Subgroups — C₅×#13266 #11935986

Order 1280.

342 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2960	C₂	1	1	yes	maximal, characteristic
2	C₅×#2150	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₅×#1379	C₂	1	4	yes	maximal
5	C₅×#533	C₄	1	1	yes	characteristic
6	C₂×C₅×M₃₂	C₂²	1	2	yes	—
7	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
8	C₅×#446	—	2	2	no	—
9	C₅×C₈×Q₈	—	2	2	no	—
10	C₅×#424	C₄	1	2	yes	—
11	C₅×#423	C₄	1	1	yes	characteristic
12	C₅×#405	C₂²	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×C₄⋊C₁₆	—	2	2	no	—
15	C₄×C₅×C₁₆	—	2	2	no	—
16	#13266	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
20	C₅×C₄⋊Q₈	C₈	1	2	yes	—
21	C₅×#125	C₈	1	2	yes	—
22	C₅×C₂²⋊Q₈	—	2	2	no	—
23	C₄×C₅×Q₈	—	2	2	no	—
24	C₅×#117	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₅×M₃₂	—	2	4	no	—
27	C₂×C₅×C₁₆	—	2	2	no	—
28	C₂×C₅×C₁₆	—	2	2	no	—
29	C₂×C₅×C₁₆	—	2	2	no	—
30	C₂×C₅×C₁₆	Q₈	1	4	yes	complemented
31	C₅×C₄⋊C₈	—	2	4	no	—
32	C₅×C₄⋊C₈	—	2	2	no	—
33	C₅×#97	—	2	2	no	—
34	C₄×C₅×C₈	—	2	2	no	—
35	C₄×C₅×C₈	D₄	1	2	yes	—
36	C₄×C₅×C₈	D₄	1	2	yes	—
37	#2960	C₂×C₅	1	1	yes	characteristic
38	#2150	C₂×C₅	1	1	yes	characteristic
39	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic
40	#1379	C₂×C₅	1	4	yes	—
41	C₂×C₅×Q₈	—	2	2	no	—
42	C₂²×C₄×C₅	M₁₆	1	2	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₅×M₁₆	—	2	4	no	—
45	C₂×C₅×C₈	—	2	2	no	—
46	C₂×C₅×C₈	—	2	2	no	—
47	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
48	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
51	C₂×C₅×C₈	#41	1	2	yes	—
52	C₅×C₄⋊C₄	—	2	4	no	—
53	C₅×C₄⋊C₄	—	2	2	no	—
54	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
55	C₅×C₂²⋊C₄	—	2	2	no	—
56	C₄²×C₅	—	2	2	no	—
57	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
60	C₅×C₁₆	—	2	4	no	—
61	C₅×C₁₆	—	4	2	no	—
62	#533	C₄×C₅	1	1	yes	characteristic
63	C₂×M₃₂	C₂²×C₅	1	2	yes	—
64	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
65	#446	—	2	2	no	—
66	C₈×Q₈	—	2	2	no	—
67	#424	C₄×C₅	1	2	yes	—
68	#423	C₄×C₅	1	1	yes	characteristic
69	#405	C₂²×C₅	1	2	yes	—
70	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
71	C₄⋊C₁₆	—	2	2	no	—
72	C₄×C₁₆	—	2	2	no	—
73	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
74	C₅×Q₈	—	4	4	no	—
75	C₂×C₄×C₅	—	2	4	no	—
76	C₂×C₄×C₅	—	2	4	no	—
77	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
78	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
79	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
81	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₅	#131	1	2	yes	—
83	C₅×C₈	—	2	2	no	—
84	C₅×C₈	—	4	2	no	—
85	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
86	C₅×C₈	#124	1	1	yes	characteristic
87	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
89	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
90	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
91	C₄⋊Q₈	C₅×C₈	1	2	yes	—
92	#125	C₅×C₈	1	2	yes	—
93	C₂²⋊Q₈	—	2	2	no	—
94	C₄×Q₈	—	2	2	no	—
95	#117	C₂×C₄×C₅	1	2	yes	—
96	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
97	M₃₂	—	2	4	no	—
98	C₂×C₁₆	—	2	2	no	—
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	2	2	no	—
101	C₂×C₁₆	C₅×Q₈	1	4	yes	complemented
102	C₄⋊C₈	—	2	4	no	—
103	C₄⋊C₈	—	2	2	no	—
104	#97	—	2	2	no	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	C₅×D₄	1	2	yes	—
107	C₄×C₈	C₅×D₄	1	2	yes	—
108	C₂²×C₅	—	2	2	no	—
109	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
110	C₂²×C₅	#446	1	2	yes	—
111	C₄×C₅	—	2	2	no	—
112	C₄×C₅	—	4	4	no	—
113	C₄×C₅	#350	1	4	yes	—
114	C₄×C₅	#391	1	1	yes	characteristic
115	C₄×C₅	#408	1	1	yes	characteristic
116	C₄×C₅	C₈×Q₈	1	2	yes	—
117	C₂×Q₈	—	2	2	no	—
118	C₂²×C₄	C₅×M₁₆	1	2	yes	—
119	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
120	M₁₆	—	2	4	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
124	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
125	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
126	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
127	C₂×C₈	C₅×#41	1	2	yes	—
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	C₂×C₅×C₈	1	4	yes	—
131	C₂²⋊C₄	—	2	2	no	—
132	C₄²	—	2	2	no	—
133	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
134	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
135	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
136	C₁₆	—	2	4	no	—
137	C₁₆	—	4	2	no	—
138	C₂×C₅	—	2	2	no	—
139	C₂×C₅	#1984	1	1	yes	characteristic
140	C₂×C₅	C₂×#350	1	2	yes	—
141	C₂³	C₂×C₅×M₁₆	1	1	yes	characteristic
142	Q₈	—	4	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
146	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
147	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
149	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
150	C₂×C₄	C₅×#131	1	2	yes	—
151	C₈	—	2	2	no	—
152	C₈	—	4	2	no	—
153	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
154	C₈	C₅×#124	1	1	yes	characteristic
155	C₈	C₅×C₄⋊Q₈	1	1	yes	characteristic
156	C₅	#13266	1	1	yes	characteristic, direct product factor, Sylow
157	C₂²	—	2	2	no	—
158	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
159	C₂²	C₅×#446	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	C₅×#350	1	4	yes	—
163	C₄	C₅×#391	1	1	yes	characteristic
164	C₄	C₅×#408	1	1	yes	characteristic
165	C₄	C₅×C₈×Q₈	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	C₅×#1984	1	1	yes	characteristic
168	C₂	C₂×C₅×#350	1	2	yes	—

	subgroup	G/H
center	C₂×C₅×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂³×C₅	C₂×M₁₆
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
16. #13266	C₅	direct (N × H)
30. C₂×C₅×C₁₆	Q₈	semidirect
101. C₂×C₁₆	C₅×Q₈	semidirect
156. C₅	#13266	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×#2960	2
C₅×#2150	2
C₂×C₄×C₅×C₁₆	2
C₅×#1379	2
#13266	5

Subgroups — C5×#13266 #11935986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13266 #11935986