Subgroups — #28742155

Order 15104.

632 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837845	C₂	1	1	yes	maximal, characteristic
2	#21433012	C₂	1	1	yes	maximal, characteristic
3	#21432963	C₂	1	1	yes	maximal, characteristic, complemented
4	#21425986	C₂	1	8	yes	maximal, complemented
5	C₂×#16837217	C₂	1	1	yes	maximal, characteristic, complemented
6	#21425831	C₂	1	1	yes	maximal, characteristic
7	#21425700	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#2155	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14273749	C₂²	1	1	yes	characteristic
10	C₂²×Dic₂₃₆	C₂²	1	1	yes	characteristic
11	#16838168	—	2	1	no	—
12	#16837879	—	2	1	no	—
13	#16837845	C₄	1	4	yes	complemented
14	#16837845	C₂²	1	4	yes	—
15	C₂×Dic₄₇₂	C₄	1	2	yes	complemented
16	C₂×#14273672	C₄	1	2	yes	complemented
17	C₂×#14273667	C₂²	1	1	yes	characteristic
18	#16837523	C₂²	1	4	yes	—
19	#16837518	—	2	2	no	—
20	#16837508	C₂²	1	4	yes	complemented
21	#16837496	—	2	2	no	—
22	#16837418	—	2	1	no	—
23	#16837410	—	2	1	no	—
24	#16837357	C₂²	1	1	yes	characteristic
25	#16837217	C₂²	1	4	yes	complemented
26	#16837211	C₂²	1	4	yes	—
27	#16837179	C₂²	1	4	yes	complemented
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	4	yes	complemented
32	C₂×#11745685	—	2	1	no	—
33	C₂²×Dic₁₁₈	—	2	1	no	—
34	#14273749	—	2	1	no	—
35	#14273749	—	2	1	no	—
36	#14273749	C₂×C₄	1	2	yes	—
37	#14273749	C₂³	1	2	yes	—
38	C₂×Dic₂₃₆	—	2	1	no	—
39	C₂×Dic₂₃₆	—	2	1	no	—
40	C₂×Dic₂₃₆	—	2	1	no	—
41	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
42	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
43	C₂×Dic₂₃₆	C₂×C₄	1	1	yes	characteristic
44	C₂×Dic₂₃₆	C₂³	1	2	yes	—
45	C₂×#11745660	—	2	1	no	—
46	C₂×D₄₇₂	C₂×C₄	1	1	yes	characteristic
47	#14273735	—	2	4	no	—
48	#14273733	—	4	2	no	—
49	C₂×#11745664	D₄	1	1	yes	characteristic
50	M₁₆×D₅₉	—	4	2	no	—
51	C₄×Dic₁₁₈	—	4	2	no	—
52	#14273692	—	2	4	no	—
53	#14273686	D₄	1	1	yes	characteristic
54	#14273683	—	4	2	no	—
55	Dic₄₇₂	C₂×C₄	1	8	yes	—
56	#14273672	C₂×C₄	1	8	yes	—
57	#14273667	—	2	1	no	—
58	#14273667	C₂³	1	2	yes	—
59	C₂×C₅₉⋊C₁₆	—	2	1	no	—
60	#14273653	—	2	4	no	—
61	#14273650	—	2	4	no	—
62	#14273637	C₂³	1	2	yes	—
63	C₈×Dic₅₉	D₄	1	2	yes	—
64	C₂×M₁₆×C₅₉	—	2	1	no	—
65	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
66	#117×C₅₉	—	2	1	no	—
67	M₃₂×C₅₉	C₂×C₄	1	2	yes	—
68	M₃₂×C₅₉	C₂³	1	2	yes	—
69	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
70	#108×C₅₉	—	2	1	no	—
71	#108×C₅₉	C₂³	1	2	yes	—
72	#107×C₅₉	—	2	1	no	—
73	#106×C₅₉	C₂³	1	2	yes	—
74	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₂²⋊D₅₉	—	4	1	no	—
76	C₂²×Dic₅₉	—	4	1	no	—
77	#11745685	—	2	2	no	—
78	#11745685	—	2	2	no	—
79	#11745685	—	4	1	no	—
80	#11745685	—	4	1	no	—
81	C₂×D₂₃₆	—	2	1	no	—
82	C₂×C₄×D₅₉	—	4	1	no	—
