Subgroups — #28504799

Order 15104.

400 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837518	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837222	C₂	1	1	yes	maximal, characteristic, complemented
3	#21425846	C₂	1	1	yes	maximal, characteristic
4	#21425843	C₂	1	1	yes	maximal, characteristic
5	#21424266	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422199	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1385	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273749	C₂²	1	1	yes	characteristic
9	C₂×#14273735	—	2	1	no	—
10	C₂×#14273667	C₂²	1	1	yes	characteristic
11	#16837518	—	2	1	no	—
12	#16837518	—	2	1	no	—
13	#16837518	C₄	1	2	yes	—
14	#16837518	C₄	1	2	yes	—
15	C₂×#14273663	C₂²	1	1	yes	characteristic
16	C₂×#14273662	—	2	1	no	—
17	#16837425	—	2	1	no	—
18	C₂×#14273653	—	2	1	no	—
19	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
20	#16837222	—	2	1	no	—
21	#16837222	—	2	1	no	—
22	#16837219	—	2	1	no	—
23	#16837217	—	2	1	no	—
24	#16837216	—	2	1	no	—
25	#16837214	—	2	1	no	—
26	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
29	C₂×#11745685	—	2	1	no	—
30	#14273749	—	2	1	no	—
31	#14273749	—	2	1	no	—
32	#14273749	C₂×C₄	1	2	yes	—
33	#14273749	Q₈	1	2	yes	—
34	C₂×Dic₂₃₆	D₄	1	1	yes	characteristic
35	C₂×#11745660	—	2	1	no	—
36	C₂×D₄₇₂	D₄	1	1	yes	characteristic
37	#14273735	—	2	2	no	—
38	#14273735	—	2	2	no	—
39	#14273735	—	4	1	no	—
40	#14273735	—	4	1	no	—
41	C₂×#11745664	—	2	1	no	—
42	C₂×#11745659	—	4	1	no	—
43	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
44	C₂×C₈×D₅₉	—	4	1	no	—
45	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
46	#14273667	—	2	1	no	—
47	#14273667	—	2	1	no	—
48	C₂×C₅₉⋊C₁₆	—	2	1	no	—
49	#14273663	—	2	1	no	—
50	#14273663	—	2	1	no	—
51	#14273662	—	4	1	no	—
52	#14273662	—	4	1	no	—
53	Dic₅₉⋊C₈	—	4	2	no	—
54	#14273653	—	2	2	no	—
55	#14273653	—	2	2	no	—
56	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
57	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
58	C₈×Dic₅₉	D₄	1	1	yes	characteristic
59	C₈×Dic₅₉	D₄	1	1	yes	characteristic
60	C₂×M₁₆×C₅₉	—	2	1	no	—
61	C₂×M₁₆×C₅₉	—	2	1	no	—
62	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
63	M₃₂×C₅₉	—	2	1	no	—
64	M₃₂×C₅₉	—	2	1	no	—
65	C₂×C₁₆×C₅₉	—	2	1	no	—
66	#108×C₅₉	—	2	1	no	—
67	#108×C₅₉	—	2	1	no	—
68	#108×C₅₉	—	2	1	no	—
69	#108×C₅₉	C₂×C₄	1	2	yes	—
70	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₂²⋊D₅₉	—	4	1	no	—
72	C₂²×Dic₅₉	—	2	1	no	—
73	#11745685	—	2	2	no	—
74	#11745685	—	2	2	no	—
75	#11745685	—	4	1	no	—
76	#11745685	—	4	1	no	—
77	C₂×D₂₃₆	—	2	1	no	—
78	C₂×C₄×D₅₉	—	4	1	no	—
79	C₂×Dic₁₁₈	—	2	1	no	—
80	M₁₆×C₅₉	—	2	2	no	—
81	M₁₆×C₅₉	—	4	1	no	—
82	M₁₆×C₅₉	—	4	1	no	—
83	M₁₆×C₅₉	—	4	1	no	—
84	C₂×C₈×C₅₉	—	4	1	no	—
85	C₂×C₈×C₅₉	—	4	1	no	—
86	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	C₂×D₄	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	C₄×Dic₅₉	—	2	1	no	—
93	C₄×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
95	#11745664	—	4	1	no	—
96	#11745664	—	4	1	no	—
97	C₂×C₅₉⋊C₈	—	4	1	no	—
98	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
101	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
102	C₂×C₅₉⋊C₈	SD₁₆	1	1	yes	characteristic
103	C₂×C₅₉⋊C₈	SD₁₆	1	1	yes	characteristic
104	Dic₂₃₆	—	2	1	no	—
105	Dic₂₃₆	C₄⋊C₄	1	2	yes	—
106	D₄₇₂	—	2	1	no	—
107	D₄₇₂	C₄⋊C₄	1	2	yes	—
108	#11745660	—	2	2	no	—
