Subgroups — C₂×#21423606 #28571215

Order 15104.

758 conjugacy classes of subgroups.

The subgroup lattice has 306 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×#16837481	C₂	1	1	yes	maximal, characteristic
3	#21423606	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837153	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×#11745659	—	2	1	no	—
9	C₂×C₄×D₂₃₆	C₂²	1	1	yes	characteristic
10	#16837604	—	2	4	no	—
11	#16837604	C₂²	1	8	yes	—
12	#16837481	—	2	2	no	—
13	#16837481	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	#16837153	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	C₂×#11745659	—	2	4	no	—
31	C₂×#11745659	—	4	4	no	—
32	C₂²×C₅₉⋊C₈	—	2	2	no	—
33	C₂²×C₅₉⋊C₈	—	2	1	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	4	no	—
71	C₂×D₂₃₆	—	2	1	no	—
72	C₂×D₂₃₆	C₂×D₄	1	2	yes	—
73	C₂×C₄×D₅₉	—	2	2	no	—
74	C₂×C₄×D₅₉	—	2	2	no	—
75	C₂×C₄×D₅₉	—	4	2	no	—
76	C₂×C₄×D₅₉	—	4	2	no	—
77	C₂×C₈×C₅₉	—	2	2	no	—
78	C₂×C₈×C₅₉	SD₁₆	1	4	yes	complemented
79	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
81	C₄⋊C₄×C₅₉	—	2	4	no	—
82	C₄⋊C₄×C₅₉	—	2	4	no	—
83	C₄⋊C₄×C₅₉	—	4	2	no	—
84	C₄⋊C₄×C₅₉	—	4	2	no	—
85	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₉	#41	1	1	yes	characteristic
88	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
89	#11745671	—	4	16	no	—
90	(C₂×D₅₉)⋊C₄	—	2	2	no	—
91	(C₂×D₅₉)⋊C₄	—	2	2	no	—
92	C₄⋊Dic₅₉	—	2	1	no	—
93	C₄⋊Dic₅₉	C₂×D₄	1	2	yes	—
94	C₂×C₅₉⋊C₈	—	2	2	no	—
95	C₂×C₅₉⋊C₈	—	2	4	no	—
96	C₂×C₅₉⋊C₈	—	4	2	no	—
97	C₂×C₅₉⋊C₈	—	4	4	no	—
98	#11745659	—	4	8	no	—
99	C₂³×C₅₉	#121	1	1	yes	characteristic
100	C₂²×D₅₉	—	2	2	no	—
101	C₂²×D₅₉	—	4	2	no	—
102	C₂²×D₅₉	—	4	2	no	—
103	C₂²×D₅₉	—	4	2	no	—
104	Q₈×C₅₉	—	4	4	no	—
105	Q₈×C₅₉	—	4	4	no	—
106	Q₈×C₅₉	—	8	4	no	—
107	C₂×C₄×C₅₉	—	2	2	no	—
108	C₂×C₄×C₅₉	—	4	4	no	—
109	C₂×C₄×C₅₉	—	8	4	no	—
110	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
111	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
112	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
114	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
115	C₂×C₄×C₅₉	#137	1	2	yes	—
116	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
119	C₂×Dic₅₉	—	2	2	no	—
120	C₂×Dic₅₉	—	4	2	no	—
121	D₂₃₆	—	2	4	no	—
122	D₂₃₆	—	4	2	no	—
123	C₄×D₅₉	—	4	2	no	—
124	C₄×D₅₉	—	4	2	no	—
125	C₄×D₅₉	—	4	4	no	—
126	C₈×C₅₉	—	2	2	no	—
127	C₈×C₅₉	C₂×SD₁₆	1	4	yes	complemented
128	C₅₉⋊C₈	—	4	2	no	—
129	C₅₉⋊C₈	—	4	4	no	—
130	C₂×#1693	—	59	1	no	maximal, Sylow
131	C₂²×C₅₉	#461	1	2	yes	—
132	C₂²×C₅₉	#469	1	2	yes	—
133	C₂²×C₅₉	#473	1	2	yes	—
134	C₂²×C₅₉	C₂×#121	1	1	yes	characteristic
135	C₂×D₅₉	—	4	2	no	—
136	C₂×D₅₉	—	4	2	no	—
137	C₂×D₅₉	—	4	2	no	—
138	C₂×D₅₉	—	8	4	no	—
139	C₄×C₅₉	—	2	1	no	—
140	C₄×C₅₉	—	2	1	no	—
141	C₄×C₅₉	—	8	4	no	—
