Subgroups — #28479467

Order 15104.

590 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₂₃₆	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14273649	C₂	1	1	yes	maximal, characteristic
4	C₂×#16837131	C₂	1	1	yes	maximal, characteristic
5	#21421936	C₂	1	1	yes	maximal, characteristic, complemented
6	#21421931	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₉×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₅₉	—	2	1	no	—
9	C₂×C₄×D₂₃₆	C₂²	1	1	yes	characteristic
10	C₈×D₂₃₆	—	2	2	no	—
11	C₈×D₂₃₆	C₄	1	4	yes	complemented
12	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×#14273649	C₄	1	2	yes	—
15	C₂×(C₂×D₅₉)⋊C₈	—	2	1	no	—
16	C₂×C₈×Dic₅₉	—	2	1	no	—
17	C₄×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
18	#16837165	C₄	1	2	yes	—
19	#16837156	—	2	1	no	—
20	#16837131	C₄	1	4	yes	complemented
21	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₅₉×#322	—	2	2	no	—
25	C₂²×D₂₃₆	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₅₉	—	2	1	no	—
27	C₂²×C₅₉⋊C₈	—	2	1	no	—
28	C₂²×C₅₉⋊C₈	—	2	1	no	—
29	C₂×C₈×D₅₉	—	2	4	no	—
30	C₂×C₈×D₅₉	—	4	2	no	—
31	C₂×C₈×D₅₉	—	4	2	no	—
32	C₄×D₂₃₆	—	2	1	no	—
33	C₄×D₂₃₆	—	2	1	no	—
34	C₄×D₂₃₆	C₂×C₄	1	2	yes	—
35	C₄×D₂₃₆	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
38	C₂×(C₂×D₅₉)⋊C₄	—	2	1	no	—
39	C₂×C₄×Dic₅₉	—	2	1	no	—
40	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
41	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
42	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
43	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
44	#14273649	C₈	1	8	yes	complemented
45	(C₂×D₅₉)⋊C₈	—	2	4	no	—
46	C₈×Dic₅₉	—	4	2	no	—
47	C₅₉⋊#95	—	2	1	no	—
48	C₂²×C₈×C₅₉	—	2	1	no	—
49	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₅₉	—	2	1	no	—
54	C₄×C₈×C₅₉	—	2	1	no	—
55	C₄×C₈×C₅₉	—	2	1	no	—
56	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₅₉	D₄	1	2	yes	—
59	C₂³×D₅₉	—	2	1	no	—
60	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
63	C₂²×Dic₅₉	—	2	2	no	—
64	C₂²×Dic₅₉	—	2	1	no	—
65	C₂×D₂₃₆	—	2	1	no	—
66	C₂×D₂₃₆	—	2	2	no	—
67	C₂×D₂₃₆	C₄²	1	2	yes	—
68	C₂×D₂₃₆	C₂×C₈	1	4	yes	—
69	C₂×C₄×D₅₉	—	2	2	no	—
70	C₂×C₄×D₅₉	—	2	2	no	—
71	C₂×C₄×D₅₉	—	4	1	no	—
72	C₂×C₄×D₅₉	—	4	1	no	—
73	C₂×C₄×D₅₉	—	4	1	no	—
74	C₂×C₄×D₅₉	—	4	1	no	—
75	C₂×C₈×C₅₉	—	2	2	no	—
76	C₂×C₈×C₅₉	—	2	2	no	—
77	C₂×C₈×C₅₉	—	2	1	no	—
78	C₂×C₈×C₅₉	—	2	1	no	—
79	C₂×C₈×C₅₉	—	2	1	no	—
80	C₂×C₈×C₅₉	—	2	1	no	—
81	C₂×C₈×C₅₉	—	2	2	no	—
82	C₂×C₈×C₅₉	—	4	2	no	—
83	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	complemented
86	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅₉	#41	1	1	yes	characteristic
