Subgroups — #28743442

Order 15104.

514 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	#21434628	C₂	1	1	yes	maximal, characteristic, complemented
2	#21433012	C₂	1	1	yes	maximal, characteristic
3	#21426271	C₂	1	4	yes	maximal, complemented
4	#21426266	C₂	1	4	yes	maximal, complemented
5	C₂×#16837226	C₂	1	1	yes	maximal, characteristic, complemented
6	#21425861	C₂	1	1	yes	maximal, characteristic, complemented
7	#21425831	C₂	1	1	yes	maximal, characteristic
8	C₂×#16837213	C₂	1	1	yes	maximal, characteristic
9	C₅₉×#2155	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14273749	C₂²	1	1	yes	characteristic
11	#16838168	—	2	1	no	—
12	#16838030	—	2	1	no	—
13	C₂×#14273667	C₂²	1	1	yes	characteristic
14	#16837810	—	2	2	no	—
15	#16837802	C₂²	1	2	yes	complemented
16	#16837799	C₂²	1	2	yes	complemented
17	#16837523	—	2	2	no	—
18	#16837518	C₂²	1	4	yes	—
19	#16837445	—	2	1	no	—
20	C₂×#14273659	C₂²	1	1	yes	characteristic
21	#16837418	—	2	1	no	—
22	#16837357	C₂²	1	1	yes	characteristic
23	#16837226	C₂²	1	4	yes	complemented
24	#16837222	C₂²	1	4	yes	complemented
25	#16837213	C₂²	1	4	yes	—
26	#16837211	C₂²	1	2	yes	—
27	#16837211	C₂²	1	2	yes	—
28	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
29	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
30	C₅₉×#427	C₂²	1	1	yes	characteristic
31	C₅₉×#365	C₂²	1	2	yes	complemented
32	C₅₉×#365	C₂²	1	2	yes	complemented
33	C₂×#11745685	—	2	1	no	—
34	#14273749	—	2	1	no	—
35	#14273749	—	2	1	no	—
36	#14273749	Q₈	1	2	yes	complemented
37	#14273749	C₂³	1	2	yes	—
38	C₂×Dic₂₃₆	Q₈	1	1	yes	characteristic, complemented
39	C₂×#11745660	—	2	1	no	—
40	C₂×D₄₇₂	Q₈	1	1	yes	characteristic, complemented
41	#14273735	—	2	4	no	—
42	#14273733	—	4	2	no	—
43	C₂×#11745664	D₄	1	1	yes	characteristic
44	M₁₆×D₅₉	—	4	2	no	—
45	#14273727	—	4	1	no	—
46	#14273726	—	2	2	no	—
47	#14273719	—	4	1	no	—
48	#14273718	—	2	2	no	—
49	#14273704	—	4	2	no	—
50	#14273686	D₄	1	1	yes	characteristic
51	C₂×C₄⋊Dic₅₉	—	2	1	no	—
52	#14273667	—	2	1	no	—
53	#14273667	C₂³	1	2	yes	—
54	C₂×C₅₉⋊C₁₆	—	2	1	no	—
55	#14273660	—	2	2	no	—
56	#14273659	—	2	1	no	—
57	#14273659	C₂³	1	2	yes	—
58	#14273658	—	2	2	no	—
59	#14273653	—	2	4	no	—
60	#14273637	D₄	1	2	yes	—
61	C₈×Dic₅₉	C₂³	1	2	yes	—
62	C₂×M₁₆×C₅₉	—	2	1	no	—
63	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
64	#117×C₅₉	—	2	1	no	—
65	M₃₂×C₅₉	—	2	1	no	—
66	M₃₂×C₅₉	C₂³	1	2	yes	—
67	C₂×C₁₆×C₅₉	—	2	1	no	—
68	#108×C₅₉	—	2	1	no	—
69	#108×C₅₉	C₂³	1	2	yes	—
70	#107×C₅₉	—	2	1	no	—
71	#106×C₅₉	C₂³	1	1	yes	characteristic
72	#106×C₅₉	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₅₉	—	4	1	no	—
