Subgroups — #28746056

Order 15104.

516 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21436489	C₂	1	1	yes	maximal, characteristic, complemented
2	#21426376	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426293	C₂	1	2	yes	maximal, complemented
4	#21426279	C₂	1	2	yes	maximal, complemented
5	#21426243	C₂	1	1	yes	maximal, characteristic, complemented
6	#21426235	C₂	1	1	yes	maximal, characteristic, complemented
7	#21426190	C₂	1	2	yes	maximal, complemented
8	C₄×#14273671	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16837229	C₂	1	2	yes	maximal, complemented
11	C₅₉×#3334	C₂	1	1	yes	maximal, characteristic
12	#16838109	—	2	1	no	—
13	C₂×#14273671	C₂²	1	1	yes	characteristic
14	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
15	#16837807	C₂²	1	1	yes	characteristic
16	#16837738	—	2	2	no	—
17	#16837698	—	2	1	no	—
18	#16837643	C₂²	1	2	yes	—
19	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
20	C₂×#14273661	C₂²	1	2	yes	—
21	#16837229	C₂²	1	2	yes	—
22	#16837229	C₂²	1	1	yes	characteristic
23	#16837229	C₂²	1	2	yes	—
24	#16837229	C₂²	1	2	yes	—
25	#16837229	C₂²	1	2	yes	—
26	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
27	#16837205	C₂²	1	2	yes	—
28	#16837205	C₂²	1	1	yes	characteristic
29	#16837203	C₂²	1	2	yes	—
30	#16837203	C₂²	1	1	yes	characteristic
31	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₅₉×#547	—	2	1	no	—
34	C₅₉×#500	C₂²	1	1	yes	characteristic
35	C₅₉×#480	—	2	2	no	—
36	C₅₉×#474	—	2	1	no	—
37	C₅₉×#466	C₂²	1	2	yes	—
38	C₄×D₈×C₅₉	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₅₉	C₂²	1	1	yes	characteristic
40	C₂×#107×C₅₉	C₂²	1	2	yes	—
41	C₂×Dic₂₃₆	C₂³	1	1	yes	characteristic
42	#14273727	—	2	1	no	—
43	#14273718	—	4	1	no	—
44	#14273706	—	2	2	no	—
45	#14273704	—	4	2	no	—
46	C₄×Dic₁₁₈	—	2	1	no	—
47	D₄×Dic₅₉	—	4	1	no	—
48	C₂×C₄⋊Dic₅₉	—	2	2	no	—
49	#14273671	—	2	2	no	—
50	C₂×C₅₉⋊C₁₆	—	2	4	no	—
51	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
53	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
54	#14273661	—	2	2	no	—
55	#14273661	—	2	1	no	—
56	#14273661	C₂³	1	2	yes	—
57	#14273661	C₂³	1	1	yes	characteristic
58	#14273652	—	2	2	no	—
59	#14273652	—	2	1	no	—
60	C₂×#41×C₅₉	—	4	1	no	—
61	C₂×D₈×C₅₉	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₅₉	—	2	1	no	—
64	#123×C₅₉	—	4	2	no	—
65	C₂²⋊Q₈×C₅₉	—	4	1	no	—
66	#121×C₅₉	—	2	2	no	—
67	C₄×D₄×C₅₉	—	2	1	no	—
68	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
70	#107×C₅₉	—	2	2	no	—
71	#107×C₅₉	—	2	1	no	—
72	#107×C₅₉	C₂³	1	1	yes	characteristic
73	#107×C₅₉	C₂³	1	2	yes	—