109	#11745660	—	2	2	no	—
110	#11745659	—	4	2	no	—
111	#11745659	—	4	2	no	—
112	C₈×D₅₉	—	4	2	no	—
113	C₈×D₅₉	—	4	2	no	—
114	C₁₆×C₅₉	—	4	1	no	—
115	C₅₉⋊C₁₆	—	4	1	no	—
116	C₂³×C₅₉	#137	1	1	yes	characteristic
117	C₂²×D₅₉	—	4	1	no	—
118	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂×D₈	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	C₂²⋊D₅₉	—	4	2	no	—
125	C₂²⋊D₅₉	—	4	2	no	—
126	C₂×Dic₅₉	—	2	1	no	—
127	C₂×Dic₅₉	—	2	1	no	—
128	C₂×Dic₅₉	—	4	2	no	—
129	C₂×Dic₅₉	—	4	1	no	—
130	D₂₃₆	—	2	2	no	—
131	D₂₃₆	—	4	1	no	—
132	C₄×D₅₉	—	4	2	no	—
133	C₄×D₅₉	—	4	2	no	—
134	Dic₁₁₈	—	2	2	no	—
135	Dic₁₁₈	—	4	1	no	—
136	C₈×C₅₉	—	4	2	no	—
137	C₈×C₅₉	—	8	1	no	—
138	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₈×C₅₉	#117	1	1	yes	characteristic
140	C₈×C₅₉	#121	1	1	yes	characteristic
141	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
142	C₅₉⋊C₈	—	8	1	no	—
143	C₅₉⋊C₈	#102	1	1	yes	characteristic
144	C₅₉⋊C₈	#102	1	1	yes	characteristic
145	C₅₉⋊C₈	#102	1	1	yes	characteristic
146	C₅₉⋊C₈	#102	1	1	yes	characteristic
147	#12825	—	59	1	no	maximal, Sylow
148	C₂²×C₅₉	—	4	1	no	—
149	C₂²×C₅₉	#417	1	1	yes	characteristic
150	C₂²×C₅₉	#461	1	1	yes	characteristic
151	C₂²×C₅₉	#474	1	1	yes	characteristic
152	C₂×D₅₉	—	4	2	no	—
153	C₂×D₅₉	—	8	2	no	—
154	C₄×C₅₉	#380	1	1	yes	characteristic
155	C₄×C₅₉	C₂×#102	1	1	yes	characteristic
156	C₄×C₅₉	#459	1	1	yes	characteristic
157	C₄×C₅₉	#476	1	1	yes	characteristic
158	Dic₅₉	—	2	2	no	—
159	Dic₅₉	—	4	1	no	—
160	Dic₅₉	—	4	2	no	—
161	C₂×#443	—	59	1	no	—
162	C₂×#364	—	59	1	no	—
163	#2148	—	59	1	no	—
164	#2148	—	59	1	no	—
165	#1848	—	59	1	no	—
166	#1385	—	59	1	no	—
167	#1385	D₅₉	1	1	yes	characteristic, complemented
168	C₂×C₅₉	—	2	1	no	—
169	C₂×C₅₉	—	2	1	no	—
170	C₂×C₅₉	—	2	1	no	—
171	C₂×C₅₉	#1802	1	1	yes	characteristic
172	D₅₉	—	8	2	no	—
173	C₂×#135	—	59	1	no	—
174	C₂×#131	—	118	1	no	—
175	C₂×M₃₂	—	59	1	no	—
176	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
177	#443	—	59	2	no	—
178	#443	—	59	2	no	—
179	#443	—	118	1	no	—
180	#443	—	118	1	no	—
181	C₂×#108	—	59	1	no	—
182	C₂×#108	—	118	1	no	—
183	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
184	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
185	#423	—	118	1	no	—
186	C₂×#104	—	118	1	no	—
187	C₂×C₄×C₈	—	59	1	no	—
188	#364	—	118	1	no	—
189	#364	—	118	1	no	—
190	#362	—	118	1	no	—
191	#362	—	118	1	no	—
192	#361	—	118	1	no	—
193	#361	—	118	1	no	—
194	C₅₉	#12825	1	1	yes	characteristic, complemented, Sylow
195	C₂×#41	—	118	1	no	—
196	#135	—	59	2	no	—
197	#135	—	59	2	no	—
198	#135	—	118	1	no	—
199	#135	—	118	1	no	—
200	C₂×Q₁₆	—	59	1	no	—
201	C₂×SD₁₆	—	118	1	no	—
202	C₂×D₈	—	59	1	no	—
203	#131	—	118	2	no	—
204	#131	—	118	2	no	—
205	#131	—	236	1	no	—
206	#131	—	236	1	no	—
207	C₂×M₁₆	—	2	1	no	—
208	C₂×M₁₆	—	2	1	no	—
209	C₂×M₁₆	—	118	1	no	—
210	C₂×M₁₆	—	236	1	no	—
211	C₂²×C₈	—	59	1	no	—
212	C₂²×C₈	—	236	1	no	—
213	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
214	C₂×C₄²	—	59	1	no	—
215	M₃₂	—	2	1	no	—
216	M₃₂	—	2	1	no	—
217	M₃₂	—	118	1	no	—
218	M₃₂	—	118	1	no	—
219	C₂×C₁₆	—	2	1	no	—
220	C₂×C₁₆	—	118	1	no	—
221	#108	—	2	1	no	—
222	#108	—	2	1	no	—
223	#108	—	2	1	no	—
224	#108	—	118	1	no	—
225	#108	—	118	1	no	—
226	#108	—	236	1	no	—
227	#108	—	236	1	no	—
228	#108	C₄×D₅₉	1	2	yes	—
229	C₄⋊C₈	—	236	2	no	—
230	#104	—	118	2	no	—
231	#104	—	118	2	no	—
