Subgroups — C₅×#21709 #11944429

Order 1280.

1006 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	C₅×#3150	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×#1974	C₂	1	4	yes	maximal, complemented
5	C₅×#1963	C₂	1	2	yes	maximal, complemented
6	C₅×#1849	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1726	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×#497	C₄	1	4	yes	complemented
12	C₅×#496	C₄	1	4	yes	complemented
13	C₅×#444	—	2	16	no	—
14	C₅×#436	—	2	4	no	—
15	C₅×C₈×D₄	—	2	4	no	—
16	C₂×C₅×#108	C₂²	1	4	yes	—
17	C₅×#423	C₂²	1	2	yes	—
18	C₂×C₅×#104	C₂²	1	8	yes	—
19	C₅×#408	C₂²	1	4	yes	—
20	C₂×C₅×#97	C₂²	1	1	yes	characteristic
21	C₂×C₅×#97	C₂²	1	2	yes	—
22	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₅×#376	C₂²	1	1	yes	characteristic
25	#21709	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅×#41	C₂³	1	4	yes	—
27	C₅×#135	—	2	8	no	—
28	C₅×#135	—	2	8	no	—
29	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
30	C₂×C₅×SD₁₆	C₂×C₄	1	8	yes	—
31	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
32	C₅×#131	—	2	16	no	—
33	C₂×C₅×M₁₆	—	2	4	no	—
34	C₂×C₅×M₁₆	C₂³	1	4	yes	—
35	C₂²×C₅×C₈	—	2	4	no	—
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₅×#127	C₂×C₄	1	2	yes	—
41	C₅×#124	C₂×C₄	1	4	yes	—
42	C₅×#121	—	2	4	no	—
43	C₄×C₅×D₄	—	2	4	no	—
44	C₂×C₄²×C₅	C₂³	1	2	yes	—
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₅×#108	—	2	8	no	—
47	C₅×C₄⋊C₈	—	2	4	no	—
48	C₅×#104	—	2	16	no	—
49	C₅×C₂²⋊C₈	—	2	8	no	—
50	C₅×#97	C₂×C₄	1	4	yes	—
51	C₅×#97	D₄	1	8	yes	complemented
52	C₄×C₅×C₈	—	2	4	no	—
53	C₄×C₅×C₈	D₄	1	4	yes	—
54	C₅×#95	—	2	2	no	—
55	#3150	C₂×C₅	1	1	yes	characteristic
56	C₂×#444	C₂×C₅	1	4	yes	complemented
57	#2931	C₂×C₅	1	2	yes	complemented
58	#1974	C₂×C₅	1	4	yes	complemented
59	#1963	C₂×C₅	1	2	yes	complemented
60	#1849	C₂×C₅	1	1	yes	characteristic, complemented
61	#1726	C₂×C₅	1	1	yes	characteristic, complemented
62	C₅×#41	—	2	8	no	—
63	C₅×#41	—	2	8	no	—
64	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
65	C₂×C₅×D₄	—	2	4	no	—
66	C₂×C₅×D₄	—	2	4	no	—
67	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₅	—	2	2	no	—
69	C₂²×C₄×C₅	—	2	4	no	—
70	C₂²×C₄×C₅	#41	1	2	yes	—
71	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
72	C₅×Q₁₆	—	2	8	no	—
73	C₅×SD₁₆	—	2	16	no	—
74	C₅×D₈	—	2	8	no	—
75	C₅×M₁₆	—	2	8	no	—
76	C₅×M₁₆	—	4	8	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	—	4	8	no	—
80	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
81	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
82	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
83	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
84	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
85	C₂×C₅×C₈	#41	1	4	yes	—
86	C₅×C₄⋊C₄	—	2	4	no	—
87	C₅×C₂²⋊C₄	—	2	8	no	—
88	C₄²×C₅	—	2	2	no	—
89	C₄²×C₅	C₂²×C₄	1	2	yes	—
90	C₄²×C₅	C₂×D₄	1	4	yes	—
91	C₄²×C₅	C₂×D₄	1	2	yes	—
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	#444	—	2	16	no	—
98	#436	—	2	4	no	—
99	C₈×D₄	—	2	4	no	—
100	C₂×#108	C₂²×C₅	1	4	yes	—
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×#97	C₂²×C₅	1	2	yes	—
106	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
107	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
108	#376	C₂²×C₅	1	1	yes	characteristic
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	8	no	—
114	C₅×D₄	—	4	8	no	—
115	C₂×C₄×C₅	—	2	4	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	C₄×D₄	1	4	yes	—
122	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
124	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×#41	1	2	yes	—
126	C₅×C₈	—	2	2	no	—
127	C₅×C₈	—	4	4	no	—
128	C₅×C₈	C₄×D₄	1	8	yes	—
129	C₅×C₈	#127	1	4	yes	—
130	C₂×#41	C₂³×C₅	1	4	yes	—
131	#135	—	2	8	no	—