90	(C₂×D₅₉)⋊C₄	—	2	2	no	—
91	(C₂×D₅₉)⋊C₄	—	2	2	no	—
92	C₄⋊Dic₅₉	—	2	1	no	—
93	C₄⋊Dic₅₉	—	2	2	no	—
94	C₄⋊Dic₅₉	C₄²	1	2	yes	—
95	C₄⋊Dic₅₉	C₂×C₈	1	4	yes	complemented
96	C₄×Dic₅₉	—	4	1	no	—
97	C₄×Dic₅₉	—	4	1	no	—
98	C₂×C₅₉⋊C₈	—	2	2	no	—
99	C₂×C₅₉⋊C₈	—	2	2	no	—
100	C₂×C₅₉⋊C₈	—	4	1	no	—
101	C₂×C₅₉⋊C₈	—	4	1	no	—
102	C₂×C₅₉⋊C₈	—	4	1	no	—
103	C₂×C₅₉⋊C₈	—	4	1	no	—
104	C₈×D₅₉	—	4	8	no	—
105	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
106	C₂²×D₅₉	—	2	2	no	—
107	C₂²×D₅₉	—	2	4	no	—
108	C₂²×D₅₉	—	4	1	no	—
109	C₂²×D₅₉	—	4	1	no	—
110	C₂²×D₅₉	—	4	2	no	—
111	C₂×C₄×C₅₉	—	2	2	no	—
112	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₅₉	#131	1	2	yes	—
120	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
124	C₂×Dic₅₉	—	2	2	no	—
125	C₂×Dic₅₉	—	2	4	no	—
126	C₂×Dic₅₉	—	4	1	no	—
127	C₂×Dic₅₉	—	4	2	no	—
128	C₂×Dic₅₉	—	4	1	no	—
129	C₂×Dic₅₉	—	4	2	no	—
130	D₂₃₆	—	2	2	no	—
131	D₂₃₆	—	2	4	no	—
132	D₂₃₆	C₄×C₈	1	4	yes	complemented
133	C₄×D₅₉	—	4	2	no	—
134	C₄×D₅₉	—	4	2	no	—
135	C₄×D₅₉	—	4	2	no	—
136	C₄×D₅₉	—	4	2	no	—
137	C₈×C₅₉	—	2	2	no	—
138	C₈×C₅₉	—	2	2	no	—
139	C₈×C₅₉	—	2	4	no	—
140	C₈×C₅₉	—	4	2	no	—
141	C₅₉⋊C₈	—	4	2	no	—
142	C₅₉⋊C₈	—	4	2	no	—
143	#8200	—	59	1	no	maximal, Sylow
144	C₂²×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₅₉	C₈×D₄	1	2	yes	—
146	C₂²×C₅₉	#440	1	1	yes	characteristic
147	C₂²×C₅₉	#442	1	1	yes	characteristic
148	C₂²×C₅₉	#444	1	2	yes	—
149	C₂×D₅₉	—	2	4	no	—
150	C₂×D₅₉	—	4	4	no	—
151	C₂×D₅₉	—	4	4	no	—
152	C₂×D₅₉	—	4	2	no	—
153	C₂×D₅₉	—	4	2	no	—
154	C₄×C₅₉	—	2	1	no	—
155	C₄×C₅₉	—	2	1	no	—
156	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
157	C₄×C₅₉	#405	1	1	yes	characteristic
158	C₄×C₅₉	#417	1	1	yes	characteristic
159	C₄×C₅₉	#418	1	1	yes	characteristic
160	C₄×C₅₉	C₈×D₄	1	2	yes	—
161	C₄×C₅₉	#440	1	1	yes	characteristic
162	C₄×C₅₉	#442	1	1	yes	characteristic
163	Dic₅₉	—	4	4	no	—
164	Dic₅₉	—	4	2	no	—
165	C₂×C₈×D₄	—	59	1	no	—
166	C₈×C₄⋊C₄	—	59	1	no	—
167	C₄×#102	—	59	1	no	—
168	C₂×D₄⋊C₈	—	59	1	no	—
169	C₂×#322	D₅₉	1	1	yes	characteristic, complemented
170	#1279	—	59	1	no	—
171	#1277	—	59	1	no	—
172	C₂×C₅₉	SD₁₆⋊C₈	1	2	yes	—
173	C₂×C₅₉	#1580	1	2	yes	—
174	C₂×C₅₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₅₉	#1762	1	1	yes	characteristic
176	C₂×C₅₉	#1765	1	1	yes	characteristic
177	D₅₉	—	4	4	no	—
178	C₂³×C₈	—	118	1	no	—
179	C₂×C₄×D₄	—	59	1	no	—
180	C₈×D₄	—	59	4	no	—
181	C₈×D₄	—	118	2	no	—
182	C₂×C₄⋊C₈	—	59	1	no	—
183	C₂×C₄⋊C₈	—	59	1	no	—
184	C₂×#102	—	59	2	no	—
185	C₂×C₂²⋊C₈	—	118	1	no	—
186	C₂×C₄×C₈	—	118	1	no	—
187	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
