Subgroups — C₅×#3040 #10616635

Order 640.

500 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×#41	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#464	C₂	1	4	yes	maximal, complemented
6	C₅×#463	C₂	1	4	yes	maximal, complemented
7	C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#419	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#418	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#41	C₂²	1	1	yes	characteristic
11	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₅×#137	—	2	4	no	—
14	C₅×AGL(1,ℤ/8)	—	2	4	no	—
15	C₂×C₅×Q₁₆	C₂²	1	2	yes	complemented
16	C₂×C₅×SD₁₆	C₂²	1	2	yes	—
17	C₂×C₅×SD₁₆	C₂²	1	2	yes	complemented
18	C₂×C₅×D₈	C₂²	1	2	yes	—
19	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
21	C₅×#124	—	2	1	no	—
22	C₅×#124	—	2	1	no	—
23	C₅×#124	C₂²	1	4	yes	—
24	C₄×C₅×Q₈	C₂²	1	2	yes	complemented
25	C₄×C₅×D₄	—	2	2	no	—
26	C₄×C₅×D₄	C₂²	1	2	yes	complemented
27	C₅×#117	—	2	1	no	—
28	C₅×#117	C₂²	1	1	yes	characteristic
29	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
30	C₂×C₄²×C₅	—	2	1	no	—
31	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
32	C₅×C₄⋊C₈	C₂²	1	4	yes	complemented
33	C₅×#103	C₂²	1	4	yes	complemented
34	C₅×#102	C₂²	1	4	yes	—
35	#3040	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂⁴×C₅	—	2	1	no	—
37	C₅×#41	—	2	2	no	—
38	C₅×#41	D₄	1	4	yes	—
39	C₂×C₅×Q₈	—	2	1	no	—
40	C₂×C₅×Q₈	—	2	2	no	—
41	C₂×C₅×Q₈	—	2	2	no	—
42	C₂×C₅×Q₈	C₂³	1	2	yes	—
43	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
44	C₂×C₅×D₄	—	2	1	no	—
45	C₂×C₅×D₄	—	2	2	no	—
46	C₂×C₅×D₄	—	2	1	no	—
47	C₂×C₅×D₄	—	2	2	no	—
48	C₂×C₅×D₄	C₂³	1	2	yes	—
49	C₂×C₅×D₄	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₅	—	2	1	no	—
51	C₂²×C₄×C₅	—	2	1	no	—
52	C₂²×C₄×C₅	—	2	2	no	—
53	C₂²×C₄×C₅	D₄	1	2	yes	—
54	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₅×Q₁₆	—	2	4	no	—
56	C₅×SD₁₆	—	2	4	no	—
57	C₅×SD₁₆	—	2	4	no	—
58	C₅×D₈	—	2	4	no	—
59	C₅×M₁₆	—	2	2	no	—
60	C₅×M₁₆	—	2	2	no	—
61	C₂×C₅×C₈	C₂³	1	2	yes	—
62	C₂×C₅×C₈	C₂³	1	2	yes	—
63	C₅×C₄⋊C₄	—	2	2	no	—
64	C₅×C₄⋊C₄	C₂³	1	2	yes	—
65	C₅×C₂²⋊C₄	—	2	1	no	—
66	C₅×C₂²⋊C₄	—	2	4	no	—
67	C₅×C₂²⋊C₄	—	2	4	no	—
68	C₅×C₂²⋊C₄	—	2	2	no	—
69	C₄²×C₅	—	2	1	no	—
70	C₄²×C₅	—	2	2	no	—
71	C₄²×C₅	D₄	1	1	yes	characteristic
72	C₄²×C₅	D₄	1	1	yes	characteristic
73	C₄²×C₅	C₂³	1	2	yes	—
74	C₂×#137	C₂×C₅	1	1	yes	characteristic, complemented
75	C₂×AGL(1,ℤ/8)	C₂×C₅	1	1	yes	characteristic
76	C₂×#124	C₂×C₅	1	1	yes	characteristic, complemented
77	C₄×#41	C₂×C₅	1	1	yes	characteristic, complemented
78	#464	C₂×C₅	1	4	yes	complemented
79	#463	C₂×C₅	1	4	yes	complemented
80	#423	C₂×C₅	1	1	yes	characteristic, complemented
81	#419	C₂×C₅	1	1	yes	characteristic, complemented
82	#418	C₂×C₅	1	1	yes	characteristic, complemented
