Subgroups — C₅×#21708 #11944428

Order 1280.

1006 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	C₂×C₅×#443	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×#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×#443	—	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	2	yes	—
21	C₂×C₅×#97	C₂²	1	1	yes	characteristic
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	#21708	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	—	2	4	no	—
51	C₅×#97	C₂×C₄	1	4	yes	—
52	C₄×C₅×C₈	D₄	1	2	yes	—
53	C₄×C₅×C₈	D₄	1	2	yes	—
54	C₄×C₅×C₈	D₄	1	8	yes	complemented
55	C₅×#95	—	2	2	no	—
56	#3150	C₂×C₅	1	1	yes	characteristic
57	C₂×#443	C₂×C₅	1	4	yes	complemented
58	#2931	C₂×C₅	1	2	yes	complemented
59	#1975	C₂×C₅	1	4	yes	complemented
60	#1963	C₂×C₅	1	2	yes	complemented
61	#1849	C₂×C₅	1	1	yes	characteristic, complemented
62	#1726	C₂×C₅	1	1	yes	characteristic, complemented
63	C₅×#41	—	2	8	no	—
64	C₅×#41	—	2	8	no	—
65	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
66	C₂×C₅×D₄	—	2	4	no	—
67	C₂×C₅×D₄	—	2	4	no	—
68	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₅	—	2	2	no	—
70	C₂²×C₄×C₅	—	2	4	no	—
71	C₂²×C₄×C₅	#41	1	2	yes	—
72	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
73	C₅×Q₁₆	—	2	8	no	—
74	C₅×SD₁₆	—	2	16	no	—
75	C₅×D₈	—	2	8	no	—
76	C₅×M₁₆	—	2	8	no	—
77	C₅×M₁₆	—	4	8	no	—
78	C₂×C₅×C₈	—	2	2	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	4	yes	—
84	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
86	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂×C₅×C₈	#41	1	4	yes	—
88	C₅×C₄⋊C₄	—	2	4	no	—
89	C₅×C₂²⋊C₄	—	2	8	no	—
90	C₄²×C₅	—	2	2	no	—
91	C₄²×C₅	C₂²×C₄	1	2	yes	—
92	C₄²×C₅	C₂×D₄	1	4	yes	—
93	C₄²×C₅	C₂×D₄	1	2	yes	—
94	C₂×#135	C₂²×C₅	1	4	yes	—
95	C₂×#131	C₂²×C₅	1	4	yes	—
96	#532	C₂²×C₅	1	2	yes	—
97	#497	C₄×C₅	1	4	yes	complemented
98	#496	C₄×C₅	1	4	yes	complemented
99	#443	—	2	16	no	—
100	#436	—	2	4	no	—
101	C₈×D₄	—	2	4	no	—
102	C₂×#108	C₂²×C₅	1	4	yes	—
103	#423	C₂²×C₅	1	2	yes	—
104	C₂×#104	C₂²×C₅	1	8	yes	—
105	#408	C₂²×C₅	1	4	yes	—
106	C₂×#97	C₂²×C₅	1	2	yes	—
107	C₂×#97	C₂²×C₅	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
110	#376	C₂²×C₅	1	1	yes	characteristic
111	C₂³×C₅	—	2	4	no	—
112	C₂³×C₅	C₂×#41	1	1	yes	characteristic
113	C₅×Q₈	—	2	8	no	—
114	C₅×D₄	—	2	8	no	—
115	C₅×D₄	—	4	8	no	—
116	C₅×D₄	—	4	8	no	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	4	no	—
119	C₂×C₄×C₅	—	4	2	no	—
120	C₂×C₄×C₅	—	4	4	no	—
121	C₂×C₄×C₅	—	4	4	no	—
122	C₂×C₄×C₅	—	4	4	no	—
123	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
124	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
126	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×#41	1	2	yes	—
128	C₅×C₈	—	2	2	no	—
129	C₅×C₈	—	4	4	no	—
130	C₅×C₈	C₄×D₄	1	8	yes	—
131	C₅×C₈	#127	1	2	yes	—
132	C₅×C₈	#127	1	2	yes	—
133	C₂×#41	C₂³×C₅	1	4	yes	—
134	#135	—	2	8	no	—
135	#135	—	2	8	no	—
136	C₂×Q₁₆	C₂×C₄×C₅	1	4	yes	—
137	C₂×SD₁₆	C₂×C₄×C₅	1	8	yes	—
138	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
139	#131	—	2	16	no	—
140	C₂×M₁₆	—	2	4	no	—
141	C₂×M₁₆	C₂³×C₅	1	4	yes	—
142	C₂²×C₈	—	2	4	no	—
143	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
144	C₂²×C₈	C₂³×C₅	1	2	yes	—
145	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
146	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
147	#127	C₂×C₄×C₅	1	2	yes	—
148	#124	C₂×C₄×C₅	1	4	yes	—
149	#121	—	2	4	no	—
150	C₄×D₄	—	2	4	no	—
151	C₂×C₄²	C₂³×C₅	1	2	yes	—