142	C₄×C₅₉	#461	1	2	yes	—
143	C₄×C₅₉	#469	1	2	yes	—
144	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
145	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
146	C₄×C₅₉	C₂×#137	1	1	yes	characteristic
147	Dic₅₉	—	4	2	no	—
148	C₂×#500	D₅₉	1	1	yes	characteristic, complemented
149	C₂×#461	—	59	2	no	—
150	C₂×#436	—	59	1	no	—
151	#1693	—	59	8	no	—
152	C₂×#334	—	59	1	no	—
153	C₂×#332	—	59	2	no	—
154	C₂×C₅₉	#1693	1	4	yes	complemented
155	C₂×C₅₉	C₂×#461	1	1	yes	characteristic
156	C₂×C₅₉	C₂×#469	1	1	yes	characteristic
157	C₂×C₅₉	C₂×#473	1	1	yes	characteristic
158	D₅₉	—	8	2	no	—
159	C₂²×SD₁₆	—	118	2	no	—
160	C₂²×M₁₆	—	118	1	no	—
161	C₂×C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
162	C₂×C₄×D₄	—	59	1	no	—
163	#500	—	2	2	no	—
164	#500	C₂×D₅₉	1	4	yes	complemented
165	#461	—	59	8	no	—
166	#461	—	118	4	no	—
167	#436	—	59	4	no	—
168	#436	—	118	2	no	—
169	C₂×#107	—	2	1	no	—
170	C₂×#107	—	2	1	no	—
171	C₂×C₄⋊C₈	—	59	1	no	—
172	C₂×C₄⋊C₈	—	59	2	no	—
173	C₂×#103	—	118	2	no	—
174	C₂×C₂²⋊C₈	—	118	1	no	—
175	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
176	#334	—	59	4	no	—
177	#332	—	59	8	no	—
178	C₅₉	C₂×#1693	1	1	yes	characteristic, complemented, Sylow
179	C₂²×Q₈	—	2	2	no	—
180	C₂²×D₄	—	59	1	no	—
181	C₂³×C₄	—	118	1	no	—
182	C₂×SD₁₆	—	118	8	no	—
183	C₂×SD₁₆	—	236	8	no	—
184	C₂×M₁₆	—	118	4	no	—
185	C₂×M₁₆	—	236	4	no	—
186	C₂²×C₈	—	118	2	no	—
187	C₂²×C₈	—	118	1	no	—
188	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
189	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
190	C₄⋊Q₈	—	2	2	no	—
191	C₄⋊Q₈	—	2	2	no	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	C₂²×D₅₉	1	4	yes	—
194	C₄×D₄	—	59	2	no	—
195	C₄×D₄	—	59	2	no	—
196	C₄×D₄	—	118	1	no	—
197	C₄×D₄	—	118	1	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	59	1	no	—
201	C₂×C₂²⋊C₄	—	118	1	no	—
202	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
203	#107	—	2	4	no	—
204	#107	—	2	4	no	—
205	C₄⋊C₈	—	59	2	no	—
206	C₄⋊C₈	—	59	4	no	—
207	C₄⋊C₈	—	118	2	no	—
208	C₄⋊C₈	—	118	1	no	—
209	#103	—	118	8	no	—
210	C₂²⋊C₈	—	118	4	no	—
211	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
212	C₄×C₈	C₂²×D₅₉	1	2	yes	—
213	C₂⁴	—	118	1	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	—	4	4	no	—
217	C₂×Q₈	—	4	4	no	—
218	C₂×D₄	—	59	2	no	—
219	C₂×D₄	—	118	1	no	—
220	C₂×D₄	—	118	4	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	118	1	no	—
223	C₂²×C₄	—	118	2	no	—
224	C₂²×C₄	—	118	2	no	—
225	C₂²×C₄	—	236	2	no	—
226	C₂²×C₄	—	236	2	no	—
227	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
228	C₂²×C₄	#11745687	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
230	SD₁₆	—	236	16	no	—
231	M₁₆	—	236	8	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	118	4	no	—
234	C₂×C₈	—	118	2	no	—
