Subgroups — C₂×#10604702 #13043101

Order 1280.

918 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#3998	C₂	1	2	yes	maximal
2	C₂²×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	#10604702	C₂	1	8	yes	maximal, direct product factor
4	C₂×#64886	C₂	1	2	yes	maximal, complemented
5	C₂×#64885	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₅	C₄	1	1	yes	characteristic
8	C₂³×#168	C₂²	1	1	yes	characteristic
9	C₂×#3998	C₄	1	16	yes	complemented
10	C₂×#3998	C₂²	1	8	yes	—
11	C₂²×#674	C₂²	1	4	yes	—
12	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅	C₄	1	4	yes	—
14	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₅	C₂²	1	4	yes	—
18	#64886	C₂²	1	8	yes	complemented
19	#64885	C₂²	1	8	yes	complemented
20	C₂×C₄×M₃₂	—	5	1	no	maximal, Sylow
21	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂³×Dic₅	C₈	1	2	yes	—
23	C₂²×C₅×C₈	C₈	1	8	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₅	C₈	1	4	yes	—
28	C₂×C₄×Dic₅	C₈	1	2	yes	—
29	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₅	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
32	C₂²×#168	C₂×C₄	1	2	yes	—
33	C₂²×#168	C₂×C₄	1	1	yes	characteristic
34	C₂²×#168	C₂×C₄	1	8	yes	—
35	C₂²×#168	C₂³	1	1	yes	characteristic
36	C₂²×#168	C₂³	1	2	yes	—
37	#3998	C₂×C₄	1	32	yes	complemented
38	C₂×#674	C₂×C₄	1	16	yes	complemented
39	C₂×#674	C₂³	1	8	yes	—
40	C₈×Dic₅	C₂×C₄	1	2	yes	—
41	C₈×Dic₅	C₂×C₄	1	2	yes	—
42	C₈×Dic₅	C₂×C₄	1	4	yes	—
43	C₈×Dic₅	C₂×C₄	1	4	yes	—
44	C₈×Dic₅	C₂³	1	2	yes	—
45	C₈×Dic₅	C₂³	1	2	yes	—
46	C₂²×M₃₂	—	5	2	no	—
47	C₂²×C₄×C₈	—	5	1	no	—
48	C₄×M₃₂	—	5	8	no	—
49	C₂×#346	—	5	2	no	—
50	C₂×C₄×C₁₆	—	5	2	no	—
51	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₄²	1	4	yes	—
53	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
56	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₅	—	2	8	no	—
59	C₂²×Dic₅	C₂×C₈	1	2	yes	—
60	C₂²×Dic₅	C₂×C₈	1	8	yes	—
61	C₂²×Dic₅	C₂×C₈	1	2	yes	—
62	C₂×C₅×C₈	C₂×C₈	1	16	yes	—
63	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
64	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
65	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
66	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
67	C₄×Dic₅	C₂×C₈	1	2	yes	—
68	C₄×Dic₅	C₂×C₈	1	8	yes	—
69	C₄×Dic₅	C₂×C₈	1	2	yes	—
70	C₄×Dic₅	C₂²×C₄	1	2	yes	—
71	C₄×Dic₅	C₂²×C₄	1	2	yes	—
72	C₂×#168	C₄²	1	8	yes	—
73	C₂×#168	C₄²	1	4	yes	—
74	C₂×#168	C₂²×C₄	1	1	yes	characteristic
75	C₂×#168	C₂²×C₄	1	2	yes	—
76	C₂×#168	C₂²×C₄	1	4	yes	—
77	C₂×#168	C₂²×C₄	1	8	yes	—
78	C₂×#168	C₂⁴	1	1	yes	characteristic
79	#674	C₂²×C₄	1	16	yes	complemented
80	C₂³×C₈	—	5	1	no	—
81	C₂³×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
