Subgroups — C₅×#12640 #11935360

Order 1280.

442 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	C₄×C₅×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#359	C₂	1	2	yes	maximal
3	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1382	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#135	C₂²	1	1	yes	characteristic
7	C₄×C₅×#41	—	2	1	no	—
8	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
9	C₄×C₅×Q₁₆	C₄	1	2	yes	—
10	C₄×C₅×SD₁₆	—	2	2	no	—
11	C₄×C₅×D₈	C₄	1	2	yes	—
12	C₂×C₅×#108	C₂²	1	2	yes	—
13	C₅×#427	C₂²	1	1	yes	characteristic
14	C₅×#416	—	2	1	no	—
15	C₄×C₅×M₁₆	—	2	1	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#359	C₄	1	8	yes	complemented
18	C₅×#346	—	2	2	no	—
19	C₅×#341	—	2	1	no	—
20	#12640	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₅×#41	—	2	1	no	—
22	C₅×#135	—	2	2	no	—
23	C₅×#135	C₂×C₄	1	4	yes	—
24	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅×SD₁₆	—	2	1	no	—
26	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₄×C₅×Q₈	—	2	2	no	—
31	C₄×C₅×D₄	—	2	2	no	—
32	C₄×C₅×D₄	—	4	2	no	—
33	C₅×#117	—	2	1	no	—
34	C₅×#117	—	4	1	no	—
35	C₂×C₄²×C₅	—	4	1	no	—
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₅×C₁₆	—	2	2	no	—
38	C₂×C₅×C₁₆	—	2	2	no	—
39	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
40	C₅×#108	—	2	2	no	—
41	C₅×#108	C₂×C₄	1	4	yes	complemented
42	C₅×#107	C₂×C₄	1	2	yes	—
43	C₅×#106	—	2	1	no	—
44	C₅×#103	—	2	2	no	—
45	C₅×#102	—	2	2	no	—
46	C₅×#97	—	4	1	no	—
47	C₄×C₅×C₈	—	4	1	no	—
48	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
49	C₄×C₅×C₈	D₄	1	2	yes	—
50	C₄×#135	C₂×C₅	1	1	yes	characteristic
51	C₂×#359	C₂×C₅	1	2	yes	—
52	C₂×#346	C₂×C₅	1	1	yes	characteristic, complemented
53	C₄×#108	C₂×C₅	1	1	yes	characteristic, complemented
54	#1382	C₂×C₅	1	2	yes	complemented
55	C₅×#41	—	2	4	no	—
56	C₅×#41	—	4	2	no	—
57	C₂×C₅×Q₈	—	2	1	no	—
58	C₂×C₅×D₄	—	2	1	no	—
59	C₂×C₅×D₄	—	4	1	no	—
60	C₂²×C₄×C₅	—	4	1	no	—
61	C₂²×C₄×C₅	—	4	2	no	—
62	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₅×Q₁₆	—	2	1	no	—
65	C₅×Q₁₆	C₄²	1	2	yes	—
66	C₅×SD₁₆	—	2	4	no	—
67	C₅×D₈	—	2	1	no	—
68	C₅×D₈	C₄²	1	2	yes	—
69	C₅×M₁₆	—	2	4	no	—
70	C₅×M₁₆	—	4	2	no	—
71	C₂×C₅×C₈	—	2	2	no	—
72	C₂×C₅×C₈	—	4	2	no	—
73	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
74	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
76	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₅×C₈	#41	1	2	yes	—
78	C₅×C₄⋊C₄	—	2	2	no	—
79	C₅×C₄⋊C₄	—	4	2	no	—
80	C₅×C₂²⋊C₄	—	4	1	no	—
81	C₅×C₂²⋊C₄	—	4	2	no	—
82	C₄²×C₅	—	4	2	no	—
83	C₄²×C₅	—	4	1	no	—
84	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₅	D₈	1	1	yes	characteristic
86	C₄²×C₅	D₈	1	1	yes	characteristic
87	C₅×C₁₆	—	2	4	no	—
88	C₂×#135	C₂²×C₅	1	1	yes	characteristic
89	C₄×#41	—	2	1	no	—
90	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
91	C₄×Q₁₆	C₄×C₅	1	2	yes	—
92	C₄×SD₁₆	—	2	2	no	—
93	C₄×D₈	C₄×C₅	1	2	yes	—
94	C₂×#108	C₂²×C₅	1	2	yes	—
95	#427	C₂²×C₅	1	1	yes	characteristic
96	#416	—	2	1	no	—
97	C₄×M₁₆	—	2	1	no	—
98	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
99	#359	C₄×C₅	1	8	yes	complemented
100	#346	—	2	2	no	—
101	#341	—	2	1	no	—
102	C₂³×C₅	—	4	1	no	—
103	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
104	C₅×Q₈	—	2	2	no	—
105	C₅×Q₈	—	4	1	no	—
106	C₅×D₄	—	2	2	no	—
107	C₅×D₄	—	4	4	no	—
108	C₅×D₄	—	4	1	no	—
109	C₂×C₄×C₅	—	4	2	no	—
110	C₂×C₄×C₅	—	4	1	no	—
111	C₂×C₄×C₅	—	4	4	no	—
112	C₂×C₄×C₅	—	4	2	no	—
113	C₂×C₄×C₅	—	8	4	no	—
114	C₂×C₄×C₅	#102	1	2	yes	—
115	C₂×C₄×C₅	#102	1	2	yes	—
116	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₅	#135	1	2	yes	—
120	C₅×C₈	—	2	2	no	—
121	C₅×C₈	—	4	4	no	—
122	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
123	C₅×C₈	#117	1	1	yes	characteristic
124	C₅×C₈	C₄×D₄	1	2	yes	—
125	C₂×#41	—	2	1	no	—
126	#135	—	2	2	no	—
127	#135	C₂×C₄×C₅	1	4	yes	—
128	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
129	C₂×SD₁₆	—	2	1	no	—
130	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
