Subgroups — #28494336

Order 15104.

642 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837357	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837174	C₂	1	2	yes	maximal
3	#21423813	C₂	1	1	yes	maximal, characteristic
4	C₂×#16837159	C₂	1	2	yes	maximal
5	C₂×C₅₉×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11745664	C₂²	1	1	yes	characteristic
7	C₂×#14273686	C₂²	1	2	yes	—
8	C₂×#14273686	C₂²	1	1	yes	characteristic
9	#16837357	—	2	4	no	—
10	C₂×#14273637	C₄	1	2	yes	complemented
11	C₂×C₈×Dic₅₉	C₄	1	2	yes	complemented
12	#16837174	C₄	1	8	yes	complemented
13	#16837165	—	2	2	no	—
14	#16837159	—	2	4	no	—
15	C₂²×M₁₆×C₅₉	C₂²	1	2	yes	—
16	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#341	—	2	2	no	—
18	C₂×#11745664	—	2	2	no	—
19	C₂×#11745664	—	2	1	no	—
20	C₂×#11745664	—	2	1	no	—
21	C₂×#11745664	C₂×C₄	1	4	yes	—
22	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
23	#14273686	—	2	2	no	—
24	#14273686	—	2	2	no	—
25	#14273686	—	2	2	no	—
26	#14273686	—	2	2	no	—
27	#14273686	C₂×C₄	1	4	yes	—
28	#14273686	D₄	1	4	yes	—
29	C₂×C₄⋊Dic₅₉	—	2	1	no	—
30	C₂×C₄⋊Dic₅₉	—	2	2	no	—
31	C₂×C₂²⋊Dic₅₉	—	2	1	no	—
32	C₂×C₂²⋊Dic₅₉	—	2	2	no	—
33	C₂×C₄×Dic₅₉	—	2	2	no	—
34	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
35	#14273637	—	2	4	no	—
36	C₈×Dic₅₉	—	2	4	no	—
37	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
43	C₂×M₁₆×C₅₉	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₅₉	—	2	2	no	—
45	C₂²×C₈×C₅₉	—	2	2	no	—
46	C₂²×C₈×C₅₉	—	2	2	no	—
47	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	D₄	1	2	yes	—
50	C₂²×C₈×C₅₉	Q₈	1	2	yes	—
51	C₂⁴×C₅₉	#41	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₉	#41	1	2	yes	—
58	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	#41	1	2	yes	—
62	C₂²×Dic₅₉	—	2	2	no	—
63	C₂²×Dic₅₉	—	4	2	no	—
64	M₁₆×C₅₉	—	2	4	no	—
65	M₁₆×C₅₉	—	2	4	no	—
66	M₁₆×C₅₉	—	4	2	no	—
67	M₁₆×C₅₉	—	4	2	no	—
68	C₂×C₈×C₅₉	—	2	1	no	—
69	C₂×C₈×C₅₉	—	2	1	no	—
70	C₂×C₈×C₅₉	—	2	1	no	—
71	C₂×C₈×C₅₉	—	2	1	no	—
72	C₂×C₈×C₅₉	—	2	4	no	—
73	C₂×C₈×C₅₉	—	2	4	no	—
74	C₂×C₈×C₅₉	—	2	4	no	—
75	C₂×C₈×C₅₉	—	4	2	no	—
76	C₂×C₈×C₅₉	—	4	2	no	—
77	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
78	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
79	C₂²⋊Dic₅₉	—	2	4	no	—
80	C₂²⋊Dic₅₉	—	2	4	no	—
81	C₂²⋊Dic₅₉	—	4	2	no	—
82	C₄⋊Dic₅₉	—	2	4	no	—
83	C₄⋊Dic₅₉	—	2	4	no	—
84	C₄⋊Dic₅₉	—	4	2	no	—
85	C₄×Dic₅₉	—	2	2	no	—
86	C₄×Dic₅₉	—	2	4	no	—
87	C₄×Dic₅₉	—	4	2	no	—
88	C₄×Dic₅₉	C₂²⋊C₄	1	4	yes	—
89	#11745664	—	2	4	no	—
90	#11745664	—	2	4	no	—
91	C₂×C₅₉⋊C₈	—	2	2	no	—
92	C₂×C₅₉⋊C₈	—	2	2	no	—
93	C₂×C₅₉⋊C₈	C₄²	1	4	yes	—
94	C₂³×C₅₉	—	4	2	no	—
95	C₂³×C₅₉	#117	1	1	yes	characteristic
96	C₂³×C₅₉	#117	1	2	yes	—
97	C₂³×C₅₉	#123	1	2	yes	—
98	C₂³×C₅₉	#124	1	1	yes	characteristic
99	C₂³×C₅₉	#125	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
104	C₂×C₄×C₅₉	#117	1	2	yes	—
