Subgroups — #28488623

Order 15104.

460 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425074	C₂	1	1	yes	maximal, characteristic, complemented
2	#21424925	C₂	1	1	yes	maximal, characteristic
3	#21423841	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422855	C₂	1	1	yes	maximal, characteristic, complemented
5	#21422551	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422547	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#3167	C₂	1	1	yes	maximal, characteristic
8	#16837426	C₂²	1	1	yes	characteristic
9	#16837426	C₂²	1	1	yes	characteristic
10	C₂×#14273654	—	2	2	no	—
11	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
13	#16837161	—	2	2	no	—
14	#16837156	—	2	2	no	—
15	#16837155	—	2	2	no	—
16	Q₈⋊(C₅₉⋊C₈)	—	2	2	no	—
17	D₄⋊(C₅₉⋊C₈)	—	2	2	no	—
18	C₅₉×#533	C₂²	1	1	yes	characteristic
19	C₅₉×#532	C₂²	1	1	yes	characteristic
20	C₅₉×#498	C₄	1	1	yes	characteristic, complemented
21	C₅₉×#492	C₄	1	1	yes	characteristic, complemented
22	C₅₉×#487	C₄	1	1	yes	characteristic, complemented
23	C₅₉×#486	C₄	1	1	yes	characteristic, complemented
24	C₅₉×#465	—	2	2	no	—
25	C₄×SD₁₆×C₅₉	—	2	2	no	—
26	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#11745664	—	2	1	no	—
28	C₂×#11745664	—	2	1	no	—
29	C₂²×C₅₉⋊C₈	—	2	1	no	—
30	C₂×C₄×Dic₅₉	—	2	1	no	—
31	C₂×C₄×Dic₅₉	—	2	1	no	—
32	C₂×C₄×Dic₅₉	D₄	1	2	yes	—
33	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
34	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
35	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
36	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
37	#14273654	—	4	4	no	—
38	C₄×C₅₉⋊C₈	—	2	1	no	—
39	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
40	C₂×#41×C₅₉	—	2	1	no	—
41	C₂×SD₁₆×C₅₉	—	2	2	no	—
42	C₂²×C₈×C₅₉	D₄	1	2	yes	—
43	C₄⋊Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
44	C₄⋊Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
45	C₄⋊Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
46	#127×C₅₉	C₂×C₄	1	1	yes	characteristic
47	#125×C₅₉	—	2	1	no	—
48	#124×C₅₉	—	2	1	no	—
49	C₂²⋊Q₈×C₅₉	—	2	1	no	—
50	C₂²⋊Q₈×C₅₉	—	2	1	no	—
51	#121×C₅₉	—	2	1	no	—
52	#121×C₅₉	—	2	1	no	—
53	C₄×Q₈×C₅₉	—	2	1	no	—
54	C₄×Q₈×C₅₉	—	2	1	no	—
55	C₄×D₄×C₅₉	—	2	1	no	—
56	C₄×D₄×C₅₉	—	2	1	no	—
57	#117×C₅₉	—	2	1	no	—
58	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
59	#106×C₅₉	—	2	2	no	—
60	#103×C₅₉	—	2	2	no	—
61	#102×C₅₉	—	2	2	no	—
62	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₉	D₄	1	2	yes	—
65	#41×C₅₉	—	4	2	no	—
66	#41×C₅₉	—	4	2	no	—
67	C₂×Q₈×C₅₉	—	2	1	no	—
68	C₂×Q₈×C₅₉	—	2	1	no	—
69	C₂×Q₈×C₅₉	—	2	1	no	—
70	C₂×D₄×C₅₉	—	2	1	no	—
71	C₂×D₄×C₅₉	—	2	1	no	—
72	C₂×D₄×C₅₉	—	2	1	no	—
73	C₂×D₄×C₅₉	—	4	1	no	—
74	C₂²×C₄×C₅₉	—	4	1	no	—
75	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
78	C₂²×Dic₅₉	—	2	2	no	—
79	SD₁₆×C₅₉	—	4	4	no	—
80	C₂×C₈×C₅₉	—	4	2	no	—
81	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
83	C₄⋊C₄×C₅₉	—	2	1	no	—
84	C₄⋊C₄×C₅₉	—	2	1	no	—
85	C₄⋊C₄×C₅₉	—	2	1	no	—
86	C₄⋊C₄×C₅₉	—	2	1	no	—
87	C₄⋊C₄×C₅₉	—	2	1	no	—
88	C₄⋊C₄×C₅₉	—	2	1	no	—
89	C₄⋊C₄×C₅₉	—	4	1	no	—
90	C₄⋊C₄×C₅₉	—	4	1	no	—
91	C₂²⋊C₄×C₅₉	—	4	1	no	—
92	C₂²⋊C₄×C₅₉	—	4	1	no	—
93	C₂²⋊C₄×C₅₉	—	4	1	no	—
94	C₄²×C₅₉	—	4	1	no	—
95	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₅₉	#41	1	1	yes	characteristic
99	C₄×Dic₅₉	—	2	4	no	—
100	C₄×Dic₅₉	—	2	2	no	—
101	C₄×Dic₅₉	—	2	2	no	—
102	C₄×Dic₅₉	—	4	2	no	—
103	#11745664	—	4	2	no	—
104	#11745664	—	4	2	no	—
105	C₂×C₅₉⋊C₈	—	2	2	no	—
