Subgroups — C₅×#22281 #11945001

Order 1280.

818 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3150	C₂	1	1	yes	maximal, characteristic
2	C₅×#2931	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×#1838	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×#497	C₄	1	4	yes	complemented
11	C₅×#496	C₄	1	4	yes	complemented
12	C₅×#436	—	2	2	no	—
13	C₅×#436	—	2	2	no	—
14	C₅×C₈×D₄	—	2	2	no	—
15	C₅×C₈×D₄	—	2	2	no	—
16	C₅×#423	C₂²	1	2	yes	—
17	C₂×C₅×#104	C₂²	1	8	yes	—
18	C₅×#408	C₂²	1	4	yes	—
19	C₂×C₅×#97	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₅×#376	C₂²	1	1	yes	characteristic
22	C₅×#341	C₂²	1	4	yes	—
23	C₅×#336	C₂²	1	4	yes	—
24	#22281	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₅×#41	C₂³	1	4	yes	—
26	C₅×#135	—	4	8	no	—
27	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
28	C₂×C₅×SD₁₆	C₂×C₄	1	8	yes	—
29	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
30	C₅×#131	—	4	8	no	—
31	C₂×C₅×M₁₆	—	2	4	no	—
32	C₂×C₅×M₁₆	C₂³	1	4	yes	—
33	C₂²×C₅×C₈	—	2	4	no	—
34	C₂²×C₅×C₈	C₂³	1	2	yes	—
35	C₂²×C₅×C₈	C₂³	1	2	yes	—
36	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
37	C₅×#127	C₂×C₄	1	2	yes	—
38	C₅×#124	C₂×C₄	1	4	yes	—
39	C₅×#121	—	2	2	no	—
40	C₅×#121	—	2	2	no	—
41	C₄×C₅×D₄	—	2	2	no	—
42	C₄×C₅×D₄	—	2	2	no	—
43	C₂×C₄²×C₅	C₂³	1	2	yes	—
44	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₅×C₄⋊C₈	—	2	2	no	—
46	C₅×C₄⋊C₈	—	2	2	no	—
47	C₅×#104	—	4	8	no	—
48	C₅×C₂²⋊C₈	—	2	4	no	—
49	C₅×C₂²⋊C₈	—	2	4	no	—
50	C₅×#97	C₂×C₄	1	4	yes	—
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	C₄×C₅×C₈	D₄	1	2	yes	—
53	C₄×C₅×C₈	D₄	1	1	yes	characteristic
54	C₅×#95	—	2	2	no	—
55	#3150	C₂×C₅	1	1	yes	characteristic
56	#2931	C₂×C₅	1	2	yes	complemented
57	#1963	C₂×C₅	1	2	yes	complemented
58	#1876	C₂×C₅	1	8	yes	complemented
59	#1838	C₂×C₅	1	1	yes	characteristic, complemented
60	#1746	C₂×C₅	1	1	yes	characteristic, complemented
61	C₅×#41	—	2	8	no	—
62	C₅×#41	—	2	8	no	—
63	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
64	C₂×C₅×D₄	—	2	4	no	—
65	C₂×C₅×D₄	—	2	2	no	—
66	C₂×C₅×D₄	—	2	2	no	—
67	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₅	—	2	4	no	—
69	C₂²×C₄×C₅	—	2	2	no	—
70	C₂²×C₄×C₅	#41	1	2	yes	—
71	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
72	C₅×Q₁₆	—	4	4	no	—
73	C₅×SD₁₆	—	4	8	no	—
74	C₅×D₈	—	4	4	no	—
75	C₅×M₁₆	—	2	8	no	—
76	C₅×M₁₆	—	4	8	no	—
77	C₂×C₅×C₈	—	2	4	no	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	4	8	no	—
81	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
82	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
83	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
84	C₅×C₄⋊C₄	—	2	2	no	—
85	C₅×C₄⋊C₄	—	2	2	no	—
86	C₅×C₂²⋊C₄	—	2	4	no	—
87	C₅×C₂²⋊C₄	—	2	4	no	—
88	C₄²×C₅	—	2	4	no	—
89	C₄²×C₅	C₂²×C₄	1	2	yes	—
90	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂×#135	C₂²×C₅	1	4	yes	—
93	C₂×#131	C₂²×C₅	1	4	yes	—
94	#532	C₂²×C₅	1	2	yes	—
95	#497	C₄×C₅	1	4	yes	complemented
96	#496	C₄×C₅	1	4	yes	complemented
97	#436	—	2	2	no	—
98	#436	—	2	2	no	—
99	C₈×D₄	—	2	2	no	—
100	C₈×D₄	—	2	2	no	—
101	#423	C₂²×C₅	1	2	yes	—
102	C₂×#104	C₂²×C₅	1	8	yes	—
103	#408	C₂²×C₅	1	4	yes	—
104	C₂×#97	C₂²×C₅	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
106	#376	C₂²×C₅	1	1	yes	characteristic
107	#341	C₂²×C₅	1	4	yes	—
108	#336	C₂²×C₅	1	4	yes	—
109	C₂³×C₅	—	2	4	no	—
110	C₂³×C₅	C₂×#41	1	1	yes	characteristic
111	C₅×Q₈	—	2	8	no	—
112	C₅×D₄	—	2	8	no	—
113	C₅×D₄	—	4	4	no	—
114	C₅×D₄	—	4	4	no	—
115	C₅×D₄	—	4	8	no	—
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	—	4	4	no	—
119	C₂×C₄×C₅	—	4	4	no	—
120	C₂×C₄×C₅	—	4	4	no	—
121	C₂×C₄×C₅	—	4	2	no	—
122	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
127	C₅×C₈	—	2	4	no	—
128	C₅×C₈	—	2	2	no	—
129	C₅×C₈	—	2	2	no	—
