Subgroups — C₅×C₈×H(5,𝔽₂) #11983023

Order 1280.

2422 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₅×#2988	C₂	1	6	yes	maximal, complemented
3	C₅×#2972	C₂	1	9	yes	maximal, complemented
4	C₅×C₈×#41	C₂	1	6	yes	maximal, complemented
5	C₂×C₅×C₈×D₄	C₂	1	9	yes	maximal, complemented
6	C₂×C₅×H(5,𝔽₂)	C₄	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	6	yes	complemented
8	C₅×#520	C₄	1	6	yes	—
9	C₅×#518	C₄	1	9	yes	—
10	C₄×C₅×#41	C₂²	1	6	yes	—
11	C₂×C₄×C₅×D₄	C₂²	1	9	yes	—
12	C₅×C₈×Q₈	C₂²	1	6	yes	complemented
13	C₅×C₈×Q₈	C₂²	1	2	yes	complemented
14	C₅×C₈×D₄	C₂²	1	36	yes	complemented
15	C₅×C₈×D₄	C₂²	1	18	yes	complemented
16	C₅×C₈×D₄	C₂²	1	18	yes	complemented
17	C₅×#431	C₂²	1	18	yes	complemented
18	C₂×C₅×C₄⋊C₈	C₂²	1	9	yes	complemented
19	C₂×C₅×C₂²⋊C₈	C₂²	1	18	yes	complemented
20	C₂×C₄×C₅×C₈	C₂²	1	9	yes	complemented
21	C₈×H(5,𝔽₂)	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₅×H(5,𝔽₂)	C₈	1	16	yes	direct product factor
23	C₂×C₅×#41	C₂×C₄	1	6	yes	—
24	C₂²×C₅×D₄	C₂×C₄	1	9	yes	—
25	C₂³×C₄×C₅	C₂³	1	6	yes	—
26	C₂²×C₅×C₈	—	4	12	no	—
27	C₂²×C₅×C₈	C₂³	1	9	yes	—
28	C₂²×C₅×C₈	C₂³	1	9	yes	—
29	C₂²×C₅×C₈	C₂³	1	18	yes	—
30	C₄×C₅×Q₈	C₂×C₄	1	6	yes	—
31	C₄×C₅×Q₈	C₂³	1	2	yes	—
32	C₄×C₅×D₄	C₂×C₄	1	36	yes	—
33	C₄×C₅×D₄	C₂×C₄	1	18	yes	—
34	C₄×C₅×D₄	C₂³	1	18	yes	—
35	C₅×#117	C₂×C₄	1	18	yes	—
36	C₂×C₅×C₄⋊C₄	C₂×C₄	1	9	yes	—
37	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	18	yes	—
38	C₂×C₄²×C₅	C₂³	1	9	yes	—
39	C₅×C₄⋊C₈	C₂³	1	18	yes	—
40	C₅×C₄⋊C₈	C₂³	1	6	yes	—
41	C₅×C₂²⋊C₈	C₂³	1	36	yes	—
42	C₄×C₅×C₈	C₂³	1	18	yes	—
43	C₄×C₅×C₈	C₂³	1	6	yes	—
44	C₄×H(5,𝔽₂)	C₂×C₅	1	1	yes	characteristic
45	#2988	C₂×C₅	1	6	yes	complemented
46	#2972	C₂×C₅	1	9	yes	complemented
47	C₈×#41	C₂×C₅	1	6	yes	complemented
48	C₂×C₈×D₄	C₂×C₅	1	9	yes	complemented
49	C₂⁴×C₅	C₂²×C₄	1	6	yes	—
50	C₅×#41	C₂×C₈	1	48	yes	complemented
51	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
52	C₂×C₅×D₄	C₂×C₈	1	72	yes	complemented
53	C₂×C₅×D₄	C₂²×C₄	1	18	yes	—
54	C₂²×C₄×C₅	—	4	6	no	—
55	C₂²×C₄×C₅	—	4	6	no	—
