Subgroups — #28487626

Order 15104.

498 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	#21422844	C₂	1	1	yes	maximal, characteristic, complemented
2	#21422487	C₂	1	2	yes	maximal, complemented
3	#21422417	C₂	1	2	yes	maximal, complemented
4	#21422388	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×D₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	#16837455	C₂²	1	1	yes	characteristic
8	#16837455	C₂²	1	1	yes	characteristic
9	#16837155	—	2	1	no	—
10	#16837153	—	2	1	no	—
11	D₄⋊(C₅₉⋊C₈)	—	2	4	no	—
12	#16837126	—	2	4	no	—
13	#16837124	—	2	1	no	—
14	#16837124	—	2	1	no	—
15	C₂²×D₈×C₅₉	C₄	1	1	yes	characteristic
16	C₂×C₄×D₄×C₅₉	C₂²	1	2	yes	—
17	C₄×D₈×C₅₉	—	2	1	no	—
18	C₄×D₈×C₅₉	—	2	1	no	—
19	C₄×D₈×C₅₉	—	2	1	no	—
20	C₄×D₈×C₅₉	—	2	1	no	—
21	C₂×#107×C₅₉	C₄	1	1	yes	characteristic
22	C₂×#102×C₅₉	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	C₄⋊(C₅₉⋊C₈)	D₄	1	2	yes	—
27	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
28	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
29	C₂²⋊(C₅₉⋊C₈)	D₄	1	2	yes	—
30	C₄×C₅₉⋊C₈	—	2	1	no	—
31	C₄×C₅₉⋊C₈	—	2	1	no	—
32	C₄×C₅₉⋊C₈	—	2	2	no	—
33	C₂²×D₄×C₅₉	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₅₉	—	2	2	no	—
35	C₂×D₈×C₅₉	—	2	4	no	—
36	C₂×D₈×C₅₉	—	2	1	no	—
37	C₂×D₈×C₅₉	—	2	1	no	—
38	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₄×D₄×C₅₉	—	2	2	no	—
41	C₄×D₄×C₅₉	—	2	2	no	—
42	C₄×D₄×C₅₉	—	2	2	no	—
43	C₄×D₄×C₅₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
46	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
47	#107×C₅₉	—	2	1	no	—
48	#107×C₅₉	—	2	1	no	—
49	#102×C₅₉	—	2	2	no	—
50	#102×C₅₉	—	2	2	no	—
51	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
54	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
55	C₂⁴×C₅₉	—	2	2	no	—
56	C₂×D₄×C₅₉	—	2	2	no	—
57	C₂×D₄×C₅₉	—	2	2	no	—
58	C₂×D₄×C₅₉	—	2	2	no	—
59	C₂×D₄×C₅₉	—	2	2	no	—
60	C₂×D₄×C₅₉	—	2	2	no	—
61	C₂×D₄×C₅₉	M₁₆	1	4	yes	—
62	C₂²×C₄×C₅₉	—	2	2	no	—
63	C₂²×C₄×C₅₉	—	4	2	no	—
64	C₂²×C₄×C₅₉	—	4	2	no	—
65	C₂²×C₄×C₅₉	—	8	4	no	—
66	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
69	D₈×C₅₉	—	2	4	no	—
70	D₈×C₅₉	—	2	4	no	—
71	C₂×C₈×C₅₉	—	2	2	no	—
72	C₂×C₈×C₅₉	—	2	2	no	—
73	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₄⋊C₄×C₅₉	—	2	2	no	—
78	C₄⋊C₄×C₅₉	—	2	2	no	—
79	C₂²⋊C₄×C₅₉	—	4	2	no	—
80	C₂²⋊C₄×C₅₉	—	4	2	no	—
81	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₉	#41	1	1	yes	characteristic
85	C₂×C₅₉⋊C₈	—	2	2	no	—
86	C₂×C₅₉⋊C₈	—	2	2	no	—
87	C₂×C₅₉⋊C₈	—	4	2	no	—
88	C₂³×C₅₉	—	2	4	no	—
89	C₂³×C₅₉	—	4	2	no	—
90	C₂³×C₅₉	—	4	2	no	—
91	C₂³×C₅₉	—	8	2	no	—
92	C₂³×C₅₉	—	8	2	no	—
93	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
94	D₄×C₅₉	—	2	4	no	—
95	D₄×C₅₉	—	2	4	no	—
96	D₄×C₅₉	—	4	2	no	—
97	D₄×C₅₉	—	4	2	no	—
98	C₂×C₄×C₅₉	—	2	2	no	—