235	C₂×C₈	—	236	2	no	—
236	C₂×C₈	—	236	4	no	—
237	C₂×C₈	#11745671	1	4	yes	complemented
238	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
239	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	59	2	no	—
245	C₄⋊C₄	—	118	1	no	—
246	C₂²⋊C₄	—	118	2	no	—
247	C₂²⋊C₄	—	118	2	no	—
248	C₄²	D₄×D₅₉	1	1	yes	characteristic
249	C₄²	#11745687	1	1	yes	characteristic
250	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
251	C₄²	C₂³×D₅₉	1	1	yes	characteristic
252	C₂³	—	118	2	no	—
253	C₂³	—	236	2	no	—
254	C₂³	—	236	2	no	—
255	C₂³	—	236	2	no	—
256	C₂³	#14273698	1	1	yes	characteristic
257	Q₈	—	4	4	no	—
258	Q₈	—	4	4	no	—
259	Q₈	—	8	4	no	—
260	D₄	—	118	4	no	—
261	D₄	—	236	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	—	118	2	no	—
266	C₂×C₄	—	236	2	no	—
267	C₂×C₄	—	236	2	no	—
268	C₂×C₄	—	236	2	no	—
269	C₂×C₄	—	236	4	no	—
270	C₂×C₄	#14273698	1	1	yes	characteristic
271	C₂×C₄	#14273698	1	1	yes	characteristic
272	C₂×C₄	#14273698	1	1	yes	characteristic
273	C₂×C₄	C₂×#11745671	1	2	yes	—
274	C₂×C₄	#14273751	1	4	yes	—
275	C₂×C₄	#14273758	1	2	yes	—
276	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
278	C₂×C₄	C₂×#11745687	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	236	2	no	—
281	C₈	—	236	4	no	—
282	C₈	C₂×#11745671	1	4	yes	complemented
283	C₂²	—	472	4	no	—
284	C₂²	—	236	2	no	—
285	C₂²	—	236	2	no	—
286	C₂²	—	236	2	no	—
287	C₂²	#16837604	1	2	yes	—
288	C₂²	#16837652	1	2	yes	—
289	C₂²	#16837674	1	2	yes	—
290	C₂²	C₂×#14273698	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	8	4	no	—
294	C₄	—	236	2	no	—
295	C₄	#16837604	1	2	yes	—
296	C₄	#16837652	1	2	yes	—
297	C₄	C₂²×#11745671	1	1	yes	characteristic
298	C₄	C₂×#14273751	1	2	yes	—
299	C₄	C₂×#14273758	1	1	yes	characteristic
300	C₂	—	472	2	no	—
301	C₂	#21423606	1	4	yes	direct product factor
302	C₂	C₂×#16837604	1	1	yes	characteristic
303	C₂	C₂×#16837652	1	1	yes	characteristic
304	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×#1693
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. #21423606	C₂	direct (N × H)
4. C₂×#16837153	C₂	semidirect
5. C₂×#16837148	C₂	semidirect
6. C₂×C₅₉×#500	C₂	semidirect
18. #16837153	C₂²	semidirect
19. #16837148	C₂²	semidirect
22. C₅₉×#500	C₂²	semidirect
78. C₂×C₈×C₅₉	SD₁₆	semidirect
127. C₈×C₅₉	C₂×SD₁₆	semidirect
148. C₂×#500	D₅₉	semidirect
154. C₂×C₅₉	#1693	semidirect
164. #500	C₂×D₅₉	semidirect
178. C₅₉	C₂×#1693	semidirect
237. C₂×C₈	#11745671	semidirect
282. C₈	C₂×#11745671	semidirect
301. C₂	#21423606	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×#16837481	2
#21423606	2
C₂×#16837153	2
C₂×#16837148	2
C₂×C₅₉×#500	2
C₂×#1693	59

Subgroups — C2×#21423606 #28571215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21423606 #28571215