75	C₂²×Dic₅₉	—	4	1	no	—
76	#11745685	—	2	2	no	—
77	#11745685	—	2	2	no	—
78	#11745685	—	4	1	no	—
79	#11745685	—	4	1	no	—
80	C₂×D₂₃₆	—	2	1	no	—
81	C₂×C₄×D₅₉	—	4	1	no	—
82	C₂×Dic₁₁₈	—	2	1	no	—
83	C₂×Dic₁₁₈	—	4	2	no	—
84	M₁₆×C₅₉	—	2	2	no	—
85	M₁₆×C₅₉	—	4	1	no	—
86	C₂×C₈×C₅₉	—	4	1	no	—
87	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₅₉	C₂×Q₈	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	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₅₉	—	2	1	no	—
94	C₄⋊C₄×C₅₉	—	2	1	no	—
95	C₂²⋊C₄×C₅₉	—	4	1	no	—
96	C₄²×C₅₉	—	4	1	no	—
97	C₂²⋊Dic₅₉	—	2	1	no	—
98	C₄⋊Dic₅₉	—	2	1	no	—
99	C₄⋊Dic₅₉	—	2	2	no	—
100	C₄⋊Dic₅₉	—	4	2	no	—
101	C₄⋊Dic₅₉	—	4	1	no	—
102	Dic₅₉⋊C₄	—	4	2	no	—
103	Dic₅₉⋊C₄	—	4	2	no	—
104	C₄×Dic₅₉	C₂×D₄	1	2	yes	—
105	#11745664	SD₁₆	1	2	yes	—
106	#11745664	SD₁₆	1	2	yes	—
107	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
108	Dic₂₃₆	—	2	1	no	—
109	Dic₂₃₆	C₂×Q₈	1	2	yes	—
110	D₄₇₂	—	2	1	no	—
111	D₄₇₂	C₂×Q₈	1	2	yes	—
112	#11745660	—	2	2	no	—
113	#11745660	—	2	2	no	—
114	#11745659	—	4	4	no	—
115	C₈×D₅₉	—	4	4	no	—
116	C₁₆×C₅₉	—	2	2	no	—
117	C₅₉⋊C₁₆	—	2	2	no	—
118	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂²×D₅₉	—	4	1	no	—
120	C₂×C₄×C₅₉	—	4	1	no	—
121	C₂×C₄×C₅₉	—	4	1	no	—
122	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
128	C₂²⋊D₅₉	—	4	2	no	—
129	C₂²⋊D₅₉	—	4	2	no	—
130	C₂×Dic₅₉	—	2	2	no	—
131	C₂×Dic₅₉	—	4	1	no	—
132	C₂×Dic₅₉	—	4	2	no	—
133	C₂×Dic₅₉	—	16	1	no	—
134	D₂₃₆	—	2	2	no	—
135	D₂₃₆	—	4	1	no	—
136	C₄×D₅₉	—	4	2	no	—
137	C₄×D₅₉	—	4	2	no	—
138	Dic₁₁₈	—	2	2	no	—
139	Dic₁₁₈	—	4	1	no	—
140	Dic₁₁₈	—	16	2	no	—
141	C₈×C₅₉	—	4	2	no	—
142	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
143	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
145	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
146	C₅₉⋊C₈	C₂×SD₁₆	1	4	yes	—
147	#33638	—	59	1	no	maximal, Sylow
148	C₂²×C₅₉	—	4	1	no	—
149	C₂²×C₅₉	#476	1	1	yes	characteristic
150	C₂²×C₅₉	#476	1	1	yes	characteristic
151	C₂²×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×D₅₉	—	4	2	no	—
153	C₂×D₅₉	—	16	1	no	—
154	C₄×C₅₉	—	8	1	no	—
155	C₄×C₅₉	—	8	1	no	—
156	C₄×C₅₉	#476	1	1	yes	characteristic
157	C₄×C₅₉	#476	1	1	yes	characteristic
158	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
159	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
160	Dic₅₉	—	4	2	no	—
161	Dic₅₉	—	4	2	no	—
162	Dic₅₉	—	8	2	no	—
163	#3163	—	59	1	no	—
164	#2957	—	59	1	no	—
165	#2231	—	59	4	no	—
166	#2230	—	59	4	no	—
167	#2155	D₅₉	1	1	yes	characteristic, complemented
168	C₂×#365	—	59	1	no	—
169	#2153	—	59	1	no	—
170	#2145	—	59	1	no	—
171	C₂×#360	—	59	1	no	—
