Subgroups — C₅×#22253 #11944973

Order 1280.

674 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#498	C₂	1	1	yes	maximal, characteristic
2	C₅×#2331	C₂	1	2	yes	maximal
3	C₅×#1993	C₂	1	2	yes	maximal
4	C₅×#1920	C₂	1	4	yes	maximal
5	C₅×#1919	C₂	1	4	yes	maximal
6	C₅×#1835	C₂	1	1	yes	maximal, characteristic
7	C₅×#1746	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₅×#498	C₄	1	8	yes	complemented
10	C₂×C₅×#106	C₂²	1	4	yes	—
11	C₂×C₅×#97	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#391	—	2	4	no	—
14	C₅×#389	C₂²	1	4	yes	—
15	C₅×#387	—	2	4	no	—
16	C₅×#382	C₂²	1	2	yes	—
17	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
18	C₅×#376	C₂²	1	1	yes	characteristic
19	C₅×#340	C₂²	1	4	yes	—
20	C₅×#340	C₂²	1	4	yes	—
21	C₅×#340	C₂²	1	4	yes	—
22	C₅×#336	C₂²	1	2	yes	—
23	C₅×#336	C₂²	1	2	yes	—
24	#22253	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×Q₈	—	2	2	no	—
26	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂³	1	2	yes	—
29	C₅×C₄⋊Q₈	—	2	4	no	—
30	C₅×C₄⋊Q₈	C₂×C₄	1	8	yes	—
31	C₂×C₅×C₄⋊C₄	—	2	2	no	—
32	C₂×C₅×C₄⋊C₄	—	2	4	no	—
33	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
34	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
35	C₂×C₄²×C₅	—	2	4	no	—
36	C₂×C₄²×C₅	C₂³	1	2	yes	—
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₅×#106	C₂×C₄	1	16	yes	—
39	C₅×#97	—	2	2	no	—
40	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
41	C₅×#95	—	2	4	no	—
42	C₅×#95	—	2	2	no	—
43	C₅×#95	—	2	2	no	—
44	C₅×#95	—	2	4	no	—
45	C₅×#95	D₄	1	4	yes	—
46	C₂×#498	C₂×C₅	1	1	yes	characteristic
47	#2331	C₂×C₅	1	2	yes	—
48	#1993	C₂×C₅	1	2	yes	—
49	#1920	C₂×C₅	1	4	yes	—
50	#1919	C₂×C₅	1	4	yes	—
51	#1835	C₂×C₅	1	1	yes	characteristic
52	#1746	C₂×C₅	1	1	yes	characteristic
53	C₂×C₅×Q₈	—	2	8	no	—
54	C₂×C₅×Q₈	—	8	4	no	—
55	C₂²×C₄×C₅	—	2	4	no	—
56	C₂²×C₄×C₅	—	2	4	no	—
57	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	—	2	2	no	—
62	C₂×C₅×C₈	—	2	1	no	—
63	C₂×C₅×C₈	—	2	1	no	—
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
67	C₅×C₄⋊C₄	—	2	8	no	—
68	C₅×C₄⋊C₄	—	2	4	no	—
69	C₅×C₄⋊C₄	—	2	4	no	—
70	C₅×C₄⋊C₄	—	2	4	no	—
71	C₅×C₄⋊C₄	—	8	4	no	—
72	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
73	C₄²×C₅	—	2	2	no	—
74	C₄²×C₅	—	2	2	no	—
75	C₄²×C₅	—	8	4	no	—
76	C₄²×C₅	C₂²×C₄	1	2	yes	—
77	C₄²×C₅	#41	1	2	yes	—
78	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
79	#498	C₄×C₅	1	8	yes	complemented
80	C₂×#106	C₂²×C₅	1	4	yes	—
81	C₂×#97	C₂²×C₅	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
83	#391	—	2	4	no	—
84	#389	C₂²×C₅	1	4	yes	—
85	#387	—	2	4	no	—
86	#382	C₂²×C₅	1	2	yes	—
87	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
88	#376	C₂²×C₅	1	1	yes	characteristic
89	#340	C₂²×C₅	1	4	yes	—
90	#340	C₂²×C₅	1	4	yes	—
91	#340	C₂²×C₅	1	4	yes	—
92	#336	C₂²×C₅	1	2	yes	—
93	#336	C₂²×C₅	1	2	yes	—
94	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
95	C₅×Q₈	—	8	8	no	—
96	C₂×C₄×C₅	—	2	4	no	—
97	C₂×C₄×C₅	—	2	8	no	—
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	4	no	—
100	C₂×C₄×C₅	—	2	4	no	—
101	C₂×C₄×C₅	—	4	4	no	—
102	C₂×C₄×C₅	—	4	4	no	—
103	C₂×C₄×C₅	—	4	4	no	—
104	C₂×C₄×C₅	—	4	4	no	—
105	C₂×C₄×C₅	—	4	4	no	—
