Subgroups — #28481932

Order 15104.

430 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21432807	C₂	1	1	yes	maximal, characteristic
2	#21424267	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16837152	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422774	C₂	1	1	yes	maximal, characteristic
5	#21422771	C₂	1	1	yes	maximal, characteristic
6	#21421970	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1292	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273735	—	2	1	no	—
9	#16837952	C₂²	1	1	yes	characteristic
10	#16837479	—	2	1	no	—
11	#16837479	C₄	1	2	yes	—
12	#16837467	—	2	1	no	—
13	#16837467	C₄	1	2	yes	—
14	C₂×#14273662	—	2	1	no	—
15	C₂×#14273662	—	2	1	no	—
16	#16837426	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
18	#16837370	—	2	1	no	—
19	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
20	#16837152	—	2	1	no	—
21	#16837152	—	2	1	no	—
22	#16837149	—	2	1	no	—
23	#16837147	—	2	1	no	—
24	#16837145	—	2	1	no	—
25	#16837145	—	2	1	no	—
26	C₅₉×#423	C₂²	1	1	yes	characteristic
27	C₅₉×#423	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
29	C₂×#11745685	—	2	1	no	—
30	#14273735	—	4	2	no	—
31	#14273735	—	4	2	no	—
32	C₂×#11745664	—	2	1	no	—
33	C₂×#11745659	—	2	2	no	—
34	C₂²×C₅₉⋊C₈	—	2	1	no	—
35	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
36	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
37	C₂×C₈×D₅₉	—	2	2	no	—
38	#14273727	D₄	1	1	yes	characteristic
39	#14273724	D₄	1	1	yes	characteristic
40	#14273715	—	2	1	no	—
41	#14273701	—	2	1	no	—
42	#14273701	Q₈	1	2	yes	—
43	C₄×D₂₃₆	—	2	1	no	—
44	C₄×D₂₃₆	C₂×C₄	1	2	yes	—
45	#14273662	—	4	2	no	—
46	#14273662	—	4	2	no	—
47	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
48	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
49	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
50	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
51	(C₂×D₅₉)⋊C₈	—	2	2	no	—
52	(C₂×D₅₉)⋊C₈	—	2	2	no	—
53	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
56	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₅₉	—	2	1	no	—
58	C₂×M₁₆×C₅₉	—	2	1	no	—
59	C₂²×C₈×C₅₉	—	2	1	no	—
60	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₅₉	—	2	1	no	—
62	C₄⋊C₈×C₅₉	—	2	1	no	—
63	C₄⋊C₈×C₅₉	—	2	1	no	—
64	C₄⋊C₈×C₅₉	—	2	1	no	—
65	C₄⋊C₈×C₅₉	—	2	1	no	—
66	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
67	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
70	C₂×C₂²⋊D₅₉	—	2	2	no	—
71	#11745685	—	4	1	no	—
72	#11745685	—	4	1	no	—
73	#11745685	—	4	1	no	—
74	#11745685	—	4	1	no	—
75	C₂×D₂₃₆	—	2	1	no	—
76	C₂×D₂₃₆	—	2	1	no	—
77	C₂×D₂₃₆	C₄⋊C₄	1	2	yes	—
78	C₂×C₄×D₅₉	—	2	2	no	—
79	C₂×Dic₁₁₈	—	2	1	no	—
80	M₁₆×C₅₉	—	4	1	no	—
81	M₁₆×C₅₉	—	4	1	no	—
82	M₁₆×C₅₉	—	4	1	no	—
83	M₁₆×C₅₉	—	4	1	no	—
84	C₂×C₈×C₅₉	—	2	2	no	—
85	C₂×C₈×C₅₉	—	4	1	no	—
86	C₂×C₈×C₅₉	—	4	1	no	—
87	C₂×C₈×C₅₉	—	8	1	no	—
88	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₉	#41	1	1	yes	characteristic
92	(C₂×D₅₉)⋊C₄	—	2	2	no	—
93	(C₂×D₅₉)⋊C₄	—	2	2	no	—
94	C₄⋊Dic₅₉	—	2	1	no	—
95	C₄⋊Dic₅₉	C₄⋊C₄	1	2	yes	—
96	#11745664	—	4	1	no	—
97	#11745664	—	4	1	no	—
98	C₂×C₅₉⋊C₈	—	2	1	no	—
99	C₂×C₅₉⋊C₈	—	2	1	no	—
100	C₂×C₅₉⋊C₈	—	4	1	no	—
101	C₂×C₅₉⋊C₈	—	4	1	no	—
102	C₂×C₅₉⋊C₈	—	8	1	no	—
103	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₅₉⋊C₈	D₈	1	2	yes	—
108	C₂×C₅₉⋊C₈	SD₁₆	1	2	yes	—
109	#11745659	—	4	4	no	—
110	C₈×D₅₉	—	4	4	no	—
111	C₂³×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂²×D₅₉	—	2	2	no	—
