Subgroups — #28481793

Order 15104.

494 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424224	C₂	1	1	yes	maximal, characteristic, complemented
2	#21424096	C₂	1	2	yes	maximal
3	C₂×#16837155	C₂	1	1	yes	maximal, characteristic, complemented
4	#21421980	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₉×#436	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14273648	—	2	2	no	—
8	#16837272	C₂²	1	2	yes	—
9	#16837165	—	2	1	no	—
10	#16837165	—	2	1	no	—
11	#16837164	—	2	2	no	—
12	#16837155	—	2	2	no	—
13	C₂²×M₁₆×C₅₉	—	2	1	no	—
14	C₂×C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#436	—	2	2	no	—
16	C₅₉×#436	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₅₉	—	2	1	no	—
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₅₉⋊C₈	—	2	2	no	—
21	C₂×C₄⋊Dic₅₉	—	2	2	no	—
22	C₂×C₄⋊Dic₅₉	—	2	2	no	—
23	C₂×C₄⋊Dic₅₉	—	2	2	no	—
24	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	#14273648	—	4	4	no	—
27	C₂²×D₄×C₅₉	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₉	—	2	1	no	—
29	C₂×M₁₆×C₅₉	—	2	4	no	—
30	C₂×M₁₆×C₅₉	—	4	4	no	—
31	C₂²×C₈×C₅₉	—	2	1	no	—
32	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
34	C₄×D₄×C₅₉	—	2	1	no	—
35	C₄×D₄×C₅₉	—	2	1	no	—
36	C₄×D₄×C₅₉	C₂×C₄	1	2	yes	—
37	C₄×D₄×C₅₉	Q₈	1	2	yes	—
38	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
40	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₉	—	2	1	no	—
42	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
43	C₂²⋊C₈×C₅₉	—	2	4	no	—
44	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₅₉	D₄	1	2	yes	—
46	C₂⁴×C₅₉	—	2	1	no	—
47	C₂×D₄×C₅₉	—	2	1	no	—
48	C₂×D₄×C₅₉	—	2	2	no	—
49	C₂×D₄×C₅₉	—	2	2	no	—
50	C₂×D₄×C₅₉	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₉	—	2	1	no	—
52	C₂²×C₄×C₅₉	—	2	2	no	—
53	C₂²×C₄×C₅₉	—	2	2	no	—
54	C₂²×C₄×C₅₉	—	4	2	no	—
55	C₂²×C₄×C₅₉	—	4	2	no	—
56	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
59	C₂²×Dic₅₉	—	4	2	no	—
60	M₁₆×C₅₉	—	4	8	no	—
61	C₂×C₈×C₅₉	—	2	2	no	—
62	C₂×C₈×C₅₉	—	2	2	no	—
63	C₂×C₈×C₅₉	—	2	2	no	—
64	C₂×C₈×C₅₉	—	4	2	no	—
65	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
67	C₄⋊C₄×C₅₉	—	2	1	no	—
68	C₄⋊C₄×C₅₉	C₄⋊C₄	1	2	yes	—
69	C₂²⋊C₄×C₅₉	—	2	2	no	—
70	C₂²⋊C₄×C₅₉	—	2	2	no	—
71	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₅₉	#41	1	1	yes	characteristic
75	C₄⋊Dic₅₉	—	4	2	no	—
76	C₄⋊Dic₅₉	—	4	2	no	—
77	C₄⋊Dic₅₉	—	4	2	no	—
78	C₄⋊Dic₅₉	—	4	2	no	—
79	C₄⋊Dic₅₉	—	8	2	no	—
80	C₂×C₅₉⋊C₈	—	4	2	no	—
81	C₂×C₅₉⋊C₈	—	4	2	no	—
82	C₂×C₅₉⋊C₈	—	4	2	no	—
83	C₂³×C₅₉	—	2	2	no	—
84	C₂³×C₅₉	—	4	2	no	—
85	C₂³×C₅₉	—	4	2	no	—
86	C₂³×C₅₉	—	4	1	no	—
87	C₂³×C₅₉	—	4	1	no	—
88	C₂³×C₅₉	#121	1	1	yes	characteristic
89	D₄×C₅₉	—	2	4	no	—
90	D₄×C₅₉	—	4	1	no	—
91	D₄×C₅₉	—	4	1	no	—
92	C₂×C₄×C₅₉	—	2	1	no	—
93	C₂×C₄×C₅₉	—	2	1	no	—
94	C₂×C₄×C₅₉	—	2	2	no	—
95	C₂×C₄×C₅₉	—	4	2	no	—
96	C₂×C₄×C₅₉	—	4	2	no	—
97	C₂×C₄×C₅₉	—	4	2	no	—
98	C₂×C₄×C₅₉	—	4	4	no	—
99	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