83	C₂×Dic₁₁₈	—	2	1	no	—
84	C₂×Dic₁₁₈	—	2	2	no	—
85	C₂×Dic₁₁₈	—	2	2	no	—
86	C₂×Dic₁₁₈	—	2	2	no	—
87	C₂×Dic₁₁₈	—	4	1	no	—
88	C₂×Dic₁₁₈	—	4	1	no	—
89	C₂×Dic₁₁₈	—	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	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₉	C₂×D₄	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	C₄⋊C₄×C₅₉	—	2	2	no	—
100	C₂²⋊C₄×C₅₉	—	4	1	no	—
101	C₄²×C₅₉	—	4	1	no	—
102	C₂²⋊Dic₅₉	—	2	1	no	—
103	C₄⋊Dic₅₉	—	2	1	no	—
104	Dic₅₉⋊C₄	—	4	4	no	—
105	C₄×Dic₅₉	—	4	4	no	—
106	C₄×Dic₅₉	C₂×D₄	1	2	yes	—
107	#11745664	—	2	1	no	—
108	#11745664	—	2	1	no	—
109	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
110	Dic₂₃₆	—	2	2	no	—
111	Dic₂₃₆	—	2	2	no	—
112	Dic₂₃₆	—	2	1	no	—
113	Dic₂₃₆	—	2	2	no	—
114	Dic₂₃₆	C₂²×C₄	1	4	yes	—
115	Dic₂₃₆	C₂²×C₄	1	2	yes	—
116	D₄₇₂	—	2	1	no	—
117	D₄₇₂	C₂²×C₄	1	2	yes	—
118	#11745660	—	2	2	no	—
119	#11745660	—	2	2	no	—
120	#11745659	—	4	4	no	—
121	C₈×D₅₉	—	4	4	no	—
122	C₁₆×C₅₉	C₂²×C₄	1	4	yes	—
123	C₅₉⋊C₁₆	—	2	2	no	—
124	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂²×D₅₉	—	4	1	no	—
126	C₂×C₄×C₅₉	—	4	2	no	—
127	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅₉	AGL(1,ℤ/8)	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	—	4	2	no	—
134	C₂²⋊D₅₉	—	4	2	no	—
135	C₂×Dic₅₉	—	2	2	no	—
136	C₂×Dic₅₉	—	4	2	no	—
137	C₂×Dic₅₉	—	4	2	no	—
138	C₂×Dic₅₉	—	4	2	no	—
139	C₂×Dic₅₉	—	4	1	no	—
140	D₂₃₆	—	2	2	no	—
141	D₂₃₆	—	4	1	no	—
142	C₄×D₅₉	—	4	2	no	—
143	C₄×D₅₉	—	4	2	no	—
144	Dic₁₁₈	—	2	2	no	—
145	Dic₁₁₈	—	2	4	no	—
146	Dic₁₁₈	—	4	2	no	—
147	Dic₁₁₈	—	4	2	no	—
148	Dic₁₁₈	—	4	2	no	—
149	Dic₁₁₈	—	4	1	no	—
150	C₈×C₅₉	—	4	2	no	—
151	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
152	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
153	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
154	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
155	C₅₉⋊C₈	—	2	2	no	—
156	#33470	—	59	1	no	maximal, Sylow
157	C₂²×C₅₉	—	4	1	no	—
158	C₂²×C₅₉	#442	1	1	yes	characteristic
159	C₂²×C₅₉	#442	1	1	yes	characteristic
160	C₂²×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂×D₅₉	—	4	2	no	—
162	C₂×D₅₉	—	16	1	no	—
163	C₄×C₅₉	—	8	2	no	—
164	C₄×C₅₉	#442	1	1	yes	characteristic
165	C₄×C₅₉	#442	1	1	yes	characteristic
166	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
167	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
168	Dic₅₉	—	4	2	no	—
169	Dic₅₉	—	4	4	no	—
170	Dic₅₉	—	4	2	no	—
171	C₂×#510	—	59	1	no	—
172	#2957	—	59	1	no	—
173	#2950	—	59	1	no	—
174	#2180	—	59	8	no	—
175	#2155	D₅₉	1	1	yes	characteristic, complemented
176	C₂×#362	—	59	1	no	—
177	#2145	—	59	1	no	—
178	#2110	—	59	1	no	—
179	C₂×C₅₉	—	2	1	no	—
180	C₂×C₅₉	—	2	1	no	—
181	C₂×C₅₉	—	2	1	no	—
182	C₂×C₅₉	C₂×#442	1	1	yes	characteristic
183	D₅₉	—	8	2	no	—