56	C₂²×C₄×C₅	C₂²×C₄	1	9	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	18	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	9	yes	—
59	C₂²×C₄×C₅	C₂⁴	1	9	yes	—
60	C₂×C₅×C₈	—	2	18	no	—
61	C₂×C₅×C₈	—	4	18	no	—
62	C₂×C₅×C₈	—	4	9	no	—
63	C₂×C₅×C₈	C₂⁴	1	9	yes	—
64	C₂×C₅×C₈	C₂⁴	1	6	yes	—
65	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
66	C₅×C₄⋊C₄	C₂²×C₄	1	18	yes	—
67	C₅×C₄⋊C₄	C₂²×C₄	1	6	yes	—
68	C₅×C₂²⋊C₄	C₂²×C₄	1	36	yes	—
69	C₄²×C₅	C₂²×C₄	1	18	yes	—
70	C₄²×C₅	C₂⁴	1	6	yes	—
71	C₂×H(5,𝔽₂)	C₄×C₅	1	1	yes	characteristic
72	C₂³×C₈	C₂²×C₅	1	6	yes	complemented
73	#520	C₄×C₅	1	6	yes	—
74	#518	C₄×C₅	1	9	yes	—
75	C₄×#41	C₂²×C₅	1	6	yes	—
76	C₂×C₄×D₄	C₂²×C₅	1	9	yes	—
77	C₈×Q₈	C₂²×C₅	1	2	yes	complemented
78	C₈×Q₈	C₂²×C₅	1	6	yes	complemented
79	C₈×D₄	C₂²×C₅	1	18	yes	complemented
80	C₈×D₄	C₂²×C₅	1	36	yes	complemented
81	C₈×D₄	C₂²×C₅	1	18	yes	complemented
82	#431	C₂²×C₅	1	18	yes	complemented
83	C₂×C₄⋊C₈	C₂²×C₅	1	9	yes	complemented
84	C₂×C₂²⋊C₈	C₂²×C₅	1	18	yes	complemented
85	C₂×C₄×C₈	C₂²×C₅	1	9	yes	complemented
86	C₂³×C₅	—	4	6	no	—
87	C₂³×C₅	—	4	6	no	—
88	C₂³×C₅	C₂²×C₈	1	24	yes	complemented
89	C₂³×C₅	C₂³×C₄	1	9	yes	—
90	C₅×Q₈	C₂²×C₈	1	8	yes	complemented
91	C₅×D₄	C₂²×C₈	1	72	yes	complemented
92	C₂×C₄×C₅	—	2	9	no	—
93	C₂×C₄×C₅	—	2	9	no	—
94	C₂×C₄×C₅	—	4	18	no	—
95	C₂×C₄×C₅	—	4	9	no	—
96	C₂×C₄×C₅	C₂²×C₈	1	36	yes	complemented
97	C₂×C₄×C₅	C₂³×C₄	1	6	yes	—
98	C₂×C₄×C₅	C₂³×C₄	1	9	yes	—
99	C₂×C₄×C₅	C₂³×C₄	1	6	yes	—
100	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
101	C₅×C₈	—	2	9	no	—
102	C₅×C₈	—	2	6	no	—
103	C₅×C₈	H(5,𝔽₂)	1	2	yes	direct product factor
104	H(5,𝔽₂)	C₅×C₈	1	16	yes	direct product factor
105	C₂×#41	C₂×C₄×C₅	1	6	yes	—
106	C₂²×D₄	C₂×C₄×C₅	1	9	yes	—
107	C₂³×C₄	C₂³×C₅	1	6	yes	—
108	C₂²×C₈	—	4	12	no	—
109	C₂²×C₈	C₂³×C₅	1	9	yes	—
110	C₂²×C₈	C₂³×C₅	1	18	yes	—
111	C₂²×C₈	C₂³×C₅	1	9	yes	—
112	C₄×Q₈	C₂×C₄×C₅	1	6	yes	—
113	C₄×Q₈	C₂³×C₅	1	2	yes	—
114	C₄×D₄	C₂×C₄×C₅	1	36	yes	—
115	C₄×D₄	C₂×C₄×C₅	1	18	yes	—
116	C₄×D₄	C₂³×C₅	1	18	yes	—
117	#117	C₂×C₄×C₅	1	18	yes	—
118	C₂×C₄⋊C₄	C₂×C₄×C₅	1	9	yes	—
119	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	18	yes	—
120	C₂×C₄²	C₂³×C₅	1	9	yes	—
