Subgroups — C₅×#20807 #11943527

Order 1280.

954 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#430	C₂	1	2	yes	maximal
2	C₂²×C₅×#108	C₂	1	2	yes	maximal
3	C₅×#2905	C₂	1	2	yes	maximal
4	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
5	C₅×#1849	C₂	1	4	yes	maximal, complemented
6	C₅×#1848	C₂	1	4	yes	maximal, complemented
7	C₂²×C₅×M₁₆	C₂²	1	4	yes	—
8	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#117	C₂²	1	1	yes	characteristic
11	C₅×#430	—	2	8	no	—
12	C₂×C₅×#108	—	2	8	no	—
13	C₂×C₅×#108	C₂²	1	4	yes	—
14	C₂×C₅×#108	C₂²	1	8	yes	—
15	C₂×C₅×#108	C₂²	1	4	yes	—
16	C₅×#424	—	2	8	no	—
17	C₅×#423	C₂²	1	8	yes	—
18	C₅×#405	C₄	1	8	yes	—
19	C₂×C₅×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
21	#20807	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₅×M₁₆	—	2	4	no	—
24	C₂×C₅×M₁₆	—	2	8	no	—
25	C₂×C₅×M₁₆	—	2	8	no	—
26	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
27	C₂×C₅×M₁₆	D₄	1	4	yes	—
28	C₂×C₅×M₁₆	Q₈	1	4	yes	—
29	C₂×C₅×M₁₆	C₂³	1	8	yes	—
30	C₂²×C₅×C₈	—	2	4	no	—
31	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
32	C₂²×C₅×C₈	D₄	1	4	yes	—
33	C₂²×C₅×C₈	Q₈	1	4	yes	—
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	2	yes	—
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₅×#117	C₂×C₄	1	4	yes	—
38	C₅×#117	D₄	1	4	yes	—
39	C₂×C₅×C₄⋊C₄	Q₈	1	2	yes	—
40	C₂×C₅×C₂²⋊C₄	Q₈	1	2	yes	—
41	C₂×C₄²×C₅	C₂³	1	2	yes	—
42	C₅×#108	—	2	8	no	—
43	C₅×#108	—	2	8	no	—
44	C₅×#108	—	2	16	no	—
45	C₅×C₄⋊C₈	—	2	16	no	—
46	C₅×#97	C₂×C₄	1	8	yes	—
47	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
48	C₂×#430	C₂×C₅	1	2	yes	—
49	C₂²×#108	C₂×C₅	1	2	yes	—
50	#2905	C₂×C₅	1	2	yes	—
51	C₂×#405	C₂×C₅	1	1	yes	characteristic
52	#1849	C₂×C₅	1	4	yes	complemented
53	#1848	C₂×C₅	1	4	yes	complemented
54	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
60	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
64	C₅×M₁₆	—	2	8	no	—
65	C₅×M₁₆	—	2	8	no	—
66	C₅×M₁₆	—	4	8	no	—
67	C₅×M₁₆	C₄⋊C₄	1	16	yes	—
68	C₂×C₅×C₈	—	2	8	no	—
69	C₂×C₅×C₈	—	4	8	no	—
70	C₂×C₅×C₈	C₄⋊C₄	1	16	yes	—
71	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
72	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
73	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
74	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
75	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
76	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
77	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
78	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
79	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
80	C₅×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
81	C₅×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
82	C₄²×C₅	C₂²×C₄	1	4	yes	—
83	C₄²×C₅	C₂×D₄	1	4	yes	—
84	C₂²×M₁₆	C₂²×C₅	1	4	yes	—
85	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
86	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
87	C₂×#117	C₂²×C₅	1	1	yes	characteristic
88	#430	—	2	8	no	—
89	C₂×#108	—	2	8	no	—
90	C₂×#108	C₂²×C₅	1	4	yes	—
91	C₂×#108	C₂²×C₅	1	8	yes	—
92	C₂×#108	C₂²×C₅	1	4	yes	—
93	#424	—	2	8	no	—
94	#423	C₂²×C₅	1	8	yes	—
95	#405	C₄×C₅	1	8	yes	—
96	C₂×#97	C₂²×C₅	1	2	yes	—
97	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
98	C₂³×C₅	—	4	2	no	—
99	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
100	C₂³×C₅	C₄⋊Q₈	1	2	yes	—
101	C₂³×C₅	C₄⋊Q₈	1	2	yes	—
102	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₅	—	4	4	no	—
104	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
105	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₅	#127	1	4	yes	—
109	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
110	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
111	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
112	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
118	C₅×C₈	—	4	8	no	—
119	C₅×C₈	C₂×C₄⋊C₄	1	4	yes	—
120	C₅×C₈	C₂×C₄⋊C₄	1	4	yes	—
121	C₅×C₈	C₂×C₄⋊C₄	1	8	yes	—
122	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
123	C₂×M₁₆	—	2	4	no	—
124	C₂×M₁₆	—	2	8	no	—
125	C₂×M₁₆	—	2	8	no	—
126	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
127	C₂×M₁₆	C₅×D₄	1	4	yes	—
128	C₂×M₁₆	C₅×Q₈	1	4	yes	—
129	C₂×M₁₆	C₂³×C₅	1	8	yes	—
130	C₂²×C₈	—	2	4	no	—
131	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
132	C₂²×C₈	C₅×D₄	1	4	yes	—
133	C₂²×C₈	C₅×Q₈	1	4	yes	—
134	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
135	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
136	C₂²×C₈	C₂³×C₅	1	2	yes	—