106	C₂×C₅₉⋊C₈	—	4	2	no	—
107	C₂×C₅₉⋊C₈	—	4	1	no	—
108	C₂×C₅₉⋊C₈	—	4	1	no	—
109	C₂³×C₅₉	—	4	1	no	—
110	C₂³×C₅₉	#121	1	1	yes	characteristic
111	Q₈×C₅₉	—	4	2	no	—
112	Q₈×C₅₉	—	4	2	no	—
113	Q₈×C₅₉	—	4	2	no	—
114	D₄×C₅₉	—	4	2	no	—
115	D₄×C₅₉	—	4	2	no	—
116	D₄×C₅₉	—	4	2	no	—
117	D₄×C₅₉	—	8	2	no	—
118	C₂×C₄×C₅₉	—	2	2	no	—
119	C₂×C₄×C₅₉	—	2	2	no	—
120	C₂×C₄×C₅₉	—	4	1	no	—
121	C₂×C₄×C₅₉	—	4	1	no	—
122	C₂×C₄×C₅₉	—	4	1	no	—
123	C₂×C₄×C₅₉	—	8	1	no	—
124	C₂×C₄×C₅₉	—	8	1	no	—
125	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
129	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
130	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
131	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
132	C₂×C₄×C₅₉	#137	1	2	yes	—
133	C₂×Dic₅₉	—	2	4	no	—
134	C₂×Dic₅₉	—	4	2	no	—
135	C₂×Dic₅₉	—	4	2	no	—
136	C₈×C₅₉	—	2	2	no	—
137	C₈×C₅₉	—	2	2	no	—
138	C₅₉⋊C₈	—	4	2	no	—
139	C₅₉⋊C₈	—	8	1	no	—
140	C₅₉⋊C₈	—	8	1	no	—
141	#9854	—	59	1	no	maximal, Sylow
142	C₂²×C₅₉	—	2	1	no	—
143	C₂²×C₅₉	—	2	1	no	—
144	C₂²×C₅₉	—	8	1	no	—
145	C₂²×C₅₉	—	8	1	no	—
146	C₂²×C₅₉	—	8	1	no	—
147	C₂²×C₅₉	#390	1	1	yes	characteristic
148	C₂²×C₅₉	#468	1	2	yes	—
149	C₄×C₅₉	—	2	1	no	—
150	C₄×C₅₉	—	2	1	no	—
151	C₄×C₅₉	—	8	1	no	—
152	C₄×C₅₉	—	8	1	no	—
153	C₄×C₅₉	—	8	1	no	—
154	C₄×C₅₉	#421	1	2	yes	—
155	C₄×C₅₉	#440	1	2	yes	—
156	C₄×C₅₉	#443	1	2	yes	—
157	C₄×C₅₉	#496	1	1	yes	characteristic
158	C₄×C₅₉	#497	1	1	yes	characteristic
159	Dic₅₉	—	4	4	no	—
160	#3167	D₅₉	1	1	yes	characteristic
161	#1994	—	59	1	no	—
162	#1962	—	59	1	no	—
163	#1744	—	59	1	no	—
164	#1567	—	59	1	no	—
165	#1493	—	59	1	no	—
166	#1492	—	59	1	no	—
167	C₂×C₅₉	—	2	2	no	—
168	C₂×C₅₉	—	16	1	no	—
169	C₂×C₅₉	#1963	1	1	yes	characteristic
170	C₂×C₅₉	#1973	1	1	yes	characteristic
171	C₂×C₅₉	#1974	1	1	yes	characteristic
172	#533	C₂×D₅₉	1	1	yes	characteristic
173	#532	C₂×D₅₉	1	1	yes	characteristic
174	#498	Dic₅₉	1	1	yes	characteristic, complemented
175	#492	Dic₅₉	1	1	yes	characteristic, complemented
176	#487	Dic₅₉	1	1	yes	characteristic, complemented
177	#486	Dic₅₉	1	1	yes	characteristic, complemented
178	#465	—	2	2	no	—
179	C₄×SD₁₆	—	2	2	no	—
180	#423	—	59	1	no	—
181	#423	—	59	1	no	—
182	C₂×#104	—	118	2	no	—
183	C₂×C₄×C₈	—	59	1	no	—
184	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
185	C₄³	—	59	1	no	—
186	#339	—	118	2	no	—
187	#336	—	118	2	no	—
188	#335	—	118	2	no	—
189	Q₈⋊C₈	—	118	2	no	—
190	D₄⋊C₈	—	118	2	no	—
191	C₅₉	#9854	1	1	yes	characteristic, complemented, Sylow
192	C₂×#41	—	2	1	no	—
193	C₂×SD₁₆	—	2	2	no	—
194	C₂×M₁₆	—	118	1	no	—
195	C₂×M₁₆	—	118	1	no	—
196	C₂²×C₈	—	118	1	no	—
197	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
198	C₄⋊Q₈	C₂×Dic₅₉	1	1	yes	characteristic
199	C₄⋊Q₈	C₂×Dic₅₉	1	1	yes	characteristic
200	C₄⋊Q₈	C₂×Dic₅₉	1	1	yes	characteristic
201	#127	C₂×Dic₅₉	1	1	yes	characteristic
202	#125	—	2	1	no	—
203	#124	—	2	1	no	—
204	C₂²⋊Q₈	—	2	1	no	—
205	C₂²⋊Q₈	—	2	1	no	—
206	#121	—	2	1	no	—
207	#121	—	2	1	no	—
208	C₄×Q₈	—	2	1	no	—
209	C₄×Q₈	—	2	1	no	—
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	—	2	1	no	—
212	#117	—	2	1	no	—
213	C₂×C₄²	—	59	2	no	—
214	C₂×C₄²	—	118	1	no	—
215	C₂×C₄²	—	118	1	no	—
216	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
217	#106	—	2	2	no	—
218	C₄⋊C₈	—	118	1	no	—
219	C₄⋊C₈	—	118	1	no	—
220	C₄⋊C₈	—	118	1	no	—
221	C₄⋊C₈	—	118	1	no	—
222	#104	—	236	4	no	—
223	#103	—	2	2	no	—
224	#102	—	2	2	no	—
225	C₄×C₈	—	59	2	no	—
226	C₄×C₈	—	118	1	no	—
