Subgroups — #28608451

Order 15104.

766 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	#21427061	C₂	1	2	yes	maximal
2	#21424958	C₂	1	2	yes	maximal
3	#21424732	C₂	1	4	yes	maximal
4	#21424260	C₂	1	1	yes	maximal, characteristic
5	#21424111	C₂	1	2	yes	maximal
6	#107×Dic₅₉	C₂	1	2	yes	maximal
7	C₈×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#14273661	C₂²	1	2	yes	—
10	C₂×#14273659	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#14273651	C₂²	1	4	yes	—
13	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
14	#16837309	C₂²	1	4	yes	—
15	#16837288	—	2	4	no	—
16	#16837270	—	2	2	no	—
17	#16837268	C₂²	1	2	yes	—
18	C₄⋊C₄×Dic₅₉	C₂²	1	4	yes	—
19	C₄×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
20	#16837248	—	2	2	no	—
21	#16837164	C₂²	1	4	yes	—
22	#16837156	C₂²	1	2	yes	—
23	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₅₉	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₅₉	C₄	1	8	yes	complemented
26	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#103×C₅₉	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
29	C₂²×C₅₉⋊C₈	C₂³	1	2	yes	—
30	C₂×C₄⋊Dic₅₉	D₄	1	2	yes	—
31	C₂×C₄⋊Dic₅₉	C₂³	1	4	yes	—
32	C₂×C₄×Dic₅₉	—	2	4	no	—
33	C₂×C₄×Dic₅₉	C₂³	1	2	yes	—
34	#14273661	—	2	4	no	—
35	#14273659	—	4	2	no	—
36	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
37	#14273651	—	4	4	no	—
38	C₈×Dic₅₉	—	2	4	no	—
39	C₅₉⋊#95	—	2	2	no	—
40	C₅₉⋊#95	—	2	2	no	—
41	C₅₉⋊#95	—	2	4	no	—
42	C₅₉⋊#95	—	2	4	no	—
43	C₅₉⋊#95	—	2	4	no	—
44	C₅₉⋊#95	D₄	1	1	yes	characteristic
45	C₅₉⋊#95	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₅₉	C₂³	1	2	yes	—
47	C₂×Q₁₆×C₅₉	—	2	2	no	—
48	C₂×Q₁₆×C₅₉	—	2	2	no	—
49	C₂×Q₁₆×C₅₉	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
52	C₄×Q₈×C₅₉	—	2	4	no	—
53	C₄×Q₈×C₅₉	C₂×C₄	1	8	yes	—
54	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
55	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
56	C₂×C₄²×C₅₉	—	2	2	no	—
57	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
58	#107×C₅₉	C₂×C₄	1	4	yes	—
59	#103×C₅₉	C₂×C₄	1	8	yes	—
60	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
61	C₂×Q₈×C₅₉	—	2	2	no	—
62	C₂×Q₈×C₅₉	—	2	2	no	—
63	C₂×Q₈×C₅₉	—	2	2	no	—
64	C₂×Q₈×C₅₉	—	2	2	no	—
65	C₂×Q₈×C₅₉	—	2	2	no	—
66	C₂×Q₈×C₅₉	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₅₉	—	2	2	no	—
68	C₂²×C₄×C₅₉	—	2	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	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₅₉	—	2	4	no	—
73	C₂²×Dic₅₉	C₂×D₄	1	2	yes	—
74	C₂²×Dic₅₉	C₂×D₄	1	2	yes	—
75	Q₁₆×C₅₉	—	2	8	no	—
76	C₂×C₈×C₅₉	—	2	1	no	—
77	C₂×C₈×C₅₉	—	2	1	no	—
78	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₅₉	#41	1	2	yes	—
81	C₂×C₈×C₅₉	#41	1	2	yes	—
82	C₄⋊C₄×C₅₉	—	2	8	no	—
83	C₄⋊C₄×C₅₉	—	2	2	no	—
84	C₄⋊C₄×C₅₉	C₂²×C₄	1	4	yes	—
85	C₄²×C₅₉	—	2	8	no	—
86	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
87	C₄²×C₅₉	#41	1	2	yes	—
88	C₄⋊Dic₅₉	—	2	4	no	—
89	C₄⋊Dic₅₉	—	2	4	no	—
90	C₄⋊Dic₅₉	—	2	4	no	—
91	C₄×Dic₅₉	—	2	2	no	—
92	C₄×Dic₅₉	—	2	2	no	—
93	C₄×Dic₅₉	—	4	8	no	—
