Subgroups — C₂×#66103 #10619085

Order 640.

1512 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4149	C₂	1	2	yes	maximal, complemented
2	C₂×C₅×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#4095	C₂	1	2	yes	maximal, complemented
4	#66103	C₂	1	16	yes	maximal, direct product factor
5	C₂×#4065	C₂	1	2	yes	maximal, complemented
6	C₂×#4060	C₂	1	4	yes	maximal, complemented
7	C₂×D₅×D₈	C₂	1	2	yes	maximal, complemented
8	C₂²×D₄₀	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#4047	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#41	C₂²	1	2	yes	complemented
11	#4149	—	2	8	no	—
12	#4149	C₂²	1	16	yes	complemented
13	C₂×#713	C₂²	1	4	yes	complemented
14	C₂×D₄×D₅	—	2	4	no	—
15	C₂×D₄×D₅	C₂²	1	4	yes	complemented
16	C₂×#709	—	2	2	no	—
17	C₂×#709	C₂²	1	1	yes	characteristic
18	C₂²×D₂₀	—	2	2	no	—
19	C₂²×D₂₀	C₂²	1	2	yes	complemented
20	C₅×#135	C₂²	1	8	yes	complemented
21	C₂×C₅×Q₁₆	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₅×SD₁₆	C₂²	1	2	yes	complemented
23	C₂×C₅×D₈	C₂²	1	1	yes	characteristic, complemented
24	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
25	#4095	C₂²	1	16	yes	complemented
26	C₂×#688	C₂²	1	4	yes	complemented
27	C₂×#686	C₂²	1	4	yes	complemented
28	C₂×#681	C₂²	1	1	yes	characteristic, complemented
29	#4065	C₂²	1	16	yes	complemented
30	#4060	C₂²	1	32	yes	complemented
31	D₅×D₈	C₂²	1	16	yes	complemented
32	C₂×D₄₀	C₂²	1	2	yes	complemented
33	C₂×D₄₀	C₂²	1	2	yes	complemented
34	C₂×D₄₀	C₂²	1	8	yes	complemented
35	#4047	C₂²	1	8	yes	complemented
36	C₂×#676	C₂²	1	2	yes	complemented
37	C₂×C₈×D₅	C₂²	1	2	yes	complemented
38	C₂×#576	—	5	1	no	maximal, Sylow
39	C₂³×D₅	—	2	2	no	—
40	C₂³×D₅	—	2	4	no	—
41	C₅×#41	—	2	4	no	—
42	C₅×#41	C₂³	1	8	yes	complemented
43	C₂×C₅×Q₈	C₂³	1	2	yes	—
44	C₂×C₅×D₄	—	2	2	no	—
45	C₂×C₅×D₄	C₂³	1	2	yes	—
46	C₂²×C₄×C₅	—	2	2	no	—
47	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₅	—	2	4	no	—
49	C₂×C₂²⋊D₅	D₄	1	2	yes	—
50	#713	—	2	8	no	—
51	#713	C₂³	1	16	yes	complemented
52	D₄×D₅	—	2	32	no	—
53	D₄×D₅	—	2	8	no	—
54	D₄×D₅	C₂³	1	16	yes	complemented
55	#709	—	2	16	no	—
56	#709	D₄	1	4	yes	—
57	#709	C₂³	1	4	yes	—
58	C₂×D₂₀	—	2	4	no	—
59	C₂×D₂₀	—	2	2	no	—
60	C₂×D₂₀	—	2	4	no	—
61	C₂×D₂₀	—	2	16	no	—
62	C₂×D₂₀	D₄	1	1	yes	characteristic
63	C₂×D₂₀	C₂³	1	4	yes	—
64	C₂×D₂₀	C₂³	1	8	yes	complemented
65	C₂×C₄×D₅	—	2	4	no	—
66	C₂×C₄×D₅	C₂³	1	2	yes	—
67	C₂×Dic₁₀	D₄	1	1	yes	characteristic
68	C₅×Q₁₆	C₂³	1	4	yes	complemented
69	C₅×SD₁₆	C₂³	1	8	yes	complemented
70	C₅×D₈	C₂³	1	4	yes	complemented
