Subgroups — C₅×#22258 #11944978

Order 1280.

578 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3160	C₂	1	1	yes	maximal, characteristic
2	C₅×#2961	C₂	1	2	yes	maximal
3	C₅×#1995	C₂	1	2	yes	maximal
4	C₅×#1922	C₂	1	8	yes	maximal
5	C₅×#1837	C₂	1	1	yes	maximal, characteristic
6	C₅×#1746	C₂	1	1	yes	maximal, characteristic
7	C₅×#533	C₂²	1	2	yes	—
8	C₅×#500	C₄	1	2	yes	complemented
9	C₅×#499	C₄	1	4	yes	complemented
10	C₅×#498	C₄	1	2	yes	complemented
11	C₅×#446	—	2	8	no	—
12	C₅×#427	C₂²	1	4	yes	—
13	C₅×#424	C₂²	1	4	yes	—
14	C₅×#423	C₂²	1	2	yes	—
15	C₅×#405	C₂²	1	4	yes	—
16	C₂×C₅×#97	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×#376	C₂²	1	1	yes	characteristic
19	C₅×#341	C₂²	1	4	yes	—
20	C₅×#339	C₂²	1	8	yes	—
21	C₅×#336	C₂²	1	4	yes	—
22	#22258	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂×C₅×M₁₆	C₂³	1	4	yes	—
24	C₂²×C₅×C₈	C₂³	1	2	yes	—
25	C₂²×C₅×C₈	C₂³	1	2	yes	—
26	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
27	C₅×#125	C₂×C₄	1	4	yes	—
28	C₅×C₂²⋊Q₈	—	2	4	no	—
29	C₄×C₅×Q₈	—	2	4	no	—
30	C₅×#117	C₂³	1	4	yes	—
31	C₂×C₄²×C₅	C₂³	1	2	yes	—
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₅×#107	C₂×C₄	1	8	yes	—
34	C₅×#106	C₂×C₄	1	8	yes	—
35	C₅×C₄⋊C₈	—	2	8	no	—
36	C₅×C₄⋊C₈	—	2	4	no	—
37	C₅×#97	—	2	4	no	—
38	C₅×#97	D₄	1	4	yes	—
39	C₄×C₅×C₈	—	2	4	no	—
40	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₅×#95	—	2	2	no	—
44	#3160	C₂×C₅	1	1	yes	characteristic
45	#2961	C₂×C₅	1	2	yes	—
46	#1995	C₂×C₅	1	2	yes	—
47	#1922	C₂×C₅	1	8	yes	—
48	#1837	C₂×C₅	1	1	yes	characteristic
49	#1746	C₂×C₅	1	1	yes	characteristic
50	C₂×C₅×Q₈	—	2	4	no	—
51	C₂²×C₄×C₅	—	2	2	no	—
52	C₂²×C₄×C₅	#41	1	2	yes	—
53	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
54	C₅×M₁₆	—	2	8	no	—
55	C₂×C₅×C₈	—	2	4	no	—
56	C₂×C₅×C₈	—	2	4	no	—
57	C₂×C₅×C₈	—	2	4	no	—
58	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
59	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
60	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
61	C₅×C₄⋊C₄	—	2	8	no	—
62	C₅×C₄⋊C₄	—	2	4	no	—
63	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
64	C₅×C₂²⋊C₄	—	2	4	no	—
65	C₄²×C₅	—	2	4	no	—
66	C₄²×C₅	—	2	4	no	—
67	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₅	C₂×D₄	1	2	yes	—
70	#533	C₂²×C₅	1	2	yes	—
71	#500	C₄×C₅	1	2	yes	complemented
72	#499	C₄×C₅	1	4	yes	complemented
73	#498	C₄×C₅	1	2	yes	complemented
74	#446	—	2	8	no	—
75	#427	C₂²×C₅	1	4	yes	—
76	#424	C₂²×C₅	1	4	yes	—
77	#423	C₂²×C₅	1	2	yes	—
78	#405	C₂²×C₅	1	4	yes	—
79	C₂×#97	C₂²×C₅	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
81	#376	C₂²×C₅	1	1	yes	characteristic
82	#341	C₂²×C₅	1	4	yes	—
83	#339	C₂²×C₅	1	8	yes	—
84	#336	C₂²×C₅	1	4	yes	—
85	C₂³×C₅	C₂×#41	1	1	yes	characteristic
86	C₅×Q₈	—	8	4	no	—
87	C₂×C₄×C₅	—	2	8	no	—
88	C₂×C₄×C₅	—	4	2	no	—
89	C₂×C₄×C₅	—	4	4	no	—
90	C₂×C₄×C₅	—	4	2	no	—
91	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅	2₋¹⁺⁴	1	4	yes	—
96	C₅×C₈	—	2	2	no	—
97	C₅×C₈	—	2	2	no	—
