Subgroups — C₅×#22276 #11944996

Order 1280.

1050 conjugacy classes of subgroups.

The subgroup lattice has 206 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×#2328	C₂	1	2	yes	maximal, complemented
3	C₅×#1961	C₂	1	2	yes	maximal, complemented
4	C₅×#1879	C₂	1	8	yes	maximal, complemented
5	C₅×#1835	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1746	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₅×D₈	C₂²	1	4	yes	—
8	C₂×C₅×#127	C₂²	1	2	yes	—
9	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
10	C₅×#493	C₄	1	8	yes	complemented
11	C₂×C₅×#102	C₂²	1	8	yes	—
12	C₂×C₅×#97	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×#390	C₂²	1	4	yes	—
15	C₅×#386	—	2	8	no	—
16	C₅×#382	C₂²	1	2	yes	—
17	C₅×#376	C₂²	1	1	yes	characteristic
18	C₅×#340	C₂²	1	4	yes	—
19	C₅×#336	C₂²	1	4	yes	—
20	#22276	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×D₄	—	2	4	no	—
22	C₂²×C₅×D₄	C₂³	1	4	yes	—
23	C₂³×C₄×C₅	—	2	4	no	—
24	C₂×C₅×D₈	—	4	8	no	—
25	C₂×C₅×D₈	C₂×C₄	1	16	yes	complemented
26	C₂²×C₅×C₈	C₂³	1	2	yes	—
27	C₂²×C₅×C₈	C₂³	1	2	yes	—
28	C₅×#127	—	2	4	no	—
29	C₅×#127	C₂×C₄	1	8	yes	complemented
30	C₄×C₅×D₄	—	4	8	no	—
31	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
32	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
33	C₂×C₅×C₂²⋊C₄	—	2	8	no	—
34	C₂×C₄²×C₅	C₂³	1	2	yes	—
35	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₅×#102	—	4	8	no	—
37	C₅×#97	—	2	2	no	—
38	C₄×C₅×C₈	C₂×C₄	1	4	yes	complemented
39	C₅×#95	—	2	4	no	—
40	C₅×#95	D₄	1	4	yes	—
41	C₂×#493	C₂×C₅	1	1	yes	characteristic
42	#2328	C₂×C₅	1	2	yes	complemented
43	#1961	C₂×C₅	1	2	yes	complemented
44	#1879	C₂×C₅	1	8	yes	complemented
45	#1835	C₂×C₅	1	1	yes	characteristic, complemented
46	#1746	C₂×C₅	1	1	yes	characteristic, complemented
47	C₂⁴×C₅	—	2	4	no	—
48	C₂×C₅×D₄	—	2	16	no	—
49	C₂×C₅×D₄	—	2	4	no	—
50	C₂×C₅×D₄	—	2	8	no	—
51	C₂×C₅×D₄	—	2	8	no	—
52	C₂×C₅×D₄	—	8	8	no	—
53	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
54	C₂²×C₄×C₅	—	2	4	no	—
55	C₂²×C₄×C₅	—	4	8	no	—
56	C₂²×C₄×C₅	—	8	8	no	—
57	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
60	C₅×D₈	—	4	16	no	—
61	C₂×C₅×C₈	—	2	4	no	—
62	C₂×C₅×C₈	—	2	2	no	—
63	C₂×C₅×C₈	—	2	4	no	—
64	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
65	C₅×C₄⋊C₄	—	2	8	no	—
66	C₅×C₂²⋊C₄	—	4	8	no	—
67	C₅×C₂²⋊C₄	—	8	8	no	—
68	C₄²×C₅	—	2	4	no	—
69	C₄²×C₅	C₂²×C₄	1	2	yes	—
70	C₄²×C₅	#41	1	2	yes	—
71	C₂²×D₈	C₂²×C₅	1	4	yes	—
72	C₂×#127	C₂²×C₅	1	2	yes	—
73	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
74	#493	C₄×C₅	1	8	yes	complemented
75	C₂×#102	C₂²×C₅	1	8	yes	—
76	C₂×#97	C₂²×C₅	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
78	#390	C₂²×C₅	1	4	yes	—
79	#386	—	2	8	no	—
80	#382	C₂²×C₅	1	2	yes	—
81	#376	C₂²×C₅	1	1	yes	characteristic
82	#340	C₂²×C₅	1	4	yes	—
83	#336	C₂²×C₅	1	4	yes	—
84	C₂³×C₅	—	2	8	no	—
85	C₂³×C₅	—	4	4	no	—
86	C₂³×C₅	—	4	4	no	—
87	C₂³×C₅	—	4	8	no	—
88	C₂³×C₅	—	4	8	no	—
89	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
90	C₅×D₄	—	2	16	no	—
91	C₅×D₄	—	4	8	no	—
92	C₅×D₄	—	8	16	no	—
93	C₂×C₄×C₅	—	2	4	no	—
94	C₂×C₄×C₅	—	2	4	no	—
95	C₂×C₄×C₅	—	2	8	no	—
96	C₂×C₄×C₅	—	4	4	no	—
97	C₂×C₄×C₅	—	4	8	no	—
98	C₂×C₄×C₅	—	8	16	no	—
99	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
102	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
103	C₅×C₈	—	2	4	no	—
104	C₅×C₈	—	4	2	no	—
105	C₂²×D₄	—	2	4	no	—
106	C₂²×D₄	C₂³×C₅	1	4	yes	—
107	C₂³×C₄	—	2	4	no	—
108	C₂×D₈	—	4	8	no	—
109	C₂×D₈	C₂×C₄×C₅	1	16	yes	complemented
