Subgroups — C₅×#20769 #11943489

Order 1280.

818 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#423	C₂	1	2	yes	maximal
2	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#1838	C₂	1	4	yes	maximal, complemented
4	C₅×#1837	C₂	1	4	yes	maximal, complemented
5	C₅×#1738	C₂	1	4	yes	maximal
6	C₂²×C₅×M₁₆	C₂²	1	4	yes	—
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₅×#423	—	2	4	no	—
10	C₅×#423	C₂²	1	4	yes	—
11	C₅×#423	C₂²	1	4	yes	—
12	C₂×C₅×C₄⋊C₈	—	2	4	no	—
13	C₄×C₅×M₁₆	C₄	1	4	yes	—
14	C₄×C₅×M₁₆	C₄	1	4	yes	—
15	C₂×C₅×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
17	C₅×#341	C₂²	1	16	yes	—
18	#20769	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂³×C₄×C₅	Q₈	1	2	yes	—
20	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂×C₅×M₁₆	—	2	8	no	—
22	C₂×C₅×M₁₆	—	2	4	no	—
23	C₂×C₅×M₁₆	—	2	16	no	—
24	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
25	C₂×C₅×M₁₆	C₂³	1	8	yes	—
26	C₂²×C₅×C₈	—	2	4	no	—
27	C₂²×C₅×C₈	C₂³	1	4	yes	—
28	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅	D₄	1	4	yes	—
31	C₂×C₄²×C₅	Q₈	1	2	yes	—
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₅×C₄⋊C₈	—	2	8	no	—
35	C₅×C₄⋊C₈	—	2	8	no	—
36	C₅×#97	C₂×C₄	1	8	yes	—
37	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
38	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
39	C₂×#423	C₂×C₅	1	2	yes	—
40	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
41	#1838	C₂×C₅	1	4	yes	complemented
42	#1837	C₂×C₅	1	4	yes	complemented
43	#1738	C₂×C₅	1	4	yes	—
44	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₅	—	4	4	no	—
46	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
52	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₅	#41	1	8	yes	—
54	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
55	C₅×M₁₆	—	2	16	no	—
56	C₅×M₁₆	—	2	16	no	—
57	C₅×M₁₆	—	4	8	no	—
58	C₂×C₅×C₈	—	2	8	no	—
59	C₂×C₅×C₈	—	2	8	no	—
60	C₂×C₅×C₈	—	4	8	no	—
61	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
62	C₄²×C₅	C₄⋊C₄	1	4	yes	—
63	C₄²×C₅	C₄⋊C₄	1	4	yes	—
64	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₅	C₂²×C₄	1	2	yes	—
67	C₄²×C₅	C₂×D₄	1	2	yes	—
68	C₄²×C₅	C₂×D₄	1	2	yes	—
69	C₂²×M₁₆	C₂²×C₅	1	4	yes	—
70	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
71	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
72	#423	—	2	4	no	—
73	#423	C₂²×C₅	1	4	yes	—
74	#423	C₂²×C₅	1	4	yes	—
75	C₂×C₄⋊C₈	—	2	4	no	—
76	C₄×M₁₆	C₄×C₅	1	4	yes	—
77	C₄×M₁₆	C₄×C₅	1	4	yes	—
78	C₂×#97	C₂²×C₅	1	2	yes	—
79	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
80	#341	C₂²×C₅	1	16	yes	—
81	C₂³×C₅	—	4	2	no	—
82	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
83	C₂³×C₅	#125	1	4	yes	—
84	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₅	—	2	8	no	—
86	C₂×C₄×C₅	—	2	4	no	—
87	C₂×C₄×C₅	—	4	4	no	—
88	C₂×C₄×C₅	—	4	2	no	—
89	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₅	#117	1	8	yes	—
94	C₂×C₄×C₅	#124	1	4	yes	—
95	C₂×C₄×C₅	#124	1	4	yes	—
96	C₂×C₄×C₅	#125	1	4	yes	—
97	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×#41	1	4	yes	—
100	C₅×C₈	—	2	8	no	—
101	C₅×C₈	—	4	8	no	—
102	C₂³×C₄	C₅×Q₈	1	2	yes	—
103	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
104	C₂×M₁₆	—	2	8	no	—
105	C₂×M₁₆	—	2	4	no	—
106	C₂×M₁₆	—	2	16	no	—
107	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
108	C₂×M₁₆	C₂³×C₅	1	8	yes	—
109	C₂²×C₈	—	2	4	no	—
110	C₂²×C₈	C₂³×C₅	1	4	yes	—
111	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
112	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
