Subgroups — #28487272

Order 15104.

404 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424248	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837153	C₂	1	2	yes	maximal
4	#21421931	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅₉⋊C₈	—	2	1	no	—
12	#16837157	—	2	1	no	—
13	#16837153	—	2	4	no	—
14	C₂×C₄⋊C₈×C₅₉	C₄	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₅₉	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
17	C₄³×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#336	C₄	1	2	yes	—
19	C₂²×C₅₉⋊C₈	—	2	1	no	—
20	C₂²×C₅₉⋊C₈	—	2	1	no	—
21	C₂²×C₅₉⋊C₈	—	2	2	no	—
22	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
24	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
27	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
28	C₄⋊(C₅₉⋊C₈)	D₄	1	2	yes	—
29	C₄⋊(C₅₉⋊C₈)	Q₈	1	2	yes	—
30	C₄×C₅₉⋊C₈	—	4	2	no	—
31	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₉	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₅₉	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₉	—	2	2	no	—
41	C₄⋊C₈×C₅₉	C₈	1	4	yes	complemented
42	C₄×C₈×C₅₉	—	2	2	no	—
43	C₄×C₈×C₅₉	—	2	2	no	—
44	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₅₉	—	2	2	no	—
52	C₂×C₈×C₅₉	—	2	2	no	—
53	C₂×C₈×C₅₉	—	2	2	no	—
54	C₂×C₈×C₅₉	—	2	2	no	—
55	C₂×C₈×C₅₉	—	2	2	no	—
56	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
57	C₄²×C₅₉	—	2	2	no	—
58	C₄²×C₅₉	—	2	1	no	—
59	C₄²×C₅₉	—	2	1	no	—
60	C₄²×C₅₉	—	2	1	no	—
61	C₄²×C₅₉	—	2	1	no	—
62	C₄²×C₅₉	—	4	1	no	—
63	C₄²×C₅₉	C₂×C₈	1	2	yes	—
64	C₄²×C₅₉	M₁₆	1	2	yes	—
65	C₄²×C₅₉	SD₁₆	1	2	yes	—
66	C₄²×C₅₉	SD₁₆	1	2	yes	—
67	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
69	C₄²×C₅₉	#41	1	1	yes	characteristic
70	C₄²×C₅₉	#41	1	1	yes	characteristic
71	C₂×C₅₉⋊C₈	—	2	2	no	—
72	C₂×C₅₉⋊C₈	—	2	2	no	—
73	C₂×C₅₉⋊C₈	—	4	1	no	—
74	C₂×C₅₉⋊C₈	—	4	1	no	—
75	C₂×C₅₉⋊C₈	—	4	1	no	—
76	C₂×C₅₉⋊C₈	—	4	1	no	—
77	C₂×C₅₉⋊C₈	—	4	2	no	—
78	C₂×C₅₉⋊C₈	—	4	2	no	—
79	C₂×C₅₉⋊C₈	—	4	2	no	—
80	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₉	—	2	1	no	—
82	C₂×C₄×C₅₉	—	2	1	no	—
83	C₂×C₄×C₅₉	—	2	1	no	—
84	C₂×C₄×C₅₉	—	2	1	no	—
85	C₂×C₄×C₅₉	—	2	1	no	—
86	C₂×C₄×C₅₉	—	2	1	no	—
87	C₂×C₄×C₅₉	—	2	1	no	—
88	C₂×C₄×C₅₉	—	2	1	no	—
89	C₂×C₄×C₅₉	—	2	1	no	—
90	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
91	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
92	C₂×C₄×C₅₉	#106	1	2	yes	—
93	C₂×C₄×C₅₉	#106	1	2	yes	—
94	C₂×C₄×C₅₉	#108	1	2	yes	—
95	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
98	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₅₉	#125	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
104	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
109	C₈×C₅₉	—	2	4	no	—
110	C₈×C₅₉	—	4	2	no	—
111	C₅₉⋊C₈	—	4	2	no	—
112	C₅₉⋊C₈	—	4	2	no	—
113	C₅₉⋊C₈	—	8	2	no	—
114	#9602	—	59	1	no	maximal, Sylow
115	C₂²×C₅₉	#389	1	1	yes	characteristic
116	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