110	C₂²×C₈	C₂³×C₅	1	2	yes	—
111	C₂²×C₈	C₂³×C₅	1	2	yes	—
112	#127	—	2	4	no	—
113	#127	C₂×C₄×C₅	1	8	yes	complemented
114	C₄×D₄	—	4	8	no	—
115	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
116	C₂×C₂²⋊C₄	—	2	4	no	—
117	C₂×C₂²⋊C₄	—	2	8	no	—
118	C₂×C₄²	C₂³×C₅	1	2	yes	—
119	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
120	#102	—	4	8	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	C₂×C₄×C₅	1	4	yes	complemented
123	#95	—	2	4	no	—
124	#95	C₅×D₄	1	4	yes	—
125	C₂²×C₅	—	4	8	no	—
126	C₂²×C₅	—	4	16	no	—
127	C₂²×C₅	—	8	4	no	—
128	C₂²×C₅	—	8	4	no	—
129	C₂²×C₅	—	8	8	no	—
130	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₅	#535	1	2	yes	—
132	C₂²×C₅	#576	1	4	yes	—
133	C₄×C₅	—	2	4	no	—
134	C₄×C₅	—	4	4	no	—
135	C₄×C₅	—	8	4	no	—
136	C₄×C₅	#518	1	2	yes	—
137	C₄×C₅	#540	1	2	yes	—
138	C₂⁴	—	2	4	no	—
139	C₂×D₄	—	2	16	no	—
140	C₂×D₄	—	2	4	no	—
141	C₂×D₄	—	2	8	no	—
142	C₂×D₄	—	2	8	no	—
143	C₂×D₄	—	8	8	no	—
144	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	4	8	no	—
147	C₂²×C₄	—	8	8	no	—
148	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
149	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
150	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
151	D₈	—	4	16	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
156	C₄⋊C₄	—	2	8	no	—
157	C₂²⋊C₄	—	4	8	no	—
158	C₂²⋊C₄	—	8	8	no	—
159	C₄²	—	2	4	no	—
160	C₄²	C₂²×C₄×C₅	1	2	yes	—
161	C₄²	C₅×#41	1	2	yes	—
162	C₂×C₅	—	8	8	no	—
163	C₂×C₅	#2328	1	1	yes	characteristic
164	C₂×C₅	#2948	1	2	yes	—
165	C₂×C₅	#3220	1	4	yes	—
166	C₂³	—	2	8	no	—
167	C₂³	—	4	4	no	—
168	C₂³	—	4	4	no	—
169	C₂³	—	4	8	no	—
170	C₂³	—	4	8	no	—
171	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
172	D₄	—	2	16	no	—
173	D₄	—	4	8	no	—
174	D₄	—	8	16	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	4	8	no	—
180	C₂×C₄	—	8	16	no	—
181	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
182	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
183	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
184	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
185	C₈	—	2	4	no	—
186	C₈	—	4	2	no	—
187	C₅	#22276	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²	—	4	8	no	—
189	C₂²	—	4	16	no	—
190	C₂²	—	8	4	no	—
191	C₂²	—	8	4	no	—
192	C₂²	—	8	8	no	—
193	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
194	C₂²	C₅×#535	1	2	yes	—
195	C₂²	C₅×#576	1	4	yes	—
196	C₄	—	2	4	no	—
197	C₄	—	4	4	no	—
198	C₄	—	8	4	no	—
199	C₄	C₅×#518	1	2	yes	—
200	C₄	C₅×#540	1	2	yes	—
201	C₂	—	8	8	no	—
202	C₂	C₅×#2328	1	1	yes	characteristic
203	C₂	C₅×#2948	1	2	yes	—
204	C₂	C₅×#3220	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#493	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2328	C₂	semidirect
3. C₅×#1961	C₂	semidirect
4. C₅×#1879	C₂	semidirect
5. C₅×#1835	C₂	semidirect
6. C₅×#1746	C₂	semidirect
10. C₅×#493	C₄	semidirect
20. #22276	C₅	direct (N × H)
25. C₂×C₅×D₈	C₂×C₄	semidirect
29. C₅×#127	C₂×C₄	semidirect
38. C₄×C₅×C₈	C₂×C₄	semidirect
42. #2328	C₂×C₅	semidirect
43. #1961	C₂×C₅	semidirect
44. #1879	C₂×C₅	semidirect
45. #1835	C₂×C₅	semidirect
46. #1746	C₂×C₅	semidirect
74. #493	C₄×C₅	semidirect
109. C₂×D₈	C₂×C₄×C₅	semidirect
113. #127	C₂×C₄×C₅	semidirect
122. C₄×C₈	C₂×C₄×C₅	semidirect
187. C₅	#22276	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×#2328	2
C₅×#1961	2
C₅×#1879	2
C₅×#1835	2
C₅×#1746	2
#22276	5

Subgroups — C5×#22276 #11944996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22276 #11944996