Subgroups — C₈×#14273642 #28479230

Order 15104.

614 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14273642	C₂	1	1	yes	maximal, characteristic
2	C₈×C₂²⋊Dic₅₉	C₂	1	2	yes	maximal, complemented
3	#21422414	C₂	1	1	yes	maximal, characteristic
4	#21421943	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273642	C₄	1	2	yes	—
7	C₂×C₂²⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₈×Dic₅₉	—	2	2	no	—
9	C₄×C₂²⋊Dic₅₉	C₂²	1	2	yes	—
10	#16837169	C₄	1	2	yes	—
11	#16837156	—	2	2	no	—
12	#16837128	C₄	1	2	yes	complemented
13	#16837126	C₄	1	2	yes	complemented
14	C₂³×C₈×C₅₉	C₄	1	1	yes	characteristic, complemented
15	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
17	C₈×D₄×C₅₉	—	2	2	no	—
18	C₈×D₄×C₅₉	—	2	2	no	—
19	C₂×C₄⋊C₈×C₅₉	C₄	1	1	yes	characteristic, complemented
20	C₂×C₂²⋊C₈×C₅₉	C₄	1	1	yes	characteristic, complemented
21	C₂×C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₉	C₄	1	1	yes	characteristic, complemented
23	C₂²×C₅₉⋊C₈	—	2	2	no	—
24	C₂×C₂²⋊Dic₅₉	C₂×C₄	1	4	yes	—
25	C₂×C₄×Dic₅₉	—	2	2	no	—
26	#14273642	C₈	1	8	yes	direct product factor
27	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
28	C₂²⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
29	C₈×Dic₅₉	—	4	4	no	—
30	C₅₉⋊#95	—	2	2	no	—
31	C₂²×D₄×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₅₉	—	2	1	no	—
35	C₂²×C₈×C₅₉	—	2	1	no	—
36	C₂²×C₈×C₅₉	—	4	2	no	—
37	C₂²×C₈×C₅₉	—	4	2	no	—
38	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₉	D₄	1	2	yes	—
43	C₂²×C₈×C₅₉	D₄	1	2	yes	—
44	C₄×D₄×C₅₉	—	2	2	no	—
45	C₄×D₄×C₅₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₈×C₅₉	—	2	2	no	—
51	C₂²⋊C₈×C₅₉	—	2	2	no	—
52	C₂²⋊C₈×C₅₉	—	2	2	no	—
53	C₄×C₈×C₅₉	—	2	1	no	—
54	C₄×C₈×C₅₉	—	2	1	no	—
55	C₂⁴×C₅₉	C₄²	1	1	yes	characteristic
56	C₂⁴×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×D₄×C₅₉	—	2	2	no	—
58	C₂×D₄×C₅₉	—	2	2	no	—
59	C₂×D₄×C₅₉	C₂×C₈	1	4	yes	—
60	C₂²×C₄×C₅₉	—	2	1	no	—
61	C₂²×C₄×C₅₉	—	2	1	no	—
62	C₂²×C₄×C₅₉	—	4	1	no	—
63	C₂²×C₄×C₅₉	—	4	1	no	—
64	C₂²×C₄×C₅₉	—	4	1	no	—
65	C₂²×C₄×C₅₉	—	4	1	no	—
66	C₂²×C₄×C₅₉	C₄²	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₄²	1	1	yes	characteristic
68	C₂²×C₄×C₅₉	C₄²	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₅₉	#41	1	2	yes	—
73	C₂²×Dic₅₉	—	2	4	no	—
74	C₂×C₈×C₅₉	—	2	2	no	—
75	C₂×C₈×C₅₉	—	2	4	no	—
76	C₂×C₈×C₅₉	—	2	1	no	—
77	C₂×C₈×C₅₉	—	2	1	no	—
78	C₂×C₈×C₅₉	—	4	1	no	—
79	C₂×C₈×C₅₉	—	4	2	no	—
80	C₂×C₈×C₅₉	—	4	2	no	—
81	C₂×C₈×C₅₉	—	4	2	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	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₅₉	—	2	2	no	—
