Subgroups — C₅×#22243 #11944963

Order 1280.

666 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#446	C₂	1	2	yes	maximal
2	C₅×#2399	C₂	1	1	yes	maximal, characteristic
3	C₅×#1985	C₂	1	2	yes	maximal
4	C₅×#1917	C₂	1	8	yes	maximal
5	C₅×#1832	C₂	1	1	yes	maximal, characteristic
6	C₅×#1746	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
8	C₅×#446	—	2	8	no	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
11	C₂×C₅×#97	C₂²	1	4	yes	—
12	C₂×C₅×#97	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×#398	C₄	1	4	yes	—
15	C₅×#395	C₄	1	4	yes	—
16	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
17	C₅×#376	C₂²	1	1	yes	characteristic
18	C₅×#336	C₂²	1	4	yes	—
19	C₅×#336	C₂²	1	8	yes	—
20	C₅×#336	C₂²	1	4	yes	—
21	#22243	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
23	C₂²×C₅×C₈	C₂³	1	2	yes	—
24	C₂²×C₅×C₈	C₂³	1	4	yes	—
25	C₂²×C₅×C₈	C₂³	1	2	yes	—
26	C₄×C₅×Q₈	—	2	4	no	—
27	C₄×C₅×Q₈	—	2	4	no	—
28	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
29	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₅	C₂³	1	2	yes	—
32	C₂×C₄²×C₅	C₂³	1	4	yes	—
33	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₅×C₄⋊C₈	—	2	8	no	—
35	C₅×C₄⋊C₈	—	2	4	no	—
36	C₅×#97	—	2	8	no	—
37	C₅×#97	—	2	2	no	—
38	C₄×C₅×C₈	D₄	1	4	yes	—
39	C₅×#95	C₂×C₄	1	4	yes	—
40	C₅×#95	C₂×C₄	1	4	yes	—
41	C₅×#95	C₂×C₄	1	4	yes	—
42	C₂×#446	C₂×C₅	1	2	yes	—
43	#2399	C₂×C₅	1	1	yes	characteristic
44	#1985	C₂×C₅	1	2	yes	—
45	#1917	C₂×C₅	1	8	yes	—
46	#1832	C₂×C₅	1	1	yes	characteristic
47	#1746	C₂×C₅	1	1	yes	characteristic
48	C₂×C₅×Q₈	—	2	4	no	—
49	C₂×C₅×Q₈	—	4	4	no	—
50	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
55	C₂×C₅×C₈	—	2	4	no	—
56	C₂×C₅×C₈	—	2	4	no	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	2	8	no	—
59	C₂×C₅×C₈	—	2	4	no	—
60	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
61	C₅×C₄⋊C₄	—	2	4	no	—
62	C₅×C₄⋊C₄	—	2	8	no	—
63	C₅×C₄⋊C₄	—	4	8	no	—
64	C₄²×C₅	—	2	4	no	—
65	C₄²×C₅	—	2	4	no	—
66	C₄²×C₅	—	2	4	no	—
67	C₄²×C₅	C₂×D₄	1	2	yes	—
68	C₄²×C₅	#41	1	2	yes	—
69	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
70	#446	—	2	8	no	—
71	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
72	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
73	C₂×#97	C₂²×C₅	1	4	yes	—
74	C₂×#97	C₂²×C₅	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
76	#398	C₄×C₅	1	4	yes	—
77	#395	C₄×C₅	1	4	yes	—
78	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
79	#376	C₂²×C₅	1	1	yes	characteristic
80	#336	C₂²×C₅	1	4	yes	—
81	#336	C₂²×C₅	1	8	yes	—
82	#336	C₂²×C₅	1	4	yes	—
83	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
84	C₅×Q₈	—	4	8	no	—
85	C₂×C₄×C₅	—	2	4	no	—
86	C₂×C₄×C₅	—	2	4	no	—
87	C₂×C₄×C₅	—	2	4	no	—
88	C₂×C₄×C₅	—	2	4	no	—
89	C₂×C₄×C₅	—	2	4	no	—
90	C₂×C₄×C₅	—	2	8	no	—
91	C₂×C₄×C₅	—	2	8	no	—
