Subgroups — C₅×#14713 #11937433

Order 1280.

3342 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#390	C₂	1	12	yes	maximal, complemented
3	C₂×C₅×#384	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄³×C₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₅×D₄	C₂²	1	6	yes	—
6	C₂×C₅×#127	—	2	4	no	—
7	C₂×C₅×#127	—	2	4	no	—
8	C₂×C₅×#127	C₄	1	8	yes	complemented
9	C₂²×C₅×C₄⋊C₄	C₂²	1	6	yes	—
10	C₂²×C₅×C₂²⋊C₄	C₂²	1	16	yes	—
11	C₂²×C₄²×C₅	C₂²	1	6	yes	—
12	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₅×#390	—	2	48	no	—
14	C₅×#384	—	2	4	no	—
15	C₄³×C₅	—	2	4	no	—
16	#14713	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂⁵×C₅	C₂³	1	4	yes	—
18	C₂²×C₅×D₄	—	2	24	no	—
19	C₂²×C₅×D₄	—	2	12	no	—
20	C₂²×C₅×D₄	—	2	12	no	—
21	C₂²×C₅×D₄	—	2	24	no	—
22	C₂²×C₅×D₄	C₂×C₄	1	24	yes	—
23	C₂³×C₄×C₅	C₂³	1	4	yes	—
24	C₂³×C₄×C₅	C₂³	1	4	yes	—
25	C₂³×C₄×C₅	C₂³	1	3	yes	—
26	C₅×#127	—	2	32	no	—
27	C₂×C₅×C₄⋊C₄	—	2	12	no	—
28	C₂×C₅×C₄⋊C₄	—	2	24	no	—
29	C₂×C₅×C₂²⋊C₄	—	2	96	no	—
30	C₂×C₄²×C₅	—	2	12	no	—
31	C₂×C₄²×C₅	—	2	12	no	—
32	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₅	D₄	1	4	yes	—
34	C₂×C₄²×C₅	D₄	1	4	yes	—
35	C₂×C₄²×C₅	D₄	1	24	yes	complemented
36	C₂²×#127	C₂×C₅	1	1	yes	characteristic
37	C₂×#390	C₂×C₅	1	12	yes	complemented
38	C₂×#384	C₂×C₅	1	1	yes	characteristic, complemented
39	C₂×C₄³	C₂×C₅	1	1	yes	characteristic, complemented
40	C₂⁴×C₅	—	2	24	no	—
41	C₂⁴×C₅	—	2	4	no	—
42	C₂⁴×C₅	—	2	4	no	—
43	C₂⁴×C₅	—	2	24	no	—
44	C₂⁴×C₅	C₂²×C₄	1	8	yes	—
45	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
46	C₂×C₅×D₄	—	2	48	no	—
47	C₂×C₅×D₄	—	2	96	no	—
48	C₂×C₅×D₄	—	4	48	no	—
49	C₂×C₅×D₄	—	4	24	no	—
50	C₂×C₅×D₄	—	4	24	no	—
51	C₂²×C₄×C₅	—	2	4	no	—
52	C₂²×C₄×C₅	—	2	12	no	—
53	C₂²×C₄×C₅	—	2	24	no	—
54	C₂²×C₄×C₅	—	2	4	no	—
55	C₂²×C₄×C₅	C₂²×C₄	1	6	yes	—
56	C₂²×C₄×C₅	C₂×D₄	1	24	yes	—
57	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
58	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
59	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
60	C₂²×C₄×C₅	#41	1	12	yes	—
61	C₅×C₄⋊C₄	—	4	24	no	—
62	C₅×C₂²⋊C₄	—	4	64	no	—
63	C₄²×C₅	—	2	48	no	—
64	C₄²×C₅	C₂²⋊C₄	1	16	yes	complemented
65	C₂³×D₄	C₂²×C₅	1	6	yes	—
66	C₂×#127	—	2	4	no	—
67	C₂×#127	—	2	4	no	—
68	C₂×#127	C₄×C₅	1	8	yes	complemented
69	C₂²×C₄⋊C₄	C₂²×C₅	1	6	yes	—
70	C₂²×C₂²⋊C₄	C₂²×C₅	1	16	yes	—
71	C₂²×C₄²	C₂²×C₅	1	6	yes	—
72	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