74	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
75	#102×C₅₉	—	2	1	no	—
76	#102×C₅₉	—	2	2	no	—
77	C₂²⋊C₈×C₅₉	D₄	1	2	yes	—
78	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
79	#41×C₅₉	—	8	4	no	—
80	C₂×Q₈×C₅₉	—	4	1	no	—
81	C₂×D₄×C₅₉	—	2	1	no	—
82	C₂×D₄×C₅₉	—	4	2	no	—
83	C₂×D₄×C₅₉	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₅₉	—	4	2	no	—
85	C₂²×C₄×C₅₉	—	4	1	no	—
86	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
87	C₂²×Dic₅₉	—	4	2	no	—
88	C₂×Dic₁₁₈	—	2	1	no	—
89	C₂×Dic₁₁₈	—	4	1	no	—
90	D₈×C₅₉	—	2	1	no	—
91	D₈×C₅₉	—	2	1	no	—
92	C₂×C₈×C₅₉	—	2	2	no	—
93	C₂×C₈×C₅₉	—	2	2	no	—
94	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₅₉	—	2	1	no	—
98	C₄⋊C₄×C₅₉	—	2	2	no	—
99	C₄⋊C₄×C₅₉	—	4	2	no	—
100	C₄⋊C₄×C₅₉	—	4	1	no	—
101	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
102	C₂²⋊C₄×C₅₉	—	4	1	no	—
103	C₂²⋊C₄×C₅₉	—	4	2	no	—
104	C₂²⋊C₄×C₅₉	D₈	1	2	yes	—
105	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
106	C₂²⋊Dic₅₉	—	4	2	no	—
107	C₄⋊Dic₅₉	—	2	1	no	—
108	C₄⋊Dic₅₉	—	2	2	no	—
109	C₄⋊Dic₅₉	—	4	2	no	—
110	C₄⋊Dic₅₉	—	4	1	no	—
111	Dic₅₉⋊C₄	—	4	1	no	—
112	Dic₅₉⋊C₄	—	4	2	no	—
113	C₄×Dic₅₉	—	4	1	no	—
114	Dic₂₃₆	—	2	1	no	—
115	Dic₂₃₆	—	2	1	no	—
116	C₅₉⋊C₁₆	—	2	2	no	—
117	C₅₉⋊C₁₆	—	2	1	no	—
118	C₅₉⋊C₁₆	#41	1	2	yes	complemented
119	C₂³×C₅₉	—	4	1	no	—
120	C₂³×C₅₉	C₂×D₈	1	2	yes	—
121	Q₈×C₅₉	—	8	2	no	—
122	D₄×C₅₉	—	4	1	no	—
123	D₄×C₅₉	—	4	1	no	—
124	D₄×C₅₉	—	8	4	no	—
125	D₄×C₅₉	SD₃₂	1	4	yes	complemented
126	C₂×C₄×C₅₉	—	2	2	no	—
127	C₂×C₄×C₅₉	—	2	2	no	—
128	C₂×C₄×C₅₉	—	4	1	no	—
129	C₂×C₄×C₅₉	—	4	2	no	—
130	C₂×C₄×C₅₉	—	8	4	no	—
131	C₂×C₄×C₅₉	—	8	2	no	—
132	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₅₉	—	4	1	no	—
137	C₂×Dic₅₉	—	4	1	no	—
138	C₂×Dic₅₉	—	4	2	no	—
139	C₂×Dic₅₉	—	16	2	no	—
140	Dic₁₁₈	—	4	1	no	—
141	Dic₁₁₈	—	4	1	no	—
142	Dic₁₁₈	—	16	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	#33902	—	59	1	no	maximal, Sylow
148	C₂²×C₅₉	—	2	2	no	—
149	C₂²×C₅₉	—	2	2	no	—
150	C₂²×C₅₉	—	8	1	no	—
151	C₂²×C₅₉	—	8	1	no	—
152	C₂²×C₅₉	—	8	1	no	—
153	C₂²×C₅₉	C₂×SD₃₂	1	4	yes	—
154	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
155	C₄×C₅₉	—	2	2	no	—
156	C₄×C₅₉	—	2	1	no	—
157	C₄×C₅₉	—	8	1	no	—
158	C₄×C₅₉	—	8	2	no	—
159	C₄×C₅₉	C₂×SD₃₂	1	2	yes	—
160	C₄×C₅₉	#547	1	1	yes	characteristic
161	C₄×C₅₉	#578	1	1	yes	characteristic
162	Dic₅₉	—	8	1	no	—
163	Dic₅₉	—	8	2	no	—
164	Dic₅₉	—	8	1	no	—
165	#3338	—	59	1	no	—
166	#3334	D₅₉	1	1	yes	characteristic
167	#2254	—	59	1	no	—
168	#2235	—	59	2	no	—
169	#2233	—	59	2	no	—
170	#2226	—	59	1	no	—
171	#2225	—	59	1	no	—
172	#2217	—	59	2	no	—
173	C₄×SD₃₂	—	59	1	no	—
