Subgroups — #28746128

Order 15104.

468 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21436528	C₂	1	1	yes	maximal, characteristic
2	#21436526	C₂	1	1	yes	maximal, characteristic
3	#21426369	C₂	1	1	yes	maximal, characteristic, complemented
4	#21426307	C₂	1	2	yes	maximal, complemented
5	#21426302	C₂	1	2	yes	maximal, complemented
6	#21426239	C₂	1	1	yes	maximal, characteristic, complemented
7	#21426186	C₂	1	2	yes	maximal, complemented
8	C₄×#14273673	C₂	1	1	yes	maximal, characteristic, complemented
9	#21425915	C₂	1	1	yes	maximal, characteristic
10	#21425879	C₂	1	2	yes	maximal, complemented
11	C₅₉×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	#16838126	—	2	1	no	—
13	#16838125	—	2	1	no	—
14	C₂×#14273673	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₅₉	C₂²	1	2	yes	—
16	#16837807	C₂²	1	1	yes	characteristic
17	#16837745	—	2	2	no	—
18	#16837742	—	2	2	no	—
19	#16837687	—	2	1	no	—
20	#16837684	—	2	1	no	—
21	#16837644	C₂²	1	2	yes	—
22	#16837640	C₂²	1	2	yes	—
23	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
24	C₈×D₂₃₆	C₂²	1	1	yes	characteristic
25	#16837439	C₂²	1	2	yes	—
26	#16837439	C₂²	1	2	yes	—
27	#16837230	C₂²	1	2	yes	—
28	#16837227	C₂²	1	2	yes	—
29	#16837227	C₂²	1	1	yes	characteristic
30	#16837227	C₂²	1	2	yes	—
31	#16837207	C₂²	1	2	yes	—
32	#16837206	C₂²	1	1	yes	characteristic
33	#16837205	C₂²	1	1	yes	characteristic
34	(C₂×D₅₉)⋊C₁₆	C₂²	1	2	yes	—
35	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₅₉×#500	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
38	C₅₉×#367	C₂²	1	2	yes	—
39	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₅₉×#358	C₂²	1	2	yes	—
41	C₂×Dic₂₃₆	C₂³	1	1	yes	characteristic
42	C₂×C₈×D₅₉	C₂³	1	2	yes	—
43	#14273727	—	2	1	no	—
44	#14273722	—	4	2	no	—
45	#14273722	—	4	2	no	—
46	#14273716	—	4	1	no	—
47	#14273715	—	4	1	no	—
48	#14273707	—	2	2	no	—
49	#14273703	—	2	2	no	—
50	Q₈×Dic₅₉	—	4	1	no	—
51	C₄×Dic₁₁₈	—	2	1	no	—
52	C₄×Dic₁₁₈	—	4	1	no	—
53	C₄×D₂₃₆	D₄	1	1	yes	characteristic
54	#14273685	—	2	2	no	—
55	#14273685	—	2	2	no	—
56	#14273673	—	2	2	no	—
57	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₁₆×D₅₉	—	2	4	no	—
60	#14273661	—	2	1	no	—
61	#14273661	C₂³	1	2	yes	—
62	#14273661	C₂³	1	1	yes	characteristic
63	#14273658	—	2	2	no	—
64	#14273658	—	2	2	no	—
65	C₄⋊(C₅₉⋊C₈)	D₄	1	1	yes	characteristic
66	#14273652	—	2	1	no	—
67	#14273651	—	2	2	no	—
68	#14273650	—	2	2	no	—
69	(C₂×D₅₉)⋊C₈	D₄	1	2	yes	—
70	C₂×Q₁₆×C₅₉	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₅₉	—	2	1	no	—
