Subgroups — #28746513

Order 15104.

486 conjugacy classes of subgroups.

The subgroup lattice has 334 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
2	#21426335	C₂	1	2	yes	maximal, complemented
3	#21426321	C₂	1	1	yes	maximal, characteristic
4	#21426245	C₂	1	1	yes	maximal, characteristic, complemented
5	#21426230	C₂	1	2	yes	maximal, complemented
6	#21426158	C₂	1	2	yes	maximal, complemented
7	#21426136	C₂	1	1	yes	maximal, characteristic
8	#21426005	C₂	1	1	yes	maximal, characteristic, complemented
9	#21425958	C₂	1	2	yes	maximal, complemented
10	C₄×#14273673	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₈×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16838137	—	2	1	no	—
13	C₂×#14273673	C₂²	1	1	yes	characteristic
14	C₂×#14273670	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	#16837227	C₂²	1	1	yes	characteristic
22	#16837227	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	1	yes	characteristic
27	#16837205	C₂²	1	2	yes	—
28	#16837202	C₂²	1	2	yes	—
29	#16837202	C₂²	1	2	yes	—
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	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	#14273673	—	2	2	no	—
53	#14273670	—	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×Q₁₆×C₅₉	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₅₉	—	2	2	no	—
62	C₄⋊Q₈×C₅₉	—	2	1	no	—
63	C₄⋊Q₈×C₅₉	—	4	1	no	—
64	C₄⋊Q₈×C₅₉	—	4	2	no	—
65	C₄×Q₈×C₅₉	—	2	2	no	—
66	C₄×Q₈×C₅₉	—	2	1	no	—
67	C₄×Q₈×C₅₉	—	4	1	no	—
68	C₄×Q₈×C₅₉	D₄	1	1	yes	characteristic
69	#107×C₅₉	—	2	2	no	—
70	#107×C₅₉	—	2	1	no	—
71	#107×C₅₉	C₂³	1	2	yes	—
72	#107×C₅₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
75	#103×C₅₉	—	2	2	no	—
76	#103×C₅₉	—	2	1	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	1	no	—
81	C₂×Q₈×C₅₉	—	4	2	no	—
82	C₂×Q₈×C₅₉	D₈	1	1	yes	characteristic
83	C₂×D₂₃₆	—	2	2	no	—
84	C₂×D₂₃₆	—	4	1	no	—
85	C₂×D₂₃₆	—	4	1	no	—
86	C₂×C₄×D₅₉	—	4	2	no	—
87	C₂×Dic₁₁₈	—	2	1	no	—
88	Q₁₆×C₅₉	—	2	1	no	—
89	Q₁₆×C₅₉	#41	1	2	yes	—
90	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₅₉	—	2	2	no	—
94	C₄⋊C₄×C₅₉	—	2	1	no	—
95	C₄⋊C₄×C₅₉	—	4	1	no	—
96	C₄⋊C₄×C₅₉	—	4	2	no	—
97	C₄⋊C₄×C₅₉	—	4	2	no	—
98	C₄⋊C₄×C₅₉	—	4	1	no	—
99	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
100	C₄⋊C₄×C₅₉	D₈	1	2	yes	—
101	C₄²×C₅₉	—	4	2	no	—
102	C₄²×C₅₉	—	4	1	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	C₂²×D₅₉	—	4	2	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	1	yes	characteristic
130	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
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	1	no	—
136	D₂₃₆	—	4	2	no	—
137	D₂₃₆	—	4	2	no	—
138	D₂₃₆	—	8	2	no	—
139	D₂₃₆	—	16	1	no	—
140	C₄×D₅₉	—	16	2	no	—
141	Dic₁₁₈	—	4	1	no	—
142	Dic₁₁₈	—	4	1	no	—
143	C₈×C₅₉	—	2	2	no	—
144	C₈×C₅₉	—	2	1	no	—
