Subgroups — #28496963

Order 15104.

634 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273661	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837170	C₂	1	2	yes	maximal
3	#21423816	C₂	1	2	yes	maximal
4	#21423792	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₉×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11745664	C₂²	1	1	yes	characteristic
7	C₂×#14273686	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₅₉	C₂²	1	2	yes	—
9	C₂×#14273661	—	2	4	no	—
10	C₂×#14273661	—	2	2	no	—
11	#16837170	—	2	4	no	—
12	#16837165	—	2	4	no	—
13	#16837156	—	2	2	no	—
14	C₂²×M₁₆×C₅₉	C₂²	1	2	yes	—
15	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#341	C₄	1	4	yes	complemented
17	C₂³×Dic₅₉	—	2	2	no	—
18	C₂×#11745664	—	2	1	no	—
19	C₂×#11745664	—	2	2	no	—
20	C₂×#11745664	—	2	2	no	—
21	C₂×#11745664	Q₈	1	2	yes	—
22	C₂²×C₅₉⋊C₈	—	2	1	no	—
23	#14273686	—	2	2	no	—
24	#14273686	—	2	2	no	—
25	C₂×C₄⋊Dic₅₉	—	2	2	no	—
26	C₂×C₄⋊Dic₅₉	—	2	2	no	—
27	C₂×C₄⋊Dic₅₉	—	2	2	no	—
28	C₂×C₄⋊Dic₅₉	—	2	2	no	—
29	C₂×C₄⋊Dic₅₉	—	2	2	no	—
30	C₂×C₄⋊Dic₅₉	D₄	1	2	yes	—
31	C₂×C₄⋊Dic₅₉	D₄	1	4	yes	—
32	C₂×C₂²⋊Dic₅₉	—	2	1	no	—
33	C₂×C₄×Dic₅₉	—	2	1	no	—
34	#14273661	—	2	8	no	—
35	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₅₉	—	2	2	no	—
37	C₂×M₁₆×C₅₉	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₅₉	—	2	2	no	—
43	C₂²×C₈×C₅₉	—	2	2	no	—
44	C₂²×C₈×C₅₉	—	2	2	no	—
45	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₅₉	D₄	1	2	yes	—
47	C₂²×C₈×C₅₉	Q₈	1	2	yes	—
48	C₂⁴×C₅₉	#41	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	#41	1	2	yes	—
56	C₂²×C₄×C₅₉	#41	1	2	yes	—
57	C₂²×C₄×C₅₉	#41	1	2	yes	—
58	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
59	C₂²×Dic₅₉	—	2	2	no	—
60	C₂²×Dic₅₉	—	2	4	no	—
61	C₂²×Dic₅₉	—	4	2	no	—
62	C₂²×Dic₅₉	—	4	2	no	—
63	C₂²×Dic₅₉	—	8	4	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	4	no	—
69	C₂×C₈×C₅₉	—	2	2	no	—
70	C₂×C₈×C₅₉	—	2	4	no	—
71	C₂×C₈×C₅₉	—	2	2	no	—
72	C₂×C₈×C₅₉	—	2	4	no	—
73	C₂×C₈×C₅₉	—	4	2	no	—
74	C₂×C₈×C₅₉	—	4	2	no	—
75	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
76	C₂²⋊Dic₅₉	—	4	2	no	—
77	C₄⋊Dic₅₉	—	2	4	no	—
78	C₄⋊Dic₅₉	—	2	4	no	—
79	C₄⋊Dic₅₉	—	2	4	no	—
80	C₄⋊Dic₅₉	—	4	2	no	—
81	C₄⋊Dic₅₉	—	4	2	no	—
82	C₄×Dic₅₉	—	4	1	no	—
83	C₄×Dic₅₉	—	4	1	no	—
84	#11745664	—	2	4	no	—
85	#11745664	—	4	2	no	—
86	C₂×C₅₉⋊C₈	—	2	2	no	—
87	C₂×C₅₉⋊C₈	—	4	1	no	—
88	C₂×C₅₉⋊C₈	—	4	1	no	—
89	C₂³×C₅₉	—	2	2	no	—
90	C₂³×C₅₉	—	2	2	no	—
91	C₂³×C₅₉	C₄×Q₈	1	1	yes	characteristic
92	C₂³×C₅₉	#121	1	1	yes	characteristic
93	C₂³×C₅₉	#123	1	2	yes	—
94	C₂³×C₅₉	#125	1	1	yes	characteristic
95	C₂³×C₅₉	#126	1	2	yes	—
96	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₉	#117	1	2	yes	—
98	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
101	C₂×C₄×C₅₉	#121	1	2	yes	—
102	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅₉	#123	1	2	yes	—
106	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
107	C₂×C₄×C₅₉	#124	1	2	yes	—
108	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
