Subgroups — #28743591

Order 15104.

566 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433740	C₂	1	1	yes	maximal, characteristic
2	#21432704	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426309	C₂	1	8	yes	maximal, complemented
4	#21425870	C₂	1	1	yes	maximal, characteristic, complemented
5	#21425869	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16837215	C₂	1	1	yes	maximal, characteristic, complemented
7	#21425716	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₉×#361	C₂	1	1	yes	maximal, characteristic, complemented
9	#16838168	—	2	1	no	—
10	C₂×#14273733	C₂²	1	1	yes	characteristic
11	#16837969	—	2	1	no	—
12	C₂×#14273667	C₂²	1	1	yes	characteristic
13	#16837652	C₂²	1	4	yes	—
14	#16837635	—	2	2	no	—
15	#16837450	—	2	1	no	—
16	#16837449	C₂²	1	4	yes	complemented
17	C₂×#14273663	C₂²	1	1	yes	characteristic
18	#108×D₅₉	—	2	2	no	—
19	#16837440	C₂²	1	1	yes	characteristic
20	#16837407	—	2	1	no	—
21	#16837226	C₂²	1	4	yes	complemented
22	#16837225	C₂²	1	4	yes	complemented
23	#16837215	C₂²	1	4	yes	—
24	#16837183	C₂²	1	4	yes	complemented
25	C₂²×D₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
28	C₅₉×#361	C₂²	1	4	yes	—
29	C₂×#11745685	D₄	1	1	yes	characteristic
30	#14273735	—	4	2	no	—
31	#14273733	—	2	1	no	—
32	#14273733	—	2	1	no	—
33	#14273733	—	2	1	no	—
34	#14273733	—	2	1	no	—
35	#14273733	—	4	2	no	—
36	C₂×#11745664	—	2	1	no	—
37	C₂×#11745664	D₄	1	1	yes	characteristic
38	C₂×#11745659	C₂³	1	2	yes	—
39	M₁₆×D₅₉	—	2	4	no	—
40	C₂×C₈×D₅₉	D₄	1	2	yes	—
41	#14273723	—	4	1	no	—
42	#14273714	—	4	1	no	—
43	#14273698	—	2	4	no	—
44	#14273696	—	4	2	no	—
45	#14273686	—	2	1	no	—
46	#14273667	—	2	1	no	—
47	#14273667	C₂³	1	2	yes	—
48	C₂×C₅₉⋊C₁₆	—	2	1	no	—
49	#14273663	—	2	1	no	—
50	#14273663	C₂³	1	2	yes	—
51	#14273662	—	2	4	no	—
52	#14273661	—	2	1	no	—
53	#14273659	C₂³	1	2	yes	—
54	#14273648	—	2	4	no	—
55	C₂²×D₄×C₅₉	—	2	1	no	—
56	C₂×D₈×C₅₉	—	2	1	no	—
57	C₂×D₈×C₅₉	—	2	1	no	—
58	C₂×D₈×C₅₉	—	2	1	no	—
59	C₂×D₈×C₅₉	—	2	1	no	—
60	C₂×D₈×C₅₉	—	2	1	no	—
61	C₂×D₈×C₅₉	C₂³	1	2	yes	—
62	C₂×M₁₆×C₅₉	—	2	1	no	—
63	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
64	M₃₂×C₅₉	—	2	1	no	—
65	M₃₂×C₅₉	C₂³	1	2	yes	—
66	C₂×C₁₆×C₅₉	—	2	1	no	—
67	#108×C₅₉	—	2	1	no	—
68	#108×C₅₉	C₂³	1	2	yes	—
69	C₂⁴×C₅₉	—	4	1	no	—
70	C₂×D₄×C₅₉	—	2	2	no	—
71	C₂×D₄×C₅₉	—	4	1	no	—
72	C₂×D₄×C₅₉	—	4	1	no	—
73	C₂×D₄×C₅₉	—	4	1	no	—
74	C₂×D₄×C₅₉	—	4	1	no	—
75	C₂×D₄×C₅₉	—	4	1	no	—
76	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₂²⋊D₅₉	—	4	4	no	—
78	C₂×C₂²⋊D₅₉	SD₁₆	1	2	yes	—
79	#11745685	—	2	1	no	—
80	#11745685	—	2	1	no	—
81	#11745685	—	2	1	no	—
82	#11745685	—	2	1	no	—
83	C₂×D₂₃₆	SD₁₆	1	1	yes	characteristic