232	C₄×C₈	—	59	1	no	—
233	C₄×C₈	—	59	1	no	—
234	C₄×C₈	—	59	1	no	—
235	C₄×C₈	—	59	1	no	—
236	#41	—	118	2	no	—
237	#41	—	118	2	no	—
238	#41	—	236	1	no	—
239	#41	—	236	1	no	—
240	C₂×Q₈	—	118	1	no	—
241	C₂×D₄	—	118	1	no	—
242	C₂×D₄	—	236	1	no	—
243	C₂²×C₄	—	118	1	no	—
244	C₂²×C₄	—	236	1	no	—
245	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
246	Q₁₆	—	59	2	no	—
247	Q₁₆	—	118	1	no	—
248	SD₁₆	—	118	2	no	—
249	SD₁₆	—	118	2	no	—
250	D₈	—	59	2	no	—
251	D₈	—	118	1	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	4	1	no	—
254	M₁₆	—	4	1	no	—
255	M₁₆	—	4	1	no	—
256	M₁₆	—	236	2	no	—
257	M₁₆	—	236	2	no	—
258	M₁₆	—	236	1	no	—
259	M₁₆	—	236	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	59	1	no	—
263	C₂×C₈	—	59	1	no	—
264	C₂×C₈	—	59	1	no	—
265	C₂×C₈	—	59	1	no	—
266	C₂×C₈	—	59	1	no	—
267	C₂×C₈	—	59	1	no	—
268	C₂×C₈	—	236	2	no	—
269	C₂×C₈	—	236	2	no	—
270	C₂×C₈	—	236	1	no	—
271	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
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	D₄×D₅₉	1	1	yes	characteristic
276	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
277	C₄²	—	59	1	no	—
278	C₄²	—	59	1	no	—
279	C₄²	—	118	1	no	—
280	C₁₆	—	4	1	no	—
281	C₁₆	—	236	1	no	—
282	C₂³	—	236	1	no	—
283	C₂³	#14273759	1	1	yes	characteristic
284	Q₈	—	118	2	no	—
285	Q₈	—	236	1	no	—
286	D₄	—	118	2	no	—
287	D₄	—	236	2	no	—
288	D₄	—	236	2	no	—
289	D₄	—	236	1	no	—
290	C₂×C₄	—	118	1	no	—
291	C₂×C₄	—	118	1	no	—
292	C₂×C₄	—	236	2	no	—
293	C₂×C₄	—	236	1	no	—
294	C₂×C₄	—	236	2	no	—
295	C₂×C₄	—	236	2	no	—
296	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
297	C₂×C₄	#14273700	1	1	yes	characteristic
298	C₂×C₄	#14273705	1	1	yes	characteristic
299	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
300	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
301	C₂×C₄	#14273754	1	1	yes	characteristic
302	C₈	—	4	2	no	—
303	C₈	—	8	1	no	—
304	C₈	—	59	1	no	—
305	C₈	—	59	1	no	—
306	C₈	—	59	1	no	—
307	C₈	—	59	1	no	—
308	C₈	—	472	1	no	—
309	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
310	C₈	#14273684	1	1	yes	characteristic
311	C₈	#14273700	1	1	yes	characteristic
312	C₈	#14273705	1	1	yes	characteristic
313	C₂²	—	4	1	no	—
314	C₂²	—	472	2	no	—
315	C₂²	—	236	2	no	—
316	C₂²	#16837402	1	1	yes	characteristic
317	C₂²	#16837606	1	1	yes	characteristic
318	C₂²	#16837681	1	1	yes	characteristic
319	C₄	—	118	2	no	—
320	C₄	—	236	1	no	—
321	C₄	—	236	2	no	—
322	C₄	#16837261	1	1	yes	characteristic
323	C₄	#102×D₅₉	1	1	yes	characteristic
324	C₄	#16837596	1	1	yes	characteristic
325	C₄	#16837691	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	472	2	no	—
330	C₂	#21424095	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16837691
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1385	C₂
socle	C₂×C₅₉	#1802
prime radical	C₂×#14273749	C₂²
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1385	D₅₉
hypocenter	C₅₉	#12825
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

Complements

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

N	H ≅ G/N	type
2. C₂×#16837222	C₂	semidirect
5. #21424266	C₂	semidirect
6. #21422199	C₂	semidirect
7. C₅₉×#1385	C₂	semidirect
167. #1385	D₅₉	semidirect
194. C₅₉	#12825	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837518	2
C₂×#16837222	2
#21425846	2
#21425843	2
#21424266	2
#21422199	2
C₅₉×#1385	2
#12825	59