Subgroups — C₅×#9638 #11932358

Order 1280.

774 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₅×#1842	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×D₄⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×#127	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×D₄	—	2	2	no	—
7	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
8	C₂×C₅×C₄⋊C₈	—	2	2	no	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	—	2	2	no	—
11	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
12	C₅×#390	—	2	2	no	—
13	C₅×#390	C₄	1	2	yes	—
14	C₅×#384	C₄	1	1	yes	characteristic
15	C₄³×C₅	C₂²	1	1	yes	characteristic
16	C₅×D₄⋊C₈	—	2	8	no	—
17	#9638	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×D₄	—	2	2	no	—
19	C₂²×C₅×D₄	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₅	—	2	2	no	—
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₅×#127	—	2	2	no	—
24	C₅×#127	C₈	1	4	yes	complemented
25	C₄×C₅×D₄	—	2	4	no	—
26	C₄×C₅×D₄	—	2	4	no	—
27	C₄×C₅×D₄	—	4	4	no	—
28	C₄×C₅×D₄	—	4	4	no	—
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
32	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	D₄	1	2	yes	—
36	C₂×C₄²×C₅	D₄	1	2	yes	—
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₅×C₄⋊C₈	—	2	4	no	—
39	C₅×C₄⋊C₈	—	4	4	no	—
40	C₄×C₅×C₈	—	2	8	no	—
41	C₄×C₅×C₈	—	2	2	no	—
42	C₄×C₅×C₈	D₄	1	4	yes	complemented
43	C₄×#127	C₂×C₅	1	1	yes	characteristic
44	#1842	C₂×C₅	1	1	yes	characteristic, complemented
45	C₄²×C₈	C₂×C₅	1	1	yes	characteristic, complemented
46	C₂×D₄⋊C₈	C₂×C₅	1	4	yes	complemented
47	C₂⁴×C₅	—	2	2	no	—
48	C₂×C₅×D₄	—	2	8	no	—
49	C₂×C₅×D₄	—	2	4	no	—
50	C₂×C₅×D₄	—	2	2	no	—
51	C₂×C₅×D₄	—	2	4	no	—
52	C₂×C₅×D₄	—	4	4	no	—
53	C₂×C₅×D₄	—	4	4	no	—
54	C₂×C₅×D₄	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₅	—	2	2	no	—
56	C₂²×C₄×C₅	—	4	2	no	—
57	C₂²×C₄×C₅	—	4	2	no	—
58	C₂²×C₄×C₅	—	4	4	no	—
59	C₂²×C₄×C₅	—	4	4	no	—
60	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×C₅×C₈	—	2	8	no	—
66	C₂×C₅×C₈	—	2	4	no	—
67	C₂×C₅×C₈	—	4	2	no	—
68	C₂×C₅×C₈	—	4	4	no	—
69	C₅×C₄⋊C₄	—	2	4	no	—
70	C₅×C₄⋊C₄	—	4	4	no	—
71	C₅×C₂²⋊C₄	—	4	8	no	—
72	C₅×C₂²⋊C₄	—	4	4	no	—
73	C₄²×C₅	—	2	4	no	—
74	C₄²×C₅	—	2	2	no	—
75	C₄²×C₅	C₂×C₈	1	2	yes	—
76	C₄²×C₅	M₁₆	1	2	yes	—
77	C₄²×C₅	D₈	1	4	yes	—
78	C₄²×C₅	SD₁₆	1	4	yes	—
79	C₄²×C₅	C₂×D₄	1	2	yes	—
80	C₄²×C₅	#41	1	2	yes	—
81	C₂×#127	C₄×C₅	1	1	yes	characteristic
82	C₂×C₄×D₄	—	2	2	no	—
83	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
84	C₂×C₄⋊C₈	—	2	2	no	—
85	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
86	C₂×C₄×C₈	—	2	2	no	—
87	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
88	#390	—	2	2	no	—
89	#390	C₄×C₅	1	2	yes	—
90	#384	C₄×C₅	1	1	yes	characteristic
91	C₄³	C₂²×C₅	1	1	yes	characteristic
92	D₄⋊C₈	—	2	8	no	—
93	C₂³×C₅	—	2	4	no	—