72	C₄×Q₈×C₅₉	—	2	1	no	—
73	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
74	#107×C₅₉	—	2	1	no	—
75	#107×C₅₉	C₂³	1	2	yes	—
76	#107×C₅₉	C₂³	1	1	yes	characteristic
77	#103×C₅₉	—	2	1	no	—
78	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₉	—	2	1	no	—
80	C₂×Q₈×C₅₉	—	4	1	no	—
81	C₂×D₂₃₆	D₈	1	1	yes	characteristic
82	C₂×C₄×D₅₉	C₂×D₄	1	2	yes	—
83	C₂×Dic₁₁₈	—	2	1	no	—
84	C₂×Dic₁₁₈	—	4	1	no	—
85	Q₁₆×C₅₉	—	2	1	no	—
86	Q₁₆×C₅₉	—	2	1	no	—
87	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₅₉	—	2	1	no	—
90	C₄⋊C₄×C₅₉	—	2	2	no	—
91	C₄⋊C₄×C₅₉	—	4	1	no	—
92	C₄⋊C₄×C₅₉	—	4	1	no	—
93	C₄²×C₅₉	—	4	1	no	—
94	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
95	(C₂×D₅₉)⋊C₄	—	4	2	no	—
96	(C₂×D₅₉)⋊C₄	—	4	2	no	—
97	(C₂×D₅₉)⋊C₄	—	4	2	no	—
98	(C₂×D₅₉)⋊C₄	—	4	2	no	—
99	(C₂×D₅₉)⋊C₄	D₈	1	2	yes	—
100	C₄⋊Dic₅₉	—	2	2	no	—
101	C₄⋊Dic₅₉	—	2	1	no	—
102	C₄⋊Dic₅₉	—	4	1	no	—
103	C₄⋊Dic₅₉	—	4	2	no	—
104	C₄⋊Dic₅₉	D₈	1	1	yes	characteristic
105	Dic₅₉⋊C₄	—	4	1	no	—
106	Dic₅₉⋊C₄	—	4	2	no	—
107	C₄×Dic₅₉	—	4	1	no	—
108	C₄×Dic₅₉	—	4	2	no	—
109	C₄×Dic₅₉	—	4	2	no	—
110	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
111	Dic₂₃₆	—	2	1	no	—
112	Dic₂₃₆	—	2	1	no	—
113	C₈×D₅₉	—	2	2	no	—
114	C₈×D₅₉	—	2	2	no	—
115	C₁₆×C₅₉	—	2	2	no	—
116	C₅₉⋊C₁₆	—	2	1	no	—
117	C₅₉⋊C₁₆	#41	1	2	yes	—
118	C₂²×D₅₉	C₂×D₈	1	2	yes	—
119	Q₈×C₅₉	—	4	1	no	—
120	Q₈×C₅₉	—	4	1	no	—
121	Q₈×C₅₉	—	8	2	no	—
122	C₂×C₄×C₅₉	—	4	1	no	—
123	C₂×C₄×C₅₉	—	4	2	no	—
124	C₂×C₄×C₅₉	—	4	1	no	—
125	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₅₉	—	4	1	no	—
129	C₂×Dic₅₉	—	4	2	no	—
130	C₂×Dic₅₉	—	4	1	no	—
131	C₂×Dic₅₉	C₂×D₈	1	2	yes	—
132	D₂₃₆	—	2	2	no	—
133	C₄×D₅₉	—	2	2	no	—
134	C₄×D₅₉	—	2	2	no	—
135	Dic₁₁₈	—	4	1	no	—
136	Dic₁₁₈	—	4	1	no	—
137	Dic₁₁₈	—	8	2	no	—
138	C₈×C₅₉	—	2	1	no	—
139	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
140	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
141	C₅₉⋊C₈	—	2	2	no	—
142	#33909	—	59	1	no	maximal, Sylow
143	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
144	C₂×D₅₉	—	2	2	no	—
145	C₂×D₅₉	—	2	2	no	—
146	C₂×D₅₉	—	2	2	no	—
147	C₄×C₅₉	—	2	1	no	—
148	C₄×C₅₉	—	8	1	no	—
149	C₄×C₅₉	—	8	2	no	—
150	C₄×C₅₉	—	8	1	no	—
151	C₄×C₅₉	#508	1	2	yes	—
152	C₄×C₅₉	#547	1	1	yes	characteristic
153	C₄×C₅₉	#578	1	1	yes	characteristic
154	Dic₅₉	—	2	2	no	—
155	Dic₅₉	—	8	1	no	—
156	Dic₅₉	—	8	2	no	—
157	Dic₅₉	—	8	1	no	—
158	#3342	—	59	1	no	—
159	#3342	—	59	1	no	—
160	#2252	—	59	1	no	—
161	#2237	—	59	2	no	—
