Subgroups — C₅×#22282 #11945002

Order 1280.

810 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3147	C₂	1	1	yes	maximal, characteristic
2	C₅×#2932	C₂	1	2	yes	maximal, complemented
3	C₅×#1963	C₂	1	2	yes	maximal, complemented
4	C₅×#1876	C₂	1	8	yes	maximal, complemented
5	C₅×#1837	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1746	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#135	C₂²	1	4	yes	—
8	C₂×C₅×#131	C₂²	1	4	yes	—
9	C₅×#532	C₂²	1	2	yes	—
10	C₅×#495	C₄	1	4	yes	complemented
11	C₅×#494	C₄	1	1	yes	characteristic, complemented
12	C₅×#493	C₄	1	1	yes	characteristic, complemented
13	C₅×#492	C₄	1	2	yes	complemented
14	C₅×#435	—	2	8	no	—
15	C₅×#423	C₂²	1	2	yes	—
16	C₂×C₅×#104	C₂²	1	8	yes	—
17	C₅×#408	C₂²	1	4	yes	—
18	C₂×C₅×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×#376	C₂²	1	1	yes	characteristic
21	C₅×#341	C₂²	1	4	yes	—
22	C₅×#336	C₂²	1	4	yes	—
23	#22282	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×#41	C₂³	1	4	yes	—
25	C₅×#135	—	4	8	no	—
26	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
27	C₂×C₅×SD₁₆	C₂×C₄	1	8	yes	—
28	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
29	C₅×#131	—	4	8	no	—
30	C₂×C₅×M₁₆	—	2	4	no	—
31	C₂×C₅×M₁₆	C₂³	1	4	yes	—
32	C₂²×C₅×C₈	—	2	4	no	—
33	C₂²×C₅×C₈	C₂³	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	2	yes	—
35	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₅×#127	C₂×C₄	1	2	yes	—
37	C₅×#124	C₂×C₄	1	4	yes	—
38	C₅×#121	—	2	4	no	—
39	C₄×C₅×D₄	—	2	4	no	—
40	C₂×C₄²×C₅	C₂³	1	2	yes	—
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×C₄⋊C₈	—	2	4	no	—
43	C₅×#104	—	4	8	no	—
44	C₅×C₂²⋊C₈	—	2	8	no	—
45	C₅×#97	D₄	1	4	yes	—
46	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
49	C₅×#95	—	2	2	no	—
50	#3147	C₂×C₅	1	1	yes	characteristic
51	#2932	C₂×C₅	1	2	yes	complemented
52	#1963	C₂×C₅	1	2	yes	complemented
53	#1876	C₂×C₅	1	8	yes	complemented
54	#1837	C₂×C₅	1	1	yes	characteristic, complemented
55	#1746	C₂×C₅	1	1	yes	characteristic, complemented
56	C₅×#41	—	2	8	no	—
57	C₅×#41	—	2	8	no	—
58	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
59	C₂×C₅×D₄	—	2	4	no	—
60	C₂×C₅×D₄	—	2	4	no	—
61	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	—	2	2	no	—
64	C₂²×C₄×C₅	#41	1	2	yes	—
65	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
66	C₅×Q₁₆	—	4	4	no	—
67	C₅×SD₁₆	—	4	8	no	—
68	C₅×D₈	—	4	4	no	—
69	C₅×M₁₆	—	2	8	no	—
70	C₅×M₁₆	—	4	8	no	—
71	C₂×C₅×C₈	—	2	4	no	—
72	C₂×C₅×C₈	—	2	4	no	—
73	C₂×C₅×C₈	—	2	4	no	—
74	C₂×C₅×C₈	—	4	8	no	—
75	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
76	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
77	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
78	C₅×C₄⋊C₄	—	2	4	no	—
79	C₅×C₂²⋊C₄	—	2	8	no	—
80	C₄²×C₅	—	2	4	no	—
81	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅	C₂×D₄	1	2	yes	—
84	C₂×#135	C₂²×C₅	1	4	yes	—
85	C₂×#131	C₂²×C₅	1	4	yes	—
86	#532	C₂²×C₅	1	2	yes	—
87	#495	C₄×C₅	1	4	yes	complemented
88	#494	C₄×C₅	1	1	yes	characteristic, complemented
89	#493	C₄×C₅	1	1	yes	characteristic, complemented
90	#492	C₄×C₅	1	2	yes	complemented
91	#435	—	2	8	no	—
92	#423	C₂²×C₅	1	2	yes	—
93	C₂×#104	C₂²×C₅	1	8	yes	—
94	#408	C₂²×C₅	1	4	yes	—
95	C₂×#97	C₂²×C₅	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
97	#376	C₂²×C₅	1	1	yes	characteristic
98	#341	C₂²×C₅	1	4	yes	—
99	#336	C₂²×C₅	1	4	yes	—
100	C₂³×C₅	—	2	4	no	—
101	C₂³×C₅	C₂×#41	1	1	yes	characteristic
102	C₅×Q₈	—	2	8	no	—
103	C₅×D₄	—	2	8	no	—
104	C₅×D₄	—	4	8	no	—
105	C₅×D₄	—	8	4	no	—
106	C₂×C₄×C₅	—	2	4	no	—
107	C₂×C₄×C₅	—	4	2	no	—
108	C₂×C₄×C₅	—	4	4	no	—
109	C₂×C₄×C₅	—	4	4	no	—
110	C₂×C₄×C₅	—	4	4	no	—
111	C₂×C₄×C₅	—	4	2	no	—
112	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
117	C₅×C₈	—	2	2	no	—
118	C₅×C₈	—	2	2	no	—
119	C₅×C₈	—	2	4	no	—
120	C₅×C₈	—	4	4	no	—