109	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
110	C₂×C₄×C₅₉	#125	1	2	yes	—
111	C₂×C₄×C₅₉	#125	1	2	yes	—
112	C₂×C₄×C₅₉	#126	1	2	yes	—
113	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×Dic₅₉	—	4	4	no	—
115	C₂×Dic₅₉	—	4	2	no	—
116	C₂×Dic₅₉	—	4	2	no	—
117	C₂×Dic₅₉	—	4	2	no	—
118	C₂×Dic₅₉	—	8	4	no	—
119	C₂×Dic₅₉	—	8	4	no	—
120	C₂×Dic₅₉	—	8	4	no	—
121	C₈×C₅₉	—	2	4	no	—
122	C₈×C₅₉	—	4	4	no	—
123	C₅₉⋊C₈	—	4	2	no	—
124	#11270	—	59	1	no	maximal, Sylow
125	C₂²×C₅₉	—	2	4	no	—
126	C₂²×C₅₉	—	2	2	no	—
127	C₂²×C₅₉	—	2	2	no	—
128	C₂²×C₅₉	—	2	2	no	—
129	C₂²×C₅₉	—	2	2	no	—
130	C₂²×C₅₉	#387	1	2	yes	—
131	C₂²×C₅₉	#389	1	1	yes	characteristic
132	C₂²×C₅₉	#394	1	1	yes	characteristic
133	C₂²×C₅₉	#397	1	1	yes	characteristic
134	C₂²×C₅₉	#400	1	2	yes	—
135	C₂²×C₅₉	#472	1	2	yes	—
136	C₂²×C₅₉	#473	1	2	yes	—
137	C₄×C₅₉	#382	1	1	yes	characteristic
138	C₄×C₅₉	#385	1	1	yes	characteristic
139	C₄×C₅₉	#388	1	2	yes	—
140	C₄×C₅₉	#393	1	1	yes	characteristic
141	C₄×C₅₉	#398	1	2	yes	—
142	C₄×C₅₉	#400	1	1	yes	characteristic
143	Dic₅₉	—	8	2	no	—
144	Dic₅₉	—	8	4	no	—
145	C₂²×#107	—	59	1	no	—
146	C₂×#343	—	59	2	no	—
147	C₂×#341	D₅₉	1	1	yes	characteristic
148	#1736	—	59	2	no	—
149	#1729	—	59	1	no	—
150	C₂×C₅₉	—	2	2	no	—
151	C₂×C₅₉	—	2	2	no	—
152	C₂×C₅₉	#1442	1	1	yes	characteristic
153	C₂×C₅₉	#1952	1	2	yes	—
154	C₂×C₅₉	#2045	1	1	yes	characteristic
155	C₂×C₅₉	#2047	1	1	yes	characteristic
156	C₂×C₅₉	#2078	1	2	yes	—
157	C₂²×M₁₆	—	59	1	no	—
158	C₂²×M₁₆	C₂×D₅₉	1	2	yes	—
159	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
160	C₂×#117	—	59	1	no	—
161	C₂²×C₄⋊C₄	—	59	2	no	—
162	C₂×#107	—	118	4	no	—
163	C₂×#107	—	118	2	no	—
164	#343	—	118	4	no	—
165	#341	Dic₅₉	1	4	yes	complemented
166	#340	—	118	4	no	—
167	#336	—	118	2	no	—
168	C₅₉	#11270	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	—	118	2	no	—
170	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
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	—	2	2	no	—
176	C₂×M₁₆	—	59	2	no	—
177	C₂×M₁₆	—	118	1	no	—
178	C₂×M₁₆	—	118	2	no	—
179	C₂×M₁₆	—	118	2	no	—
180	C₂×M₁₆	C₂×Dic₅₉	1	4	yes	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	118	1	no	—
185	C₂²×C₈	Dic₁₁₈	1	2	yes	—
186	C₂²×C₈	D₂₃₆	1	2	yes	—
187	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
188	#117	—	118	2	no	—
189	#117	—	118	2	no	—
190	C₂×C₄⋊C₄	—	59	4	no	—
191	C₂×C₄⋊C₄	—	59	2	no	—
192	C₂×C₄⋊C₄	—	118	2	no	—
193	C₂×C₄⋊C₄	—	118	2	no	—
194	C₂×C₄⋊C₄	—	118	2	no	—
195	C₂×C₄⋊C₄	—	118	2	no	—
196	C₂×C₄⋊C₄	—	118	2	no	—
197	C₂×C₂²⋊C₄	—	118	1	no	—
198	C₂×C₄²	—	118	1	no	—
199	#107	—	118	8	no	—
200	C₂⁴	#11745689	1	1	yes	characteristic
201	C₂²×C₄	—	118	4	no	—
202	C₂²×C₄	—	118	2	no	—
203	C₂²×C₄	—	472	4	no	—
204	C₂²×C₄	—	236	2	no	—
205	C₂²×C₄	—	236	2	no	—
206	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
208	C₂²×C₄	#11745685	1	2	yes	—
209	C₂²×C₄	#11745685	1	2	yes	—
210	C₂²×C₄	D₄×D₅₉	1	2	yes	—
211	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
