Subgroups — #28743600

Order 15104.

534 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433752	C₂	1	1	yes	maximal, characteristic
2	#21432704	C₂	1	1	yes	maximal, characteristic
3	#21426317	C₂	1	8	yes	maximal, complemented
4	#21425868	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837219	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16837217	C₂	1	1	yes	maximal, characteristic, complemented
7	#21425716	C₂	1	1	yes	maximal, characteristic
8	C₅₉×#2155	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₂₃₆	C₂²	1	1	yes	characteristic
10	#16838168	—	2	1	no	—
11	C₂×#14273733	C₂²	1	1	yes	characteristic
12	#16837978	—	2	1	no	—
13	C₂×#14273667	C₂²	1	1	yes	characteristic
14	#16837661	C₂²	1	4	yes	—
15	#16837644	—	2	2	no	—
16	#16837450	—	2	1	no	—
17	#16837449	C₂²	1	4	yes	—
18	C₂×#14273663	C₂²	1	1	yes	characteristic
19	#108×D₅₉	—	2	2	no	—
20	#16837410	—	2	1	no	—
21	#16837224	C₂²	1	4	yes	—
22	#16837219	C₂²	1	4	yes	—
23	#16837217	C₂²	1	4	yes	—
24	#16837183	C₂²	1	4	yes	—
25	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
27	C₅₉×#427	C₂²	1	1	yes	characteristic
28	C₅₉×#365	C₂²	1	4	yes	—
29	C₂×#11745685	D₄	1	1	yes	characteristic
30	C₂²×Dic₁₁₈	—	2	1	no	—
31	C₂×Dic₂₃₆	—	2	1	no	—
32	C₂×Dic₂₃₆	—	2	1	no	—
33	C₂×Dic₂₃₆	—	2	1	no	—
34	C₂×Dic₂₃₆	—	2	1	no	—
35	C₂×Dic₂₃₆	—	2	1	no	—
36	C₂×Dic₂₃₆	C₂³	1	2	yes	—
37	#14273735	—	4	2	no	—
38	#14273733	—	2	1	no	—
39	#14273733	—	2	1	no	—
40	#14273733	—	2	1	no	—
41	#14273733	—	2	1	no	—
42	#14273733	—	4	2	no	—
43	C₂×#11745664	—	2	1	no	—
44	C₂×#11745664	D₄	1	1	yes	characteristic
45	C₂×#11745659	C₂³	1	2	yes	—
46	M₁₆×D₅₉	—	2	4	no	—
47	C₂×C₈×D₅₉	D₄	1	2	yes	—
48	#14273727	—	4	1	no	—
49	#14273715	—	4	1	no	—
50	#14273711	—	4	2	no	—
51	#14273707	—	2	4	no	—
52	#14273667	—	2	1	no	—
53	#14273667	C₂³	1	2	yes	—
54	C₂×C₅₉⋊C₁₆	—	2	1	no	—
55	#14273663	—	2	1	no	—
56	#14273663	C₂³	1	2	yes	—
57	#14273662	—	2	4	no	—
58	#14273650	—	2	4	no	—
59	C₂×M₁₆×C₅₉	—	2	1	no	—
60	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
61	#117×C₅₉	—	2	1	no	—
62	M₃₂×C₅₉	—	2	1	no	—
63	M₃₂×C₅₉	C₂³	1	2	yes	—
64	C₂×C₁₆×C₅₉	—	2	1	no	—
65	#108×C₅₉	—	2	1	no	—
66	#108×C₅₉	C₂³	1	2	yes	—
67	#107×C₅₉	—	2	1	no	—
68	#106×C₅₉	C₂³	1	2	yes	—
69	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₂²⋊D₅₉	SD₁₆	1	2	yes	—
71	C₂²×Dic₅₉	—	4	1	no	—
72	#11745685	—	2	1	no	—
73	#11745685	—	2	1	no	—
74	#11745685	—	2	1	no	—
75	#11745685	—	2	1	no	—
76	C₂×D₂₃₆	SD₁₆	1	1	yes	characteristic
77	C₂×C₄×D₅₉	C₂×D₄	1	2	yes	—
78	C₂×Dic₁₁₈	—	2	2	no	—
79	C₂×Dic₁₁₈	—	4	1	no	—
80	C₂×Dic₁₁₈	—	4	1	no	—
81	C₂×Dic₁₁₈	—	4	1	no	—
82	C₂×Dic₁₁₈	—	4	1	no	—