71	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂³	1	4	yes	complemented
74	#688	C₂³	1	16	yes	complemented
75	#686	C₂³	1	16	yes	complemented
76	#681	C₂³	1	4	yes	complemented
77	C₂×#168	C₂³	1	2	yes	—
78	D₄₀	C₂³	1	8	yes	complemented
79	D₄₀	C₂³	1	8	yes	complemented
80	#676	C₂³	1	8	yes	complemented
81	C₈×D₅	C₂³	1	8	yes	complemented
82	C₂×H(5,𝔽₂)	—	5	2	no	—
83	#576	—	5	16	no	—
84	C₂×AGL(1,ℤ/8)	—	5	2	no	—
85	C₂×AGL(1,ℤ/8)	—	5	4	no	—
86	C₂×#135	—	5	2	no	—
87	C₂×#135	D₅	1	1	yes	characteristic, complemented
88	C₂²×D₈	—	5	2	no	—
89	C₂²×D₈	—	5	1	no	—
90	C₂×#131	—	5	1	no	—
91	C₂³×C₅	—	2	2	no	—
92	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
93	C₂²×D₅	—	2	4	no	—
94	C₂²×D₅	—	2	8	no	—
95	C₂²×D₅	—	2	16	no	—
96	C₂²×D₅	—	8	8	no	—
97	C₂²×D₅	—	8	4	no	—
98	C₂²×D₅	C₂×D₄	1	2	yes	—
99	C₅×Q₈	—	2	2	no	—
100	C₅×Q₈	C₂⁴	1	4	yes	—
101	C₅×D₄	—	2	8	no	—
102	C₅×D₄	—	2	2	no	—
103	C₅×D₄	C₂⁴	1	4	yes	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	8	no	—
106	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂⁴	1	2	yes	—
110	C₂²⋊D₅	—	2	16	no	—
111	C₂²⋊D₅	C₂×D₄	1	8	yes	—
112	C₂×Dic₅	C₂×D₄	1	2	yes	—
113	D₂₀	—	2	16	no	—
114	D₂₀	—	2	8	no	—
115	D₂₀	—	2	4	no	—
116	D₂₀	C₂×D₄	1	4	yes	—
117	D₂₀	C₂⁴	1	8	yes	—
118	C₄×D₅	—	2	16	no	—
119	C₄×D₅	C₂×D₄	1	4	yes	—
120	C₄×D₅	C₂⁴	1	4	yes	—
121	Dic₁₀	C₂×D₄	1	4	yes	—
122	C₅×C₈	C₂⁴	1	2	yes	—
123	C₅×C₈	C₂⁴	1	2	yes	—
124	#168	C₂⁴	1	4	yes	—
125	H(5,𝔽₂)	—	5	16	no	—
126	H(5,𝔽₂)	—	10	8	no	—
127	C₂×#41	—	5	1	no	—
128	C₂×#41	—	5	4	no	—
129	C₂×#41	—	10	2	no	—
130	C₂×#41	C₂×D₅	1	2	yes	complemented
131	C₂²×D₄	—	5	4	no	—
132	C₂²×D₄	—	5	2	no	—
133	C₂²×D₄	—	10	2	no	—
134	C₂²×D₄	—	10	4	no	—
135	AGL(1,ℤ/8)	—	5	16	no	—
136	AGL(1,ℤ/8)	—	5	32	no	—
137	#135	—	5	16	no	—
138	#135	C₂×D₅	1	8	yes	complemented
139	C₂×Q₁₆	C₂×D₅	1	1	yes	characteristic, complemented
140	C₂×SD₁₆	—	5	4	no	—
141	C₂×SD₁₆	C₂×D₅	1	2	yes	complemented
142	C₂×D₈	—	5	4	no	—
143	C₂×D₈	—	5	16	no	—
144	C₂×D₈	—	5	2	no	—
145	C₂×D₈	—	5	2	no	—
146	C₂×D₈	—	5	8	no	—
147	C₂×D₈	C₂×D₅	1	1	yes	characteristic, complemented
148	#131	—	5	8	no	—
149	C₂×M₁₆	—	5	1	no	—
150	C₂×M₁₆	—	5	2	no	—
151	C₂²×C₈	—	5	2	no	—
152	C₂²×C₈	C₂×D₅	1	1	yes	characteristic, complemented
153	C₂²×C₅	—	2	2	no	—
154	C₂²×C₅	—	2	4	no	—
155	C₂²×C₅	—	4	4	no	—
156	C₂²×C₅	C₂²×D₄	1	2	yes	—
157	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
158	C₂×D₅	—	2	4	no	—
159	C₂×D₅	—	2	8	no	—
160	C₂×D₅	—	4	8	no	—
161	C₂×D₅	—	8	16	no	—
162	C₂×D₅	—	8	8	no	—
163	C₂×D₅	C₂²×D₄	1	4	yes	—
