Subgroups — #28504784

Order 15104.

400 conjugacy classes of subgroups.

The subgroup lattice has 328 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	#21425825	C₂	1	1	yes	maximal, characteristic
4	#21425823	C₂	1	1	yes	maximal, characteristic
5	#21424266	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422209	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1388	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	#16837210	—	2	2	no	—
23	#16837208	—	2	2	no	—
24	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#11745685	—	2	1	no	—
28	#14273749	—	2	1	no	—
29	#14273749	—	2	1	no	—
30	#14273749	C₂×C₄	1	2	yes	—
31	#14273749	D₄	1	2	yes	—
32	C₂×Dic₂₃₆	Q₈	1	1	yes	characteristic
33	C₂×#11745660	—	2	1	no	—
34	C₂×D₄₇₂	Q₈	1	1	yes	characteristic
35	#14273735	—	2	2	no	—
36	#14273735	—	2	2	no	—
37	#14273735	—	4	1	no	—
38	#14273735	—	4	1	no	—
39	C₂×#11745664	—	2	1	no	—
40	C₂×#11745659	—	4	1	no	—
41	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
42	C₂×C₈×D₅₉	—	4	1	no	—
43	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
44	#14273667	—	2	1	no	—
45	#14273667	—	2	1	no	—
46	C₂×C₅₉⋊C₁₆	—	2	1	no	—
47	#14273663	—	2	1	no	—
48	#14273663	—	2	1	no	—
49	#14273662	—	4	1	no	—
50	#14273662	—	4	1	no	—
51	Dic₅₉⋊C₈	—	4	2	no	—
52	#14273653	—	2	2	no	—
53	#14273653	—	2	2	no	—
54	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₅₉	D₄	1	1	yes	characteristic
57	C₈×Dic₅₉	D₄	1	1	yes	characteristic
58	C₂×M₁₆×C₅₉	—	2	1	no	—
59	C₂×M₁₆×C₅₉	—	2	1	no	—
60	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
61	M₃₂×C₅₉	—	2	1	no	—
62	M₃₂×C₅₉	—	2	1	no	—
63	C₂×C₁₆×C₅₉	—	2	1	no	—
64	#108×C₅₉	—	2	1	no	—
65	#108×C₅₉	—	2	1	no	—
66	#108×C₅₉	—	2	1	no	—
67	#108×C₅₉	C₂×C₄	1	2	yes	—
68	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₅₉	—	4	1	no	—
70	C₂²×Dic₅₉	—	2	1	no	—
71	#11745685	—	2	2	no	—
72	#11745685	—	2	2	no	—
73	#11745685	—	4	1	no	—
74	#11745685	—	4	1	no	—
75	C₂×D₂₃₆	—	2	1	no	—
76	C₂×C₄×D₅₉	—	4	1	no	—
77	C₂×Dic₁₁₈	—	2	1	no	—
78	M₁₆×C₅₉	—	2	2	no	—
79	M₁₆×C₅₉	—	4	1	no	—
80	M₁₆×C₅₉	—	4	1	no	—
81	M₁₆×C₅₉	—	4	1	no	—
82	C₂×C₈×C₅₉	—	4	1	no	—
83	C₂×C₈×C₅₉	—	4	1	no	—
84	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
89	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
90	C₄×Dic₅₉	—	2	1	no	—
91	C₄×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
93	#11745664	—	4	1	no	—
94	#11745664	—	4	1	no	—
95	C₂×C₅₉⋊C₈	—	4	1	no	—
96	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
99	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
100	C₂×C₅₉⋊C₈	SD₁₆	1	1	yes	characteristic
101	C₂×C₅₉⋊C₈	SD₁₆	1	1	yes	characteristic
102	Dic₂₃₆	—	2	1	no	—
103	Dic₂₃₆	C₄⋊C₄	1	2	yes	—
104	D₄₇₂	—	2	1	no	—
105	D₄₇₂	C₄⋊C₄	1	2	yes	—
106	#11745660	—	2	2	no	—
107	#11745660	—	2	2	no	—
108	#11745659	—	4	2	no	—
109	#11745659	—	4	2	no	—
110	C₈×D₅₉	—	4	2	no	—
111	C₈×D₅₉	—	4	2	no	—
112	C₁₆×C₅₉	—	4	1	no	—
113	C₅₉⋊C₁₆	—	4	1	no	—
114	C₂³×C₅₉	#137	1	1	yes	characteristic
115	C₂²×D₅₉	—	4	1	no	—
116	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂²⋊D₅₉	—	4	2	no	—
123	C₂²⋊D₅₉	—	4	2	no	—
124	C₂×Dic₅₉	—	2	1	no	—
125	C₂×Dic₅₉	—	2	1	no	—