117	C₂²×C₅₉	#424	1	1	yes	characteristic
118	C₂²×C₅₉	C₂×#106	1	1	yes	characteristic
119	C₂²×C₅₉	#428	1	1	yes	characteristic
120	C₂²×C₅₉	C₂×#108	1	1	yes	characteristic
121	C₂²×C₅₉	#430	1	1	yes	characteristic
122	C₄×C₅₉	—	2	2	no	—
123	C₄×C₅₉	—	2	2	no	—
124	C₄×C₅₉	—	2	1	no	—
125	C₄×C₅₉	—	2	1	no	—
126	C₄×C₅₉	—	4	1	no	—
127	C₄×C₅₉	#334	1	4	yes	complemented
128	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
129	C₄×C₅₉	#436	1	1	yes	characteristic
130	C₄×C₅₉	C₈×Q₈	1	1	yes	characteristic
131	C₄×C₅₉	#446	1	1	yes	characteristic
132	C₄×C₅₉	#460	1	1	yes	characteristic
133	C₄×C₅₉	#461	1	1	yes	characteristic
134	C₄×C₅₉	#475	1	1	yes	characteristic
135	C₄×C₅₉	#476	1	1	yes	characteristic
136	#1842	—	59	1	no	—
137	C₄×C₄⋊C₈	—	59	1	no	—
138	C₄×C₄⋊C₈	D₅₉	1	1	yes	characteristic
139	C₂×#334	—	59	2	no	—
140	#1277	—	59	2	no	—
141	C₂×C₅₉	C₂×#334	1	1	yes	characteristic
142	C₂×C₅₉	#1565	1	1	yes	characteristic
143	C₂×C₅₉	#1570	1	1	yes	characteristic
144	C₂×C₅₉	#1574	1	1	yes	characteristic
145	C₂×C₅₉	#1939	1	1	yes	characteristic
146	C₂×C₅₉	#1948	1	1	yes	characteristic
147	C₂×C₅₉	#1950	1	1	yes	characteristic
148	C₂×C₄⋊C₈	—	59	1	no	—
149	C₂×C₄⋊C₈	—	59	1	no	—
150	C₂×C₄⋊C₈	—	59	2	no	—
151	C₂×C₄⋊C₈	—	118	1	no	—
152	C₂×C₄⋊C₈	—	118	1	no	—
153	C₂×C₄⋊C₈	Dic₅₉	1	1	yes	characteristic
154	C₂×C₄⋊C₈	Dic₅₉	1	1	yes	characteristic
155	C₂×C₄×C₈	—	118	1	no	—
156	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
157	C₄³	C₂×D₅₉	1	1	yes	characteristic
158	#336	—	118	1	no	—
159	#336	Dic₅₉	1	2	yes	—
160	#334	—	118	4	no	—
161	C₅₉	#9602	1	1	yes	characteristic, complemented, Sylow
162	C₂²×C₈	—	118	1	no	—
163	C₂²×C₈	—	118	1	no	—
164	C₂²×C₈	—	118	2	no	—
165	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
166	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
167	C₂×C₄²	Dic₁₁₈	1	1	yes	characteristic
168	C₂×C₄²	Dic₁₁₈	1	1	yes	characteristic
169	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
170	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
171	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
172	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	59	2	no	—
176	C₄⋊C₈	—	59	2	no	—
177	C₄⋊C₈	—	118	2	no	—
178	C₄⋊C₈	—	118	1	no	—
179	C₄⋊C₈	—	118	1	no	—
180	C₄⋊C₈	—	118	2	no	—
181	C₄⋊C₈	—	236	2	no	—
182	C₄⋊C₈	—	236	2	no	—
183	C₄⋊C₈	C₅₉⋊C₈	1	4	yes	complemented
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	—	236	2	no	—
187	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
188	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
189	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
191	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
193	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	118	2	no	—
200	C₂×C₈	—	118	2	no	—
201	C₂×C₈	—	236	2	no	—
202	C₂×C₈	—	236	2	no	—
203	C₂×C₈	—	236	2	no	—
204	C₂×C₈	—	236	1	no	—
205	C₂×C₈	—	236	1	no	—
206	C₂×C₈	—	236	1	no	—