82	C₂²×C₄²	—	5	1	no	—
83	C₂×M₃₂	—	5	8	no	—
84	C₂×M₃₂	—	5	16	no	—
85	C₂²×C₁₆	—	5	4	no	—
86	C₂×C₄×C₈	—	5	1	no	—
87	C₂×C₄×C₈	—	5	4	no	—
88	C₂×C₄×C₈	—	5	4	no	—
89	C₂×C₄×C₈	—	5	1	no	—
90	C₂×C₄×C₈	—	5	1	no	—
91	C₂×C₄×C₈	—	5	1	no	—
92	#346	—	5	8	no	—
93	C₄×C₁₆	—	5	8	no	—
94	C₂³×C₅	—	2	4	no	—
95	C₂³×C₅	C₄×C₈	1	4	yes	—
96	C₂³×C₅	C₂²×C₈	1	2	yes	—
97	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₅	C₄×C₈	1	8	yes	—
99	C₂×C₄×C₅	C₄×C₈	1	4	yes	—
100	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
101	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
103	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
105	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
107	C₂×Dic₅	—	2	4	no	—
108	C₂×Dic₅	—	2	16	no	—
109	C₂×Dic₅	M₃₂	1	8	yes	complemented
110	C₂×Dic₅	C₂²×C₈	1	4	yes	—
111	C₂×Dic₅	C₂²×C₈	1	4	yes	—
112	C₅×C₈	C₂²×C₈	1	4	yes	—
113	C₅×C₈	C₂²×C₈	1	4	yes	—
114	#168	C₂×C₄²	1	2	yes	—
115	#168	C₂×C₄²	1	4	yes	—
116	#168	C₂³×C₄	1	2	yes	—
117	C₂³×C₄	—	5	2	no	—
118	C₂³×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
119	C₂²×C₈	—	5	8	no	—
120	C₂²×C₈	—	5	2	no	—
121	C₂²×C₈	—	5	1	no	—
122	C₂²×C₈	—	5	1	no	—
123	C₂²×C₈	—	5	2	no	—
124	C₂²×C₈	#170	1	8	yes	—
125	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
126	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	4	yes	—
127	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
128	C₂×C₄²	—	5	2	no	—
129	C₂×C₄²	—	5	4	no	—
130	C₂×C₄²	—	5	1	no	—
131	C₂×C₄²	—	5	4	no	—
132	C₂×C₄²	—	5	1	no	—
133	M₃₂	—	5	32	no	—
134	C₂×C₁₆	—	5	8	no	—
135	C₂×C₁₆	—	5	16	no	—
136	C₄×C₈	—	5	2	no	—
137	C₄×C₈	—	5	2	no	—
138	C₄×C₈	—	5	2	no	—
139	C₄×C₈	—	5	4	no	—
140	C₄×C₈	—	5	4	no	—
141	C₄×C₈	—	5	2	no	—
142	C₂²×C₅	—	2	8	no	—
143	C₂²×C₅	—	2	4	no	—
144	C₂²×C₅	C₂×C₄×C₈	1	4	yes	—
145	C₂²×C₅	C₂×C₄×C₈	1	2	yes	—
146	C₂²×C₅	C₂×M₃₂	1	4	yes	—
147	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
148	C₄×C₅	C₂×C₄×C₈	1	2	yes	—
149	C₄×C₅	C₂×C₄×C₈	1	4	yes	—
150	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
151	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
152	Dic₅	—	2	4	no	—
153	Dic₅	C₂×M₃₂	1	8	yes	complemented
154	C₂⁴	D₅⋊C₈	1	1	yes	characteristic
155	C₂²×C₄	—	5	2	no	—
156	C₂²×C₄	—	5	8	no	—
157	C₂²×C₄	—	5	2	no	—
158	C₂²×C₄	—	10	8	no	—
159	C₂²×C₄	D₅⋊C₈	1	2	yes	—
160	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
161	C₂²×C₄	C₄×AGL(1,𝔽₅)	1	4	yes	—
162	C₂²×C₄	C₂×#170	1	4	yes	—
163	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
164	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
165	C₂×C₈	—	5	4	no	—
166	C₂×C₈	—	5	8	no	—
167	C₂×C₈	—	5	8	no	—
168	C₂×C₈	—	5	4	no	—
169	C₂×C₈	—	5	1	no	—
