Subgroups — #28479802

Order 15104.

406 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433026	C₂	1	1	yes	maximal, characteristic
2	#21423961	C₂	1	1	yes	maximal, characteristic
3	#21422754	C₂	1	1	yes	maximal, characteristic
4	#21422596	C₂	1	1	yes	maximal, characteristic
5	#21422389	C₂	1	1	yes	maximal, characteristic
6	#21421997	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1301	C₂	1	1	yes	maximal, characteristic
8	#16837959	C₂²	1	1	yes	characteristic
9	#16837529	—	2	1	no	—
10	#16837529	C₄	1	2	yes	complemented
11	#16837526	—	2	1	no	—
12	#16837526	C₄	1	2	yes	complemented
13	#16837455	C₂²	1	1	yes	characteristic
14	#16837430	—	2	1	no	—
15	#16837360	—	2	1	no	—
16	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₅₉	—	2	1	no	—
18	C₄×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
19	#16837156	—	2	1	no	—
20	#16837150	C₄	1	2	yes	complemented
21	#16837147	C₄	1	2	yes	complemented
22	#16837140	C₄	1	4	yes	—
23	#16837124	—	2	1	no	—
24	#16837123	—	2	1	no	—
25	C₅₉×#431	C₂²	1	1	yes	characteristic
26	C₅₉×#423	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
28	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
29	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
30	#14273727	C₂×C₄	1	1	yes	characteristic
31	#14273727	C₂×C₄	1	1	yes	characteristic
32	#14273719	—	2	1	no	—
33	#14273706	—	2	1	no	—
34	#14273706	C₂×C₄	1	2	yes	—
35	C₄×Dic₁₁₈	—	2	1	no	—
36	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
37	#14273686	—	2	1	no	—
38	C₂×C₄⋊Dic₅₉	—	2	1	no	—
39	C₂×C₄×Dic₅₉	—	2	1	no	—
40	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
41	Dic₅₉⋊C₈	—	2	2	no	—
42	Dic₅₉⋊C₈	—	2	2	no	—
43	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
44	#14273637	—	2	2	no	—
45	C₄×C₅₉⋊C₈	—	2	1	no	—
46	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
49	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
50	C₈×Dic₅₉	—	2	2	no	—
51	C₈×Dic₅₉	—	4	2	no	—
52	C₅₉⋊#95	C₂×C₄	1	2	yes	—
53	C₂×M₁₆×C₅₉	—	2	1	no	—
54	C₂²×C₈×C₅₉	—	2	1	no	—
55	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₅₉	—	2	1	no	—
57	C₄⋊C₈×C₅₉	—	2	1	no	—
58	C₄⋊C₈×C₅₉	—	2	1	no	—
59	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
60	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
61	C₂²⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₅₉	—	2	1	no	—
63	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
66	C₂²×Dic₅₉	—	2	2	no	—
67	C₂×Dic₁₁₈	—	2	1	no	—
68	C₂×Dic₁₁₈	C₄²	1	2	yes	—
69	M₁₆×C₅₉	—	4	1	no	—
70	M₁₆×C₅₉	—	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	C₂×C₈×C₅₉	—	4	1	no	—
75	C₂×C₈×C₅₉	—	4	1	no	—
76	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₉	#41	1	1	yes	characteristic
80	C₂²⋊Dic₅₉	—	2	2	no	—
81	C₄⋊Dic₅₉	—	2	4	no	—
82	C₄⋊Dic₅₉	—	2	1	no	—
83	C₄⋊Dic₅₉	—	2	1	no	—
84	C₄⋊Dic₅₉	—	2	1	no	—
85	C₄⋊Dic₅₉	C₄²	1	2	yes	—
86	Dic₅₉⋊C₄	—	2	2	no	—
87	Dic₅₉⋊C₄	—	2	2	no	—
88	C₄×Dic₅₉	—	2	2	no	—
89	C₄×Dic₅₉	—	4	1	no	—
90	C₄×Dic₅₉	—	4	1	no	—
91	C₂×C₅₉⋊C₈	—	2	1	no	—
92	C₂×C₅₉⋊C₈	—	2	1	no	—
