Subgroups — C₅×#8785 #11931505

Order 1280.

382 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1831	C₂	1	2	yes	maximal
2	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#1277	C₂	1	4	yes	maximal
4	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
5	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
6	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
9	C₄³×C₅	C₂²	1	1	yes	characteristic
10	C₅×#336	—	2	4	no	—
11	#8785	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂²×C₅×C₈	—	2	4	no	—
13	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
14	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₅	D₄	1	2	yes	—
18	C₂×C₄²×C₅	Q₈	1	2	yes	—
19	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
20	C₅×C₄⋊C₈	—	2	8	no	—
21	C₄×C₅×C₈	—	2	4	no	—
22	C₄×C₅×C₈	—	2	2	no	—
23	C₄×C₅×C₈	—	2	4	no	—
24	C₄×C₅×C₈	C₈	1	4	yes	complemented
25	#1831	C₂×C₅	1	2	yes	—
26	C₄²×C₈	C₂×C₅	1	1	yes	characteristic
27	#1277	C₂×C₅	1	4	yes	—
28	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
29	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
30	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₅×C₈	—	2	2	no	—
34	C₂×C₅×C₈	—	2	2	no	—
35	C₂×C₅×C₈	—	2	2	no	—
36	C₂×C₅×C₈	—	2	2	no	—
37	C₂×C₅×C₈	—	4	4	no	—
38	C₂×C₅×C₈	—	4	8	no	—
39	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
40	C₂×C₅×C₈	M₁₆	1	4	yes	—
41	C₄²×C₅	—	2	4	no	—
42	C₄²×C₅	—	2	2	no	—
43	C₄²×C₅	—	2	4	no	—
44	C₄²×C₅	C₂×C₈	1	2	yes	—
45	C₄²×C₅	M₁₆	1	2	yes	—
46	C₄²×C₅	#41	1	4	yes	—
47	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
48	C₂×C₄×C₈	C₄×C₅	1	2	yes	—
49	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
50	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
52	C₄³	C₂²×C₅	1	1	yes	characteristic
53	#336	—	2	4	no	—
54	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅	—	2	2	no	—
56	C₂×C₄×C₅	—	2	2	no	—
57	C₂×C₄×C₅	—	2	2	no	—
58	C₂×C₄×C₅	—	2	2	no	—
59	C₂×C₄×C₅	—	2	2	no	—
60	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
61	C₂×C₄×C₅	#108	1	4	yes	—
62	C₂×C₄×C₅	#117	1	2	yes	—
63	C₂×C₄×C₅	#124	1	2	yes	—
64	C₂×C₄×C₅	#125	1	2	yes	—
65	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅	#135	1	4	yes	—
69	C₅×C₈	—	2	4	no	—
70	C₅×C₈	—	2	4	no	—
71	C₅×C₈	—	8	4	no	—
72	C₂²×C₈	—	2	4	no	—
73	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
74	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
75	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
76	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
77	C₂×C₄²	C₅×D₄	1	2	yes	—
78	C₂×C₄²	C₅×Q₈	1	2	yes	—
79	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
80	C₄⋊C₈	—	2	8	no	—
81	C₄×C₈	—	2	4	no	—
82	C₄×C₈	—	2	2	no	—
83	C₄×C₈	—	2	4	no	—
84	C₄×C₈	C₅×C₈	1	4	yes	complemented
85	C₂²×C₅	#382	1	1	yes	characteristic
86	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂²×C₅	#423	1	1	yes	characteristic
88	C₂²×C₅	#427	1	2	yes	—
89	C₂²×C₅	C₂×#108	1	2	yes	—
90	C₄×C₅	—	2	4	no	—
91	C₄×C₅	—	2	4	no	—
92	C₄×C₅	—	2	2	no	—
93	C₄×C₅	#431	1	2	yes	—
94	C₄×C₅	#432	1	2	yes	—
95	C₄×C₅	#488	1	4	yes	—
96	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
97	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
98	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
99	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
100	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	4	4	no	—
106	C₂×C₈	—	4	8	no	—
107	C₂×C₈	C₂×C₅×C₈	1	4	yes	—
108	C₂×C₈	C₅×M₁₆	1	4	yes	—
109	C₄²	—	2	2	no	—
110	C₄²	—	2	4	no	—
111	C₄²	—	2	4	no	—
112	C₄²	C₂×C₅×C₈	1	2	yes	—
113	C₄²	C₅×M₁₆	1	2	yes	—
114	C₄²	C₅×#41	1	4	yes	—
115	C₂×C₅	#1564	1	2	yes	—
116	C₂×C₅	#1568	1	2	yes	—
117	C₂×C₅	#1831	1	1	yes	characteristic
118	C₂×C₅	#1835	1	1	yes	characteristic
119	C₂×C₅	#1837	1	1	yes	characteristic
120	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	C₅×C₄⋊C₈	1	4	yes	—
127	C₂×C₄	C₅×#108	1	4	yes	—
128	C₂×C₄	C₅×#117	1	2	yes	—
129	C₂×C₄	C₅×#124	1	2	yes	—
130	C₂×C₄	C₅×#125	1	2	yes	—
131	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
132	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
133	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
134	C₂×C₄	C₅×#135	1	4	yes	—
135	C₈	—	2	4	no	—
136	C₈	—	2	4	no	—
137	C₈	—	8	4	no	—
138	C₅	#8785	1	1	yes	characteristic, direct product factor, Sylow
139	C₂²	C₅×#382	1	1	yes	characteristic
140	C₂²	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
141	C₂²	C₅×#423	1	1	yes	characteristic
142	C₂²	C₅×#427	1	2	yes	—
143	C₂²	C₂×C₅×#108	1	2	yes	—
144	C₄	—	2	4	no	—
145	C₄	—	2	4	no	—
146	C₄	—	2	2	no	—
147	C₄	C₅×#431	1	2	yes	—
148	C₄	C₅×#432	1	2	yes	—
149	C₄	C₅×#488	1	4	yes	—
150	C₂	C₅×#1564	1	2	yes	—
151	C₂	C₅×#1568	1	2	yes	—
152	C₂	C₅×#1831	1	1	yes	characteristic
153	C₂	C₅×#1835	1	1	yes	characteristic
154	C₂	C₅×#1837	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
11. #8785	C₅	direct (N × H)
24. C₄×C₅×C₈	C₈	semidirect
84. C₄×C₈	C₅×C₈	semidirect
138. C₅	#8785	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#1831	2
C₄²×C₅×C₈	2
C₅×#1277	2
#8785	5

Subgroups — C5×#8785 #11931505

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8785 #11931505