164	C₄×C₅	—	2	4	no	—
165	C₄×C₅	C₂²×D₄	1	2	yes	—
166	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
167	C₄×C₅	C₂⁵	1	1	yes	characteristic
168	Dic₅	C₂²×D₄	1	4	yes	—
169	C₂⁴	—	10	4	no	—
170	C₂⁴	—	10	2	no	—
171	#41	—	2	4	no	—
172	#41	—	5	4	no	—
173	#41	—	5	4	no	—
174	#41	—	5	16	no	—
175	#41	—	10	16	no	—
176	#41	—	10	8	no	—
177	#41	C₂²×D₅	1	8	yes	complemented
178	C₂×Q₈	—	5	1	no	—
179	C₂×Q₈	C₂²×D₅	1	2	yes	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	5	2	no	—
182	C₂×D₄	—	5	16	no	—
183	C₂×D₄	—	5	1	no	—
184	C₂×D₄	—	5	4	no	—
185	C₂×D₄	—	5	8	no	—
186	C₂×D₄	—	10	16	no	—
187	C₂×D₄	—	10	4	no	—
188	C₂×D₄	—	10	8	no	—
189	C₂×D₄	—	10	4	no	—
190	C₂×D₄	—	10	32	no	—
191	C₂×D₄	—	10	2	no	—
192	C₂×D₄	—	10	4	no	—
193	C₂×D₄	C₂²×D₅	1	2	yes	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	5	2	no	—
196	C₂²×C₄	—	10	4	no	—
197	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
198	Q₁₆	C₂²×D₅	1	4	yes	complemented
199	SD₁₆	—	5	16	no	—
200	SD₁₆	C₂²×D₅	1	8	yes	complemented
201	D₈	—	5	16	no	—
202	D₈	—	5	8	no	—
203	D₈	—	5	8	no	—
204	D₈	C₂²×D₅	1	4	yes	complemented
205	M₁₆	—	5	4	no	—
206	M₁₆	—	5	8	no	—
207	C₂×C₈	—	5	2	no	—
208	C₂×C₈	—	5	8	no	—
209	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
211	C₂×C₈	C₂²×D₅	1	4	yes	complemented
212	C₂×C₅	—	2	2	no	—
213	C₂×C₅	—	4	4	no	—
214	C₂×C₅	#576	1	2	yes	complemented
215	C₂×C₅	C₂³×D₄	1	1	yes	characteristic
216	D₅	—	2	4	no	—
217	D₅	—	4	8	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	5	2	no	—
220	C₂³	—	10	16	no	—
221	C₂³	—	10	4	no	—
222	C₂³	—	10	8	no	—
223	C₂³	—	40	4	no	—
224	C₂³	—	40	8	no	—
225	C₂³	D₄×D₅	1	1	yes	characteristic
226	Q₈	—	2	2	no	—
227	Q₈	—	5	4	no	—
228	Q₈	C₂³×D₅	1	4	yes	—
229	D₄	—	2	8	no	—
230	D₄	—	2	2	no	—
231	D₄	—	5	8	no	—
232	D₄	—	5	4	no	—
233	D₄	—	5	8	no	—
234	D₄	—	10	16	no	—
235	D₄	—	10	16	no	—
236	D₄	—	10	8	no	—
237	D₄	—	10	4	no	—
238	D₄	C₂³×D₅	1	4	yes	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	5	2	no	—
242	C₂×C₄	—	5	4	no	—
243	C₂×C₄	—	5	4	no	—
244	C₂×C₄	—	10	16	no	—
245	C₂×C₄	D₄×D₅	1	2	yes	—
246	C₂×C₄	D₄×D₅	1	1	yes	characteristic
247	C₂×C₄	C₂³×D₅	1	2	yes	—
248	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
249	C₈	—	5	4	no	—
250	C₈	C₂³×D₅	1	2	yes	—
251	C₈	C₂³×D₅	1	2	yes	—
252	C₅	C₂×#576	1	1	yes	characteristic, complemented, Sylow
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	5	4	no	—
257	C₂²	—	10	4	no	—
258	C₂²	—	10	8	no	—
259	C₂²	—	20	8	no	—
260	C₂²	—	40	8	no	—
261	C₂²	—	40	16	no	—
262	C₂²	C₂×D₄×D₅	1	2	yes	—