174	D₄×C₁₆	—	59	1	no	—
175	C₂×#367	—	59	2	no	—
176	C₂×C₅₉	—	2	4	no	—
177	C₂×C₅₉	—	16	1	no	—
178	C₂×C₅₉	#3334	1	1	yes	characteristic
179	C₂×C₅₉	C₂²×SD₃₂	1	1	yes	characteristic
180	C₂×C₅₉	#3416	1	1	yes	characteristic
181	#554	—	118	1	no	—
182	#547	—	2	1	no	—
183	C₂×SD₃₂	—	59	1	no	—
184	C₂²×C₁₆	—	59	2	no	—
185	#500	—	59	1	no	—
186	#500	C₂×D₅₉	1	1	yes	characteristic
187	#484	—	118	2	no	—
188	#480	—	2	2	no	—
189	#477	—	118	1	no	—
190	#474	—	2	1	no	—
191	#467	—	59	2	no	—
192	#466	C₂×D₅₉	1	2	yes	—
193	C₄×Q₁₆	—	59	1	no	—
194	C₄×D₈	C₂×D₅₉	1	1	yes	characteristic
195	C₈×D₄	C₂×D₅₉	1	1	yes	characteristic
196	C₂×#107	—	59	2	no	—
197	C₂×#107	C₂×D₅₉	1	2	yes	—
198	#367	—	59	1	no	—
199	#367	—	59	2	no	—
200	#367	—	59	2	no	—
201	#367	—	59	2	no	—
202	#367	—	59	2	no	—
203	C₄⋊C₁₆	—	59	1	no	—
204	#358	—	59	1	no	—
205	#358	—	59	2	no	—
206	#357	—	59	1	no	—
207	#357	—	59	2	no	—
208	C₂²⋊C₁₆	—	59	2	no	—
209	C₄×C₁₆	—	59	1	no	—
210	C₅₉	#33902	1	1	yes	characteristic, complemented, Sylow
211	C₂×#41	—	4	1	no	—
212	C₂×Q₁₆	—	59	1	no	—
213	C₂×D₈	C₂²×D₅₉	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	118	1	no	—
217	#125	—	236	1	no	—
218	#123	—	4	2	no	—
219	C₂²⋊Q₈	—	4	1	no	—
220	C₂²⋊Q₈	—	118	2	no	—
221	C₂²⋊Q₈	—	236	2	no	—
222	#121	—	2	2	no	—
223	C₄×Q₈	—	118	1	no	—
224	C₄×D₄	—	2	1	no	—
225	C₄×D₄	—	236	1	no	—
226	C₄×D₄	C₂²⋊D₅₉	1	1	yes	characteristic
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	118	2	no	—
229	SD₃₂	—	118	2	no	—
230	C₂×C₁₆	—	59	1	no	—
231	C₂×C₁₆	—	59	1	no	—
232	C₂×C₁₆	—	59	2	no	—
233	C₂×C₁₆	—	118	4	no	—
234	#107	—	2	2	no	—
235	#107	—	2	1	no	—
236	#107	—	59	1	no	—
237	#107	—	59	2	no	—
238	#107	—	118	2	no	—
239	#107	—	118	1	no	—
240	#107	C₂²×D₅₉	1	1	yes	characteristic
241	#107	C₂²×D₅₉	1	2	yes	—
242	C₄⋊C₈	C₂²⋊D₅₉	1	1	yes	characteristic
243	#103	—	118	1	no	—
244	#103	—	118	2	no	—
245	#102	—	2	1	no	—
246	#102	—	2	2	no	—
247	C₂²⋊C₈	C₂²⋊D₅₉	1	2	yes	—
248	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
249	#41	—	8	4	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	118	1	no	—
252	C₂×Q₈	—	236	1	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	#11745669	1	1	yes	characteristic
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	236	2	no	—
259	C₂²×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
260	Q₁₆	—	118	1	no	—
261	Q₁₆	—	118	1	no	—
262	D₈	—	2	1	no	—
263	D₈	—	2	1	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
267	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
268	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	118	1	no	—
274	C₄⋊C₄	—	118	2	no	—
275	C₄⋊C₄	—	236	2	no	—
