Subgroups — C₅×#12597 #11935317

Order 1280.

258 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×#2110	C₂	1	1	yes	maximal, characteristic
2	C₅×#2108	C₂	1	1	yes	maximal, characteristic
3	C₅×#1747	C₂	1	1	yes	maximal, characteristic
4	C₅×#1379	C₂	1	2	yes	maximal
5	C₅×#1378	C₂	1	2	yes	maximal
6	C₂×C₅×M₃₂	C₂²	1	2	yes	—
7	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
8	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
9	C₅×#405	C₂²	1	2	yes	—
10	C₅×#405	C₂²	1	1	yes	characteristic
11	C₅×#347	C₄	1	4	yes	complemented
12	C₅×#346	C₄	1	2	yes	complemented
13	C₄×C₅×C₁₆	C₄	1	2	yes	complemented
14	C₅×#338	C₄	1	1	yes	characteristic
15	C₅×H(3,ℤ/4)	C₄	1	1	yes	characteristic
16	C₅×H(3,ℤ/4)	C₄	1	2	yes	—
17	#12597	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₅×#117	C₂×C₄	1	2	yes	—
22	C₅×#117	C₂×C₄	1	1	yes	characteristic
23	C₅×M₃₂	—	2	2	no	—
24	C₅×M₃₂	—	2	2	no	—
25	C₅×M₃₂	C₂×C₄	1	4	yes	—
26	C₂×C₅×C₁₆	—	2	2	no	—
27	C₂×C₅×C₁₆	—	2	2	no	—
28	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₅×#97	—	2	2	no	—
32	C₅×#97	C₂×C₄	1	2	yes	—
33	C₄×C₅×C₈	—	2	2	no	—
34	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
35	#2110	C₂×C₅	1	1	yes	characteristic
36	#2108	C₂×C₅	1	1	yes	characteristic
37	#1747	C₂×C₅	1	1	yes	characteristic
38	#1379	C₂×C₅	1	2	yes	—
39	#1378	C₂×C₅	1	2	yes	—
40	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₅×M₁₆	—	2	2	no	—
42	C₅×M₁₆	—	2	2	no	—
43	C₅×M₁₆	—	2	2	no	—
44	C₂×C₅×C₈	—	2	2	no	—
45	C₂×C₅×C₈	C₄²	1	2	yes	—
46	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₅×C₈	#41	1	2	yes	—
49	C₂×C₅×C₈	#41	1	2	yes	—
50	C₅×C₄⋊C₄	—	2	1	no	—
51	C₅×C₄⋊C₄	—	2	2	no	—
52	C₅×C₂²⋊C₄	—	2	1	no	—
53	C₅×C₂²⋊C₄	—	2	2	no	—
54	C₄²×C₅	—	2	2	no	—
55	C₄²×C₅	C₄²	1	2	yes	—
56	C₅×C₁₆	—	2	1	no	—
57	C₅×C₁₆	—	2	1	no	—
58	C₅×C₁₆	—	2	4	no	—
59	C₅×C₁₆	—	2	2	no	—
60	C₂×M₃₂	C₂²×C₅	1	2	yes	—
61	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
62	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
63	#405	C₂²×C₅	1	2	yes	—
64	#405	C₂²×C₅	1	1	yes	characteristic
65	#347	C₄×C₅	1	4	yes	complemented
66	#346	C₄×C₅	1	2	yes	complemented
67	C₄×C₁₆	C₄×C₅	1	2	yes	complemented
68	#338	C₄×C₅	1	1	yes	characteristic
69	H(3,ℤ/4)	C₄×C₅	1	1	yes	characteristic
70	H(3,ℤ/4)	C₄×C₅	1	2	yes	—
71	C₂³×C₅	#131	1	1	yes	characteristic
72	C₂×C₄×C₅	—	2	2	no	—
73	C₂×C₄×C₅	—	4	2	no	—
74	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₅	#117	1	2	yes	—
76	C₂×C₄×C₅	#131	1	2	yes	—
77	C₂×C₄×C₅	#131	1	1	yes	characteristic
78	C₅×C₈	—	2	2	no	—
79	C₅×C₈	—	4	2	no	—
80	C₅×C₈	#117	1	1	yes	characteristic
81	C₅×C₈	#117	1	2	yes	—
82	C₅×C₈	#117	1	1	yes	characteristic
83	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
84	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