88	C₂²⋊C₄×C₅₉	—	2	2	no	—
89	C₂²⋊C₄×C₅₉	—	2	2	no	—
90	C₄²×C₅₉	—	2	1	no	—
91	C₄²×C₅₉	—	2	1	no	—
92	C₂²⋊Dic₅₉	—	2	4	no	—
93	C₂²⋊Dic₅₉	C₂×C₈	1	8	yes	complemented
94	C₄×Dic₅₉	—	4	2	no	—
95	C₄×Dic₅₉	—	4	2	no	—
96	C₂×C₅₉⋊C₈	—	2	4	no	—
97	C₂×C₅₉⋊C₈	—	4	2	no	—
98	C₂×C₅₉⋊C₈	—	4	2	no	—
99	C₂³×C₅₉	—	2	1	no	—
100	C₂³×C₅₉	—	2	1	no	—
101	C₂³×C₅₉	—	4	1	no	—
102	C₂³×C₅₉	—	4	1	no	—
103	C₂³×C₅₉	—	4	1	no	—
104	C₂³×C₅₉	—	4	1	no	—
105	C₂³×C₅₉	C₄×C₈	1	2	yes	complemented
106	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
107	C₂³×C₅₉	C₄×D₄	1	2	yes	—
108	C₂³×C₅₉	C₂²×C₈	1	2	yes	—
109	C₂³×C₅₉	#131	1	2	yes	—
110	D₄×C₅₉	—	2	8	no	—
111	C₂×C₄×C₅₉	—	2	1	no	—
112	C₂×C₄×C₅₉	—	2	1	no	—
113	C₂×C₄×C₅₉	—	2	2	no	—
114	C₂×C₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	—	2	2	no	—
116	C₂×C₄×C₅₉	—	2	1	no	—
117	C₂×C₄×C₅₉	—	2	1	no	—
118	C₂×C₄×C₅₉	—	2	1	no	—
119	C₂×C₄×C₅₉	—	4	1	no	—
120	C₂×C₄×C₅₉	—	4	2	no	—
121	C₂×C₄×C₅₉	—	4	2	no	—
122	C₂×C₄×C₅₉	—	4	1	no	—
123	C₂×C₄×C₅₉	—	4	1	no	—
124	C₂×C₄×C₅₉	—	4	1	no	—
125	C₂×C₄×C₅₉	—	4	1	no	—
126	C₂×C₄×C₅₉	C₄×C₈	1	2	yes	complemented
127	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
130	C₂×Dic₅₉	—	2	8	no	—
131	C₂×Dic₅₉	—	4	4	no	—
132	C₂×Dic₅₉	—	4	4	no	—
133	C₈×C₅₉	—	2	1	no	—
134	C₈×C₅₉	—	2	2	no	—
135	C₈×C₅₉	—	4	1	no	—
136	C₈×C₅₉	—	4	1	no	—
137	C₈×C₅₉	#99	1	2	yes	complemented
138	C₅₉⋊C₈	—	4	4	no	—
139	C₈×#99	—	59	1	no	maximal, Sylow
140	C₂²×C₅₉	—	2	2	no	—
141	C₂²×C₅₉	—	2	1	no	—
142	C₂²×C₅₉	—	2	2	no	—
143	C₂²×C₅₉	—	2	1	no	—
144	C₂²×C₅₉	—	2	2	no	—
145	C₂²×C₅₉	—	4	2	no	—
146	C₂²×C₅₉	—	4	2	no	—
147	C₂²×C₅₉	—	4	2	no	—
148	C₂²×C₅₉	—	4	2	no	—
149	C₂²×C₅₉	—	4	1	no	—
150	C₂²×C₅₉	—	4	1	no	—
151	C₂²×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
153	C₂²×C₅₉	#405	1	1	yes	characteristic
154	C₂²×C₅₉	C₈×D₄	1	4	yes	—
155	C₄×C₅₉	—	2	2	no	—
156	C₄×C₅₉	—	2	2	no	—
157	C₄×C₅₉	—	2	1	no	—
158	C₄×C₅₉	—	4	1	no	—
159	C₄×C₅₉	—	4	1	no	—
160	C₄×C₅₉	C₂×#99	1	1	yes	characteristic
161	C₄×C₅₉	#410	1	1	yes	characteristic
162	Dic₅₉	—	4	8	no	—
163	C₂×C₈×D₄	D₅₉	1	1	yes	characteristic
164	C₄×#99	—	59	1	no	—
165	C₈×C₂²⋊C₄	—	59	2	no	—
166	#1459	—	59	1	no	—
167	#1282	—	59	2	no	—
168	C₂×C₅₉	—	2	4	no	—
169	C₂×C₅₉	—	2	1	no	—
170	C₂×C₅₉	—	2	1	no	—
171	C₂×C₅₉	—	4	2	no	—
172	C₂×C₅₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₅₉	C₄×#99	1	1	yes	characteristic
174	C₂×C₅₉	#1757	1	1	yes	characteristic
175	C₂³×C₈	Dic₅₉	1	1	yes	characteristic, complemented
176	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
177	C₂×C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
