Subgroups — C₂×#334×D₅₉ #28557653

Order 15104.

942 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₈×D₅₉	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₅₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837153	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16837152	C₂	1	2	yes	maximal, complemented
5	#334×D₅₉	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅₉×#334	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₅₉	—	2	2	no	—
8	C₂²×C₈×D₅₉	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₅₉	C₄	1	1	yes	characteristic
10	C₂×C₄²×D₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
12	C₂×Dic₅₉⋊C₈	—	2	2	no	—
13	C₄⋊C₈×D₅₉	—	2	4	no	—
14	C₄⋊C₈×D₅₉	C₂²	1	8	yes	—
15	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₅₉	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₅₉	C₄	1	1	yes	characteristic
18	C₄×C₈×D₅₉	C₄	1	4	yes	—
19	C₄×C₈×D₅₉	C₂²	1	4	yes	—
20	#16837153	C₂²	1	4	yes	complemented
21	#16837152	C₂²	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₅₉×#334	C₂²	1	4	yes	complemented
25	C₂²×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₅₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₅₉	Q₈	1	1	yes	characteristic
28	C₂²×C₅₉⋊C₈	—	2	2	no	—
29	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₅₉	—	2	4	no	—
32	C₂×C₈×D₅₉	—	2	8	no	—
33	C₂×C₈×D₅₉	—	4	8	no	—
34	C₂×C₈×D₅₉	C₈	1	8	yes	complemented
35	C₂×C₈×D₅₉	C₂×C₄	1	4	yes	—
36	C₂×C₈×D₅₉	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
39	C₂×C₄×Dic₅₉	Q₈	1	1	yes	characteristic
40	C₄²×D₅₉	C₂×C₄	1	2	yes	—
41	C₄²×D₅₉	D₄	1	2	yes	—
42	C₄²×D₅₉	Q₈	1	2	yes	—
43	C₄²×D₅₉	C₂³	1	2	yes	—
44	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
45	C₄⋊(C₅₉⋊C₈)	C₂³	1	4	yes	—
46	Dic₅₉⋊C₈	—	2	8	no	—
47	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
48	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
49	C₈×Dic₅₉	C₂×C₄	1	4	yes	—
50	C₈×Dic₅₉	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₅₉	—	2	2	no	—
52	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₉	—	2	2	no	—
56	C₄⋊C₈×C₅₉	C₂³	1	4	yes	—
57	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₅₉	C₂³	1	2	yes	—
59	C₂³×D₅₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
63	C₂²×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×D₅₉	—	2	4	no	—
67	C₂×C₄×D₅₉	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×D₅₉	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×D₅₉	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×D₅₉	C₂×C₈	1	4	yes	—
71	C₂×C₄×D₅₉	M₁₆	1	4	yes	—
72	C₂×C₄×D₅₉	SD₁₆	1	8	yes	—
73	C₂×C₄×D₅₉	C₂²×C₄	1	2	yes	—
74	C₂×C₄×D₅₉	C₂×D₄	1	2	yes	—
75	C₂×C₄×D₅₉	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₅₉	—	2	2	no	—
77	C₂×C₈×C₅₉	—	2	4	no	—
78	C₂×C₈×C₅₉	—	4	4	no	—
79	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	complemented
