Subgroups — #28745984

Order 15104.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 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	Q₁₆×D₂₃₆	C₂	1	1	yes	maximal, characteristic
3	#21426324	C₂	1	1	yes	maximal, characteristic, complemented
4	#21426306	C₂	1	2	yes	maximal, complemented
5	#21426238	C₂	1	1	yes	maximal, characteristic, complemented
6	#21426203	C₂	1	2	yes	maximal, complemented
7	#21426146	C₂	1	1	yes	maximal, characteristic, complemented
8	#21426028	C₂	1	2	yes	maximal, complemented
9	#21425929	C₂	1	1	yes	maximal, characteristic
10	#21425814	C₂	1	2	yes	maximal, complemented
11	C₄×Q₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₅₉	C₂²	1	2	yes	—
13	#16838125	—	2	1	no	—
14	Q₈×D₂₃₆	—	2	1	no	—
15	C₂×#14273673	C₂²	1	2	yes	—
16	C₂×#14273666	C₂²	1	2	yes	—
17	#16837807	C₂²	1	1	yes	characteristic
18	#16837782	C₂²	1	1	yes	characteristic
19	#16837742	—	2	2	no	—
20	#16837687	—	2	1	no	—
21	#16837644	C₂²	1	2	yes	—
22	#16837640	C₂²	1	2	yes	—
23	#16837612	—	2	1	no	—
24	#16837559	—	2	2	no	—
25	C₈×D₂₃₆	C₂²	1	1	yes	characteristic
26	#16837439	C₂²	1	2	yes	—
27	#16837229	C₂²	1	2	yes	—
28	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
29	#16837207	C₂²	1	2	yes	—
30	#16837207	C₂²	1	2	yes	—
31	#16837206	C₂²	1	2	yes	—
32	#16837205	C₂²	1	2	yes	—
33	(C₂×D₅₉)⋊C₁₆	C₂²	1	2	yes	—
34	C₂×Q₃₂×C₅₉	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
37	C₅₉×#366	C₂²	1	1	yes	characteristic
38	C₅₉×#358	C₂²	1	1	yes	characteristic
39	C₅₉×#358	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₅₉	—	2	2	no	—
42	C₂×Dic₂₃₆	—	2	1	no	—
43	C₂×Dic₂₃₆	C₂³	1	2	yes	—
44	C₂×#11745672	—	2	2	no	—
45	Q₁₆×D₅₉	—	2	4	no	—
46	Q₁₆×D₅₉	—	4	2	no	—
47	C₂×C₈×D₅₉	C₂³	1	2	yes	—
48	#14273727	—	2	1	no	—
49	#14273727	—	2	1	no	—
50	#14273727	—	4	1	no	—
51	#14273722	—	4	2	no	—
52	#14273715	—	4	1	no	—
53	#14273707	—	2	2	no	—
54	#14273707	—	4	2	no	—
55	#14273703	—	2	2	no	—
56	C₄×Dic₁₁₈	—	4	1	no	—
57	C₄×D₂₃₆	—	4	1	no	—
58	C₄×D₂₃₆	D₄	1	1	yes	characteristic
59	#14273685	—	2	2	no	—
60	#14273673	—	4	2	no	—
61	#14273666	—	4	2	no	—
62	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
63	#14273661	—	2	1	no	—
64	#14273661	C₂³	1	2	yes	—
65	#14273658	—	2	2	no	—
66	C₄⋊(C₅₉⋊C₈)	D₄	1	1	yes	characteristic
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	C₂×Q₁₆×C₅₉	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₅₉	—	2	1	no	—
73	C₄×Q₈×C₅₉	—	2	1	no	—
74	Q₃₂×C₅₉	—	2	2	no	—
75	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
77	#107×C₅₉	C₂³	1	1	yes	characteristic
78	#107×C₅₉	C₂³	1	1	yes	characteristic
79	#103×C₅₉	—	2	1	no	—
