Subgroups — #28479404

Order 15104.

498 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11745685	C₂	1	1	yes	maximal, characteristic
2	C₄×#14273655	C₂	1	1	yes	maximal, characteristic
3	#21422479	C₂	1	1	yes	maximal, characteristic
4	C₈×#11745664	C₂	1	1	yes	maximal, characteristic, complemented
5	#21421933	C₂	1	1	yes	maximal, characteristic, complemented
6	#21421929	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1744	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11745685	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₁₈	C₄	1	2	yes	complemented
10	C₈×D₂₃₆	C₄	1	2	yes	complemented
11	C₈×C₂²⋊D₅₉	—	2	2	no	—
12	#16837455	C₂²	1	1	yes	characteristic
13	#16837455	C₂²	1	1	yes	characteristic
14	#16837453	—	2	1	no	—
15	C₂×#14273655	C₄	1	2	yes	—
16	C₄×#11745664	C₂²	1	1	yes	characteristic
17	C₄×C₈×D₅₉	—	2	1	no	—
18	#16837162	C₄	1	2	yes	—
19	#16837134	C₄	1	2	yes	complemented
20	#16837131	C₄	1	2	yes	complemented
21	#16837124	—	2	1	no	—
22	C₈×C₅₉⋊C₈	—	2	1	no	—
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₄³×C₅₉	C₂²	1	1	yes	characteristic
26	C₅₉×#336	—	2	2	no	—
27	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
28	C₂×#11745664	C₂×C₄	1	2	yes	—
29	C₂×C₈×D₅₉	—	2	2	no	—
30	C₄×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
32	C₄×D₂₃₆	C₂×C₄	1	1	yes	characteristic
33	C₄×D₂₃₆	C₂×C₄	1	1	yes	characteristic
34	C₄×C₂²⋊D₅₉	—	2	1	no	—
35	C₄×C₂²⋊D₅₉	—	2	1	no	—
36	#14273686	C₂×C₄	1	1	yes	characteristic
37	#14273684	—	2	1	no	—
38	C₄²×D₅₉	—	2	1	no	—
39	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
40	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
41	Dic₅₉⋊C₈	—	2	2	no	—
42	#14273655	C₈	1	8	yes	complemented
43	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
44	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
45	(C₂×D₅₉)⋊C₈	—	2	2	no	—
46	#14273638	—	2	1	no	—
47	C₄×C₅₉⋊C₈	—	2	1	no	—
48	C₄×C₅₉⋊C₈	—	2	1	no	—
49	C₄×C₅₉⋊C₈	—	2	1	no	—
50	C₈×Dic₅₉	—	2	2	no	—
51	C₂²×C₈×C₅₉	—	2	1	no	—
52	C₂²×C₈×C₅₉	D₄	1	2	yes	—
53	C₂×C₄²×C₅₉	—	2	1	no	—
54	C₂×C₄²×C₅₉	—	2	1	no	—
55	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₅₉	—	2	1	no	—
58	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
61	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	—	2	2	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	C₂×C₂²⋊D₅₉	—	2	1	no	—
67	#11745685	—	2	2	no	—
68	#11745685	C₂×C₈	1	4	yes	—
69	C₂×D₂₃₆	C₄²	1	1	yes	characteristic
70	C₂×C₄×D₅₉	—	2	1	no	—
71	C₂×C₄×D₅₉	—	2	1	no	—
72	C₂×C₄×D₅₉	—	2	1	no	—
73	C₂×Dic₁₁₈	C₄²	1	1	yes	characteristic
74	C₂×C₈×C₅₉	—	2	2	no	—
75	C₂×C₈×C₅₉	—	4	2	no	—
76	C₂×C₈×C₅₉	—	4	2	no	—
77	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₅₉	—	2	2	no	—
80	C₄²×C₅₉	—	4	2	no	—
81	C₄²×C₅₉	—	4	2	no	—
82	C₄²×C₅₉	C₂×C₈	1	4	yes	complemented
83	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₉	#41	1	1	yes	characteristic
87	C₂²⋊Dic₅₉	—	2	1	no	—
88	(C₂×D₅₉)⋊C₄	—	2	1	no	—
89	(C₂×D₅₉)⋊C₄	—	2	1	no	—
90	C₄⋊Dic₅₉	C₄²	1	1	yes	characteristic
91	C₄⋊Dic₅₉	C₄²	1	1	yes	characteristic
92	Dic₅₉⋊C₄	—	2	1	no	—
93	Dic₅₉⋊C₄	—	2	1	no	—
94	C₄×Dic₅₉	—	2	1	no	—
95	C₄×Dic₅₉	—	2	1	no	—
96	C₄×Dic₅₉	—	2	1	no	—
97	#11745664	—	2	2	no	—
98	#11745664	C₂×C₈	1	4	yes	complemented
99	C₂×C₅₉⋊C₈	—	2	2	no	—
100	C₂×C₅₉⋊C₈	—	2	2	no	—
101	C₂×C₅₉⋊C₈	—	2	2	no	—
102	C₈×D₅₉	—	4	4	no	—
103	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
104	C₂²×D₅₉	—	2	1	no	—