131	C₂×M₁₆	—	2	2	no	—
132	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
133	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
134	C₄×Q₈	—	2	2	no	—
135	C₄×D₄	—	2	2	no	—
136	C₄×D₄	—	4	2	no	—
137	#117	—	2	1	no	—
138	#117	—	4	1	no	—
139	C₂×C₄²	—	4	1	no	—
140	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	complemented
144	#108	—	2	2	no	—
145	#108	C₂×C₄×C₅	1	4	yes	complemented
146	#107	C₂×C₄×C₅	1	2	yes	—
147	#106	—	2	1	no	—
148	#103	—	2	2	no	—
149	#102	—	2	2	no	—
150	#97	—	4	1	no	—
151	C₄×C₈	—	4	1	no	—
152	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
153	C₄×C₈	C₅×D₄	1	2	yes	—
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	—	4	2	no	—
156	C₂²×C₅	—	8	2	no	—
157	C₂²×C₅	C₂×#102	1	1	yes	characteristic
158	C₂²×C₅	C₄×SD₁₆	1	2	yes	—
159	C₄×C₅	—	2	2	no	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	4	2	no	—
162	C₄×C₅	—	8	2	no	—
163	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
164	C₄×C₅	#416	1	1	yes	characteristic
165	C₄×C₅	C₄×D₈	1	2	yes	—
166	#41	—	2	4	no	—
167	#41	—	4	2	no	—
168	C₂×Q₈	—	2	1	no	—
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	—	4	1	no	—
171	C₂²×C₄	—	4	1	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
174	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
175	Q₁₆	—	2	1	no	—
176	Q₁₆	C₄²×C₅	1	2	yes	—
177	SD₁₆	—	2	4	no	—
178	D₈	—	2	1	no	—
179	D₈	C₄²×C₅	1	2	yes	—
180	M₁₆	—	2	4	no	—
181	M₁₆	—	4	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
185	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
186	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
187	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
188	C₂×C₈	C₅×#41	1	2	yes	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₂²⋊C₄	—	4	1	no	—
192	C₂²⋊C₄	—	4	2	no	—
193	C₄²	—	4	1	no	—
194	C₄²	—	4	2	no	—
195	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
196	C₄²	C₅×D₈	1	1	yes	characteristic
197	C₄²	C₅×D₈	1	1	yes	characteristic
198	C₁₆	—	2	4	no	—
199	C₂×C₅	—	2	2	no	—
200	C₂×C₅	—	8	2	no	—
201	C₂×C₅	C₄×#102	1	1	yes	characteristic
202	C₂×C₅	#2143	1	2	yes	—
203	C₂³	—	4	1	no	—
204	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
205	Q₈	—	2	2	no	—
206	Q₈	—	4	1	no	—
207	D₄	—	2	2	no	—
208	D₄	—	4	4	no	—
209	D₄	—	4	1	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	8	4	no	—
215	C₂×C₄	C₅×#102	1	2	yes	—
216	C₂×C₄	C₅×#102	1	2	yes	—
217	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
218	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
219	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
220	C₂×C₄	C₅×#135	1	2	yes	—
221	C₈	—	2	2	no	—
222	C₈	—	4	4	no	—
223	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
224	C₈	C₅×#117	1	1	yes	characteristic
225	C₈	C₄×C₅×D₄	1	2	yes	—
226	C₅	#12640	1	1	yes	characteristic, direct product factor, Sylow
227	C₂²	—	2	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	8	2	no	—
230	C₂²	C₂×C₅×#102	1	1	yes	characteristic
231	C₂²	C₄×C₅×SD₁₆	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	2	no	—
236	C₄	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
237	C₄	C₅×#416	1	1	yes	characteristic
238	C₄	C₄×C₅×D₈	1	2	yes	—
239	C₂	—	2	2	no	—
240	C₂	—	8	2	no	—
241	C₂	C₄×C₅×#102	1	1	yes	characteristic
242	C₂	C₅×#2143	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#102
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#346	C₂	semidirect
4. C₄×C₅×#108	C₂	semidirect
5. C₅×#1382	C₂	semidirect
17. C₅×#359	C₄	semidirect
20. #12640	C₅	direct (N × H)
39. C₂×C₅×C₁₆	C₂×C₄	semidirect
41. C₅×#108	C₂×C₄	semidirect
52. C₂×#346	C₂×C₅	semidirect
53. C₄×#108	C₂×C₅	semidirect
54. #1382	C₂×C₅	semidirect
99. #359	C₄×C₅	semidirect
143. C₂×C₁₆	C₂×C₄×C₅	semidirect
145. #108	C₂×C₄×C₅	semidirect
226. C₅	#12640	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×#135	2
C₂×C₅×#359	2
C₂×C₅×#346	2
C₄×C₅×#108	2
C₅×#1382	2
#12640	5

Subgroups — C5×#12640 #11935360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12640 #11935360