121	C₂×#41	C₂³×C₅	1	4	yes	—
122	#135	—	4	8	no	—
123	C₂×Q₁₆	C₂×C₄×C₅	1	4	yes	—
124	C₂×SD₁₆	C₂×C₄×C₅	1	8	yes	—
125	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
126	#131	—	4	8	no	—
127	C₂×M₁₆	—	2	4	no	—
128	C₂×M₁₆	C₂³×C₅	1	4	yes	—
129	C₂²×C₈	—	2	4	no	—
130	C₂²×C₈	C₂³×C₅	1	2	yes	—
131	C₂²×C₈	C₂³×C₅	1	2	yes	—
132	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
133	#127	C₂×C₄×C₅	1	2	yes	—
134	#124	C₂×C₄×C₅	1	4	yes	—
135	#121	—	2	4	no	—
136	C₄×D₄	—	2	4	no	—
137	C₂×C₄²	C₂³×C₅	1	2	yes	—
138	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
139	C₄⋊C₈	—	2	4	no	—
140	#104	—	4	8	no	—
141	C₂²⋊C₈	—	2	8	no	—
142	#97	C₅×D₄	1	4	yes	—
143	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
144	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
145	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
146	#95	—	2	2	no	—
147	C₂²×C₅	—	2	2	no	—
148	C₂²×C₅	—	4	4	no	—
149	C₂²×C₅	—	4	8	no	—
150	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂²×C₅	#540	1	2	yes	—
152	C₄×C₅	—	2	2	no	—
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	—	4	4	no	—
155	C₄×C₅	—	8	2	no	—
156	C₄×C₅	#518	1	2	yes	—
157	C₄×C₅	#535	1	2	yes	—
158	#41	—	2	8	no	—
159	#41	—	2	8	no	—
160	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
161	C₂×D₄	—	2	4	no	—
162	C₂×D₄	—	2	4	no	—
163	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	C₅×#41	1	2	yes	—
167	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
168	Q₁₆	—	4	4	no	—
169	SD₁₆	—	4	8	no	—
170	D₈	—	4	4	no	—
171	M₁₆	—	2	8	no	—
172	M₁₆	—	4	8	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	4	8	no	—
177	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
178	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
179	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
180	C₄⋊C₄	—	2	4	no	—
181	C₂²⋊C₄	—	2	8	no	—
182	C₄²	—	2	4	no	—
183	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
184	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
185	C₄²	C₂×C₅×D₄	1	2	yes	—
186	C₂×C₅	—	2	2	no	—
187	C₂×C₅	—	8	4	no	—
188	C₂×C₅	#2328	1	1	yes	characteristic
189	C₂×C₅	#2957	1	2	yes	—
190	C₂³	—	2	4	no	—
191	C₂³	C₂×C₅×#41	1	1	yes	characteristic
192	Q₈	—	2	8	no	—
193	D₄	—	2	8	no	—
194	D₄	—	4	8	no	—
195	D₄	—	8	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
203	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
204	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
206	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	2	4	no	—
210	C₈	—	4	4	no	—
211	C₅	#22282	1	1	yes	characteristic, direct product factor, Sylow
212	C₂²	—	2	2	no	—
213	C₂²	—	4	4	no	—
214	C₂²	—	4	8	no	—
215	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
216	C₂²	C₅×#540	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	4	no	—
220	C₄	—	8	2	no	—
221	C₄	C₅×#518	1	2	yes	—
222	C₄	C₅×#535	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	8	4	no	—
225	C₂	C₅×#2328	1	1	yes	characteristic
226	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	C₅×#3147	C₂
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
2. C₅×#2932	C₂	semidirect
3. C₅×#1963	C₂	semidirect
4. C₅×#1876	C₂	semidirect
5. C₅×#1837	C₂	semidirect
6. C₅×#1746	C₂	semidirect
10. C₅×#495	C₄	semidirect
11. C₅×#494	C₄	semidirect
12. C₅×#493	C₄	semidirect
13. C₅×#492	C₄	semidirect
23. #22282	C₅	direct (N × H)
51. #2932	C₂×C₅	semidirect
52. #1963	C₂×C₅	semidirect
53. #1876	C₂×C₅	semidirect
54. #1837	C₂×C₅	semidirect
55. #1746	C₂×C₅	semidirect
87. #495	C₄×C₅	semidirect
88. #494	C₄×C₅	semidirect
89. #493	C₄×C₅	semidirect
90. #492	C₄×C₅	semidirect
211. C₅	#22282	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×#3147	2
C₅×#2932	2
C₅×#1963	2
C₅×#1876	2
C₅×#1837	2
C₅×#1746	2
#22282	5

Subgroups — C5×#22282 #11945002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22282 #11945002