Subgroups — #28746439

Order 15104.

474 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	#21437085	C₂	1	1	yes	maximal, characteristic
2	#21426335	C₂	1	2	yes	maximal
3	#21426325	C₂	1	1	yes	maximal, characteristic, complemented
4	#21426240	C₂	1	1	yes	maximal, characteristic
5	#21426235	C₂	1	2	yes	maximal, complemented
6	#21426158	C₂	1	2	yes	maximal
7	#21426146	C₂	1	1	yes	maximal, characteristic, complemented
8	#21426005	C₂	1	1	yes	maximal, characteristic, complemented
9	#21425965	C₂	1	2	yes	maximal, complemented
10	C₄×#14273670	C₂	1	1	yes	maximal, characteristic
11	Q₈×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16838125	—	2	1	no	—
13	C₂×#14273673	C₂²	1	2	yes	—
14	C₂×#14273670	C₂²	1	1	yes	characteristic
15	#16837786	C₂²	1	2	yes	—
16	#16837782	C₂²	1	1	yes	characteristic
17	#16837625	—	2	2	no	—
18	#16837608	—	2	1	no	—
19	C₄×Dic₂₃₆	C₂²	1	2	yes	—
20	C₄×D₄₇₂	C₂²	1	1	yes	characteristic
21	#16837229	C₂²	1	1	yes	characteristic
22	#16837229	C₂²	1	2	yes	—
23	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
24	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
25	#16837206	C₂²	1	1	yes	characteristic
26	#16837205	C₂²	1	2	yes	—
27	#16837205	C₂²	1	2	yes	—
28	#16837205	C₂²	1	2	yes	—
29	#16837202	C₂²	1	1	yes	characteristic
30	#16837202	C₂²	1	2	yes	—
31	#16837178	C₂²	1	2	yes	—
32	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
33	Q₈²×C₅₉	—	2	1	no	—
34	C₅₉×#500	C₂²	1	2	yes	—
35	C₅₉×#500	C₂²	1	1	yes	characteristic
36	C₅₉×#477	—	2	1	no	—
37	C₅₉×#477	—	2	2	no	—
38	C₈×Q₈×C₅₉	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
41	C₂×Dic₂₃₆	—	2	1	no	—
42	C₂×Dic₂₃₆	C₂³	1	2	yes	—
43	C₂×#11745660	—	2	2	no	—
44	C₂×D₄₇₂	C₂³	1	1	yes	characteristic
45	#14273727	—	2	1	no	—
46	#14273722	—	4	2	no	—
47	#14273715	—	2	2	no	—
48	#14273703	—	4	1	no	—
49	C₄×Dic₁₁₈	—	2	2	no	—
50	C₄×D₂₃₆	—	2	1	no	—
51	#14273685	—	4	1	no	—
52	#14273673	—	2	4	no	—
53	#14273670	—	2	2	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	2	no	—
59	#14273649	—	2	1	no	—
60	C₂×Q₁₆×C₅₉	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₅₉	—	2	1	no	—
62	C₄⋊Q₈×C₅₉	—	2	2	no	—
63	C₄⋊Q₈×C₅₉	—	4	1	no	—
64	C₄⋊Q₈×C₅₉	—	4	2	no	—
65	C₄×Q₈×C₅₉	—	2	1	no	—
66	C₄×Q₈×C₅₉	—	2	2	no	—
67	C₄×Q₈×C₅₉	—	4	1	no	—
68	C₄×Q₈×C₅₉	D₄	1	1	yes	characteristic
69	#107×C₅₉	—	2	1	no	—
70	#107×C₅₉	—	2	2	no	—
71	#107×C₅₉	C₂³	1	1	yes	characteristic
72	#107×C₅₉	C₂³	1	2	yes	—
73	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
75	#103×C₅₉	—	2	1	no	—
76	#103×C₅₉	—	2	2	no	—
77	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₅₉	C₂³	1	2	yes	—
79	C₂×Q₈×C₅₉	—	2	1	no	—
80	C₂×Q₈×C₅₉	—	4	2	no	—
81	C₂×Q₈×C₅₉	—	4	1	no	—
82	C₂×Q₈×C₅₉	D₈	1	1	yes	characteristic
83	C₂×D₂₃₆	—	2	1	no	—
84	C₂×C₄×D₅₉	—	4	1	no	—
85	C₂×Dic₁₁₈	—	2	2	no	—
86	Q₁₆×C₅₉	—	2	1	no	—
87	Q₁₆×C₅₉	#41	1	2	yes	—
88	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₅₉	—	2	1	no	—
92	C₄⋊C₄×C₅₉	—	2	2	no	—
93	C₄⋊C₄×C₅₉	—	4	2	no	—
94	C₄⋊C₄×C₅₉	—	4	1	no	—
95	C₄⋊C₄×C₅₉	—	4	2	no	—
96	C₄⋊C₄×C₅₉	—	4	1	no	—
97	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
98	C₄⋊C₄×C₅₉	D₈	1	2	yes	—
99	C₄²×C₅₉	—	4	2	no	—
100	C₄²×C₅₉	—	4	1	no	—
