Subgroups — #28503963

Order 15104.

486 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837515	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837214	C₂	1	1	yes	maximal, characteristic
3	#21425811	C₂	1	1	yes	maximal, characteristic
4	C₂×#16837180	C₂	1	1	yes	maximal, characteristic, complemented
5	#21422201	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422196	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1773	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₇₂	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₂₃₆	—	2	1	no	—
10	C₂×#14273667	C₂²	1	1	yes	characteristic
11	C₂×#14273667	C₂²	1	1	yes	characteristic
12	#16837515	—	2	2	no	—
13	#16837515	C₄	1	4	yes	complemented
14	C₂×#14273659	C₂²	1	1	yes	characteristic
15	C₂×#14273649	—	2	1	no	—
16	#16837214	C₄	1	4	yes	complemented
17	#16837202	C₄	1	4	yes	complemented
18	#16837180	—	2	2	no	—
19	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
20	C₅₉×#427	C₂²	1	1	yes	characteristic
21	C₅₉×#405	—	2	1	no	—
22	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
23	C₅₉×#339	—	2	1	no	—
24	C₂²×D₂₃₆	—	2	1	no	—
25	C₂²×C₄×D₅₉	—	4	1	no	—
26	C₂×D₄₇₂	—	2	1	no	—
27	C₂×D₄₇₂	—	2	1	no	—
28	C₂×D₄₇₂	—	2	1	no	—
29	C₂×D₄₇₂	C₂×C₄	1	2	yes	—
30	C₂×D₄₇₂	C₂×C₄	1	2	yes	—
31	C₂×D₄₇₂	C₂×C₄	1	2	yes	—
32	C₄×D₂₃₆	—	2	4	no	—
33	C₄×D₂₃₆	—	4	2	no	—
34	C₂×C₄⋊Dic₅₉	—	2	1	no	—
35	C₂×(C₂×D₅₉)⋊C₄	—	4	1	no	—
36	#14273667	—	2	1	no	—
37	#14273667	—	2	1	no	—
38	#14273667	—	2	1	no	—
39	#14273667	C₂×C₄	1	2	yes	complemented
40	C₂×C₅₉⋊C₁₆	—	2	1	no	—
41	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	complemented
42	#14273659	—	2	1	no	—
43	#14273659	C₂×C₄	1	2	yes	—
44	#14273649	—	2	4	no	—
45	C₂×M₁₆×C₅₉	—	2	1	no	—
46	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
48	#117×C₅₉	—	2	1	no	—
49	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
50	#108×C₅₉	—	2	1	no	—
51	#108×C₅₉	C₂×C₄	1	2	yes	complemented
52	#107×C₅₉	C₂×C₄	1	2	yes	complemented
53	#106×C₅₉	—	2	1	no	—
54	#97×C₅₉	—	4	1	no	—
55	#97×C₅₉	C₂×C₄	1	2	yes	—
56	#97×C₅₉	D₄	1	2	yes	—
57	C₄×C₈×C₅₉	—	4	1	no	—
58	C₂³×D₅₉	—	4	1	no	—
59	C₂²×C₄×C₅₉	—	2	1	no	—
60	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₅₉	—	4	1	no	—
62	C₂×D₂₃₆	—	2	2	no	—
63	C₂×D₂₃₆	—	2	2	no	—
64	C₂×D₂₃₆	—	2	2	no	—
65	C₂×D₂₃₆	—	4	1	no	—
66	C₂×D₂₃₆	—	4	1	no	—
67	C₂×D₂₃₆	—	4	1	no	—
68	C₂×C₄×D₅₉	—	4	4	no	—
69	C₂×C₄×D₅₉	—	8	4	no	—
70	M₁₆×C₅₉	—	2	2	no	—
71	M₁₆×C₅₉	—	4	1	no	—
72	C₂×C₈×C₅₉	—	2	1	no	—
73	C₂×C₈×C₅₉	—	2	1	no	—
74	C₂×C₈×C₅₉	—	4	1	no	—
75	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
81	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
82	C₄⋊C₄×C₅₉	—	2	2	no	—
83	C₂²⋊C₄×C₅₉	—	4	1	no	—
84	C₄²×C₅₉	—	2	1	no	—
85	C₄²×C₅₉	—	4	1	no	—
86	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
87	(C₂×D₅₉)⋊C₄	—	4	4	no	—
88	C₄⋊Dic₅₉	—	2	2	no	—
89	C₄⋊Dic₅₉	—	4	1	no	—
90	D₄₇₂	—	2	2	no	—
91	D₄₇₂	—	2	2	no	—
92	D₄₇₂	—	2	2	no	—
93	D₄₇₂	C₄²	1	4	yes	—
94	C₅₉⋊C₁₆	—	2	2	no	—
95	C₅₉⋊C₁₆	—	2	2	no	—
96	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂²×D₅₉	—	4	2	no	—
98	C₂²×D₅₉	—	4	2	no	—
99	C₂²×D₅₉	—	4	2	no	—