93	C₂×C₅₉⋊C₈	—	2	1	no	—
94	C₂×C₅₉⋊C₈	—	2	1	no	—
95	C₂×C₅₉⋊C₈	—	2	2	no	—
96	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	—	2	2	no	—
102	C₂×C₄×C₅₉	—	4	1	no	—
103	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
111	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
112	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
113	C₂×Dic₅₉	—	2	4	no	—
114	C₂×Dic₅₉	—	2	2	no	—
115	C₂×Dic₅₉	—	2	2	no	—
116	C₂×Dic₅₉	—	4	2	no	—
117	C₂×Dic₅₉	—	4	2	no	—
118	Dic₁₁₈	—	4	2	no	—
119	Dic₁₁₈	—	4	2	no	—
120	C₈×C₅₉	—	4	2	no	—
121	C₈×C₅₉	—	4	2	no	—
122	C₈×C₅₉	—	4	2	no	—
123	C₅₉⋊C₈	—	2	1	no	—
124	C₅₉⋊C₈	—	2	1	no	—
125	C₅₉⋊C₈	—	2	1	no	—
126	C₅₉⋊C₈	—	2	1	no	—
127	C₅₉⋊C₈	—	4	2	no	—
128	#8258	—	59	1	no	maximal, Sylow
129	C₂²×C₅₉	—	2	1	no	—
130	C₂²×C₅₉	—	2	1	no	—
131	C₂²×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₅₉	#440	1	1	yes	characteristic
133	C₂²×C₅₉	#441	1	1	yes	characteristic
134	C₄×C₅₉	—	2	1	no	—
135	C₄×C₅₉	—	2	1	no	—
136	C₄×C₅₉	—	2	1	no	—
137	C₄×C₅₉	#417	1	1	yes	characteristic
138	C₄×C₅₉	#419	1	1	yes	characteristic
139	C₄×C₅₉	#440	1	1	yes	characteristic
140	C₄×C₅₉	#441	1	1	yes	characteristic
141	C₄×C₅₉	#443	1	2	yes	—
142	Dic₅₉	—	4	4	no	—
143	Dic₅₉	—	4	2	no	—
144	Dic₅₉	—	4	2	no	—
145	#2960	—	59	1	no	—
146	C₈×C₄⋊C₄	—	59	1	no	—
147	#1542	—	59	1	no	—
148	#1503	—	59	1	no	—
149	#1452	—	59	1	no	—
150	#1303	—	59	1	no	—
151	#1301	D₅₉	1	1	yes	characteristic
152	C₂×C₅₉	—	2	1	no	—
153	C₂×C₅₉	—	2	1	no	—
154	C₂×C₅₉	#1757	1	1	yes	characteristic
155	C₂×C₅₉	#1763	1	1	yes	characteristic
156	C₂×C₅₉	#1764	1	1	yes	characteristic
157	#533	—	59	1	no	—
158	#446	—	59	2	no	—
159	#446	—	118	1	no	—
160	C₈×Q₈	—	59	2	no	—
161	C₈×Q₈	—	118	1	no	—
162	#431	—	59	1	no	—
163	#431	C₂×D₅₉	1	1	yes	characteristic
164	#424	—	118	1	no	—
165	#423	C₂×D₅₉	1	1	yes	characteristic
166	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
167	#405	—	118	1	no	—
168	C₂×C₄×C₈	—	59	1	no	—
169	C₂×C₄×C₈	—	118	1	no	—
170	C₄×C₄⋊C₄	—	59	1	no	—
171	#336	—	118	1	no	—
172	#333	—	59	2	no	—
173	#332	—	59	2	no	—
174	#328	—	59	4	no	—
175	#322	—	118	1	no	—
176	#322	—	118	1	no	—
177	C₅₉	#8258	1	1	yes	characteristic, complemented, Sylow
178	C₂×M₁₆	—	2	1	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	59	1	no	—
181	C₂²×C₈	—	59	1	no	—
182	C₄⋊Q₈	—	59	1	no	—
183	C₄⋊Q₈	—	59	1	no	—
184	#125	—	118	1	no	—
185	C₂²⋊Q₈	—	59	2	no	—
186	C₂²⋊Q₈	—	118	1	no	—
187	C₄×Q₈	—	59	2	no	—
188	C₄×Q₈	—	118	1	no	—
189	#117	—	118	1	no	—
190	C₂×C₄⋊C₄	—	118	1	no	—
191	C₂×C₄²	—	118	1	no	—
192	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	59	2	no	—
197	C₄⋊C₈	—	118	2	no	—
198	C₄⋊C₈	—	118	2	no	—
199	C₄⋊C₈	C₄×D₅₉	1	2	yes	—
200	C₄⋊C₈	C₄×D₅₉	1	2	yes	—
201	C₂²⋊C₈	—	118	1	no	—
202	C₂²⋊C₈	C₄×D₅₉	1	2	yes	—
203	#97	—	118	2	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	59	1	no	—
206	C₄×C₈	—	59	1	no	—
207	C₄×C₈	—	59	1	no	—
208	C₄×C₈	—	59	1	no	—
209	C₄×C₈	—	118	2	no	—
210	C₄×C₈	—	118	1	no	—
211	C₄×C₈	—	236	2	no	—
212	#95	—	59	2	no	—
213	C₂×Q₈	—	59	2	no	—