101	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₅₉	C₂×D₄	1	2	yes	—
103	(C₂×D₅₉)⋊C₄	—	4	2	no	—
104	(C₂×D₅₉)⋊C₄	—	4	1	no	—
105	(C₂×D₅₉)⋊C₄	—	4	2	no	—
106	C₄⋊Dic₅₉	—	2	1	no	—
107	C₄⋊Dic₅₉	—	4	1	no	—
108	C₄⋊Dic₅₉	—	4	1	no	—
109	Dic₅₉⋊C₄	—	4	2	no	—
110	C₄×Dic₅₉	—	4	2	no	—
111	Dic₂₃₆	—	2	2	no	—
112	Dic₂₃₆	—	2	2	no	—
113	Dic₂₃₆	—	8	1	no	—
114	D₄₇₂	—	2	1	no	—
115	D₄₇₂	—	2	1	no	—
116	#11745660	—	8	2	no	—
117	C₅₉⋊C₁₆	—	2	1	no	—
118	C₅₉⋊C₁₆	—	2	2	no	—
119	C₅₉⋊C₁₆	—	2	1	no	—
120	C₂²×D₅₉	—	4	1	no	—
121	Q₈×C₅₉	—	2	2	no	—
122	Q₈×C₅₉	—	2	2	no	—
123	Q₈×C₅₉	—	4	1	no	—
124	Q₈×C₅₉	—	16	2	no	—
125	Q₈×C₅₉	—	16	1	no	—
126	C₂×C₄×C₅₉	—	4	1	no	—
127	C₂×C₄×C₅₉	—	4	2	no	—
128	C₂×C₄×C₅₉	—	4	1	no	—
129	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×Dic₅₉	—	4	1	no	—
135	C₂×Dic₅₉	—	4	2	no	—
136	D₂₃₆	—	4	1	no	—
137	D₂₃₆	—	4	1	no	—
138	C₄×D₅₉	—	8	2	no	—
139	Dic₁₁₈	—	4	2	no	—
140	Dic₁₁₈	—	4	2	no	—
141	C₈×C₅₉	—	2	2	no	—
142	C₈×C₅₉	—	2	1	no	—
143	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
144	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
145	#33936	—	59	1	no	maximal, Sylow
146	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₅₉	—	8	1	no	—
148	C₂×D₅₉	—	8	1	no	—
149	C₂×D₅₉	—	8	1	no	—
150	C₄×C₅₉	—	2	2	no	—
151	C₄×C₅₉	—	2	2	no	—
152	C₄×C₅₉	—	2	1	no	—
153	C₄×C₅₉	—	4	2	no	—
154	C₄×C₅₉	—	8	2	no	—
155	C₄×C₅₉	—	8	1	no	—
156	C₄×C₅₉	#510	1	2	yes	—
157	C₄×C₅₉	#548	1	1	yes	characteristic
158	C₄×C₅₉	#577	1	1	yes	characteristic
159	Dic₅₉	—	8	2	no	—
160	Dic₅₉	—	8	1	no	—
161	#3393	—	59	1	no	—
162	Q₈×Q₁₆	D₅₉	1	1	yes	characteristic, complemented
163	#2243	—	59	2	no	—
164	#2241	—	59	1	no	—
165	#2226	—	59	1	no	—
166	#2225	—	59	2	no	—
167	#2211	—	59	2	no	—
168	#2209	—	59	1	no	—
169	#2183	—	59	1	no	—
170	#2176	—	59	2	no	—
171	C₄×SD₃₂	—	59	1	no	—
172	C₂×C₅₉	#3358	1	1	yes	characteristic
173	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
174	C₂×C₅₉	#3416	1	1	yes	characteristic
175	D₅₉	—	16	1	no	—
176	Q₈²	—	2	1	no	—
177	#556	—	118	1	no	—
178	C₂×Q₃₂	—	59	2	no	—
179	C₂×SD₃₂	—	59	1	no	—
180	#500	C₂×D₅₉	1	2	yes	—
181	#500	C₂×D₅₉	1	1	yes	characteristic
182	#495	—	59	2	no	—
183	#494	—	59	1	no	—
184	#477	—	2	1	no	—
185	#477	—	2	2	no	—
186	#464	—	118	2	no	—
187	#461	—	118	1	no	—
188	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
189	C₄×Q₁₆	—	59	2	no	—
190	C₄×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
191	C₄×Q₁₆	C₂×D₅₉	1	2	yes	—
192	C₄×D₈	—	59	1	no	—
193	#367	—	59	1	no	—
194	#367	—	59	2	no	—
195	C₄⋊C₁₆	—	59	1	no	—
196	C₄⋊C₁₆	—	59	2	no	—
197	#358	—	59	1	no	—
198	#358	—	59	2	no	—
199	#358	—	59	2	no	—
200	#358	—	59	2	no	—
201	#357	—	59	1	no	—
202	#357	—	59	2	no	—
203	#346	—	59	2	no	—
204	C₄×C₁₆	—	59	1	no	—
205	C₅₉	#33936	1	1	yes	characteristic, complemented, Sylow
206	C₂×Q₁₆	—	59	2	no	—
207	C₂×Q₁₆	—	118	1	no	—
208	C₂×Q₁₆	C₂²×D₅₉	1	1	yes	characteristic
209	C₂×SD₁₆	—	118	2	no	—
210	C₂×D₈	—	59	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	2	no	—
213	C₄⋊Q₈	—	4	1	no	—
214	C₄⋊Q₈	—	4	2	no	—
215	C₄⋊Q₈	—	118	1	no	—
216	#126	—	236	2	no	—
217	#124	—	118	2	no	—
218	C₂²⋊Q₈	—	236	1	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×Q₈	—	4	1	no	—
222	C₄×Q₈	—	118	2	no	—
