Subgroups — C₅×#20829 #11943549

Order 1280.

718 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2921	C₂	1	2	yes	maximal
2	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
3	C₅×#1849	C₂	1	2	yes	maximal, complemented
4	C₅×#1837	C₂	1	2	yes	maximal, complemented
5	C₅×#1811	C₂	1	2	yes	maximal
6	C₅×#1810	C₂	1	2	yes	maximal
7	C₅×#1778	C₂	1	2	yes	maximal, complemented
8	C₄×C₅×#108	C₂	1	2	yes	maximal, complemented
9	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×#117	C₂²	1	1	yes	characteristic
12	C₅×#433	—	2	8	no	—
13	C₂×C₅×#108	C₂²	1	4	yes	—
14	C₂×C₅×#108	C₂²	1	4	yes	—
15	C₅×#423	C₂²	1	4	yes	—
16	C₅×#407	C₂²	1	4	yes	—
17	C₅×#405	C₄	1	8	yes	—
18	C₄×C₅×M₁₆	C₂²	1	4	yes	—
19	C₂×C₅×#97	C₂²	1	1	yes	characteristic
20	C₂×C₅×#97	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	C₅×#341	C₂²	1	4	yes	—
24	C₅×#341	C₂²	1	4	yes	—
25	#20829	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂×C₅×M₁₆	—	2	4	no	—
28	C₂×C₅×M₁₆	C₂×C₄	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	C₂³	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×#117	C₂×C₄	1	4	yes	—
37	C₅×#117	D₄	1	4	yes	—
38	C₂×C₅×C₄⋊C₄	Q₈	1	2	yes	—
39	C₂×C₅×C₂²⋊C₄	Q₈	1	2	yes	—
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×#108	—	2	8	no	—
43	C₅×#108	—	2	8	no	—
44	C₅×C₄⋊C₈	—	2	8	no	—
45	C₅×C₂²⋊C₈	—	2	8	no	—
46	C₅×#97	—	2	4	no	—
47	C₅×#97	C₂×C₄	1	4	yes	—
48	C₅×#97	C₂×C₄	1	4	yes	—
49	C₄×C₅×C₈	—	2	4	no	—
50	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
51	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
52	#2921	C₂×C₅	1	2	yes	—
53	C₂×#405	C₂×C₅	1	1	yes	characteristic
54	#1849	C₂×C₅	1	2	yes	complemented
55	#1837	C₂×C₅	1	2	yes	complemented
56	#1811	C₂×C₅	1	2	yes	—
57	#1810	C₂×C₅	1	2	yes	—
58	#1778	C₂×C₅	1	2	yes	complemented
59	C₄×#108	C₂×C₅	1	2	yes	complemented
60	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	—	2	2	no	—
62	C₂²×C₄×C₅	—	2	2	no	—
63	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
69	C₅×M₁₆	—	2	8	no	—
70	C₅×M₁₆	—	2	8	no	—
71	C₅×M₁₆	—	2	8	no	—
72	C₂×C₅×C₈	—	2	2	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	2	8	no	—
75	C₂×C₅×C₈	—	2	8	no	—
76	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
77	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
78	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
79	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₅×C₈	#41	1	4	yes	—
81	C₂×C₅×C₈	#41	1	4	yes	—
82	C₅×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
83	C₅×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
84	C₄²×C₅	C₂²×C₄	1	2	yes	—
85	C₄²×C₅	C₂²×C₄	1	2	yes	—
86	C₄²×C₅	C₂×D₄	1	2	yes	—
87	C₄²×C₅	C₂×D₄	1	2	yes	—
88	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
89	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
90	C₂×#117	C₂²×C₅	1	1	yes	characteristic
91	#433	—	2	8	no	—
92	C₂×#108	C₂²×C₅	1	4	yes	—
93	C₂×#108	C₂²×C₅	1	4	yes	—
94	#423	C₂²×C₅	1	4	yes	—
95	#407	C₂²×C₅	1	4	yes	—
96	#405	C₄×C₅	1	8	yes	—
97	C₄×M₁₆	C₂²×C₅	1	4	yes	—
98	C₂×#97	C₂²×C₅	1	1	yes	characteristic
99	C₂×#97	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
102	#341	C₂²×C₅	1	4	yes	—
103	#341	C₂²×C₅	1	4	yes	—
104	C₂³×C₅	—	2	2	no	—
105	C₂³×C₅	—	4	2	no	—
106	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
107	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	2	4	no	—
112	C₂×C₄×C₅	—	4	2	no	—
113	C₂×C₄×C₅	—	4	2	no	—
114	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×#41	1	2	yes	—
120	C₂×C₄×C₅	C₂×#41	1	2	yes	—
121	C₅×C₈	—	2	2	no	—
122	C₅×C₈	—	2	2	no	—
123	C₅×C₈	—	4	8	no	—
124	C₅×C₈	#117	1	4	yes	—
125	C₅×C₈	#117	1	4	yes	—
126	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
127	C₂×M₁₆	—	2	4	no	—
128	C₂×M₁₆	C₂×C₄×C₅	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	C₂³×C₅	1	1	yes	characteristic