214	C₂×Q₈	—	118	1	no	—
215	C₂²×C₄	—	118	2	no	—
216	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
217	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
218	C₂²×C₄	#11745689	1	1	yes	characteristic
219	M₁₆	—	4	1	no	—
220	M₁₆	—	4	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	59	1	no	—
227	C₂×C₈	—	59	1	no	—
228	C₂×C₈	—	59	1	no	—
229	C₂×C₈	—	59	1	no	—
230	C₂×C₈	—	118	1	no	—
231	C₂×C₈	—	118	1	no	—
232	C₂×C₈	—	118	1	no	—
233	C₂×C₈	—	118	1	no	—
234	C₂×C₈	—	118	2	no	—
235	C₄⋊C₄	—	59	2	no	—
236	C₄⋊C₄	—	118	4	no	—
237	C₄⋊C₄	—	118	2	no	—
238	C₄⋊C₄	—	118	1	no	—
239	C₄⋊C₄	—	118	1	no	—
240	C₄⋊C₄	—	118	2	no	—
241	C₄⋊C₄	—	118	1	no	—
242	C₂²⋊C₄	—	118	2	no	—
243	C₄²	—	118	2	no	—
244	C₄²	—	236	1	no	—
245	C₄²	—	236	1	no	—
246	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
247	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
248	C₄²	D₄×D₅₉	1	1	yes	characteristic
249	C₄²	#11745687	1	1	yes	characteristic
250	C₂³	#14273689	1	1	yes	characteristic
251	Q₈	—	236	2	no	—
252	Q₈	—	236	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	118	4	no	—
256	C₂×C₄	—	118	2	no	—
257	C₂×C₄	—	118	2	no	—
258	C₂×C₄	—	236	2	no	—
259	C₂×C₄	—	236	2	no	—
260	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
261	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
262	C₂×C₄	#14273685	1	1	yes	characteristic
263	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
264	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
265	C₂×C₄	#14273689	1	1	yes	characteristic
266	C₂×C₄	#14273751	1	1	yes	characteristic
267	C₂×C₄	#14273756	1	1	yes	characteristic
268	C₂×C₄	#14273756	1	1	yes	characteristic
269	C₂×C₄	#14273759	1	1	yes	characteristic
270	C₈	—	4	2	no	—
271	C₈	—	4	2	no	—
272	C₈	—	4	2	no	—
273	C₈	—	118	1	no	—
274	C₈	—	118	1	no	—
275	C₈	—	118	1	no	—
276	C₈	—	118	1	no	—
277	C₈	—	236	2	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	#16837249	1	1	yes	characteristic
281	C₂²	#16837501	1	1	yes	characteristic
282	C₂²	#16837508	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	236	4	no	—
287	C₄	—	236	2	no	—
288	C₄	—	236	2	no	—
289	C₄	#16837400	1	1	yes	characteristic
290	C₄	#16837409	1	1	yes	characteristic
291	C₄	#16837499	1	1	yes	characteristic
292	C₄	#16837506	1	1	yes	characteristic
293	C₄	#16837518	1	2	yes	—
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	#21423895	1	1	yes	characteristic
297	C₂	#21423925	1	1	yes	characteristic
298	C₂	#21423932	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1301	C₂
socle	C₂²×C₅₉	C₄×C₂²⋊C₄
prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1301	D₅₉
hypocenter	C₅₉	#8258
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
10. #16837529	C₄	semidirect
12. #16837526	C₄	semidirect
20. #16837150	C₄	semidirect
21. #16837147	C₄	semidirect
177. C₅₉	#8258	semidirect

Maximal subgroups

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

subgroup	index
#21433026	2
#21423961	2
#21422754	2
#21422596	2
#21422389	2
#21421997	2
C₅₉×#1301	2
#8258	59