Subgroups — C₂×C₅×#1370 #11934723

Order 1280.

286 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×C₅×C₈²	C₂	1	1	yes	maximal, characteristic
3	C₅×#1370	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₅×C₁₆	—	2	2	no	—
5	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×C₄⋊C₁₆	—	2	2	no	—
9	C₅×C₄⋊C₁₆	C₂²	1	4	yes	—
10	C₅×C₈²	C₄	1	2	yes	—
11	C₅×C₈²	C₂²	1	2	yes	—
12	C₂×#1370	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₅×C₈	—	2	1	no	—
14	C₂²×C₅×C₈	C₈	1	1	yes	characteristic
15	C₂²×C₅×C₈	C₈	1	1	yes	characteristic
16	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₅×C₁₆	—	2	4	no	—
20	C₂×C₅×C₁₆	—	4	4	no	—
21	C₄×C₅×C₈	—	2	1	no	—
22	C₄×C₅×C₈	C₈	1	2	yes	—
23	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
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×C₄⋊C₁₆	C₂×C₅	1	2	yes	—
30	C₂×C₈²	C₂×C₅	1	1	yes	characteristic
31	#1370	C₂×C₅	1	4	yes	direct product factor
32	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
35	C₂×C₅×C₈	—	2	4	no	—
36	C₂×C₅×C₈	—	2	2	no	—
37	C₂×C₅×C₈	—	2	2	no	—
38	C₂×C₅×C₈	—	2	2	no	—
39	C₂×C₅×C₈	C₁₆	1	4	yes	complemented
40	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
43	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
44	C₂×C₅×C₈	D₈	1	2	yes	—
45	C₂×C₅×C₈	Q₁₆	1	2	yes	—
46	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₅	M₁₆	1	1	yes	characteristic
51	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₅×C₁₆	—	4	4	no	—
53	C₂²×C₁₆	—	2	2	no	—
54	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
55	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
57	C₄⋊C₁₆	—	2	2	no	—
58	C₄⋊C₁₆	C₂²×C₅	1	4	yes	—
59	C₈²	C₄×C₅	1	2	yes	—
60	C₈²	C₂²×C₅	1	2	yes	—
61	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₅	—	2	2	no	—
63	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
66	C₂×C₄×C₅	#107	1	2	yes	—
67	C₂×C₄×C₅	#108	1	2	yes	—
68	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
69	C₂×C₄×C₅	M₃₂	1	2	yes	—
70	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
73	C₅×C₈	—	2	4	no	—
74	C₅×C₈	—	2	1	no	—
75	C₅×C₈	—	2	1	no	—
76	C₅×C₈	—	2	2	no	—
77	C₅×C₈	#107	1	2	yes	—
78	C₅×C₈	C₂×C₁₆	1	4	yes	complemented
79	C₅×C₈	C₂×D₈	1	1	yes	characteristic
80	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
81	C₂²×C₈	—	2	1	no	—
82	C₂²×C₈	C₅×C₈	1	1	yes	characteristic
83	C₂²×C₈	C₅×C₈	1	1	yes	characteristic
84	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
85	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
86	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
87	C₂×C₁₆	—	2	4	no	—
88	C₂×C₁₆	—	4	4	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	C₅×C₈	1	2	yes	—
91	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
92	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
93	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
94	C₄×C₈	C₅×D₄	1	1	yes	characteristic
95	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
96	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
97	C₂²×C₅	#335	1	2	yes	—
98	C₂²×C₅	C₄⋊C₁₆	1	2	yes	—
99	C₂²×C₅	#364	1	2	yes	—
100	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₄×C₅	—	2	1	no	—
102	C₄×C₅	—	2	1	no	—