83	C₂×Dic₁₁₈	—	4	1	no	—
84	C₂×Dic₁₁₈	SD₁₆	1	1	yes	characteristic
85	M₁₆×C₅₉	—	2	2	no	—
86	M₁₆×C₅₉	—	4	1	no	—
87	C₂×C₈×C₅₉	—	4	1	no	—
88	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
90	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
91	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
92	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
93	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₅₉	—	2	2	no	—
95	C₂²⋊C₄×C₅₉	—	4	1	no	—
96	C₄²×C₅₉	—	4	1	no	—
97	(C₂×D₅₉)⋊C₄	—	4	4	no	—
98	C₄⋊Dic₅₉	—	4	4	no	—
99	#11745664	—	2	1	no	—
100	#11745664	—	2	1	no	—
101	#11745664	—	2	2	no	—
102	#11745664	—	4	1	no	—
103	C₂×C₅₉⋊C₈	—	4	1	no	—
104	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
105	Dic₂₃₆	—	2	2	no	—
106	Dic₂₃₆	—	2	2	no	—
107	Dic₂₃₆	—	4	1	no	—
108	Dic₂₃₆	—	4	1	no	—
109	#11745659	—	2	2	no	—
110	#11745659	—	2	2	no	—
111	#11745659	—	4	4	no	—
112	C₈×D₅₉	—	2	2	no	—
113	C₈×D₅₉	—	2	2	no	—
114	C₈×D₅₉	—	4	4	no	—
115	C₁₆×C₅₉	—	2	2	no	—
116	C₅₉⋊C₁₆	—	2	2	no	—
117	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
118	C₂²×D₅₉	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₅₉	—	4	2	no	—
120	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
121	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
122	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
126	C₂²⋊D₅₉	—	2	2	no	—
127	C₂²⋊D₅₉	—	2	2	no	—
128	C₂×Dic₅₉	—	4	2	no	—
129	C₂×Dic₅₉	—	8	1	no	—
130	C₂×Dic₅₉	—	8	1	no	—
131	C₂×Dic₅₉	C₂×SD₁₆	1	2	yes	—
132	D₂₃₆	—	2	1	no	—
133	D₂₃₆	—	2	1	no	—
134	C₄×D₅₉	—	2	2	no	—
135	C₄×D₅₉	—	2	2	no	—
136	Dic₁₁₈	—	2	1	no	—
137	Dic₁₁₈	—	2	1	no	—
138	Dic₁₁₈	—	4	2	no	—
139	Dic₁₁₈	—	4	2	no	—
140	Dic₁₁₈	—	8	1	no	—
141	Dic₁₁₈	—	8	1	no	—
142	C₈×C₅₉	—	4	2	no	—
143	C₈×C₅₉	#123	1	1	yes	characteristic
144	C₈×C₅₉	#123	1	1	yes	characteristic
145	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
146	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
147	C₅₉⋊C₈	—	2	2	no	—
148	C₅₉⋊C₈	—	4	2	no	—
149	#33658	—	59	1	no	maximal, Sylow
150	C₂²×C₅₉	—	4	1	no	—
151	C₂²×C₅₉	#481	1	1	yes	characteristic
152	C₂²×C₅₉	#481	1	1	yes	characteristic
153	C₂²×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
154	C₂×D₅₉	—	2	2	no	—
155	C₂×D₅₉	—	4	2	no	—
156	C₄×C₅₉	—	8	2	no	—
157	C₄×C₅₉	#481	1	1	yes	characteristic
158	C₄×C₅₉	#481	1	1	yes	characteristic
159	C₄×C₅₉	C₂×#123	1	1	yes	characteristic
160	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	Dic₅₉	—	2	2	no	—
162	Dic₅₉	—	8	2	no	—
163	#3059	—	59	1	no	—
164	#2916	—	59	1	no	—
165	#2239	—	59	8	no	—
166	#2155	—	59	1	no	—
167	#2155	D₅₉	1	1	yes	characteristic, complemented
168	C₂×#362	—	59	1	no	—