207	C₂×C₈	—	236	1	no	—
208	C₂×C₈	C₂×C₅₉⋊C₈	1	4	yes	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	4	1	no	—
215	C₄²	#11745660	1	2	yes	—
216	C₄²	#11745660	1	2	yes	—
217	C₄²	C₂×C₅₉⋊C₈	1	2	yes	—
218	C₄²	#11745664	1	2	yes	—
219	C₄²	D₄×D₅₉	1	1	yes	characteristic
220	C₄²	#11745687	1	1	yes	characteristic
221	C₄²	Q₈×D₅₉	1	1	yes	characteristic
222	C₄²	#11745689	1	1	yes	characteristic
223	C₂³	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
234	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
235	C₂×C₄	#14273659	1	2	yes	—
236	C₂×C₄	#14273659	1	2	yes	—
237	C₂×C₄	#14273663	1	2	yes	—
238	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
239	C₂×C₄	Q₈×Dic₅₉	1	1	yes	characteristic
240	C₂×C₄	#14273702	1	1	yes	characteristic
241	C₂×C₄	#14273707	1	1	yes	characteristic
242	C₂×C₄	#14273718	1	1	yes	characteristic
243	C₂×C₄	#14273726	1	1	yes	characteristic
244	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
245	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
246	C₂×C₄	#14273734	1	1	yes	characteristic
247	C₂×C₄	#14273734	1	1	yes	characteristic
248	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
249	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
250	C₂×C₄	#14273755	1	1	yes	characteristic
251	C₂×C₄	#14273760	1	1	yes	characteristic
252	C₈	—	2	4	no	—
253	C₈	—	4	2	no	—
254	C₈	—	472	2	no	—
255	C₈	—	236	2	no	—
256	C₈	—	236	2	no	—
257	C₂²	#16837301	1	1	yes	characteristic
258	C₂²	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
259	C₂²	#16837429	1	1	yes	characteristic
260	C₂²	C₂×#14273659	1	1	yes	characteristic
261	C₂²	#16837444	1	1	yes	characteristic
262	C₂²	C₂×#14273663	1	1	yes	characteristic
263	C₂²	#16837451	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	4	1	no	—
269	C₄	#16837153	1	4	yes	complemented
270	C₄	D₄×C₅₉⋊C₈	1	1	yes	characteristic
271	C₄	#16837480	1	1	yes	characteristic
272	C₄	Q₈×C₅₉⋊C₈	1	1	yes	characteristic
273	C₄	#16837533	1	1	yes	characteristic
274	C₄	#16837603	1	1	yes	characteristic
275	C₄	#16837608	1	1	yes	characteristic
276	C₄	#16837688	1	1	yes	characteristic
277	C₄	#16837693	1	1	yes	characteristic
278	C₂	C₂×#16837153	1	1	yes	characteristic
279	C₂	#21422848	1	1	yes	characteristic
280	C₂	#21422867	1	1	yes	characteristic
281	C₂	#21422885	1	1	yes	characteristic
282	C₂	#21424815	1	1	yes	characteristic
283	C₂	#21424860	1	1	yes	characteristic
284	C₂	#21424872	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₄×C₄⋊C₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₅₉
hypocenter	C₅₉	#9602
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
41. C₄⋊C₈×C₅₉	C₈	semidirect
127. C₄×C₅₉	#334	semidirect
161. C₅₉	#9602	semidirect
183. C₄⋊C₈	C₅₉⋊C₈	semidirect
269. C₄	#16837153	semidirect

Maximal subgroups

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

subgroup	index
#21424248	2
C₄×C₄⋊(C₅₉⋊C₈)	2
C₂×#16837153	2
#21421931	2
C₄×C₄⋊C₈×C₅₉	2
#9602	59