105	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
106	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₅₉	#121	1	2	yes	—
112	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₅₉	#123	1	2	yes	—
114	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
115	C₂×C₄×C₅₉	#125	1	1	yes	characteristic
116	C₂×C₄×C₅₉	#126	1	2	yes	—
117	C₂×C₄×C₅₉	#126	1	2	yes	—
118	C₂×Dic₅₉	—	2	4	no	—
119	C₂×Dic₅₉	—	4	4	no	—
120	C₂×Dic₅₉	—	4	4	no	—
121	C₂×Dic₅₉	—	8	4	no	—
122	C₈×C₅₉	—	2	2	no	—
123	C₈×C₅₉	—	2	2	no	—
124	C₈×C₅₉	—	4	4	no	—
125	C₅₉⋊C₈	—	2	4	no	—
126	#10804	—	59	1	no	maximal, Sylow
127	C₂²×C₅₉	—	2	2	no	—
128	C₂²×C₅₉	—	2	2	no	—
129	C₂²×C₅₉	—	2	2	no	—
130	C₂²×C₅₉	—	4	2	no	—
131	C₂²×C₅₉	—	4	2	no	—
132	C₂²×C₅₉	#376	1	1	yes	characteristic
133	C₂²×C₅₉	#381	1	1	yes	characteristic
134	C₂²×C₅₉	#382	1	1	yes	characteristic
135	C₂²×C₅₉	#387	1	2	yes	—
136	C₂²×C₅₉	#388	1	2	yes	—
137	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₅₉	#380	1	1	yes	characteristic
140	C₄×C₅₉	#383	1	1	yes	characteristic
141	C₄×C₅₉	#387	1	2	yes	—
142	C₄×C₅₉	#388	1	2	yes	—
143	Dic₅₉	—	4	4	no	—
144	Dic₅₉	—	8	4	no	—
145	C₂×#405	—	59	1	no	—
146	C₂×#344	—	59	2	no	—
147	C₂×#341	D₅₉	1	1	yes	characteristic
148	#1735	—	59	1	no	—
149	C₂×H(3,ℤ/4)	—	59	2	no	—
150	C₂×C₅₉	—	2	1	no	—
151	C₂×C₅₉	—	2	2	no	—
152	C₂×C₅₉	—	2	1	no	—
153	C₂×C₅₉	—	2	2	no	—
154	C₂×C₅₉	#1434	1	1	yes	characteristic
155	C₂×C₅₉	#1833	1	2	yes	—
156	C₂²×M₁₆	—	59	1	no	—
157	C₂²×M₁₆	C₂×D₅₉	1	2	yes	—
158	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
159	C₂×#117	—	59	1	no	—
160	C₂×#117	—	59	2	no	—
161	#405	—	118	4	no	—
162	C₂×#97	—	59	2	no	—
163	C₂×C₄×C₈	—	59	2	no	—
164	#344	—	59	8	no	—
165	#341	—	2	2	no	—
166	#340	—	118	2	no	—
167	H(3,ℤ/4)	—	118	4	no	—
168	C₅₉	#10804	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	59	4	no	—
176	C₂×M₁₆	—	118	2	no	—
177	C₂×M₁₆	—	118	1	no	—
178	C₂×M₁₆	—	118	1	no	—
179	C₂×M₁₆	C₄×D₅₉	1	4	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	59	2	no	—
184	C₂²×C₈	Dic₁₁₈	1	2	yes	—
185	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
187	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
188	#117	—	59	4	no	—
189	#117	—	59	4	no	—
190	#117	—	118	2	no	—
191	#117	—	118	2	no	—
192	#117	—	118	2	no	—
193	#117	—	118	2	no	—
194	C₂×C₄⋊C₄	—	118	1	no	—
195	C₂×C₄⋊C₄	—	118	2	no	—
196	C₂×C₂²⋊C₄	—	118	1	no	—
197	C₂×C₂²⋊C₄	—	118	2	no	—
198	C₂×C₄²	—	59	2	no	—
199	C₂×C₄²	—	118	2	no	—
200	#97	—	118	4	no	—
201	C₄×C₈	—	118	4	no	—
202	C₂⁴	#11745689	1	1	yes	characteristic
203	C₂²×C₄	—	118	2	no	—
204	C₂²×C₄	—	236	2	no	—
205	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
208	C₂²×C₄	#11745685	1	2	yes	—
209	C₂²×C₄	D₄×D₅₉	1	2	yes	—
210	C₂²×C₄	#11745687	1	2	yes	—
211	C₂²×C₄	#11745687	1	1	yes	characteristic
212	C₂²×C₄	#11745687	1	1	yes	characteristic
213	C₂²×C₄	#11745689	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	118	4	no	—