152	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
153	#108	—	2	8	no	—
154	C₄⋊C₈	—	2	4	no	—
155	#104	—	2	16	no	—
156	C₂²⋊C₈	—	2	8	no	—
157	#97	—	2	4	no	—
158	#97	C₂×C₄×C₅	1	4	yes	—
159	C₄×C₈	C₅×D₄	1	2	yes	—
160	C₄×C₈	C₅×D₄	1	2	yes	—
161	C₄×C₈	C₅×D₄	1	8	yes	complemented
162	#95	—	2	2	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	4	4	no	—
165	C₂²×C₅	—	4	8	no	—
166	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₅	#532	1	2	yes	—
168	C₄×C₅	—	2	4	no	—
169	C₄×C₅	—	4	4	no	—
170	C₄×C₅	—	4	4	no	—
171	C₄×C₅	C₂×C₄×D₄	1	2	yes	—
172	C₄×C₅	C₂×#127	1	1	yes	characteristic
173	C₄×C₅	#532	1	1	yes	characteristic
174	#41	—	2	8	no	—
175	#41	—	2	8	no	—
176	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
177	C₂×D₄	—	2	4	no	—
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	C₅×#41	1	2	yes	—
183	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
184	Q₁₆	—	2	8	no	—
185	SD₁₆	—	2	16	no	—
186	D₈	—	2	8	no	—
187	M₁₆	—	2	8	no	—
188	M₁₆	—	4	8	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	4	8	no	—
192	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
193	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
194	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
195	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
196	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
197	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
198	C₂×C₈	C₅×#41	1	4	yes	—
199	C₄⋊C₄	—	2	4	no	—
200	C₂²⋊C₄	—	2	8	no	—
201	C₄²	—	2	2	no	—
202	C₄²	C₂²×C₄×C₅	1	2	yes	—
203	C₄²	C₂×C₅×D₄	1	4	yes	—
204	C₄²	C₂×C₅×D₄	1	2	yes	—
205	C₂×C₅	—	2	2	no	—
206	C₂×C₅	—	8	4	no	—
207	C₂×C₅	C₄×#127	1	1	yes	characteristic
208	C₂×C₅	#2955	1	2	yes	—
209	C₂³	—	2	4	no	—
210	C₂³	C₂×C₅×#41	1	1	yes	characteristic
211	Q₈	—	2	8	no	—
212	D₄	—	2	8	no	—
213	D₄	—	4	8	no	—
214	D₄	—	4	8	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
222	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
224	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
225	C₂×C₄	C₂×C₅×#41	1	2	yes	—
226	C₈	—	2	2	no	—
227	C₈	—	4	4	no	—
228	C₈	C₄×C₅×D₄	1	8	yes	—
229	C₈	C₅×#127	1	2	yes	—
230	C₈	C₅×#127	1	2	yes	—
231	C₅	#21708	1	1	yes	characteristic, direct product factor, Sylow
232	C₂²	—	2	2	no	—
233	C₂²	—	4	4	no	—
234	C₂²	—	4	8	no	—
235	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
236	C₂²	C₅×#532	1	2	yes	—
237	C₄	—	2	4	no	—
238	C₄	—	4	4	no	—
239	C₄	—	4	4	no	—
240	C₄	C₂×C₄×C₅×D₄	1	2	yes	—
241	C₄	C₂×C₅×#127	1	1	yes	characteristic
242	C₄	C₅×#532	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	8	4	no	—
245	C₂	C₄×C₅×#127	1	1	yes	characteristic
246	C₂	C₅×#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₅×C₈	C₂×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×C₅×C₈ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#443	C₂	semidirect
3. C₅×#2931	C₂	semidirect
4. C₅×#1975	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. #21708	C₅	direct (N × H)
54. C₄×C₅×C₈	D₄	semidirect
57. C₂×#443	C₂×C₅	semidirect
58. #2931	C₂×C₅	semidirect
59. #1975	C₂×C₅	semidirect
60. #1963	C₂×C₅	semidirect
61. #1849	C₂×C₅	semidirect
62. #1726	C₂×C₅	semidirect
97. #497	C₄×C₅	semidirect
98. #496	C₄×C₅	semidirect
161. C₄×C₈	C₅×D₄	semidirect
231. C₅	#21708	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×#443	2
C₅×#2931	2
C₅×#1975	2
C₅×#1963	2
C₅×#1849	2
C₅×#1726	2
#21708	5

Subgroups — C5×#21708 #11944428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21708 #11944428