Subgroups — C₅×#12909 #11935629

Order 1280.

314 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#368	C₂	1	4	yes	maximal
2	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#1848	C₂	1	2	yes	maximal
4	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
5	C₂×C₅×#108	—	2	2	no	—
6	C₂×C₅×#108	C₂²	1	4	yes	—
7	C₅×#423	—	2	2	no	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₅×#368	—	2	8	no	—
10	C₄×C₅×C₁₆	C₄	1	2	yes	—
11	C₄×C₅×C₁₆	C₄	1	2	yes	—
12	#12909	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₅×M₁₆	—	2	4	no	—
14	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
15	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
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×#108	—	2	8	no	—
21	C₅×#108	—	4	4	no	—
22	C₅×C₄⋊C₈	—	4	4	no	—
23	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
26	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
27	C₂×#368	C₂×C₅	1	4	yes	—
28	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic
29	#1848	C₂×C₅	1	2	yes	—
30	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
31	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
32	C₅×M₁₆	—	4	4	no	—
33	C₅×M₁₆	—	4	4	no	—
34	C₂×C₅×C₈	—	4	4	no	—
35	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₅×C₈	D₈	1	2	yes	—
38	C₂×C₅×C₈	Q₁₆	1	2	yes	—
39	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
41	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
42	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₅	D₈	1	2	yes	—
46	C₅×C₁₆	C₄⋊C₄	1	8	yes	—
47	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
48	C₂×#108	—	2	2	no	—
49	C₂×#108	C₂²×C₅	1	4	yes	—
50	#423	—	2	2	no	—
51	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
52	#368	—	2	8	no	—
53	C₄×C₁₆	C₄×C₅	1	2	yes	—
54	C₄×C₁₆	C₄×C₅	1	2	yes	—
55	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
56	C₂×C₄×C₅	—	2	4	no	—
57	C₂×C₄×C₅	#107	1	2	yes	—
58	C₂×C₄×C₅	#107	1	2	yes	—
59	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅	#127	1	1	yes	characteristic
61	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
65	C₅×C₈	—	8	4	no	—
66	C₅×C₈	#107	1	2	yes	—
67	C₅×C₈	#107	1	2	yes	—
68	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
69	C₅×C₈	C₄⋊Q₈	1	2	yes	—
70	C₂×M₁₆	—	2	4	no	—
71	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
72	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
73	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
74	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
75	C₂×C₁₆	C₅×D₄	1	4	yes	—
76	C₂×C₁₆	C₅×Q₈	1	4	yes	—
77	#108	—	2	8	no	—
78	#108	—	4	4	no	—
79	C₄⋊C₈	—	4	4	no	—
80	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
81	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
82	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
83	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
84	C₂²×C₅	—	2	2	no	—
85	C₂²×C₅	C₂×#107	1	1	yes	characteristic
86	C₂²×C₅	#494	1	1	yes	characteristic
87	C₂²×C₅	#500	1	1	yes	characteristic
88	C₄×C₅	—	2	2	no	—
89	C₄×C₅	#368	1	4	yes	complemented
90	C₄×C₅	#384	1	1	yes	characteristic
91	C₄×C₅	C₂×#107	1	1	yes	characteristic
92	C₄×C₅	#493	1	1	yes	characteristic
93	C₄×C₅	#500	1	1	yes	characteristic
94	C₂²×C₄	C₅×Q₁₆	1	2	yes	—
95	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
96	M₁₆	—	4	4	no	—
97	M₁₆	—	4	4	no	—
98	C₂×C₈	—	4	4	no	—
99	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₈	C₅×D₈	1	2	yes	—
102	C₂×C₈	C₅×Q₁₆	1	2	yes	—
103	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
104	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
105	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
106	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
107	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
108	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
109	C₄²	C₅×D₈	1	2	yes	—
110	C₁₆	C₅×C₄⋊C₄	1	8	yes	—
111	C₂×C₅	—	2	1	no	—
112	C₂×C₅	—	2	1	no	—
113	C₂×C₅	#1845	1	1	yes	characteristic
114	C₂×C₅	C₂×#368	1	2	yes	—
115	C₂³	C₂×C₅×Q₁₆	1	1	yes	characteristic
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	C₅×#107	1	2	yes	—
118	C₂×C₄	C₅×#107	1	2	yes	—
119	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄	C₅×#127	1	1	yes	characteristic
121	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
123	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
124	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
125	C₈	—	8	4	no	—
126	C₈	C₅×#107	1	2	yes	—
127	C₈	C₅×#107	1	2	yes	—
128	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
129	C₈	C₅×C₄⋊Q₈	1	2	yes	—
130	C₅	#12909	1	1	yes	characteristic, direct product factor, Sylow
131	C₂²	—	2	2	no	—
132	C₂²	C₂×C₅×#107	1	1	yes	characteristic
133	C₂²	C₅×#494	1	1	yes	characteristic
134	C₂²	C₅×#500	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	C₅×#368	1	4	yes	complemented
137	C₄	C₅×#384	1	1	yes	characteristic
138	C₄	C₂×C₅×#107	1	1	yes	characteristic
139	C₄	C₅×#493	1	1	yes	characteristic
140	C₄	C₅×#500	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	C₅×#1845	1	1	yes	characteristic
144	C₂	C₂×C₅×#368	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#107
prime radical	C₂³×C₅	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
12. #12909	C₅	direct (N × H)
89. C₄×C₅	#368	semidirect
130. C₅	#12909	direct (N × H)
136. C₄	C₅×#368	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#368	2
C₂×C₄×C₅×C₁₆	2
C₅×#1848	2
#12909	5

Subgroups — C5×#12909 #11935629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12909 #11935629