94	C₂³×C₅	—	4	2	no	—
95	C₂³×C₅	—	4	2	no	—
96	C₂³×C₅	—	4	4	no	—
97	C₂³×C₅	—	4	4	no	—
98	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₅×D₄	—	2	8	no	—
100	C₅×D₄	—	4	16	no	—
101	C₅×D₄	—	4	4	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	4	2	no	—
106	C₂×C₄×C₅	—	4	4	no	—
107	C₂×C₄×C₅	—	8	4	no	—
108	C₂×C₄×C₅	—	8	2	no	—
109	C₂×C₄×C₅	—	8	2	no	—
110	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
111	C₂×C₄×C₅	#102	1	4	yes	—
112	C₂×C₄×C₅	#102	1	4	yes	—
113	C₂×C₄×C₅	#104	1	4	yes	—
114	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₅	#121	1	2	yes	—
116	C₂×C₄×C₅	#124	1	1	yes	characteristic
117	C₂×C₄×C₅	#127	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅	#131	1	2	yes	—
121	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
123	C₅×C₈	—	2	8	no	—
124	C₅×C₈	—	8	2	no	—
125	C₂²×D₄	—	2	2	no	—
126	C₂²×D₄	C₂×C₄×C₅	1	2	yes	—
127	C₂³×C₄	—	2	2	no	—
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	—	2	2	no	—
130	#127	—	2	2	no	—
131	#127	C₅×C₈	1	4	yes	complemented
132	C₄×D₄	—	2	4	no	—
133	C₄×D₄	—	2	4	no	—
134	C₄×D₄	—	4	4	no	—
135	C₄×D₄	—	4	4	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
138	C₂×C₂²⋊C₄	—	2	4	no	—
139	C₂×C₂²⋊C₄	—	2	2	no	—
140	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
141	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
142	C₂×C₄²	C₅×D₄	1	2	yes	—
143	C₂×C₄²	C₅×D₄	1	2	yes	—
144	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
145	C₄⋊C₈	—	2	4	no	—
146	C₄⋊C₈	—	4	4	no	—
147	C₄×C₈	—	2	8	no	—
148	C₄×C₈	—	2	2	no	—
149	C₄×C₈	C₅×D₄	1	4	yes	complemented
150	C₂²×C₅	—	4	4	no	—
151	C₂²×C₅	—	4	8	no	—
152	C₂²×C₅	—	8	2	no	—
153	C₂²×C₅	—	8	2	no	—
154	C₂²×C₅	—	8	4	no	—
155	C₂²×C₅	#390	1	1	yes	characteristic
156	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
157	C₂²×C₅	#408	1	1	yes	characteristic
158	C₂²×C₅	C₂×#102	1	2	yes	—
159	C₂²×C₅	C₂×#104	1	2	yes	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	2	4	no	—
162	C₄×C₅	—	8	2	no	—
163	C₄×C₅	D₄⋊C₈	1	8	yes	complemented
164	C₄×C₅	C₈×D₄	1	2	yes	—
165	C₄×C₅	#436	1	2	yes	—
166	C₄×C₅	#486	1	2	yes	—
167	C₄×C₅	#492	1	1	yes	characteristic
168	C₄×C₅	#493	1	1	yes	characteristic
169	C₂⁴	—	2	2	no	—
170	C₂×D₄	—	2	4	no	—
171	C₂×D₄	—	2	2	no	—
172	C₂×D₄	—	2	4	no	—
173	C₂×D₄	—	2	8	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂×D₄	—	4	4	no	—
176	C₂×D₄	C₂×C₅×C₈	1	4	yes	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	4	no	—
181	C₂²×C₄	—	4	4	no	—
182	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
183	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
184	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
187	C₂×C₈	—	2	8	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	4	4	no	—
193	C₂²⋊C₄	—	4	4	no	—
194	C₂²⋊C₄	—	4	8	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	4	no	—