184	C₂×#135	—	59	1	no	—
185	C₂²×Q₁₆	—	59	1	no	—
186	#568	—	118	1	no	—
187	#519	—	118	1	no	—
188	#510	—	59	4	no	—
189	#510	—	59	4	no	—
190	C₂×Q₃₂	—	59	2	no	—
191	C₂×SD₃₂	—	59	2	no	—
192	C₂×M₃₂	—	59	1	no	—
193	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
194	#444	—	59	4	no	—
195	#443	—	118	2	no	—
196	#441	—	59	4	no	—
197	C₄×Q₁₆	—	118	2	no	—
198	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
199	#427	C₂×D₅₉	1	1	yes	characteristic
200	#421	—	118	1	no	—
201	#419	—	118	1	no	—
202	#405	—	59	1	no	—
203	#365	C₂×D₅₉	1	4	yes	complemented
204	#362	—	59	4	no	—
205	#360	—	59	4	no	—
206	#347	—	59	4	no	—
207	C₅₉	#33470	1	1	yes	characteristic, complemented, Sylow
208	C₂×#41	—	118	1	no	—
209	C₂²×Q₈	—	118	1	no	—
210	#135	—	59	2	no	—
211	#135	—	59	2	no	—
212	#135	—	118	1	no	—
213	#135	—	118	1	no	—
214	C₂×Q₁₆	—	59	2	no	—
215	C₂×Q₁₆	—	59	2	no	—
216	C₂×Q₁₆	—	59	1	no	—
217	C₂×Q₁₆	—	59	2	no	—
218	C₂×Q₁₆	—	118	1	no	—
219	C₂×Q₁₆	—	118	1	no	—
220	C₂×Q₁₆	—	118	1	no	—
221	C₂×SD₁₆	—	118	1	no	—
222	C₂×D₈	—	59	1	no	—
223	#131	—	118	4	no	—
224	#131	—	236	2	no	—
225	C₂×M₁₆	—	2	1	no	—
226	C₂×M₁₆	—	59	1	no	—
227	C₂×M₁₆	—	236	2	no	—
228	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
229	C₄×Q₈	—	118	4	no	—
230	C₄×Q₈	—	236	2	no	—
231	#117	—	2	1	no	—
232	#117	—	59	1	no	—
233	#117	—	236	2	no	—
234	Q₃₂	—	59	8	no	—
235	SD₃₂	—	59	8	no	—
236	M₃₂	—	59	2	no	—
237	M₃₂	—	118	1	no	—
238	M₃₂	C₄×D₅₉	1	2	yes	—
239	M₃₂	C₂²×D₅₉	1	2	yes	—
240	C₂×C₁₆	—	118	1	no	—
241	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
242	#108	—	2	1	no	—
243	#108	C₂²×D₅₉	1	2	yes	—
244	#107	—	2	1	no	—
245	#106	C₂²×D₅₉	1	2	yes	—
246	#104	—	118	4	no	—
247	#103	—	118	4	no	—
248	#97	—	59	2	no	—
249	C₄×C₈	—	59	2	no	—
250	#41	—	118	2	no	—
251	#41	—	118	2	no	—
252	#41	—	236	1	no	—
253	#41	—	236	1	no	—
254	C₂×Q₈	—	118	1	no	—
255	C₂×Q₈	—	118	2	no	—
256	C₂×Q₈	—	118	2	no	—
257	C₂×Q₈	—	118	2	no	—
258	C₂×Q₈	—	236	1	no	—
259	C₂×Q₈	—	236	1	no	—
260	C₂×Q₈	—	236	1	no	—
261	C₂×D₄	—	118	1	no	—
262	C₂×D₄	—	236	1	no	—
263	C₂²×C₄	—	236	1	no	—
264	C₂²×C₄	—	236	1	no	—
265	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
266	Q₁₆	—	59	4	no	—
267	Q₁₆	—	59	2	no	—
268	Q₁₆	—	118	2	no	—
269	Q₁₆	—	118	2	no	—
270	Q₁₆	—	118	1	no	—
271	Q₁₆	—	118	2	no	—
272	SD₁₆	—	118	2	no	—
273	SD₁₆	—	118	2	no	—
274	D₈	—	59	2	no	—
275	D₈	—	118	1	no	—
276	M₁₆	—	2	2	no	—
277	M₁₆	—	4	1	no	—
278	M₁₆	—	118	1	no	—
279	M₁₆	—	118	1	no	—
280	M₁₆	—	236	4	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	59	2	no	—
283	C₂×C₈	—	236	4	no	—
284	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
285	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
286	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
287	C₂×C₈	#11745689	1	1	yes	characteristic
