Subgroups — #28746359

Order 15104.

498 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21437109	C₂	1	1	yes	maximal, characteristic, complemented
2	#21426330	C₂	1	2	yes	maximal, complemented
3	#21426321	C₂	1	1	yes	maximal, characteristic, complemented
4	#21426250	C₂	1	2	yes	maximal, complemented
5	#21426245	C₂	1	1	yes	maximal, characteristic, complemented
6	#21426151	C₂	1	2	yes	maximal, complemented
7	#21426131	C₂	1	1	yes	maximal, characteristic
8	Q₈×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14273671	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#14273668	C₂	1	2	yes	maximal, complemented
11	C₅₉×#3380	C₂	1	1	yes	maximal, characteristic, complemented
12	#16838137	—	2	1	no	—
13	C₂×#14273671	C₂²	1	1	yes	characteristic
14	C₂×#14273668	C₂²	1	2	yes	—
15	#16837786	C₂²	1	2	yes	—
16	#16837779	C₂²	1	1	yes	characteristic
17	#16837618	—	2	2	no	—
18	#16837603	—	2	1	no	—
19	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
20	C₄×D₄₇₂	C₂²	1	2	yes	—
21	#16837229	C₂²	1	1	yes	characteristic
22	#16837227	C₂²	1	2	yes	—
23	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
24	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
25	#16837205	C₂²	1	1	yes	characteristic
26	#16837205	C₂²	1	2	yes	—
27	#16837203	C₂²	1	1	yes	characteristic
28	#16837202	C₂²	1	2	yes	—
29	#16837202	C₂²	1	2	yes	—
30	#16837202	C₂²	1	2	yes	—
31	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	—
32	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₅₉×#554	—	2	1	no	—
34	C₅₉×#500	C₂²	1	1	yes	characteristic
35	C₅₉×#499	C₂²	1	2	yes	—
36	C₅₉×#478	—	2	2	no	—
37	C₅₉×#476	—	2	1	no	—
38	C₈×Q₈×C₅₉	C₂²	1	1	yes	characteristic
39	C₄×D₈×C₅₉	C₂²	1	2	yes	—
40	C₄×D₈×C₅₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₃₆	C₂³	1	1	yes	characteristic
42	C₂×#11745660	—	2	2	no	—
43	C₂×D₄₇₂	—	2	1	no	—
44	C₂×D₄₇₂	C₂³	1	2	yes	—
45	#14273724	—	2	1	no	—
46	#14273716	—	4	1	no	—
47	#14273715	—	2	2	no	—
48	#14273712	—	4	2	no	—
49	C₄×Dic₁₁₈	—	2	1	no	—
50	C₄×D₂₃₆	—	2	2	no	—
51	#14273689	—	4	1	no	—
52	#14273671	—	2	2	no	—
53	#14273668	—	2	4	no	—
54	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
57	#14273661	C₂³	1	1	yes	characteristic
58	#14273652	—	2	1	no	—
59	#14273649	—	2	2	no	—
60	C₂×D₈×C₅₉	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₅₉	—	2	1	no	—
62	#125×C₅₉	—	2	2	no	—
63	C₂²⋊Q₈×C₅₉	—	4	1	no	—
64	C₂²⋊Q₈×C₅₉	—	4	2	no	—
65	C₄×Q₈×C₅₉	D₄	1	1	yes	characteristic
66	C₄×D₄×C₅₉	—	2	1	no	—
67	C₄×D₄×C₅₉	—	2	2	no	—
68	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
69	#107×C₅₉	—	2	1	no	—
70	#107×C₅₉	C₂³	1	2	yes	—