105	C₂×C₄×C₅₉	—	2	2	no	—
106	C₂×C₄×C₅₉	—	2	2	no	—
107	C₂×C₄×C₅₉	—	2	4	no	—
108	C₂×C₄×C₅₉	—	4	2	no	—
109	C₂×C₄×C₅₉	—	4	2	no	—
110	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₅₉	#131	1	2	yes	—
118	C₂²⋊D₅₉	—	2	4	no	—
119	C₂×Dic₅₉	—	2	1	no	—
120	C₂×Dic₅₉	—	2	1	no	—
121	C₂×Dic₅₉	—	2	1	no	—
122	D₂₃₆	—	2	1	no	—
123	D₂₃₆	C₄×C₈	1	2	yes	—
124	C₄×D₅₉	—	2	4	no	—
125	C₄×D₅₉	—	4	2	no	—
126	C₄×D₅₉	—	4	2	no	—
127	Dic₁₁₈	—	2	1	no	—
128	Dic₁₁₈	C₄×C₈	1	2	yes	—
129	C₈×C₅₉	—	2	2	no	—
130	C₈×C₅₉	—	2	2	no	—
131	C₈×C₅₉	—	4	2	no	—
132	C₅₉⋊C₈	—	2	4	no	—
133	C₅₉⋊C₈	—	4	2	no	—
134	C₅₉⋊C₈	—	4	2	no	—
135	#8191	—	59	1	no	maximal, Sylow
136	C₂²×C₅₉	—	2	1	no	—
137	C₂²×C₅₉	—	2	1	no	—
138	C₂²×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₅₉	C₈×D₄	1	2	yes	—
140	C₂×D₅₉	—	2	2	no	—
141	C₂×D₅₉	—	4	1	no	—
142	C₂×D₅₉	—	4	1	no	—
143	C₄×C₅₉	—	2	1	no	—
144	C₄×C₅₉	—	2	1	no	—
145	C₄×C₅₉	—	4	4	no	—
146	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
147	C₄×C₅₉	#405	1	1	yes	characteristic
148	C₄×C₅₉	#421	1	2	yes	—
149	C₄×C₅₉	C₈×D₄	1	2	yes	—
150	C₄×C₅₉	#444	1	2	yes	—
151	Dic₅₉	—	2	2	no	—
152	Dic₅₉	—	4	1	no	—
153	Dic₅₉	—	4	1	no	—
154	C₈×#41	—	59	1	no	—
155	C₄×#104	—	59	1	no	—
156	#1744	D₅₉	1	1	yes	characteristic, complemented
157	#1476	—	59	1	no	—
158	C₈×M₁₆	—	59	1	no	—
159	#1278	—	59	1	no	—
160	#1276	—	59	1	no	—
161	C₂×C₅₉	—	2	2	no	—
162	C₂×C₅₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₅₉	#1759	1	1	yes	characteristic
164	C₂×C₅₉	#1765	1	1	yes	characteristic
165	D₅₉	—	4	2	no	—
166	C₄×#41	—	59	1	no	—
167	C₈×Q₈	—	59	2	no	—
168	C₈×D₄	—	59	2	no	—
169	C₈×D₄	—	118	2	no	—
170	#431	—	59	1	no	—
171	#431	—	59	1	no	—
172	#431	—	118	1	no	—
173	C₂×#104	—	59	2	no	—
174	C₄×M₁₆	—	59	1	no	—
175	C₂×C₄×C₈	—	118	1	no	—
176	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
177	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
178	C₄³	C₂×D₅₉	1	1	yes	characteristic
179	#339	—	59	2	no	—
180	#336	—	2	2	no	—
181	Q₈⋊C₈	—	59	2	no	—
182	D₄⋊C₈	—	59	2	no	—
183	#322	—	118	1	no	—
184	C₈²	—	118	1	no	—
185	C₅₉	#8191	1	1	yes	characteristic, complemented, Sylow
186	C₂×#41	—	59	1	no	—
187	C₂×M₁₆	—	59	2	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	118	2	no	—
190	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
191	C₄×Q₈	—	59	1	no	—
192	C₄×Q₈	—	59	1	no	—
193	C₄×D₄	—	59	1	no	—
194	C₄×D₄	—	59	1	no	—
195	C₄×D₄	—	118	1	no	—
196	C₄×D₄	—	118	1	no	—
197	#117	—	59	1	no	—
198	#117	—	118	1	no	—
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	118	1	no	—
202	C₂×C₄²	C₄×D₅₉	1	2	yes	—
203	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
204	C₄⋊C₈	—	59	2	no	—
205	C₄⋊C₈	—	59	2	no	—
206	C₄⋊C₈	—	118	2	no	—
207	#104	—	59	8	no	—
208	C₂²⋊C₈	—	118	1	no	—
209	C₂²⋊C₈	—	118	1	no	—
210	C₂²⋊C₈	—	118	2	no	—
211	#97	—	118	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	118	2	no	—
214	C₄×C₈	—	118	1	no	—
215	C₄×C₈	—	118	1	no	—
216	C₄×C₈	—	118	1	no	—
217	C₄×C₈	C₄×D₅₉	1	2	yes	—
218	C₄×C₈	C₄×D₅₉	1	2	yes	—
219	C₄×C₈	D₂₃₆	1	1	yes	characteristic
220	C₄×C₈	D₂₃₆	1	1	yes	characteristic
221	#41	—	59	4	no	—
222	#41	—	118	2	no	—
223	C₂×Q₈	—	59	1	no	—
224	C₂×D₄	—	59	1	no	—
225	C₂×D₄	—	118	1	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	118	1	no	—
228	C₂²×C₄	—	118	1	no	—
229	C₂²×C₄	—	118	1	no	—
230	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
231	C₂²×C₄	#11745685	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
