Subgroups — #28497124

Order 15104.

574 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837173	C₂	1	2	yes	maximal
2	C₂×#16837166	C₂	1	2	yes	maximal
3	C₂×C₅₉⋊#338	C₂	1	1	yes	maximal, characteristic
4	#21423794	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₉×#430	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11745664	C₂²	1	2	yes	—
7	C₂³×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×#14273686	C₂²	1	1	yes	characteristic
9	#16837173	—	2	4	no	—
10	#16837166	—	2	4	no	—
11	C₅₉⋊#338	C₄	1	4	yes	complemented
12	#16837156	—	2	2	no	—
13	C₂²×M₁₆×C₅₉	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#430	—	2	4	no	—
16	C₂×#108×C₅₉	—	2	2	no	—
17	C₂×#11745664	—	2	2	no	—
18	C₂×#11745664	—	2	2	no	—
19	C₂×#11745664	—	2	2	no	—
20	C₂×#11745664	—	2	2	no	—
21	C₂×#11745664	—	2	2	no	—
22	C₂×#11745664	C₂×C₄	1	4	yes	—
23	C₂²×C₅₉⋊C₈	—	2	1	no	—
24	C₂²×C₅₉⋊C₈	—	2	1	no	—
25	C₂²×C₅₉⋊C₈	—	2	2	no	—
26	C₂²×C₅₉⋊C₈	—	2	2	no	—
27	C₂²×C₅₉⋊C₈	—	2	2	no	—
28	C₂²×C₅₉⋊C₈	D₄	1	2	yes	—
29	#14273686	—	2	2	no	—
30	#14273686	—	2	2	no	—
31	C₂×C₄⋊Dic₅₉	—	2	1	no	—
32	C₂×C₂²⋊Dic₅₉	Q₈	1	2	yes	—
33	C₂×C₄×Dic₅₉	—	2	1	no	—
34	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₅₉	—	2	2	no	—
36	C₂×M₁₆×C₅₉	—	2	2	no	—
37	C₂×M₁₆×C₅₉	—	2	2	no	—
38	C₂×M₁₆×C₅₉	—	2	2	no	—
39	C₂×M₁₆×C₅₉	—	2	2	no	—
40	C₂×M₁₆×C₅₉	—	2	2	no	—
41	C₂×M₁₆×C₅₉	—	2	2	no	—
42	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
43	C₂×M₁₆×C₅₉	D₄	1	4	yes	—
44	C₂×M₁₆×C₅₉	Q₈	1	2	yes	—
45	C₂²×C₈×C₅₉	—	2	2	no	—
46	C₂²×C₈×C₅₉	D₄	1	2	yes	—
47	#108×C₅₉	—	2	8	no	—
48	C₂⁴×C₅₉	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	#41	1	2	yes	—
55	C₂²×C₄×C₅₉	#41	1	2	yes	—
56	C₂²×C₄×C₅₉	#41	1	2	yes	—
57	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
58	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
59	C₂²×Dic₅₉	—	2	2	no	—
60	M₁₆×C₅₉	—	2	4	no	—
61	M₁₆×C₅₉	—	2	4	no	—
62	M₁₆×C₅₉	—	2	4	no	—
63	M₁₆×C₅₉	—	2	4	no	—
64	M₁₆×C₅₉	—	4	2	no	—
65	M₁₆×C₅₉	—	4	2	no	—
66	C₂×C₈×C₅₉	—	2	2	no	—
67	C₂×C₈×C₅₉	—	2	2	no	—
68	C₂×C₈×C₅₉	—	2	4	no	—
69	C₂×C₈×C₅₉	—	4	2	no	—
70	C₂×C₈×C₅₉	—	4	2	no	—
71	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
72	C₂²⋊Dic₅₉	—	2	4	no	—
73	C₄⋊Dic₅₉	—	4	2	no	—
74	C₄×Dic₅₉	—	4	1	no	—
75	C₄×Dic₅₉	—	4	1	no	—
76	#11745664	—	2	4	no	—
77	#11745664	—	2	4	no	—
78	#11745664	—	4	2	no	—
79	#11745664	—	4	2	no	—
80	C₂×C₅₉⋊C₈	—	2	4	no	—
81	C₂×C₅₉⋊C₈	—	2	2	no	—
82	C₂×C₅₉⋊C₈	—	2	4	no	—
83	C₂×C₅₉⋊C₈	—	4	2	no	—
84	C₂×C₅₉⋊C₈	—	4	1	no	—
85	C₂×C₅₉⋊C₈	—	4	1	no	—
86	C₂×C₅₉⋊C₈	—	4	2	no	—
87	C₂×C₅₉⋊C₈	—	4	2	no	—
88	C₂³×C₅₉	—	4	2	no	—
89	C₂³×C₅₉	C₄×Q₈	1	1	yes	characteristic
90	C₂³×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂³×C₅₉	C₂²⋊Q₈	1	2	yes	—
92	C₂³×C₅₉	#125	1	2	yes	—
93	C₂³×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₉	#117	1	2	yes	—