80	#103×C₅₉	—	2	1	no	—
81	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₅₉	—	2	1	no	—
83	C₂×Q₈×C₅₉	—	2	1	no	—
84	Q₈×D₅₉	—	4	4	no	—
85	Q₈×D₅₉	—	8	2	no	—
86	C₂×D₂₃₆	D₈	1	1	yes	characteristic
87	C₂×C₄×D₅₉	—	4	2	no	—
88	C₂×C₄×D₅₉	C₂×D₄	1	2	yes	—
89	C₂×Dic₁₁₈	—	2	2	no	—
90	C₂×Dic₁₁₈	—	4	1	no	—
91	C₂×Dic₁₁₈	—	4	2	no	—
92	Q₁₆×C₅₉	—	2	1	no	—
93	Q₁₆×C₅₉	—	2	1	no	—
94	Q₁₆×C₅₉	—	2	1	no	—
95	Q₁₆×C₅₉	#41	1	2	yes	—
96	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₅₉	—	2	1	no	—
99	C₄⋊C₄×C₅₉	—	2	1	no	—
100	C₄⋊C₄×C₅₉	—	4	1	no	—
101	C₄⋊C₄×C₅₉	—	4	1	no	—
102	C₄²×C₅₉	—	4	1	no	—
103	C₄²×C₅₉	—	4	1	no	—
104	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
105	#11745672	—	4	4	no	—
106	(C₂×D₅₉)⋊C₄	—	4	2	no	—
107	(C₂×D₅₉)⋊C₄	—	4	2	no	—
108	(C₂×D₅₉)⋊C₄	—	4	2	no	—
109	(C₂×D₅₉)⋊C₄	D₈	1	2	yes	—
110	C₄⋊Dic₅₉	—	2	2	no	—
111	C₄⋊Dic₅₉	—	4	1	no	—
112	C₄⋊Dic₅₉	—	4	1	no	—
113	C₄⋊Dic₅₉	—	4	1	no	—
114	C₄⋊Dic₅₉	—	4	2	no	—
115	C₄⋊Dic₅₉	D₈	1	1	yes	characteristic
116	Dic₅₉⋊C₄	—	4	2	no	—
117	C₄×Dic₅₉	—	4	2	no	—
118	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
119	Dic₂₃₆	—	2	2	no	—
120	Dic₂₃₆	—	2	2	no	—
121	Dic₂₃₆	—	8	1	no	—
122	C₈×D₅₉	—	2	2	no	—
123	C₈×D₅₉	—	2	2	no	—
124	C₁₆×C₅₉	—	2	1	no	—
125	C₁₆×C₅₉	—	2	1	no	—
126	C₅₉⋊C₁₆	—	2	2	no	—
127	C₂²×D₅₉	C₂×D₈	1	2	yes	—
128	Q₈×C₅₉	—	2	2	no	—
129	Q₈×C₅₉	—	4	1	no	—
130	Q₈×C₅₉	—	4	1	no	—
131	Q₈×C₅₉	—	4	1	no	—
132	C₂×C₄×C₅₉	—	4	1	no	—
133	C₂×C₄×C₅₉	—	4	1	no	—
134	C₂×C₄×C₅₉	—	4	1	no	—
135	C₂×C₄×C₅₉	—	4	1	no	—
136	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₅₉	—	4	2	no	—
140	C₂×Dic₅₉	—	4	2	no	—
141	C₂×Dic₅₉	C₂×D₈	1	2	yes	—
142	D₂₃₆	—	2	2	no	—
143	C₄×D₅₉	—	2	2	no	—
144	C₄×D₅₉	—	2	2	no	—
145	C₄×D₅₉	—	8	4	no	—
146	Dic₁₁₈	—	4	2	no	—
147	Dic₁₁₈	—	4	2	no	—
148	Dic₁₁₈	—	8	4	no	—
149	Dic₁₁₈	—	8	2	no	—
150	C₈×C₅₉	—	2	1	no	—
151	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
152	C₈×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
153	C₅₉⋊C₈	—	2	2	no	—
154	#33895	—	59	1	no	maximal, Sylow
155	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₅₉	—	2	2	no	—
157	C₂×D₅₉	—	2	2	no	—
158	C₂×D₅₉	—	2	2	no	—
159	C₄×C₅₉	—	2	1	no	—
160	C₄×C₅₉	—	4	2	no	—
161	C₄×C₅₉	—	8	1	no	—
162	C₄×C₅₉	—	8	1	no	—
163	C₄×C₅₉	—	8	1	no	—
164	C₄×C₅₉	#510	1	2	yes	—
165	C₄×C₅₉	#546	1	1	yes	characteristic
166	C₄×C₅₉	#577	1	1	yes	characteristic
167	Dic₅₉	—	2	2	no	—
168	Dic₅₉	—	8	2	no	—
169	Dic₅₉	—	8	2	no	—
170	#3342	—	59	1	no	—
171	D₄×Q₁₆	—	59	1	no	—
172	#2241	—	59	1	no	—
173	#2237	—	59	2	no	—
174	#2225	—	59	1	no	—
175	#2219	—	59	2	no	—
176	#2209	—	59	1	no	—
177	#2188	—	59	2	no	—