85	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
86	#117	C₂×C₄×C₅	1	2	yes	—
87	#117	C₂×C₄×C₅	1	1	yes	characteristic
88	M₃₂	—	2	2	no	—
89	M₃₂	—	2	2	no	—
90	M₃₂	C₂×C₄×C₅	1	4	yes	—
91	C₂×C₁₆	—	2	2	no	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
94	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
95	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
96	#97	—	2	2	no	—
97	#97	C₂×C₄×C₅	1	2	yes	—
98	C₄×C₈	—	2	2	no	—
99	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
100	C₂²×C₅	—	4	1	no	—
101	C₂²×C₅	#405	1	1	yes	characteristic
102	C₂²×C₅	#433	1	2	yes	—
103	C₄×C₅	—	4	2	no	—
104	C₄×C₅	—	8	2	no	—
105	C₄×C₅	#376	1	1	yes	characteristic
106	C₄×C₅	#405	1	1	yes	characteristic
107	C₄×C₅	#433	1	2	yes	—
108	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	M₁₆	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	C₄²×C₅	1	2	yes	—
114	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
115	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
116	C₂×C₈	C₅×#41	1	2	yes	—
117	C₂×C₈	C₅×#41	1	2	yes	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	2	1	no	—
121	C₂²⋊C₄	—	2	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	C₄²×C₅	1	2	yes	—
124	C₁₆	—	2	1	no	—
125	C₁₆	—	2	1	no	—
126	C₁₆	—	2	4	no	—
127	C₁₆	—	2	2	no	—
128	C₂×C₅	—	2	1	no	—
129	C₂×C₅	—	2	2	no	—
130	C₂×C₅	#1746	1	1	yes	characteristic
131	C₂³	C₅×#131	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
135	C₂×C₄	C₅×#117	1	2	yes	—
136	C₂×C₄	C₅×#131	1	2	yes	—
137	C₂×C₄	C₅×#131	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	—	4	2	no	—
140	C₈	C₅×#117	1	1	yes	characteristic
141	C₈	C₅×#117	1	2	yes	—
142	C₈	C₅×#117	1	1	yes	characteristic
143	C₅	#12597	1	1	yes	characteristic, direct product factor, Sylow
144	C₂²	—	4	1	no	—
145	C₂²	C₅×#405	1	1	yes	characteristic
146	C₂²	C₅×#433	1	2	yes	—
147	C₄	—	4	2	no	—
148	C₄	—	8	2	no	—
149	C₄	C₅×#376	1	1	yes	characteristic
150	C₄	C₅×#405	1	1	yes	characteristic
151	C₄	C₅×#433	1	2	yes	—
152	C₂	—	2	1	no	—
153	C₂	—	2	2	no	—
154	C₂	C₅×#1746	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	#117
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1746
prime radical	C₂³×C₅	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₅×C₈ ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
11. C₅×#347	C₄	semidirect
12. C₅×#346	C₄	semidirect
13. C₄×C₅×C₁₆	C₄	semidirect
17. #12597	C₅	direct (N × H)
65. #347	C₄×C₅	semidirect
66. #346	C₄×C₅	semidirect
67. C₄×C₁₆	C₄×C₅	semidirect
143. C₅	#12597	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×#2110	2
C₅×#2108	2
C₅×#1747	2
C₅×#1379	2
C₅×#1378	2
#12597	5

Subgroups — C5×#12597 #11935317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12597 #11935317