83	C₂³×C₅	—	2	1	no	—
84	C₂³×C₅	—	2	2	no	—
85	C₂³×C₅	—	4	4	no	—
86	C₂³×C₅	—	4	2	no	—
87	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
88	C₅×Q₈	—	2	1	no	—
89	C₅×Q₈	—	2	2	no	—
90	C₅×Q₈	—	2	2	no	—
91	C₅×Q₈	—	4	2	no	—
92	C₅×Q₈	C₂×D₄	1	2	yes	—
93	C₅×D₄	—	2	4	no	—
94	C₅×D₄	—	2	2	no	—
95	C₅×D₄	—	2	1	no	—
96	C₅×D₄	—	2	2	no	—
97	C₅×D₄	—	4	2	no	—
98	C₅×D₄	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₅	—	2	1	no	—
100	C₂×C₄×C₅	—	2	4	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	4	2	no	—
104	C₂×C₄×C₅	—	4	4	no	—
105	C₂×C₄×C₅	—	4	2	no	—
106	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₅	#41	1	2	yes	—
111	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
112	C₅×C₈	—	2	2	no	—
113	C₅×C₈	—	2	2	no	—
114	C₂×#41	C₂²×C₅	1	1	yes	characteristic
115	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
116	C₂²×D₄	C₂²×C₅	1	1	yes	characteristic
117	#137	—	2	4	no	—
118	AGL(1,ℤ/8)	—	2	4	no	—
119	C₂×Q₁₆	C₂²×C₅	1	2	yes	complemented
120	C₂×SD₁₆	C₂²×C₅	1	2	yes	—
121	C₂×SD₁₆	C₂²×C₅	1	2	yes	complemented
122	C₂×D₈	C₂²×C₅	1	2	yes	—
123	C₂×M₁₆	C₂²×C₅	1	1	yes	characteristic
124	C₂×M₁₆	C₂²×C₅	1	1	yes	characteristic
125	#124	—	2	1	no	—
126	#124	—	2	1	no	—
127	#124	C₂²×C₅	1	4	yes	—
128	C₄×Q₈	C₂²×C₅	1	2	yes	complemented
129	C₄×D₄	—	2	2	no	—
130	C₄×D₄	C₂²×C₅	1	2	yes	complemented
131	#117	—	2	1	no	—
132	#117	C₂²×C₅	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	—	2	2	no	—
134	C₂×C₄²	—	2	1	no	—
135	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
136	C₄⋊C₈	C₂²×C₅	1	4	yes	complemented
137	#103	C₂²×C₅	1	4	yes	complemented
138	#102	C₂²×C₅	1	4	yes	—
139	C₂²×C₅	—	2	2	no	—
140	C₂²×C₅	—	2	2	no	—
141	C₂²×C₅	—	4	2	no	—
142	C₂²×C₅	—	4	2	no	—
143	C₂²×C₅	—	4	4	no	—
144	C₂²×C₅	—	8	4	no	—
145	C₂²×C₅	#121	1	2	yes	—
146	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
147	C₄×C₅	—	2	2	no	—
148	C₄×C₅	—	2	2	no	—
149	C₄×C₅	—	2	2	no	—
150	C₄×C₅	—	4	2	no	—
151	C₄×C₅	—	4	2	no	—
152	C₄×C₅	#121	1	2	yes	—
153	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₄×C₅	C₂×#41	1	1	yes	characteristic
155	C₂⁴	—	2	1	no	—
156	#41	—	2	2	no	—
157	#41	C₅×D₄	1	4	yes	—
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	C₂³×C₅	1	2	yes	—
162	C₂×Q₈	C₂³×C₅	1	1	yes	characteristic
163	C₂×D₄	—	2	1	no	—
164	C₂×D₄	—	2	2	no	—
165	C₂×D₄	—	2	1	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	C₂³×C₅	1	2	yes	—
168	C₂×D₄	C₂³×C₅	1	1	yes	characteristic
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	C₅×D₄	1	2	yes	—
173	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
174	Q₁₆	—	2	4	no	—
175	SD₁₆	—	2	4	no	—
176	SD₁₆	—	2	4	no	—
177	D₈	—	2	4	no	—
178	M₁₆	—	2	2	no	—
179	M₁₆	—	2	2	no	—
180	C₂×C₈	C₂³×C₅	1	2	yes	—