188	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
190	C₄×C₄⋊C₄	—	59	1	no	—
191	#340	—	59	2	no	—
192	#336	—	118	1	no	—
193	D₄⋊C₈	—	59	4	no	—
194	#322	—	2	2	no	—
195	C₅₉	#8200	1	1	yes	characteristic, complemented, Sylow
196	C₂²×D₄	—	59	1	no	—
197	C₂³×C₄	—	118	1	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	118	1	no	—
200	C₂²×C₈	—	118	1	no	—
201	C₂²×C₈	—	118	4	no	—
202	C₂²×C₈	—	236	2	no	—
203	C₂²×C₈	—	236	2	no	—
204	C₂²×C₈	C₄×D₅₉	1	2	yes	—
205	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
206	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
207	C₄×D₄	—	59	2	no	—
208	C₄×D₄	—	59	2	no	—
209	C₄×D₄	—	118	1	no	—
210	C₄×D₄	—	118	1	no	—
211	C₂×C₄⋊C₄	—	59	1	no	—
212	C₂×C₄⋊C₄	—	59	2	no	—
213	C₂×C₂²⋊C₄	—	118	1	no	—
214	C₂×C₄²	—	118	1	no	—
215	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
216	C₄⋊C₈	—	59	2	no	—
217	C₄⋊C₈	—	59	2	no	—
218	C₄⋊C₈	—	118	1	no	—
219	C₄⋊C₈	—	118	1	no	—
220	#102	—	59	8	no	—
221	C₂²⋊C₈	—	118	4	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	236	2	no	—
226	C₄×C₈	C₄×D₅₉	1	2	yes	—
227	C₄×C₈	C₄×D₅₉	1	2	yes	—
228	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
229	#95	—	118	1	no	—
230	C₂⁴	—	118	1	no	—
231	C₂×D₄	—	59	2	no	—
232	C₂×D₄	—	59	4	no	—
233	C₂×D₄	—	118	1	no	—
234	C₂×D₄	—	118	2	no	—
235	C₂²×C₄	—	118	2	no	—
236	C₂²×C₄	—	118	1	no	—
237	C₂²×C₄	—	118	2	no	—
238	C₂²×C₄	—	118	2	no	—
239	C₂²×C₄	—	236	1	no	—
240	C₂²×C₄	—	236	1	no	—
241	C₂²×C₄	—	236	1	no	—
242	C₂²×C₄	—	236	1	no	—
243	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
245	C₂²×C₄	#11745685	1	1	yes	characteristic
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	118	2	no	—
255	C₂×C₈	—	118	2	no	—
256	C₂×C₈	—	236	8	no	—
257	C₂×C₈	—	236	1	no	—
258	C₂×C₈	—	236	1	no	—
259	C₂×C₈	—	236	1	no	—
260	C₂×C₈	—	236	1	no	—
261	C₂×C₈	C₈×D₅₉	1	4	yes	complemented
262	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
263	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
264	C₄⋊C₄	—	59	2	no	—
265	C₄⋊C₄	—	59	4	no	—
266	C₄⋊C₄	—	118	1	no	—
267	C₄⋊C₄	—	118	2	no	—
268	C₂²⋊C₄	—	118	2	no	—
269	C₂²⋊C₄	—	118	2	no	—
270	C₄²	—	236	1	no	—
271	C₄²	—	236	1	no	—
272	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
274	C₄²	#11745685	1	1	yes	characteristic
275	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
276	C₂³	—	118	4	no	—
277	C₂³	—	118	2	no	—
278	C₂³	—	236	1	no	—
279	C₂³	—	236	1	no	—
280	C₂³	—	236	2	no	—
281	C₂³	C₄×D₂₃₆	1	1	yes	characteristic
282	D₄	—	59	4	no	—
283	D₄	—	118	4	no	—
284	D₄	—	118	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	118	4	no	—
287	C₂×C₄	—	118	2	no	—