71	#107×C₅₉	C₂³	1	1	yes	characteristic
72	#106×C₅₉	—	2	2	no	—
73	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
75	#102×C₅₉	—	2	2	no	—
76	#102×C₅₉	—	2	1	no	—
77	C₄×C₈×C₅₉	C₂³	1	2	yes	—
78	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₉	—	4	1	no	—
80	C₂×Q₈×C₅₉	D₈	1	1	yes	characteristic
81	C₂×D₄×C₅₉	—	2	1	no	—
82	C₂²×C₄×C₅₉	—	4	1	no	—
83	C₂²×C₄×C₅₉	—	4	2	no	—
84	C₂×D₂₃₆	—	2	2	no	—
85	C₂×D₂₃₆	—	4	1	no	—
86	C₂×D₂₃₆	—	4	1	no	—
87	C₂×C₄×D₅₉	—	4	2	no	—
88	C₂×Dic₁₁₈	—	2	1	no	—
89	D₈×C₅₉	—	2	1	no	—
90	D₈×C₅₉	#41	1	2	yes	—
91	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₅₉	—	2	1	no	—
95	C₄⋊C₄×C₅₉	—	2	2	no	—
96	C₄⋊C₄×C₅₉	—	4	2	no	—
97	C₄⋊C₄×C₅₉	—	4	1	no	—
98	C₄⋊C₄×C₅₉	—	4	2	no	—
99	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
100	C₄⋊C₄×C₅₉	D₈	1	2	yes	—
101	C₂²⋊C₄×C₅₉	—	4	1	no	—
102	C₂²⋊C₄×C₅₉	—	4	2	no	—
103	C₄²×C₅₉	C₂×D₄	1	2	yes	—
104	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
105	(C₂×D₅₉)⋊C₄	—	4	2	no	—
106	(C₂×D₅₉)⋊C₄	—	4	2	no	—
107	(C₂×D₅₉)⋊C₄	—	4	2	no	—
108	C₄⋊Dic₅₉	—	2	1	no	—
109	Dic₅₉⋊C₄	—	4	1	no	—
110	C₄×Dic₅₉	—	4	1	no	—
111	Dic₂₃₆	—	2	1	no	—
112	Dic₂₃₆	—	2	1	no	—
113	D₄₇₂	—	2	2	no	—
114	D₄₇₂	—	2	2	no	—
115	D₄₇₂	—	8	1	no	—
116	#11745660	—	8	2	no	—
117	C₅₉⋊C₁₆	—	2	2	no	—
118	C₅₉⋊C₁₆	—	2	1	no	—
119	C₅₉⋊C₁₆	—	2	1	no	—
120	C₂³×C₅₉	—	4	1	no	—
121	C₂²×D₅₉	—	4	2	no	—
122	Q₈×C₅₉	—	2	2	no	—
123	Q₈×C₅₉	—	16	1	no	—
124	D₄×C₅₉	—	2	2	no	—
125	D₄×C₅₉	—	4	1	no	—
126	C₂×C₄×C₅₉	—	4	1	no	—
127	C₂×C₄×C₅₉	—	4	2	no	—
128	C₂×C₄×C₅₉	—	8	2	no	—
129	C₂×C₄×C₅₉	—	16	2	no	—
130	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₅₉	—	4	1	no	—
136	C₂×Dic₅₉	—	4	1	no	—
137	D₂₃₆	—	4	2	no	—
138	D₂₃₆	—	4	2	no	—
139	D₂₃₆	—	8	2	no	—
140	D₂₃₆	—	16	1	no	—
141	C₄×D₅₉	—	16	2	no	—
142	Dic₁₁₈	—	4	1	no	—
143	Dic₁₁₈	—	4	1	no	—
144	C₈×C₅₉	—	2	2	no	—
145	C₈×C₅₉	—	2	1	no	—
146	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
147	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
148	#33928	—	59	1	no	maximal, Sylow
149	C₂²×C₅₉	—	4	2	no	—
150	C₂²×C₅₉	—	8	1	no	—
151	C₂²×C₅₉	—	8	1	no	—
152	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
153	C₂×D₅₉	—	8	2	no	—
154	C₂×D₅₉	—	8	2	no	—
155	C₂×D₅₉	—	8	2	no	—
156	C₄×C₅₉	—	2	2	no	—
157	C₄×C₅₉	—	2	2	no	—
158	C₄×C₅₉	—	2	1	no	—
159	C₄×C₅₉	—	8	1	no	—
160	C₄×C₅₉	—	8	2	no	—
161	C₄×C₅₉	#508	1	2	yes	—
162	C₄×C₅₉	#548	1	1	yes	characteristic
163	C₄×C₅₉	#577	1	1	yes	characteristic
164	Dic₅₉	—	8	1	no	—
165	Dic₅₉	—	8	1	no	—
166	#3396	—	59	1	no	—
167	#3380	D₅₉	1	1	yes	characteristic, complemented
168	#2242	—	59	2	no	—
169	#2240	—	59	1	no	—
170	#2228	—	59	2	no	—