106	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
107	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
110	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
111	C₅×C₈	—	2	4	no	—
112	C₅×C₈	—	4	1	no	—
113	C₅×C₈	—	4	1	no	—
114	C₂²×Q₈	—	2	2	no	—
115	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
116	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
117	C₂²×C₈	C₂³×C₅	1	2	yes	—
118	C₄⋊Q₈	—	2	4	no	—
119	C₄⋊Q₈	C₂×C₄×C₅	1	8	yes	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	4	no	—
122	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
123	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
124	C₂×C₄²	—	2	4	no	—
125	C₂×C₄²	C₂³×C₅	1	2	yes	—
126	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
127	#106	C₂×C₄×C₅	1	16	yes	—
128	#97	—	2	2	no	—
129	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
130	#95	—	2	4	no	—
131	#95	—	2	2	no	—
132	#95	—	2	2	no	—
133	#95	—	2	4	no	—
134	#95	C₅×D₄	1	4	yes	—
135	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₅	#536	1	2	yes	—
137	C₂²×C₅	#577	1	4	yes	—
138	C₄×C₅	—	2	4	no	—
139	C₄×C₅	—	4	8	no	—
140	C₄×C₅	—	4	4	no	—
141	C₄×C₅	—	8	4	no	—
142	C₄×C₅	#519	1	2	yes	—
143	C₄×C₅	#541	1	1	yes	characteristic
144	C₄×C₅	#541	1	1	yes	characteristic
145	C₂×Q₈	—	2	8	no	—
146	C₂×Q₈	—	8	4	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
150	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
151	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
159	C₄⋊C₄	—	2	8	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	8	4	no	—
164	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
165	C₄²	—	2	2	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	8	4	no	—
168	C₄²	C₂²×C₄×C₅	1	2	yes	—
169	C₄²	C₅×#41	1	2	yes	—
170	C₂×C₅	#2327	1	1	yes	characteristic
171	C₂×C₅	#2949	1	2	yes	—
172	C₂×C₅	#3244	1	4	yes	—
173	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
174	Q₈	—	8	8	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
186	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
187	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
188	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
189	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
190	C₈	—	2	4	no	—
191	C₈	—	4	1	no	—
192	C₈	—	4	1	no	—
193	C₅	#22253	1	1	yes	characteristic, direct product factor, Sylow
194	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
195	C₂²	C₅×#536	1	2	yes	—
196	C₂²	C₅×#577	1	4	yes	—
197	C₄	—	2	4	no	—
198	C₄	—	4	4	no	—
199	C₄	—	4	8	no	—
200	C₄	—	8	4	no	—
201	C₄	C₅×#519	1	2	yes	—
202	C₄	C₅×#541	1	1	yes	characteristic
203	C₄	C₅×#541	1	1	yes	characteristic
204	C₂	C₅×#2327	1	1	yes	characteristic
205	C₂	C₅×#2949	1	2	yes	—
206	C₂	C₅×#3244	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#376 ◁ G

Complements

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

N	H ≅ G/N	type
9. C₅×#498	C₄	semidirect
24. #22253	C₅	direct (N × H)
79. #498	C₄×C₅	semidirect
193. C₅	#22253	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×#498	2
C₅×#2331	2
C₅×#1993	2
C₅×#1920	2
C₅×#1919	2
C₅×#1835	2
C₅×#1746	2
#22253	5

Subgroups — C5×#22253 #11944973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22253 #11944973