233	M₁₆	—	59	4	no	—
234	M₁₆	—	118	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	118	2	no	—
239	C₂×C₈	—	118	2	no	—
240	C₂×C₈	—	118	2	no	—
241	C₂×C₈	—	236	4	no	—
242	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
243	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
244	C₄⋊C₄	—	59	1	no	—
245	C₄⋊C₄	—	59	1	no	—
246	C₄⋊C₄	—	118	1	no	—
247	C₄⋊C₄	—	118	1	no	—
248	C₂²⋊C₄	—	118	1	no	—
249	C₂²⋊C₄	—	118	1	no	—
250	C₂²⋊C₄	—	118	1	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	4	2	no	—
253	C₄²	—	4	2	no	—
254	C₄²	—	118	1	no	—
255	C₄²	—	118	1	no	—
256	C₄²	—	118	1	no	—
257	C₄²	C₈×D₅₉	1	4	yes	complemented
258	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
259	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
260	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
261	C₄²	#11745685	1	1	yes	characteristic
262	C₂³	—	118	1	no	—
263	C₂³	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
264	Q₈	—	59	2	no	—
265	Q₈	—	118	1	no	—
266	D₄	—	59	2	no	—
267	D₄	—	118	4	no	—
268	D₄	—	118	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	118	1	no	—
275	C₂×C₄	—	118	4	no	—
276	C₂×C₄	—	118	1	no	—
277	C₂×C₄	—	118	1	no	—
278	C₂×C₄	—	236	2	no	—
279	C₂×C₄	—	236	2	no	—
280	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
281	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
282	C₂×C₄	#14273684	1	1	yes	characteristic
283	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
284	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
285	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
286	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
287	C₂×C₄	#14273733	1	2	yes	—
288	C₈	—	2	2	no	—
289	C₈	—	2	2	no	—
290	C₈	—	4	2	no	—
291	C₈	—	118	4	no	—
292	C₈	—	236	2	no	—
293	C₈	—	236	2	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	118	2	no	—
297	C₂²	—	236	1	no	—
298	C₂²	—	236	1	no	—
299	C₂²	C₄×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
300	C₂²	C₈×C₂²⋊D₅₉	1	2	yes	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	4	no	—
304	C₄	—	118	2	no	—
305	C₄	—	236	1	no	—
306	C₄	—	236	1	no	—
307	C₄	C₄×C₈×D₅₉	1	1	yes	characteristic
308	C₄	#16837356	1	1	yes	characteristic
309	C₄	#16837420	1	2	yes	—
310	C₄	C₈×D₂₃₆	1	2	yes	—
311	C₄	#16837525	1	2	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	236	2	no	—
314	C₂	C₈×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
315	C₂	#21423907	1	1	yes	characteristic
316	C₂	#21423940	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₃₆
commutator	C₄×C₅₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1744	C₂
socle	C₂²×C₅₉	C₄×C₂²⋊C₄
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1744	D₅₉
hypocenter	C₅₉	#8191
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1744

Complements

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

N	H ≅ G/N	type
4. C₈×#11745664	C₂	semidirect
5. #21421933	C₂	semidirect
6. #21421929	C₂	semidirect
7. C₅₉×#1744	C₂	semidirect
9. C₈×Dic₁₁₈	C₄	semidirect
10. C₈×D₂₃₆	C₄	semidirect
19. #16837134	C₄	semidirect
20. #16837131	C₄	semidirect
42. #14273655	C₈	semidirect
82. C₄²×C₅₉	C₂×C₈	semidirect
98. #11745664	C₂×C₈	semidirect
156. #1744	D₅₉	semidirect
185. C₅₉	#8191	semidirect
257. C₄²	C₈×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11745685	2
C₄×#14273655	2
#21422479	2
C₈×#11745664	2
#21421933	2
#21421929	2
C₅₉×#1744	2
#8191	59