132	#135	—	2	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	—	2	16	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	1	yes	characteristic
141	C₂²×C₈	C₂³×C₅	1	2	yes	—
142	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
143	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
144	#127	C₂×C₄×C₅	1	2	yes	—
145	#124	C₂×C₄×C₅	1	4	yes	—
146	#121	—	2	4	no	—
147	C₄×D₄	—	2	4	no	—
148	C₂×C₄²	C₂³×C₅	1	2	yes	—
149	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
150	#108	—	2	8	no	—
151	C₄⋊C₈	—	2	4	no	—
152	#104	—	2	16	no	—
153	C₂²⋊C₈	—	2	8	no	—
154	#97	C₂×C₄×C₅	1	4	yes	—
155	#97	C₅×D₄	1	8	yes	complemented
156	C₄×C₈	—	2	4	no	—
157	C₄×C₈	C₅×D₄	1	4	yes	—
158	#95	—	2	2	no	—
159	C₂²×C₅	—	2	2	no	—
160	C₂²×C₅	—	4	4	no	—
161	C₂²×C₅	—	4	8	no	—
162	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂²×C₅	#532	1	2	yes	—
164	C₄×C₅	—	2	4	no	—
165	C₄×C₅	—	4	4	no	—
166	C₄×C₅	—	4	4	no	—
167	C₄×C₅	C₂×C₄×D₄	1	2	yes	—
168	C₄×C₅	C₂×#127	1	1	yes	characteristic
169	C₄×C₅	#532	1	1	yes	characteristic
170	#41	—	2	8	no	—
171	#41	—	2	8	no	—
172	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	C₅×#41	1	2	yes	—
179	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
180	Q₁₆	—	2	8	no	—
181	SD₁₆	—	2	16	no	—
182	D₈	—	2	8	no	—
183	M₁₆	—	2	8	no	—
184	M₁₆	—	4	8	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	4	8	no	—
188	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
189	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
190	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
191	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
192	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
193	C₂×C₈	C₅×#41	1	4	yes	—
194	C₄⋊C₄	—	2	4	no	—
195	C₂²⋊C₄	—	2	8	no	—
196	C₄²	—	2	2	no	—
197	C₄²	C₂²×C₄×C₅	1	2	yes	—
198	C₄²	C₂×C₅×D₄	1	4	yes	—
199	C₄²	C₂×C₅×D₄	1	2	yes	—
200	C₂×C₅	—	2	2	no	—
201	C₂×C₅	—	8	4	no	—
202	C₂×C₅	C₄×#127	1	1	yes	characteristic
203	C₂×C₅	#2955	1	2	yes	—
204	C₂³	—	2	4	no	—
205	C₂³	C₂×C₅×#41	1	1	yes	characteristic
206	Q₈	—	2	8	no	—
207	D₄	—	2	8	no	—
208	D₄	—	4	8	no	—
209	D₄	—	4	8	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
217	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
218	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
219	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
220	C₂×C₄	C₂×C₅×#41	1	2	yes	—
221	C₈	—	2	2	no	—
222	C₈	—	4	4	no	—
223	C₈	C₄×C₅×D₄	1	8	yes	—
224	C₈	C₅×#127	1	4	yes	—
225	C₅	#21709	1	1	yes	characteristic, direct product factor, Sylow
226	C₂²	—	2	2	no	—
227	C₂²	—	4	4	no	—
228	C₂²	—	4	8	no	—
229	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
230	C₂²	C₅×#532	1	2	yes	—
231	C₄	—	2	4	no	—
232	C₄	—	4	4	no	—
233	C₄	—	4	4	no	—
234	C₄	C₂×C₄×C₅×D₄	1	2	yes	—
235	C₄	C₂×C₅×#127	1	1	yes	characteristic
236	C₄	C₅×#532	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	—	8	4	no	—
239	C₂	C₄×C₅×#127	1	1	yes	characteristic
240	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. C₂×C₅×#444	C₂	semidirect
3. C₅×#2931	C₂	semidirect
4. C₅×#1974	C₂	semidirect
5. C₅×#1963	C₂	semidirect
6. C₅×#1849	C₂	semidirect
7. C₅×#1726	C₂	semidirect
11. C₅×#497	C₄	semidirect
12. C₅×#496	C₄	semidirect
25. #21709	C₅	direct (N × H)
51. C₅×#97	D₄	semidirect
56. C₂×#444	C₂×C₅	semidirect
57. #2931	C₂×C₅	semidirect
58. #1974	C₂×C₅	semidirect
59. #1963	C₂×C₅	semidirect
60. #1849	C₂×C₅	semidirect
61. #1726	C₂×C₅	semidirect
95. #497	C₄×C₅	semidirect
96. #496	C₄×C₅	semidirect
155. #97	C₅×D₄	semidirect
225. C₅	#21709	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
C₂×C₅×#444	2
C₅×#2931	2
C₅×#1974	2
C₅×#1963	2
C₅×#1849	2
C₅×#1726	2
#21709	5

Subgroups — C5×#21709 #11944429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21709 #11944429