Subgroups — #28493736

Order 15104.

510 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424902	C₂	1	1	yes	maximal, characteristic
2	#21424901	C₂	1	1	yes	maximal, characteristic
3	#21424901	C₂	1	1	yes	maximal, characteristic
4	#21424255	C₂	1	1	yes	maximal, characteristic, complemented
5	#21424252	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273724	C₂²	1	1	yes	characteristic
9	C₂×#14273660	—	2	2	no	—
10	C₂×#14273658	—	2	2	no	—
11	C₂×#14273649	—	2	2	no	—
12	C₂×#14273647	—	2	2	no	—
13	C₂×#14273647	—	2	2	no	—
14	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
16	#16837272	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#384	C₂²	1	1	yes	characteristic
19	C₅₉×#384	C₂²	1	1	yes	characteristic
20	C₅₉×#340	—	2	2	no	—
21	C₂²×D₂₃₆	—	2	1	no	—
22	C₂²×D₂₃₆	—	2	1	no	—
23	C₂²×D₂₃₆	—	2	1	no	—
24	C₂²×C₅₉⋊C₈	D₄	1	2	yes	—
25	C₂²×C₅₉⋊C₈	D₄	1	2	yes	—
26	#14273724	—	2	1	no	—
27	#14273724	—	2	1	no	—
28	#14273724	—	2	1	no	—
29	#14273724	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₅₉	—	2	1	no	—
31	C₂×C₄⋊Dic₅₉	—	2	1	no	—
32	C₂×C₄⋊Dic₅₉	—	2	1	no	—
33	#14273660	—	4	4	no	—
34	#14273658	—	4	4	no	—
35	#14273649	—	4	4	no	—
36	#14273647	—	4	4	no	—
37	#14273647	—	4	4	no	—
38	C₄×C₅₉⋊C₈	—	2	1	no	—
39	C₄×C₅₉⋊C₈	—	2	1	no	—
40	C₄×C₅₉⋊C₈	—	2	1	no	—
41	C₄×C₅₉⋊C₈	—	2	1	no	—
42	C₂²×C₈×C₅₉	D₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
44	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
47	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
48	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
49	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₅₉	—	2	1	no	—
51	C₄×C₈×C₅₉	—	2	1	no	—
52	C₂³×D₅₉	—	4	1	no	—
53	C₂²×C₄×C₅₉	—	4	1	no	—
54	C₂²×C₄×C₅₉	—	4	1	no	—
55	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₅₉	—	4	1	no	—
59	C₂×D₂₃₆	—	2	2	no	—
60	C₂×D₂₃₆	—	2	2	no	—
61	C₂×D₂₃₆	—	2	2	no	—
62	C₂×D₂₃₆	—	4	2	no	—
63	C₂×D₂₃₆	—	4	2	no	—
64	C₂×D₂₃₆	—	4	2	no	—
65	C₂×D₂₃₆	—	4	2	no	—
66	C₂×D₂₃₆	—	4	2	no	—
67	C₂×D₂₃₆	—	4	2	no	—
68	C₂×D₂₃₆	—	4	1	no	—
69	C₂×D₂₃₆	—	4	1	no	—
70	C₂×D₂₃₆	—	4	1	no	—
71	C₂×C₈×C₅₉	—	2	2	no	—
72	C₂×C₈×C₅₉	—	2	2	no	—
73	C₂×C₈×C₅₉	—	2	2	no	—
74	C₄⋊C₄×C₅₉	—	4	2	no	—
75	C₄⋊C₄×C₅₉	—	4	2	no	—
76	C₄⋊C₄×C₅₉	—	4	2	no	—
77	C₄⋊C₄×C₅₉	—	4	2	no	—
78	C₄⋊C₄×C₅₉	—	4	2	no	—
79	C₄⋊C₄×C₅₉	—	4	2	no	—
