Subgroups — C₅×#10841 #11933561

Order 1280.

1126 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#384	C₂	1	2	yes	maximal
3	C₂×C₅×#340	C₂	1	4	yes	maximal
4	C₂³×C₅×C₈	C₂²	1	2	yes	—
5	C₂²×C₅×C₄⋊C₄	—	2	4	no	—
6	C₂²×C₅×C₄⋊C₄	C₂²	1	4	yes	—
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	—	2	4	no	—
9	C₂×C₄×C₅×C₈	C₄	1	4	yes	complemented
10	C₅×#384	—	2	4	no	—
11	C₅×#384	C₄	1	8	yes	complemented
12	C₅×#340	—	2	8	no	—
13	#10841	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂³×C₄×C₅	—	2	4	no	—
15	C₂³×C₄×C₅	D₄	1	2	yes	—
16	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
17	C₂²×C₅×C₈	—	2	4	no	—
18	C₂²×C₅×C₈	—	2	4	no	—
19	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₅×C₈	D₄	1	4	yes	—
21	C₂²×C₅×C₈	Q₈	1	4	yes	—
22	C₂×C₅×C₄⋊C₄	—	2	8	no	—
23	C₂×C₅×C₄⋊C₄	—	2	8	no	—
24	C₂×C₅×C₄⋊C₄	—	2	16	no	—
25	C₂×C₅×C₄⋊C₄	—	2	4	no	—
26	C₂×C₅×C₄⋊C₄	—	4	8	no	—
27	C₂×C₅×C₄⋊C₄	—	4	8	no	—
28	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅	D₄	1	4	yes	—
32	C₂×C₄²×C₅	Q₈	1	2	yes	—
33	C₄×C₅×C₈	—	2	8	no	—
34	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
35	C₂×#384	C₂×C₅	1	2	yes	—
36	C₂×#340	C₂×C₅	1	4	yes	—
37	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅	—	2	8	no	—
39	C₂²×C₄×C₅	—	4	4	no	—
40	C₂²×C₄×C₅	—	4	4	no	—
41	C₂²×C₄×C₅	—	4	8	no	—
42	C₂²×C₄×C₅	—	4	8	no	—
43	C₂²×C₄×C₅	C₂²⋊C₄	1	8	yes	—
44	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₅	D₈	1	4	yes	—
46	C₂²×C₄×C₅	SD₁₆	1	8	yes	—
47	C₂²×C₄×C₅	Q₁₆	1	4	yes	—
48	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
54	C₂²×C₄×C₅	#41	1	4	yes	—
55	C₂×C₅×C₈	—	2	4	no	—
56	C₂×C₅×C₈	—	2	4	no	—
57	C₂×C₅×C₈	—	2	8	no	—
58	C₂×C₅×C₈	—	2	8	no	—
59	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	complemented
60	C₅×C₄⋊C₄	—	2	16	no	—
61	C₅×C₄⋊C₄	—	4	32	no	—
62	C₅×C₄⋊C₄	—	4	8	no	—
63	C₄²×C₅	C₄²	1	4	yes	—
64	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
65	C₄²×C₅	C₄⋊C₄	1	8	yes	—
66	C₂³×C₈	C₂²×C₅	1	2	yes	—
67	C₂²×C₄⋊C₄	—	2	4	no	—
68	C₂²×C₄⋊C₄	C₂²×C₅	1	4	yes	—
69	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
70	C₂×C₄×C₈	—	2	4	no	—
71	C₂×C₄×C₈	C₄×C₅	1	4	yes	complemented
72	#384	—	2	4	no	—
73	#384	C₄×C₅	1	8	yes	complemented
74	#340	—	2	8	no	—
75	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
76	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂³×C₅	#124	1	2	yes	—
78	C₂³×C₅	#127	1	1	yes	characteristic
79	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
80	C₂³×C₅	C₂×D₈	1	2	yes	—
81	C₂³×C₅	C₂×SD₁₆	1	4	yes	—
82	C₂³×C₅	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₅	—	4	8	no	—
84	C₂×C₄×C₅	—	4	16	no	—
85	C₂×C₄×C₅	—	8	8	no	—
86	C₂×C₄×C₅	—	8	4	no	—
87	C₂×C₄×C₅	—	8	4	no	—
88	C₂×C₄×C₅	#95	1	8	yes	—
89	C₂×C₄×C₅	#102	1	16	yes	—
90	C₂×C₄×C₅	#103	1	16	yes	—
91	C₂×C₄×C₅	#106	1	8	yes	—
