Subgroups — C₅×#21706 #11944426

Order 1280.

1022 conjugacy classes of subgroups.

The subgroup lattice has 262 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	C₂×C₅×#444	C₂	1	4	yes	maximal, complemented
3	C₅×#2931	C₂	1	2	yes	maximal, complemented
4	C₅×#1975	C₂	1	4	yes	maximal, complemented
5	C₅×#1962	C₂	1	2	yes	maximal, complemented
6	C₅×#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	C₂²	1	4	yes	—
9	C₂×C₅×#131	C₂²	1	4	yes	—
10	C₅×#532	C₂²	1	2	yes	—
11	C₅×#495	C₄	1	4	yes	complemented
12	C₅×#494	C₄	1	1	yes	characteristic, complemented
13	C₅×#493	C₄	1	1	yes	characteristic, complemented
14	C₅×#492	C₄	1	2	yes	complemented
15	C₅×#444	—	2	16	no	—
16	C₅×#436	—	2	4	no	—
17	C₅×C₈×D₄	—	2	4	no	—
18	C₂×C₅×#108	C₂²	1	4	yes	—
19	C₅×#423	C₂²	1	2	yes	—
20	C₂×C₅×#104	C₂²	1	8	yes	—
21	C₅×#408	C₂²	1	4	yes	—
22	C₂×C₅×#97	C₂²	1	2	yes	—
23	C₂×C₅×#97	C₂²	1	2	yes	—
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
26	C₄³×C₅	C₂²	1	1	yes	characteristic
27	#21706	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₅×#41	C₂³	1	4	yes	—
29	C₅×#135	—	2	8	no	—
30	C₅×#135	—	2	8	no	—
31	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
32	C₂×C₅×SD₁₆	C₂×C₄	1	8	yes	—
33	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
34	C₅×#131	—	2	16	no	—
35	C₂×C₅×M₁₆	—	2	4	no	—
36	C₂×C₅×M₁₆	C₂³	1	4	yes	—
37	C₂²×C₅×C₈	—	2	4	no	—
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂³	1	2	yes	—
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
42	C₅×#127	C₂×C₄	1	2	yes	—
43	C₅×#124	C₂×C₄	1	4	yes	—
44	C₅×#121	—	2	4	no	—
45	C₄×C₅×D₄	—	2	4	no	—
46	C₂×C₄²×C₅	—	2	2	no	—
47	C₂×C₄²×C₅	C₂³	1	2	yes	—
48	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₅×#108	—	2	8	no	—
50	C₅×C₄⋊C₈	—	2	4	no	—
51	C₅×#104	—	2	16	no	—
52	C₅×C₂²⋊C₈	—	2	8	no	—
53	C₅×#97	—	2	4	no	—
54	C₅×#97	D₄	1	8	yes	complemented
55	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
56	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
57	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
58	C₄×C₅×C₈	D₄	1	4	yes	—
59	#3147	C₂×C₅	1	1	yes	characteristic
60	C₂×#444	C₂×C₅	1	4	yes	complemented
61	#2931	C₂×C₅	1	2	yes	complemented
62	#1975	C₂×C₅	1	4	yes	complemented
63	#1962	C₂×C₅	1	2	yes	complemented
64	#1848	C₂×C₅	1	1	yes	characteristic, complemented
65	C₄×#97	C₂×C₅	1	1	yes	characteristic, complemented
66	C₅×#41	—	2	8	no	—
67	C₅×#41	—	2	8	no	—
68	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
69	C₂×C₅×D₄	—	2	4	no	—
70	C₂×C₅×D₄	—	2	4	no	—
71	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₅	—	2	2	no	—
73	C₂²×C₄×C₅	—	2	4	no	—
74	C₂²×C₄×C₅	#41	1	2	yes	—
75	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
76	C₅×Q₁₆	—	2	8	no	—
77	C₅×SD₁₆	—	2	16	no	—
78	C₅×D₈	—	2	8	no	—
79	C₅×M₁₆	—	2	8	no	—
80	C₅×M₁₆	—	4	8	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	—	4	8	no	—
84	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
87	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
88	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