227	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
228	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
229	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
230	#41	—	4	2	no	—
231	#41	—	4	2	no	—
232	C₂×Q₈	—	2	1	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	118	2	no	—
241	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
242	C₂²×C₄	#11745687	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
244	SD₁₆	—	4	4	no	—
245	M₁₆	—	236	2	no	—
246	M₁₆	—	236	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	118	2	no	—
249	C₂×C₈	—	236	1	no	—
250	C₂×C₈	—	236	1	no	—
251	C₂×C₈	—	236	2	no	—
252	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
253	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	1	no	—
263	C₂²⋊C₄	—	4	1	no	—
264	C₂²⋊C₄	—	4	1	no	—
265	C₄²	—	4	1	no	—
266	C₄²	—	118	4	no	—
267	C₄²	—	118	2	no	—
268	C₄²	—	118	2	no	—
269	C₄²	—	236	2	no	—
270	C₄²	D₄×D₅₉	1	1	yes	characteristic
271	C₄²	#11745687	1	1	yes	characteristic
272	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
273	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
274	C₂³	—	4	1	no	—
275	C₂³	#14273698	1	1	yes	characteristic
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	4	2	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	8	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	8	1	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	118	4	no	—
291	C₂×C₄	—	236	2	no	—
292	C₂×C₄	—	236	2	no	—
293	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
294	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
295	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
296	C₂×C₄	#14273698	1	1	yes	characteristic
297	C₂×C₄	#14273714	1	1	yes	characteristic
298	C₂×C₄	#14273723	1	1	yes	characteristic
299	C₂×C₄	#14273754	1	2	yes	—
300	C₂×C₄	#14273756	1	2	yes	—
301	C₈	—	2	2	no	—
302	C₈	—	2	2	no	—
303	C₈	—	472	1	no	—
304	C₈	—	472	1	no	—
305	C₈	—	236	2	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	8	1	no	—
309	C₂²	—	8	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	#16837305	1	1	yes	characteristic
312	C₂²	#16837645	1	2	yes	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	8	1	no	—
316	C₄	—	8	1	no	—
317	C₄	—	8	1	no	—
318	C₄	—	236	4	no	—
319	C₄	#16837419	1	2	yes	—
320	C₄	#16837503	1	2	yes	—
321	C₄	#16837519	1	2	yes	—
322	C₄	#16837788	1	1	yes	characteristic
323	C₄	#16837793	1	1	yes	characteristic
324	C₂	—	2	2	no	—
325	C₂	—	16	1	no	—
326	C₂	#21424929	1	1	yes	characteristic
327	C₂	#21424979	1	1	yes	characteristic
328	C₂	#21424985	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273723
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#3167	C₂
socle	C₂²×C₅₉	#390
prime radical	C₅₉×#532	C₂²
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#3167	D₅₉
hypocenter	C₅₉	#9854
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3167

Complements

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

N	H ≅ G/N	type
1. #21425074	C₂	semidirect
3. #21423841	C₂	semidirect
4. #21422855	C₂	semidirect
5. #21422551	C₂	semidirect
20. C₅₉×#498	C₄	semidirect
21. C₅₉×#492	C₄	semidirect
22. C₅₉×#487	C₄	semidirect
23. C₅₉×#486	C₄	semidirect
174. #498	Dic₅₉	semidirect
175. #492	Dic₅₉	semidirect
176. #487	Dic₅₉	semidirect
177. #486	Dic₅₉	semidirect
191. C₅₉	#9854	semidirect

Maximal subgroups

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

subgroup	index
#21425074	2
#21424925	2
#21423841	2
#21422855	2
#21422551	2
#21422547	2
C₅₉×#3167	2
#9854	59