263	C₂²	C₂×D₄×D₅	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	5	4	no	—
266	C₄	C₂×D₄×D₅	1	2	yes	—
267	C₄	C₂×D₄×D₅	1	1	yes	characteristic
268	C₄	C₂⁴×D₅	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	4	4	no	—
271	C₂	—	10	4	no	—
272	C₂	—	20	8	no	—
273	C₂	#66103	1	2	yes	direct product factor
274	C₂	C₂²×D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅
commutator, cosocle	C₄×C₅	C₂⁵
Frattini	C₄	C₂⁴×D₅
Fitting	C₂×C₅×#135	C₂
socle	C₂²×C₅	C₂²×D₄
hypercenter	C₂×#135	D₅
hypocenter	C₅	C₂×#576
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#4149	C₂	semidirect
2. C₂×C₅×#135	C₂	semidirect
3. C₂×#4095	C₂	semidirect
4. #66103	C₂	direct (N × H)
5. C₂×#4065	C₂	semidirect
6. C₂×#4060	C₂	semidirect
7. C₂×D₅×D₈	C₂	semidirect
8. C₂²×D₄₀	C₂	semidirect
9. C₂×#4047	C₂	semidirect
10. C₂×C₅×#41	C₂²	semidirect
12. #4149	C₂²	semidirect
13. C₂×#713	C₂²	semidirect
15. C₂×D₄×D₅	C₂²	semidirect
19. C₂²×D₂₀	C₂²	semidirect
20. C₅×#135	C₂²	semidirect
21. C₂×C₅×Q₁₆	C₂²	semidirect
22. C₂×C₅×SD₁₆	C₂²	semidirect
23. C₂×C₅×D₈	C₂²	semidirect
24. C₂²×C₅×C₈	C₂²	semidirect
25. #4095	C₂²	semidirect
26. C₂×#688	C₂²	semidirect
27. C₂×#686	C₂²	semidirect
28. C₂×#681	C₂²	semidirect
29. #4065	C₂²	semidirect
30. #4060	C₂²	semidirect
31. D₅×D₈	C₂²	semidirect
32. C₂×D₄₀	C₂²	semidirect
33. C₂×D₄₀	C₂²	semidirect
34. C₂×D₄₀	C₂²	semidirect
35. #4047	C₂²	semidirect
36. C₂×#676	C₂²	semidirect
37. C₂×C₈×D₅	C₂²	semidirect
42. C₅×#41	C₂³	semidirect
51. #713	C₂³	semidirect
54. D₄×D₅	C₂³	semidirect
64. C₂×D₂₀	C₂³	semidirect
68. C₅×Q₁₆	C₂³	semidirect
69. C₅×SD₁₆	C₂³	semidirect
70. C₅×D₈	C₂³	semidirect
73. C₂×C₅×C₈	C₂³	semidirect
74. #688	C₂³	semidirect
75. #686	C₂³	semidirect
76. #681	C₂³	semidirect
78. D₄₀	C₂³	semidirect
79. D₄₀	C₂³	semidirect
80. #676	C₂³	semidirect
81. C₈×D₅	C₂³	semidirect
87. C₂×#135	D₅	semidirect
130. C₂×#41	C₂×D₅	semidirect
138. #135	C₂×D₅	semidirect
139. C₂×Q₁₆	C₂×D₅	semidirect
141. C₂×SD₁₆	C₂×D₅	semidirect
147. C₂×D₈	C₂×D₅	semidirect
152. C₂²×C₈	C₂×D₅	semidirect
177. #41	C₂²×D₅	semidirect
198. Q₁₆	C₂²×D₅	semidirect
200. SD₁₆	C₂²×D₅	semidirect
204. D₈	C₂²×D₅	semidirect
211. C₂×C₈	C₂²×D₅	semidirect
214. C₂×C₅	#576	semidirect
252. C₅	C₂×#576	semidirect
273. C₂	#66103	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×#4149	2
C₂×C₅×#135	2
C₂×#4095	2
#66103	2
C₂×#4065	2
C₂×#4060	2
C₂×D₅×D₈	2
C₂²×D₄₀	2
C₂×#4047	2
C₂×#576	5

Subgroups — C2×#66103 #10619085

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#66103 #10619085