Subgroups — C₅×#12906 #11935626

Order 1280.

374 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#367	C₂	1	4	yes	maximal
2	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#1845	C₂	1	2	yes	maximal
4	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
5	C₂×C₅×#107	—	2	2	no	—
6	C₂×C₅×#107	C₂²	1	4	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×#384	—	2	2	no	—
9	C₅×#367	—	2	8	no	—
10	C₄×C₅×C₁₆	C₄	1	4	yes	complemented
11	#12906	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
13	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
14	C₂×C₅×C₄⋊C₄	—	2	4	no	—
15	C₂×C₅×C₄⋊C₄	—	4	4	no	—
16	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
17	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
18	C₂×C₅×C₁₆	D₄	1	4	yes	—
19	C₂×C₅×C₁₆	Q₈	1	4	yes	—
20	C₅×#107	—	2	8	no	—
21	C₅×#107	—	4	4	no	—
22	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₄×C₅×C₈	Q₈	1	2	yes	—
24	C₂×#367	C₂×C₅	1	4	yes	—
25	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic
26	#1845	C₂×C₅	1	2	yes	—
27	C₂²×C₄×C₅	—	4	4	no	—
28	C₂²×C₄×C₅	D₈	1	2	yes	—
29	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
30	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
31	C₂×C₅×C₈	D₈	1	2	yes	—
32	C₂×C₅×C₈	Q₁₆	1	2	yes	—
33	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
35	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
36	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
37	C₅×C₄⋊C₄	—	4	4	no	—
38	C₅×C₄⋊C₄	—	4	4	no	—
39	C₅×C₄⋊C₄	—	8	8	no	—
40	C₄²×C₅	C₄⋊C₄	1	2	yes	—
41	C₄²×C₅	Q₁₆	1	2	yes	—
42	C₅×C₁₆	C₄⋊C₄	1	8	yes	complemented
43	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
44	C₂×#107	—	2	2	no	—
45	C₂×#107	C₂²×C₅	1	4	yes	—
46	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
47	#384	—	2	2	no	—
48	#367	—	2	8	no	—
49	C₄×C₁₆	C₄×C₅	1	4	yes	complemented
50	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₅	—	8	4	no	—
52	C₂×C₄×C₅	—	8	8	no	—
53	C₂×C₄×C₅	#107	1	4	yes	—
54	C₂×C₄×C₅	SD₃₂	1	8	yes	—
55	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅	#127	1	1	yes	characteristic
57	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
60	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
61	C₅×C₈	#107	1	4	yes	—
62	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
63	C₅×C₈	C₄⋊Q₈	1	2	yes	—
64	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
65	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
66	C₂×C₄⋊C₄	—	2	4	no	—
67	C₂×C₄⋊C₄	—	4	4	no	—
68	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
69	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
70	C₂×C₁₆	C₅×D₄	1	4	yes	—
71	C₂×C₁₆	C₅×Q₈	1	4	yes	—
72	#107	—	2	8	no	—
73	#107	—	4	4	no	—
74	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
75	C₄×C₈	C₅×Q₈	1	2	yes	—
76	C₂²×C₅	C₂×#107	1	1	yes	characteristic
77	C₂²×C₅	#493	1	1	yes	characteristic
78	C₂²×C₅	#500	1	1	yes	characteristic
79	C₂²×C₅	C₂×SD₃₂	1	4	yes	—
80	C₄×C₅	—	16	4	no	—
81	C₄×C₅	#367	1	8	yes	complemented
82	C₄×C₅	#384	1	1	yes	characteristic
83	C₄×C₅	C₂×#107	1	1	yes	characteristic
84	C₄×C₅	#494	1	1	yes	characteristic
85	C₄×C₅	#500	1	1	yes	characteristic
86	C₂²×C₄	—	4	4	no	—
87	C₂²×C₄	C₅×D₈	1	2	yes	—
88	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
89	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
90	C₂×C₈	C₅×D₈	1	2	yes	—
91	C₂×C₈	C₅×Q₁₆	1	2	yes	—
92	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
93	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
94	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
95	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
96	C₄⋊C₄	—	4	4	no	—
97	C₄⋊C₄	—	4	4	no	—
98	C₄⋊C₄	—	8	8	no	—
99	C₄²	C₅×C₄⋊C₄	1	2	yes	—
100	C₄²	C₅×Q₁₆	1	2	yes	—
101	C₁₆	C₅×C₄⋊C₄	1	8	yes	complemented
102	C₂×C₅	#1845	1	1	yes	characteristic
103	C₂×C₅	C₂×#367	1	2	yes	—
104	C₂×C₅	#2248	1	2	yes	—
105	C₂×C₅	#2254	1	2	yes	—
106	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
107	C₂×C₄	—	8	4	no	—
108	C₂×C₄	—	8	8	no	—
109	C₂×C₄	C₅×#107	1	4	yes	—
110	C₂×C₄	C₅×SD₃₂	1	8	yes	—
111	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄	C₅×#127	1	1	yes	characteristic
113	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
114	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
115	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
116	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
117	C₈	C₅×#107	1	4	yes	—
118	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
119	C₈	C₅×C₄⋊Q₈	1	2	yes	—
120	C₅	#12906	1	1	yes	characteristic, direct product factor, Sylow
121	C₂²	C₂×C₅×#107	1	1	yes	characteristic
122	C₂²	C₅×#493	1	1	yes	characteristic
123	C₂²	C₅×#500	1	1	yes	characteristic
124	C₂²	C₂×C₅×SD₃₂	1	4	yes	—
125	C₄	—	16	4	no	—
126	C₄	C₅×#367	1	8	yes	complemented
127	C₄	C₅×#384	1	1	yes	characteristic
128	C₄	C₂×C₅×#107	1	1	yes	characteristic
129	C₄	C₅×#494	1	1	yes	characteristic
130	C₄	C₅×#500	1	1	yes	characteristic
131	C₂	C₅×#1845	1	1	yes	characteristic
132	C₂	C₂×C₅×#367	1	2	yes	—
133	C₂	C₅×#2248	1	2	yes	—
134	C₂	C₅×#2254	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₄×C₅×C₁₆	C₄	semidirect
11. #12906	C₅	direct (N × H)
42. C₅×C₁₆	C₄⋊C₄	semidirect
49. C₄×C₁₆	C₄×C₅	semidirect
81. C₄×C₅	#367	semidirect
101. C₁₆	C₅×C₄⋊C₄	semidirect
120. C₅	#12906	direct (N × H)
126. C₄	C₅×#367	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#367	2
C₂×C₄×C₅×C₁₆	2
C₅×#1845	2
#12906	5

Subgroups — C5×#12906 #11935626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12906 #11935626