89	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
90	C₂×C₅×C₈	#41	1	4	yes	—
91	C₅×C₄⋊C₄	—	2	4	no	—
92	C₅×C₂²⋊C₄	—	2	8	no	—
93	C₄²×C₅	—	2	2	no	—
94	C₄²×C₅	—	4	4	no	—
95	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅	C₂×D₄	1	4	yes	—
98	C₄²×C₅	C₂×D₄	1	2	yes	—
99	C₂×#135	C₂²×C₅	1	4	yes	—
100	C₂×#131	C₂²×C₅	1	4	yes	—
101	#532	C₂²×C₅	1	2	yes	—
102	#495	C₄×C₅	1	4	yes	complemented
103	#494	C₄×C₅	1	1	yes	characteristic, complemented
104	#493	C₄×C₅	1	1	yes	characteristic, complemented
105	#492	C₄×C₅	1	2	yes	complemented
106	#444	—	2	16	no	—
107	#436	—	2	4	no	—
108	C₈×D₄	—	2	4	no	—
109	C₂×#108	C₂²×C₅	1	4	yes	—
110	#423	C₂²×C₅	1	2	yes	—
111	C₂×#104	C₂²×C₅	1	8	yes	—
112	#408	C₂²×C₅	1	4	yes	—
113	C₂×#97	C₂²×C₅	1	2	yes	—
114	C₂×#97	C₂²×C₅	1	2	yes	—
115	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
117	C₄³	C₂²×C₅	1	1	yes	characteristic
118	C₂³×C₅	—	2	4	no	—
119	C₂³×C₅	C₂×#41	1	1	yes	characteristic
120	C₅×Q₈	—	2	8	no	—
121	C₅×D₄	—	2	8	no	—
122	C₅×D₄	—	4	8	no	—
123	C₅×D₄	—	4	8	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	4	no	—
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	4	4	no	—
128	C₂×C₄×C₅	—	4	4	no	—
129	C₂×C₄×C₅	—	4	4	no	—
130	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
134	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×#41	1	2	yes	—
136	C₅×C₈	—	2	2	no	—
137	C₅×C₈	—	4	4	no	—
138	C₅×C₈	C₄×D₄	1	4	yes	—
139	C₅×C₈	C₄×D₄	1	4	yes	—
140	C₅×C₈	#127	1	4	yes	—
141	C₂×#41	C₂³×C₅	1	4	yes	—
142	#135	—	2	8	no	—
143	#135	—	2	8	no	—
144	C₂×Q₁₆	C₂×C₄×C₅	1	4	yes	—
145	C₂×SD₁₆	C₂×C₄×C₅	1	8	yes	—
146	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
147	#131	—	2	16	no	—
148	C₂×M₁₆	—	2	4	no	—
149	C₂×M₁₆	C₂³×C₅	1	4	yes	—
150	C₂²×C₈	—	2	4	no	—
151	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
152	C₂²×C₈	C₂³×C₅	1	2	yes	—
153	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
154	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
155	#127	C₂×C₄×C₅	1	2	yes	—
156	#124	C₂×C₄×C₅	1	4	yes	—
157	#121	—	2	4	no	—
158	C₄×D₄	—	2	4	no	—
159	C₂×C₄²	—	2	2	no	—
160	C₂×C₄²	C₂³×C₅	1	2	yes	—
161	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
162	#108	—	2	8	no	—
163	C₄⋊C₈	—	2	4	no	—
164	#104	—	2	16	no	—
165	C₂²⋊C₈	—	2	8	no	—
166	#97	—	2	4	no	—
167	#97	C₅×D₄	1	8	yes	complemented
168	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
169	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
170	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
171	C₄×C₈	C₅×D₄	1	4	yes	—
172	C₂²×C₅	—	2	2	no	—
173	C₂²×C₅	—	4	4	no	—
174	C₂²×C₅	—	4	8	no	—
175	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂²×C₅	#532	1	2	yes	—
177	C₄×C₅	—	2	2	no	—
178	C₄×C₅	—	4	4	no	—
179	C₄×C₅	—	4	4	no	—
180	C₄×C₅	#444	1	4	yes	complemented
181	C₄×C₅	C₂×C₄×D₄	1	2	yes	—
182	C₄×C₅	C₂×#127	1	1	yes	characteristic
183	C₄×C₅	#532	1	1	yes	characteristic
184	#41	—	2	8	no	—
185	#41	—	2	8	no	—
186	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	C₅×#41	1	2	yes	—
193	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