80	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
81	C₄²×C₅₉	#41	1	1	yes	characteristic
82	C₄²×C₅₉	#41	1	1	yes	characteristic
83	C₄²×C₅₉	#41	1	1	yes	characteristic
84	C₄⋊Dic₅₉	—	4	2	no	—
85	C₄⋊Dic₅₉	—	4	2	no	—
86	C₄⋊Dic₅₉	—	4	2	no	—
87	C₂×C₅₉⋊C₈	—	2	2	no	—
88	C₂×C₅₉⋊C₈	—	2	2	no	—
89	C₂×C₅₉⋊C₈	—	2	2	no	—
90	C₂×C₅₉⋊C₈	—	2	2	no	—
91	C₂×C₅₉⋊C₈	D₈	1	4	yes	—
92	C₂×C₅₉⋊C₈	SD₁₆	1	4	yes	—
93	C₂³×C₅₉	#127	1	1	yes	characteristic
94	C₂²×D₅₉	—	4	2	no	—
95	C₂²×D₅₉	—	8	1	no	—
96	C₂²×D₅₉	—	8	1	no	—
97	C₂²×D₅₉	—	8	1	no	—
98	C₂²×D₅₉	—	8	1	no	—
99	C₂²×D₅₉	—	8	1	no	—
100	C₂²×D₅₉	—	8	1	no	—
101	C₂×C₄×C₅₉	—	8	1	no	—
102	C₂×C₄×C₅₉	—	8	1	no	—
103	C₂×C₄×C₅₉	—	8	1	no	—
104	C₂×C₄×C₅₉	—	8	1	no	—
105	C₂×C₄×C₅₉	—	8	1	no	—
106	C₂×C₄×C₅₉	—	8	1	no	—
107	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
111	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
112	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
115	C₂×C₄×C₅₉	#135	1	2	yes	—
116	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
118	C₂×C₄×C₅₉	#137	1	2	yes	—
119	C₂×Dic₅₉	—	8	1	no	—
120	C₂×Dic₅₉	—	8	1	no	—
121	C₂×Dic₅₉	—	8	1	no	—
122	D₂₃₆	—	4	4	no	—
123	D₂₃₆	—	4	4	no	—
124	D₂₃₆	—	4	4	no	—
125	D₂₃₆	—	8	2	no	—
126	D₂₃₆	—	8	2	no	—
127	D₂₃₆	—	8	2	no	—
128	C₈×C₅₉	—	4	2	no	—
129	C₅₉⋊C₈	—	2	4	no	—
130	C₅₉⋊C₈	—	2	4	no	—
131	#10672	—	59	1	no	maximal, Sylow
132	C₂²×C₅₉	#399	1	1	yes	characteristic
133	C₂²×C₅₉	#492	1	1	yes	characteristic
134	C₂²×C₅₉	#493	1	1	yes	characteristic
135	C₂²×C₅₉	#495	1	1	yes	characteristic
136	C₂²×C₅₉	#496	1	1	yes	characteristic
137	C₂²×C₅₉	#496	1	1	yes	characteristic
138	C₂²×C₅₉	#497	1	1	yes	characteristic
139	C₂×D₅₉	—	8	2	no	—
140	C₂×D₅₉	—	8	2	no	—
141	C₂×D₅₉	—	8	2	no	—
142	C₂×D₅₉	—	16	1	no	—
143	C₂×D₅₉	—	16	1	no	—
144	C₂×D₅₉	—	16	1	no	—
145	C₂×D₅₉	—	16	1	no	—
146	C₄×C₅₉	—	16	1	no	—
147	C₄×C₅₉	—	16	1	no	—
148	C₄×C₅₉	#474	1	2	yes	—
149	C₄×C₅₉	#476	1	2	yes	—
150	C₄×C₅₉	#478	1	2	yes	—
151	C₄×C₅₉	#480	1	2	yes	—
152	C₄×C₅₉	#481	1	2	yes	—
153	C₄×C₅₉	#482	1	2	yes	—
154	Dic₅₉	—	16	1	no	—
155	#1957	—	59	1	no	—
156	#1957	—	59	1	no	—
157	#1957	—	59	1	no	—
158	#1845	—	59	1	no	—
159	#1844	—	59	1	no	—
160	#1834	D₅₉	1	1	yes	characteristic, complemented
161	#1273	—	59	1	no	—
162	C₂×C₅₉	#1706	1	1	yes	characteristic
163	C₂×C₅₉	#1714	1	1	yes	characteristic
164	C₂×C₅₉	#1720	1	1	yes	characteristic
