Subgroups — #28481789

Order 15104.

434 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837532	C₂	1	1	yes	maximal, characteristic
2	#21424256	C₂	1	1	yes	maximal, characteristic
3	#21424102	C₂	1	2	yes	maximal
4	#21421980	C₂	1	2	yes	maximal
5	C₂×C₅₉×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	#16837532	—	2	2	no	—
8	#16837532	C₄	1	4	yes	complemented
9	C₂×#14273661	—	2	1	no	—
10	C₂×#14273661	—	2	1	no	—
11	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×Dic₅₉⋊C₈	—	2	1	no	—
13	C₂×#14273652	—	2	2	no	—
14	C₂×#14273637	—	2	1	no	—
15	#16837272	C₂²	1	2	yes	—
16	#16837165	—	2	2	no	—
17	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₅₉×#334	—	2	2	no	—
20	C₂²×Dic₁₁₈	D₄	1	1	yes	characteristic
21	C₂²×C₅₉⋊C₈	—	2	1	no	—
22	C₄×Dic₁₁₈	—	2	1	no	—
23	C₄×Dic₁₁₈	—	2	1	no	—
24	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₁₈	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₅₉	—	2	2	no	—
27	C₂×C₄⋊Dic₅₉	—	2	2	no	—
28	C₂×C₄⋊Dic₅₉	—	2	2	no	—
29	C₂×C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
30	C₂×Dic₅₉⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₅₉	—	2	1	no	—
32	#14273661	—	4	2	no	—
33	#14273661	—	4	2	no	—
34	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
35	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
36	Dic₅₉⋊C₈	—	2	4	no	—
37	#14273652	—	4	4	no	—
38	#14273637	—	2	4	no	—
39	C₂²×C₈×C₅₉	—	2	2	no	—
40	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₅₉	—	2	2	no	—
44	C₄⋊C₈×C₅₉	—	2	2	no	—
45	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₅₉	D₄	1	2	yes	—
47	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
50	C₂²×Dic₅₉	—	2	1	no	—
51	C₂²×Dic₅₉	—	2	1	no	—
52	C₂²×Dic₅₉	—	4	2	no	—
53	C₂×Dic₁₁₈	—	2	2	no	—
54	C₂×Dic₁₁₈	—	2	2	no	—
55	C₂×Dic₁₁₈	—	2	1	no	—
56	C₂×Dic₁₁₈	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₅₉	—	2	1	no	—
58	C₂×C₈×C₅₉	—	2	1	no	—
59	C₂×C₈×C₅₉	—	4	2	no	—
60	C₂×C₈×C₅₉	—	4	2	no	—
61	C₂×C₈×C₅₉	—	4	2	no	—
62	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
65	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
66	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
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	C₄⋊Dic₅₉	—	2	1	no	—
71	C₄⋊Dic₅₉	—	4	2	no	—
72	C₄⋊Dic₅₉	—	4	2	no	—
73	C₄⋊Dic₅₉	—	4	2	no	—
74	C₄⋊Dic₅₉	—	4	2	no	—
75	C₄⋊Dic₅₉	—	8	2	no	—
76	C₄⋊Dic₅₉	C₄⋊C₄	1	2	yes	—
77	Dic₅₉⋊C₄	—	2	2	no	—
78	Dic₅₉⋊C₄	—	2	2	no	—
79	C₄×Dic₅₉	—	2	2	no	—
80	C₄×Dic₅₉	—	2	2	no	—
81	C₂×C₅₉⋊C₈	—	2	2	no	—
82	C₂×C₅₉⋊C₈	—	4	1	no	—
83	C₂×C₅₉⋊C₈	—	4	1	no	—
84	C₂³×C₅₉	#121	1	1	yes	characteristic
85	C₂×C₄×C₅₉	—	2	1	no	—
86	C₂×C₄×C₅₉	—	2	1	no	—
87	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
90	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
91	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₅₉	#137	1	2	yes	—
98	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
99	C₂×Dic₅₉	—	2	2	no	—
100	C₂×Dic₅₉	—	2	2	no	—
101	C₂×Dic₅₉	—	4	1	no	—
102	C₂×Dic₅₉	—	4	1	no	—
103	C₂×Dic₅₉	—	4	1	no	—
104	C₂×Dic₅₉	—	4	1	no	—