276	C₄⋊C₄	—	236	1	no	—
277	C₄⋊C₄	—	236	1	no	—
278	C₄⋊C₄	—	236	2	no	—
279	C₄⋊C₄	#11745669	1	1	yes	characteristic
280	C₂²⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	236	2	no	—
283	C₂²⋊C₄	#11745669	1	2	yes	—
284	C₄²	—	236	1	no	—
285	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
286	C₁₆	—	59	2	no	—
287	C₁₆	—	118	2	no	—
288	C₁₆	—	118	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	C₂×#11745669	1	2	yes	—
291	Q₈	—	8	2	no	—
292	Q₈	—	944	1	no	—
293	Q₈	—	236	1	no	—
294	Q₈	—	236	1	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	1	no	—
297	D₄	—	8	4	no	—
298	D₄	#14273671	1	4	yes	complemented
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	4	no	—
304	C₂×C₄	—	8	2	no	—
305	C₂×C₄	—	944	2	no	—
306	C₂×C₄	—	236	1	no	—
307	C₂×C₄	—	236	1	no	—
308	C₂×C₄	—	236	2	no	—
309	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
310	C₂×C₄	C₂×#11745669	1	2	yes	—
311	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
312	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	2	2	no	—
315	C₈	#41×D₅₉	1	1	yes	characteristic
316	C₈	#14273774	1	1	yes	characteristic
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	C₂×#14273671	1	4	yes	—
323	C₂²	C₂²×#11745669	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	2	1	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	2	no	—
328	C₄	—	472	1	no	—
329	C₄	—	472	1	no	—
330	C₄	—	472	2	no	—
331	C₄	C₂×#14273671	1	2	yes	—
332	C₄	#16838051	1	1	yes	characteristic
333	C₄	#16838217	1	1	yes	characteristic
334	C₂	—	2	4	no	—
335	C₂	—	16	1	no	—
336	C₂	#21436439	1	1	yes	characteristic
337	C₂	C₂²×#14273671	1	1	yes	characteristic
338	C₂	#21437240	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting, prime radical	C₅₉×#3334	C₂
socle	C₂²×C₅₉	C₂²×D₈
hypercenter	#3334	D₅₉
hypocenter	C₅₉	#33902
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3334

Complements

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

N	H ≅ G/N	type
1. #21436489	C₂	semidirect
2. #21426376	C₂	semidirect
3. #21426293	C₂	semidirect
4. #21426279	C₂	semidirect
5. #21426243	C₂	semidirect
6. #21426235	C₂	semidirect
7. #21426190	C₂	semidirect
8. C₄×#14273671	C₂	semidirect
9. D₄×C₅₉⋊C₁₆	C₂	semidirect
10. C₂×#16837229	C₂	semidirect
118. C₅₉⋊C₁₆	#41	semidirect
125. D₄×C₅₉	SD₃₂	semidirect
210. C₅₉	#33902	semidirect
298. D₄	#14273671	semidirect

Maximal subgroups

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

subgroup	index
#21436489	2
#21426376	2
#21426293	2
#21426279	2
#21426243	2
#21426235	2
#21426190	2
C₄×#14273671	2
D₄×C₅₉⋊C₁₆	2
C₂×#16837229	2
C₅₉×#3334	2
#33902	59