113	C₂×C₄×C₅₉	—	2	1	no	—
114	C₂×C₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	—	4	1	no	—
116	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
119	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
120	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
126	C₂²⋊D₅₉	—	4	2	no	—
127	C₂²⋊D₅₉	—	4	2	no	—
128	C₂×Dic₅₉	—	2	2	no	—
129	D₂₃₆	—	4	2	no	—
130	D₂₃₆	—	4	1	no	—
131	D₂₃₆	—	4	1	no	—
132	D₂₃₆	—	4	1	no	—
133	D₂₃₆	—	4	1	no	—
134	C₄×D₅₉	—	4	2	no	—
135	C₄×D₅₉	—	4	2	no	—
136	Dic₁₁₈	—	4	1	no	—
137	Dic₁₁₈	—	4	1	no	—
138	C₈×C₅₉	—	4	2	no	—
139	C₈×C₅₉	—	8	1	no	—
140	C₈×C₅₉	—	8	1	no	—
141	C₅₉⋊C₈	—	2	1	no	—
142	C₅₉⋊C₈	—	2	1	no	—
143	C₅₉⋊C₈	—	8	1	no	—
144	C₅₉⋊C₈	—	8	1	no	—
145	C₅₉⋊C₈	#102	1	2	yes	—
146	C₅₉⋊C₈	#102	1	2	yes	—
147	#8682	—	59	1	no	maximal, Sylow
148	C₂²×C₅₉	—	2	1	no	—
149	C₂²×C₅₉	—	2	1	no	—
150	C₂²×C₅₉	#380	1	1	yes	characteristic
151	C₂²×C₅₉	#474	1	1	yes	characteristic
152	C₂²×C₅₉	#476	1	1	yes	characteristic
153	C₂×D₅₉	—	4	2	no	—
154	C₂×D₅₉	—	4	2	no	—
155	C₂×D₅₉	—	4	2	no	—
156	C₄×C₅₉	—	2	1	no	—
157	C₄×C₅₉	—	2	1	no	—
158	C₄×C₅₉	—	2	1	no	—
159	C₄×C₅₉	C₂×#102	1	1	yes	characteristic
160	C₄×C₅₉	#417	1	1	yes	characteristic
161	C₄×C₅₉	#459	1	1	yes	characteristic
162	C₄×C₅₉	#461	1	1	yes	characteristic
163	C₄×C₅₉	#471	1	1	yes	characteristic
164	C₄×C₅₉	#471	1	1	yes	characteristic
165	Dic₅₉	—	4	2	no	—
166	#2931	—	59	1	no	—
167	#1848	—	59	1	no	—
168	C₂×#334	—	59	1	no	—
169	#1547	—	59	1	no	—
170	#1547	—	59	1	no	—
171	#1292	—	59	1	no	—
172	#1292	D₅₉	1	1	yes	characteristic, complemented
173	C₂×C₅₉	—	2	1	no	—
174	C₂×C₅₉	—	2	1	no	—
175	C₂×C₅₉	#1802	1	1	yes	characteristic
176	C₂×C₅₉	#1813	1	1	yes	characteristic
177	C₂×C₅₉	#1815	1	1	yes	characteristic
178	D₅₉	—	8	2	no	—
179	C₂×#131	—	118	1	no	—
180	#532	—	59	1	no	—
181	#436	—	59	2	no	—
182	#436	—	118	1	no	—
183	C₈×D₄	—	59	2	no	—
184	C₈×D₄	—	118	1	no	—
185	C₂×#108	—	118	1	no	—
186	C₂×#108	—	118	1	no	—
187	#423	—	59	1	no	—
188	#423	C₂×D₅₉	1	1	yes	characteristic
189	#423	C₂×D₅₉	1	1	yes	characteristic
190	C₂×C₄⋊C₈	—	59	1	no	—
191	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
192	#408	—	118	1	no	—
193	C₂×C₄×C₈	—	59	1	no	—
194	#334	—	118	1	no	—
195	#334	—	118	1	no	—
196	#332	—	118	1	no	—
197	#332	—	118	1	no	—
198	#331	—	118	1	no	—
199	#331	—	118	1	no	—
200	C₅₉	#8682	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	118	1	no	—
202	#131	—	236	2	no	—
203	#131	—	236	2	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	118	2	no	—
207	C₂×M₁₆	—	118	1	no	—
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	—	59	1	no	—
210	C₂²×C₈	—	59	1	no	—
211	C₂²×C₈	—	118	2	no	—
212	C₂²×C₈	—	118	1	no	—
213	C₄⋊Q₈	—	59	1	no	—
214	#127	—	59	1	no	—
215	#124	—	118	1	no	—
216	#121	—	59	2	no	—
217	#121	—	118	1	no	—
218	C₄×D₄	—	59	2	no	—
219	C₄×D₄	—	118	1	no	—
220	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
221	#108	—	236	2	no	—
222	#108	—	236	2	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	118	1	no	—
229	C₄⋊C₈	—	118	1	no	—
230	C₄⋊C₈	—	118	1	no	—
231	C₄⋊C₈	—	118	1	no	—
232	C₄⋊C₈	C₄×D₅₉	1	2	yes	—
233	C₂²⋊C₈	—	118	2	no	—
234	C₂²⋊C₈	—	118	2	no	—
235	C₄×C₈	—	59	1	no	—
236	C₄×C₈	—	59	1	no	—
237	C₄×C₈	—	59	1	no	—
238	C₄×C₈	—	59	1	no	—
239	#41	—	236	1	no	—
240	#41	—	236	1	no	—
241	#41	—	236	1	no	—
242	#41	—	236	1	no	—
243	C₂×Q₈	—	118	1	no	—
244	C₂×D₄	—	59	2	no	—
245	C₂×D₄	—	118	2	no	—
246	C₂×D₄	—	118	1	no	—
247	C₂×D₄	—	118	1	no	—
