Subgroups — C₅×#9824 #11932544

Order 1280.

726 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#493	C₂	1	1	yes	maximal, characteristic
2	C₅×#1957	C₂	1	4	yes	maximal, complemented
3	C₅×#1845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₅×D₈	—	2	2	no	—
6	C₂×C₅×#127	C₂²	1	2	yes	—
7	C₅×#493	—	2	2	no	—
8	C₅×#493	C₄	1	4	yes	complemented
9	C₂×C₅×#107	—	2	2	no	—
10	C₂×C₅×#102	—	2	8	no	—
11	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#384	C₂²	1	2	yes	—
14	C₅×C₈²	—	2	2	no	—
15	#9824	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₅×D₄	—	2	4	no	—
17	C₂²×C₅×D₄	—	2	2	no	—
18	C₂×C₅×D₈	—	2	8	no	—
19	C₂×C₅×D₈	—	4	4	no	—
20	C₂×C₅×D₈	—	4	4	no	—
21	C₂²×C₅×C₈	D₄	1	4	yes	—
22	C₂²×C₅×C₈	D₄	1	2	yes	—
23	C₅×#127	—	2	2	no	—
24	C₅×#127	—	2	4	no	—
25	C₅×#127	C₂×C₄	1	4	yes	—
26	C₂×C₅×C₄⋊C₄	—	2	4	no	—
27	C₂×C₅×C₄⋊C₄	—	2	2	no	—
28	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₅×#107	—	4	4	no	—
30	C₅×#102	—	4	16	no	—
31	C₄×C₅×C₈	—	2	2	no	—
32	C₄×C₅×C₈	—	2	2	no	—
33	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₄×C₅×C₈	D₄	1	2	yes	—
35	C₂×#493	C₂×C₅	1	1	yes	characteristic
36	#1957	C₂×C₅	1	4	yes	complemented
37	#1845	C₂×C₅	1	1	yes	characteristic, complemented
38	C₂×C₈²	C₂×C₅	1	1	yes	characteristic, complemented
39	C₂⁴×C₅	—	4	2	no	—
40	C₂×C₅×D₄	—	2	4	no	—
41	C₂×C₅×D₄	—	2	8	no	—
42	C₂×C₅×D₄	—	4	4	no	—
43	C₂×C₅×D₄	—	4	2	no	—
44	C₂×C₅×D₄	—	4	4	no	—
45	C₂×C₅×D₄	—	4	8	no	—
46	C₂×C₅×D₄	—	4	4	no	—
47	C₂×C₅×D₄	—	4	8	no	—
48	C₂²×C₄×C₅	—	4	2	no	—
49	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
51	C₅×D₈	—	4	16	no	—
52	C₂×C₅×C₈	—	2	4	no	—
53	C₂×C₅×C₈	—	2	4	no	—
54	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
55	C₂×C₅×C₈	D₈	1	4	yes	—
56	C₂×C₅×C₈	D₈	1	8	yes	complemented
57	C₂×C₅×C₈	SD₁₆	1	4	yes	—
58	C₅×C₄⋊C₄	—	4	8	no	—
59	C₅×C₄⋊C₄	—	4	4	no	—
60	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₅	#41	1	2	yes	—
63	C₂²×D₈	—	2	2	no	—
64	C₂×#127	C₂²×C₅	1	2	yes	—
65	#493	—	2	2	no	—
66	#493	C₄×C₅	1	4	yes	complemented
67	C₂×#107	—	2	2	no	—
68	C₂×#102	—	2	8	no	—
69	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
70	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
71	#384	C₂²×C₅	1	2	yes	—
72	C₈²	—	2	2	no	—
73	C₂³×C₅	—	4	4	no	—
74	C₂³×C₅	—	8	2	no	—
75	C₂³×C₅	—	8	2	no	—
76	C₂³×C₅	—	8	4	no	—
77	C₂³×C₅	—	8	4	no	—
78	C₂³×C₅	#127	1	1	yes	characteristic
79	C₅×D₄	—	4	8	no	—
80	C₅×D₄	—	4	16	no	—
81	C₅×D₄	—	8	8	no	—
82	C₅×D₄	—	8	4	no	—
83	C₂×C₄×C₅	—	8	2	no	—
84	C₂×C₄×C₅	—	8	4	no	—
85	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₅	#121	1	2	yes	—
88	C₂×C₄×C₅	#124	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
91	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
92	C₂×C₄×C₅	#135	1	4	yes	—
93	C₅×C₈	—	2	8	no	—
94	C₅×C₈	#102	1	8	yes	complemented
95	C₂²×D₄	—	2	2	no	—
96	C₂²×D₄	—	2	4	no	—
97	C₂×D₈	—	2	8	no	—
98	C₂×D₈	—	4	4	no	—
99	C₂×D₈	—	4	4	no	—
100	C₂²×C₈	C₅×D₄	1	4	yes	—
101	C₂²×C₈	C₅×D₄	1	2	yes	—
102	#127	—	2	2	no	—
103	#127	—	2	4	no	—
104	#127	C₂×C₄×C₅	1	4	yes	—
105	C₂×C₄⋊C₄	—	2	4	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
108	#107	—	4	4	no	—
109	#102	—	4	16	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
113	C₄×C₈	C₅×D₄	1	2	yes	—
114	C₂²×C₅	—	8	4	no	—
115	C₂²×C₅	—	8	8	no	—