100	C₂²×D₅₉	—	8	2	no	—
101	C₂²×D₅₉	—	8	2	no	—
102	C₂×C₄×C₅₉	—	2	2	no	—
103	C₂×C₄×C₅₉	—	4	2	no	—
104	C₂×C₄×C₅₉	—	4	2	no	—
105	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	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	AGL(1,ℤ/8)	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×Dic₅₉	—	4	2	no	—
112	C₂×Dic₅₉	—	8	2	no	—
113	D₂₃₆	—	2	4	no	—
114	D₂₃₆	—	4	2	no	—
115	D₂₃₆	—	4	2	no	—
116	D₂₃₆	—	4	2	no	—
117	C₄×D₅₉	—	8	8	no	—
118	C₈×C₅₉	—	2	2	no	—
119	C₈×C₅₉	—	4	2	no	—
120	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
121	C₈×C₅₉	#117	1	1	yes	characteristic
122	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
123	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
124	#12647	—	59	1	no	maximal, Sylow
125	C₂²×C₅₉	—	2	2	no	—
126	C₂²×C₅₉	#418	1	1	yes	characteristic
127	C₂²×C₅₉	#440	1	1	yes	characteristic
128	C₂²×C₅₉	#442	1	1	yes	characteristic
129	C₂×D₅₉	—	4	4	no	—
130	C₂×D₅₉	—	8	2	no	—
131	C₂×D₅₉	—	8	2	no	—
132	C₂×D₅₉	—	8	4	no	—
133	C₂×D₅₉	—	8	4	no	—
134	C₄×C₅₉	—	4	2	no	—
135	C₄×C₅₉	—	8	2	no	—
136	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₅₉	#417	1	1	yes	characteristic
138	C₄×C₅₉	#440	1	1	yes	characteristic
139	C₄×C₅₉	#442	1	1	yes	characteristic
140	Dic₅₉	—	8	2	no	—
141	C₂×#442	—	59	1	no	—
142	C₂×#361	—	59	1	no	—
143	#2140	—	59	1	no	—
144	C₂×#347	—	59	1	no	—
145	#1773	D₅₉	1	1	yes	characteristic, complemented
146	#1386	—	59	1	no	—
147	#1383	—	59	1	no	—
148	C₂×C₅₉	—	2	1	no	—
149	C₂×C₅₉	—	2	1	no	—
150	C₂×C₅₉	#1762	1	1	yes	characteristic
151	C₂×C₅₉	AGL(1,ℤ/16)	1	2	yes	—
152	D₅₉	—	8	4	no	—
153	C₂²×D₈	—	59	1	no	—
154	C₂×C₄×D₄	—	118	1	no	—
155	C₂×M₃₂	—	59	1	no	—
156	C₂×M₃₂	—	59	1	no	—
157	#442	—	59	4	no	—
158	#442	—	118	2	no	—
159	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
160	#427	C₂×D₅₉	1	1	yes	characteristic
161	C₂×#106	—	59	1	no	—
162	C₂×#102	—	118	1	no	—
163	#405	—	2	1	no	—
164	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
165	#361	—	59	4	no	—
166	#357	—	59	4	no	—
167	#347	—	118	2	no	—
168	#339	—	2	1	no	—
169	C₅₉	#12647	1	1	yes	characteristic, complemented, Sylow
170	C₂²×D₄	—	118	1	no	—
171	C₂³×C₄	—	236	1	no	—
172	C₂×D₈	—	59	2	no	—
173	C₂×D₈	—	59	2	no	—
174	C₂×D₈	—	59	2	no	—
175	C₂×D₈	—	118	1	no	—
176	C₂×D₈	—	118	1	no	—
177	C₂×D₈	—	118	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
181	C₄×D₄	—	118	4	no	—
182	C₄×D₄	—	236	2	no	—
183	#117	—	2	1	no	—
184	C₂×C₄⋊C₄	—	118	1	no	—
185	C₂×C₂²⋊C₄	—	236	1	no	—
186	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
187	M₃₂	—	59	2	no	—
188	M₃₂	—	118	1	no	—
189	M₃₂	—	118	1	no	—
190	M₃₂	—	118	1	no	—
191	C₂×C₁₆	—	59	2	no	—
192	C₂×C₁₆	—	118	1	no	—
193	#108	—	2	1	no	—
194	#108	C₄×D₅₉	1	2	yes	complemented
195	#107	C₄×D₅₉	1	2	yes	complemented
196	#106	—	2	1	no	—
197	#106	—	59	2	no	—
198	#106	—	118	1	no	—
199	#102	—	118	4	no	—
200	#97	—	4	1	no	—
201	#97	C₄×D₅₉	1	2	yes	—
202	#97	D₂₃₆	1	2	yes	—
203	C₄×C₈	—	4	1	no	—
204	C₂⁴	—	236	1	no	—
205	C₂×D₄	—	118	2	no	—
206	C₂×D₄	—	118	2	no	—
207	C₂×D₄	—	118	2	no	—
208	C₂×D₄	—	236	1	no	—
209	C₂×D₄	—	236	1	no	—
210	C₂×D₄	—	236	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	472	4	no	—
