Subgroups — C₅×#21952 #11944672

Order 1280.

826 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#431	C₂	1	2	yes	maximal
2	C₅×#2285	C₂	1	1	yes	maximal, characteristic
3	C₅×#1939	C₂	1	8	yes	maximal, complemented
4	C₅×#1798	C₂	1	4	yes	maximal
5	C₂×C₅×#117	C₂²	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₅×#431	—	2	8	no	—
8	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	8	yes	—
10	C₂×C₅×C₂²⋊C₈	C₂²	1	4	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
12	C₅×#384	C₄	1	2	yes	—
13	C₅×#382	C₄	1	2	yes	—
14	C₅×#380	C₄	1	4	yes	—
15	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
16	C₅×#336	C₂²	1	8	yes	—
17	#21952	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂³	1	8	yes	—
21	C₅×#117	—	4	4	no	—
22	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
23	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
24	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	D₄	1	4	yes	—
28	C₂×C₄²×C₅	Q₈	1	4	yes	—
29	C₂×C₄²×C₅	C₂³	1	4	yes	—
30	C₂×C₄²×C₅	C₂³	1	2	yes	—
31	C₅×C₄⋊C₈	—	2	8	no	—
32	C₅×C₄⋊C₈	—	4	8	no	—
33	C₅×C₂²⋊C₈	—	2	8	no	—
34	C₄×C₅×C₈	—	2	8	no	—
35	C₅×#95	C₂×C₄	1	8	yes	—
36	C₂×#431	C₂×C₅	1	2	yes	—
37	#2285	C₂×C₅	1	1	yes	characteristic
38	#1939	C₂×C₅	1	8	yes	complemented
39	#1798	C₂×C₅	1	4	yes	—
40	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅	—	2	8	no	—
42	C₂²×C₄×C₅	—	2	2	no	—
43	C₂²×C₄×C₅	—	2	2	no	—
44	C₂²×C₄×C₅	C₄⋊C₄	1	8	yes	—
45	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
46	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂²×C₄	1	2	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	C₂⁴	1	1	yes	characteristic
52	C₂×C₅×C₈	—	2	8	no	—
53	C₂×C₅×C₈	—	2	8	no	—
54	C₂×C₅×C₈	—	2	8	no	—
55	C₅×C₄⋊C₄	—	2	16	no	—
56	C₅×C₄⋊C₄	—	4	4	no	—
57	C₅×C₂²⋊C₄	—	4	4	no	—
58	C₄²×C₅	—	2	8	no	—
59	C₄²×C₅	C₄⋊C₄	1	4	yes	—
60	C₄²×C₅	C₄⋊C₄	1	4	yes	—
61	C₄²×C₅	C₂×D₄	1	4	yes	—
62	C₄²×C₅	C₂×Q₈	1	4	yes	—
63	C₂×#117	C₂²×C₅	1	2	yes	—
64	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
65	#431	—	2	8	no	—
66	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
67	C₂×C₄⋊C₈	C₂²×C₅	1	8	yes	—
68	C₂×C₂²⋊C₈	C₂²×C₅	1	4	yes	—
69	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
70	#384	C₄×C₅	1	2	yes	—
71	#382	C₄×C₅	1	2	yes	—
72	#380	C₄×C₅	1	4	yes	—
73	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
74	#336	C₂²×C₅	1	8	yes	—
75	C₂³×C₅	—	2	1	no	—
76	C₂³×C₅	—	2	1	no	—
77	C₂³×C₅	—	2	4	no	—
78	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
79	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₅	—	2	1	no	—
81	C₂×C₄×C₅	—	2	1	no	—
82	C₂×C₄×C₅	—	2	4	no	—
83	C₂×C₄×C₅	—	2	4	no	—
84	C₂×C₄×C₅	—	2	4	no	—
85	C₂×C₄×C₅	—	2	16	no	—
86	C₂×C₄×C₅	—	2	8	no	—
87	C₂×C₄×C₅	—	2	16	no	—
88	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₅	#131	1	8	yes	—
92	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
95	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
96	C₅×C₈	—	4	8	no	—
97	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
98	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
99	C₂²×C₈	C₂³×C₅	1	8	yes	—
100	#117	—	4	4	no	—
101	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
102	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
103	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