172	C₂×C₅₉	—	2	1	no	—
173	C₂×C₅₉	—	2	1	no	—
174	C₂×C₅₉	—	2	1	no	—
175	C₂×C₅₉	C₂×#476	1	1	yes	characteristic
176	D₅₉	—	8	2	no	—
177	C₂×#135	—	59	1	no	—
178	#568	—	118	1	no	—
179	#544	—	118	1	no	—
180	C₂×M₃₂	—	59	1	no	—
181	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
182	#501	—	118	2	no	—
183	#499	—	59	2	no	—
184	#498	—	59	2	no	—
185	#444	—	118	2	no	—
186	#443	—	59	4	no	—
187	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
188	#428	—	118	1	no	—
189	#427	C₂×D₅₉	1	1	yes	characteristic
190	C₂×#106	—	59	1	no	—
191	#421	—	118	1	no	—
192	#405	—	59	1	no	—
193	#365	—	59	4	no	—
194	#365	C₂×D₅₉	1	2	yes	complemented
195	#365	C₂×D₅₉	1	2	yes	complemented
196	#364	—	59	4	no	—
197	#360	—	59	4	no	—
198	#360	—	59	2	no	—
199	#360	—	59	2	no	—
200	C₅₉	#33638	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	118	1	no	—
202	#135	—	59	2	no	—
203	#135	—	59	2	no	—
204	#135	—	118	1	no	—
205	#135	—	118	1	no	—
206	C₂×Q₁₆	—	59	1	no	—
207	C₂×SD₁₆	—	118	1	no	—
208	C₂×D₈	—	59	1	no	—
209	#131	—	118	4	no	—
210	#131	—	236	2	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	—	59	1	no	—
213	C₂×M₁₆	—	236	2	no	—
214	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
215	C₄⋊Q₈	—	118	2	no	—
216	C₄⋊Q₈	—	236	1	no	—
217	#125	—	118	2	no	—
218	#125	—	236	1	no	—
219	C₂²⋊Q₈	—	236	2	no	—
220	#117	—	2	1	no	—
221	#117	—	59	1	no	—
222	C₂×C₄⋊C₄	—	118	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	C₂²×D₅₉	1	2	yes	—
231	#107	—	2	1	no	—
232	#107	—	118	2	no	—
233	#106	—	59	2	no	—
234	#106	—	118	2	no	—
235	#106	—	118	1	no	—
236	#106	C₂²×D₅₉	1	1	yes	characteristic
237	#106	C₂²×D₅₉	1	1	yes	characteristic
238	#104	—	118	4	no	—
239	#97	—	59	2	no	—
240	C₄×C₈	—	59	2	no	—
241	#41	—	118	2	no	—
242	#41	—	118	2	no	—
243	#41	—	236	1	no	—
244	#41	—	236	1	no	—
245	C₂×Q₈	—	118	1	no	—
246	C₂×Q₈	—	236	2	no	—
247	C₂×D₄	—	118	1	no	—
248	C₂×D₄	—	236	1	no	—
249	C₂²×C₄	—	236	1	no	—
250	C₂²×C₄	—	236	1	no	—
251	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
252	Q₁₆	—	59	2	no	—
253	Q₁₆	—	118	1	no	—
254	SD₁₆	—	118	2	no	—
255	SD₁₆	—	118	2	no	—
256	D₈	—	59	2	no	—
257	D₈	—	118	1	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	59	2	no	—
261	M₁₆	—	59	2	no	—
262	M₁₆	—	236	4	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	59	2	no	—
265	C₂×C₈	—	236	4	no	—
266	C₂×C₈	#11745685	1	1	yes	characteristic
267	C₂×C₈	#11745685	1	1	yes	characteristic
268	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
269	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
270	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	118	1	no	—
275	C₄⋊C₄	—	118	2	no	—
276	C₄⋊C₄	—	236	2	no	—
