Subgroups — C₅×#12019 #11934739

Order 1280.

226 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	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic
2	C₅×#1568	C₂	1	1	yes	maximal, characteristic
3	C₅×#1566	C₂	1	1	yes	maximal, characteristic
4	C₅×#1368	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×C₁₆	C₄	1	2	yes	—
6	C₅×#432	—	2	1	no	—
7	C₅×#423	—	2	1	no	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₅×#346	C₄	1	1	yes	characteristic
10	C₅×#346	C₄	1	1	yes	characteristic
11	C₅×#346	C₂²	1	2	yes	—
12	C₅×#335	—	2	1	no	—
13	C₅×#334	—	2	1	no	—
14	C₅×#334	C₂²	1	2	yes	—
15	C₅×#334	C₂²	1	2	yes	—
16	#12019	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂×C₅×M₁₆	—	4	1	no	—
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
20	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
21	C₂×C₅×C₁₆	—	2	4	no	—
22	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
24	C₅×C₄⋊C₈	—	2	2	no	—
25	C₅×C₄⋊C₈	—	2	2	no	—
26	C₅×C₄⋊C₈	—	4	1	no	—
27	C₅×C₂²⋊C₈	—	4	1	no	—
28	C₅×#97	—	4	1	no	—
29	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×C₅×C₈	D₄	1	1	yes	characteristic
31	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
32	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₂×#346	C₂×C₅	1	1	yes	characteristic
34	#1568	C₂×C₅	1	1	yes	characteristic
35	#1566	C₂×C₅	1	1	yes	characteristic
36	#1368	C₂×C₅	1	4	yes	complemented
37	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
40	C₅×M₁₆	—	8	2	no	—
41	C₂×C₅×C₈	—	2	2	no	—
42	C₂×C₅×C₈	—	2	2	no	—
43	C₂×C₅×C₈	—	4	2	no	—
44	C₂×C₅×C₈	—	4	2	no	—
45	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₅	SD₁₆	1	1	yes	characteristic
51	C₄²×C₅	SD₁₆	1	1	yes	characteristic
52	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
53	C₅×C₁₆	—	2	2	no	—
54	C₅×C₁₆	M₁₆	1	4	yes	complemented
55	C₂²×C₁₆	C₄×C₅	1	2	yes	—
56	#432	—	2	1	no	—
57	#423	—	2	1	no	—
58	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
59	#346	C₄×C₅	1	1	yes	characteristic
60	#346	C₄×C₅	1	1	yes	characteristic
61	#346	C₂²×C₅	1	2	yes	—
62	#335	—	2	1	no	—
63	#334	—	2	1	no	—
64	#334	C₂²×C₅	1	2	yes	—
65	#334	C₂²×C₅	1	2	yes	—
66	C₂³×C₅	#106	1	1	yes	characteristic
67	C₂×C₄×C₅	—	2	2	no	—
68	C₂×C₄×C₅	—	2	1	no	—
69	C₂×C₄×C₅	—	2	1	no	—
70	C₂×C₄×C₅	—	4	1	no	—
71	C₂×C₄×C₅	#106	1	1	yes	characteristic
72	C₂×C₄×C₅	#106	1	1	yes	characteristic
73	C₂×C₄×C₅	#106	1	1	yes	characteristic
74	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
77	C₅×C₈	—	2	1	no	—
78	C₅×C₈	—	2	1	no	—
79	C₅×C₈	—	8	2	no	—
80	C₅×C₈	—	8	2	no	—
81	C₅×C₈	#108	1	2	yes	—
82	C₅×C₈	C₂×M₁₆	1	2	yes	—
83	C₂×M₁₆	—	4	1	no	—
84	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
85	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
86	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
87	C₂×C₁₆	—	2	4	no	—
88	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
89	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
90	C₄⋊C₈	—	2	2	no	—
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	—	4	1	no	—
93	C₂²⋊C₈	—	4	1	no	—
94	#97	—	4	1	no	—
95	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
96	C₄×C₈	C₅×D₄	1	1	yes	characteristic
97	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
98	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
99	C₂²×C₅	—	2	1	no	—
100	C₂²×C₅	—	2	1	no	—
101	C₂²×C₅	#365	1	2	yes	—
102	C₂²×C₅	C₂×#106	1	1	yes	characteristic
103	C₄×C₅	—	2	1	no	—
104	C₄×C₅	—	2	1	no	—
105	C₄×C₅	—	2	1	no	—
106	C₄×C₅	—	2	1	no	—
107	C₄×C₅	—	2	1	no	—
108	C₄×C₅	#423	1	1	yes	characteristic
109	C₄×C₅	C₂×#108	1	1	yes	characteristic
110	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
112	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
113	M₁₆	—	8	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
121	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
122	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
123	C₄²	C₅×SD₁₆	1	1	yes	characteristic
124	C₄²	C₅×SD₁₆	1	1	yes	characteristic
125	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
126	C₁₆	—	2	2	no	—
127	C₁₆	C₅×M₁₆	1	4	yes	complemented
128	C₂×C₅	—	2	2	no	—
129	C₂×C₅	#1566	1	1	yes	characteristic
130	C₂×C₅	C₂×#365	1	1	yes	characteristic
131	C₂×C₅	#2155	1	1	yes	characteristic
132	C₂³	C₅×#106	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	C₅×#106	1	1	yes	characteristic
138	C₂×C₄	C₅×#106	1	1	yes	characteristic
139	C₂×C₄	C₅×#106	1	1	yes	characteristic
140	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
142	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	8	2	no	—
146	C₈	—	8	2	no	—
147	C₈	C₅×#108	1	2	yes	—
148	C₈	C₂×C₅×M₁₆	1	2	yes	—
149	C₅	#12019	1	1	yes	characteristic, direct product factor, Sylow
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	C₅×#365	1	2	yes	—
153	C₂²	C₂×C₅×#106	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	C₅×#423	1	1	yes	characteristic
160	C₄	C₂×C₅×#108	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	C₅×#1566	1	1	yes	characteristic
163	C₂	C₂×C₅×#365	1	1	yes	characteristic
164	C₂	C₅×#2155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂³×C₅	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
4. C₅×#1368	C₂	semidirect
16. #12019	C₅	direct (N × H)
36. #1368	C₂×C₅	semidirect
54. C₅×C₁₆	M₁₆	semidirect
127. C₁₆	C₅×M₁₆	semidirect
149. C₅	#12019	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×#346	2
C₅×#1568	2
C₅×#1566	2
C₅×#1368	2
#12019	5

Subgroups — C5×#12019 #11934739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12019 #11934739