288	C₂×C₄	—	236	1	no	—
289	C₂×C₄	—	236	2	no	—
290	C₂×C₄	—	236	1	no	—
291	C₂×C₄	—	236	2	no	—
292	C₂×C₄	—	236	2	no	—
293	C₂×C₄	—	236	2	no	—
294	C₂×C₄	—	236	2	no	—
295	C₂×C₄	—	236	2	no	—
296	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
297	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
298	C₂×C₄	#14273684	1	1	yes	characteristic
299	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
300	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
301	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
302	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
303	C₂×C₄	#14273733	1	2	yes	—
304	C₂×C₄	#14273755	1	1	yes	characteristic
305	C₂×C₄	#14273755	1	1	yes	characteristic
306	C₂×C₄	#14273755	1	1	yes	characteristic
307	C₂×C₄	#14273760	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	—	2	4	no	—
311	C₈	—	4	2	no	—
312	C₈	—	236	2	no	—
313	C₈	—	236	2	no	—
314	C₂²	—	118	4	no	—
315	C₂²	—	236	2	no	—
316	C₂²	—	236	4	no	—
317	C₂²	—	236	2	no	—
318	C₂²	—	236	4	no	—
319	C₂²	C₄×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
320	C₂²	C₈×D₂₃₆	1	2	yes	—
321	C₂²	#16837504	1	1	yes	characteristic
322	C₂²	#16837515	1	1	yes	characteristic
323	C₂²	#16837525	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	236	2	no	—
327	C₄	—	236	4	no	—
328	C₄	C₄×C₈×D₅₉	1	1	yes	characteristic
329	C₄	#16837356	1	1	yes	characteristic
330	C₄	#16837404	1	1	yes	characteristic
331	C₄	#16837408	1	1	yes	characteristic
332	C₄	C₈×C₂²⋊D₅₉	1	2	yes	—
333	C₄	#16837504	1	1	yes	characteristic
334	C₄	#16837515	1	1	yes	characteristic
335	C₂	—	236	4	no	—
336	C₂	#21422910	1	2	yes	—
337	C₂	#21422921	1	2	yes	—
338	C₂	C₈×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
339	C₂	#21423922	1	1	yes	characteristic
340	C₂	#21423940	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃₆
commutator	C₄×C₅₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#322	C₂
socle	C₂³×C₅₉	C₄×D₄
prime radical	C₂²×D₂₃₆	C₂×C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#322	D₅₉
hypocenter	C₅₉	#8200
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. C₈×C₄⋊Dic₅₉	C₂	semidirect
5. #21421936	C₂	semidirect
6. #21421931	C₂	semidirect
7. C₂×C₅₉×#322	C₂	semidirect
11. C₈×D₂₃₆	C₄	semidirect
20. #16837131	C₄	semidirect
44. #14273649	C₈	semidirect
85. C₂×C₈×C₅₉	C₂×C₈	semidirect
95. C₄⋊Dic₅₉	C₂×C₈	semidirect
132. D₂₃₆	C₄×C₈	semidirect
169. C₂×#322	D₅₉	semidirect
195. C₅₉	#8200	semidirect
261. C₂×C₈	C₈×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₃₆	2
C₈×C₄⋊Dic₅₉	2
C₄×#14273649	2
C₂×#16837131	2
#21421936	2
#21421931	2
C₂×C₅₉×#322	2
#8200	59