116	C₂²×C₅	—	16	4	no	—
117	C₂²×C₅	—	16	2	no	—
118	C₂²×C₅	—	16	2	no	—
119	C₂²×C₅	#390	1	1	yes	characteristic
120	C₂²×C₅	#492	1	1	yes	characteristic
121	C₂²×C₅	#493	1	1	yes	characteristic
122	C₂²×C₅	#493	1	2	yes	—
123	C₂²×C₅	#495	1	2	yes	—
124	C₄×C₅	—	16	2	no	—
125	C₄×C₅	C₂×#102	1	2	yes	—
126	C₄×C₅	C₄×D₈	1	4	yes	—
127	C₄×C₅	#466	1	4	yes	—
128	C₄×C₅	#486	1	2	yes	—
129	C₂⁴	—	4	2	no	—
130	C₂×D₄	—	2	4	no	—
131	C₂×D₄	—	2	8	no	—
132	C₂×D₄	—	4	4	no	—
133	C₂×D₄	—	4	2	no	—
134	C₂×D₄	—	4	8	no	—
135	C₂×D₄	—	4	4	no	—
136	C₂×D₄	—	4	4	no	—
137	C₂×D₄	—	4	8	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
141	D₈	—	4	16	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
145	C₂×C₈	C₅×D₈	1	4	yes	—
146	C₂×C₈	C₅×D₈	1	8	yes	complemented
147	C₂×C₈	C₅×SD₁₆	1	4	yes	—
148	C₄⋊C₄	—	4	8	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
151	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
152	C₄²	C₅×#41	1	2	yes	—
153	C₂×C₅	—	16	4	no	—
154	C₂×C₅	#1706	1	2	yes	—
155	C₂×C₅	#1709	1	1	yes	characteristic
156	C₂×C₅	#1712	1	1	yes	characteristic
157	C₂×C₅	#1957	1	1	yes	characteristic
158	C₂×C₅	#1970	1	2	yes	—
159	C₂³	—	4	4	no	—
160	C₂³	—	8	2	no	—
161	C₂³	—	8	2	no	—
162	C₂³	—	8	4	no	—
163	C₂³	—	8	4	no	—
164	C₂³	C₅×#127	1	1	yes	characteristic
165	D₄	—	4	8	no	—
166	D₄	—	4	16	no	—
167	D₄	—	8	8	no	—
168	D₄	—	8	4	no	—
169	C₂×C₄	—	8	2	no	—
170	C₂×C₄	—	8	4	no	—
171	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
172	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
173	C₂×C₄	C₅×#121	1	2	yes	—
174	C₂×C₄	C₅×#124	1	1	yes	characteristic
175	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
176	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
177	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
178	C₂×C₄	C₅×#135	1	4	yes	—
179	C₈	—	2	8	no	—
180	C₈	C₅×#102	1	8	yes	complemented
181	C₅	#9824	1	1	yes	characteristic, direct product factor, Sylow
182	C₂²	—	8	4	no	—
183	C₂²	—	8	8	no	—
184	C₂²	—	16	4	no	—
185	C₂²	—	16	2	no	—
186	C₂²	—	16	2	no	—
187	C₂²	C₅×#390	1	1	yes	characteristic
188	C₂²	C₅×#492	1	1	yes	characteristic
189	C₂²	C₅×#493	1	1	yes	characteristic
190	C₂²	C₅×#493	1	2	yes	—
191	C₂²	C₅×#495	1	2	yes	—
192	C₄	—	16	2	no	—
193	C₄	C₂×C₅×#102	1	2	yes	—
194	C₄	C₄×C₅×D₈	1	4	yes	—
195	C₄	C₅×#466	1	4	yes	—
196	C₄	C₅×#486	1	2	yes	—
197	C₂	—	16	4	no	—
198	C₂	C₅×#1706	1	2	yes	—
199	C₂	C₅×#1709	1	1	yes	characteristic
200	C₂	C₅×#1712	1	1	yes	characteristic
201	C₂	C₅×#1957	1	1	yes	characteristic
202	C₂	C₅×#1970	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	#127
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂×C₅×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1957	C₂	semidirect
3. C₅×#1845	C₂	semidirect
4. C₂×C₅×C₈²	C₂	semidirect
8. C₅×#493	C₄	semidirect
15. #9824	C₅	direct (N × H)
36. #1957	C₂×C₅	semidirect
37. #1845	C₂×C₅	semidirect
38. C₂×C₈²	C₂×C₅	semidirect
56. C₂×C₅×C₈	D₈	semidirect
66. #493	C₄×C₅	semidirect
94. C₅×C₈	#102	semidirect
146. C₂×C₈	C₅×D₈	semidirect
180. C₈	C₅×#102	semidirect
181. C₅	#9824	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#493	2
C₅×#1957	2
C₅×#1845	2
C₂×C₅×C₈²	2
#9824	5

Subgroups — C5×#9824 #11932544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9824 #11932544