213	C₂²×C₄	—	236	4	no	—
214	C₂²×C₄	—	236	1	no	—
215	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
216	D₈	—	59	4	no	—
217	D₈	—	118	2	no	—
218	D₈	—	118	2	no	—
219	D₈	—	118	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	4	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
226	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
229	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
230	C₂×C₈	#11745685	1	1	yes	characteristic
231	C₂×C₈	#11745685	1	1	yes	characteristic
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	118	2	no	—
234	C₄⋊C₄	—	236	1	no	—
235	C₂²⋊C₄	—	4	1	no	—
236	C₂²⋊C₄	—	236	4	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
240	C₁₆	—	118	2	no	—
241	C₁₆	—	118	2	no	—
242	C₂³	—	472	2	no	—
243	C₂³	—	472	2	no	—
244	C₂³	—	236	2	no	—
245	C₂³	—	236	2	no	—
246	C₂³	—	236	2	no	—
247	C₂³	#14273752	1	1	yes	characteristic
248	D₄	—	118	4	no	—
249	D₄	—	236	2	no	—
250	D₄	—	236	2	no	—
251	D₄	—	236	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	472	8	no	—
256	C₂×C₄	—	472	2	no	—
257	C₂×C₄	—	236	2	no	—
258	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
260	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
261	C₂×C₄	#14273752	1	1	yes	characteristic
262	C₂×C₄	#14273753	1	1	yes	characteristic
263	C₂×C₄	#14273758	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	C₄²×D₅₉	1	1	yes	characteristic
267	C₈	#14273684	1	1	yes	characteristic
268	C₈	C₄×D₂₃₆	1	1	yes	characteristic
269	C₈	C₄×D₂₃₆	1	1	yes	characteristic
270	C₂²	—	2	2	no	—
271	C₂²	—	472	2	no	—
272	C₂²	—	472	2	no	—
273	C₂²	—	472	4	no	—
274	C₂²	—	472	4	no	—
275	C₂²	—	236	4	no	—
276	C₂²	#16837406	1	1	yes	characteristic
277	C₂²	#16837500	1	1	yes	characteristic
278	C₂²	#16837512	1	1	yes	characteristic
279	C₄	—	4	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	472	2	no	—
282	C₄	C₄×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
283	C₄	#16837401	1	1	yes	characteristic
284	C₄	#16837500	1	1	yes	characteristic
285	C₄	#16837512	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	472	4	no	—
289	C₂	#21423924	1	1	yes	characteristic
290	C₂	#21425994	1	2	yes	—

	subgroup	G/H
center	C₂²	#16837512
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1773	C₂
socle	C₂²×C₅₉	#442
prime radical	C₂²×D₄₇₂	C₂²
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1773	D₅₉
hypocenter	C₅₉	#12647
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1773

Complements

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

N	H ≅ G/N	type
4. C₂×#16837180	C₂	semidirect
5. #21422201	C₂	semidirect
6. #21422196	C₂	semidirect
7. C₅₉×#1773	C₂	semidirect
13. #16837515	C₄	semidirect
16. #16837214	C₄	semidirect
17. #16837202	C₄	semidirect
39. #14273667	C₂×C₄	semidirect
41. C₂×C₅₉⋊C₁₆	C₂×C₄	semidirect
51. #108×C₅₉	C₂×C₄	semidirect
52. #107×C₅₉	C₂×C₄	semidirect
145. #1773	D₅₉	semidirect
169. C₅₉	#12647	semidirect
194. #108	C₄×D₅₉	semidirect
195. #107	C₄×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837515	2
C₂×#16837214	2
#21425811	2
C₂×#16837180	2
#21422201	2
#21422196	2
C₅₉×#1773	2
#12647	59