Subgroups — C₅×#8806 #11931526

Order 1280.

462 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1842	C₂	1	2	yes	maximal
2	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#335	C₂	1	4	yes	maximal
4	C₂×C₅×C₄⋊C₈	—	2	4	no	—
5	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
6	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
10	C₄³×C₅	C₂²	1	1	yes	characteristic
11	C₅×#335	—	2	8	no	—
12	#8806	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₅×C₈	—	2	4	no	—
14	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
15	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₅	D₄	1	2	yes	—
19	C₂×C₄²×C₅	Q₈	1	2	yes	—
20	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
21	C₅×C₄⋊C₈	—	2	8	no	—
22	C₅×C₄⋊C₈	—	4	8	no	—
23	C₄×C₅×C₈	—	2	4	no	—
24	C₄×C₅×C₈	—	2	2	no	—
25	C₄×C₅×C₈	C₈	1	4	yes	complemented
26	C₄×C₅×C₈	D₄	1	4	yes	—
27	C₄×C₅×C₈	Q₈	1	4	yes	—
28	#1842	C₂×C₅	1	2	yes	—
29	C₄²×C₈	C₂×C₅	1	1	yes	characteristic
30	C₂×#335	C₂×C₅	1	4	yes	—
31	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₅×C₈	—	2	2	no	—
37	C₂×C₅×C₈	—	2	2	no	—
38	C₂×C₅×C₈	—	4	4	no	—
39	C₂×C₅×C₈	—	4	8	no	—
40	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
41	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
42	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
43	C₂×C₅×C₈	M₁₆	1	4	yes	—
44	C₄²×C₅	—	2	4	no	—
45	C₄²×C₅	—	2	2	no	—
46	C₄²×C₅	C₂×C₈	1	2	yes	—
47	C₄²×C₅	M₁₆	1	2	yes	—
48	C₄²×C₅	D₈	1	4	yes	—
49	C₄²×C₅	Q₁₆	1	4	yes	—
50	C₄²×C₅	C₂×D₄	1	2	yes	—
51	C₄²×C₅	C₂×Q₈	1	2	yes	—
52	C₂×C₄⋊C₈	—	2	4	no	—
53	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
54	C₂×C₄×C₈	C₄×C₅	1	2	yes	—
55	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
56	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
58	C₄³	C₂²×C₅	1	1	yes	characteristic
59	#335	—	2	8	no	—
60	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅	—	2	2	no	—
62	C₂×C₄×C₅	—	2	2	no	—
63	C₂×C₄×C₅	—	2	2	no	—
64	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
65	C₂×C₄×C₅	#107	1	4	yes	—
66	C₂×C₄×C₅	#107	1	4	yes	—
67	C₂×C₄×C₅	#108	1	4	yes	—
68	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₅	#127	1	1	yes	characteristic
70	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
71	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
76	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
77	C₅×C₈	—	2	4	no	—
78	C₅×C₈	—	8	4	no	—
79	C₅×C₈	C₄⋊C₈	1	8	yes	complemented
80	C₂²×C₈	—	2	4	no	—
81	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
82	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
83	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
84	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
85	C₂×C₄²	C₅×D₄	1	2	yes	—
86	C₂×C₄²	C₅×Q₈	1	2	yes	—
87	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
88	C₄⋊C₈	—	2	8	no	—
89	C₄⋊C₈	—	4	8	no	—
90	C₄×C₈	—	2	4	no	—
91	C₄×C₈	—	2	2	no	—
92	C₄×C₈	C₅×C₈	1	4	yes	complemented
93	C₄×C₈	C₅×D₄	1	4	yes	—
94	C₄×C₈	C₅×Q₈	1	4	yes	—
95	C₂²×C₅	#384	1	1	yes	characteristic
96	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₂²×C₅	#423	1	1	yes	characteristic
98	C₂²×C₅	C₂×#107	1	2	yes	—
99	C₂²×C₅	C₂×#108	1	2	yes	—
100	C₄×C₅	—	2	4	no	—
101	C₄×C₅	—	2	2	no	—
102	C₄×C₅	#335	1	8	yes	complemented
103	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
104	C₄×C₅	#423	1	2	yes	—
105	C₄×C₅	#493	1	1	yes	characteristic
106	C₄×C₅	#494	1	1	yes	characteristic
107	C₄×C₅	#500	1	2	yes	—
108	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
109	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
110	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
111	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	4	4	no	—
116	C₂×C₈	—	4	8	no	—
117	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
118	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
119	C₂×C₈	C₂×C₅×C₈	1	4	yes	—
120	C₂×C₈	C₅×M₁₆	1	4	yes	—
121	C₄²	—	2	2	no	—
122	C₄²	—	2	4	no	—
123	C₄²	C₂×C₅×C₈	1	2	yes	—
124	C₄²	C₅×M₁₆	1	2	yes	—
125	C₄²	C₅×D₈	1	4	yes	—
126	C₄²	C₅×Q₁₆	1	4	yes	—
127	C₄²	C₂×C₅×D₄	1	2	yes	—
128	C₄²	C₂×C₅×Q₈	1	2	yes	—
129	C₂×C₅	C₂×#335	1	2	yes	—
130	C₂×C₅	#1567	1	2	yes	—
131	C₂×C₅	#1842	1	1	yes	characteristic
132	C₂×C₅	#1845	1	1	yes	characteristic
133	C₂×C₅	#1848	1	1	yes	characteristic
134	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₅×C₄⋊C₈	1	4	yes	—
139	C₂×C₄	C₅×#107	1	4	yes	—
140	C₂×C₄	C₅×#107	1	4	yes	—
141	C₂×C₄	C₅×#108	1	4	yes	—
142	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
143	C₂×C₄	C₅×#127	1	1	yes	characteristic
144	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
145	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
146	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
147	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
148	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
149	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
150	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
151	C₈	—	2	4	no	—
152	C₈	—	8	4	no	—
153	C₈	C₅×C₄⋊C₈	1	8	yes	complemented
154	C₅	#8806	1	1	yes	characteristic, direct product factor, Sylow
155	C₂²	C₅×#384	1	1	yes	characteristic
156	C₂²	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
157	C₂²	C₅×#423	1	1	yes	characteristic
158	C₂²	C₂×C₅×#107	1	2	yes	—
159	C₂²	C₂×C₅×#108	1	2	yes	—
160	C₄	—	2	4	no	—
161	C₄	—	2	2	no	—
162	C₄	C₅×#335	1	8	yes	complemented
163	C₄	C₂×C₅×C₄⋊C₈	1	2	yes	—
164	C₄	C₅×#423	1	2	yes	—
165	C₄	C₅×#493	1	1	yes	characteristic
166	C₄	C₅×#494	1	1	yes	characteristic
167	C₄	C₅×#500	1	2	yes	—
168	C₂	C₂×C₅×#335	1	2	yes	—
169	C₂	C₅×#1567	1	2	yes	—
170	C₂	C₅×#1842	1	1	yes	characteristic
171	C₂	C₅×#1845	1	1	yes	characteristic
172	C₂	C₅×#1848	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
12. #8806	C₅	direct (N × H)
25. C₄×C₅×C₈	C₈	semidirect
79. C₅×C₈	C₄⋊C₈	semidirect
92. C₄×C₈	C₅×C₈	semidirect
102. C₄×C₅	#335	semidirect
153. C₈	C₅×C₄⋊C₈	semidirect
154. C₅	#8806	direct (N × H)
162. C₄	C₅×#335	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1842	2
C₄²×C₅×C₈	2
C₂×C₅×#335	2
#8806	5

Subgroups — C5×#8806 #11931526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8806 #11931526