80	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
82	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
86	C₄×Dic₅₉	C₄⋊C₄	1	2	yes	—
87	C₄×Dic₅₉	C₄⋊C₄	1	2	yes	—
88	C₄×Dic₅₉	C₄⋊C₄	1	2	yes	—
89	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
90	C₄×Dic₅₉	C₂×D₄	1	2	yes	—
91	C₄×Dic₅₉	C₂×Q₈	1	2	yes	—
92	C₂×C₅₉⋊C₈	—	2	2	no	—
93	C₂×C₅₉⋊C₈	—	2	4	no	—
94	C₂×C₅₉⋊C₈	—	4	4	no	—
95	C₂×C₅₉⋊C₈	C₂×C₈	1	4	yes	complemented
96	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
98	C₈×D₅₉	—	2	8	no	—
99	C₈×D₅₉	—	4	16	no	—
100	C₈×D₅₉	C₂×C₈	1	16	yes	complemented
101	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×D₅₉	C₄⋊C₈	1	4	yes	—
103	C₂²×D₅₉	#106	1	4	yes	—
104	C₂²×D₅₉	#108	1	4	yes	—
105	C₂²×D₅₉	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₅₉	—	2	2	no	—
107	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
112	C₂×C₄×C₅₉	C₂×SD₁₆	1	4	yes	—
113	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
116	C₂×Dic₅₉	—	2	2	no	—
117	C₂×Dic₅₉	C₄⋊C₈	1	4	yes	—
118	C₂×Dic₅₉	#106	1	4	yes	—
119	C₂×Dic₅₉	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×Dic₅₉	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×Dic₅₉	C₂×C₄⋊C₄	1	2	yes	—
122	C₄×D₅₉	—	2	4	no	—
123	C₄×D₅₉	—	2	4	no	—
124	C₄×D₅₉	C₄⋊C₈	1	8	yes	—
125	C₄×D₅₉	#106	1	8	yes	—
126	C₄×D₅₉	C₂²×C₈	1	4	yes	—
127	C₄×D₅₉	C₂×M₁₆	1	4	yes	—
128	C₄×D₅₉	C₂×SD₁₆	1	8	yes	—
129	C₈×C₅₉	—	2	2	no	—
130	C₈×C₅₉	—	4	4	no	—
131	C₈×C₅₉	C₂²×C₈	1	4	yes	complemented
132	C₅₉⋊C₈	—	2	2	no	—
133	C₅₉⋊C₈	—	4	4	no	—
134	C₅₉⋊C₈	C₂²×C₈	1	4	yes	complemented
135	C₂²×#334	—	59	1	no	maximal, Sylow
136	C₂²×C₅₉	C₂×C₄⋊C₈	1	2	yes	—
137	C₂²×C₅₉	C₂×#106	1	2	yes	—
138	C₂²×C₅₉	C₂×#108	1	2	yes	—
139	C₂²×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₂×D₅₉	#334	1	16	yes	direct product factor
141	C₂×D₅₉	C₂×C₄⋊C₈	1	4	yes	—
142	C₂×D₅₉	C₂×#106	1	4	yes	—
143	C₂×D₅₉	C₂×#108	1	4	yes	—
144	C₄×C₅₉	—	2	1	no	—
145	C₄×C₅₉	—	2	1	no	—
146	C₄×C₅₉	C₂×C₄⋊C₈	1	2	yes	—
147	C₄×C₅₉	C₂×#106	1	2	yes	—
148	C₄×C₅₉	C₂³×C₈	1	1	yes	characteristic
149	C₄×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
150	C₄×C₅₉	C₂²×SD₁₆	1	2	yes	—
151	Dic₅₉	—	2	2	no	—
152	Dic₅₉	C₂×C₄⋊C₈	1	4	yes	—
153	Dic₅₉	C₂×#106	1	4	yes	—
154	C₂²×C₄⋊C₈	—	59	2	no	—
155	C₂²×C₄×C₈	—	59	1	no	—
156	C₂×#334	—	59	2	no	—
157	C₂×#334	—	59	1	no	—
158	C₂×#334	—	59	8	no	—
159	C₂×#334	D₅₉	1	1	yes	characteristic, direct product factor
160	C₂×C₅₉	C₂×#334	1	4	yes	complemented
161	C₂×C₅₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
162	C₂×C₅₉	C₂²×#106	1	1	yes	characteristic
163	C₂×C₅₉	C₂²×#108	1	1	yes	characteristic
164	D₅₉	C₂×#334	1	8	yes	direct product factor
165	C₂³×C₈	—	59	1	no	—
166	C₂³×C₈	—	59	1	no	—
167	C₂³×C₈	—	118	2	no	—
168	C₂²×C₄²	—	59	1	no	—
169	C₂×C₄⋊C₈	—	59	2	no	—