165	C₂×C₅₉	#1721	1	1	yes	characteristic
166	C₂×C₅₉	#2092	1	1	yes	characteristic
167	C₂×C₅₉	#2093	1	1	yes	characteristic
168	C₂×C₅₉	#2094	1	1	yes	characteristic
169	D₅₉	—	16	2	no	—
170	C₂×#127	—	59	1	no	—
171	C₂×#107	—	118	2	no	—
172	C₂×#106	—	118	2	no	—
173	C₂×#102	—	118	2	no	—
174	C₂×#102	—	118	2	no	—
175	C₂×#102	—	118	2	no	—
176	C₂×C₄×C₈	—	59	1	no	—
177	C₂×C₄×C₈	—	59	1	no	—
178	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
179	#384	—	59	1	no	—
180	#384	C₂×D₅₉	1	1	yes	characteristic
181	#384	C₂×D₅₉	1	1	yes	characteristic
182	#340	—	2	2	no	—
183	C₅₉	#10672	1	1	yes	characteristic, complemented, Sylow
184	C₂²×D₄	—	118	1	no	—
185	C₂²×D₄	—	118	1	no	—
186	C₂²×D₄	—	118	1	no	—
187	C₂²×C₈	—	59	2	no	—
188	C₂²×C₈	—	59	2	no	—
189	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
190	#127	—	59	2	no	—
191	#127	—	118	1	no	—
192	#127	—	118	1	no	—
193	#127	—	118	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	118	1	no	—
201	C₂×C₄⋊C₄	—	118	1	no	—
202	C₂×C₄⋊C₄	—	118	1	no	—
203	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
204	#107	—	236	4	no	—
205	#106	—	236	4	no	—
206	#102	—	236	4	no	—
207	#102	—	236	4	no	—
208	#102	—	236	4	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	118	1	no	—
212	C₄×C₈	—	118	1	no	—
213	C₄×C₈	—	118	1	no	—
214	C₄×C₈	—	118	1	no	—
215	C₂⁴	—	236	1	no	—
216	C₂×D₄	—	118	2	no	—
217	C₂×D₄	—	118	2	no	—
218	C₂×D₄	—	118	2	no	—
219	C₂×D₄	—	236	1	no	—
220	C₂×D₄	—	236	1	no	—
221	C₂×D₄	—	236	1	no	—
222	C₂×D₄	—	236	2	no	—
223	C₂×D₄	—	236	2	no	—
224	C₂×D₄	—	236	2	no	—
225	C₂×D₄	—	236	2	no	—
226	C₂×D₄	—	236	2	no	—
227	C₂×D₄	—	236	2	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	—	236	1	no	—
231	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	59	4	no	—
238	C₂×C₈	—	59	4	no	—
239	C₂×C₈	—	118	2	no	—
240	C₂×C₈	—	118	2	no	—
241	C₂×C₈	—	118	2	no	—
242	C₂×C₈	—	118	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	236	2	no	—
250	C₄⋊C₄	—	236	2	no	—
251	C₄⋊C₄	—	236	2	no	—
252	C₄²	#11745685	1	1	yes	characteristic
253	C₄²	#11745685	1	1	yes	characteristic
254	C₄²	Q₈×D₅₉	1	1	yes	characteristic
255	C₄²	#11745689	1	1	yes	characteristic
256	C₂³	—	472	1	no	—
257	C₂³	—	472	1	no	—
258	C₂³	—	472	1	no	—
259	C₂³	—	472	1	no	—
260	C₂³	—	472	1	no	—
261	C₂³	—	472	1	no	—
262	C₂³	—	236	2	no	—
263	C₂³	#14273723	1	1	yes	characteristic