73	#390	—	2	48	no	—
74	#384	—	2	4	no	—
75	C₄³	—	2	4	no	—
76	C₂³×C₅	—	2	8	no	—
77	C₂³×C₅	—	2	48	no	—
78	C₂³×C₅	—	4	24	no	—
79	C₂³×C₅	—	4	12	no	—
80	C₂³×C₅	—	4	16	no	—
81	C₂³×C₅	—	4	12	no	—
82	C₂³×C₅	—	4	48	no	—
83	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
84	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
85	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
86	C₂³×C₅	C₂²×D₄	1	6	yes	—
87	C₂³×C₅	C₂×#41	1	6	yes	—
88	C₅×D₄	—	4	96	no	—
89	C₂×C₄×C₅	—	2	24	no	—
90	C₂×C₄×C₅	—	2	24	no	—
91	C₂×C₄×C₅	—	4	16	no	—
92	C₂×C₄×C₅	—	4	12	no	—
93	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	12	yes	—
94	C₂×C₄×C₅	C₄×D₄	1	24	yes	—
95	C₂×C₄×C₅	#121	1	24	yes	—
96	C₂×C₄×C₅	#124	1	12	yes	—
97	C₂×C₄×C₅	#127	1	12	yes	—
98	C₂×C₄×C₅	#127	1	16	yes	complemented
99	C₂⁵	C₂³×C₅	1	4	yes	—
100	C₂²×D₄	—	2	12	no	—
101	C₂²×D₄	—	2	12	no	—
102	C₂²×D₄	—	2	24	no	—
103	C₂²×D₄	—	2	24	no	—
104	C₂²×D₄	C₂×C₄×C₅	1	24	yes	—
105	C₂³×C₄	C₂³×C₅	1	4	yes	—
106	C₂³×C₄	C₂³×C₅	1	4	yes	—
107	C₂³×C₄	C₂³×C₅	1	3	yes	—
108	#127	—	2	32	no	—
109	C₂×C₄⋊C₄	—	2	12	no	—
110	C₂×C₄⋊C₄	—	2	24	no	—
111	C₂×C₂²⋊C₄	—	2	96	no	—
112	C₂×C₄²	—	2	12	no	—
113	C₂×C₄²	—	2	12	no	—
114	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
115	C₂×C₄²	C₅×D₄	1	4	yes	—
116	C₂×C₄²	C₅×D₄	1	4	yes	—
117	C₂×C₄²	C₅×D₄	1	24	yes	complemented
118	C₂²×C₅	—	4	32	no	—
119	C₂²×C₅	—	4	24	no	—
120	C₂²×C₅	—	8	16	no	—
121	C₂²×C₅	—	8	12	no	—
122	C₂²×C₅	—	8	4	no	—
123	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₅	C₂×C₄×D₄	1	6	yes	—
125	C₂²×C₅	C₂×#121	1	6	yes	—
126	C₂²×C₅	C₂×#124	1	3	yes	—
127	C₂²×C₅	C₂×#127	1	4	yes	—
128	C₂²×C₅	C₂×#127	1	3	yes	—
129	C₂²×C₅	#532	1	12	yes	—
130	C₄×C₅	—	2	16	no	—
131	C₄×C₅	—	8	4	no	—
132	C₄×C₅	#390	1	24	yes	complemented
133	C₂⁴	—	2	24	no	—
134	C₂⁴	—	2	4	no	—
135	C₂⁴	—	2	4	no	—
136	C₂⁴	—	2	24	no	—
137	C₂⁴	C₂²×C₄×C₅	1	8	yes	—
138	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
139	C₂×D₄	—	2	48	no	—
140	C₂×D₄	—	2	96	no	—
141	C₂×D₄	—	4	48	no	—
142	C₂×D₄	—	4	24	no	—
143	C₂×D₄	—	4	24	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	2	12	no	—
146	C₂²×C₄	—	2	24	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	C₂²×C₄×C₅	1	6	yes	—
149	C₂²×C₄	C₂×C₅×D₄	1	24	yes	—
150	C₂²×C₄	C₂×C₅×D₄	1	6	yes	—
151	C₂²×C₄	C₂×C₅×D₄	1	6	yes	—
152	C₂²×C₄	C₂×C₅×D₄	1	12	yes	—
153	C₂²×C₄	C₅×#41	1	12	yes	—
154	C₄⋊C₄	—	4	24	no	—
155	C₂²⋊C₄	—	4	64	no	—
156	C₄²	—	2	48	no	—