94	C₂×C₅₉⋊C₈	—	2	2	no	—
95	C₂×C₅₉⋊C₈	—	2	2	no	—
96	C₂×C₅₉⋊C₈	—	2	2	no	—
97	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₅₉	—	2	4	no	—
99	Q₈×C₅₉	—	2	4	no	—
100	Q₈×C₅₉	—	4	2	no	—
101	Q₈×C₅₉	—	4	2	no	—
102	C₂×C₄×C₅₉	—	2	2	no	—
103	C₂×C₄×C₅₉	—	2	2	no	—
104	C₂×C₄×C₅₉	—	2	4	no	—
105	C₂×C₄×C₅₉	—	2	4	no	—
106	C₂×C₄×C₅₉	—	4	2	no	—
107	C₂×C₄×C₅₉	—	4	2	no	—
108	C₂×C₄×C₅₉	—	4	4	no	—
109	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
117	C₂×Dic₅₉	—	2	2	no	—
118	C₂×Dic₅₉	—	2	2	no	—
119	C₂×Dic₅₉	—	2	2	no	—
120	C₂×Dic₅₉	—	2	2	no	—
121	C₂×Dic₅₉	—	2	2	no	—
122	C₂×Dic₅₉	—	4	4	no	—
123	C₂×Dic₅₉	—	4	4	no	—
124	C₂×Dic₅₉	—	4	4	no	—
125	C₂×Dic₅₉	#135	1	4	yes	—
126	C₈×C₅₉	—	2	2	no	—
127	C₈×C₅₉	#117	1	4	yes	—
128	C₅₉⋊C₈	—	4	2	no	—
129	#22809	—	59	1	no	maximal, Sylow
130	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₅₉	#547	1	1	yes	characteristic
132	C₂²×C₅₉	#550	1	1	yes	characteristic
133	C₂²×C₅₉	C₂×#135	1	2	yes	—
134	C₂²×C₅₉	#578	1	2	yes	—
135	C₄×C₅₉	—	2	2	no	—
136	C₄×C₅₉	—	2	2	no	—
137	C₄×C₅₉	—	4	4	no	—
138	C₄×C₅₉	—	4	4	no	—
139	C₄×C₅₉	C₂×#117	1	1	yes	characteristic
140	C₄×C₅₉	#520	1	1	yes	characteristic
141	C₄×C₅₉	#555	1	1	yes	characteristic
142	C₄×C₅₉	#556	1	1	yes	characteristic
143	Dic₅₉	—	2	4	no	—
144	Dic₅₉	—	4	2	no	—
145	Dic₅₉	—	8	4	no	—
146	C₂×C₄×Q₁₆	D₅₉	1	1	yes	characteristic
147	#2356	—	59	2	no	—
148	#1969	—	59	2	no	—
149	#1927	—	59	4	no	—
150	#1846	—	59	1	no	—
151	#1805	—	59	2	no	—
152	C₄×#107	—	59	2	no	—
153	C₈×C₄⋊C₄	—	59	1	no	—
154	C₂×C₅₉	#2354	1	1	yes	characteristic
155	C₂×C₅₉	C₄×#135	1	1	yes	characteristic
156	C₂×C₅₉	#2950	1	1	yes	characteristic
157	C₂×C₅₉	#3336	1	2	yes	—
158	C₂×C₅₉	#3393	1	2	yes	—
159	C₂²×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
160	C₂×C₄×Q₈	C₂×D₅₉	1	2	yes	—
161	C₄×Q₁₆	Dic₅₉	1	8	yes	complemented
162	C₂×#107	—	59	2	no	—
163	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
164	C₂×#106	—	59	2	no	—
165	C₂×C₄⋊C₈	—	59	1	no	—
166	C₂×#103	—	59	4	no	—
167	C₂×#103	C₂×D₅₉	1	2	yes	—
168	C₂×C₄×C₈	—	59	2	no	—
169	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
170	#391	—	59	4	no	—
171	#387	—	118	4	no	—
172	#383	—	118	2	no	—
173	#382	—	59	2	no	—
174	C₄×C₄⋊C₄	—	59	1	no	—
175	C₄×C₄⋊C₄	—	59	4	no	—
176	#376	—	118	2	no	—
177	#340	—	59	4	no	—
178	#336	—	59	2	no	—
179	C₅₉	#22809	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	C₂²×D₅₉	1	2	yes	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	C₂×Dic₅₉	1	4	yes	—
184	C₂²×C₈	—	59	2	no	—
185	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
187	C₄×Q₈	—	2	4	no	—
188	C₄×Q₈	C₂×Dic₅₉	1	8	yes	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	59	2	no	—
191	C₂×C₄⋊C₄	—	59	4	no	—
192	C₂×C₄⋊C₄	C₂²×D₅₉	1	2	yes	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	59	2	no	—
195	C₂×C₄²	—	118	4	no	—
196	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
197	#107	—	118	4	no	—
198	#107	C₂×Dic₅₉	1	4	yes	—
199	#106	—	236	2	no	—
200	C₄⋊C₈	—	118	2	no	—
201	#103	—	236	4	no	—
202	#103	C₂×Dic₅₉	1	8	yes	—
203	C₄×C₈	—	118	4	no	—