92	C₂×C₄×C₅	#107	1	8	yes	—
93	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₅	#117	1	2	yes	—
98	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
99	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
100	C₂×C₄×C₅	#121	1	4	yes	—
101	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
102	C₂×C₄×C₅	#125	1	2	yes	—
103	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
104	C₅×C₈	—	2	8	no	—
105	C₂³×C₄	—	2	4	no	—
106	C₂³×C₄	C₅×D₄	1	2	yes	—
107	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
108	C₂²×C₈	—	2	4	no	—
109	C₂²×C₈	—	2	4	no	—
110	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
111	C₂²×C₈	C₅×D₄	1	4	yes	—
112	C₂²×C₈	C₅×Q₈	1	4	yes	—
113	C₂×C₄⋊C₄	—	2	8	no	—
114	C₂×C₄⋊C₄	—	2	8	no	—
115	C₂×C₄⋊C₄	—	2	16	no	—
116	C₂×C₄⋊C₄	—	2	4	no	—
117	C₂×C₄⋊C₄	—	4	8	no	—
118	C₂×C₄⋊C₄	—	4	8	no	—
119	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
120	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
121	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
122	C₂×C₄²	C₅×D₄	1	4	yes	—
123	C₂×C₄²	C₅×Q₈	1	2	yes	—
124	C₄×C₈	—	2	8	no	—
125	C₂²×C₅	C₂×#95	1	1	yes	characteristic
126	C₂²×C₅	#382	1	1	yes	characteristic
127	C₂²×C₅	#384	1	1	yes	characteristic
128	C₂²×C₅	#390	1	2	yes	—
129	C₂²×C₅	#391	1	2	yes	—
130	C₂²×C₅	C₂×#102	1	4	yes	—
131	C₂²×C₅	C₂×#103	1	4	yes	—
132	C₂²×C₅	C₂×#106	1	2	yes	—
133	C₂²×C₅	C₂×#107	1	2	yes	—
134	C₂²×C₅	#486	1	4	yes	—
135	C₂²×C₅	#487	1	4	yes	—
136	C₂²×C₅	#492	1	2	yes	—
137	C₂²×C₅	#493	1	1	yes	characteristic
138	C₂²×C₅	#494	1	1	yes	characteristic
139	C₂²×C₅	#498	1	2	yes	—
140	C₂²×C₅	#500	1	2	yes	—
141	C₄×C₅	—	8	8	no	—
142	C₄×C₅	#340	1	16	yes	complemented
143	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
144	C₄×C₅	#380	1	2	yes	—
145	C₄×C₅	#389	1	4	yes	—
146	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
147	C₂²×C₄	—	2	8	no	—
148	C₂²×C₄	—	4	4	no	—
149	C₂²×C₄	—	4	4	no	—
150	C₂²×C₄	—	4	8	no	—
151	C₂²×C₄	—	4	8	no	—
152	C₂²×C₄	C₅×C₂²⋊C₄	1	8	yes	—
153	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
154	C₂²×C₄	C₅×D₈	1	4	yes	—
155	C₂²×C₄	C₅×SD₁₆	1	8	yes	—
156	C₂²×C₄	C₅×Q₁₆	1	4	yes	—
157	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
158	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
159	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
160	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
161	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
163	C₂²×C₄	C₅×#41	1	4	yes	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	8	no	—
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	complemented
169	C₄⋊C₄	—	2	16	no	—
170	C₄⋊C₄	—	4	8	no	—
171	C₄⋊C₄	—	4	32	no	—
172	C₄²	C₄²×C₅	1	4	yes	—
173	C₄²	C₅×C₂²⋊C₄	1	4	yes	—
174	C₄²	C₅×C₄⋊C₄	1	8	yes	—
175	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