121	C₄⋊C₈	C₂³×C₅	1	18	yes	—
122	C₄⋊C₈	C₂³×C₅	1	6	yes	—
123	C₂²⋊C₈	C₂³×C₅	1	36	yes	—
124	C₄×C₈	C₂³×C₅	1	6	yes	—
125	C₄×C₈	C₂³×C₅	1	18	yes	—
126	C₂²×C₅	—	2	9	no	—
127	C₂²×C₅	—	2	9	no	—
128	C₂²×C₅	—	4	24	no	—
129	C₂²×C₅	C₂³×C₈	1	18	yes	—
130	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
131	C₄×C₅	—	2	9	no	—
132	C₄×C₅	—	2	6	no	—
133	C₄×C₅	C₂³×C₈	1	12	yes	—
134	C₄×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
135	C₄×C₅	#585	1	1	yes	characteristic
136	C₂⁴	C₂²×C₄×C₅	1	6	yes	—
137	#41	C₂×C₅×C₈	1	48	yes	complemented
138	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
139	C₂×D₄	C₂×C₅×C₈	1	72	yes	complemented
140	C₂×D₄	C₂²×C₄×C₅	1	18	yes	—
141	C₂²×C₄	—	4	6	no	—
142	C₂²×C₄	—	4	6	no	—
143	C₂²×C₄	C₂²×C₄×C₅	1	9	yes	—
144	C₂²×C₄	C₂²×C₄×C₅	1	18	yes	—
145	C₂²×C₄	C₂²×C₄×C₅	1	9	yes	—
146	C₂²×C₄	C₂⁴×C₅	1	9	yes	—
147	C₂×C₈	—	2	18	no	—
148	C₂×C₈	—	4	18	no	—
149	C₂×C₈	—	4	9	no	—
150	C₂×C₈	C₂⁴×C₅	1	6	yes	—
151	C₂×C₈	C₂⁴×C₅	1	9	yes	—
152	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
153	C₄⋊C₄	C₂²×C₄×C₅	1	18	yes	—
154	C₄⋊C₄	C₂²×C₄×C₅	1	6	yes	—
155	C₂²⋊C₄	C₂²×C₄×C₅	1	36	yes	—
156	C₄²	C₂²×C₄×C₅	1	18	yes	—
157	C₄²	C₂⁴×C₅	1	6	yes	—
158	C₂×C₅	—	2	18	no	—
159	C₂×C₅	C₄×H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×C₅	C₂⁴×C₈	1	1	yes	characteristic
161	C₂×C₅	#3573	1	1	yes	characteristic
162	C₂³	—	4	6	no	—
163	C₂³	—	4	6	no	—
164	C₂³	C₂²×C₅×C₈	1	24	yes	complemented
165	C₂³	C₂³×C₄×C₅	1	9	yes	—
166	Q₈	C₂²×C₅×C₈	1	8	yes	complemented
167	D₄	C₂²×C₅×C₈	1	72	yes	complemented
168	C₂×C₄	—	2	9	no	—
169	C₂×C₄	—	2	9	no	—
170	C₂×C₄	—	4	18	no	—
171	C₂×C₄	—	4	9	no	—
172	C₂×C₄	C₂²×C₅×C₈	1	36	yes	complemented
173	C₂×C₄	C₂³×C₄×C₅	1	6	yes	—
174	C₂×C₄	C₂³×C₄×C₅	1	9	yes	—
175	C₂×C₄	C₂³×C₄×C₅	1	6	yes	—
176	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
177	C₈	—	2	9	no	—
178	C₈	—	2	6	no	—
179	C₈	C₅×H(5,𝔽₂)	1	2	yes	direct product factor
180	C₅	C₈×H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
181	C₂²	—	2	9	no	—
182	C₂²	—	2	9	no	—
183	C₂²	—	4	24	no	—
184	C₂²	C₂³×C₅×C₈	1	18	yes	—
185	C₂²	C₂⁴×C₄×C₅	1	1	yes	characteristic