103	C₄×C₅	#335	1	2	yes	—
104	C₄×C₅	C₄⋊C₁₆	1	2	yes	—
105	C₄×C₅	C₂×#107	1	1	yes	characteristic
106	C₄×C₅	C₂×#108	1	1	yes	characteristic
107	C₄×C₅	C₂²×C₁₆	1	1	yes	characteristic
108	C₄×C₅	C₂×M₃₂	1	1	yes	characteristic
109	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
111	C₂²×C₄	C₅×M₁₆	1	1	yes	characteristic
112	C₂×C₈	—	2	4	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	C₅×C₁₆	1	4	yes	complemented
117	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
120	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
121	C₂×C₈	C₅×D₈	1	2	yes	—
122	C₂×C₈	C₅×Q₁₆	1	2	yes	—
123	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
124	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
125	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
126	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
127	C₄²	C₅×M₁₆	1	1	yes	characteristic
128	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
129	C₁₆	—	4	4	no	—
130	C₂×C₅	#1370	1	4	yes	direct product factor
131	C₂×C₅	C₂×#335	1	1	yes	characteristic
132	C₂×C₅	C₂×C₄⋊C₁₆	1	1	yes	characteristic
133	C₂×C₅	C₂×#364	1	1	yes	characteristic
134	C₂³	C₅×C₄⋊C₈	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	C₅×C₄⋊C₈	1	1	yes	characteristic
137	C₂×C₄	C₅×C₄⋊C₈	1	1	yes	characteristic
138	C₂×C₄	C₅×C₄⋊C₈	1	1	yes	characteristic
139	C₂×C₄	C₅×#107	1	2	yes	—
140	C₂×C₄	C₅×#108	1	2	yes	—
141	C₂×C₄	C₂×C₅×C₁₆	1	2	yes	—
142	C₂×C₄	C₅×M₃₂	1	2	yes	—
143	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
145	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
146	C₈	—	2	4	no	—
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	2	no	—
150	C₈	C₅×#107	1	2	yes	—
151	C₈	C₂×C₅×C₁₆	1	4	yes	complemented
152	C₈	C₂×C₅×D₈	1	1	yes	characteristic
153	C₈	C₂×C₅×Q₁₆	1	1	yes	characteristic
154	C₅	C₂×#1370	1	1	yes	characteristic, direct product factor, Sylow
155	C₂²	C₅×#335	1	2	yes	—
156	C₂²	C₅×C₄⋊C₁₆	1	2	yes	—
157	C₂²	C₅×#364	1	2	yes	—
158	C₂²	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	C₅×#335	1	2	yes	—
162	C₄	C₅×C₄⋊C₁₆	1	2	yes	—
163	C₄	C₂×C₅×#107	1	1	yes	characteristic
164	C₄	C₂×C₅×#108	1	1	yes	characteristic
165	C₄	C₂²×C₅×C₁₆	1	1	yes	characteristic
166	C₄	C₂×C₅×M₃₂	1	1	yes	characteristic
167	C₂	C₅×#1370	1	4	yes	direct product factor
168	C₂	C₂×C₅×#335	1	1	yes	characteristic
169	C₂	C₂×C₅×C₄⋊C₁₆	1	1	yes	characteristic
170	C₂	C₂×C₅×#364	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅×C₈	D₄
commutator	C₄	C₂²×C₅×C₁₆
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle, prime radical	C₂³×C₅	C₄⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ 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×#1370	C₂	direct (N × H)
12. C₂×#1370	C₅	direct (N × H)
31. #1370	C₂×C₅	direct (N × H)
39. C₂×C₅×C₈	C₁₆	semidirect
78. C₅×C₈	C₂×C₁₆	semidirect
116. C₂×C₈	C₅×C₁₆	semidirect
130. C₂×C₅	#1370	direct (N × H)
151. C₈	C₂×C₅×C₁₆	semidirect
154. C₅	C₂×#1370	direct (N × H)
167. C₂	C₅×#1370	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×C₄⋊C₁₆	2
C₂×C₅×C₈²	2
C₅×#1370	2
C₂×#1370	5

Subgroups — C2×C5×#1370 #11934723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1370 #11934723