98	C₅×C₈	—	2	4	no	—
99	C₅×C₈	—	4	4	no	—
100	C₂×M₁₆	C₂³×C₅	1	4	yes	—
101	C₂²×C₈	C₂³×C₅	1	2	yes	—
102	C₂²×C₈	C₂³×C₅	1	2	yes	—
103	C₄⋊Q₈	C₂×C₄×C₅	1	4	yes	—
104	#125	C₂×C₄×C₅	1	4	yes	—
105	C₂²⋊Q₈	—	2	4	no	—
106	C₄×Q₈	—	2	4	no	—
107	#117	C₂³×C₅	1	4	yes	—
108	C₂×C₄²	C₂³×C₅	1	2	yes	—
109	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
110	#107	C₂×C₄×C₅	1	8	yes	—
111	#106	C₂×C₄×C₅	1	8	yes	—
112	C₄⋊C₈	—	2	8	no	—
113	C₄⋊C₈	—	2	4	no	—
114	#97	—	2	4	no	—
115	#97	C₅×D₄	1	4	yes	—
116	C₄×C₈	—	2	4	no	—
117	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
118	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
119	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
120	#95	—	2	2	no	—
121	C₂²×C₅	—	2	2	no	—
122	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
123	C₂²×C₅	#541	1	2	yes	—
124	C₄×C₅	—	2	2	no	—
125	C₄×C₅	—	2	2	no	—
126	C₄×C₅	—	4	8	no	—
127	C₄×C₅	—	8	2	no	—
128	C₄×C₅	#519	1	2	yes	—
129	C₄×C₅	#536	1	2	yes	—
130	C₂×Q₈	—	2	4	no	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	C₅×#41	1	2	yes	—
133	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
134	M₁₆	—	2	8	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
139	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
140	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
144	C₂²⋊C₄	—	2	4	no	—
145	C₄²	—	2	4	no	—
146	C₄²	—	2	4	no	—
147	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
148	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
149	C₄²	C₂×C₅×D₄	1	2	yes	—
150	C₂×C₅	—	2	2	no	—
151	C₂×C₅	#2327	1	1	yes	characteristic
152	C₂×C₅	#2957	1	2	yes	—
153	C₂³	C₂×C₅×#41	1	1	yes	characteristic
154	Q₈	—	8	4	no	—
155	C₂×C₄	—	2	8	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
160	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
161	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
162	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
163	C₂×C₄	C₅×2₋¹⁺⁴	1	4	yes	—
164	C₈	—	2	2	no	—
165	C₈	—	2	2	no	—
166	C₈	—	2	4	no	—
167	C₈	—	4	4	no	—
168	C₅	#22258	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²	—	2	2	no	—
170	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
171	C₂²	C₅×#541	1	2	yes	—
172	C₄	—	2	2	no	—
173	C₄	—	2	2	no	—
174	C₄	—	4	8	no	—
175	C₄	—	8	2	no	—
176	C₄	C₅×#519	1	2	yes	—
177	C₄	C₅×#536	1	2	yes	—
178	C₂	—	2	2	no	—
179	C₂	C₅×#2327	1	1	yes	characteristic
180	C₂	C₅×#2957	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	C₂×C₄×D₄
prime radical	C₂³×C₅	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅×#500	C₄	semidirect
9. C₅×#499	C₄	semidirect
10. C₅×#498	C₄	semidirect
22. #22258	C₅	direct (N × H)
71. #500	C₄×C₅	semidirect
72. #499	C₄×C₅	semidirect
73. #498	C₄×C₅	semidirect
168. C₅	#22258	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×#3160	2
C₅×#2961	2
C₅×#1995	2
C₅×#1922	2
C₅×#1837	2
C₅×#1746	2
#22258	5

Subgroups — C5×#22258 #11944978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22258 #11944978