Subgroups — C₂×#21423482 #28570943

Order 15104.

790 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837604	C₂	1	2	yes	maximal
2	C₂×C₈×D₂₃₆	C₂	1	1	yes	maximal, characteristic
3	#21423482	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837155	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837148	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₉×#500	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11745671	—	2	2	no	—
8	C₂²×C₈×D₅₉	—	2	1	no	—
9	C₂×C₄×D₂₃₆	C₂²	1	1	yes	characteristic
10	#16837604	—	2	4	no	—
11	#16837604	C₂²	1	8	yes	—
12	C₈×D₂₃₆	—	2	2	no	—
13	C₈×D₂₃₆	C₂²	1	4	yes	—
14	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
16	C₂×#14273651	—	2	2	no	—
17	C₂×(C₂×D₅₉)⋊C₈	—	2	1	no	—
18	#16837155	C₂²	1	4	yes	complemented
19	#16837148	C₂²	1	8	yes	complemented
20	C₂×C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
21	C₅₉×#500	—	2	2	no	—
22	C₅₉×#500	C₂²	1	4	yes	complemented
23	C₂×#107×C₅₉	—	2	1	no	—
24	C₂×#107×C₅₉	—	2	1	no	—
25	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₂²×D₂₃₆	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₅₉	—	2	1	no	—
28	C₂×#11745671	—	2	8	no	—
29	C₂×#11745671	—	4	8	no	—
30	C₂²×C₅₉⋊C₈	—	2	1	no	—
31	C₂²×C₅₉⋊C₈	—	2	2	no	—
32	C₂×C₈×D₅₉	—	2	4	no	—
33	C₂×C₈×D₅₉	—	4	4	no	—
34	C₄×D₂₃₆	—	2	1	no	—
35	C₄×D₂₃₆	—	2	1	no	—
36	C₄×D₂₃₆	D₄	1	2	yes	—
37	C₄×D₂₃₆	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
39	C₂×(C₂×D₅₉)⋊C₄	—	2	1	no	—
40	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
41	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
42	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
43	C₄⋊(C₅₉⋊C₈)	C₂³	1	4	yes	—
44	#14273651	—	2	8	no	—
45	(C₂×D₅₉)⋊C₈	—	2	4	no	—
46	C₂²×Q₈×C₅₉	—	2	2	no	—
47	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
49	C₄⋊Q₈×C₅₉	—	2	2	no	—
50	C₄⋊Q₈×C₅₉	—	2	2	no	—
51	C₄⋊Q₈×C₅₉	—	2	2	no	—
52	C₄⋊Q₈×C₅₉	C₂³	1	4	yes	—
53	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
54	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
55	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
56	#107×C₅₉	—	2	4	no	—
57	#107×C₅₉	—	2	4	no	—
58	C₄×C₈×C₅₉	D₄	1	2	yes	—
59	C₄×C₈×C₅₉	C₂³	1	2	yes	—
60	C₂³×D₅₉	—	2	1	no	—
61	C₂×Q₈×C₅₉	—	2	4	no	—
62	C₂×Q₈×C₅₉	—	4	2	no	—
63	C₂×Q₈×C₅₉	—	4	4	no	—
64	C₂×Q₈×C₅₉	—	4	4	no	—
65	C₂²×C₄×C₅₉	—	4	2	no	—
66	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
69	C₂²×Dic₅₉	—	2	1	no	—
70	C₂×D₂₃₆	—	2	1	no	—
71	C₂×D₂₃₆	—	2	2	no	—
72	C₂×D₂₃₆	D₈	1	4	yes	—
73	C₂×D₂₃₆	C₂×D₄	1	2	yes	—
74	C₂×C₄×D₅₉	—	2	2	no	—
75	C₂×C₄×D₅₉	—	2	2	no	—
76	C₂×C₄×D₅₉	—	4	2	no	—
