Subgroups — C₅×#12004 #11934724

Order 1280.

270 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2150	C₂	1	2	yes	maximal
2	C₂×C₅×C₈²	C₂	1	1	yes	maximal, characteristic
3	C₅×#1370	C₂	1	4	yes	maximal, complemented
4	C₂×C₅×M₃₂	—	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	#12004	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₅×C₈	—	2	1	no	—
14	C₂²×C₅×C₈	C₈	1	2	yes	—
15	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
16	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₅×M₃₂	—	4	4	no	—
19	C₂×C₅×C₁₆	—	2	4	no	—
20	C₄×C₅×C₈	—	2	1	no	—
21	C₄×C₅×C₈	C₈	1	1	yes	characteristic
22	C₄×C₅×C₈	C₈	1	1	yes	characteristic
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	#2150	C₂×C₅	1	2	yes	—
30	C₂×C₈²	C₂×C₅	1	1	yes	characteristic
31	#1370	C₂×C₅	1	4	yes	complemented
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	4	no	—
39	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
42	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
43	C₂×C₅×C₈	D₈	1	2	yes	—
44	C₂×C₅×C₈	Q₁₆	1	2	yes	—
45	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
46	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₅	M₁₆	1	1	yes	characteristic
50	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₅×C₁₆	—	4	4	no	—
52	C₂×M₃₂	—	2	2	no	—
53	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
54	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
56	C₄⋊C₁₆	—	2	2	no	—
57	C₄⋊C₁₆	C₂²×C₅	1	4	yes	—
58	C₈²	C₄×C₅	1	2	yes	—
59	C₈²	C₂²×C₅	1	2	yes	—
60	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₅	—	2	2	no	—
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×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
71	C₅×C₈	—	2	4	no	—
72	C₅×C₈	—	2	1	no	—
73	C₅×C₈	—	2	1	no	—
74	C₅×C₈	—	2	2	no	—
75	C₅×C₈	#107	1	2	yes	—
76	C₅×C₈	M₃₂	1	4	yes	complemented
77	C₅×C₈	C₂×D₈	1	1	yes	characteristic
78	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
79	C₂²×C₈	—	2	1	no	—
80	C₂²×C₈	C₅×C₈	1	2	yes	—
81	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
82	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
83	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
84	M₃₂	—	4	4	no	—
85	C₂×C₁₆	—	2	4	no	—
86	C₄×C₈	—	2	1	no	—
87	C₄×C₈	C₅×C₈	1	1	yes	characteristic
88	C₄×C₈	C₅×C₈	1	1	yes	characteristic
89	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
90	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
91	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
92	C₄×C₈	C₅×D₄	1	1	yes	characteristic
93	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
94	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
95	C₂²×C₅	—	2	1	no	—
96	C₂²×C₅	—	2	1	no	—
97	C₂²×C₅	#335	1	2	yes	—
98	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₄×C₅	—	2	1	no	—
100	C₄×C₅	—	2	1	no	—
101	C₄×C₅	#335	1	2	yes	—
102	C₄×C₅	#364	1	2	yes	—
103	C₄×C₅	C₂×#107	1	1	yes	characteristic
104	C₄×C₅	C₂×#108	1	1	yes	characteristic
105	C₄×C₅	C₂×M₃₂	1	2	yes	—
106	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
108	C₂²×C₄	C₅×M₁₆	1	1	yes	characteristic
109	C₂×C₈	—	2	4	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	4	no	—
113	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
116	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
117	C₂×C₈	C₅×D₈	1	2	yes	—
118	C₂×C₈	C₅×Q₁₆	1	2	yes	—
119	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
120	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
121	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
122	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
123	C₄²	C₅×M₁₆	1	1	yes	characteristic
124	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
125	C₁₆	—	4	4	no	—
126	C₂×C₅	—	2	2	no	—
127	C₂×C₅	C₂×#335	1	1	yes	characteristic
128	C₂×C₅	#2150	1	1	yes	characteristic
129	C₂×C₅	C₂×#364	1	1	yes	characteristic
130	C₂³	C₅×C₄⋊C₈	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	C₅×C₄⋊C₈	1	1	yes	characteristic
134	C₂×C₄	C₅×C₄⋊C₈	1	1	yes	characteristic
135	C₂×C₄	C₅×C₄⋊C₈	1	1	yes	characteristic
136	C₂×C₄	C₅×#107	1	2	yes	—
137	C₂×C₄	C₅×#108	1	2	yes	—
138	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
140	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
141	C₈	—	2	4	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	2	no	—
145	C₈	C₅×#107	1	2	yes	—
146	C₈	C₅×M₃₂	1	4	yes	complemented
147	C₈	C₂×C₅×D₈	1	1	yes	characteristic
148	C₈	C₂×C₅×Q₁₆	1	1	yes	characteristic
149	C₅	#12004	1	1	yes	characteristic, direct product factor, Sylow
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	C₅×#335	1	2	yes	—
153	C₂²	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	C₅×#335	1	2	yes	—
157	C₄	C₅×#364	1	2	yes	—
158	C₄	C₂×C₅×#107	1	1	yes	characteristic
159	C₄	C₂×C₅×#108	1	1	yes	characteristic
160	C₄	C₂×C₅×M₃₂	1	2	yes	—
161	C₂	—	2	2	no	—
162	C₂	C₂×C₅×#335	1	1	yes	characteristic
163	C₂	C₅×#2150	1	1	yes	characteristic
164	C₂	C₂×C₅×#364	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×C₄⋊C₈
prime radical	C₂³×C₅	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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	semidirect
12. #12004	C₅	direct (N × H)
31. #1370	C₂×C₅	semidirect
76. C₅×C₈	M₃₂	semidirect
146. C₈	C₅×M₃₂	semidirect
149. C₅	#12004	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×#2150	2
C₂×C₅×C₈²	2
C₅×#1370	2
#12004	5

Subgroups — C5×#12004 #11934724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12004 #11934724