277	C₄⋊C₄	—	236	2	no	—
278	C₄⋊C₄	—	236	2	no	—
279	C₄⋊C₄	—	236	1	no	—
280	C₂²⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	118	1	no	—
282	C₄²	—	4	1	no	—
283	C₄²	—	59	2	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	118	2	no	—
286	C₂³	—	236	1	no	—
287	C₂³	#14273754	1	1	yes	characteristic
288	Q₈	—	118	2	no	—
289	Q₈	—	944	2	no	—
290	Q₈	—	236	1	no	—
291	D₄	—	118	2	no	—
292	D₄	—	236	2	no	—
293	D₄	—	236	2	no	—
294	D₄	—	236	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	118	2	no	—
298	C₂×C₄	—	944	1	no	—
299	C₂×C₄	—	236	2	no	—
300	C₂×C₄	—	236	1	no	—
301	C₂×C₄	—	236	2	no	—
302	C₂×C₄	—	236	2	no	—
303	C₂×C₄	#14273705	1	1	yes	characteristic
304	C₂×C₄	#14273705	1	1	yes	characteristic
305	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
307	C₂×C₄	#14273754	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	—	59	4	no	—
311	C₈	#14273705	1	1	yes	characteristic
312	C₈	#14273705	1	1	yes	characteristic
313	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
314	C₈	C₂×#11745685	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	944	1	no	—
317	C₂²	—	236	2	no	—
318	C₂²	#16837691	1	1	yes	characteristic
319	C₂²	#16837691	1	1	yes	characteristic
320	C₂²	C₂×#14273754	1	1	yes	characteristic
321	C₄	—	8	1	no	—
322	C₄	—	8	1	no	—
323	C₄	—	472	2	no	—
324	C₄	—	236	2	no	—
325	C₄	—	236	2	no	—
326	C₄	#16837691	1	1	yes	characteristic
327	C₄	#16837691	1	1	yes	characteristic
328	C₄	C₂×#14273705	1	1	yes	characteristic
329	C₄	C₂×SD₁₆×D₅₉	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	472	2	no	—
334	C₂	C₂×#16837691	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16837691
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2155	C₂
socle	C₂×C₅₉	C₂×#476
prime radical	C₂×#14273749	C₂²
hypercenter	#2155	D₅₉
hypocenter	C₅₉	#33638
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
1. #21434628	C₂	semidirect
3. #21426271	C₂	semidirect
4. #21426266	C₂	semidirect
5. C₂×#16837226	C₂	semidirect
6. #21425861	C₂	semidirect
9. C₅₉×#2155	C₂	semidirect
15. #16837802	C₂²	semidirect
16. #16837799	C₂²	semidirect
23. #16837226	C₂²	semidirect
24. #16837222	C₂²	semidirect
31. C₅₉×#365	C₂²	semidirect
32. C₅₉×#365	C₂²	semidirect
36. #14273749	Q₈	semidirect
38. C₂×Dic₂₃₆	Q₈	semidirect
40. C₂×D₄₇₂	Q₈	semidirect
167. #2155	D₅₉	semidirect
194. #365	C₂×D₅₉	semidirect
195. #365	C₂×D₅₉	semidirect
200. C₅₉	#33638	semidirect

Maximal subgroups

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

subgroup	index
#21434628	2
#21433012	2
#21426271	2
#21426266	2
C₂×#16837226	2
#21425861	2
#21425831	2
C₂×#16837213	2
C₅₉×#2155	2
#33638	59