96	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
99	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
100	C₂×C₄×C₅₉	#121	1	2	yes	—
101	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₅₉	#123	1	2	yes	—
103	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
104	C₂×C₄×C₅₉	#123	1	2	yes	—
105	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
106	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
107	C₂×C₄×C₅₉	#124	1	2	yes	—
108	C₂×C₄×C₅₉	#125	1	2	yes	—
109	C₂×C₄×C₅₉	#125	1	1	yes	characteristic
110	C₂×C₄×C₅₉	#126	1	2	yes	—
111	C₂×C₄×C₅₉	#126	1	2	yes	—
112	C₂×Dic₅₉	—	4	4	no	—
113	C₂×Dic₅₉	—	4	2	no	—
114	C₈×C₅₉	—	2	4	no	—
115	C₈×C₅₉	—	4	4	no	—
116	C₅₉⋊C₈	—	4	4	no	—
117	C₅₉⋊C₈	—	4	2	no	—
118	#11295	—	59	1	no	maximal, Sylow
119	C₂²×C₅₉	—	2	2	no	—
120	C₂²×C₅₉	—	2	4	no	—
121	C₂²×C₅₉	—	4	1	no	—
122	C₂²×C₅₉	—	4	2	no	—
123	C₂²×C₅₉	—	4	1	no	—
124	C₂²×C₅₉	#387	1	2	yes	—
125	C₂²×C₅₉	#389	1	1	yes	characteristic
126	C₂²×C₅₉	#393	1	1	yes	characteristic
127	C₂²×C₅₉	#398	1	2	yes	—
128	C₂²×C₅₉	#400	1	1	yes	characteristic
129	C₄×C₅₉	#382	1	1	yes	characteristic
130	C₄×C₅₉	#385	1	1	yes	characteristic
131	C₄×C₅₉	#388	1	2	yes	—
132	C₄×C₅₉	#394	1	1	yes	characteristic
133	C₄×C₅₉	#397	1	1	yes	characteristic
134	C₄×C₅₉	#400	1	2	yes	—
135	Dic₅₉	—	8	2	no	—
136	C₂×#430	D₅₉	1	1	yes	characteristic
137	C₂×#344	—	59	2	no	—
138	C₂×#341	—	59	2	no	—
139	C₂×#338	—	59	1	no	—
140	#1729	—	59	1	no	—
141	C₂×C₅₉	—	2	1	no	—
142	C₂×C₅₉	—	2	2	no	—
143	C₂×C₅₉	—	2	1	no	—
144	C₂×C₅₉	—	2	2	no	—
145	C₂×C₅₉	#1442	1	1	yes	characteristic
146	C₂×C₅₉	#2103	1	2	yes	—
147	C₂²×M₁₆	—	59	2	no	—
148	C₂²×M₁₆	C₂×D₅₉	1	2	yes	—
149	C₂²×M₁₆	C₂×D₅₉	1	1	yes	characteristic
150	C₂³×C₈	—	59	1	no	—
151	C₂×#117	—	59	1	no	—
152	#430	—	2	4	no	—
153	C₂×#108	—	2	2	no	—
154	#344	—	118	4	no	—
155	#341	—	118	4	no	—
156	#338	—	59	4	no	—
157	#336	—	118	2	no	—
158	C₅₉	#11295	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	59	4	no	—
168	C₂×M₁₆	—	118	2	no	—
169	C₂×M₁₆	—	118	2	no	—
170	C₂×M₁₆	—	118	2	no	—
171	C₂×M₁₆	—	118	2	no	—
172	C₂×M₁₆	—	118	2	no	—
173	C₂×M₁₆	Dic₁₁₈	1	2	yes	—
174	C₂×M₁₆	C₄×D₅₉	1	2	yes	—
175	C₂×M₁₆	C₂²⋊D₅₉	1	4	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	59	2	no	—
178	C₂²×C₈	—	118	2	no	—
179	C₂²×C₈	—	118	1	no	—
180	C₂²×C₈	—	118	2	no	—
181	C₂²×C₈	—	118	2	no	—
182	C₂²×C₈	—	118	1	no	—
183	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
184	#117	—	118	2	no	—
185	#117	—	118	2	no	—
186	C₂×C₄⋊C₄	—	118	1	no	—
187	C₂×C₂²⋊C₄	—	59	2	no	—
188	C₂×C₄²	—	118	1	no	—
189	#108	—	2	8	no	—
190	C₂⁴	Q₈×D₅₉	1	1	yes	characteristic
191	C₂²×C₄	—	118	2	no	—
192	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
194	C₂²×C₄	#11745685	1	2	yes	—
195	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
