Subgroups — C₅×#12668 #11935388

Order 1280.

308 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2155	C₂	1	2	yes	maximal
2	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₅×#1772	C₂	1	2	yes	maximal
4	C₅×#1385	C₂	1	1	yes	maximal, characteristic
5	C₅×#1384	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₅×#108	C₂²	1	2	yes	—
9	C₅×#427	C₂²	1	2	yes	—
10	C₅×#405	—	2	2	no	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#365	C₄	1	8	yes	complemented
13	C₅×#346	C₄	1	2	yes	complemented
14	C₄×C₅×C₁₆	C₄	1	2	yes	complemented
15	C₅×#339	—	2	2	no	—
16	#12668	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂×C₅×M₁₆	—	2	2	no	—
18	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
20	C₅×#117	—	2	2	no	—
21	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
22	C₅×M₃₂	—	2	2	no	—
23	C₅×M₃₂	C₂×C₄	1	4	yes	—
24	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
26	C₅×#108	—	2	2	no	—
27	C₅×#108	C₂×C₄	1	4	yes	—
28	C₅×#107	—	2	2	no	—
29	C₅×#106	C₂×C₄	1	4	yes	—
30	C₅×#97	—	4	2	no	—
31	C₄×C₅×C₈	—	4	2	no	—
32	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
35	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
36	#2155	C₂×C₅	1	2	yes	—
37	C₄×M₃₂	C₂×C₅	1	1	yes	characteristic
38	#1772	C₂×C₅	1	2	yes	—
39	#1385	C₂×C₅	1	1	yes	characteristic
40	#1384	C₂×C₅	1	1	yes	characteristic
41	C₂²×C₄×C₅	—	2	1	no	—
42	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	C₅×M₁₆	—	2	4	no	—
44	C₅×M₁₆	—	4	2	no	—
45	C₂×C₅×C₈	—	4	2	no	—
46	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₅×C₈	SD₁₆	1	2	yes	—
49	C₂×C₅×C₈	SD₁₆	1	2	yes	—
50	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
52	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₅×C₈	#41	1	2	yes	—
54	C₅×C₄⋊C₄	—	2	4	no	—
55	C₅×C₂²⋊C₄	—	4	2	no	—
56	C₄²×C₅	—	2	1	no	—
57	C₄²×C₅	—	4	2	no	—
58	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
60	C₅×C₁₆	—	2	2	no	—
61	C₅×C₁₆	C₄²	1	4	yes	—
62	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
63	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
64	C₂×#108	C₂²×C₅	1	2	yes	—
65	#427	C₂²×C₅	1	2	yes	—
66	#405	—	2	2	no	—
67	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
68	#365	C₄×C₅	1	8	yes	complemented
69	#346	C₄×C₅	1	2	yes	complemented
70	C₄×C₁₆	C₄×C₅	1	2	yes	complemented
71	#339	—	2	2	no	—
72	C₂³×C₅	#135	1	1	yes	characteristic
73	C₂×C₄×C₅	—	2	1	no	—
74	C₂×C₄×C₅	—	2	1	no	—
75	C₂×C₄×C₅	—	4	2	no	—
76	C₂×C₄×C₅	—	4	4	no	—
77	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
80	C₂×C₄×C₅	#135	1	1	yes	characteristic
81	C₅×C₈	—	4	4	no	—
82	C₅×C₈	#106	1	2	yes	—
83	C₅×C₈	#106	1	1	yes	characteristic
84	C₅×C₈	#106	1	1	yes	characteristic
85	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
86	C₅×C₈	#117	1	1	yes	characteristic
87	C₅×C₈	C₄×Q₈	1	2	yes	—
88	C₂×M₁₆	—	2	2	no	—
89	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
90	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
91	#117	—	2	2	no	—
92	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
93	M₃₂	—	2	2	no	—
94	M₃₂	C₂×C₄×C₅	1	4	yes	—
95	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
96	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
97	#108	—	2	2	no	—
98	#108	C₂×C₄×C₅	1	4	yes	—
99	#107	—	2	2	no	—
100	#106	C₂×C₄×C₅	1	4	yes	—
101	#97	—	4	2	no	—
102	C₄×C₈	—	4	2	no	—
103	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
104	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
105	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
106	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
107	C₂²×C₅	—	4	1	no	—
108	C₂²×C₅	#427	1	1	yes	characteristic
109	C₂²×C₅	C₄×SD₁₆	1	2	yes	—
110	C₄×C₅	—	2	2	no	—
111	C₄×C₅	—	4	1	no	—
112	C₄×C₅	—	8	4	no	—
113	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
114	C₄×C₅	C₂×#106	1	1	yes	characteristic
115	C₄×C₅	C₄×SD₁₆	1	2	yes	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
118	M₁₆	—	2	4	no	—
119	M₁₆	—	4	2	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₈	C₅×SD₁₆	1	2	yes	—
124	C₂×C₈	C₅×SD₁₆	1	2	yes	—
125	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
126	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
127	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
128	C₂×C₈	C₅×#41	1	2	yes	—
129	C₄⋊C₄	—	2	4	no	—
130	C₂²⋊C₄	—	4	2	no	—
131	C₄²	—	2	1	no	—
132	C₄²	—	4	2	no	—
133	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
134	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
135	C₁₆	—	2	2	no	—
136	C₁₆	C₄²×C₅	1	4	yes	—
137	C₂×C₅	—	2	1	no	—
138	C₂×C₅	—	2	2	no	—
139	C₂×C₅	C₄×#106	1	1	yes	characteristic
140	C₂³	C₅×#135	1	1	yes	characteristic
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	4	4	no	—
145	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
147	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
148	C₂×C₄	C₅×#135	1	1	yes	characteristic
149	C₈	—	4	4	no	—
150	C₈	C₅×#106	1	2	yes	—
151	C₈	C₅×#106	1	1	yes	characteristic
152	C₈	C₅×#106	1	1	yes	characteristic
153	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
154	C₈	C₅×#117	1	1	yes	characteristic
155	C₈	C₄×C₅×Q₈	1	2	yes	—
156	C₅	#12668	1	1	yes	characteristic, direct product factor, Sylow
157	C₂²	—	4	1	no	—
158	C₂²	C₅×#427	1	1	yes	characteristic
159	C₂²	C₄×C₅×SD₁₆	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	1	no	—
162	C₄	—	8	4	no	—
163	C₄	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
164	C₄	C₂×C₅×#106	1	1	yes	characteristic
165	C₄	C₄×C₅×SD₁₆	1	2	yes	—
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	C₄×C₅×#106	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	#106
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	C₄×#106
prime radical	C₂³×C₅	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₅×C₈ ◁ C₂²×C₅×C₈ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₅×#365	C₄	semidirect
13. C₅×#346	C₄	semidirect
14. C₄×C₅×C₁₆	C₄	semidirect
16. #12668	C₅	direct (N × H)
68. #365	C₄×C₅	semidirect
69. #346	C₄×C₅	semidirect
70. C₄×C₁₆	C₄×C₅	semidirect
156. C₅	#12668	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×#2155	2
C₄×C₅×M₃₂	2
C₅×#1772	2
C₅×#1385	2
C₅×#1384	2
#12668	5

Subgroups — C5×#12668 #11935388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12668 #11935388