248	C₂²×C₄	—	118	2	no	—
249	C₂²×C₄	#11745685	1	1	yes	characteristic
250	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
251	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
252	M₁₆	—	4	1	no	—
253	M₁₆	—	4	1	no	—
254	M₁₆	—	4	1	no	—
255	M₁₆	—	4	1	no	—
256	M₁₆	—	236	4	no	—
257	M₁₆	—	236	1	no	—
258	M₁₆	—	236	1	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	8	1	no	—
263	C₂×C₈	—	59	1	no	—
264	C₂×C₈	—	59	1	no	—
265	C₂×C₈	—	59	2	no	—
266	C₂×C₈	—	59	2	no	—
267	C₂×C₈	—	59	1	no	—
268	C₂×C₈	—	59	1	no	—
269	C₂×C₈	—	118	1	no	—
270	C₂×C₈	—	118	1	no	—
271	C₂×C₈	—	472	1	no	—
272	C₂×C₈	—	236	4	no	—
273	C₂×C₈	—	236	1	no	—
274	C₂×C₈	—	236	1	no	—
275	C₄⋊C₄	—	59	2	no	—
276	C₄⋊C₄	—	118	1	no	—
277	C₂²⋊C₄	—	118	2	no	—
278	C₂²⋊C₄	—	118	2	no	—
279	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
280	C₄²	#11745685	1	1	yes	characteristic
281	C₄²	D₄×D₅₉	1	1	yes	characteristic
282	C₄²	D₄×D₅₉	1	1	yes	characteristic
283	C₂³	—	118	2	no	—
284	C₂³	#14273705	1	1	yes	characteristic
285	Q₈	—	236	1	no	—
286	Q₈	—	236	1	no	—
287	D₄	—	236	2	no	—
288	D₄	—	236	2	no	—
289	D₄	—	236	2	no	—
290	D₄	—	236	1	no	—
291	D₄	—	236	1	no	—
292	D₄	—	236	1	no	—
293	D₄	—	236	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	118	2	no	—
298	C₂×C₄	—	236	2	no	—
299	C₂×C₄	—	236	2	no	—
300	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
301	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
302	C₂×C₄	#14273684	1	1	yes	characteristic
303	C₂×C₄	#14273700	1	1	yes	characteristic
304	C₂×C₄	#14273700	1	1	yes	characteristic
305	C₂×C₄	#14273705	1	1	yes	characteristic
306	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
307	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
308	C₂×C₄	#14273754	1	1	yes	characteristic
309	C₂×C₄	#14273759	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	8	1	no	—
312	C₈	—	8	1	no	—
313	C₈	—	59	2	no	—
314	C₈	—	59	2	no	—
315	C₈	—	118	1	no	—
316	C₈	—	118	1	no	—
317	C₈	—	472	1	no	—
318	C₈	—	472	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	236	2	no	—
322	C₂²	—	236	2	no	—
323	C₂²	—	236	2	no	—
324	C₂²	#16837261	1	1	yes	characteristic
325	C₂²	#16837681	1	1	yes	characteristic
326	C₂²	#16837691	1	1	yes	characteristic
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	236	2	no	—
331	C₄	#102×D₅₉	1	1	yes	characteristic
332	C₄	#16837402	1	1	yes	characteristic
333	C₄	#16837596	1	1	yes	characteristic
334	C₄	#16837606	1	1	yes	characteristic
335	C₄	#16837664	1	1	yes	characteristic
336	C₄	#16837668	1	1	yes	characteristic
337	C₂	—	2	1	no	—
338	C₂	—	2	1	no	—
339	C₂	—	472	2	no	—
340	C₂	#21424095	1	1	yes	characteristic
341	C₂	#21424145	1	1	yes	characteristic
342	C₂	#21424154	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837261
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1292	C₂
socle	C₂²×C₅₉	#380
prime radical	#16837952	C₂²
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1292	D₅₉
hypocenter	C₅₉	#8682
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1292

Complements

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

N	H ≅ G/N	type
2. #21424267	C₂	semidirect
3. C₂×#16837152	C₂	semidirect
6. #21421970	C₂	semidirect
7. C₅₉×#1292	C₂	semidirect
172. #1292	D₅₉	semidirect
200. C₅₉	#8682	semidirect

Maximal subgroups

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

subgroup	index
#21432807	2
#21424267	2
C₂×#16837152	2
#21422774	2
#21422771	2
#21421970	2
C₅₉×#1292	2
#8682	59