126	C₂×Dic₅₉	—	4	2	no	—
127	C₂×Dic₅₉	—	4	1	no	—
128	D₂₃₆	—	2	2	no	—
129	D₂₃₆	—	4	1	no	—
130	C₄×D₅₉	—	4	2	no	—
131	C₄×D₅₉	—	4	2	no	—
132	Dic₁₁₈	—	2	2	no	—
133	Dic₁₁₈	—	4	1	no	—
134	C₈×C₅₉	—	4	2	no	—
135	C₈×C₅₉	—	8	1	no	—
136	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₈×C₅₉	#117	1	1	yes	characteristic
138	C₈×C₅₉	#121	1	1	yes	characteristic
139	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
140	C₅₉⋊C₈	—	8	1	no	—
141	C₅₉⋊C₈	#102	1	1	yes	characteristic
142	C₅₉⋊C₈	#102	1	1	yes	characteristic
143	C₅₉⋊C₈	#102	1	1	yes	characteristic
144	C₅₉⋊C₈	#102	1	1	yes	characteristic
145	#12822	—	59	1	no	maximal, Sylow
146	C₂²×C₅₉	—	4	1	no	—
147	C₂²×C₅₉	#417	1	1	yes	characteristic
148	C₂²×C₅₉	#461	1	1	yes	characteristic
149	C₂²×C₅₉	#474	1	1	yes	characteristic
150	C₂×D₅₉	—	4	2	no	—
151	C₂×D₅₉	—	8	2	no	—
152	C₄×C₅₉	#380	1	1	yes	characteristic
153	C₄×C₅₉	C₂×#102	1	1	yes	characteristic
154	C₄×C₅₉	#459	1	1	yes	characteristic
155	C₄×C₅₉	#476	1	1	yes	characteristic
156	Dic₅₉	—	2	2	no	—
157	Dic₅₉	—	4	1	no	—
158	Dic₅₉	—	4	2	no	—
159	C₂×#443	—	59	1	no	—
160	C₂×#364	—	59	1	no	—
161	#2143	—	59	1	no	—
162	#2143	—	59	1	no	—
163	#1848	—	59	1	no	—
164	#1388	—	59	1	no	—
165	#1388	D₅₉	1	1	yes	characteristic, complemented
166	C₂×C₅₉	—	2	1	no	—
167	C₂×C₅₉	—	2	1	no	—
168	C₂×C₅₉	—	2	1	no	—
169	C₂×C₅₉	#1802	1	1	yes	characteristic
170	D₅₉	—	8	2	no	—
171	C₂×#135	—	59	1	no	—
172	C₂×#131	—	118	1	no	—
173	C₂×M₃₂	—	59	1	no	—
174	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
175	#443	—	59	2	no	—
176	#443	—	59	2	no	—
177	#443	—	118	1	no	—
178	#443	—	118	1	no	—
179	C₂×#108	—	59	1	no	—
180	C₂×#108	—	118	1	no	—
181	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
182	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
183	#423	—	118	1	no	—
184	C₂×#104	—	118	1	no	—
185	C₂×C₄×C₈	—	59	1	no	—
186	#364	—	118	1	no	—
187	#364	—	118	1	no	—
188	#359	—	118	2	no	—
189	#359	—	118	2	no	—
190	C₅₉	#12822	1	1	yes	characteristic, complemented, Sylow
191	C₂×#41	—	118	1	no	—
192	#135	—	59	2	no	—
193	#135	—	59	2	no	—
194	#135	—	118	1	no	—
195	#135	—	118	1	no	—
196	C₂×Q₁₆	—	59	1	no	—
197	C₂×SD₁₆	—	118	1	no	—
198	C₂×D₈	—	59	1	no	—
199	#131	—	118	2	no	—
200	#131	—	118	2	no	—
201	#131	—	236	1	no	—
202	#131	—	236	1	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	118	1	no	—
206	C₂×M₁₆	—	236	1	no	—
207	C₂²×C₈	—	59	1	no	—
208	C₂²×C₈	—	236	1	no	—
209	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
210	C₂×C₄²	—	59	1	no	—
211	M₃₂	—	2	1	no	—
212	M₃₂	—	2	1	no	—
213	M₃₂	—	118	1	no	—
214	M₃₂	—	118	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	118	1	no	—
217	#108	—	2	1	no	—
218	#108	—	2	1	no	—
219	#108	—	2	1	no	—
220	#108	—	118	1	no	—
221	#108	—	118	1	no	—
222	#108	—	236	1	no	—
223	#108	—	236	1	no	—
224	#108	C₄×D₅₉	1	2	yes	—
225	C₄⋊C₈	—	236	2	no	—
226	#104	—	118	2	no	—
227	#104	—	118	2	no	—
228	C₄×C₈	—	59	1	no	—
229	C₄×C₈	—	59	1	no	—
230	C₄×C₈	—	59	1	no	—
231	C₄×C₈	—	59	1	no	—
232	#41	—	118	2	no	—
233	#41	—	118	2	no	—
234	#41	—	236	1	no	—
235	#41	—	236	1	no	—
236	C₂×Q₈	—	118	1	no	—
237	C₂×D₄	—	118	1	no	—
238	C₂×D₄	—	236	1	no	—