220	M₁₆	—	118	4	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	59	4	no	—
231	C₂×C₈	—	118	2	no	—
232	C₂×C₈	—	118	2	no	—
233	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
234	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
235	C₄⋊C₄	—	118	4	no	—
236	C₄⋊C₄	—	118	4	no	—
237	C₄⋊C₄	—	236	2	no	—
238	C₂²⋊C₄	—	118	4	no	—
239	C₂²⋊C₄	—	118	4	no	—
240	C₂²⋊C₄	—	236	2	no	—
241	C₄²	—	59	4	no	—
242	C₄²	—	118	2	no	—
243	C₄²	—	118	4	no	—
244	C₄²	—	236	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	#14273685	1	1	yes	characteristic
247	C₂³	#14273685	1	2	yes	—
248	C₂³	#14273712	1	2	yes	—
249	C₂³	#14273716	1	1	yes	characteristic
250	C₂³	#14273718	1	1	yes	characteristic
251	C₂×C₄	—	118	4	no	—
252	C₂×C₄	—	472	4	no	—
253	C₂×C₄	—	236	4	no	—
254	C₂×C₄	—	236	4	no	—
255	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
256	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
257	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
258	C₂×C₄	#14273683	1	1	yes	characteristic
259	C₂×C₄	#14273683	1	2	yes	—
260	C₂×C₄	#14273684	1	1	yes	characteristic
261	C₂×C₄	#14273685	1	1	yes	characteristic
262	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	2	yes	—
263	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
264	C₂×C₄	#14273689	1	2	yes	—
265	C₂×C₄	C₄×Dic₁₁₈	1	2	yes	—
266	C₂×C₄	#14273699	1	2	yes	—
267	C₂×C₄	#14273704	1	2	yes	—
268	C₂×C₄	#14273709	1	2	yes	—
269	C₂×C₄	#14273714	1	1	yes	characteristic
270	C₂×C₄	#14273718	1	1	yes	characteristic
271	C₂×C₄	#14273720	1	2	yes	—
272	C₂×C₄	#14273722	1	2	yes	—
273	C₈	—	2	2	no	—
274	C₈	—	2	2	no	—
275	C₈	—	4	4	no	—
276	C₈	—	118	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	#16837247	1	1	yes	characteristic
283	C₂²	#16837264	1	1	yes	characteristic
284	C₂²	#16837267	1	1	yes	characteristic
285	C₂²	#16837289	1	2	yes	—
286	C₂²	#16837292	1	2	yes	—
287	C₄	—	472	4	no	—
288	C₄	—	236	4	no	—
289	C₄	#16837249	1	1	yes	characteristic
290	C₄	C₄×Dic₅₉⋊C₄	1	1	yes	characteristic
291	C₄	#16837260	1	1	yes	characteristic
292	C₄	#16837269	1	1	yes	characteristic
293	C₄	#16837285	1	2	yes	—
294	C₄	#16837296	1	2	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	2	1	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	#21422322	1	1	yes	characteristic
300	C₂	#21424221	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14273685
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₉
Fitting	C₂×C₅₉×#341	C₂
socle	C₂²×C₅₉	#382
prime radical	C₂⁴×C₅₉	#41
cosocle	C₂³×C₄×C₅₉	C₂³
hypercenter	C₂×#341	D₅₉
hypocenter	C₅₉	#10804
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
10. C₂×#14273637	C₄	semidirect
11. C₂×C₈×Dic₅₉	C₄	semidirect
12. #16837174	C₄	semidirect
168. C₅₉	#10804	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837357	2
C₂×#16837174	2
#21423813	2
C₂×#16837159	2
C₂×C₅₉×#341	2
#10804	59