99	C₂×C₄×C₅₉	—	4	1	no	—
100	C₂×C₄×C₅₉	—	4	2	no	—
101	C₂×C₄×C₅₉	—	8	4	no	—
102	C₂×C₄×C₅₉	—	8	4	no	—
103	C₂×C₄×C₅₉	—	8	2	no	—
104	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
107	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
108	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
109	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₅₉	#131	1	2	yes	—
112	C₈×C₅₉	—	2	1	no	—
113	C₈×C₅₉	—	2	1	no	—
114	C₈×C₅₉	—	2	1	no	—
115	C₈×C₅₉	—	2	1	no	—
116	C₅₉⋊C₈	—	4	2	no	—
117	C₅₉⋊C₈	—	4	2	no	—
118	C₅₉⋊C₈	—	8	2	no	—
119	#9670	—	59	1	no	maximal, Sylow
120	C₂²×C₅₉	—	2	1	no	—
121	C₂²×C₅₉	—	2	1	no	—
122	C₂²×C₅₉	—	4	4	no	—
123	C₂²×C₅₉	—	8	4	no	—
124	C₂²×C₅₉	—	8	4	no	—
125	C₂²×C₅₉	—	8	2	no	—
126	C₂²×C₅₉	—	8	2	no	—
127	C₂²×C₅₉	#390	1	1	yes	characteristic
128	C₂²×C₅₉	#435	1	2	yes	—
129	C₄×C₅₉	—	2	1	no	—
130	C₄×C₅₉	—	2	1	no	—
131	C₄×C₅₉	—	2	1	no	—
132	C₄×C₅₉	—	8	2	no	—
133	C₄×C₅₉	#407	1	1	yes	characteristic
134	C₄×C₅₉	#408	1	1	yes	characteristic
135	C₄×C₅₉	#435	1	2	yes	—
136	C₄×C₅₉	#472	1	2	yes	—
137	C₂×C₄×D₈	D₅₉	1	1	yes	characteristic
138	#1564	—	59	1	no	—
139	#1478	—	59	2	no	—
140	#1460	—	59	2	no	—
141	#1452	—	59	1	no	—
142	C₂×C₅₉	—	2	2	no	—
143	C₂×C₅₉	—	8	4	no	—
144	C₂×C₅₉	#1955	1	1	yes	characteristic
145	C₂×C₅₉	#1976	1	1	yes	characteristic
146	C₂×C₅₉	#1978	1	1	yes	characteristic
147	C₂²×D₈	Dic₅₉	1	1	yes	characteristic
148	C₂×C₄×D₄	C₂×D₅₉	1	2	yes	—
149	C₄×D₈	—	2	1	no	—
150	C₄×D₈	—	2	1	no	—
151	C₄×D₈	—	2	1	no	—
152	C₄×D₈	—	2	1	no	—
153	#431	—	59	1	no	—
154	#431	—	59	1	no	—
155	#431	—	59	2	no	—
156	C₂×#107	Dic₅₉	1	1	yes	characteristic
157	C₂×#102	Dic₅₉	1	2	yes	—
158	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
159	#335	—	118	1	no	—
160	#334	—	118	1	no	—
161	D₄⋊C₈	—	118	4	no	—
162	#323	—	118	4	no	—
163	#322	—	118	1	no	—
164	#322	—	118	1	no	—
165	C₅₉	#9670	1	1	yes	characteristic, complemented, Sylow
166	C₂²×D₄	C₂×Dic₅₉	1	2	yes	—
167	C₂³×C₄	—	2	2	no	—
168	C₂×D₈	—	2	4	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	—	2	1	no	—
171	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
172	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
173	C₄×D₄	—	2	2	no	—
174	C₄×D₄	—	2	2	no	—
175	C₄×D₄	—	2	2	no	—
176	C₄×D₄	—	2	2	no	—
177	C₂×C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
178	C₂×C₂²⋊C₄	—	2	2	no	—
179	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
180	#107	—	2	1	no	—
181	#107	—	2	1	no	—
182	C₄⋊C₈	—	59	2	no	—
183	C₄⋊C₈	—	118	1	no	—
184	C₄⋊C₈	—	118	2	no	—
185	#102	—	2	2	no	—
186	#102	—	2	2	no	—
187	C₂²⋊C₈	—	59	2	no	—
188	C₂²⋊C₈	—	118	1	no	—
189	C₂²⋊C₈	—	118	2	no	—
190	C₄×C₈	—	118	2	no	—
191	C₄×C₈	—	118	1	no	—
192	C₄×C₈	—	118	1	no	—
193	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
194	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