77	C₂×C₄×D₅₉	—	4	2	no	—
78	C₂×C₈×C₅₉	—	2	2	no	—
79	C₂×C₈×C₅₉	SD₁₆	1	4	yes	complemented
80	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
82	C₄⋊C₄×C₅₉	—	2	4	no	—
83	C₄⋊C₄×C₅₉	—	2	4	no	—
84	C₄⋊C₄×C₅₉	—	4	2	no	—
85	C₄⋊C₄×C₅₉	—	4	2	no	—
86	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₅₉	#41	1	1	yes	characteristic
89	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
90	#11745671	—	4	16	no	—
91	(C₂×D₅₉)⋊C₄	—	2	2	no	—
92	(C₂×D₅₉)⋊C₄	—	2	2	no	—
93	C₄⋊Dic₅₉	—	2	1	no	—
94	C₄⋊Dic₅₉	C₂×D₄	1	2	yes	—
95	C₂×C₅₉⋊C₈	—	2	2	no	—
96	C₂×C₅₉⋊C₈	—	2	4	no	—
97	C₂×C₅₉⋊C₈	—	4	4	no	—
98	C₂×C₅₉⋊C₈	—	4	2	no	—
99	C₈×D₅₉	—	4	8	no	—
100	C₂³×C₅₉	#121	1	1	yes	characteristic
101	C₂²×D₅₉	—	2	2	no	—
102	C₂²×D₅₉	—	2	4	no	—
103	C₂²×D₅₉	—	4	2	no	—
104	C₂²×D₅₉	—	4	2	no	—
105	Q₈×C₅₉	—	4	4	no	—
106	Q₈×C₅₉	—	4	4	no	—
107	Q₈×C₅₉	—	8	4	no	—
108	C₂×C₄×C₅₉	—	2	2	no	—
109	C₂×C₄×C₅₉	—	4	4	no	—
110	C₂×C₄×C₅₉	—	8	4	no	—
111	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
112	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
113	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄×C₅₉	#135	1	2	yes	—
117	C₂×C₄×C₅₉	#137	1	2	yes	—
118	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
121	C₂×Dic₅₉	—	2	2	no	—
122	C₂×Dic₅₉	—	4	2	no	—
123	D₂₃₆	—	2	2	no	—
124	D₂₃₆	—	2	4	no	—
125	D₂₃₆	C₂×D₈	1	4	yes	—
126	C₄×D₅₉	—	4	2	no	—
127	C₄×D₅₉	—	4	2	no	—
128	C₄×D₅₉	—	4	4	no	—
129	C₈×C₅₉	—	2	2	no	—
130	C₈×C₅₉	C₂×SD₁₆	1	4	yes	complemented
131	C₅₉⋊C₈	—	4	4	no	—
132	C₅₉⋊C₈	—	4	2	no	—
133	C₂×#1673	—	59	1	no	maximal, Sylow
134	C₂²×C₅₉	#461	1	2	yes	—
135	C₂²×C₅₉	#466	1	2	yes	—
136	C₂²×C₅₉	#473	1	2	yes	—
137	C₂²×C₅₉	C₂×#121	1	1	yes	characteristic
138	C₂×D₅₉	—	2	4	no	—
139	C₂×D₅₉	—	4	8	no	—
140	C₂×D₅₉	—	4	2	no	—
141	C₂×D₅₉	—	4	2	no	—
142	C₄×C₅₉	—	2	1	no	—
143	C₄×C₅₉	—	2	1	no	—
144	C₄×C₅₉	—	8	4	no	—
145	C₄×C₅₉	#461	1	2	yes	—
146	C₄×C₅₉	#466	1	2	yes	—
147	C₄×C₅₉	C₂²×D₈	1	1	yes	characteristic
148	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
149	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
150	C₄×C₅₉	C₂×#137	1	1	yes	characteristic
151	Dic₅₉	—	4	2	no	—
152	C₂×#500	D₅₉	1	1	yes	characteristic, complemented
153	C₂×#461	—	59	2	no	—
154	C₂×C₈×D₄	—	59	1	no	—
155	#1673	—	59	8	no	—
156	C₂×#335	—	59	1	no	—
157	C₂×#332	—	59	2	no	—
158	C₂×C₅₉	#1673	1	4	yes	complemented
159	C₂×C₅₉	C₂×#461	1	1	yes	characteristic
160	C₂×C₅₉	C₂×#466	1	1	yes	characteristic
161	C₂×C₅₉	C₂×#473	1	1	yes	characteristic
162	D₅₉	—	4	4	no	—
163	C₂²×SD₁₆	—	118	2	no	—
164	C₂³×C₈	—	118	1	no	—
165	C₂×C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