194	Q₁₆	—	2	8	no	—
195	SD₁₆	—	2	16	no	—
196	D₈	—	2	8	no	—
197	M₁₆	—	2	8	no	—
198	M₁₆	—	4	8	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	4	8	no	—
202	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
203	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
204	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
205	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
206	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
207	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
208	C₂×C₈	C₅×#41	1	4	yes	—
209	C₄⋊C₄	—	2	4	no	—
210	C₂²⋊C₄	—	2	8	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	4	4	no	—
213	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
214	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
215	C₄²	C₂×C₅×D₄	1	4	yes	—
216	C₄²	C₂×C₅×D₄	1	2	yes	—
217	C₂×C₅	—	2	2	no	—
218	C₂×C₅	—	8	4	no	—
219	C₂×C₅	C₄×#127	1	1	yes	characteristic
220	C₂×C₅	C₂×#444	1	2	yes	—
221	C₂³	—	2	4	no	—
222	C₂³	C₂×C₅×#41	1	1	yes	characteristic
223	Q₈	—	2	8	no	—
224	D₄	—	2	8	no	—
225	D₄	—	4	8	no	—
226	D₄	—	4	8	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
234	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
235	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
237	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
238	C₂×C₄	C₂×C₅×#41	1	2	yes	—
239	C₈	—	2	2	no	—
240	C₈	—	4	4	no	—
241	C₈	C₄×C₅×D₄	1	4	yes	—
242	C₈	C₄×C₅×D₄	1	4	yes	—
243	C₈	C₅×#127	1	4	yes	—
244	C₅	#21706	1	1	yes	characteristic, direct product factor, Sylow
245	C₂²	—	2	2	no	—
246	C₂²	—	4	4	no	—
247	C₂²	—	4	8	no	—
248	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
249	C₂²	C₅×#532	1	2	yes	—
250	C₄	—	2	2	no	—
251	C₄	—	4	4	no	—
252	C₄	—	4	4	no	—
253	C₄	C₅×#444	1	4	yes	complemented
254	C₄	C₂×C₄×C₅×D₄	1	2	yes	—
255	C₄	C₂×C₅×#127	1	1	yes	characteristic
256	C₄	C₅×#532	1	1	yes	characteristic
257	C₂	—	2	2	no	—
258	C₂	—	8	4	no	—
259	C₂	C₄×C₅×#127	1	1	yes	characteristic
260	C₂	C₂×C₅×#444	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×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#444	C₂	semidirect
3. C₅×#2931	C₂	semidirect
4. C₅×#1975	C₂	semidirect
5. C₅×#1962	C₂	semidirect
6. C₅×#1848	C₂	semidirect
7. C₄×C₅×#97	C₂	semidirect
11. C₅×#495	C₄	semidirect
12. C₅×#494	C₄	semidirect
13. C₅×#493	C₄	semidirect
14. C₅×#492	C₄	semidirect
27. #21706	C₅	direct (N × H)
54. C₅×#97	D₄	semidirect
60. C₂×#444	C₂×C₅	semidirect
61. #2931	C₂×C₅	semidirect
62. #1975	C₂×C₅	semidirect
63. #1962	C₂×C₅	semidirect
64. #1848	C₂×C₅	semidirect
65. C₄×#97	C₂×C₅	semidirect
102. #495	C₄×C₅	semidirect
103. #494	C₄×C₅	semidirect
104. #493	C₄×C₅	semidirect
105. #492	C₄×C₅	semidirect
167. #97	C₅×D₄	semidirect
180. C₄×C₅	#444	semidirect
244. C₅	#21706	direct (N × H)
253. C₄	C₅×#444	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×#3147	2
C₂×C₅×#444	2
C₅×#2931	2
C₅×#1975	2
C₅×#1962	2
C₅×#1848	2
C₄×C₅×#97	2
#21706	5

Subgroups — C5×#21706 #11944426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21706 #11944426