102	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
103	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅₉	#135	1	2	yes	—
106	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
107	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
108	C₂×C₄×C₅₉	#137	1	2	yes	—
109	C₂×Dic₅₉	—	8	2	no	—
110	C₂×Dic₅₉	—	8	2	no	—
111	C₂×Dic₅₉	—	8	2	no	—
112	C₈×C₅₉	—	2	2	no	—
113	C₈×C₅₉	—	2	2	no	—
114	C₈×C₅₉	—	4	2	no	—
115	C₅₉⋊C₈	—	8	2	no	—
116	#8654	—	59	1	no	maximal, Sylow
117	C₂²×C₅₉	—	4	2	no	—
118	C₂²×C₅₉	—	4	2	no	—
119	C₂²×C₅₉	—	4	2	no	—
120	C₂²×C₅₉	—	8	2	no	—
121	C₂²×C₅₉	—	8	2	no	—
122	C₂²×C₅₉	#380	1	1	yes	characteristic
123	C₂²×C₅₉	#463	1	2	yes	—
124	C₂²×C₅₉	#468	1	2	yes	—
125	C₂²×C₅₉	#472	1	1	yes	characteristic
126	C₂²×C₅₉	#473	1	1	yes	characteristic
127	C₄×C₅₉	—	2	1	no	—
128	C₄×C₅₉	—	2	1	no	—
129	C₄×C₅₉	—	4	2	no	—
130	C₄×C₅₉	#416	1	1	yes	characteristic
131	C₄×C₅₉	#418	1	1	yes	characteristic
132	C₄×C₅₉	#428	1	2	yes	—
133	C₄×C₅₉	#469	1	1	yes	characteristic
134	C₄×C₅₉	#470	1	1	yes	characteristic
135	C₄×C₅₉	#478	1	2	yes	—
136	Dic₅₉	—	16	2	no	—
137	C₂×#436	D₅₉	1	1	yes	characteristic
138	#1834	—	59	1	no	—
139	#1802	—	59	2	no	—
140	C₂×#335	—	59	1	no	—
141	#1296	—	59	2	no	—
142	C₂×C₅₉	—	8	2	no	—
143	C₂×C₅₉	#1697	1	2	yes	—
144	C₂×C₅₉	#1699	1	2	yes	—
145	C₂×C₅₉	#1804	1	1	yes	characteristic
146	C₂×C₅₉	#1809	1	1	yes	characteristic
147	C₂×C₅₉	#1816	1	1	yes	characteristic
148	C₂²×M₁₆	—	2	1	no	—
149	C₂×C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
150	#436	—	2	2	no	—
151	#436	Dic₅₉	1	4	yes	complemented
152	C₂×C₄⋊C₈	—	59	2	no	—
153	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
154	C₂×#102	—	118	2	no	—
155	C₂×C₂²⋊C₈	—	2	1	no	—
156	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
157	#384	—	59	2	no	—
158	#340	—	118	1	no	—
159	#340	—	118	1	no	—
160	#340	—	118	2	no	—
161	#335	—	118	2	no	—
162	C₅₉	#8654	1	1	yes	characteristic, complemented, Sylow
163	C₂²×D₄	D₂₃₆	1	1	yes	characteristic
164	C₂³×C₄	—	2	1	no	—
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	—	4	4	no	—
167	C₂²×C₈	—	2	1	no	—
168	C₂²×C₈	—	118	2	no	—
169	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
170	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
171	C₄×D₄	—	2	1	no	—
172	C₄×D₄	—	2	1	no	—
173	C₄×D₄	Dic₁₁₈	1	2	yes	—
174	C₄×D₄	C₂×Dic₅₉	1	2	yes	—
175	C₂×C₄⋊C₄	—	118	2	no	—
176	C₂×C₄⋊C₄	—	118	2	no	—
177	C₂×C₄⋊C₄	—	118	2	no	—
178	C₂×C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
179	C₂×C₂²⋊C₄	—	2	1	no	—
180	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
181	C₄⋊C₈	—	2	1	no	—
182	C₄⋊C₈	—	118	2	no	—
183	C₄⋊C₈	—	118	2	no	—
184	C₄⋊C₈	C₂×Dic₅₉	1	2	yes	—
185	#102	—	236	4	no	—
186	C₂²⋊C₈	—	2	4	no	—
187	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
188	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
189	C₂⁴	—	2	1	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	C₄⋊Dic₅₉	1	2	yes	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	236	2	no	—