133	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
134	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
135	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
136	#117	C₂×C₄×C₅	1	4	yes	—
137	#117	C₅×D₄	1	4	yes	—
138	C₂×C₄⋊C₄	C₅×Q₈	1	2	yes	—
139	C₂×C₂²⋊C₄	C₅×Q₈	1	2	yes	—
140	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
141	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
142	#108	—	2	8	no	—
143	#108	—	2	8	no	—
144	C₄⋊C₈	—	2	8	no	—
145	C₂²⋊C₈	—	2	8	no	—
146	#97	—	2	4	no	—
147	#97	C₂×C₄×C₅	1	4	yes	—
148	#97	C₂×C₄×C₅	1	4	yes	—
149	C₄×C₈	—	2	4	no	—
150	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
151	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
152	C₂²×C₅	—	2	2	no	—
153	C₂²×C₅	—	2	4	no	—
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	—	4	4	no	—
156	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×C₅	#533	1	2	yes	—
158	C₄×C₅	—	2	4	no	—
159	C₄×C₅	—	2	4	no	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	C₂×#117	1	1	yes	characteristic
162	C₄×C₅	C₂×#117	1	1	yes	characteristic
163	C₄×C₅	#532	1	2	yes	—
164	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
168	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
169	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
171	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
172	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
173	M₁₆	—	2	8	no	—
174	M₁₆	—	2	8	no	—
175	M₁₆	—	2	8	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	8	no	—
179	C₂×C₈	—	2	8	no	—
180	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
181	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
182	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
183	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
184	C₂×C₈	C₅×#41	1	4	yes	—
185	C₂×C₈	C₅×#41	1	4	yes	—
186	C₄⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
187	C₂²⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
188	C₄²	C₂²×C₄×C₅	1	2	yes	—
189	C₄²	C₂²×C₄×C₅	1	2	yes	—
190	C₄²	C₂×C₅×D₄	1	2	yes	—
191	C₄²	C₂×C₅×D₄	1	2	yes	—
192	C₂×C₅	—	2	2	no	—
193	C₂×C₅	—	4	2	no	—
194	C₂×C₅	#2285	1	1	yes	characteristic
195	C₂×C₅	#2916	1	2	yes	—
196	C₂³	—	2	2	no	—
197	C₂³	—	4	2	no	—
198	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
199	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
207	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
208	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
209	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
211	C₂×C₄	C₂×C₅×#41	1	2	yes	—
212	C₂×C₄	C₂×C₅×#41	1	2	yes	—
213	C₈	—	2	2	no	—
214	C₈	—	2	2	no	—
215	C₈	—	4	8	no	—
216	C₈	C₅×#117	1	4	yes	—
217	C₈	C₅×#117	1	4	yes	—
218	C₅	#20829	1	1	yes	characteristic, direct product factor, Sylow
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	4	4	no	—
223	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
224	C₂²	C₅×#533	1	2	yes	—
225	C₄	—	2	2	no	—
226	C₄	—	2	4	no	—
227	C₄	—	2	4	no	—
228	C₄	C₂×C₅×#117	1	1	yes	characteristic
229	C₄	C₂×C₅×#117	1	1	yes	characteristic
230	C₄	C₅×#532	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	4	2	no	—
233	C₂	C₅×#2285	1	1	yes	characteristic
234	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
3. C₅×#1849	C₂	semidirect
4. C₅×#1837	C₂	semidirect
7. C₅×#1778	C₂	semidirect
8. C₄×C₅×#108	C₂	semidirect
25. #20829	C₅	direct (N × H)
54. #1849	C₂×C₅	semidirect
55. #1837	C₂×C₅	semidirect
58. #1778	C₂×C₅	semidirect
59. C₄×#108	C₂×C₅	semidirect
218. C₅	#20829	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×#2921	2
C₂×C₅×#405	2
C₅×#1849	2
C₅×#1837	2
C₅×#1811	2
C₅×#1810	2
C₅×#1778	2
C₄×C₅×#108	2
#20829	5

Subgroups — C5×#20829 #11943549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20829 #11943549