223	C₄×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
224	C₄×D₄	—	118	1	no	—
225	#117	—	236	1	no	—
226	Q₃₂	—	118	4	no	—
227	SD₃₂	—	118	2	no	—
228	C₂×C₁₆	—	59	1	no	—
229	C₂×C₁₆	—	59	1	no	—
230	C₂×C₁₆	—	59	2	no	—
231	#107	—	2	1	no	—
232	#107	—	2	2	no	—
233	#107	—	59	1	no	—
234	#107	C₂²×D₅₉	1	1	yes	characteristic
235	#107	C₂²×D₅₉	1	2	yes	—
236	C₄⋊C₈	C₂²⋊D₅₉	1	1	yes	characteristic
237	C₄⋊C₈	C₂²⋊D₅₉	1	2	yes	—
238	#103	—	2	1	no	—
239	#103	—	2	2	no	—
240	#103	—	118	2	no	—
241	#102	—	118	1	no	—
242	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₅₉	1	2	yes	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	118	2	no	—
248	C₂×Q₈	#11745669	1	1	yes	characteristic
249	C₂×D₄	—	118	1	no	—
250	C₂²×C₄	—	236	1	no	—
251	Q₁₆	—	2	1	no	—
252	Q₁₆	—	118	2	no	—
253	Q₁₆	—	118	2	no	—
254	Q₁₆	—	472	1	no	—
255	Q₁₆	#11745689	1	2	yes	—
256	SD₁₆	—	472	2	no	—
257	D₈	—	118	1	no	—
258	D₈	—	118	1	no	—
259	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
261	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	118	1	no	—
269	C₄⋊C₄	—	236	1	no	—
270	C₄⋊C₄	—	236	2	no	—
271	C₄⋊C₄	—	236	1	no	—
272	C₄⋊C₄	#11745669	1	1	yes	characteristic
273	C₄⋊C₄	#11745669	1	2	yes	—
274	C₂²⋊C₄	—	236	2	no	—
275	C₂²⋊C₄	—	236	1	no	—
276	C₂²⋊C₄	—	236	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	236	2	no	—
280	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
281	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
282	C₁₆	—	118	1	no	—
283	C₁₆	—	118	2	no	—
284	C₁₆	—	118	1	no	—
285	C₂³	—	236	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	16	2	no	—
290	Q₈	—	16	1	no	—
291	Q₈	—	236	2	no	—
292	Q₈	—	236	2	no	—
293	D₄	—	236	1	no	—
294	D₄	—	236	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	472	2	no	—
299	C₂×C₄	—	236	1	no	—
300	C₂×C₄	—	236	2	no	—
301	C₂×C₄	C₂×#11745669	1	2	yes	—
302	C₂×C₄	C₂×#11745669	1	2	yes	—
303	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
304	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	C₂×#11745689	1	1	yes	characteristic
309	C₈	#14273775	1	1	yes	characteristic
310	C₂²	—	472	1	no	—
311	C₂²	—	472	1	no	—
312	C₂²	—	472	1	no	—
313	C₂²	C₂²×#11745669	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	1	no	—
320	C₄	—	472	1	no	—
321	C₄	—	472	2	no	—
322	C₄	#16837843	1	2	yes	—
323	C₄	#16838063	1	1	yes	characteristic
324	C₄	#16838210	1	1	yes	characteristic
325	C₂	—	944	1	no	—
326	C₂	#21436731	1	1	yes	characteristic
327	C₂	C₂×#16837843	1	1	yes	characteristic
328	C₂	#21437243	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	Q₈×Q₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂×D₄₇₂	C₂³
hypercenter	Q₈×Q₁₆	D₅₉
hypocenter	C₅₉	#33936
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #21426325	C₂	semidirect
5. #21426235	C₂	semidirect
7. #21426146	C₂	semidirect
8. #21426005	C₂	semidirect
9. #21425965	C₂	semidirect
11. Q₈×Q₁₆×C₅₉	C₂	semidirect
162. Q₈×Q₁₆	D₅₉	semidirect
205. C₅₉	#33936	semidirect

Maximal subgroups

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

subgroup	index
#21437085	2
#21426335	2
#21426325	2
#21426240	2
#21426235	2
#21426158	2
#21426146	2
#21426005	2
#21425965	2
C₄×#14273670	2
Q₈×Q₁₆×C₅₉	2
#33936	59