170	C₂×C₈	—	5	1	no	—
171	C₂×C₈	—	5	2	no	—
172	C₂×C₈	C₂×#170	1	4	yes	—
173	C₂×C₈	C₂×#170	1	16	yes	—
174	C₂×C₈	C₂×#170	1	4	yes	—
175	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
176	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
177	C₄²	—	5	2	no	—
178	C₄²	—	5	8	no	—
179	C₄²	—	5	2	no	—
180	C₄²	—	5	2	no	—
181	C₄²	—	5	2	no	—
182	C₁₆	—	5	16	no	—
183	C₂×C₅	—	2	4	no	—
184	C₂×C₅	C₄×M₃₂	1	4	yes	complemented
185	C₂×C₅	C₂²×C₄×C₈	1	1	yes	characteristic
186	C₂×C₅	C₂²×M₃₂	1	2	yes	—
187	C₂³	—	2	4	no	—
188	C₂³	C₄×#170	1	4	yes	—
189	C₂³	C₂×D₅⋊C₈	1	2	yes	—
190	C₂³	C₂×D₅⋊C₈	1	1	yes	characteristic
191	C₂×C₄	—	5	8	no	—
192	C₂×C₄	—	5	4	no	—
193	C₂×C₄	—	5	4	no	—
194	C₂×C₄	—	10	16	no	—
195	C₂×C₄	—	10	4	no	—
196	C₂×C₄	C₄×#170	1	8	yes	—
197	C₂×C₄	C₄×#170	1	4	yes	—
198	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
199	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	4	yes	—
201	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	2	yes	—
202	C₂×C₄	C₂²×#170	1	4	yes	—
203	C₂×C₄	C₂²×#170	1	2	yes	—
204	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
205	C₈	—	5	2	no	—
206	C₈	—	5	2	no	—
207	C₈	—	5	4	no	—
208	C₈	C₂²×#170	1	4	yes	—
209	C₈	C₂²×#170	1	4	yes	—
210	C₅	C₂×C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
211	C₂²	—	2	4	no	—
212	C₂²	—	2	8	no	—
213	C₂²	D₅⋊M₃₂	1	4	yes	—
214	C₂²	C₂×C₄×#170	1	4	yes	—
215	C₂²	C₂×C₄×#170	1	2	yes	—
216	C₂²	C₂²×D₅⋊C₈	1	1	yes	characteristic
217	C₄	—	5	8	no	—
218	C₄	—	10	4	no	—
219	C₄	C₂×C₄×#170	1	4	yes	—
220	C₄	C₂×C₄×#170	1	2	yes	—
221	C₄	C₂²×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
222	C₄	C₂³×#170	1	1	yes	characteristic
223	C₂	—	2	4	no	—
224	C₂	#10604702	1	4	yes	direct product factor
225	C₂	C₂×D₅⋊M₃₂	1	2	yes	—
226	C₂	C₂²×C₄×#170	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₂³×C₅×C₈	C₄
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂×#168	C₂⁴
hypercenter	C₂³×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₄×M₃₂
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. #10604702	C₂	direct (N × H)
4. C₂×#64886	C₂	semidirect
5. C₂×#64885	C₂	semidirect
9. C₂×#3998	C₄	semidirect
18. #64886	C₂²	semidirect
19. #64885	C₂²	semidirect
37. #3998	C₂×C₄	semidirect
38. C₂×#674	C₂×C₄	semidirect
79. #674	C₂²×C₄	semidirect
109. C₂×Dic₅	M₃₂	semidirect
153. Dic₅	C₂×M₃₂	semidirect
184. C₂×C₅	C₄×M₃₂	semidirect
210. C₅	C₂×C₄×M₃₂	semidirect
224. C₂	#10604702	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×#3998	2
C₂²×C₈×Dic₅	2
#10604702	2
C₂×#64886	2
C₂×#64885	2
C₂×C₄×M₃₂	5

Subgroups — C2×#10604702 #13043101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10604702 #13043101