288	C₂×C₈	#11745689	1	1	yes	characteristic
289	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	118	1	no	—
292	C₄⋊C₄	—	236	4	no	—
293	C₂²⋊C₄	—	4	1	no	—
294	C₂²⋊C₄	—	118	1	no	—
295	C₄²	—	4	1	no	—
296	C₄²	—	59	2	no	—
297	C₄²	—	236	4	no	—
298	C₁₆	—	118	2	no	—
299	C₁₆	C₂×C₄×D₅₉	1	4	yes	—
300	C₂³	—	236	1	no	—
301	C₂³	#14273754	1	1	yes	characteristic
302	Q₈	—	118	2	no	—
303	Q₈	—	118	4	no	—
304	Q₈	—	236	2	no	—
305	Q₈	—	236	2	no	—
306	Q₈	—	236	2	no	—
307	Q₈	—	236	1	no	—
308	D₄	—	118	2	no	—
309	D₄	—	236	2	no	—
310	D₄	—	236	2	no	—
311	D₄	—	236	1	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	118	2	no	—
314	C₂×C₄	—	236	2	no	—
315	C₂×C₄	—	236	1	no	—
316	C₂×C₄	—	236	2	no	—
317	C₂×C₄	—	236	2	no	—
318	C₂×C₄	—	236	2	no	—
319	C₂×C₄	—	236	2	no	—
320	C₂×C₄	#14273689	1	1	yes	characteristic
321	C₂×C₄	#14273689	1	1	yes	characteristic
322	C₂×C₄	#14273754	1	1	yes	characteristic
323	C₂×C₄	#14273754	1	1	yes	characteristic
324	C₂×C₄	#14273754	1	1	yes	characteristic
325	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
326	C₈	—	4	2	no	—
327	C₈	—	118	2	no	—
328	C₈	#14273689	1	1	yes	characteristic
329	C₈	#14273689	1	1	yes	characteristic
330	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
331	C₈	C₂×#11745689	1	1	yes	characteristic
332	C₂²	—	4	1	no	—
333	C₂²	—	944	1	no	—
334	C₂²	—	236	2	no	—
335	C₂²	#16837513	1	1	yes	characteristic
336	C₂²	#16837513	1	1	yes	characteristic
337	C₂²	C₂×#14273754	1	1	yes	characteristic
338	C₄	—	8	2	no	—
339	C₄	—	236	2	no	—
340	C₄	—	236	2	no	—
341	C₄	—	236	4	no	—
342	C₄	#16837513	1	1	yes	characteristic
343	C₄	#16837513	1	1	yes	characteristic
344	C₄	C₂×#14273689	1	1	yes	characteristic
345	C₄	C₂×#14273754	1	1	yes	characteristic
346	C₂	—	2	1	no	—
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	472	2	no	—
350	C₂	C₂×#16837513	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16837513
commutator	C₈×C₅₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2155	C₂
socle	C₂×C₅₉	C₂×#442
prime radical	C₂×#14273749	C₂²
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	#2155	D₅₉
hypocenter	C₅₉	#33470
LCS	G ▷ 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. #21432963	C₂	semidirect
4. #21425986	C₂	semidirect
5. C₂×#16837217	C₂	semidirect
7. #21425700	C₂	semidirect
8. C₅₉×#2155	C₂	semidirect
13. #16837845	C₄	semidirect
15. C₂×Dic₄₇₂	C₄	semidirect
16. C₂×#14273672	C₄	semidirect
20. #16837508	C₂²	semidirect
25. #16837217	C₂²	semidirect
27. #16837179	C₂²	semidirect
31. C₅₉×#365	C₂²	semidirect
175. #2155	D₅₉	semidirect
203. #365	C₂×D₅₉	semidirect
207. C₅₉	#33470	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837845	2
#21433012	2
#21432963	2
#21425986	2
C₂×#16837217	2
#21425831	2
#21425700	2
C₅₉×#2155	2
#33470	59