200	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
201	C₂²×C₄	#11745685	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
203	M₁₆	—	4	8	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	236	2	no	—
209	C₂×C₈	—	236	2	no	—
210	C₂×C₈	—	236	2	no	—
211	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
212	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	472	2	no	—
215	C₄⋊C₄	—	236	2	no	—
216	C₄⋊C₄	—	236	2	no	—
217	C₄⋊C₄	—	236	2	no	—
218	C₄⋊C₄	—	236	2	no	—
219	C₄⋊C₄	C₄⋊Dic₅₉	1	2	yes	—
220	C₂²⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	—	2	2	no	—
222	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
223	C₄²	#11745685	1	1	yes	characteristic
224	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
226	C₂³	—	2	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	4	1	no	—
229	C₂³	—	4	1	no	—
230	C₂³	—	4	2	no	—
231	C₂³	#14273701	1	1	yes	characteristic
232	D₄	—	2	4	no	—
233	D₄	—	4	1	no	—
234	D₄	—	4	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	472	2	no	—
243	C₂×C₄	—	472	2	no	—
244	C₂×C₄	—	472	2	no	—
245	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
246	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
247	C₂×C₄	#14273686	1	1	yes	characteristic
248	C₂×C₄	#14273701	1	1	yes	characteristic
249	C₂×C₄	#14273706	1	1	yes	characteristic
250	C₂×C₄	#14273706	1	1	yes	characteristic
251	C₂×C₄	#14273747	1	2	yes	—
252	C₂×C₄	#14273753	1	2	yes	—
253	C₂×C₄	#14273755	1	2	yes	—
254	C₂×C₄	#14273760	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₈	—	4	2	no	—
258	C₈	—	472	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	8	2	no	—
263	C₂²	—	8	2	no	—
264	C₂²	#16837262	1	1	yes	characteristic
265	C₂²	#16837617	1	2	yes	—
266	C₂²	#16837648	1	2	yes	—
267	C₂²	#16837672	1	1	yes	characteristic
268	C₂²	#16837677	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	944	2	no	—
273	C₄	#16837398	1	1	yes	characteristic
274	C₄	#16837407	1	1	yes	characteristic
275	C₄	#16837444	1	2	yes	—
276	C₄	#16837655	1	1	yes	characteristic
277	C₄	#16837660	1	1	yes	characteristic
278	C₄	#16837702	1	2	yes	—
279	C₂	—	8	2	no	—
280	C₂	#21423629	1	2	yes	—
281	C₂	#21423643	1	2	yes	—
282	C₂	#21424106	1	1	yes	characteristic
283	C₂	#21424129	1	1	yes	characteristic
284	C₂	#21424160	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273701
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#436	C₂
socle	C₂³×C₅₉	#121
prime radical	C₂²×D₄×C₅₉	D₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#436	D₅₉
hypocenter	C₅₉	#8654
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. #21424224	C₂	semidirect
3. C₂×#16837155	C₂	semidirect
4. #21421980	C₂	semidirect
16. C₅₉×#436	C₄	semidirect
151. #436	Dic₅₉	semidirect
162. C₅₉	#8654	semidirect

Maximal subgroups

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

subgroup	index
#21424224	2
#21424096	2
C₂×#16837155	2
#21421980	2
C₂×C₅₉×#436	2
#8654	59