239	C₂²×C₄	—	118	1	no	—
240	C₂²×C₄	—	236	1	no	—
241	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
242	Q₁₆	—	59	2	no	—
243	Q₁₆	—	118	1	no	—
244	SD₁₆	—	118	2	no	—
245	SD₁₆	—	118	2	no	—
246	D₈	—	59	2	no	—
247	D₈	—	118	1	no	—
248	M₁₆	—	2	2	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	236	2	no	—
253	M₁₆	—	236	2	no	—
254	M₁₆	—	236	1	no	—
255	M₁₆	—	236	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	59	1	no	—
259	C₂×C₈	—	59	1	no	—
260	C₂×C₈	—	59	1	no	—
261	C₂×C₈	—	59	1	no	—
262	C₂×C₈	—	59	1	no	—
263	C₂×C₈	—	59	1	no	—
264	C₂×C₈	—	236	2	no	—
265	C₂×C₈	—	236	2	no	—
266	C₂×C₈	—	236	1	no	—
267	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
268	C₂×C₈	#11745685	1	1	yes	characteristic
269	C₂×C₈	#11745685	1	1	yes	characteristic
270	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
271	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
272	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
273	C₄²	—	59	1	no	—
274	C₄²	—	59	1	no	—
275	C₄²	—	118	1	no	—
276	C₁₆	—	4	1	no	—
277	C₁₆	—	236	1	no	—
278	C₂³	—	236	1	no	—
279	C₂³	#14273759	1	1	yes	characteristic
280	Q₈	—	118	2	no	—
281	Q₈	—	236	1	no	—
282	D₄	—	118	2	no	—
283	D₄	—	236	2	no	—
284	D₄	—	236	2	no	—
285	D₄	—	236	1	no	—
286	C₂×C₄	—	118	1	no	—
287	C₂×C₄	—	118	1	no	—
288	C₂×C₄	—	236	2	no	—
289	C₂×C₄	—	236	1	no	—
290	C₂×C₄	—	236	2	no	—
291	C₂×C₄	—	236	2	no	—
292	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
293	C₂×C₄	#14273700	1	1	yes	characteristic
294	C₂×C₄	#14273705	1	1	yes	characteristic
295	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
296	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
297	C₂×C₄	#14273754	1	1	yes	characteristic
298	C₈	—	4	2	no	—
299	C₈	—	8	1	no	—
300	C₈	—	59	1	no	—
301	C₈	—	59	1	no	—
302	C₈	—	59	1	no	—
303	C₈	—	59	1	no	—
304	C₈	—	472	1	no	—
305	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
306	C₈	#14273684	1	1	yes	characteristic
307	C₈	#14273700	1	1	yes	characteristic
308	C₈	#14273705	1	1	yes	characteristic
309	C₂²	—	4	1	no	—
310	C₂²	—	472	2	no	—
311	C₂²	—	236	2	no	—
312	C₂²	#16837402	1	1	yes	characteristic
313	C₂²	#16837606	1	1	yes	characteristic
314	C₂²	#16837681	1	1	yes	characteristic
315	C₄	—	118	2	no	—
316	C₄	—	236	1	no	—
317	C₄	—	236	2	no	—
318	C₄	#16837261	1	1	yes	characteristic
319	C₄	#102×D₅₉	1	1	yes	characteristic
320	C₄	#16837596	1	1	yes	characteristic
321	C₄	#16837691	1	1	yes	characteristic
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	472	2	no	—
326	C₂	#21424095	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16837596
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1388	C₂
socle	C₂×C₅₉	#1802
prime radical	C₂×#14273749	C₂²
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1388	D₅₉
hypocenter	C₅₉	#12822
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1388

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. #21422209	C₂	semidirect
7. C₅₉×#1388	C₂	semidirect
165. #1388	D₅₉	semidirect
190. C₅₉	#12822	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
#21425825	2
#21425823	2
#21424266	2
#21422209	2
C₅₉×#1388	2
#12822	59