92	C₂×C₄×C₅	—	2	4	no	—
93	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
94	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
96	C₂×C₄×C₅	2₋¹⁺⁴	1	4	yes	—
97	C₅×C₈	—	2	4	no	—
98	C₅×C₈	—	4	4	no	—
99	C₅×C₈	—	4	2	no	—
100	C₂²×Q₈	C₂×C₄×C₅	1	2	yes	—
101	C₂²×C₈	C₂³×C₅	1	2	yes	—
102	C₂²×C₈	C₂³×C₅	1	4	yes	—
103	C₂²×C₈	C₂³×C₅	1	2	yes	—
104	C₄×Q₈	—	2	4	no	—
105	C₄×Q₈	—	2	4	no	—
106	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
107	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
108	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
109	C₂×C₄²	C₂³×C₅	1	4	yes	—
110	C₂×C₄²	C₂³×C₅	1	2	yes	—
111	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
112	C₄⋊C₈	—	2	8	no	—
113	C₄⋊C₈	—	2	4	no	—
114	#97	—	2	8	no	—
115	#97	—	2	2	no	—
116	C₄×C₈	C₅×D₄	1	4	yes	—
117	#95	C₂×C₄×C₅	1	4	yes	—
118	#95	C₂×C₄×C₅	1	4	yes	—
119	#95	C₂×C₄×C₅	1	4	yes	—
120	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂²×C₅	#519	1	2	yes	—
122	C₂²×C₅	#568	1	4	yes	—
123	C₄×C₅	—	2	4	no	—
124	C₄×C₅	—	4	4	no	—
125	C₄×C₅	—	4	4	no	—
126	C₄×C₅	—	4	4	no	—
127	C₄×C₅	#536	1	2	yes	—
128	C₄×C₅	#541	1	2	yes	—
129	C₂×Q₈	—	2	4	no	—
130	C₂×Q₈	—	4	4	no	—
131	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
132	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
133	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
134	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
135	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	8	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	8	no	—
144	C₄⋊C₄	—	4	8	no	—
145	C₄²	—	2	4	no	—
146	C₄²	—	2	4	no	—
147	C₄²	—	2	4	no	—
148	C₄²	C₂×C₅×D₄	1	2	yes	—
149	C₄²	C₅×#41	1	2	yes	—
150	C₂×C₅	#2327	1	1	yes	characteristic
151	C₂×C₅	#2930	1	2	yes	—
152	C₂×C₅	#2987	1	4	yes	—
153	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
154	Q₈	—	4	8	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
164	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
165	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
166	C₂×C₄	C₅×2₋¹⁺⁴	1	4	yes	—
167	C₈	—	2	4	no	—
168	C₈	—	4	4	no	—
169	C₈	—	4	2	no	—
170	C₅	#22243	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
172	C₂²	C₅×#519	1	2	yes	—
173	C₂²	C₅×#568	1	4	yes	—
174	C₄	—	2	4	no	—
175	C₄	—	4	4	no	—
176	C₄	—	4	4	no	—
177	C₄	—	4	4	no	—
178	C₄	C₅×#536	1	2	yes	—
179	C₄	C₅×#541	1	2	yes	—
180	C₂	C₅×#2327	1	1	yes	characteristic
181	C₂	C₅×#2930	1	2	yes	—
182	C₂	C₅×#2987	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂³×C₅	C₂³×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
21. #22243	C₅	direct (N × H)
170. C₅	#22243	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×#446	2
C₅×#2399	2
C₅×#1985	2
C₅×#1917	2
C₅×#1832	2
C₅×#1746	2
#22243	5

Subgroups — C5×#22243 #11944963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22243 #11944963