178	C₄×Q₃₂	D₅₉	1	1	yes	characteristic, complemented
179	#2169	—	59	1	no	—
180	#2141	—	59	2	no	—
181	C₂×C₅₉	#3294	1	1	yes	characteristic
182	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
183	C₂×C₅₉	#3417	1	1	yes	characteristic
184	D₅₉	—	4	2	no	—
185	C₂²×Q₁₆	—	59	2	no	—
186	#556	—	118	1	no	—
187	D₄×Q₈	—	118	1	no	—
188	C₂×Q₃₂	—	59	2	no	—
189	C₂×Q₃₂	C₂×D₅₉	1	1	yes	characteristic
190	C₂×M₃₂	—	59	2	no	—
191	#500	—	59	1	no	—
192	#494	—	59	1	no	—
193	#485	—	118	2	no	—
194	#475	—	118	1	no	—
195	#467	—	59	2	no	—
196	#467	—	59	2	no	—
197	#462	—	118	1	no	—
198	#451	—	118	2	no	—
199	C₄×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
200	C₄×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
201	C₈×D₄	—	59	1	no	—
202	#427	—	59	2	no	—
203	#367	—	59	2	no	—
204	#366	C₂×D₅₉	1	1	yes	characteristic
205	C₄⋊C₁₆	—	59	1	no	—
206	#358	—	59	2	no	—
207	#358	—	59	2	no	—
208	#358	—	59	2	no	—
209	#358	—	59	2	no	—
210	#358	C₂×D₅₉	1	1	yes	characteristic
211	#358	C₂×D₅₉	1	1	yes	characteristic
212	C₂²⋊C₁₆	—	59	2	no	—
213	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
214	C₅₉	#33895	1	1	yes	characteristic, complemented, Sylow
215	C₂²×Q₈	—	118	2	no	—
216	C₂×Q₁₆	—	59	2	no	—
217	C₂×Q₁₆	—	118	2	no	—
218	C₂×Q₁₆	—	118	4	no	—
219	C₂×Q₁₆	—	118	1	no	—
220	C₂×Q₁₆	—	236	2	no	—
221	C₂×Q₁₆	C₂²×D₅₉	1	1	yes	characteristic
222	C₂×Q₁₆	C₂²×D₅₉	1	1	yes	characteristic
223	C₂²×C₈	—	59	2	no	—
224	C₄⋊Q₈	—	118	1	no	—
225	C₄⋊Q₈	—	118	1	no	—
226	C₄⋊Q₈	—	236	1	no	—
227	#126	—	236	2	no	—
228	#124	—	236	1	no	—
229	C₂²⋊Q₈	—	118	2	no	—
230	C₂²⋊Q₈	—	118	2	no	—
231	C₂²⋊Q₈	—	236	2	no	—
232	C₄×Q₈	—	2	1	no	—
233	C₄×Q₈	—	2	1	no	—
234	C₄×Q₈	—	236	1	no	—
235	C₄×D₄	—	59	1	no	—
236	C₄×D₄	—	236	1	no	—
237	#117	—	118	2	no	—
238	Q₃₂	—	2	2	no	—
239	Q₃₂	—	236	2	no	—
240	M₃₂	—	236	2	no	—
241	C₂×C₁₆	—	59	2	no	—
242	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
243	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
244	#107	—	59	2	no	—
245	#107	—	118	1	no	—
246	#107	C₂²×D₅₉	1	1	yes	characteristic
247	#107	C₂²×D₅₉	1	1	yes	characteristic
248	#106	—	118	2	no	—
249	C₄⋊C₈	—	59	1	no	—
250	#103	—	2	1	no	—
251	#103	—	2	1	no	—
252	#103	—	118	2	no	—
253	#103	—	118	2	no	—
254	C₂²⋊C₈	—	59	2	no	—
255	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	118	2	no	—
259	C₂×Q₈	—	472	2	no	—
260	C₂×Q₈	—	236	2	no	—
261	C₂×Q₈	—	236	4	no	—
262	C₂×Q₈	—	236	1	no	—
263	C₂×D₄	—	59	1	no	—
264	C₂²×C₄	—	59	2	no	—
265	C₂²×C₄	—	236	2	no	—
266	Q₁₆	—	2	1	no	—
267	Q₁₆	—	2	1	no	—
268	Q₁₆	—	2	1	no	—
269	Q₁₆	—	118	2	no	—
270	Q₁₆	—	118	2	no	—
271	Q₁₆	—	472	1	no	—
272	Q₁₆	—	236	4	no	—
