Subgroups — C₂×C₅×#463 #10616626

Order 640.

564 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₂²×C₅×D₈	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×SD₁₆	C₂	semidirect
3. C₂×C₅×#124	C₂	semidirect
4. C₂×C₄×C₅×D₄	C₂	semidirect
5. C₅×#463	C₂	direct (N × H)
6. C₂×C₅×C₄⋊C₈	C₂	semidirect
7. C₂×C₅×#103	C₂	semidirect
8. C₂×C₅×#102	C₂	semidirect
14. C₂×C₅×SD₁₆	C₂²	semidirect
20. C₅×#124	C₂²	semidirect
22. C₄×C₅×D₄	C₂²	semidirect
28. C₅×C₄⋊C₈	C₂²	semidirect
29. C₅×#103	C₂²	semidirect
30. C₅×#102	C₂²	semidirect
31. C₂×#463	C₅	direct (N × H)
66. C₂²×SD₁₆	C₂×C₅	semidirect
68. C₂×#124	C₂×C₅	semidirect
69. C₂×C₄×D₄	C₂×C₅	semidirect
70. #463	C₂×C₅	direct (N × H)
71. C₂×C₄⋊C₈	C₂×C₅	semidirect
72. C₂×#103	C₂×C₅	semidirect
73. C₂×#102	C₂×C₅	semidirect
112. C₂×SD₁₆	C₂²×C₅	semidirect
118. #124	C₂²×C₅	semidirect
120. C₄×D₄	C₂²×C₅	semidirect
126. C₄⋊C₈	C₂²×C₅	semidirect
127. #103	C₂²×C₅	semidirect
128. #102	C₂²×C₅	semidirect
184. C₂×C₅	#463	direct (N × H)
221. C₅	C₂×#463	direct (N × H)
243. C₂	C₅×#463	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×SD₁₆	2
C₂²×C₅×D₈	2
C₂×C₅×#124	2
C₂×C₄×C₅×D₄	2
C₅×#463	2
C₂×C₅×C₄⋊C₈	2
C₂×C₅×#103	2
C₂×C₅×#102	2
C₂×#463	5

Subgroups — C2×C5×#463 #10616626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#463 #10616626