Subgroups — C₅×#21670 #11944390

Order 1280.

656 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2401	C₂	1	1	yes	maximal, characteristic
2	C₅×#1924	C₂	1	3	yes	maximal
3	C₅×#1917	C₂	1	3	yes	maximal, complemented
4	C₅×#1840	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1771	C₂	1	3	yes	maximal, complemented
6	C₅×#1753	C₂	1	3	yes	maximal
7	C₅×#1726	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊C₈	C₂²	1	3	yes	—
10	C₂×C₅×C₂²⋊C₈	C₂²	1	3	yes	—
11	C₂×C₅×#97	C₂²	1	3	yes	—
12	C₂×C₅×#97	C₂²	1	3	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	3	yes	—
14	C₅×#401	C₄	1	1	yes	characteristic
15	C₅×#400	C₄	1	3	yes	—
16	C₅×#397	C₄	1	3	yes	—
17	C₅×#396	C₄	1	1	yes	characteristic
18	C₄×C₅×C₄⋊C₄	C₂²	1	3	yes	—
19	C₄×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
20	C₅×#376	C₂²	1	1	yes	characteristic
21	C₅×#336	C₂²	1	3	yes	—
22	C₅×#336	C₂²	1	3	yes	—
23	C₅×#336	C₂²	1	3	yes	—
24	C₅×#336	C₂²	1	3	yes	—
25	#21670	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂×C₅×M₁₆	—	2	6	no	—
28	C₂²×C₅×C₈	C₂³	1	3	yes	—
29	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂³	1	3	yes	—
31	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	3	yes	—
33	C₂×C₅×C₄⋊C₄	C₂×C₄	1	3	yes	—
34	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	3	yes	—
35	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	3	yes	—
36	C₂×C₄²×C₅	C₂³	1	3	yes	—
37	C₂×C₄²×C₅	C₂³	1	3	yes	—
38	C₅×C₄⋊C₈	—	2	6	no	—
39	C₅×C₂²⋊C₈	—	2	6	no	—
40	C₅×#97	—	2	6	no	—
41	C₅×#97	—	2	6	no	—
42	C₄×C₅×C₈	—	2	6	no	—
43	C₅×#95	C₂×C₄	1	1	yes	characteristic
44	C₅×#95	C₂×C₄	1	3	yes	—
45	C₅×#95	C₂×C₄	1	3	yes	—
46	C₅×#95	C₂×C₄	1	3	yes	—
47	C₅×#95	C₂×C₄	1	3	yes	—
48	C₅×#95	C₂×C₄	1	3	yes	—
49	#2401	C₂×C₅	1	1	yes	characteristic
50	#1924	C₂×C₅	1	3	yes	—
51	#1917	C₂×C₅	1	3	yes	complemented
52	#1840	C₂×C₅	1	1	yes	characteristic, complemented
53	#1771	C₂×C₅	1	3	yes	complemented
54	#1753	C₂×C₅	1	3	yes	—
55	#1726	C₂×C₅	1	1	yes	characteristic, complemented
56	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	—	2	3	no	—
58	C₂²×C₄×C₅	—	2	3	no	—
59	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
60	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
63	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
65	C₅×M₁₆	—	4	4	no	—
66	C₂×C₅×C₈	—	2	3	no	—
67	C₂×C₅×C₈	—	2	3	no	—
68	C₂×C₅×C₈	—	2	3	no	—
69	C₂×C₅×C₈	—	2	3	no	—
70	C₂×C₅×C₈	—	2	3	no	—
71	C₂×C₅×C₈	—	2	3	no	—
72	C₂×C₅×C₈	#41	1	6	yes	—
73	C₂×C₅×C₈	#41	1	6	yes	—
74	C₅×C₄⋊C₄	—	4	3	no	—
75	C₅×C₄⋊C₄	—	4	3	no	—
