Subgroups — C₅×#21971 #11944691

Order 1280.

662 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2921	C₂	1	2	yes	maximal
2	C₂×C₅×#424	C₂	1	1	yes	maximal, characteristic
3	C₅×#1950	C₂	1	8	yes	maximal, complemented
4	C₅×#1811	C₂	1	2	yes	maximal
5	C₅×#1810	C₂	1	2	yes	maximal
6	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#117	C₂²	1	1	yes	characteristic
9	C₅×#433	—	2	8	no	—
10	C₂×C₅×#108	C₂²	1	4	yes	—
11	C₂×C₅×#108	C₂²	1	4	yes	—
12	C₅×#424	C₄	1	8	yes	—
13	C₅×#423	C₂²	1	4	yes	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
15	C₅×#407	C₂²	1	4	yes	—
16	C₄×C₅×M₁₆	C₂²	1	4	yes	—
17	C₅×#341	C₂²	1	4	yes	—
18	C₅×#341	C₂²	1	4	yes	—
19	#21971	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂×C₅×M₁₆	—	4	2	no	—
22	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
23	C₂×C₅×M₁₆	C₂³	1	8	yes	—
24	C₂²×C₅×C₈	—	4	2	no	—
25	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₅×C₈	C₂³	1	2	yes	—
27	C₂²×C₅×C₈	C₂³	1	2	yes	—
28	C₅×#117	C₂×C₄	1	4	yes	—
29	C₅×#117	D₄	1	4	yes	—
30	C₂×C₅×C₄⋊C₄	Q₈	1	2	yes	—
31	C₂×C₅×C₂²⋊C₄	Q₈	1	2	yes	—
32	C₂×C₄²×C₅	C₂³	1	2	yes	—
33	C₅×#108	—	4	4	no	—
34	C₅×#108	—	4	4	no	—
35	C₅×C₄⋊C₈	—	2	8	no	—
36	C₅×C₄⋊C₈	C₂×C₄	1	16	yes	—
37	C₅×C₂²⋊C₈	—	2	8	no	—
38	C₅×#97	—	2	4	no	—
39	C₄×C₅×C₈	—	2	4	no	—
40	#2921	C₂×C₅	1	2	yes	—
41	C₂×#424	C₂×C₅	1	1	yes	characteristic
42	#1950	C₂×C₅	1	8	yes	complemented
43	#1811	C₂×C₅	1	2	yes	—
44	#1810	C₂×C₅	1	2	yes	—
45	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₅	—	2	2	no	—
47	C₂²×C₄×C₅	—	2	2	no	—
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	1	yes	characteristic
51	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
52	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
53	C₅×M₁₆	—	2	8	no	—
54	C₅×M₁₆	—	2	8	no	—
55	C₅×M₁₆	—	4	4	no	—
56	C₂×C₅×C₈	—	2	4	no	—
57	C₂×C₅×C₈	—	2	4	no	—
58	C₂×C₅×C₈	—	2	8	no	—
59	C₂×C₅×C₈	—	4	4	no	—
60	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
61	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
62	C₅×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
63	C₅×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
64	C₄²×C₅	C₂²×C₄	1	4	yes	—
65	C₄²×C₅	C₂×D₄	1	4	yes	—
66	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
67	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
68	C₂×#117	C₂²×C₅	1	1	yes	characteristic
69	#433	—	2	8	no	—
70	C₂×#108	C₂²×C₅	1	4	yes	—
71	C₂×#108	C₂²×C₅	1	4	yes	—
72	#424	C₄×C₅	1	8	yes	—
73	#423	C₂²×C₅	1	4	yes	—
74	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
75	#407	C₂²×C₅	1	4	yes	—
76	C₄×M₁₆	C₂²×C₅	1	4	yes	—
77	#341	C₂²×C₅	1	4	yes	—
78	#341	C₂²×C₅	1	4	yes	—
79	C₂³×C₅	—	2	2	no	—
80	C₂³×C₅	—	4	2	no	—
81	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
82	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	—	2	4	no	—
87	C₂×C₄×C₅	—	4	4	no	—
88	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
89	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
93	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
94	C₅×C₈	—	2	4	no	—
95	C₅×C₈	—	2	4	no	—
96	C₅×C₈	—	4	8	no	—
97	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
98	C₂×M₁₆	—	4	2	no	—
99	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
100	C₂×M₁₆	C₂³×C₅	1	8	yes	—
101	C₂²×C₈	—	4	2	no	—
102	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
103	C₂²×C₈	C₂³×C₅	1	2	yes	—
104	C₂²×C₈	C₂³×C₅	1	2	yes	—
105	#117	C₂×C₄×C₅	1	4	yes	—
106	#117	C₅×D₄	1	4	yes	—
107	C₂×C₄⋊C₄	C₅×Q₈	1	2	yes	—
108	C₂×C₂²⋊C₄	C₅×Q₈	1	2	yes	—
109	C₂×C₄²	C₂³×C₅	1	2	yes	—
110	#108	—	4	4	no	—
111	#108	—	4	4	no	—
112	C₄⋊C₈	—	2	8	no	—
113	C₄⋊C₈	C₂×C₄×C₅	1	16	yes	—
114	C₂²⋊C₈	—	2	8	no	—
115	#97	—	2	4	no	—
116	C₄×C₈	—	2	4	no	—
117	C₂²×C₅	—	2	2	no	—
118	C₂²×C₅	—	2	2	no	—
119	C₂²×C₅	—	2	4	no	—
120	C₂²×C₅	—	4	4	no	—
121	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₅	#544	1	2	yes	—
123	C₄×C₅	—	2	2	no	—
124	C₄×C₅	—	2	8	no	—
125	C₄×C₅	#518	1	1	yes	characteristic
126	C₄×C₅	#519	1	1	yes	characteristic
127	C₄×C₅	#537	1	2	yes	—
128	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
132	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
133	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
135	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
136	M₁₆	—	2	8	no	—
137	M₁₆	—	2	8	no	—
138	M₁₆	—	4	4	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	8	no	—
142	C₂×C₈	—	4	4	no	—
143	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
144	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
145	C₄⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
146	C₂²⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
147	C₄²	C₂²×C₄×C₅	1	4	yes	—
148	C₄²	C₂×C₅×D₄	1	4	yes	—
149	C₂×C₅	—	2	2	no	—
150	C₂×C₅	—	4	2	no	—
151	C₂×C₅	#2316	1	1	yes	characteristic
152	C₂×C₅	#2916	1	2	yes	—
153	C₂³	—	2	2	no	—
154	C₂³	—	4	2	no	—
155	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
156	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	C₂×C₅×C₄⋊C₄	1	8	yes	—
163	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
164	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
165	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
166	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
167	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
168	C₈	—	2	4	no	—
169	C₈	—	2	4	no	—
170	C₈	—	4	8	no	—
171	C₅	#21971	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	4	no	—
175	C₂²	—	4	4	no	—
176	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
177	C₂²	C₅×#544	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	2	8	no	—
180	C₄	C₅×#518	1	1	yes	characteristic
181	C₄	C₅×#519	1	1	yes	characteristic
182	C₄	C₅×#537	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	4	2	no	—
185	C₂	C₅×#2316	1	1	yes	characteristic
186	C₂	C₅×#2916	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 ◁ C₂×C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1950	C₂	semidirect
19. #21971	C₅	direct (N × H)
42. #1950	C₂×C₅	semidirect
171. C₅	#21971	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×#2921	2
C₂×C₅×#424	2
C₅×#1950	2
C₅×#1811	2
C₅×#1810	2
#21971	5

Subgroups — C5×#21971 #11944691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21971 #11944691