166	C₂×C₄×D₄	—	59	1	no	—
167	#500	—	2	2	no	—
168	#500	C₂×D₅₉	1	4	yes	complemented
169	#461	—	59	8	no	—
170	#461	—	118	4	no	—
171	C₈×D₄	—	59	4	no	—
172	C₈×D₄	—	118	2	no	—
173	C₂×#107	—	2	1	no	—
174	C₂×#107	—	2	1	no	—
175	C₂×C₄⋊C₈	—	59	1	no	—
176	C₂×C₄⋊C₈	—	59	2	no	—
177	C₂×#103	—	118	2	no	—
178	C₂×C₂²⋊C₈	—	118	1	no	—
179	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
180	#335	—	59	4	no	—
181	#332	—	59	8	no	—
182	C₅₉	C₂×#1673	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₈	—	2	2	no	—
184	C₂²×D₄	—	59	1	no	—
185	C₂³×C₄	—	118	1	no	—
186	C₂×SD₁₆	—	118	8	no	—
187	C₂×SD₁₆	—	236	8	no	—
188	C₂²×C₈	—	118	2	no	—
189	C₂²×C₈	—	118	1	no	—
190	C₂²×C₈	—	118	4	no	—
191	C₂²×C₈	—	236	4	no	—
192	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
193	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	2	2	no	—
197	C₄⋊Q₈	C₂²×D₅₉	1	4	yes	—
198	C₄×D₄	—	59	2	no	—
199	C₄×D₄	—	59	2	no	—
200	C₄×D₄	—	118	1	no	—
201	C₄×D₄	—	118	1	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	59	1	no	—
205	C₂×C₂²⋊C₄	—	118	1	no	—
206	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
207	#107	—	2	4	no	—
208	#107	—	2	4	no	—
209	C₄⋊C₈	—	59	2	no	—
210	C₄⋊C₈	—	59	4	no	—
211	C₄⋊C₈	—	118	2	no	—
212	C₄⋊C₈	—	118	1	no	—
213	#103	—	118	8	no	—
214	C₂²⋊C₈	—	118	4	no	—
215	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
216	C₄×C₈	C₂²×D₅₉	1	2	yes	—
217	C₂⁴	—	118	1	no	—
218	C₂×Q₈	—	2	4	no	—
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	4	4	no	—
221	C₂×Q₈	—	4	4	no	—
222	C₂×D₄	—	59	4	no	—
223	C₂×D₄	—	59	2	no	—
224	C₂×D₄	—	118	1	no	—
225	C₂×D₄	—	118	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	118	1	no	—
228	C₂²×C₄	—	118	2	no	—
229	C₂²×C₄	—	118	2	no	—
230	C₂²×C₄	—	236	2	no	—
231	C₂²×C₄	—	236	2	no	—
232	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
233	C₂²×C₄	#11745687	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
235	SD₁₆	—	236	16	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	118	4	no	—
238	C₂×C₈	—	118	2	no	—
239	C₂×C₈	—	236	4	no	—
240	C₂×C₈	—	236	2	no	—
241	C₂×C₈	—	236	8	no	—
242	C₂×C₈	#11745671	1	4	yes	complemented
243	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
244	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	59	2	no	—
250	C₄⋊C₄	—	118	1	no	—
251	C₂²⋊C₄	—	118	2	no	—
252	C₂²⋊C₄	—	118	2	no	—
253	C₄²	D₄×D₅₉	1	1	yes	characteristic
254	C₄²	#11745687	1	1	yes	characteristic
255	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
256	C₄²	C₂³×D₅₉	1	1	yes	characteristic
257	C₂³	—	118	4	no	—
258	C₂³	—	118	2	no	—
259	C₂³	—	236	2	no	—
260	C₂³	—	236	2	no	—