84	C₂×C₄×D₅₉	C₂×D₄	1	2	yes	—
85	C₂×Dic₁₁₈	SD₁₆	1	1	yes	characteristic
86	D₈×C₅₉	—	2	2	no	—
87	D₈×C₅₉	—	2	2	no	—
88	D₈×C₅₉	—	4	1	no	—
89	D₈×C₅₉	—	4	1	no	—
90	M₁₆×C₅₉	—	2	2	no	—
91	M₁₆×C₅₉	—	4	1	no	—
92	C₂×C₈×C₅₉	—	4	1	no	—
93	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
95	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
96	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
97	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
98	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
99	C₂²⋊Dic₅₉	—	4	1	no	—
100	C₂²⋊Dic₅₉	—	4	4	no	—
101	C₄⋊Dic₅₉	—	2	2	no	—
102	C₄×Dic₅₉	—	4	1	no	—
103	#11745664	—	2	1	no	—
104	#11745664	—	2	1	no	—
105	#11745664	—	2	2	no	—
106	#11745664	—	4	1	no	—
107	C₂×C₅₉⋊C₈	—	4	1	no	—
108	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
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	—	4	2	no	—
118	C₂³×C₅₉	—	8	1	no	—
119	C₂³×C₅₉	—	8	1	no	—
120	C₂³×C₅₉	—	16	2	no	—
121	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
122	C₂²×D₅₉	C₂×SD₁₆	1	2	yes	—
123	D₄×C₅₉	—	4	2	no	—
124	D₄×C₅₉	—	4	2	no	—
125	D₄×C₅₉	—	8	1	no	—
126	D₄×C₅₉	—	8	1	no	—
127	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
128	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
133	C₂²⋊D₅₉	—	2	2	no	—
134	C₂²⋊D₅₉	—	2	2	no	—
135	C₂²⋊D₅₉	—	16	4	no	—
136	C₂×Dic₅₉	—	4	2	no	—
137	C₂×Dic₅₉	C₂×SD₁₆	1	2	yes	—
138	D₂₃₆	—	2	1	no	—
139	D₂₃₆	—	2	1	no	—
140	C₄×D₅₉	—	2	2	no	—
141	C₄×D₅₉	—	2	2	no	—
142	Dic₁₁₈	—	2	1	no	—
143	Dic₁₁₈	—	2	1	no	—
144	C₈×C₅₉	—	4	2	no	—
145	C₈×C₅₉	#123	1	1	yes	characteristic
146	C₈×C₅₉	#123	1	1	yes	characteristic
147	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
148	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
149	C₅₉⋊C₈	—	2	2	no	—
150	C₅₉⋊C₈	—	4	2	no	—
151	#33657	—	59	1	no	maximal, Sylow
152	C₂²×C₅₉	—	4	1	no	—
153	C₂²×C₅₉	—	8	4	no	—
154	C₂²×C₅₉	—	8	2	no	—
155	C₂²×C₅₉	—	16	2	no	—
156	C₂²×C₅₉	—	16	2	no	—
157	C₂²×C₅₉	#481	1	1	yes	characteristic
158	C₂²×C₅₉	#481	1	1	yes	characteristic
159	C₂²×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
160	C₂×D₅₉	—	2	2	no	—
161	C₂×D₅₉	—	4	2	no	—
162	C₄×C₅₉	#481	1	1	yes	characteristic
163	C₄×C₅₉	#481	1	1	yes	characteristic
164	C₄×C₅₉	C₂×#123	1	1	yes	characteristic
165	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	Dic₅₉	—	2	2	no	—
167	Dic₅₉	—	8	2	no	—
168	#3058	—	59	1	no	—
169	#2916	—	59	1	no	—
170	#2238	—	59	8	no	—
171	#2155	—	59	1	no	—
172	#2155	—	59	1	no	—
173	C₂×#361	—	59	1	no	—
174	C₂×#361	D₅₉	1	1	yes	characteristic, complemented
175	#2115	—	59	1	no	—
176	C₂×C₅₉	—	2	1	no	—
177	C₂×C₅₉	—	2	1	no	—
178	C₂×C₅₉	—	2	1	no	—
179	C₂×C₅₉	—	16	2	no	—
180	C₂×C₅₉	C₂×#481	1	1	yes	characteristic