171	#2227	—	59	1	no	—
172	#2210	—	59	2	no	—
173	#2206	—	59	1	no	—
174	Q₈×C₁₆	—	59	1	no	—
175	C₄×SD₃₂	—	59	1	no	—
176	C₄×D₁₆	—	59	2	no	—
177	C₂×C₅₉	—	8	2	no	—
178	C₂×C₅₉	#3358	1	1	yes	characteristic
179	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
180	C₂×C₅₉	#3415	1	1	yes	characteristic
181	D₅₉	—	16	2	no	—
182	#559	—	118	1	no	—
183	#554	—	2	1	no	—
184	C₂×SD₃₂	—	59	1	no	—
185	C₂×D₁₆	—	59	2	no	—
186	#500	C₂×D₅₉	1	1	yes	characteristic
187	#499	C₂×D₅₉	1	2	yes	—
188	#495	—	59	2	no	—
189	#493	—	59	1	no	—
190	#478	—	2	2	no	—
191	#476	—	2	1	no	—
192	#463	—	118	2	no	—
193	#460	—	118	1	no	—
194	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
195	C₄×Q₁₆	—	59	1	no	—
196	C₄×D₈	—	59	2	no	—
197	C₄×D₈	C₂×D₅₉	1	2	yes	—
198	C₄×D₈	C₂×D₅₉	1	1	yes	characteristic
199	#367	—	59	1	no	—
200	#366	—	59	2	no	—
201	C₄⋊C₁₆	—	59	1	no	—
202	C₄⋊C₁₆	—	59	2	no	—
203	#358	—	59	1	no	—
204	#358	—	59	2	no	—
205	#357	—	59	1	no	—
206	#357	—	59	2	no	—
207	#357	—	59	2	no	—
208	#357	—	59	2	no	—
209	C₄×C₁₆	—	59	2	no	—
210	C₄×C₁₆	—	59	1	no	—
211	C₅₉	#33928	1	1	yes	characteristic, complemented, Sylow
212	C₂×Q₁₆	—	59	1	no	—
213	C₂×SD₁₆	—	118	2	no	—
214	C₂×D₈	—	59	2	no	—
215	C₂×D₈	—	118	1	no	—
216	C₂×D₈	C₂²×D₅₉	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	1	no	—
218	#127	—	118	1	no	—
219	#125	—	2	2	no	—
220	#124	—	118	2	no	—
221	#124	—	236	1	no	—
222	#123	—	236	2	no	—
223	C₂²⋊Q₈	—	4	1	no	—
224	C₂²⋊Q₈	—	4	2	no	—
225	C₄×Q₈	—	118	1	no	—
226	C₄×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
227	C₄×D₄	—	2	1	no	—
228	C₄×D₄	—	2	2	no	—
229	C₄×D₄	—	118	2	no	—
230	C₄×D₄	—	236	1	no	—
231	C₂×C₄⋊C₄	—	4	1	no	—
232	SD₃₂	—	118	2	no	—
233	D₁₆	—	118	4	no	—
234	C₂×C₁₆	—	59	1	no	—
235	C₂×C₁₆	—	59	1	no	—
236	C₂×C₁₆	—	59	2	no	—
237	#107	—	2	1	no	—
238	#107	—	59	1	no	—
239	#107	C₂²×D₅₉	1	2	yes	—
240	#107	C₂²×D₅₉	1	1	yes	characteristic
241	#106	—	2	2	no	—
242	C₄⋊C₈	C₂²⋊D₅₉	1	1	yes	characteristic
243	C₄⋊C₈	C₂²⋊D₅₉	1	2	yes	—
244	#103	—	118	1	no	—
245	#102	—	2	2	no	—
246	#102	—	2	1	no	—
247	#102	—	118	2	no	—
248	C₄×C₈	C₂²×D₅₉	1	2	yes	—
249	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	118	1	no	—
252	C₂×Q₈	#11745669	1	1	yes	characteristic
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	118	2	no	—
255	C₂×D₄	—	236	1	no	—
256	C₂×D₄	—	236	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	236	2	no	—
260	Q₁₆	—	118	1	no	—
261	Q₁₆	—	118	1	no	—
262	SD₁₆	—	472	2	no	—
263	D₈	—	2	1	no	—
264	D₈	—	118	2	no	—
265	D₈	—	118	2	no	—
266	D₈	—	472	1	no	—
267	D₈	#11745689	1	2	yes	—
268	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