273	Q₁₆	#11745685	1	2	yes	—
274	C₂×C₈	—	59	2	no	—
275	C₂×C₈	—	118	2	no	—
276	C₂×C₈	—	118	2	no	—
277	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
278	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	59	1	no	—
284	C₄⋊C₄	—	118	2	no	—
285	C₄⋊C₄	—	236	2	no	—
286	C₄⋊C₄	—	236	2	no	—
287	C₄⋊C₄	—	236	1	no	—
288	C₄⋊C₄	—	236	1	no	—
289	C₄⋊C₄	—	236	1	no	—
290	C₂²⋊C₄	—	59	2	no	—
291	C₂²⋊C₄	—	236	2	no	—
292	C₂²⋊C₄	—	236	2	no	—
293	C₂²⋊C₄	—	236	2	no	—
294	C₄²	—	4	1	no	—
295	C₄²	—	4	1	no	—
296	C₄²	—	236	2	no	—
297	C₄²	D₄×D₅₉	1	1	yes	characteristic
298	C₁₆	—	2	1	no	—
299	C₁₆	—	2	1	no	—
300	C₁₆	—	118	2	no	—
301	C₂³	—	59	2	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	4	1	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	472	4	no	—
307	Q₈	—	472	2	no	—
308	Q₈	—	236	2	no	—
309	Q₈	—	236	2	no	—
310	D₄	—	118	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	59	2	no	—
316	C₂×C₄	—	118	2	no	—
317	C₂×C₄	—	118	2	no	—
318	C₂×C₄	—	472	4	no	—
319	C₂×C₄	—	236	2	no	—
320	C₂×C₄	—	236	2	no	—
321	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
322	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
323	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	118	2	no	—
326	C₈	C₂×#11745685	1	1	yes	characteristic
327	C₈	#14273773	1	1	yes	characteristic
328	C₂²	—	118	2	no	—
329	C₂²	—	118	2	no	—
330	C₂²	—	118	2	no	—
331	C₂²	C₂×D₈×D₅₉	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	2	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	1	no	—
337	C₄	—	118	2	no	—
338	C₄	—	472	2	no	—
339	C₄	—	472	2	no	—
340	C₄	#16837844	1	2	yes	—
341	C₄	#16838048	1	1	yes	characteristic
342	C₄	#16838209	1	1	yes	characteristic
343	C₂	—	236	2	no	—
344	C₂	#21435911	1	1	yes	characteristic
345	C₂	C₂×#16837844	1	1	yes	characteristic
346	C₂	#21437246	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₉
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₄×Q₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂×D₂₃₆	D₈
hypercenter	C₄×Q₃₂	D₅₉
hypocenter	C₅₉	#33895
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
3. #21426324	C₂	semidirect
4. #21426306	C₂	semidirect
5. #21426238	C₂	semidirect
6. #21426203	C₂	semidirect
7. #21426146	C₂	semidirect
8. #21426028	C₂	semidirect
10. #21425814	C₂	semidirect
11. C₄×Q₃₂×C₅₉	C₂	semidirect
178. C₄×Q₃₂	D₅₉	semidirect
214. C₅₉	#33895	semidirect

Maximal subgroups

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

subgroup	index
#21436528	2
Q₁₆×D₂₃₆	2
#21426324	2
#21426306	2
#21426238	2
#21426203	2
#21426146	2
#21426028	2
#21425929	2
#21425814	2
C₄×Q₃₂×C₅₉	2
#33895	59