145	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
146	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
147	#33942	—	59	1	no	maximal, Sylow
148	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₅₉	—	8	2	no	—
150	C₂×D₅₉	—	8	2	no	—
151	C₂×D₅₉	—	8	2	no	—
152	C₄×C₅₉	—	2	2	no	—
153	C₄×C₅₉	—	2	2	no	—
154	C₄×C₅₉	—	2	1	no	—
155	C₄×C₅₉	—	4	2	no	—
156	C₄×C₅₉	—	8	1	no	—
157	C₄×C₅₉	—	8	2	no	—
158	C₄×C₅₉	#510	1	2	yes	—
159	C₄×C₅₉	#548	1	1	yes	characteristic
160	C₄×C₅₉	#577	1	1	yes	characteristic
161	Dic₅₉	—	8	1	no	—
162	Dic₅₉	—	8	1	no	—
163	#3396	—	59	1	no	—
164	Q₈×Q₁₆	D₅₉	1	1	yes	characteristic, complemented
165	#2243	—	59	2	no	—
166	#2240	—	59	1	no	—
167	#2227	—	59	1	no	—
168	#2224	—	59	2	no	—
169	#2211	—	59	2	no	—
170	#2207	—	59	1	no	—
171	#2183	—	59	1	no	—
172	#2175	—	59	2	no	—
173	C₄×Q₃₂	—	59	1	no	—
174	C₂×C₅₉	#3358	1	1	yes	characteristic
175	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
176	C₂×C₅₉	#3415	1	1	yes	characteristic
177	D₅₉	—	16	2	no	—
178	#559	—	118	1	no	—
179	Q₈²	—	2	1	no	—
180	C₂×Q₃₂	—	59	1	no	—
181	C₂×SD₃₂	—	59	2	no	—
182	#500	C₂×D₅₉	1	2	yes	—
183	#500	C₂×D₅₉	1	1	yes	characteristic
184	#495	—	59	2	no	—
185	#493	—	59	1	no	—
186	#477	—	2	1	no	—
187	#477	—	2	2	no	—
188	#463	—	118	2	no	—
189	#460	—	118	1	no	—
190	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
191	C₄×Q₁₆	—	59	1	no	—
192	C₄×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
193	C₄×Q₁₆	C₂×D₅₉	1	2	yes	—
194	C₄×D₈	—	59	2	no	—
195	#366	—	59	1	no	—
196	#366	—	59	2	no	—
197	C₄⋊C₁₆	—	59	1	no	—
198	C₄⋊C₁₆	—	59	2	no	—
199	#358	—	59	1	no	—
200	#358	—	59	1	no	—
201	#358	—	59	2	no	—
202	#357	—	59	2	no	—
203	#357	—	59	2	no	—
204	#357	—	59	2	no	—
205	#346	—	59	2	no	—
206	C₄×C₁₆	—	59	1	no	—
207	C₅₉	#33942	1	1	yes	characteristic, complemented, Sylow
208	C₂×Q₁₆	—	59	1	no	—
209	C₂×Q₁₆	C₂²×D₅₉	1	1	yes	characteristic
210	C₂×SD₁₆	—	118	2	no	—
211	C₂×D₈	—	59	2	no	—
212	C₂×D₈	—	118	1	no	—
213	C₄⋊Q₈	—	2	2	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	4	1	no	—
216	C₄⋊Q₈	—	4	2	no	—
217	#127	—	118	1	no	—
218	#124	—	118	2	no	—
219	#124	—	236	1	no	—
220	#123	—	236	2	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	—	4	1	no	—
224	C₄×Q₈	—	118	1	no	—
225	C₄×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
226	C₄×D₄	—	118	2	no	—
227	C₄×D₄	—	236	1	no	—
228	Q₃₂	—	118	2	no	—
229	SD₃₂	—	118	4	no	—
230	C₂×C₁₆	—	59	1	no	—
231	C₂×C₁₆	—	59	1	no	—
232	C₂×C₁₆	—	59	2	no	—
233	#107	—	2	2	no	—
234	#107	—	2	1	no	—
235	#107	—	59	1	no	—
236	#107	C₂²×D₅₉	1	2	yes	—
237	#107	C₂²×D₅₉	1	1	yes	characteristic
238	C₄⋊C₈	C₂²⋊D₅₉	1	1	yes	characteristic
239	C₄⋊C₈	C₂²⋊D₅₉	1	2	yes	—
240	#103	—	2	2	no	—
241	#103	—	2	1	no	—
242	#103	—	118	1	no	—
243	#102	—	118	2	no	—