76	C₅×C₂²⋊C₄	—	4	3	no	—
77	C₅×C₂²⋊C₄	—	4	3	no	—
78	C₄²×C₅	—	2	3	no	—
79	C₄²×C₅	—	2	3	no	—
80	C₄²×C₅	—	2	3	no	—
81	C₄²×C₅	—	2	3	no	—
82	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
83	C₂×C₄⋊C₈	C₂²×C₅	1	3	yes	—
84	C₂×C₂²⋊C₈	C₂²×C₅	1	3	yes	—
85	C₂×#97	C₂²×C₅	1	3	yes	—
86	C₂×#97	C₂²×C₅	1	3	yes	—
87	C₂×C₄×C₈	C₂²×C₅	1	3	yes	—
88	#401	C₄×C₅	1	1	yes	characteristic
89	#400	C₄×C₅	1	3	yes	—
90	#397	C₄×C₅	1	3	yes	—
91	#396	C₄×C₅	1	1	yes	characteristic
92	C₄×C₄⋊C₄	C₂²×C₅	1	3	yes	—
93	C₄×C₂²⋊C₄	C₂²×C₅	1	3	yes	—
94	#376	C₂²×C₅	1	1	yes	characteristic
95	#336	C₂²×C₅	1	3	yes	—
96	#336	C₂²×C₅	1	3	yes	—
97	#336	C₂²×C₅	1	3	yes	—
98	#336	C₂²×C₅	1	3	yes	—
99	C₂³×C₅	—	2	3	no	—
100	C₂³×C₅	—	2	1	no	—
101	C₂³×C₅	—	2	3	no	—
102	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₅	—	2	3	no	—
104	C₂×C₄×C₅	—	2	3	no	—
105	C₂×C₄×C₅	—	2	3	no	—
106	C₂×C₄×C₅	—	2	3	no	—
107	C₂×C₄×C₅	—	2	3	no	—
108	C₂×C₄×C₅	—	2	3	no	—
109	C₂×C₄×C₅	—	2	3	no	—
110	C₂×C₄×C₅	—	2	3	no	—
111	C₂×C₄×C₅	—	2	3	no	—
112	C₂×C₄×C₅	—	2	3	no	—
113	C₂×C₄×C₅	—	2	3	no	—
114	C₂×C₄×C₅	—	2	3	no	—
115	C₂×C₄×C₅	—	2	3	no	—
116	C₂×C₄×C₅	—	4	1	no	—
117	C₂×C₄×C₅	—	4	3	no	—
118	C₂×C₄×C₅	C₂×#41	1	3	yes	—
119	C₂×C₄×C₅	C₂×#41	1	3	yes	—
120	C₅×C₈	—	2	6	no	—
121	C₅×C₈	—	4	1	no	—
122	C₅×C₈	—	4	3	no	—
123	C₅×C₈	#126	1	4	yes	—
124	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
125	C₂×M₁₆	—	2	6	no	—
126	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
127	C₂²×C₈	C₂³×C₅	1	3	yes	—
128	C₂²×C₈	C₂³×C₅	1	3	yes	—
129	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
130	C₂×C₄⋊C₄	C₂×C₄×C₅	1	3	yes	—
131	C₂×C₄⋊C₄	C₂×C₄×C₅	1	3	yes	—
132	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	3	yes	—
133	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	3	yes	—
134	C₂×C₄²	C₂³×C₅	1	3	yes	—
135	C₂×C₄²	C₂³×C₅	1	3	yes	—
136	C₄⋊C₈	—	2	6	no	—
137	C₂²⋊C₈	—	2	6	no	—
138	#97	—	2	6	no	—
139	#97	—	2	6	no	—
140	C₄×C₈	—	2	6	no	—
141	#95	C₂×C₄×C₅	1	1	yes	characteristic
142	#95	C₂×C₄×C₅	1	3	yes	—
143	#95	C₂×C₄×C₅	1	3	yes	—
144	#95	C₂×C₄×C₅	1	3	yes	—
145	#95	C₂×C₄×C₅	1	3	yes	—
146	#95	C₂×C₄×C₅	1	3	yes	—
147	C₂²×C₅	—	4	1	no	—
148	C₂²×C₅	—	4	3	no	—
149	C₂²×C₅	—	4	3	no	—
150	C₂²×C₅	C₄×#41	1	3	yes	—
151	C₂²×C₅	#568	1	3	yes	—
152	C₂²×C₅	#568	1	1	yes	characteristic
153	C₄×C₅	—	4	1	no	—
154	C₄×C₅	—	4	3	no	—
155	C₄×C₅	—	4	3	no	—
156	C₄×C₅	—	4	3	no	—
157	C₄×C₅	—	4	3	no	—
158	C₄×C₅	C₂×#126	1	1	yes	characteristic
159	C₄×C₅	#529	1	3	yes	—
160	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