186	C₄	—	2	9	no	—
187	C₄	—	2	6	no	—
188	C₄	C₂³×C₅×C₈	1	12	yes	—
189	C₄	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
190	C₄	C₅×#585	1	1	yes	characteristic
191	C₂	—	2	18	no	—
192	C₂	C₄×C₅×H(5,𝔽₂)	1	1	yes	characteristic
193	C₂	C₂⁴×C₅×C₈	1	1	yes	characteristic
194	C₂	C₅×#3573	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	C₂⁴
commutator	C₂	C₂⁴×C₅×C₈
Frattini, cosocle	C₂×C₄	C₂⁵×C₅
socle	C₂²×C₅	C₂⁴×C₄
prime radical	C₂×C₅×H(5,𝔽₂)	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2988	C₂	semidirect
3. C₅×#2972	C₂	semidirect
4. C₅×C₈×#41	C₂	semidirect
5. C₂×C₅×C₈×D₄	C₂	semidirect
7. C₂³×C₅×C₈	C₂²	semidirect
12. C₅×C₈×Q₈	C₂²	semidirect
13. C₅×C₈×Q₈	C₂²	semidirect
14. C₅×C₈×D₄	C₂²	semidirect
15. C₅×C₈×D₄	C₂²	semidirect
16. C₅×C₈×D₄	C₂²	semidirect
17. C₅×#431	C₂²	semidirect
18. C₂×C₅×C₄⋊C₈	C₂²	semidirect
19. C₂×C₅×C₂²⋊C₈	C₂²	semidirect
20. C₂×C₄×C₅×C₈	C₂²	semidirect
21. C₈×H(5,𝔽₂)	C₅	direct (N × H)
22. C₅×H(5,𝔽₂)	C₈	direct (N × H)
45. #2988	C₂×C₅	semidirect
46. #2972	C₂×C₅	semidirect
47. C₈×#41	C₂×C₅	semidirect
48. C₂×C₈×D₄	C₂×C₅	semidirect
50. C₅×#41	C₂×C₈	semidirect
52. C₂×C₅×D₄	C₂×C₈	semidirect
72. C₂³×C₈	C₂²×C₅	semidirect
77. C₈×Q₈	C₂²×C₅	semidirect
78. C₈×Q₈	C₂²×C₅	semidirect
79. C₈×D₄	C₂²×C₅	semidirect
80. C₈×D₄	C₂²×C₅	semidirect
81. C₈×D₄	C₂²×C₅	semidirect
82. #431	C₂²×C₅	semidirect
83. C₂×C₄⋊C₈	C₂²×C₅	semidirect
84. C₂×C₂²⋊C₈	C₂²×C₅	semidirect
85. C₂×C₄×C₈	C₂²×C₅	semidirect
88. C₂³×C₅	C₂²×C₈	semidirect
90. C₅×Q₈	C₂²×C₈	semidirect
91. C₅×D₄	C₂²×C₈	semidirect
96. C₂×C₄×C₅	C₂²×C₈	semidirect
103. C₅×C₈	H(5,𝔽₂)	direct (N × H)
104. H(5,𝔽₂)	C₅×C₈	direct (N × H)
137. #41	C₂×C₅×C₈	semidirect
139. C₂×D₄	C₂×C₅×C₈	semidirect
164. C₂³	C₂²×C₅×C₈	semidirect
166. Q₈	C₂²×C₅×C₈	semidirect
167. D₄	C₂²×C₅×C₈	semidirect
172. C₂×C₄	C₂²×C₅×C₈	semidirect
179. C₈	C₅×H(5,𝔽₂)	direct (N × H)
180. C₅	C₈×H(5,𝔽₂)	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×H(5,𝔽₂)	2
C₅×#2988	2
C₅×#2972	2
C₅×C₈×#41	2
C₂×C₅×C₈×D₄	2
C₈×H(5,𝔽₂)	5

Subgroups — C5×C8×H(5,𝔽2) #11983023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₈×H(5,𝔽₂) #11983023