Subgroups — #28488036

Order 15104.

514 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#21422844	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422591	C₂	1	2	yes	maximal, complemented
4	#21422495	C₂	1	2	yes	maximal, complemented
5	C₅₉×#3147	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₅₉	—	2	1	no	—
9	C₄×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
10	#16837156	—	2	1	no	—
11	#16837155	—	2	1	no	—
12	#16837153	—	2	1	no	—
13	#16837136	—	2	4	no	—
14	D₄⋊(C₅₉⋊C₈)	—	2	4	no	—
15	C₂×#135×C₅₉	—	2	1	no	—
16	C₂×#135×C₅₉	—	2	1	no	—
17	C₅₉×#532	C₂²	1	2	yes	—
18	C₅₉×#495	—	2	1	no	—
19	C₅₉×#495	—	2	1	no	—
20	C₅₉×#494	C₄	1	1	yes	characteristic, complemented
21	C₅₉×#493	C₄	1	1	yes	characteristic, complemented
22	C₅₉×#492	C₄	1	2	yes	complemented
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂²×C₅₉⋊C₈	—	2	1	no	—
25	C₂×C₄⋊Dic₅₉	—	2	1	no	—
26	C₂×C₄×Dic₅₉	—	2	1	no	—
27	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
28	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
29	C₄⋊(C₅₉⋊C₈)	D₄	1	2	yes	—
30	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
31	C₄×C₅₉⋊C₈	—	2	2	no	—
32	C₈×Dic₅₉	—	4	2	no	—
33	C₅₉⋊#95	D₄	1	2	yes	—
34	C₂×#41×C₅₉	—	2	2	no	—
35	#135×C₅₉	—	4	2	no	—
36	#135×C₅₉	—	4	2	no	—
37	#135×C₅₉	—	4	2	no	—
38	#135×C₅₉	—	4	2	no	—
39	C₂×Q₁₆×C₅₉	—	2	1	no	—
40	C₂×Q₁₆×C₅₉	—	2	1	no	—
41	C₂×SD₁₆×C₅₉	—	2	2	no	—
42	C₂×SD₁₆×C₅₉	—	2	2	no	—
43	C₂×D₈×C₅₉	—	2	1	no	—
44	C₂×D₈×C₅₉	—	2	1	no	—
45	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
47	C₄⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
48	#127×C₅₉	C₂×C₄	1	2	yes	—
49	#124×C₅₉	—	2	2	no	—
50	#121×C₅₉	—	2	2	no	—
51	#121×C₅₉	—	2	2	no	—
52	C₄×D₄×C₅₉	—	2	2	no	—
53	C₄×D₄×C₅₉	—	2	2	no	—
54	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
58	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
59	#41×C₅₉	—	4	2	no	—
60	#41×C₅₉	—	4	2	no	—
61	#41×C₅₉	—	4	2	no	—
62	#41×C₅₉	—	4	2	no	—
63	C₂×Q₈×C₅₉	—	2	2	no	—
64	C₂×D₄×C₅₉	—	2	2	no	—
65	C₂×D₄×C₅₉	—	2	2	no	—
66	C₂×D₄×C₅₉	—	2	2	no	—
67	C₂×D₄×C₅₉	—	4	2	no	—
68	C₂²×C₄×C₅₉	—	4	2	no	—
69	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
72	C₂²×Dic₅₉	—	2	2	no	—
73	Q₁₆×C₅₉	—	4	2	no	—
74	Q₁₆×C₅₉	—	4	2	no	—
75	SD₁₆×C₅₉	—	4	4	no	—
76	SD₁₆×C₅₉	—	4	4	no	—
77	D₈×C₅₉	—	4	2	no	—
78	D₈×C₅₉	—	4	2	no	—
79	C₂×C₈×C₅₉	—	2	1	no	—
80	C₂×C₈×C₅₉	—	2	1	no	—
81	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₉	D₈	1	2	yes	—
86	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
87	C₄⋊C₄×C₅₉	—	2	2	no	—
88	C₄⋊C₄×C₅₉	—	2	2	no	—
89	C₂²⋊C₄×C₅₉	—	4	2	no	—
90	C₂²⋊C₄×C₅₉	—	4	2	no	—
91	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅₉	#41	1	1	yes	characteristic
95	C₄⋊Dic₅₉	—	2	4	no	—
96	C₄×Dic₅₉	—	4	1	no	—
97	C₄×Dic₅₉	—	4	1	no	—
98	C₂×C₅₉⋊C₈	—	2	2	no	—
99	C₂×C₅₉⋊C₈	—	4	1	no	—
100	C₂×C₅₉⋊C₈	—	4	1	no	—
101	C₂×C₅₉⋊C₈	—	4	2	no	—
102	C₂³×C₅₉	—	4	2	no	—
103	C₂³×C₅₉	#121	1	1	yes	characteristic
104	Q₈×C₅₉	—	4	2	no	—
105	Q₈×C₅₉	—	4	2	no	—
106	D₄×C₅₉	—	4	2	no	—
107	D₄×C₅₉	—	4	2	no	—
108	D₄×C₅₉	—	4	4	no	—
109	D₄×C₅₉	—	8	2	no	—
110	D₄×C₅₉	—	8	2	no	—
111	D₄×C₅₉	—	8	2	no	—
112	C₂×C₄×C₅₉	—	2	2	no	—
113	C₂×C₄×C₅₉	—	4	1	no	—
114	C₂×C₄×C₅₉	—	4	2	no	—