170	C₂×C₄⋊C₈	—	59	8	no	—
171	C₂×C₄⋊C₈	—	118	4	no	—
172	C₂×C₄⋊C₈	—	118	2	no	—
173	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
174	C₂×C₄×C₈	—	59	1	no	—
175	C₂×C₄×C₈	—	59	1	no	—
176	C₂×C₄×C₈	—	59	1	no	—
177	C₂×C₄×C₈	—	59	4	no	—
178	C₂×C₄×C₈	—	59	4	no	—
179	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
180	#334	—	59	8	no	—
181	#334	—	59	4	no	—
182	#334	C₂×D₅₉	1	4	yes	direct product factor
183	C₅₉	C₂²×#334	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	—	59	1	no	—
185	C₂³×C₄	—	59	1	no	—
186	C₂³×C₄	—	59	1	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	59	1	no	—
189	C₂²×C₈	—	59	8	no	—
190	C₂²×C₈	—	59	4	no	—
191	C₂²×C₈	—	59	1	no	—
192	C₂²×C₈	—	59	4	no	—
193	C₂²×C₈	—	118	4	no	—
194	C₂²×C₈	—	118	8	no	—
195	C₂²×C₈	—	118	2	no	—
196	C₂²×C₈	—	236	8	no	—
197	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
198	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
199	C₂×C₄²	—	59	1	no	—
200	C₂×C₄²	—	59	1	no	—
201	C₂×C₄²	—	59	1	no	—
202	C₂×C₄²	—	59	2	no	—
203	C₂×C₄²	—	59	2	no	—
204	C₂×C₄²	—	59	2	no	—
205	C₂×C₄²	—	59	2	no	—
206	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
207	C₄⋊C₈	—	2	2	no	—
208	C₄⋊C₈	—	59	4	no	—
209	C₄⋊C₈	—	118	8	no	—
210	C₄⋊C₈	—	118	2	no	—
211	C₄⋊C₈	C₂²×D₅₉	1	4	yes	—
212	C₄×C₈	—	59	4	no	—
213	C₄×C₈	—	59	4	no	—
214	C₄×C₈	—	59	2	no	—
215	C₄×C₈	—	59	2	no	—
216	C₄×C₈	C₄×D₅₉	1	2	yes	—
217	C₄×C₈	C₂²×D₅₉	1	2	yes	—
218	C₂⁴	—	59	1	no	—
219	C₂²×C₄	—	59	1	no	—
220	C₂²×C₄	—	59	1	no	—
221	C₂²×C₄	—	59	2	no	—
222	C₂²×C₄	—	59	2	no	—
223	C₂²×C₄	—	59	4	no	—
224	C₂²×C₄	—	59	4	no	—
225	C₂²×C₄	—	59	8	no	—
226	C₂²×C₄	—	59	2	no	—
227	C₂²×C₄	—	59	2	no	—
228	C₂²×C₄	—	59	1	no	—
229	C₂²×C₄	—	59	2	no	—
230	C₂²×C₄	—	59	2	no	—
231	C₂²×C₄	—	118	4	no	—
232	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
233	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
234	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	4	4	no	—
238	C₂×C₈	—	59	4	no	—
239	C₂×C₈	—	59	2	no	—
240	C₂×C₈	—	59	16	no	—
241	C₂×C₈	—	59	2	no	—
242	C₂×C₈	—	118	8	no	—
243	C₂×C₈	—	118	2	no	—
244	C₂×C₈	—	118	4	no	—
245	C₂×C₈	—	236	16	no	—
246	C₂×C₈	—	236	4	no	—
247	C₂×C₈	C₈×D₅₉	1	4	yes	complemented
248	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
249	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
250	C₄²	—	59	2	no	—
251	C₄²	—	59	2	no	—
252	C₄²	—	59	2	no	—
253	C₄²	—	59	2	no	—
254	C₄²	—	59	2	no	—
255	C₄²	—	59	2	no	—
256	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
257	C₄²	D₄×D₅₉	1	1	yes	characteristic
258	C₄²	Q₈×D₅₉	1	1	yes	characteristic
259	C₄²	C₂³×D₅₉	1	1	yes	characteristic
260	C₂³	—	59	4	no	—
261	C₂³	—	59	4	no	—
262	C₂³	—	59	4	no	—
263	C₂³	—	59	2	no	—
264	C₂³	C₄⋊C₄×D₅₉	1	1	yes	characteristic
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	59	4	no	—