197	C₄²	C₂×C₅×C₈	1	2	yes	—
198	C₄²	C₅×M₁₆	1	2	yes	—
199	C₄²	C₅×D₈	1	4	yes	—
200	C₄²	C₅×SD₁₆	1	4	yes	—
201	C₄²	C₂×C₅×D₄	1	2	yes	—
202	C₄²	C₅×#41	1	2	yes	—
203	C₂×C₅	—	8	4	no	—
204	C₂×C₅	C₂×D₄⋊C₈	1	2	yes	—
205	C₂×C₅	#1492	1	2	yes	—
206	C₂×C₅	#1954	1	1	yes	characteristic
207	C₂×C₅	#1957	1	1	yes	characteristic
208	C₂×C₅	#1962	1	1	yes	characteristic
209	C₂³	—	2	4	no	—
210	C₂³	—	4	2	no	—
211	C₂³	—	4	2	no	—
212	C₂³	—	4	4	no	—
213	C₂³	—	4	4	no	—
214	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
215	D₄	—	2	8	no	—
216	D₄	—	4	16	no	—
217	D₄	—	4	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	C₅×C₂²⋊C₈	1	4	yes	—
227	C₂×C₄	C₅×#102	1	4	yes	—
228	C₂×C₄	C₅×#102	1	4	yes	—
229	C₂×C₄	C₅×#104	1	4	yes	—
230	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
231	C₂×C₄	C₅×#121	1	2	yes	—
232	C₂×C₄	C₅×#124	1	1	yes	characteristic
233	C₂×C₄	C₅×#127	1	1	yes	characteristic
234	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
235	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
236	C₂×C₄	C₅×#131	1	2	yes	—
237	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
238	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
239	C₈	—	2	8	no	—
240	C₈	—	8	2	no	—
241	C₅	#9638	1	1	yes	characteristic, direct product factor, Sylow
242	C₂²	—	4	4	no	—
243	C₂²	—	4	8	no	—
244	C₂²	—	8	2	no	—
245	C₂²	—	8	2	no	—
246	C₂²	—	8	4	no	—
247	C₂²	C₅×#390	1	1	yes	characteristic
248	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
249	C₂²	C₅×#408	1	1	yes	characteristic
250	C₂²	C₂×C₅×#102	1	2	yes	—
251	C₂²	C₂×C₅×#104	1	2	yes	—
252	C₄	—	2	2	no	—
253	C₄	—	2	4	no	—
254	C₄	—	8	2	no	—
255	C₄	C₅×D₄⋊C₈	1	8	yes	complemented
256	C₄	C₅×C₈×D₄	1	2	yes	—
257	C₄	C₅×#436	1	2	yes	—
258	C₄	C₅×#486	1	2	yes	—
259	C₄	C₅×#492	1	1	yes	characteristic
260	C₄	C₅×#493	1	1	yes	characteristic
261	C₂	—	8	4	no	—
262	C₂	C₂×C₅×D₄⋊C₈	1	2	yes	—
263	C₂	C₅×#1492	1	2	yes	—
264	C₂	C₅×#1954	1	1	yes	characteristic
265	C₂	C₅×#1957	1	1	yes	characteristic
266	C₂	C₅×#1962	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂³×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₅×#127	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1842	C₂	semidirect
3. C₄²×C₅×C₈	C₂	semidirect
4. C₂×C₅×D₄⋊C₈	C₂	semidirect
17. #9638	C₅	direct (N × H)
24. C₅×#127	C₈	semidirect
42. C₄×C₅×C₈	D₄	semidirect
44. #1842	C₂×C₅	semidirect
45. C₄²×C₈	C₂×C₅	semidirect
46. C₂×D₄⋊C₈	C₂×C₅	semidirect
131. #127	C₅×C₈	semidirect
149. C₄×C₈	C₅×D₄	semidirect
163. C₄×C₅	D₄⋊C₈	semidirect
241. C₅	#9638	direct (N × H)
255. C₄	C₅×D₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#127	2
C₅×#1842	2
C₄²×C₅×C₈	2
C₂×C₅×D₄⋊C₈	2
#9638	5

Subgroups — C5×#9638 #11932358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9638 #11932358