178	C₈×D₄	—	2	2	no	—
179	C₈×D₄	—	2	2	no	—
180	C₂×C₄⋊C₈	Dic₅₉	1	1	yes	characteristic, complemented
181	C₂×#99	—	59	2	no	—
182	C₂×C₂²⋊C₈	—	59	2	no	—
183	C₂×C₂²⋊C₈	Dic₅₉	1	1	yes	characteristic, complemented
184	C₂×C₂²⋊C₈	C₂×D₅₉	1	1	yes	characteristic
185	C₂×C₄×C₈	—	118	2	no	—
186	C₂×C₄×C₈	Dic₅₉	1	1	yes	characteristic, complemented
187	C₄×C₂²⋊C₄	—	59	2	no	—
188	#342	—	59	2	no	—
189	#336	—	118	2	no	—
190	#324	—	59	2	no	—
191	#323	—	59	2	no	—
192	C₅₉	C₈×#99	1	1	yes	characteristic, complemented, Sylow
193	C₂²×D₄	C₄×D₅₉	1	1	yes	characteristic
194	C₂³×C₄	C₂×Dic₅₉	1	1	yes	characteristic
195	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
196	C₂²×C₈	—	2	1	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	—	4	2	no	—
199	C₂²×C₈	—	4	2	no	—
200	C₂²×C₈	—	118	2	no	—
201	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
202	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
203	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
204	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
205	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
206	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
207	C₄×D₄	—	2	2	no	—
208	C₄×D₄	—	2	2	no	—
209	C₂×C₄⋊C₄	C₄×D₅₉	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	—	59	4	no	—
211	C₂×C₂²⋊C₄	C₄×D₅₉	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	C₄×D₅₉	1	1	yes	characteristic
213	C₂×C₄²	—	118	2	no	—
214	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
215	C₄⋊C₈	—	2	2	no	—
216	#99	—	59	8	no	—
217	C₂²⋊C₈	—	2	2	no	—
218	C₂²⋊C₈	—	2	2	no	—
219	C₂²⋊C₈	—	59	4	no	—
220	C₂²⋊C₈	—	118	2	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	236	4	no	—
224	#95	—	118	2	no	—
225	C₂⁴	C₄×Dic₅₉	1	1	yes	characteristic
226	C₂⁴	C₂×C₄×D₅₉	1	1	yes	characteristic
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₈×D₅₉	1	4	yes	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	118	4	no	—
237	C₂²×C₄	C₄×Dic₅₉	1	1	yes	characteristic
238	C₂²×C₄	C₄×Dic₅₉	1	1	yes	characteristic
239	C₂²×C₄	C₄×Dic₅₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
241	C₂²×C₄	#11745685	1	2	yes	—
242	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	118	4	no	—
255	C₂×C₈	—	236	2	no	—
256	C₂×C₈	—	236	2	no	—
257	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
258	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
259	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₂²⋊C₄	—	2	2	no	—
263	C₂²⋊C₄	—	59	8	no	—
264	C₂²⋊C₄	—	118	4	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	236	2	no	—
268	C₄²	—	236	2	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	4	1	no	—