157	C₄²	C₅×C₂²⋊C₄	1	16	yes	complemented
158	C₂×C₅	—	8	8	no	—
159	C₂×C₅	C₂×#390	1	3	yes	—
160	C₂×C₅	C₄×#127	1	4	yes	—
161	C₂×C₅	#2856	1	4	yes	—
162	C₂×C₅	#2858	1	3	yes	—
163	C₂×C₅	#2867	1	1	yes	characteristic
164	C₂³	—	2	8	no	—
165	C₂³	—	2	48	no	—
166	C₂³	—	4	24	no	—
167	C₂³	—	4	12	no	—
168	C₂³	—	4	16	no	—
169	C₂³	—	4	12	no	—
170	C₂³	—	4	48	no	—
171	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
172	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
173	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
174	C₂³	C₂²×C₅×D₄	1	6	yes	—
175	C₂³	C₂×C₅×#41	1	6	yes	—
176	D₄	—	4	96	no	—
177	C₂×C₄	—	2	24	no	—
178	C₂×C₄	—	2	24	no	—
179	C₂×C₄	—	4	16	no	—
180	C₂×C₄	—	4	12	no	—
181	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	12	yes	—
182	C₂×C₄	C₄×C₅×D₄	1	24	yes	—
183	C₂×C₄	C₅×#121	1	24	yes	—
184	C₂×C₄	C₅×#124	1	12	yes	—
185	C₂×C₄	C₅×#127	1	12	yes	—
186	C₂×C₄	C₅×#127	1	16	yes	complemented
187	C₅	#14713	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²	—	4	24	no	—
189	C₂²	—	4	32	no	—
190	C₂²	—	8	16	no	—
191	C₂²	—	8	12	no	—
192	C₂²	—	8	4	no	—
193	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
194	C₂²	C₂×C₄×C₅×D₄	1	6	yes	—
195	C₂²	C₂×C₅×#121	1	6	yes	—
196	C₂²	C₂×C₅×#124	1	3	yes	—
197	C₂²	C₂×C₅×#127	1	3	yes	—
198	C₂²	C₂×C₅×#127	1	4	yes	—
199	C₂²	C₅×#532	1	12	yes	—
200	C₄	—	2	16	no	—
201	C₄	—	8	4	no	—
202	C₄	C₅×#390	1	24	yes	complemented
203	C₂	—	8	8	no	—
204	C₂	C₂×C₅×#390	1	3	yes	—
205	C₂	C₄×C₅×#127	1	4	yes	—
206	C₂	C₅×#2856	1	4	yes	—
207	C₂	C₅×#2858	1	3	yes	—
208	C₂	C₅×#2867	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂⁴	C₂⁴×C₅
prime radical	C₂²×C₅×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#390	C₂	semidirect
3. C₂×C₅×#384	C₂	semidirect
4. C₂×C₄³×C₅	C₂	semidirect
8. C₂×C₅×#127	C₄	semidirect
16. #14713	C₅	direct (N × H)
35. C₂×C₄²×C₅	D₄	semidirect
37. C₂×#390	C₂×C₅	semidirect
38. C₂×#384	C₂×C₅	semidirect
39. C₂×C₄³	C₂×C₅	semidirect
64. C₄²×C₅	C₂²⋊C₄	semidirect
68. C₂×#127	C₄×C₅	semidirect
98. C₂×C₄×C₅	#127	semidirect
117. C₂×C₄²	C₅×D₄	semidirect
132. C₄×C₅	#390	semidirect
157. C₄²	C₅×C₂²⋊C₄	semidirect
186. C₂×C₄	C₅×#127	semidirect
187. C₅	#14713	direct (N × H)
202. C₄	C₅×#390	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₅×#127	2
C₂×C₅×#390	2
C₂×C₅×#384	2
C₂×C₄³×C₅	2
#14713	5

Subgroups — C5×#14713 #11937433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#14713 #11937433