161	C₂²×C₄	—	2	3	no	—
162	C₂²×C₄	—	2	3	no	—
163	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
164	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
165	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
166	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
167	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
168	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
169	M₁₆	—	4	4	no	—
170	C₂×C₈	—	2	3	no	—
171	C₂×C₈	—	2	3	no	—
172	C₂×C₈	—	2	3	no	—
173	C₂×C₈	—	2	3	no	—
174	C₂×C₈	—	2	3	no	—
175	C₂×C₈	—	2	3	no	—
176	C₂×C₈	C₅×#41	1	6	yes	—
177	C₂×C₈	C₅×#41	1	6	yes	—
178	C₄⋊C₄	—	4	3	no	—
179	C₄⋊C₄	—	4	3	no	—
180	C₂²⋊C₄	—	4	3	no	—
181	C₂²⋊C₄	—	4	3	no	—
182	C₄²	—	2	3	no	—
183	C₄²	—	2	3	no	—
184	C₄²	—	2	3	no	—
185	C₄²	—	2	3	no	—
186	C₂×C₅	—	8	1	no	—
187	C₂×C₅	C₄×#126	1	1	yes	characteristic
188	C₂×C₅	#2967	1	3	yes	—
189	C₂×C₅	#2967	1	3	yes	—
190	C₂³	—	2	3	no	—
191	C₂³	—	2	1	no	—
192	C₂³	—	2	3	no	—
193	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	—	2	3	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	2	3	no	—
198	C₂×C₄	—	2	3	no	—
199	C₂×C₄	—	2	3	no	—
200	C₂×C₄	—	2	3	no	—
201	C₂×C₄	—	2	3	no	—
202	C₂×C₄	—	2	3	no	—
203	C₂×C₄	—	2	3	no	—
204	C₂×C₄	—	2	3	no	—
205	C₂×C₄	—	2	3	no	—
206	C₂×C₄	—	2	3	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	3	no	—
209	C₂×C₄	C₂×C₅×#41	1	3	yes	—
210	C₂×C₄	C₂×C₅×#41	1	3	yes	—
211	C₈	—	2	6	no	—
212	C₈	—	4	1	no	—
213	C₈	—	4	3	no	—
214	C₈	C₅×#126	1	4	yes	—
215	C₅	#21670	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²	—	4	1	no	—
217	C₂²	—	4	3	no	—
218	C₂²	—	4	3	no	—
219	C₂²	C₄×C₅×#41	1	3	yes	—
220	C₂²	C₅×#568	1	3	yes	—
221	C₂²	C₅×#568	1	1	yes	characteristic
222	C₄	—	4	1	no	—
223	C₄	—	4	3	no	—
224	C₄	—	4	3	no	—
225	C₄	—	4	3	no	—
226	C₄	—	4	3	no	—
227	C₄	C₂×C₅×#126	1	1	yes	characteristic
228	C₄	C₅×#529	1	3	yes	—
229	C₂	—	8	1	no	—
230	C₂	C₄×C₅×#126	1	1	yes	characteristic
231	C₂	C₅×#2967	1	3	yes	—
232	C₂	C₅×#2967	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1917	C₂	semidirect
4. C₅×#1840	C₂	semidirect
5. C₅×#1771	C₂	semidirect
7. C₅×#1726	C₂	semidirect
25. #21670	C₅	direct (N × H)
51. #1917	C₂×C₅	semidirect
52. #1840	C₂×C₅	semidirect
53. #1771	C₂×C₅	semidirect
55. #1726	C₂×C₅	semidirect
215. C₅	#21670	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×#2401	2
C₅×#1924	2
C₅×#1917	2
C₅×#1840	2
C₅×#1771	2
C₅×#1753	2
C₅×#1726	2
#21670	5

Subgroups — C5×#21670 #11944390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21670 #11944390