105	C₂×Dic₅₉	—	8	2	no	—
106	C₂×Dic₅₉	—	8	2	no	—
107	C₂×Dic₅₉	—	8	2	no	—
108	Dic₁₁₈	—	2	4	no	—
109	Dic₁₁₈	—	4	2	no	—
110	C₈×C₅₉	—	2	2	no	—
111	C₈×C₅₉	—	8	2	no	—
112	C₈×C₅₉	#103	1	4	yes	complemented
113	C₅₉⋊C₈	—	4	2	no	—
114	#8653	—	59	1	no	maximal, Sylow
115	C₂²×C₅₉	#380	1	1	yes	characteristic
116	C₂²×C₅₉	#460	1	1	yes	characteristic
117	C₂²×C₅₉	#462	1	1	yes	characteristic
118	C₂²×C₅₉	#469	1	1	yes	characteristic
119	C₂²×C₅₉	#470	1	1	yes	characteristic
120	C₂²×C₅₉	#472	1	1	yes	characteristic
121	C₂²×C₅₉	#473	1	1	yes	characteristic
122	C₄×C₅₉	—	2	1	no	—
123	C₄×C₅₉	—	2	1	no	—
124	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
125	C₄×C₅₉	#417	1	1	yes	characteristic
126	C₄×C₅₉	#428	1	2	yes	—
127	C₄×C₅₉	#468	1	2	yes	—
128	C₄×C₅₉	#475	1	1	yes	characteristic
129	C₄×C₅₉	#477	1	1	yes	characteristic
130	Dic₅₉	—	4	2	no	—
131	Dic₅₉	—	4	2	no	—
132	Dic₅₉	—	16	2	no	—
133	C₂×#446	—	59	1	no	—
134	#1845	—	59	1	no	—
135	#1803	—	59	2	no	—
136	C₂×#334	D₅₉	1	1	yes	characteristic
137	#1296	—	59	2	no	—
138	C₂×C₅₉	#1688	1	1	yes	characteristic
139	C₂×C₅₉	#1690	1	1	yes	characteristic
140	C₂×C₅₉	#1694	1	1	yes	characteristic
141	C₂×C₅₉	#1696	1	1	yes	characteristic
142	C₂×C₅₉	#1803	1	1	yes	characteristic
143	C₂×C₅₉	#1809	1	1	yes	characteristic
144	C₂×C₅₉	#1816	1	1	yes	characteristic
145	C₂×C₄×Q₈	—	59	1	no	—
146	#446	—	59	4	no	—
147	#446	—	118	2	no	—
148	C₂×#107	—	118	1	no	—
149	C₂×#107	—	118	1	no	—
150	C₂×C₄⋊C₈	—	59	1	no	—
151	C₂×C₄⋊C₈	—	118	1	no	—
152	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
153	C₂×#103	—	118	2	no	—
154	C₂×#97	—	118	1	no	—
155	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
156	#384	—	59	2	no	—
157	#340	—	118	2	no	—
158	#334	—	2	2	no	—
159	C₅₉	#8653	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	—	59	1	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	118	1	no	—
163	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
164	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
165	C₄×Q₈	—	59	2	no	—
166	C₄×Q₈	—	59	2	no	—
167	C₄×Q₈	—	118	1	no	—
168	C₄×Q₈	—	118	1	no	—
169	C₂×C₄⋊C₄	—	59	1	no	—
170	C₂×C₄⋊C₄	—	118	1	no	—
171	C₂×C₄⋊C₄	—	118	2	no	—
172	C₂×C₄⋊C₄	—	118	2	no	—
173	C₂×C₄⋊C₄	—	118	2	no	—
174	C₂×C₄²	—	118	1	no	—
175	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
176	#107	—	236	2	no	—
177	#107	—	236	2	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	59	2	no	—
181	C₄⋊C₈	—	118	4	no	—
182	C₄⋊C₈	—	118	1	no	—
183	#103	—	236	4	no	—
184	#97	—	118	4	no	—
185	C₄×C₈	C₄×D₅₉	1	2	yes	—
186	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
187	C₂×Q₈	—	59	2	no	—
188	C₂×Q₈	—	118	1	no	—
189	C₂×Q₈	—	118	2	no	—
190	C₂×Q₈	—	118	2	no	—
191	C₂²×C₄	—	118	1	no	—
192	C₂²×C₄	—	118	1	no	—
193	C₂²×C₄	—	236	2	no	—
194	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
195	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
196	C₂²×C₄	#11745689	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	118	2	no	—
203	C₂×C₈	—	236	1	no	—
204	C₂×C₈	—	236	1	no	—