169	C₂×#362	—	59	1	no	—
170	#2115	—	59	1	no	—
171	C₂×C₅₉	—	2	1	no	—
172	C₂×C₅₉	—	2	1	no	—
173	C₂×C₅₉	—	2	1	no	—
174	C₂×C₅₉	C₂×#481	1	1	yes	characteristic
175	D₅₉	—	4	2	no	—
176	C₂²×Q₁₆	—	59	1	no	—
177	#568	—	118	1	no	—
178	C₂×#131	—	59	1	no	—
179	#536	—	118	1	no	—
180	C₂×M₃₂	—	59	1	no	—
181	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
182	#470	—	59	4	no	—
183	#467	—	118	2	no	—
184	#430	—	59	4	no	—
185	#430	—	118	1	no	—
186	C₂×#108	—	59	1	no	—
187	C₂×#108	—	118	2	no	—
188	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
189	#427	C₂×D₅₉	1	1	yes	characteristic
190	#419	—	118	1	no	—
191	#365	—	59	4	no	—
192	#365	C₂×D₅₉	1	4	yes	—
193	#362	—	59	4	no	—
194	#362	—	59	4	no	—
195	#349	—	59	4	no	—
196	C₅₉	#33658	1	1	yes	characteristic, complemented, Sylow
197	C₂×#41	—	59	1	no	—
198	C₂²×Q₈	—	118	1	no	—
199	C₂×Q₁₆	—	59	2	no	—
200	C₂×Q₁₆	—	118	1	no	—
201	C₂×Q₁₆	—	118	1	no	—
202	C₂×Q₁₆	—	118	1	no	—
203	C₂×Q₁₆	—	118	1	no	—
204	C₂×Q₁₆	—	118	1	no	—
205	#131	—	118	1	no	—
206	#131	—	118	1	no	—
207	#131	—	118	1	no	—
208	#131	—	118	1	no	—
209	#131	—	236	2	no	—
210	#131	—	236	2	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	—	59	1	no	—
213	C₂×M₁₆	—	59	2	no	—
214	C₂×M₁₆	—	118	4	no	—
215	C₂×M₁₆	—	118	1	no	—
216	C₂²×C₈	—	59	2	no	—
217	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
218	C₄⋊Q₈	—	236	1	no	—
219	#124	—	236	1	no	—
220	#123	—	236	2	no	—
221	C₂²⋊Q₈	—	118	4	no	—
222	#117	—	2	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	59	2	no	—
225	M₃₂	—	118	1	no	—
226	M₃₂	C₂²×D₅₉	1	2	yes	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	118	1	no	—
229	#108	—	2	1	no	—
230	#108	—	59	2	no	—
231	#108	—	118	4	no	—
232	#108	—	118	1	no	—
233	#108	C₂²×D₅₉	1	2	yes	—
234	#107	—	2	1	no	—
235	#106	C₂²×D₅₉	1	2	yes	—
236	#103	—	118	4	no	—
237	#41	—	118	1	no	—
238	#41	—	118	1	no	—
239	#41	—	118	1	no	—
240	#41	—	118	1	no	—
241	C₂×Q₈	—	59	1	no	—
242	C₂×Q₈	—	118	2	no	—
243	C₂×Q₈	—	236	1	no	—
244	C₂×Q₈	—	236	1	no	—
245	C₂×Q₈	—	236	1	no	—
246	C₂×Q₈	—	236	1	no	—
247	C₂×Q₈	—	236	1	no	—
248	C₂×D₄	—	59	2	no	—
249	C₂×D₄	—	59	1	no	—
250	C₂²×C₄	—	59	2	no	—
251	C₂²×C₄	—	236	1	no	—
252	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
253	Q₁₆	—	118	2	no	—
254	Q₁₆	—	118	2	no	—
255	Q₁₆	—	236	1	no	—
256	Q₁₆	—	236	1	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	118	1	no	—
260	M₁₆	—	118	1	no	—
261	M₁₆	—	118	2	no	—
262	M₁₆	—	118	2	no	—
263	M₁₆	—	118	2	no	—
264	M₁₆	—	236	4	no	—
265	M₁₆	—	236	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	59	2	no	—
268	C₂×C₈	—	118	2	no	—
269	C₂×C₈	—	118	2	no	—
270	C₂×C₈	—	236	4	no	—
