Subgroups — C₂×C₅×#1369 #11934721

Order 1280.

222 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#364	C₂	1	2	yes	maximal
2	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic
3	C₅×#1369	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅×M₃₂	—	2	2	no	—
5	C₂²×C₅×C₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₅×C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×#364	—	2	2	no	—
9	C₅×#364	C₂²	1	4	yes	—
10	C₅×#346	C₄	1	2	yes	—
11	C₅×#346	C₂²	1	2	yes	—
12	C₂×#1369	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
15	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
16	C₅×M₃₂	—	2	4	no	—
17	C₅×M₃₂	—	4	2	no	—
18	C₂×C₅×C₁₆	—	2	1	no	—
19	C₂×C₅×C₁₆	—	2	1	no	—
20	C₂×C₅×C₁₆	—	4	2	no	—
21	C₂×C₅×C₁₆	C₈	1	2	yes	—
22	C₂×C₅×C₁₆	C₈	1	2	yes	—
23	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
25	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×C₈	D₄	1	1	yes	characteristic
27	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
28	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₂×#364	C₂×C₅	1	2	yes	—
30	C₂×#346	C₂×C₅	1	1	yes	characteristic
31	#1369	C₂×C₅	1	4	yes	direct product factor
32	C₂²×C₄×C₅	—	2	1	no	—
33	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
37	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
38	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
39	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
40	C₂×C₅×C₈	SD₁₆	1	2	yes	—
41	C₂×C₅×C₈	SD₁₆	1	2	yes	—
42	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₅	—	2	1	no	—
45	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
47	C₅×C₁₆	—	2	2	no	—
48	C₅×C₁₆	—	4	4	no	—
49	C₅×C₁₆	C₂×C₈	1	4	yes	—
50	C₂×M₃₂	—	2	2	no	—
51	C₂²×C₁₆	C₄×C₅	1	1	yes	characteristic
52	C₂²×C₁₆	C₄×C₅	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
54	#364	—	2	2	no	—
55	#364	C₂²×C₅	1	4	yes	—
56	#346	C₄×C₅	1	2	yes	—
57	#346	C₂²×C₅	1	2	yes	—
58	C₂³×C₅	#108	1	1	yes	characteristic
59	C₂×C₄×C₅	—	2	1	no	—
60	C₂×C₄×C₅	—	2	1	no	—
61	C₂×C₄×C₅	—	4	2	no	—
62	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₅	#106	1	1	yes	characteristic
65	C₂×C₄×C₅	#106	1	1	yes	characteristic
66	C₂×C₄×C₅	#108	1	1	yes	characteristic
67	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
69	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
70	C₅×C₈	#106	1	2	yes	—
71	C₅×C₈	C₂²×C₈	1	1	yes	characteristic
72	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
73	C₅×C₈	C₂×SD₁₆	1	2	yes	—
74	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
75	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
76	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
77	M₃₂	—	2	4	no	—
78	M₃₂	—	4	2	no	—
79	C₂×C₁₆	—	2	1	no	—
80	C₂×C₁₆	—	2	1	no	—
81	C₂×C₁₆	—	4	2	no	—
82	C₂×C₁₆	C₅×C₈	1	2	yes	—
83	C₂×C₁₆	C₅×C₈	1	2	yes	—
84	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
85	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
86	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
87	C₄×C₈	C₅×D₄	1	1	yes	characteristic
88	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
89	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
90	C₂²×C₅	—	2	2	no	—
91	C₂²×C₅	#334	1	1	yes	characteristic
92	C₂²×C₅	#334	1	1	yes	characteristic
93	C₂²×C₅	C₂×#108	1	1	yes	characteristic
94	C₄×C₅	—	4	1	no	—
95	C₄×C₅	—	4	1	no	—
96	C₄×C₅	#334	1	1	yes	characteristic
97	C₄×C₅	#334	1	1	yes	characteristic
98	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₄×C₅	C₂×#106	1	1	yes	characteristic
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₈	C₂×C₅×C₈	1	1	yes	characteristic
105	C₂×C₈	C₂×C₅×C₈	1	1	yes	characteristic
106	C₂×C₈	C₅×M₁₆	1	1	yes	characteristic
107	C₂×C₈	C₅×M₁₆	1	1	yes	characteristic
108	C₂×C₈	C₅×SD₁₆	1	2	yes	—
109	C₂×C₈	C₅×SD₁₆	1	2	yes	—
110	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
111	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
112	C₄²	—	2	1	no	—
113	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
114	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
115	C₁₆	—	2	2	no	—
116	C₁₆	—	4	4	no	—
117	C₁₆	C₂×C₅×C₈	1	4	yes	—
118	C₂×C₅	—	2	1	no	—
119	C₂×C₅	—	2	1	no	—
120	C₂×C₅	#1369	1	2	yes	direct product factor
121	C₂×C₅	C₂×#334	1	1	yes	characteristic
122	C₂³	C₅×#108	1	1	yes	characteristic
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	C₅×C₄⋊C₈	1	1	yes	characteristic
127	C₂×C₄	C₅×C₄⋊C₈	1	1	yes	characteristic
128	C₂×C₄	C₅×#106	1	1	yes	characteristic
129	C₂×C₄	C₅×#106	1	1	yes	characteristic
130	C₂×C₄	C₅×#108	1	1	yes	characteristic
131	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
132	C₈	C₅×C₄⋊C₈	1	1	yes	characteristic
133	C₈	C₅×C₄⋊C₈	1	1	yes	characteristic
134	C₈	C₅×#106	1	2	yes	—
135	C₈	C₂²×C₅×C₈	1	1	yes	characteristic
136	C₈	C₂×C₅×M₁₆	1	1	yes	characteristic
137	C₈	C₂×C₅×SD₁₆	1	2	yes	—
138	C₅	C₂×#1369	1	1	yes	characteristic, direct product factor, Sylow
139	C₂²	—	2	2	no	—
140	C₂²	C₅×#334	1	1	yes	characteristic
141	C₂²	C₅×#334	1	1	yes	characteristic
142	C₂²	C₂×C₅×#108	1	1	yes	characteristic
143	C₄	—	4	1	no	—
144	C₄	—	4	1	no	—
145	C₄	C₅×#334	1	1	yes	characteristic
146	C₄	C₅×#334	1	1	yes	characteristic
147	C₄	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
148	C₄	C₂×C₅×#106	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	C₅×#1369	1	2	yes	direct product factor
152	C₂	C₂×C₅×#334	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#106
commutator	C₈	C₂²×C₅×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle	C₂²×C₅	#334
prime radical	C₂³×C₅	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1369	C₂	direct (N × H)
12. C₂×#1369	C₅	direct (N × H)
31. #1369	C₂×C₅	direct (N × H)
120. C₂×C₅	#1369	direct (N × H)
138. C₅	C₂×#1369	direct (N × H)
151. C₂	C₅×#1369	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#364	2
C₂×C₅×#346	2
C₅×#1369	2
C₂×#1369	5

Subgroups — C2×C5×#1369 #11934721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1369 #11934721