115	C₂×C₄×C₅₉	—	8	2	no	—
116	C₂×C₄×C₅₉	—	8	2	no	—
117	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
121	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
122	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×Dic₅₉	—	2	4	no	—
128	C₂×Dic₅₉	—	4	2	no	—
129	C₂×Dic₅₉	—	4	2	no	—
130	C₈×C₅₉	—	2	1	no	—
131	C₈×C₅₉	—	2	1	no	—
132	C₈×C₅₉	#102	1	2	yes	—
133	C₈×C₅₉	#102	1	2	yes	—
134	C₅₉⋊C₈	—	4	2	no	—
135	C₅₉⋊C₈	—	8	2	no	—
136	#9750	—	59	1	no	maximal, Sylow
137	C₂²×C₅₉	—	2	1	no	—
138	C₂²×C₅₉	—	2	1	no	—
139	C₂²×C₅₉	—	8	2	no	—
140	C₂²×C₅₉	—	8	2	no	—
141	C₂²×C₅₉	—	8	2	no	—
142	C₂²×C₅₉	#390	1	1	yes	characteristic
143	C₂²×C₅₉	#459	1	1	yes	characteristic
144	C₂²×C₅₉	#460	1	1	yes	characteristic
145	C₄×C₅₉	—	2	1	no	—
146	C₄×C₅₉	—	2	1	no	—
147	C₄×C₅₉	—	2	1	no	—
148	C₄×C₅₉	—	8	2	no	—
149	C₄×C₅₉	C₂×#102	1	1	yes	characteristic
150	C₄×C₅₉	#418	1	1	yes	characteristic
151	C₄×C₅₉	#443	1	2	yes	—
152	C₄×C₅₉	#459	1	1	yes	characteristic
153	C₄×C₅₉	#460	1	1	yes	characteristic
154	Dic₅₉	—	4	4	no	—
155	#3147	D₅₉	1	1	yes	characteristic
156	C₈×C₄⋊C₄	—	59	1	no	—
157	#1564	—	59	1	no	—
158	#1502	—	59	2	no	—
159	#1480	—	59	2	no	—
160	C₂×C₅₉	—	2	1	no	—
161	C₂×C₅₉	—	2	1	no	—
162	C₂×C₅₉	—	16	2	no	—
163	C₂×C₅₉	#1974	1	1	yes	characteristic
164	C₂×C₅₉	#1976	1	1	yes	characteristic
165	C₂×C₅₉	#1982	1	1	yes	characteristic
166	C₂×#135	—	2	1	no	—
167	C₂×#135	—	2	1	no	—
168	#532	C₂×D₅₉	1	2	yes	—
169	#495	—	2	1	no	—
170	#495	—	2	1	no	—
171	#494	Dic₅₉	1	1	yes	characteristic, complemented
172	#493	Dic₅₉	1	1	yes	characteristic, complemented
173	#492	Dic₅₉	1	2	yes	complemented
174	#431	—	59	2	no	—
175	C₂×C₄⋊C₈	—	59	1	no	—
176	C₂×C₄×C₈	—	118	1	no	—
177	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
178	C₄×C₄⋊C₄	—	59	1	no	—
179	#336	—	118	1	no	—
180	#335	—	118	1	no	—
181	#334	—	118	1	no	—
182	#327	—	118	4	no	—
183	D₄⋊C₈	—	118	4	no	—
184	C₅₉	#9750	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	2	2	no	—
186	#135	—	4	2	no	—
187	#135	—	4	2	no	—
188	#135	—	4	2	no	—
189	#135	—	4	2	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×SD₁₆	—	2	2	no	—
193	C₂×SD₁₆	—	2	2	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂²×C₈	—	118	1	no	—
197	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
198	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
199	C₄⋊Q₈	C₂×Dic₅₉	1	2	yes	—
200	#127	C₂×Dic₅₉	1	2	yes	—
201	#124	—	2	2	no	—
202	#121	—	2	2	no	—
203	#121	—	2	2	no	—
204	C₄×D₄	—	2	2	no	—
205	C₄×D₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	118	1	no	—
207	C₂×C₄²	—	118	1	no	—
208	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
209	C₄⋊C₈	—	59	2	no	—
210	C₄⋊C₈	—	118	1	no	—
211	C₄⋊C₈	—	118	2	no	—
212	C₂²⋊C₈	—	118	2	no	—
213	C₄×C₈	—	118	2	no	—
214	C₄×C₈	—	236	2	no	—
215	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
216	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
217	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
218	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
219	#95	—	59	2	no	—
220	#41	—	4	2	no	—
221	#41	—	4	2	no	—
222	#41	—	4	2	no	—
223	#41	—	4	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	118	2	no	—