130	C₅×C₈	—	4	4	no	—
131	C₂×#41	C₂³×C₅	1	4	yes	—
132	#135	—	4	8	no	—
133	C₂×Q₁₆	C₂×C₄×C₅	1	4	yes	—
134	C₂×SD₁₆	C₂×C₄×C₅	1	8	yes	—
135	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
136	#131	—	4	8	no	—
137	C₂×M₁₆	—	2	4	no	—
138	C₂×M₁₆	C₂³×C₅	1	4	yes	—
139	C₂²×C₈	—	2	4	no	—
140	C₂²×C₈	C₂³×C₅	1	2	yes	—
141	C₂²×C₈	C₂³×C₅	1	2	yes	—
142	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
143	#127	C₂×C₄×C₅	1	2	yes	—
144	#124	C₂×C₄×C₅	1	4	yes	—
145	#121	—	2	2	no	—
146	#121	—	2	2	no	—
147	C₄×D₄	—	2	2	no	—
148	C₄×D₄	—	2	2	no	—
149	C₂×C₄²	C₂³×C₅	1	2	yes	—
150	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	#104	—	4	8	no	—
154	C₂²⋊C₈	—	2	4	no	—
155	C₂²⋊C₈	—	2	4	no	—
156	#97	C₂×C₄×C₅	1	4	yes	—
157	C₄×C₈	C₅×D₄	1	1	yes	characteristic
158	C₄×C₈	C₅×D₄	1	2	yes	—
159	C₄×C₈	C₅×D₄	1	1	yes	characteristic
160	#95	—	2	2	no	—
161	C₂²×C₅	—	2	2	no	—
162	C₂²×C₅	—	4	4	no	—
163	C₂²×C₅	—	4	4	no	—
164	C₂²×C₅	—	4	4	no	—
165	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂²×C₅	#540	1	2	yes	—
167	C₄×C₅	—	2	2	no	—
168	C₄×C₅	—	2	2	no	—
169	C₄×C₅	—	4	4	no	—
170	C₄×C₅	—	8	2	no	—
171	C₄×C₅	#518	1	2	yes	—
172	C₄×C₅	#535	1	2	yes	—
173	#41	—	2	8	no	—
174	#41	—	2	8	no	—
175	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
176	C₂×D₄	—	2	4	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	C₅×#41	1	2	yes	—
183	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
184	Q₁₆	—	4	4	no	—
185	SD₁₆	—	4	8	no	—
186	D₈	—	4	4	no	—
187	M₁₆	—	2	8	no	—
188	M₁₆	—	4	8	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	4	8	no	—
193	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
194	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
195	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₂²⋊C₄	—	2	4	no	—
199	C₂²⋊C₄	—	2	4	no	—
200	C₄²	—	2	4	no	—
201	C₄²	C₂²×C₄×C₅	1	2	yes	—
202	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
203	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
204	C₂×C₅	—	2	2	no	—
205	C₂×C₅	—	8	4	no	—
206	C₂×C₅	#2328	1	1	yes	characteristic
207	C₂×C₅	#2955	1	2	yes	—
208	C₂³	—	2	4	no	—
209	C₂³	C₂×C₅×#41	1	1	yes	characteristic
210	Q₈	—	2	8	no	—
211	D₄	—	2	8	no	—
212	D₄	—	4	4	no	—
213	D₄	—	4	4	no	—
214	D₄	—	4	8	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
222	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
223	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
225	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
226	C₈	—	2	4	no	—
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	4	4	no	—
230	C₅	#22281	1	1	yes	characteristic, direct product factor, Sylow
231	C₂²	—	2	2	no	—
232	C₂²	—	4	4	no	—
233	C₂²	—	4	4	no	—
234	C₂²	—	4	4	no	—
235	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
236	C₂²	C₅×#540	1	2	yes	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	4	4	no	—
240	C₄	—	8	2	no	—
241	C₄	C₅×#518	1	2	yes	—
242	C₄	C₅×#535	1	2	yes	—
243	C₂	—	2	2	no	—
244	C₂	—	8	4	no	—
245	C₂	C₅×#2328	1	1	yes	characteristic
246	C₂	C₅×#2955	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×#3150	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2931	C₂	semidirect
3. C₅×#1963	C₂	semidirect
4. C₅×#1876	C₂	semidirect
5. C₅×#1838	C₂	semidirect
6. C₅×#1746	C₂	semidirect
10. C₅×#497	C₄	semidirect
11. C₅×#496	C₄	semidirect
24. #22281	C₅	direct (N × H)
56. #2931	C₂×C₅	semidirect
57. #1963	C₂×C₅	semidirect
58. #1876	C₂×C₅	semidirect
59. #1838	C₂×C₅	semidirect
60. #1746	C₂×C₅	semidirect
95. #497	C₄×C₅	semidirect
96. #496	C₄×C₅	semidirect
230. C₅	#22281	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×#3150	2
C₅×#2931	2
C₅×#1963	2
C₅×#1876	2
C₅×#1838	2
C₅×#1746	2
#22281	5

Subgroups — C5×#22281 #11945001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22281 #11945001