205	C₂×C₈	#11745660	1	2	yes	—
206	C₂×C₈	Dic₂₃₆	1	2	yes	—
207	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
208	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
209	C₄⋊C₄	—	59	2	no	—
210	C₄⋊C₄	—	118	2	no	—
211	C₄⋊C₄	—	118	1	no	—
212	C₄⋊C₄	—	118	2	no	—
213	C₄⋊C₄	—	472	2	no	—
214	C₄⋊C₄	—	236	2	no	—
215	C₄⋊C₄	—	236	2	no	—
216	C₄⋊C₄	—	236	2	no	—
217	C₄⋊C₄	—	236	2	no	—
218	C₄²	—	118	2	no	—
219	C₄²	—	118	2	no	—
220	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
221	C₄²	#11745687	1	1	yes	characteristic
222	C₄²	Q₈×D₅₉	1	1	yes	characteristic
223	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
224	C₂³	#14273702	1	1	yes	characteristic
225	Q₈	—	118	4	no	—
226	Q₈	—	236	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	118	2	no	—
230	C₂×C₄	—	118	2	no	—
231	C₂×C₄	—	472	2	no	—
232	C₂×C₄	—	472	2	no	—
233	C₂×C₄	—	472	2	no	—
234	C₂×C₄	—	236	1	no	—
235	C₂×C₄	—	236	1	no	—
236	C₂×C₄	—	236	1	no	—
237	C₂×C₄	—	236	1	no	—
238	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
239	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
240	C₂×C₄	#14273685	1	1	yes	characteristic
241	C₂×C₄	#14273698	1	1	yes	characteristic
242	C₂×C₄	#14273703	1	1	yes	characteristic
243	C₂×C₄	#14273707	1	1	yes	characteristic
244	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
245	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
246	C₂×C₄	#14273754	1	2	yes	—
247	C₂×C₄	#14273755	1	1	yes	characteristic
248	C₂×C₄	#14273756	1	2	yes	—
249	C₂×C₄	#14273760	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	8	2	no	—
252	C₈	—	236	2	no	—
253	C₈	#14273652	1	4	yes	complemented
254	C₂²	#16837259	1	1	yes	characteristic
255	C₂²	#16837603	1	1	yes	characteristic
256	C₂²	#16837613	1	1	yes	characteristic
257	C₂²	#16837656	1	1	yes	characteristic
258	C₂²	#16837661	1	1	yes	characteristic
259	C₂²	#16837673	1	1	yes	characteristic
260	C₂²	#16837678	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	944	2	no	—
264	C₄	—	236	2	no	—
265	C₄	—	236	2	no	—
266	C₄	C₂×#14273652	1	1	yes	characteristic
267	C₄	#16837404	1	1	yes	characteristic
268	C₄	#16837441	1	2	yes	—
269	C₄	#16837645	1	2	yes	—
270	C₄	#16837688	1	1	yes	characteristic
271	C₄	#16837698	1	1	yes	characteristic
272	C₂	#21423581	1	1	yes	characteristic
273	C₂	#21423591	1	1	yes	characteristic
274	C₂	#21423615	1	1	yes	characteristic
275	C₂	#21423625	1	1	yes	characteristic
276	C₂	#21424102	1	1	yes	characteristic
277	C₂	#21424126	1	1	yes	characteristic
278	C₂	#21424157	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273702
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#334	C₂
socle, prime radical	C₂³×C₅₉	#121
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#334	D₅₉
hypocenter	C₅₉	#8653
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. #16837532	C₄	semidirect
112. C₈×C₅₉	#103	semidirect
159. C₅₉	#8653	semidirect
253. C₈	#14273652	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837532	2
#21424256	2
#21424102	2
#21421980	2
C₂×C₅₉×#334	2
#8653	59