196	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
197	C₂⁴	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	#11745664	1	4	yes	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	8	4	no	—
208	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
209	C₂²×C₄	#11745687	1	1	yes	characteristic
210	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
211	D₈	—	2	4	no	—
212	D₈	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	118	2	no	—
216	C₂×C₈	—	118	2	no	—
217	C₂×C₈	—	236	2	no	—
218	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
220	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
221	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₄²	D₄×D₅₉	1	1	yes	characteristic
227	C₄²	#11745687	1	1	yes	characteristic
228	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
229	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
230	C₂³	—	2	4	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	—	8	2	no	—
234	C₂³	—	8	2	no	—
235	C₂³	D₄×Dic₅₉	1	1	yes	characteristic
236	D₄	—	2	4	no	—
237	D₄	—	2	4	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	8	4	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
247	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
248	C₂×C₄	#14273698	1	1	yes	characteristic
249	C₂×C₄	#14273698	1	1	yes	characteristic
250	C₂×C₄	#14273714	1	1	yes	characteristic
251	C₂×C₄	#14273723	1	1	yes	characteristic
252	C₂×C₄	C₂×#11745664	1	2	yes	—
253	C₂×C₄	#14273734	1	2	yes	—
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	472	2	no	—
259	C₈	—	236	2	no	—
260	C₈	—	236	2	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	4	4	no	—
264	C₂²	—	8	4	no	—
265	C₂²	—	8	4	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	8	2	no	—
268	C₂²	#16837305	1	1	yes	characteristic
269	C₂²	#16837474	1	2	yes	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	8	2	no	—
274	C₄	#16837366	1	1	yes	characteristic
275	C₄	#16837369	1	1	yes	characteristic
276	C₄	#16837474	1	2	yes	—
277	C₄	#16837669	1	2	yes	—
278	C₂	—	2	2	no	—
279	C₂	—	8	4	no	—
280	C₂	#21424895	1	1	yes	characteristic
281	C₂	#21424995	1	1	yes	characteristic
282	C₂	#21425007	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273698
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₄×D₈×C₅₉	C₂
socle	C₂²×C₅₉	#390
prime radical	C₂²×D₈×C₅₉	C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×C₄×D₈	D₅₉
hypocenter	C₅₉	#9670
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #21422844	C₂	semidirect
2. #21422487	C₂	semidirect
3. #21422417	C₂	semidirect
4. #21422388	C₂	semidirect
165. C₅₉	#9670	semidirect

Maximal subgroups

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

subgroup	index
#21422844	2
#21422487	2
#21422417	2
#21422388	2
C₂×C₄×D₈×C₅₉	2
#9670	59