#21424891	1	1	yes	characteristic
287	C₂	#21424963	1	1	yes	characteristic
288	C₂	#21424985	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₄×D₈×C₅₉	C₂
socle	C₂³×C₅₉	C₄×D₄
prime radical	C₂²×D₈×C₅₉	C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×C₄×D₈	D₅₉
hypocenter	C₅₉	#9657
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #21424891	C₂	semidirect
2. C₂×#16837155	C₂	semidirect
3. C₂×D₄⋊(C₅₉⋊C₈)	C₂	semidirect
4. C₂×C₈×C₅₉⋊C₈	C₂	semidirect
34. C₂×D₈×C₅₉	C₈	semidirect
84. C₂×C₅₉⋊C₈	D₈	semidirect
114. C₈×C₅₉	C₂²⋊C₈	semidirect
164. C₅₉	#9657	semidirect
169. C₂×D₈	C₅₉⋊C₈	semidirect
262. C₈	C₂²⋊(C₅₉⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
#21424891	2
C₂×#16837155	2
C₂×D₄⋊(C₅₉⋊C₈)	2
C₂×C₈×C₅₉⋊C₈	2
C₂×C₄×D₈×C₅₉	2
#9657	59