204	C₄×C₈	C₂×Dic₅₉	1	4	yes	—
205	#95	—	59	1	no	—
206	#95	—	59	1	no	—
207	#95	—	118	4	no	—
208	#95	—	118	4	no	—
209	#95	—	118	4	no	—
210	#95	—	118	2	no	—
211	#95	—	118	2	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	C₂²×Dic₅₉	1	4	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	59	2	no	—
221	C₂²×C₄	—	59	2	no	—
222	C₂²×C₄	—	118	4	no	—
223	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
224	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
226	Q₁₆	—	2	8	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	118	2	no	—
230	C₂×C₈	—	118	2	no	—
231	C₂×C₈	—	118	2	no	—
232	C₂×C₈	#11745685	1	2	yes	—
233	C₂×C₈	#11745685	1	2	yes	—
234	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
235	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	118	4	no	—
239	C₄⋊C₄	—	118	4	no	—
240	C₄⋊C₄	—	118	4	no	—
241	C₄⋊C₄	C₂²×Dic₅₉	1	4	yes	—
242	C₄²	—	2	8	no	—
243	C₄²	—	118	2	no	—
244	C₄²	—	118	2	no	—
245	C₄²	—	236	8	no	—
246	C₄²	#11745687	1	2	yes	—
247	C₄²	C₂²×Dic₅₉	1	2	yes	—
248	C₂³	C₂×D₄×D₅₉	1	1	yes	characteristic
249	Q₈	—	2	4	no	—
250	Q₈	—	2	4	no	—
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	59	4	no	—
261	C₂×C₄	—	118	2	no	—
262	C₂×C₄	—	118	2	no	—
263	C₂×C₄	—	118	2	no	—
264	C₂×C₄	—	118	2	no	—
265	C₂×C₄	—	118	2	no	—
266	C₂×C₄	—	236	4	no	—
267	C₂×C₄	—	236	4	no	—
268	C₂×C₄	—	236	4	no	—
269	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
270	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
271	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
272	C₂×C₄	C₂×#11745687	1	1	yes	characteristic
273	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
274	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
275	C₂×C₄	#14273773	1	1	yes	characteristic
276	C₂×C₄	#14273774	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	236	2	no	—
279	C₈	#14273686	1	4	yes	—
280	C₂²	C₂×D₄×Dic₅₉	1	1	yes	characteristic
281	C₂²	#16838057	1	1	yes	characteristic
282	C₂²	#16838073	1	1	yes	characteristic
283	C₂²	#135×D₅₉	1	2	yes	—
284	C₂²	#16838215	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	4	no	—
289	C₄	—	118	4	no	—
290	C₄	—	472	4	no	—
291	C₄	—	236	2	no	—
292	C₄	C₂×#14273686	1	1	yes	characteristic
293	C₄	#16837885	1	1	yes	characteristic
294	C₄	#16838117	1	1	yes	characteristic
295	C₄	#16838122	1	1	yes	characteristic
296	C₂	#21427045	1	1	yes	characteristic
297	C₂	#135×Dic₅₉	1	1	yes	characteristic
298	C₂	#21432964	1	1	yes	characteristic
299	C₂	#21436466	1	2	yes	—
300	C₂	#21437084	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₂×C₄×Q₁₆×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂²×D₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₅₉
hypocenter	C₅₉	#22809
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
25. C₄×Q₁₆×C₅₉	C₄	semidirect
161. C₄×Q₁₆	Dic₅₉	semidirect
179. C₅₉	#22809	semidirect

Maximal subgroups

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

subgroup	index
#21427061	2
#21424958	2
#21424732	2
#21424260	2
#21424111	2
#107×Dic₅₉	2
C₈×C₄⋊Dic₅₉	2
C₂×C₄×Q₁₆×C₅₉	2
#22809	59