196	C₂²×C₄	#11745687	1	2	yes	—
197	C₂²×C₄	#11745687	1	1	yes	characteristic
198	C₂²×C₄	#11745689	1	2	yes	—
199	C₂²×C₄	#11745689	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	4	2	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	118	4	no	—
209	M₁₆	—	118	4	no	—
210	M₁₆	—	236	2	no	—
211	M₁₆	—	236	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	118	4	no	—
218	C₂×C₈	—	118	2	no	—
219	C₂×C₈	—	118	4	no	—
220	C₂×C₈	—	236	2	no	—
221	C₂×C₈	—	236	1	no	—
222	C₂×C₈	—	236	1	no	—
223	C₂×C₈	—	236	2	no	—
224	C₂×C₈	—	236	2	no	—
225	C₂×C₈	Dic₅₉⋊C₄	1	4	yes	—
226	C₄⋊C₄	—	236	2	no	—
227	C₂²⋊C₄	—	118	4	no	—
228	C₄²	—	236	1	no	—
229	C₄²	—	236	1	no	—
230	C₂³	—	4	2	no	—
231	C₂³	#14273692	1	1	yes	characteristic
232	C₂³	#14273703	1	1	yes	characteristic
233	C₂³	#14273703	1	2	yes	—
234	C₂³	#14273717	1	2	yes	—
235	C₂³	#14273726	1	1	yes	characteristic
236	C₂×C₄	—	236	4	no	—
237	C₂×C₄	—	236	2	no	—
238	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
239	C₂×C₄	#14273685	1	2	yes	—
240	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
241	C₂×C₄	#14273689	1	1	yes	characteristic
242	C₂×C₄	#14273696	1	1	yes	characteristic
243	C₂×C₄	#14273698	1	1	yes	characteristic
244	C₂×C₄	#14273699	1	2	yes	—
245	C₂×C₄	#14273706	1	2	yes	—
246	C₂×C₄	#14273708	1	2	yes	—
247	C₂×C₄	#14273710	1	1	yes	characteristic
248	C₂×C₄	#14273712	1	2	yes	—
249	C₂×C₄	#14273714	1	1	yes	characteristic
250	C₂×C₄	#14273716	1	1	yes	characteristic
251	C₂×C₄	#14273716	1	2	yes	—
252	C₂×C₄	#14273718	1	2	yes	—
253	C₂×C₄	#14273718	1	1	yes	characteristic
254	C₂×C₄	#14273722	1	2	yes	—
255	C₂×C₄	#14273722	1	2	yes	—
256	C₈	—	2	4	no	—
257	C₈	—	4	4	no	—
258	C₈	—	236	4	no	—
259	C₈	—	236	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	#16837283	1	2	yes	—
266	C₂²	#16837298	1	1	yes	characteristic
267	C₂²	#16837313	1	1	yes	characteristic
268	C₂²	#16837335	1	2	yes	—
269	C₂²	#16837340	1	1	yes	characteristic
270	C₄	—	472	2	no	—
271	C₄	#16837267	1	1	yes	characteristic
272	C₄	#16837275	1	1	yes	characteristic
273	C₄	#16837292	1	2	yes	—
274	C₄	#16837316	1	1	yes	characteristic
275	C₄	#16837328	1	1	yes	characteristic
276	C₄	#16837341	1	2	yes	—
277	C₂	—	2	1	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	2	no	—
281	C₂	#21422355	1	1	yes	characteristic
282	C₂	#21425675	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14273714
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₉
Fitting	C₂×C₅₉×#430	C₂
socle	C₂²×C₅₉	#400
prime radical	C₂⁴×C₅₉	C₂×Q₈
cosocle	C₂³×C₄×C₅₉	C₂³
hypercenter	C₂×#430	D₅₉
hypocenter	C₅₉	#11295
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
11. C₅₉⋊#338	C₄	semidirect
158. C₅₉	#11295	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837173	2
C₂×#16837166	2
C₂×C₅₉⋊#338	2
#21423794	2
C₂×C₅₉×#430	2
#11295	59