212	C₂²×C₄	#11745687	1	2	yes	—
213	C₂²×C₄	#11745687	1	1	yes	characteristic
214	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
215	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	118	4	no	—
221	M₁₆	—	236	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	118	2	no	—
230	C₂×C₈	—	236	1	no	—
231	C₂×C₈	—	236	1	no	—
232	C₂×C₈	C₄⋊Dic₅₉	1	4	yes	—
233	C₄⋊C₄	—	118	4	no	—
234	C₄⋊C₄	—	118	4	no	—
235	C₄⋊C₄	—	118	4	no	—
236	C₄⋊C₄	—	236	2	no	—
237	C₄⋊C₄	—	236	2	no	—
238	C₂²⋊C₄	—	236	2	no	—
239	C₄²	—	236	1	no	—
240	C₄²	—	236	1	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	Q₈×Dic₅₉	1	1	yes	characteristic
244	C₂³	#14273702	1	1	yes	characteristic
245	C₂³	#14273712	1	2	yes	—
246	C₂³	#14273718	1	1	yes	characteristic
247	C₂³	#14273722	1	2	yes	—
248	C₂×C₄	—	472	4	no	—
249	C₂×C₄	—	472	4	no	—
250	C₂×C₄	—	472	4	no	—
251	C₂×C₄	—	236	2	no	—
252	C₂×C₄	—	236	2	no	—
253	C₂×C₄	—	236	4	no	—
254	C₂×C₄	—	236	2	no	—
255	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
256	C₂×C₄	#14273686	1	2	yes	—
257	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
258	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
259	C₂×C₄	#14273697	1	1	yes	characteristic
260	C₂×C₄	#14273700	1	2	yes	—
261	C₂×C₄	#14273704	1	2	yes	—
262	C₂×C₄	#14273705	1	2	yes	—
263	C₂×C₄	#14273707	1	1	yes	characteristic
264	C₂×C₄	#14273709	1	2	yes	—
265	C₂×C₄	#14273711	1	1	yes	characteristic
266	C₂×C₄	#14273713	1	2	yes	—
267	C₂×C₄	#14273715	1	1	yes	characteristic
268	C₂×C₄	#14273715	1	1	yes	characteristic
269	C₂×C₄	#14273717	1	2	yes	—
270	C₂×C₄	#14273719	1	2	yes	—
271	C₂×C₄	#14273720	1	2	yes	—
272	C₂×C₄	#14273726	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	—	4	4	no	—
275	C₈	—	236	2	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	#16837288	1	2	yes	—
282	C₂²	#16837301	1	1	yes	characteristic
283	C₂²	#16837320	1	1	yes	characteristic
284	C₂²	#16837329	1	1	yes	characteristic
285	C₂²	#16837344	1	2	yes	—
286	C₂²	#16837673	1	2	yes	—
287	C₂²	#16837678	1	2	yes	—
288	C₄	—	472	2	no	—
289	C₄	—	472	4	no	—
290	C₄	#16837268	1	1	yes	characteristic
291	C₄	#16837277	1	1	yes	characteristic
292	C₄	#16837295	1	2	yes	—
293	C₄	#16837315	1	1	yes	characteristic
294	C₄	#16837336	1	2	yes	—
295	C₄	#16837342	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	2	2	no	—
298	C₂	#21422353	1	1	yes	characteristic
299	C₂	#21424882	1	2	yes	—
300	C₂	#21425353	1	1	yes	characteristic
301	C₂	#21425365	1	1	yes	characteristic
302	C₂	#21425538	1	2	yes	—

	subgroup	G/H
center	C₂³	#14273702
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₉
Fitting	C₂×C₅₉×#341	C₂
socle	C₂³×C₅₉	#121
prime radical	C₂⁴×C₅₉	#41
cosocle	C₂³×C₄×C₅₉	C₂³
hypercenter	C₂×#341	D₅₉
hypocenter	C₅₉	#11270
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
16. C₅₉×#341	C₄	semidirect
165. #341	Dic₅₉	semidirect
168. C₅₉	#11270	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14273661	2
C₂×#16837170	2
#21423816	2
#21423792	2
C₂×C₅₉×#341	2
#11270	59