264	D₄	—	472	2	no	—
265	D₄	—	472	2	no	—
266	D₄	—	472	2	no	—
267	D₄	—	236	4	no	—
268	D₄	—	236	4	no	—
269	D₄	—	236	4	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	—	472	1	no	—
277	C₂×C₄	—	472	1	no	—
278	C₂×C₄	—	472	1	no	—
279	C₂×C₄	#14273703	1	1	yes	characteristic
280	C₂×C₄	#14273705	1	1	yes	characteristic
281	C₂×C₄	#14273705	1	1	yes	characteristic
282	C₂×C₄	#14273710	1	1	yes	characteristic
283	C₂×C₄	#14273712	1	1	yes	characteristic
284	C₂×C₄	#14273712	1	1	yes	characteristic
285	C₂×C₄	D₈×D₅₉	1	2	yes	—
286	C₂×C₄	SD₁₆×D₅₉	1	2	yes	—
287	C₂×C₄	#14273747	1	2	yes	—
288	C₂×C₄	#14273752	1	2	yes	—
289	C₂×C₄	#14273754	1	2	yes	—
290	C₂×C₄	#14273759	1	2	yes	—
291	C₈	—	4	2	no	—
292	C₈	—	118	4	no	—
293	C₈	—	118	4	no	—
294	C₂²	—	944	1	no	—
295	C₂²	—	944	1	no	—
296	C₂²	—	944	1	no	—
297	C₂²	—	944	1	no	—
298	C₂²	—	472	2	no	—
299	C₂²	—	472	2	no	—
300	C₂²	—	472	2	no	—
301	C₂²	#16837337	1	1	yes	characteristic
302	C₂²	#16837773	1	1	yes	characteristic
303	C₂²	#16837777	1	1	yes	characteristic
304	C₂²	#16837784	1	1	yes	characteristic
305	C₂²	#16837788	1	1	yes	characteristic
306	C₂²	#16837792	1	1	yes	characteristic
307	C₂²	#16837795	1	1	yes	characteristic
308	C₄	—	16	1	no	—
309	C₄	—	16	1	no	—
310	C₄	—	944	1	no	—
311	C₄	#16837681	1	2	yes	—
312	C₄	#16837691	1	2	yes	—
313	C₄	#16837699	1	2	yes	—
314	C₄	#16837717	1	2	yes	—
315	C₄	#16837722	1	2	yes	—
316	C₄	#16837726	1	2	yes	—
317	C₂	—	944	2	no	—
318	C₂	#21423687	1	1	yes	characteristic
319	C₂	#21423720	1	1	yes	characteristic
320	C₂	#21423753	1	1	yes	characteristic
321	C₂	#21423764	1	1	yes	characteristic
322	C₂	#21425620	1	1	yes	characteristic
323	C₂	#21425621	1	1	yes	characteristic
324	C₂	#21425626	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273723
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1834	C₂
socle	C₂³×C₅₉	#127
prime radical	C₂×#14273724	C₂²
hypercenter	#1834	D₅₉
hypocenter	C₅₉	#10672
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
4. #21424255	C₂	semidirect
5. #21424252	C₂	semidirect
6. C₅₉⋊#1273	C₂	semidirect
7. C₅₉×#1834	C₂	semidirect
160. #1834	D₅₉	semidirect
183. C₅₉	#10672	semidirect

Maximal subgroups

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

subgroup	index
#21424902	2
#21424901	2
#21424255	2
#21424252	2
C₅₉⋊#1273	2
C₅₉×#1834	2
#10672	59