181	D₅₉	—	4	2	no	—
182	C₂²×D₈	C₂×D₅₉	1	1	yes	characteristic
183	#568	—	118	1	no	—
184	C₂×#131	—	59	1	no	—
185	#535	—	118	1	no	—
186	C₂×M₃₂	—	59	1	no	—
187	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
188	#469	—	59	4	no	—
189	#466	—	118	2	no	—
190	#430	—	59	4	no	—
191	#430	—	118	1	no	—
192	C₂×#108	—	59	1	no	—
193	C₂×#108	—	118	2	no	—
194	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
195	#427	—	59	1	no	—
196	#418	—	118	1	no	—
197	#365	—	59	4	no	—
198	#365	—	59	4	no	—
199	#361	—	59	4	no	—
200	#361	C₂×D₅₉	1	4	yes	—
201	#349	—	59	4	no	—
202	C₅₉	#33657	1	1	yes	characteristic, complemented, Sylow
203	C₂×#41	—	59	1	no	—
204	C₂²×D₄	—	2	1	no	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	—	2	1	no	—
207	C₂×D₈	—	2	1	no	—
208	C₂×D₈	—	2	1	no	—
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	C₂²×D₅₉	1	2	yes	—
211	#131	—	118	1	no	—
212	#131	—	118	1	no	—
213	#131	—	118	1	no	—
214	#131	—	118	1	no	—
215	#131	—	236	2	no	—
216	#131	—	236	2	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	59	1	no	—
219	C₂×M₁₆	—	59	2	no	—
220	C₂×M₁₆	—	118	4	no	—
221	C₂×M₁₆	—	118	1	no	—
222	C₂²×C₈	—	59	2	no	—
223	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
224	#127	—	236	1	no	—
225	#124	—	236	1	no	—
226	#121	—	118	4	no	—
227	#120	—	236	2	no	—
228	#117	—	118	1	no	—
229	M₃₂	—	2	1	no	—
230	M₃₂	—	59	2	no	—
231	M₃₂	—	118	1	no	—
232	M₃₂	C₂²×D₅₉	1	2	yes	—
233	C₂×C₁₆	—	2	1	no	—
234	C₂×C₁₆	—	118	1	no	—
235	#108	—	2	1	no	—
236	#108	—	59	2	no	—
237	#108	—	118	4	no	—
238	#108	—	118	1	no	—
239	#108	C₂²×D₅₉	1	2	yes	—
240	#107	—	118	1	no	—
241	#106	—	59	2	no	—
242	#102	—	118	4	no	—
243	C₂⁴	—	4	1	no	—
244	#41	—	118	1	no	—
245	#41	—	118	1	no	—
246	#41	—	118	1	no	—
247	#41	—	118	1	no	—
248	C₂×Q₈	—	59	1	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	59	2	no	—
256	C₂×D₄	—	59	1	no	—
257	C₂×D₄	—	236	4	no	—
258	C₂²×C₄	—	59	2	no	—
259	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
260	D₈	—	2	2	no	—
261	D₈	—	2	2	no	—
262	D₈	—	4	1	no	—
263	D₈	—	4	1	no	—
264	M₁₆	—	2	2	no	—
265	M₁₆	—	4	1	no	—
266	M₁₆	—	118	1	no	—
267	M₁₆	—	118	1	no	—
268	M₁₆	—	118	2	no	—
269	M₁₆	—	118	2	no	—
270	M₁₆	—	118	2	no	—
271	M₁₆	—	236	4	no	—
272	M₁₆	—	236	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	59	2	no	—
275	C₂×C₈	—	118	2	no	—
276	C₂×C₈	—	118	2	no	—
277	C₂×C₈	—	236	4	no	—
278	C₂×C₈	—	236	1	no	—
279	C₂×C₈	#11745685	1	1	yes	characteristic
280	C₂×C₈	#11745685	1	1	yes	characteristic
281	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
282	C₂×C₈	#11745687	1	1	yes	characteristic
283	C₂×C₈	#11745687	1	1	yes	characteristic
284	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
