Subgroups — C₂×#21422391 #28554307

Order 15104.

866 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837455	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11745664	C₂	1	1	yes	maximal, characteristic
3	#21422391	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837124	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16837123	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₉×#431	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11745664	C₄	1	2	yes	—
10	#16837455	C₄	1	4	yes	—
11	#16837455	C₂²	1	4	yes	—
12	C₂×C₄⋊(C₅₉⋊C₈)	C₄	1	2	yes	—
13	C₂×C₂²⋊(C₅₉⋊C₈)	C₄	1	2	yes	—
14	C₄×#11745664	C₄	1	4	yes	—
15	C₄×#11745664	C₂²	1	4	yes	—
16	C₂×#14273638	C₄	1	2	yes	—
17	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
21	#16837124	C₂²	1	4	yes	complemented
22	#16837124	C₂²	1	4	yes	complemented
23	#16837123	C₂²	1	4	yes	complemented
24	C₈×C₅₉⋊C₈	C₂²	1	4	yes	complemented
25	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
26	C₅₉×#431	C₄	1	4	yes	—
27	C₅₉×#431	C₂²	1	4	yes	—
28	C₂×C₄⋊C₈×C₅₉	C₄	1	2	yes	—
29	C₂×C₂²⋊C₈×C₅₉	C₄	1	2	yes	—
30	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
32	C₂×#11745664	—	2	8	no	—
33	C₂×#11745664	C₂×C₄	1	8	yes	—
34	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
37	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
38	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	8	yes	—
39	C₂²⋊(C₅₉⋊C₈)	C₂×C₄	1	8	yes	—
40	#14273638	C₂×C₄	1	8	yes	—
41	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
42	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
43	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
44	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
45	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
46	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
47	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
48	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
49	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
61	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₅₉	C₂×C₄	1	8	yes	—
63	C₂²⋊C₈×C₅₉	C₂×C₄	1	8	yes	—
64	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₅₉	C₂³	1	2	yes	—
67	C₄×C₈×C₅₉	C₂³	1	2	yes	—
68	C₂⁴×C₅₉	C₄²	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	—	2	4	no	—
70	C₂²×C₄×C₅₉	—	2	4	no	—
71	C₂²×C₄×C₅₉	—	2	4	no	—
72	C₂²×C₄×C₅₉	C₄²	1	1	yes	characteristic
73	C₂²×C₄×C₅₉	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₅₉	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₅₉	C₄²	1	2	yes	—
76	C₂²×C₄×C₅₉	C₄²	1	2	yes	—
77	C₂²×C₄×C₅₉	C₄²	1	2	yes	—
78	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₉	—	2	4	no	—
85	C₂×C₈×C₅₉	—	2	4	no	—
86	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
88	C₄²×C₅₉	C₄²	1	2	yes	—
89	C₄²×C₅₉	C₄²	1	2	yes	—
90	C₄²×C₅₉	C₄²	1	2	yes	—
91	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
94	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
95	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
97	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
98	#11745664	—	2	16	no	—
99	C₂×C₅₉⋊C₈	—	2	4	no	—
100	C₂×C₅₉⋊C₈	—	2	4	no	—
101	C₂×C₅₉⋊C₈	—	2	4	no	—
102	C₂×C₅₉⋊C₈	M₁₆	1	8	yes	—
103	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
104	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
105	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
106	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
107	C₂³×C₅₉	—	2	2	no	—
108	C₂³×C₅₉	—	2	2	no	—
109	C₂³×C₅₉	—	2	2	no	—
110	C₂³×C₅₉	C₂×C₄²	1	2	yes	—