176	C₂×C₅	C₂×#340	1	2	yes	—
177	C₂×C₅	#1834	1	2	yes	—
178	C₂×C₅	#1844	1	1	yes	characteristic
179	C₂×C₅	#1845	1	1	yes	characteristic
180	C₂×C₅	#1956	1	2	yes	—
181	C₂×C₅	#1957	1	2	yes	—
182	C₂×C₅	#1990	1	4	yes	—
183	C₂³	C₂×C₅×C₂²⋊C₄	1	2	yes	—
184	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
185	C₂³	C₅×#124	1	2	yes	—
186	C₂³	C₅×#127	1	1	yes	characteristic
187	C₂³	C₅×C₄⋊Q₈	1	1	yes	characteristic
188	C₂³	C₂×C₅×D₈	1	2	yes	—
189	C₂³	C₂×C₅×SD₁₆	1	4	yes	—
190	C₂³	C₂×C₅×Q₁₆	1	2	yes	—
191	C₂×C₄	—	4	8	no	—
192	C₂×C₄	—	4	16	no	—
193	C₂×C₄	—	8	8	no	—
194	C₂×C₄	—	8	4	no	—
195	C₂×C₄	—	8	4	no	—
196	C₂×C₄	C₅×#95	1	8	yes	—
197	C₂×C₄	C₅×#102	1	16	yes	—
198	C₂×C₄	C₅×#103	1	16	yes	—
199	C₂×C₄	C₅×#106	1	8	yes	—
200	C₂×C₄	C₅×#107	1	8	yes	—
201	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
202	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
204	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
205	C₂×C₄	C₅×#117	1	2	yes	—
206	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
207	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
208	C₂×C₄	C₅×#121	1	4	yes	—
209	C₂×C₄	C₅×C₂²⋊Q₈	1	4	yes	—
210	C₂×C₄	C₅×#125	1	2	yes	—
211	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
212	C₈	—	2	8	no	—
213	C₅	#10841	1	1	yes	characteristic, direct product factor, Sylow
214	C₂²	C₂×C₅×#95	1	1	yes	characteristic
215	C₂²	C₅×#382	1	1	yes	characteristic
216	C₂²	C₅×#384	1	1	yes	characteristic
217	C₂²	C₅×#390	1	2	yes	—
218	C₂²	C₅×#391	1	2	yes	—
219	C₂²	C₂×C₅×#102	1	4	yes	—
220	C₂²	C₂×C₅×#103	1	4	yes	—
221	C₂²	C₂×C₅×#106	1	2	yes	—
222	C₂²	C₂×C₅×#107	1	2	yes	—
223	C₂²	C₅×#486	1	4	yes	—
224	C₂²	C₅×#487	1	4	yes	—
225	C₂²	C₅×#492	1	2	yes	—
226	C₂²	C₅×#493	1	1	yes	characteristic
227	C₂²	C₅×#494	1	1	yes	characteristic
228	C₂²	C₅×#498	1	2	yes	—
229	C₂²	C₅×#500	1	2	yes	—
230	C₄	—	8	8	no	—
231	C₄	C₅×#340	1	16	yes	complemented
232	C₄	C₄×C₅×C₄⋊C₄	1	2	yes	—
233	C₄	C₅×#380	1	2	yes	—
234	C₄	C₅×#389	1	4	yes	—
235	C₂	C₅×C₄⋊#95	1	1	yes	characteristic
236	C₂	C₂×C₅×#340	1	2	yes	—
237	C₂	C₅×#1834	1	2	yes	—
238	C₂	C₅×#1844	1	1	yes	characteristic
239	C₂	C₅×#1845	1	1	yes	characteristic
240	C₂	C₅×#1956	1	2	yes	—
241	C₂	C₅×#1957	1	2	yes	—
242	C₂	C₅×#1990	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₂×C₄×C₅×C₈	C₄	semidirect
11. C₅×#384	C₄	semidirect
13. #10841	C₅	direct (N × H)
59. C₂×C₅×C₈	C₄⋊C₄	semidirect
71. C₂×C₄×C₈	C₄×C₅	semidirect
73. #384	C₄×C₅	semidirect
142. C₄×C₅	#340	semidirect
168. C₂×C₈	C₅×C₄⋊C₄	semidirect
213. C₅	#10841	direct (N × H)
231. C₄	C₅×#340	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×C₈	2
C₂×C₅×#384	2
C₂×C₅×#340	2
#10841	5

Subgroups — C5×#10841 #11933561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10841 #11933561