104	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
105	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂×C₄²	C₅×D₄	1	4	yes	—
107	C₂×C₄²	C₅×Q₈	1	4	yes	—
108	C₂×C₄²	C₂³×C₅	1	4	yes	—
109	C₂×C₄²	C₂³×C₅	1	2	yes	—
110	C₄⋊C₈	—	2	8	no	—
111	C₄⋊C₈	—	4	8	no	—
112	C₂²⋊C₈	—	2	8	no	—
113	C₄×C₈	—	2	8	no	—
114	#95	C₂×C₄×C₅	1	8	yes	—
115	C₂²×C₅	—	2	2	no	—
116	C₂²×C₅	—	2	4	no	—
117	C₂²×C₅	—	4	2	no	—
118	C₂²×C₅	—	4	2	no	—
119	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₅	#518	1	1	yes	characteristic
121	C₂²×C₅	#519	1	1	yes	characteristic
122	C₂²×C₅	C₂×#131	1	4	yes	—
123	C₄×C₅	—	2	2	no	—
124	C₄×C₅	—	2	4	no	—
125	C₄×C₅	—	4	8	no	—
126	C₄×C₅	#424	1	8	yes	—
127	C₄×C₅	#537	1	2	yes	—
128	C₄×C₅	#544	1	2	yes	—
129	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
130	C₂²×C₄	—	2	8	no	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	C₅×C₄⋊C₄	1	8	yes	—
134	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
135	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
136	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
137	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
138	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
139	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
140	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
141	C₂×C₈	—	2	8	no	—
142	C₂×C₈	—	2	8	no	—
143	C₂×C₈	—	2	8	no	—
144	C₄⋊C₄	—	2	16	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₂²⋊C₄	—	4	4	no	—
147	C₄²	—	2	8	no	—
148	C₄²	C₅×C₄⋊C₄	1	4	yes	—
149	C₄²	C₅×C₄⋊C₄	1	4	yes	—
150	C₄²	C₂×C₅×D₄	1	4	yes	—
151	C₄²	C₂×C₅×Q₈	1	4	yes	—
152	C₂×C₅	—	4	2	no	—
153	C₂×C₅	#2316	1	1	yes	characteristic
154	C₂×C₅	C₂×#424	1	2	yes	—
155	C₂×C₅	#2979	1	2	yes	—
156	C₂×C₅	#2986	1	2	yes	—
157	C₂³	—	2	1	no	—
158	C₂³	—	2	1	no	—
159	C₂³	—	2	4	no	—
160	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
161	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	16	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	16	no	—
170	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
171	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
172	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
173	C₂×C₄	C₅×#131	1	8	yes	—
174	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
176	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
177	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
178	C₈	—	4	8	no	—
179	C₅	#21952	1	1	yes	characteristic, direct product factor, Sylow
180	C₂²	—	2	2	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
185	C₂²	C₅×#518	1	1	yes	characteristic
186	C₂²	C₅×#519	1	1	yes	characteristic
187	C₂²	C₂×C₅×#131	1	4	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	4	8	no	—
191	C₄	C₅×#424	1	8	yes	—
192	C₄	C₅×#537	1	2	yes	—
193	C₄	C₅×#544	1	2	yes	—
194	C₂	—	4	2	no	—
195	C₂	C₅×#2316	1	1	yes	characteristic
196	C₂	C₂×C₅×#424	1	2	yes	—
197	C₂	C₅×#2979	1	2	yes	—
198	C₂	C₅×#2986	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₅×#1939	C₂	semidirect
17. #21952	C₅	direct (N × H)
38. #1939	C₂×C₅	semidirect
179. C₅	#21952	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×#431	2
C₅×#2285	2
C₅×#1939	2
C₅×#1798	2
#21952	5

Subgroups — C5×#21952 #11944672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21952 #11944672