181	C₂×C₈	C₂³×C₅	1	2	yes	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	C₂³×C₅	1	2	yes	—
184	C₂²⋊C₄	—	2	1	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	2	2	no	—
190	C₄²	C₅×D₄	1	1	yes	characteristic
191	C₄²	C₅×D₄	1	1	yes	characteristic
192	C₄²	C₂³×C₅	1	2	yes	—
193	C₂×C₅	—	2	2	no	—
194	C₂×C₅	—	4	2	no	—
195	C₂×C₅	—	8	2	no	—
196	C₂×C₅	C₂×#121	1	1	yes	characteristic
197	C₂×C₅	#575	1	2	yes	—
198	C₂³	—	2	1	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	4	4	no	—
201	C₂³	—	4	2	no	—
202	C₂³	C₂×C₅×D₄	1	1	yes	characteristic
203	Q₈	—	2	1	no	—
204	Q₈	—	2	2	no	—
205	Q₈	—	2	2	no	—
206	Q₈	—	4	2	no	—
207	Q₈	C₂×C₅×D₄	1	2	yes	—
208	D₄	—	2	4	no	—
209	D₄	—	2	2	no	—
210	D₄	—	2	1	no	—
211	D₄	—	2	2	no	—
212	D₄	—	4	2	no	—
213	D₄	C₂×C₅×D₄	1	2	yes	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
222	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
223	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
224	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
225	C₂×C₄	C₅×#41	1	2	yes	—
226	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₅	#3040	1	1	yes	characteristic, direct product factor, Sylow
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	4	no	—
235	C₂²	—	8	4	no	—
236	C₂²	C₅×#121	1	2	yes	—
237	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	2	no	—
243	C₄	C₅×#121	1	2	yes	—
244	C₄	C₂²×C₅×D₄	1	1	yes	characteristic
245	C₄	C₂×C₅×#41	1	1	yes	characteristic
246	C₂	—	2	2	no	—
247	C₂	—	4	2	no	—
248	C₂	—	8	2	no	—
249	C₂	C₂×C₅×#121	1	1	yes	characteristic
250	C₂	C₅×#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₂×C₅×AGL(1,ℤ/8)	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#137	C₂	semidirect
3. C₂×C₅×#124	C₂	semidirect
4. C₄×C₅×#41	C₂	semidirect
5. C₅×#464	C₂	semidirect
6. C₅×#463	C₂	semidirect
7. C₅×#423	C₂	semidirect
8. C₅×#419	C₂	semidirect
9. C₅×#418	C₂	semidirect
15. C₂×C₅×Q₁₆	C₂²	semidirect
17. C₂×C₅×SD₁₆	C₂²	semidirect
24. C₄×C₅×Q₈	C₂²	semidirect
26. C₄×C₅×D₄	C₂²	semidirect
32. C₅×C₄⋊C₈	C₂²	semidirect
33. C₅×#103	C₂²	semidirect
35. #3040	C₅	direct (N × H)
74. C₂×#137	C₂×C₅	semidirect
76. C₂×#124	C₂×C₅	semidirect
77. C₄×#41	C₂×C₅	semidirect
78. #464	C₂×C₅	semidirect
79. #463	C₂×C₅	semidirect
80. #423	C₂×C₅	semidirect
81. #419	C₂×C₅	semidirect
82. #418	C₂×C₅	semidirect
119. C₂×Q₁₆	C₂²×C₅	semidirect
121. C₂×SD₁₆	C₂²×C₅	semidirect
128. C₄×Q₈	C₂²×C₅	semidirect
130. C₄×D₄	C₂²×C₅	semidirect
136. C₄⋊C₈	C₂²×C₅	semidirect
137. #103	C₂²×C₅	semidirect
229. C₅	#3040	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×C₅×#137	2
C₂×C₅×AGL(1,ℤ/8)	2
C₂×C₅×#124	2
C₄×C₅×#41	2
C₅×#464	2
C₅×#463	2
C₅×#423	2
C₅×#419	2
C₅×#418	2
#3040	5

Subgroups — C5×#3040 #10616635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3040 #10616635