111	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₅₉	—	2	2	no	—
113	C₂×C₄×C₅₉	—	2	2	no	—
114	C₂×C₄×C₅₉	—	2	2	no	—
115	C₂×C₄×C₅₉	—	2	4	no	—
116	C₂×C₄×C₅₉	—	2	4	no	—
117	C₂×C₄×C₅₉	—	2	4	no	—
118	C₂×C₄×C₅₉	—	2	4	no	—
119	C₂×C₄×C₅₉	—	2	4	no	—
120	C₂×C₄×C₅₉	—	2	4	no	—
121	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂×M₁₆	1	4	yes	—
128	C₂×C₄×C₅₉	#131	1	4	yes	—
129	C₂×C₄×C₅₉	#131	1	4	yes	—
130	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
133	C₈×C₅₉	—	2	4	no	—
134	C₈×C₅₉	—	2	4	no	—
135	C₅₉⋊C₈	—	2	4	no	—
136	C₅₉⋊C₈	—	2	4	no	—
137	C₅₉⋊C₈	—	2	4	no	—
138	C₅₉⋊C₈	C₂×M₁₆	1	8	yes	—
139	C₂×#1453	—	59	1	no	maximal, Sylow
140	C₂²×C₅₉	—	2	2	no	—
141	C₂²×C₅₉	—	2	2	no	—
142	C₂²×C₅₉	—	2	2	no	—
143	C₂²×C₅₉	—	4	4	no	—
144	C₂²×C₅₉	C₄×M₁₆	1	2	yes	—
145	C₂²×C₅₉	#405	1	2	yes	—
146	C₂²×C₅₉	#405	1	2	yes	—
147	C₂²×C₅₉	C₂²×C₄²	1	1	yes	characteristic
148	C₄×C₅₉	—	2	2	no	—
149	C₄×C₅₉	—	2	2	no	—
150	C₄×C₅₉	—	2	2	no	—
151	C₄×C₅₉	C₄×M₁₆	1	2	yes	—
152	C₄×C₅₉	#405	1	2	yes	—
153	C₄×C₅₉	#405	1	2	yes	—
154	C₄×C₅₉	C₂²×M₁₆	1	2	yes	—
155	C₄×C₅₉	C₂×#131	1	2	yes	—
156	C₄×C₅₉	C₂×#131	1	2	yes	—
157	C₂×#431	—	59	1	no	—
158	C₂×#431	D₅₉	1	1	yes	characteristic
159	C₂×C₄×M₁₆	—	59	1	no	—
160	#1453	—	59	8	no	—
161	C₂×#322	—	59	1	no	—
162	C₂×#322	—	59	1	no	—
163	C₂×#322	—	59	1	no	—
164	C₂×C₈²	—	59	1	no	—
165	C₂×C₅₉	—	4	2	no	—
166	C₂×C₅₉	#1453	1	4	yes	complemented
167	C₂×C₅₉	C₂×C₄×M₁₆	1	1	yes	characteristic
168	C₂×C₅₉	C₂×#405	1	1	yes	characteristic
169	C₂×C₅₉	C₂×#405	1	1	yes	characteristic
170	C₂²×M₁₆	—	59	2	no	—
171	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
172	#431	—	59	4	no	—
173	#431	—	59	4	no	—
174	#431	Dic₅₉	1	4	yes	—
175	#431	C₂×D₅₉	1	4	yes	—
176	C₂×C₄⋊C₈	—	59	2	no	—
177	C₂×C₄⋊C₈	Dic₅₉	1	2	yes	—
178	C₂×C₂²⋊C₈	—	59	2	no	—
179	C₂×C₂²⋊C₈	Dic₅₉	1	2	yes	—
180	C₄×M₁₆	—	59	4	no	—
181	C₄×M₁₆	—	59	4	no	—
182	C₂×#97	—	59	2	no	—
183	C₂×C₄×C₈	—	59	1	no	—
184	C₂×C₄×C₈	—	59	1	no	—
185	C₂×C₄×C₈	—	59	1	no	—
186	C₂×C₄×C₈	—	59	1	no	—
187	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
188	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
189	#322	—	59	4	no	—
190	#322	—	59	4	no	—
191	#322	—	59	4	no	—
192	C₈²	—	59	4	no	—
193	C₅₉	C₂×#1453	1	1	yes	characteristic, complemented, Sylow
194	C₂³×C₄	C₄×D₅₉	1	1	yes	characteristic
195	C₂³×C₄	C₄×D₅₉	1	1	yes	characteristic
196	C₂³×C₄	C₂×Dic₅₉	1	1	yes	characteristic
197	C₂×M₁₆	—	59	8	no	—
198	C₂×M₁₆	—	118	8	no	—
199	C₂²×C₈	—	59	2	no	—
200	C₂²×C₈	—	59	2	no	—
201	C₂²×C₈	—	59	2	no	—
202	C₂²×C₈	—	59	2	no	—
203	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
204	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
205	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
206	C₂×C₄²	C₄×D₅₉	1	2	yes	—
207	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
208	C₂×C₄²	C₄×D₅₉	1	2	yes	—
209	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
210	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
211	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
212	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
213	C₄⋊C₈	—	59	8	no	—
214	C₄⋊C₈	C₂×Dic₅₉	1	8	yes	—
215	C₂²⋊C₈	—	59	8	no	—
216	C₂²⋊C₈	C₂×Dic₅₉	1	8	yes	—
217	#97	—	59	8	no	—
218	C₄×C₈	—	59	2	no	—
219	C₄×C₈	—	59	2	no	—
220	C₄×C₈	—	59	2	no	—
221	C₄×C₈	—	59	2	no	—
222	C₄×C₈	—	59	2	no	—
223	C₄×C₈	—	59	2	no	—
224	C₄×C₈	—	59	2	no	—
225	C₄×C₈	—	59	2	no	—
226	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
227	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
228	C₄×C₈	C₂²×D₅₉	1	2	yes	—