244	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₅₉	1	2	yes	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	118	1	no	—
250	C₂×Q₈	#11745669	1	1	yes	characteristic
251	C₂×D₄	—	118	2	no	—
252	C₂×D₄	—	236	1	no	—
253	C₂×D₄	—	236	1	no	—
254	C₂²×C₄	—	236	2	no	—
255	Q₁₆	—	2	1	no	—
256	Q₁₆	—	118	1	no	—
257	Q₁₆	—	118	1	no	—
258	Q₁₆	#11745689	1	2	yes	—
259	SD₁₆	—	472	2	no	—
260	D₈	—	118	2	no	—
261	D₈	—	118	2	no	—
262	D₈	—	472	1	no	—
263	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
264	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	118	1	no	—
273	C₄⋊C₄	—	236	1	no	—
274	C₄⋊C₄	#11745669	1	1	yes	characteristic
275	C₄⋊C₄	#11745669	1	2	yes	—
276	C₂²⋊C₄	—	236	2	no	—
277	C₂²⋊C₄	—	236	2	no	—
278	C₂²⋊C₄	—	236	2	no	—
279	C₄²	—	4	2	no	—
280	C₄²	—	4	1	no	—
281	C₄²	—	236	1	no	—
282	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
283	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
284	C₁₆	—	118	2	no	—
285	C₁₆	—	118	1	no	—
286	C₁₆	—	118	1	no	—
287	C₂³	—	236	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	4	1	no	—
291	Q₈	—	16	2	no	—
292	Q₈	—	16	1	no	—
293	Q₈	—	236	1	no	—
294	Q₈	—	236	1	no	—
295	D₄	—	944	1	no	—
296	D₄	—	472	2	no	—
297	D₄	—	236	2	no	—
298	D₄	—	236	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	944	2	no	—
303	C₂×C₄	—	236	1	no	—
304	C₂×C₄	—	236	1	no	—
305	C₂×C₄	C₂×#11745669	1	2	yes	—
306	C₂×C₄	C₂×#11745669	1	2	yes	—
307	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
308	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	2	no	—
312	C₈	C₂×#11745689	1	1	yes	characteristic
313	C₈	#14273775	1	1	yes	characteristic
314	C₂²	—	472	2	no	—
315	C₂²	—	472	2	no	—
316	C₂²	—	472	2	no	—
317	C₂²	C₂²×#11745669	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	2	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	472	1	no	—
325	C₄	—	472	1	no	—
326	C₄	#16837843	1	2	yes	—
327	C₄	#16838063	1	1	yes	characteristic
328	C₄	#16838210	1	1	yes	characteristic
329	C₂	—	944	2	no	—
330	C₂	#21436731	1	1	yes	characteristic
331	C₂	C₂×#16837843	1	1	yes	characteristic
332	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	Q₈×Q₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	#16837779	C₂²
hypercenter	Q₈×Q₁₆	D₅₉
hypocenter	C₅₉	#33942
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
2. #21426335	C₂	semidirect
4. #21426245	C₂	semidirect
5. #21426230	C₂	semidirect
6. #21426158	C₂	semidirect
8. #21426005	C₂	semidirect
9. #21425958	C₂	semidirect
10. C₄×#14273673	C₂	semidirect
11. Q₈×Q₁₆×C₅₉	C₂	semidirect
164. Q₈×Q₁₆	D₅₉	semidirect
207. C₅₉	#33942	semidirect

Maximal subgroups

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

subgroup	index
#21437109	2
#21426335	2
#21426321	2
#21426245	2
#21426230	2
#21426158	2
#21426136	2
#21426005	2
#21425958	2
C₄×#14273673	2
Q₈×Q₁₆×C₅₉	2
#33942	59