261	C₂³	#14273698	1	1	yes	characteristic
262	Q₈	—	4	4	no	—
263	Q₈	—	4	4	no	—
264	Q₈	—	8	4	no	—
265	D₄	—	59	4	no	—
266	D₄	—	118	4	no	—
267	D₄	—	118	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	8	4	no	—
271	C₂×C₄	—	118	2	no	—
272	C₂×C₄	—	236	2	no	—
273	C₂×C₄	—	236	2	no	—
274	C₂×C₄	—	236	2	no	—
275	C₂×C₄	—	236	4	no	—
276	C₂×C₄	#14273698	1	1	yes	characteristic
277	C₂×C₄	#14273698	1	1	yes	characteristic
278	C₂×C₄	#14273698	1	1	yes	characteristic
279	C₂×C₄	D₈×D₅₉	1	2	yes	—
280	C₂×C₄	C₂×#11745671	1	2	yes	—
281	C₂×C₄	#14273746	1	2	yes	—
282	C₂×C₄	#14273758	1	2	yes	—
283	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
284	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
285	C₂×C₄	C₂×#11745687	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	236	4	no	—
288	C₈	—	236	2	no	—
289	C₈	C₂×#11745671	1	4	yes	complemented
290	C₂²	—	118	4	no	—
291	C₂²	—	236	8	no	—
292	C₂²	—	236	2	no	—
293	C₂²	—	236	2	no	—
294	C₂²	#16837604	1	2	yes	—
295	C₂²	#16837635	1	2	yes	—
296	C₂²	#16837674	1	2	yes	—
297	C₂²	C₂×#14273698	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	8	4	no	—
301	C₄	—	236	2	no	—
302	C₄	#16837604	1	2	yes	—
303	C₄	#16837635	1	2	yes	—
304	C₄	C₂×D₈×D₅₉	1	1	yes	characteristic
305	C₄	C₂²×#11745671	1	1	yes	characteristic
306	C₄	C₂×#14273746	1	1	yes	characteristic
307	C₄	C₂×#14273758	1	1	yes	characteristic
308	C₂	—	236	4	no	—
309	C₂	#21423482	1	4	yes	direct product factor
310	C₂	C₂×#16837604	1	1	yes	characteristic
311	C₂	C₂×#16837635	1	1	yes	characteristic
312	C₂	C₂×#16837674	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273698
commutator, cosocle	C₄²×C₅₉	C₂⁴
Frattini	C₄²	C₂³×D₅₉
Fitting	C₂×C₅₉×#500	C₂
socle	C₂³×C₅₉	#121
prime radical	C₂²×D₂₃₆	D₄
hypercenter	C₂×#500	D₅₉
hypocenter	C₅₉	C₂×#1673
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #21423482	C₂	direct (N × H)
4. C₂×#16837155	C₂	semidirect
5. C₂×#16837148	C₂	semidirect
6. C₂×C₅₉×#500	C₂	semidirect
18. #16837155	C₂²	semidirect
19. #16837148	C₂²	semidirect
22. C₅₉×#500	C₂²	semidirect
79. C₂×C₈×C₅₉	SD₁₆	semidirect
130. C₈×C₅₉	C₂×SD₁₆	semidirect
152. C₂×#500	D₅₉	semidirect
158. C₂×C₅₉	#1673	semidirect
168. #500	C₂×D₅₉	semidirect
182. C₅₉	C₂×#1673	semidirect
242. C₂×C₈	#11745671	semidirect
289. C₈	C₂×#11745671	semidirect
309. C₂	#21423482	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×#16837604	2
C₂×C₈×D₂₃₆	2
#21423482	2
C₂×#16837155	2
C₂×#16837148	2
C₂×C₅₉×#500	2
C₂×#1673	59

Subgroups — C2×#21423482 #28570943

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21423482 #28570943