229	C₄×C₈	C₂²×D₅₉	1	2	yes	—
230	C₂⁴	C₄×Dic₅₉	1	1	yes	characteristic
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	C₄×Dic₅₉	1	1	yes	characteristic
235	C₂²×C₄	C₄×Dic₅₉	1	2	yes	—
236	C₂²×C₄	C₄×Dic₅₉	1	1	yes	characteristic
237	C₂²×C₄	C₄×Dic₅₉	1	1	yes	characteristic
238	C₂²×C₄	C₄×Dic₅₉	1	2	yes	—
239	C₂²×C₄	C₄×Dic₅₉	1	2	yes	—
240	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
241	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
242	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
244	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
245	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
246	M₁₆	—	118	16	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	59	8	no	—
250	C₂×C₈	—	59	4	no	—
251	C₂×C₈	—	59	4	no	—
252	C₂×C₈	—	59	4	no	—
253	C₂×C₈	—	59	4	no	—
254	C₂×C₈	—	118	4	no	—
255	C₂×C₈	—	118	4	no	—
256	C₂×C₈	—	118	4	no	—
257	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
258	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
259	C₄²	C₄×Dic₅₉	1	2	yes	—
260	C₄²	C₄×Dic₅₉	1	2	yes	—
261	C₄²	C₄×Dic₅₉	1	2	yes	—
262	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
263	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
264	C₄²	C₂×C₄×D₅₉	1	2	yes	—
265	C₄²	C₂×C₄×D₅₉	1	2	yes	—
266	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
267	C₄²	C₂²×Dic₅₉	1	2	yes	—
268	C₄²	C₂³×D₅₉	1	1	yes	characteristic
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	C₂×C₄×Dic₅₉	1	2	yes	—
273	C₂³	C₂×C₄×Dic₅₉	1	1	yes	characteristic
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	C₂×C₄×Dic₅₉	1	2	yes	—
284	C₂×C₄	C₂×C₄×Dic₅₉	1	2	yes	—
285	C₂×C₄	C₂×C₄×Dic₅₉	1	2	yes	—
286	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
289	C₂×C₄	M₁₆×D₅₉	1	4	yes	—
290	C₂×C₄	#14273733	1	4	yes	—
291	C₂×C₄	#14273734	1	4	yes	—
292	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
294	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	2	4	no	—
297	C₈	—	59	8	no	—
298	C₈	—	118	4	no	—
299	C₈	—	118	4	no	—
300	C₈	—	118	4	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	4	no	—
305	C₂²	M₁₆×Dic₅₉	1	2	yes	—
306	C₂²	#16837357	1	2	yes	—
307	C₂²	#16837358	1	2	yes	—
308	C₂²	C₂²×C₄×Dic₅₉	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	M₁₆×Dic₅₉	1	2	yes	—
313	C₄	#16837357	1	2	yes	—
314	C₄	#16837358	1	2	yes	—
315	C₄	C₂×M₁₆×D₅₉	1	2	yes	—
316	C₄	C₂×#14273733	1	2	yes	—
317	C₄	C₂×#14273734	1	2	yes	—
318	C₂	—	4	2	no	—
319	C₂	#21422391	1	4	yes	direct product factor
320	C₂	C₂×M₁₆×Dic₅₉	1	1	yes	characteristic
321	C₂	C₂×#16837357	1	1	yes	characteristic
322	C₂	C₂×#16837358	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂²×C₄²
Frattini	C₄²	C₂³×D₅₉
Fitting	C₂×C₅₉×#431	C₂
socle	C₂³×C₅₉	C₂×C₄²
prime radical	C₂⁴×C₅₉	C₄²
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	C₂×#431	D₅₉
hypocenter	C₅₉	C₂×#1453
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #21422391	C₂	direct (N × H)
4. C₂×#16837124	C₂	semidirect
5. C₂×#16837124	C₂	semidirect
6. C₂×#16837123	C₂	semidirect
7. C₂×C₈×C₅₉⋊C₈	C₂	semidirect
21. #16837124	C₂²	semidirect
22. #16837124	C₂²	semidirect
23. #16837123	C₂²	semidirect
24. C₈×C₅₉⋊C₈	C₂²	semidirect
166. C₂×C₅₉	#1453	semidirect
193. C₅₉	C₂×#1453	semidirect
319. C₂	#21422391	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×#16837455	2
C₂×C₄×#11745664	2
#21422391	2
C₂×#16837124	2
C₂×#16837123	2
C₂×C₈×C₅₉⋊C₈	2
C₂×C₅₉×#431	2
C₂×#1453	59

Subgroups — C2×#21422391 #28554307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422391 #28554307