137	#117	C₂×C₄×C₅	1	4	yes	—
138	#117	C₅×D₄	1	4	yes	—
139	C₂×C₄⋊C₄	C₅×Q₈	1	2	yes	—
140	C₂×C₂²⋊C₄	C₅×Q₈	1	2	yes	—
141	C₂×C₄²	C₂³×C₅	1	2	yes	—
142	#108	—	2	8	no	—
143	#108	—	2	8	no	—
144	#108	—	2	16	no	—
145	C₄⋊C₈	—	2	16	no	—
146	#97	C₂×C₄×C₅	1	8	yes	—
147	C₄×C₈	C₂×C₄×C₅	1	8	yes	—
148	C₂²×C₅	—	2	2	no	—
149	C₂²×C₅	—	2	4	no	—
150	C₂²×C₅	—	4	2	no	—
151	C₂²×C₅	—	4	2	no	—
152	C₂²×C₅	#384	1	4	yes	—
153	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
155	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
156	C₄×C₅	—	2	8	no	—
157	C₄×C₅	#384	1	4	yes	—
158	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₄×C₅	C₂×#127	1	1	yes	characteristic
161	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
162	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
163	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
164	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
165	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
166	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
167	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
168	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
170	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
171	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
172	M₁₆	—	2	8	no	—
173	M₁₆	—	2	8	no	—
174	M₁₆	—	4	8	no	—
175	M₁₆	C₅×C₄⋊C₄	1	16	yes	—
176	C₂×C₈	—	2	8	no	—
177	C₂×C₈	—	4	8	no	—
178	C₂×C₈	C₅×C₄⋊C₄	1	16	yes	—
179	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
180	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
181	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
182	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
183	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
184	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
185	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
186	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
187	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
188	C₄⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
189	C₂²⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
190	C₄²	C₂²×C₄×C₅	1	4	yes	—
191	C₄²	C₂×C₅×D₄	1	4	yes	—
192	C₂×C₅	—	2	1	no	—
193	C₂×C₅	—	2	1	no	—
194	C₂×C₅	—	2	4	no	—
195	C₂×C₅	C₂×#384	1	1	yes	characteristic
196	C₂×C₅	#2916	1	2	yes	—
197	C₂³	—	4	2	no	—
198	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
199	C₂³	C₅×C₄⋊Q₈	1	2	yes	—
200	C₂³	C₅×C₄⋊Q₈	1	2	yes	—
201	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	C₂×C₅×C₄⋊C₄	1	8	yes	—
204	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
205	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
206	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
207	C₂×C₄	C₅×#127	1	4	yes	—
208	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
209	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
210	C₂×C₄	C₅×C₄⋊Q₈	1	4	yes	—
211	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
213	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
214	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
215	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
216	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
217	C₈	—	4	8	no	—
218	C₈	C₂×C₅×C₄⋊C₄	1	4	yes	—
219	C₈	C₂×C₅×C₄⋊C₄	1	4	yes	—
220	C₈	C₂×C₅×C₄⋊C₄	1	8	yes	—
221	C₅	#20807	1	1	yes	characteristic, direct product factor, Sylow
222	C₂²	—	2	2	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₅×#384	1	4	yes	—
227	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
228	C₂²	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
229	C₂²	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
230	C₄	—	2	8	no	—
231	C₄	C₅×#384	1	4	yes	—
232	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
233	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
234	C₄	C₂×C₅×#127	1	1	yes	characteristic
235	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
236	C₂	—	2	4	no	—
237	C₂	—	2	1	no	—
238	C₂	—	2	1	no	—
239	C₂	C₂×C₅×#384	1	1	yes	characteristic
240	C₂	C₅×#2916	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	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅×#1849	C₂	semidirect
6. C₅×#1848	C₂	semidirect
21. #20807	C₅	direct (N × H)
52. #1849	C₂×C₅	semidirect
53. #1848	C₂×C₅	semidirect
221. C₅	#20807	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×#430	2
C₂²×C₅×#108	2
C₅×#2905	2
C₂×C₅×#405	2
C₅×#1849	2
C₅×#1848	2
#20807	5

Subgroups — C5×#20807 #11943527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20807 #11943527