269	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
270	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	118	1	no	—
277	C₄⋊C₄	—	236	1	no	—
278	C₄⋊C₄	#11745669	1	1	yes	characteristic
279	C₄⋊C₄	#11745669	1	2	yes	—
280	C₂²⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	236	2	no	—
283	C₂²⋊C₄	—	236	2	no	—
284	C₂²⋊C₄	—	236	2	no	—
285	C₄²	—	236	1	no	—
286	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
287	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
288	C₁₆	—	118	2	no	—
289	C₁₆	—	118	1	no	—
290	C₁₆	—	118	1	no	—
291	C₂³	—	4	1	no	—
292	C₂³	—	236	2	no	—
293	Q₈	—	2	2	no	—
294	Q₈	—	16	1	no	—
295	Q₈	—	236	1	no	—
296	Q₈	—	236	1	no	—
297	D₄	—	2	2	no	—
298	D₄	—	4	1	no	—
299	D₄	—	944	1	no	—
300	D₄	—	472	2	no	—
301	D₄	—	236	2	no	—
302	D₄	—	236	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	16	2	no	—
307	C₂×C₄	—	944	2	no	—
308	C₂×C₄	—	236	1	no	—
309	C₂×C₄	—	236	1	no	—
310	C₂×C₄	C₂×#11745669	1	2	yes	—
311	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
312	C₂×C₄	C₂×#11745669	1	2	yes	—
313	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	2	no	—
317	C₈	C₂×#11745689	1	1	yes	characteristic
318	C₈	#14273775	1	1	yes	characteristic
319	C₂²	—	4	2	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	—	472	2	no	—
323	C₂²	—	472	2	no	—
324	C₂²	—	472	2	no	—
325	C₂²	C₂²×#11745669	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	2	no	—
331	C₄	—	472	1	no	—
332	C₄	—	472	1	no	—
333	C₄	#16837832	1	2	yes	—
334	C₄	#16838063	1	1	yes	characteristic
335	C₄	#16838210	1	1	yes	characteristic
336	C₂	—	8	2	no	—
337	C₂	—	944	2	no	—
338	C₂	#21436731	1	1	yes	characteristic
339	C₂	C₂×#16837832	1	1	yes	characteristic
340	C₂	#21437235	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#3380	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	#21426131	C₂
hypercenter	#3380	D₅₉
hypocenter	C₅₉	#33928
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3380

Complements

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

N	H ≅ G/N	type
1. #21437109	C₂	semidirect
2. #21426330	C₂	semidirect
3. #21426321	C₂	semidirect
4. #21426250	C₂	semidirect
5. #21426245	C₂	semidirect
6. #21426151	C₂	semidirect
8. Q₈×C₅₉⋊C₁₆	C₂	semidirect
9. C₄×#14273671	C₂	semidirect
10. C₄×#14273668	C₂	semidirect
11. C₅₉×#3380	C₂	semidirect
167. #3380	D₅₉	semidirect
211. C₅₉	#33928	semidirect

Maximal subgroups

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

subgroup	index
#21437109	2
#21426330	2
#21426321	2
#21426250	2
#21426245	2
#21426151	2
#21426131	2
Q₈×C₅₉⋊C₁₆	2
C₄×#14273671	2
C₄×#14273668	2
C₅₉×#3380	2
#33928	59