267	C₂×C₄	—	59	4	no	—
268	C₂×C₄	—	59	2	no	—
269	C₂×C₄	—	59	2	no	—
270	C₂×C₄	—	59	4	no	—
271	C₂×C₄	—	59	4	no	—
272	C₂×C₄	—	59	8	no	—
273	C₂×C₄	—	59	8	no	—
274	C₂×C₄	—	59	8	no	—
275	C₂×C₄	—	59	2	no	—
276	C₂×C₄	—	118	4	no	—
277	C₂×C₄	—	118	4	no	—
278	C₂×C₄	—	118	2	no	—
279	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
280	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
281	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
282	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
283	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
284	C₂×C₄	SD₁₆×D₅₉	1	4	yes	—
285	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
286	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
287	C₂×C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
288	C₈	—	2	2	no	—
289	C₈	—	4	4	no	—
290	C₈	—	59	4	no	—
291	C₈	—	118	2	no	—
292	C₈	—	236	4	no	—
293	C₈	C₂×C₈×D₅₉	1	4	yes	complemented
294	C₂²	—	59	4	no	—
295	C₂²	—	59	4	no	—
296	C₂²	—	59	4	no	—
297	C₂²	—	59	16	no	—
298	C₂²	C₄⋊C₈×D₅₉	1	2	yes	—
299	C₂²	#106×D₅₉	1	2	yes	—
300	C₂²	#108×D₅₉	1	2	yes	—
301	C₂²	C₂×C₄⋊C₄×D₅₉	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	59	4	no	—
305	C₄	—	59	4	no	—
306	C₄	—	118	2	no	—
307	C₄	C₄⋊C₈×D₅₉	1	2	yes	—
308	C₄	#106×D₅₉	1	2	yes	—
309	C₄	C₂²×C₈×D₅₉	1	1	yes	characteristic
310	C₄	C₂×M₁₆×D₅₉	1	1	yes	characteristic
311	C₄	C₂×SD₁₆×D₅₉	1	2	yes	—
312	C₂	—	59	8	no	—
313	C₂	#334×D₅₉	1	4	yes	direct product factor
314	C₂	C₂×C₄⋊C₈×D₅₉	1	1	yes	characteristic
315	C₂	C₂×#106×D₅₉	1	1	yes	characteristic
316	C₂	C₂×#108×D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₉
commutator	C₄×C₅₉	C₂³×C₈
Frattini	C₄²	C₂³×D₅₉
Fitting	C₂×C₅₉×#334	C₂
socle	C₂³×C₅₉	C₂×C₄⋊C₄
prime radical	C₂³×D₅₉	C₄⋊C₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	C₂×#334	D₅₉
hypocenter	C₅₉	C₂²×#334
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. C₂×#16837153	C₂	semidirect
4. C₂×#16837152	C₂	semidirect
5. #334×D₅₉	C₂	direct (N × H)
6. C₂×C₅₉×#334	C₂	semidirect
20. #16837153	C₂²	semidirect
21. #16837152	C₂²	semidirect
24. C₅₉×#334	C₂²	semidirect
34. C₂×C₈×D₅₉	C₈	semidirect
79. C₂×C₈×C₅₉	C₂×C₈	semidirect
95. C₂×C₅₉⋊C₈	C₂×C₈	semidirect
100. C₈×D₅₉	C₂×C₈	semidirect
131. C₈×C₅₉	C₂²×C₈	semidirect
134. C₅₉⋊C₈	C₂²×C₈	semidirect
140. C₂×D₅₉	#334	direct (N × H)
159. C₂×#334	D₅₉	direct (N × H)
160. C₂×C₅₉	C₂×#334	semidirect
164. D₅₉	C₂×#334	direct (N × H)
182. #334	C₂×D₅₉	direct (N × H)
183. C₅₉	C₂²×#334	semidirect
247. C₂×C₈	C₈×D₅₉	semidirect
293. C₈	C₂×C₈×D₅₉	semidirect
313. C₂	#334×D₅₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₈×D₅₉	2
C₂×C₄×C₈×D₅₉	2
C₂×#16837153	2
C₂×#16837152	2
#334×D₅₉	2
C₂×C₅₉×#334	2
C₂²×#334	59

Subgroups — C2×#334×D59 #28557653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#334×D₅₉ #28557653