113	C₂×C₄²	C₅×D₄	1	4	yes	—
114	C₂×C₄²	C₅×Q₈	1	2	yes	—
115	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
116	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
117	C₄⋊C₈	—	2	8	no	—
118	C₄⋊C₈	—	2	8	no	—
119	#97	C₂×C₄×C₅	1	8	yes	—
120	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
121	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
122	C₂²×C₅	—	2	4	no	—
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	—	4	4	no	—
125	C₂²×C₅	#382	1	4	yes	—
126	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₅	C₂×#125	1	2	yes	—
128	C₄×C₅	—	2	4	no	—
129	C₄×C₅	—	2	2	no	—
130	C₄×C₅	#382	1	4	yes	—
131	C₄×C₅	#430	1	4	yes	—
132	C₄×C₅	C₂×#117	1	2	yes	—
133	C₄×C₅	C₂×#124	1	2	yes	—
134	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
135	C₂²×C₄	—	4	4	no	—
136	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
137	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
138	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
139	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
140	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
141	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
142	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
143	C₂²×C₄	C₅×#41	1	8	yes	—
144	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
145	M₁₆	—	2	16	no	—
146	M₁₆	—	2	16	no	—
147	M₁₆	—	4	8	no	—
148	C₂×C₈	—	2	8	no	—
149	C₂×C₈	—	2	8	no	—
150	C₂×C₈	—	4	8	no	—
151	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
152	C₄²	C₅×C₄⋊C₄	1	4	yes	—
153	C₄²	C₅×C₄⋊C₄	1	4	yes	—
154	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
155	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
156	C₄²	C₂²×C₄×C₅	1	2	yes	—
157	C₄²	C₂×C₅×D₄	1	2	yes	—
158	C₄²	C₂×C₅×D₄	1	2	yes	—
159	C₂×C₅	—	2	2	no	—
160	C₂×C₅	—	2	4	no	—
161	C₂×C₅	C₂×#382	1	1	yes	characteristic
162	C₂×C₅	C₂×#430	1	2	yes	—
163	C₂³	—	4	2	no	—
164	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
165	C₂³	C₅×#125	1	4	yes	—
166	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
172	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
173	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
174	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
175	C₂×C₄	C₅×#117	1	8	yes	—
176	C₂×C₄	C₅×#124	1	4	yes	—
177	C₂×C₄	C₅×#124	1	4	yes	—
178	C₂×C₄	C₅×#125	1	4	yes	—
179	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
180	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
181	C₂×C₄	C₂×C₅×#41	1	4	yes	—
182	C₈	—	2	8	no	—
183	C₈	—	4	8	no	—
184	C₅	#20769	1	1	yes	characteristic, direct product factor, Sylow
185	C₂²	—	2	4	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	4	4	no	—
188	C₂²	C₅×#382	1	4	yes	—
189	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
190	C₂²	C₂×C₅×#125	1	2	yes	—
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	C₅×#382	1	4	yes	—
194	C₄	C₅×#430	1	4	yes	—
195	C₄	C₂×C₅×#117	1	2	yes	—
196	C₄	C₂×C₅×#124	1	2	yes	—
197	C₂	—	2	4	no	—
198	C₂	—	2	2	no	—
199	C₂	C₂×C₅×#382	1	1	yes	characteristic
200	C₂	C₂×C₅×#430	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1838	C₂	semidirect
4. C₅×#1837	C₂	semidirect
18. #20769	C₅	direct (N × H)
41. #1838	C₂×C₅	semidirect
42. #1837	C₂×C₅	semidirect
184. C₅	#20769	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×#423	2
C₂×C₄×C₅×M₁₆	2
C₅×#1838	2
C₅×#1837	2
C₅×#1738	2
#20769	5

Subgroups — C5×#20769 #11943489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20769 #11943489