272	C₂³	—	4	1	no	—
273	C₂³	—	4	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	C₈×Dic₅₉	1	2	yes	complemented
276	C₂³	C₂×C₄×Dic₅₉	1	1	yes	characteristic
277	C₂³	C₄×C₂²⋊D₅₉	1	2	yes	—
278	C₂³	C₂×C₈×D₅₉	1	2	yes	—
279	C₂³	#14273733	1	2	yes	—
280	D₄	—	2	8	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	118	8	no	—
297	C₂×C₄	—	236	4	no	—
298	C₂×C₄	—	236	4	no	—
299	C₂×C₄	C₈×Dic₅₉	1	2	yes	complemented
300	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
301	C₂×C₄	#14273686	1	1	yes	characteristic
302	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
303	C₈	—	2	1	no	—
304	C₈	—	2	2	no	—
305	C₈	—	4	1	no	—
306	C₈	—	4	1	no	—
307	C₈	—	236	4	no	—
308	C₈	#14273642	1	2	yes	direct product factor
309	C₂²	—	2	1	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	C₄×C₂²⋊Dic₅₉	1	1	yes	characteristic
321	C₂²	C₂×C₈×Dic₅₉	1	1	yes	characteristic
322	C₂²	#16837357	1	1	yes	characteristic
323	C₂²	C₈×C₂²⋊D₅₉	1	4	yes	—
324	C₄	—	2	2	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	2	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	236	8	no	—
330	C₄	C₂×#14273642	1	1	yes	characteristic
331	C₄	#16837376	1	1	yes	characteristic
332	C₂	—	2	4	no	—
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	4	2	no	—
336	C₂	C₈×C₂²⋊Dic₅₉	1	1	yes	characteristic
337	C₂	C₄×#14273642	1	1	yes	characteristic
338	C₂	#21423896	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₅₉
commutator	C₂²×C₅₉	C₂×C₄×C₈
Frattini	C₂³×C₄	C₂²×D₅₉
Fitting	C₂×C₈×D₄×C₅₉	C₂
socle	C₂²×C₅₉	C₄×C₂²⋊C₄
prime radical	C₂²×D₄×C₅₉	C₂×C₄
cosocle	C₂³×C₄×C₅₉	C₂³
hypercenter	C₂×C₈×D₄	D₅₉
hypocenter	C₅₉	C₈×#99
LCS	G ▷ C₂²×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₈×C₂²⋊Dic₅₉	C₂	semidirect
4. #21421943	C₂	semidirect
12. #16837128	C₄	semidirect
13. #16837126	C₄	semidirect
14. C₂³×C₈×C₅₉	C₄	semidirect
19. C₂×C₄⋊C₈×C₅₉	C₄	semidirect
20. C₂×C₂²⋊C₈×C₅₉	C₄	semidirect
22. C₂×C₄×C₈×C₅₉	C₄	semidirect
26. #14273642	C₈	direct (N × H)
93. C₂²⋊Dic₅₉	C₂×C₈	semidirect
105. C₂³×C₅₉	C₄×C₈	semidirect
126. C₂×C₄×C₅₉	C₄×C₈	semidirect
137. C₈×C₅₉	#99	semidirect
175. C₂³×C₈	Dic₅₉	semidirect
180. C₂×C₄⋊C₈	Dic₅₉	semidirect
183. C₂×C₂²⋊C₈	Dic₅₉	semidirect
186. C₂×C₄×C₈	Dic₅₉	semidirect
192. C₅₉	C₈×#99	semidirect
275. C₂³	C₈×Dic₅₉	semidirect
299. C₂×C₄	C₈×Dic₅₉	semidirect
308. C₈	#14273642	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#14273642	2
C₈×C₂²⋊Dic₅₉	2
#21422414	2
#21421943	2
C₂×C₈×D₄×C₅₉	2
C₈×#99	59

Subgroups — C8×#14273642 #28479230

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14273642 #28479230