285	C₄⋊C₄	—	118	2	no	—
286	C₂²⋊C₄	—	236	4	no	—
287	C₂²⋊C₄	—	236	1	no	—
288	C₄²	—	236	1	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	—	118	2	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	8	1	no	—
293	C₂³	—	8	1	no	—
294	C₂³	—	16	2	no	—
295	C₂³	—	59	2	no	—
296	C₂³	SD₁₆×D₅₉	1	1	yes	characteristic
297	Q₈	—	118	1	no	—
298	Q₈	—	118	1	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	1	no	—
302	D₄	—	8	1	no	—
303	D₄	—	118	2	no	—
304	D₄	—	118	2	no	—
305	D₄	—	118	1	no	—
306	D₄	—	118	1	no	—
307	D₄	—	944	4	no	—
308	C₂×C₄	—	59	2	no	—
309	C₂×C₄	—	118	2	no	—
310	C₂×C₄	—	118	2	no	—
311	C₂×C₄	—	236	2	no	—
312	C₂×C₄	#14273709	1	1	yes	characteristic
313	C₂×C₄	#14273709	1	1	yes	characteristic
314	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
315	C₂×C₄	#14273751	1	1	yes	characteristic
316	C₂×C₄	#14273751	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
318	C₈	—	4	2	no	—
319	C₈	—	118	2	no	—
320	C₈	—	236	2	no	—
321	C₈	#14273709	1	1	yes	characteristic
322	C₈	#14273709	1	1	yes	characteristic
323	C₈	C₂×#11745687	1	1	yes	characteristic
324	C₈	C₂×#11745685	1	1	yes	characteristic
325	C₂²	—	4	1	no	—
326	C₂²	—	8	4	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	16	2	no	—
329	C₂²	—	16	2	no	—
330	C₂²	—	118	2	no	—
331	C₂²	—	236	2	no	—
332	C₂²	#16837719	1	1	yes	characteristic
333	C₂²	#16837719	1	1	yes	characteristic
334	C₂²	C₂×SD₁₆×D₅₉	1	1	yes	characteristic
335	C₄	—	118	2	no	—
336	C₄	—	472	2	no	—
337	C₄	#16837719	1	1	yes	characteristic
338	C₄	#16837719	1	1	yes	characteristic
339	C₄	C₂×#14273709	1	1	yes	characteristic
340	C₄	C₂×#14273751	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	—	16	2	no	—
345	C₂	—	236	2	no	—
346	C₂	C₂×#16837719	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16837719
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#361	C₂
socle	C₂×C₅₉	C₂×#481
prime radical	#21433740	C₂
hypercenter	C₂×#361	D₅₉
hypocenter	C₅₉	#33657
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#361

Complements

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

N	H ≅ G/N	type
2. #21432704	C₂	semidirect
3. #21426309	C₂	semidirect
4. #21425870	C₂	semidirect
5. #21425869	C₂	semidirect
6. C₂×#16837215	C₂	semidirect
7. #21425716	C₂	semidirect
8. C₂×C₅₉×#361	C₂	semidirect
16. #16837449	C₂²	semidirect
21. #16837226	C₂²	semidirect
22. #16837225	C₂²	semidirect
24. #16837183	C₂²	semidirect
174. C₂×#361	D₅₉	semidirect
202. C₅₉	#33657	semidirect

Maximal subgroups

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

subgroup	index
#21433740	2
#21432704	2
#21426309	2
#21425870	2
#21425869	2
C₂×#16837215	2
#21425716	2
C₂×C₅₉×#361	2
#33657	59