162	#2237	—	59	2	no	—
163	#2225	—	59	1	no	—
164	#2225	D₅₉	1	1	yes	characteristic, complemented
165	#2216	—	59	2	no	—
166	C₄×Q₃₂	—	59	1	no	—
167	D₄×C₁₆	—	59	1	no	—
168	#2158	—	59	2	no	—
169	C₂×C₅₉	#3334	1	1	yes	characteristic
170	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
171	C₂×C₅₉	#3417	1	1	yes	characteristic
172	D₅₉	—	4	2	no	—
173	#556	—	118	1	no	—
174	#556	—	118	1	no	—
175	C₂×Q₃₂	—	59	1	no	—
176	C₂²×C₁₆	—	59	2	no	—
177	#500	—	59	1	no	—
178	#500	C₂×D₅₉	1	1	yes	characteristic
179	#485	—	118	2	no	—
180	#485	—	118	2	no	—
181	#475	—	118	1	no	—
182	#475	—	118	1	no	—
183	#467	—	59	2	no	—
184	#467	—	59	2	no	—
185	C₄×Q₁₆	—	59	1	no	—
186	C₄×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
187	C₈×D₄	—	59	1	no	—
188	#427	—	59	2	no	—
189	#427	—	59	2	no	—
190	#367	—	59	2	no	—
191	#367	C₂×D₅₉	1	2	yes	—
192	#366	—	59	1	no	—
193	#366	—	59	2	no	—
194	#366	—	59	2	no	—
195	C₄⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
196	#358	—	59	1	no	—
197	#358	—	59	2	no	—
198	#358	—	59	1	no	—
199	#358	C₂×D₅₉	1	2	yes	—
200	C₂²⋊C₁₆	—	59	2	no	—
201	C₄×C₁₆	—	59	1	no	—
202	C₅₉	#33909	1	1	yes	characteristic, complemented, Sylow
203	C₂×Q₁₆	—	59	1	no	—
204	C₂×Q₁₆	C₂²×D₅₉	1	1	yes	characteristic
205	C₂²×C₈	—	59	2	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	118	1	no	—
208	#126	—	236	2	no	—
209	#126	—	236	2	no	—
210	#124	—	236	1	no	—
211	#124	—	236	1	no	—
212	C₂²⋊Q₈	—	118	2	no	—
213	C₂²⋊Q₈	—	118	2	no	—
214	C₄×Q₈	—	2	1	no	—
215	C₄×Q₈	—	118	1	no	—
216	C₄×Q₈	—	236	1	no	—
217	C₄×Q₈	—	236	1	no	—
218	C₄×D₄	—	59	1	no	—
219	#117	—	118	2	no	—
220	#117	—	118	2	no	—
221	Q₃₂	—	118	2	no	—
222	C₂×C₁₆	—	59	1	no	—
223	C₂×C₁₆	—	59	1	no	—
224	C₂×C₁₆	—	118	4	no	—
225	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
226	#107	—	2	1	no	—
227	#107	—	59	2	no	—
228	#107	—	59	1	no	—
229	#107	—	118	1	no	—
230	#107	C₂²×D₅₉	1	1	yes	characteristic
231	#107	C₂²×D₅₉	1	2	yes	—
232	#106	—	118	2	no	—
233	#106	—	118	2	no	—
234	C₄⋊C₈	—	59	1	no	—
235	#103	—	2	1	no	—
236	#103	—	118	2	no	—
237	#103	—	118	2	no	—
238	#103	—	118	1	no	—
239	C₂²⋊C₈	—	59	2	no	—
240	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	118	1	no	—
244	C₂×Q₈	—	236	1	no	—
245	C₂×D₄	—	59	1	no	—
246	C₂²×C₄	—	59	2	no	—
247	Q₁₆	—	2	1	no	—
248	Q₁₆	—	2	1	no	—
249	Q₁₆	—	118	1	no	—
250	Q₁₆	—	118	1	no	—
251	C₂×C₈	—	59	2	no	—
252	C₂×C₈	—	118	2	no	—
253	C₂×C₈	—	118	2	no	—
254	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
255	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	59	1	no	—