271	C₂×C₈	—	236	1	no	—
272	C₂×C₈	#11745685	1	1	yes	characteristic
273	C₂×C₈	#11745685	1	1	yes	characteristic
274	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
275	C₂×C₈	#11745689	1	1	yes	characteristic
276	C₂×C₈	#11745689	1	1	yes	characteristic
277	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	236	4	no	—
280	C₂²⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	236	4	no	—
282	C₄²	—	4	1	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	118	2	no	—
285	C₂³	—	59	2	no	—
286	C₂³	SD₁₆×D₅₉	1	1	yes	characteristic
287	Q₈	—	118	1	no	—
288	Q₈	—	118	1	no	—
289	Q₈	—	472	1	no	—
290	Q₈	—	472	1	no	—
291	Q₈	—	236	2	no	—
292	Q₈	—	236	2	no	—
293	D₄	—	118	2	no	—
294	D₄	—	118	2	no	—
295	D₄	—	118	1	no	—
296	D₄	—	118	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	59	2	no	—
299	C₂×C₄	—	118	2	no	—
300	C₂×C₄	—	118	2	no	—
301	C₂×C₄	—	472	1	no	—
302	C₂×C₄	—	472	1	no	—
303	C₂×C₄	—	236	2	no	—
304	C₂×C₄	#14273712	1	1	yes	characteristic
305	C₂×C₄	#14273712	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
307	C₂×C₄	#14273754	1	1	yes	characteristic
308	C₂×C₄	#14273754	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	118	2	no	—
312	C₈	—	236	2	no	—
313	C₈	#14273712	1	1	yes	characteristic
314	C₈	#14273712	1	1	yes	characteristic
315	C₈	C₂×#11745685	1	1	yes	characteristic
316	C₈	C₂×#11745689	1	1	yes	characteristic
317	C₂²	—	4	1	no	—
318	C₂²	—	118	2	no	—
319	C₂²	—	236	2	no	—
320	C₂²	#16837722	1	1	yes	characteristic
321	C₂²	#16837722	1	1	yes	characteristic
322	C₂²	C₂×SD₁₆×D₅₉	1	1	yes	characteristic
323	C₄	—	8	2	no	—
324	C₄	—	118	2	no	—
325	C₄	—	472	2	no	—
326	C₄	#16837722	1	1	yes	characteristic
327	C₄	#16837722	1	1	yes	characteristic
328	C₄	C₂×#14273712	1	1	yes	characteristic
329	C₄	C₂×#14273754	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	236	2	no	—
334	C₂	C₂×#16837722	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16837722
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2155	C₂
socle	C₂×C₅₉	C₂×#481
prime radical	C₂×#11745685	D₄
hypercenter	#2155	D₅₉
hypocenter	C₅₉	#33658
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2155

Complements

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

N	H ≅ G/N	type
3. #21426317	C₂	semidirect
4. #21425868	C₂	semidirect
5. C₂×#16837219	C₂	semidirect
6. C₂×#16837217	C₂	semidirect
8. C₅₉×#2155	C₂	semidirect
167. #2155	D₅₉	semidirect
196. C₅₉	#33658	semidirect

Maximal subgroups

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

subgroup	index
#21433752	2
#21432704	2
#21426317	2
#21425868	2
C₂×#16837219	2
C₂×#16837217	2
#21425716	2
C₅₉×#2155	2
#33658	59