231	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
232	C₂²×C₄	#11745687	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
234	Q₁₆	—	4	2	no	—
235	Q₁₆	—	4	2	no	—
236	SD₁₆	—	4	4	no	—
237	SD₁₆	—	4	4	no	—
238	D₈	—	4	2	no	—
239	D₈	—	4	2	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	118	2	no	—
243	C₂×C₈	—	236	2	no	—
244	C₂×C₈	—	236	1	no	—
245	C₂×C₈	—	236	1	no	—
246	C₂×C₈	#11745669	1	2	yes	—
247	C₂×C₈	#11745670	1	2	yes	—
248	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
251	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	118	4	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	236	1	no	—
258	C₄²	—	236	1	no	—
259	C₄²	D₄×D₅₉	1	1	yes	characteristic
260	C₄²	#11745687	1	1	yes	characteristic
261	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
262	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
263	C₂³	—	4	2	no	—
264	C₂³	#14273698	1	1	yes	characteristic
265	Q₈	—	4	2	no	—
266	Q₈	—	4	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	4	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	D₄	—	8	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	8	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	118	4	no	—
279	C₂×C₄	—	236	2	no	—
280	C₂×C₄	—	236	2	no	—
281	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
282	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
283	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
284	C₂×C₄	#14273698	1	1	yes	characteristic
285	C₂×C₄	#14273714	1	1	yes	characteristic
286	C₂×C₄	#14273723	1	1	yes	characteristic
287	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
288	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
289	C₂×C₄	#14273753	1	1	yes	characteristic
290	C₂×C₄	#14273753	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	472	2	no	—
294	C₈	—	236	2	no	—
295	C₈	#14273648	1	2	yes	—
296	C₈	#14273648	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	#16837305	1	1	yes	characteristic
303	C₂²	#16837594	1	1	yes	characteristic
304	C₂²	#16837599	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	8	2	no	—
309	C₄	—	236	4	no	—
310	C₄	C₂×#14273648	1	1	yes	characteristic
311	C₄	#16837407	1	1	yes	characteristic
312	C₄	#16837519	1	2	yes	—
313	C₄	#16837594	1	1	yes	characteristic
314	C₄	#16837599	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	16	2	no	—
318	C₂	#21424985	1	1	yes	characteristic
319	C₂	#21424995	1	1	yes	characteristic
320	C₂	#21425024	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273698
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting, prime radical	C₅₉×#3147	C₂
socle	C₂²×C₅₉	#390
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#3147	D₅₉
hypocenter	C₅₉	#9750
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₈×C₄⋊Dic₅₉	C₂	semidirect
2. #21422844	C₂	semidirect
3. #21422591	C₂	semidirect
4. #21422495	C₂	semidirect
20. C₅₉×#494	C₄	semidirect
21. C₅₉×#493	C₄	semidirect
22. C₅₉×#492	C₄	semidirect
171. #494	Dic₅₉	semidirect
172. #493	Dic₅₉	semidirect
173. #492	Dic₅₉	semidirect
184. C₅₉	#9750	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₄⋊Dic₅₉	2
#21422844	2
#21422591	2
#21422495	2
C₅₉×#3147	2
#9750	59