261	C₄⋊C₄	—	118	1	no	—
262	C₄⋊C₄	—	118	2	no	—
263	C₄⋊C₄	—	236	2	no	—
264	C₄⋊C₄	—	236	2	no	—
265	C₄⋊C₄	—	236	1	no	—
266	C₄⋊C₄	—	236	1	no	—
267	C₂²⋊C₄	—	59	2	no	—
268	C₂²⋊C₄	—	236	2	no	—
269	C₂²⋊C₄	—	236	2	no	—
270	C₂²⋊C₄	—	236	2	no	—
271	C₂²⋊C₄	—	236	2	no	—
272	C₄²	—	4	1	no	—
273	C₄²	—	236	2	no	—
274	C₄²	—	236	2	no	—
275	C₄²	—	236	1	no	—
276	C₄²	D₄×D₅₉	1	1	yes	characteristic
277	C₁₆	—	2	2	no	—
278	C₁₆	—	59	2	no	—
279	C₁₆	—	118	1	no	—
280	C₂³	—	59	2	no	—
281	Q₈	—	4	1	no	—
282	Q₈	—	4	1	no	—
283	Q₈	—	8	2	no	—
284	Q₈	—	472	2	no	—
285	Q₈	—	236	1	no	—
286	Q₈	—	236	1	no	—
287	D₄	—	118	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	59	2	no	—
292	C₂×C₄	—	118	2	no	—
293	C₂×C₄	—	118	2	no	—
294	C₂×C₄	—	236	1	no	—
295	C₂×C₄	—	236	2	no	—
296	C₂×C₄	—	236	1	no	—
297	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
298	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
299	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	118	2	no	—
302	C₈	#41×D₅₉	1	1	yes	characteristic
303	C₈	#14273775	1	1	yes	characteristic
304	C₂²	—	118	2	no	—
305	C₂²	—	118	2	no	—
306	C₂²	—	118	2	no	—
307	C₂²	C₂×D₈×D₅₉	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	8	1	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	1	no	—
312	C₄	—	118	2	no	—
313	C₄	—	472	1	no	—
314	C₄	—	472	2	no	—
315	C₄	—	472	1	no	—
316	C₄	#16837833	1	2	yes	—
317	C₄	#16838054	1	1	yes	characteristic
318	C₄	#16838218	1	1	yes	characteristic
319	C₂	—	236	2	no	—
320	C₂	#21436442	1	1	yes	characteristic
321	C₂	C₂×#16837833	1	1	yes	characteristic
322	C₂	#21437248	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₉
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2225	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂×D₂₃₆	D₈
hypercenter	#2225	D₅₉
hypocenter	C₅₉	#33909
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2225

Complements

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

N	H ≅ G/N	type
3. #21426369	C₂	semidirect
4. #21426307	C₂	semidirect
5. #21426302	C₂	semidirect
6. #21426239	C₂	semidirect
7. #21426186	C₂	semidirect
8. C₄×#14273673	C₂	semidirect
10. #21425879	C₂	semidirect
11. C₅₉×#2225	C₂	semidirect
164. #2225	D₅₉	semidirect
202. C₅₉	#33909	semidirect

Maximal subgroups

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

subgroup	index
#21436528	2
#21436526	2
#21426369	2
#21426307	2
#21426302	2
#21426239	2
#21426186	2
C₄×#14273673	2
#21425915	2
#21425879	2
C₅₉×#2225	2
#33909	59