Subgroups — C₅×#10703 #11933423

Order 1280.

1142 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₂×C₅×#340	C₂	1	4	yes	maximal
4	C₂³×C₅×C₈	C₂²	1	2	yes	—
5	C₂²×C₅×C₄⋊C₄	C₂²	1	4	yes	—
6	C₂²×C₄²×C₅	—	2	2	no	—
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₅×#97	—	2	4	no	—
9	C₂×C₅×#97	C₄	1	4	yes	complemented
10	C₄×C₅×C₄⋊C₄	—	2	4	no	—
11	C₄×C₅×C₄⋊C₄	C₄	1	8	yes	complemented
12	C₂×C₅×#95	—	2	2	no	—
13	C₅×#340	C₄	1	16	yes	complemented
14	#10703	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂³×C₄×C₅	—	2	4	no	—
16	C₂³×C₄×C₅	D₄	1	2	yes	—
17	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
18	C₂²×C₅×C₈	—	2	4	no	—
19	C₂²×C₅×C₈	—	2	4	no	—
20	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
21	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
22	C₂×C₅×C₄⋊C₄	—	2	4	no	—
23	C₂×C₅×C₄⋊C₄	—	2	8	no	—
24	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
25	C₂×C₅×C₄⋊C₄	C₂×C₄	1	16	yes	—
26	C₂×C₄²×C₅	—	2	8	no	—
27	C₂×C₄²×C₅	—	4	8	no	—
28	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅	D₄	1	4	yes	—
31	C₂×C₄²×C₅	Q₈	1	2	yes	—
32	C₅×#97	—	2	8	no	—
33	C₅×#95	—	2	8	no	—
34	C₂²×#97	C₂×C₅	1	1	yes	characteristic
35	C₂×C₄×C₄⋊C₄	C₂×C₅	1	2	yes	—
36	C₂×#340	C₂×C₅	1	4	yes	—
37	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅	—	2	4	no	—
39	C₂²×C₄×C₅	—	2	4	no	—
40	C₂²×C₄×C₅	—	2	8	no	—
41	C₂²×C₄×C₅	—	2	16	no	—
42	C₂²×C₄×C₅	—	4	8	no	—
43	C₂²×C₄×C₅	—	4	4	no	—
44	C₂²×C₄×C₅	—	4	4	no	—
45	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
46	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
47	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₅	#41	1	4	yes	—
54	C₂×C₅×C₈	—	2	4	no	—
55	C₂×C₅×C₈	—	2	4	no	—
56	C₂×C₅×C₈	—	2	16	no	—
57	C₂×C₅×C₈	C₄²	1	8	yes	complemented
58	C₅×C₄⋊C₄	—	2	8	no	—
59	C₅×C₄⋊C₄	—	2	16	no	—
60	C₅×C₄⋊C₄	C₄²	1	16	yes	complemented
61	C₄²×C₅	—	4	16	no	—
62	C₄²×C₅	C₄²	1	4	yes	—
63	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
64	C₄²×C₅	C₄⋊C₄	1	8	yes	—
65	C₂³×C₈	C₂²×C₅	1	2	yes	—
66	C₂²×C₄⋊C₄	C₂²×C₅	1	4	yes	—
67	C₂²×C₄²	—	2	2	no	—
68	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
69	C₂×#97	—	2	4	no	—
70	C₂×#97	C₄×C₅	1	4	yes	complemented
71	C₄×C₄⋊C₄	—	2	4	no	—
72	C₄×C₄⋊C₄	C₄×C₅	1	8	yes	complemented
73	C₂×#95	—	2	2	no	—
74	#340	C₄×C₅	1	16	yes	complemented
75	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
76	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂³×C₅	C₄×D₄	1	4	yes	—
78	C₂³×C₅	AGL(1,ℤ/8)	1	4	yes	—
79	C₂³×C₅	#137	1	4	yes	—
80	C₂×C₄×C₅	—	2	8	no	—
81	C₂×C₄×C₅	—	2	4	no	—
82	C₂×C₄×C₅	—	2	4	no	—
83	C₂×C₄×C₅	—	2	8	no	—
84	C₂×C₄×C₅	—	2	16	no	—
85	C₂×C₄×C₅	—	4	8	no	—
86	C₂×C₄×C₅	—	4	8	no	—
87	C₂×C₄×C₅	—	4	16	no	—
88	C₂×C₄×C₅	—	4	16	no	—
89	C₂×C₄×C₅	#95	1	8	yes	—
90	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
92	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
93	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₅	#117	1	2	yes	—
96	C₂×C₄×C₅	#117	1	4	yes	—
97	C₂×C₄×C₅	C₄×D₄	1	8	yes	—
98	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
99	C₅×C₈	—	2	8	no	—
100	C₂³×C₄	—	2	4	no	—
101	C₂³×C₄	C₅×D₄	1	2	yes	—
102	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
103	C₂²×C₈	—	2	4	no	—
104	C₂²×C₈	—	2	4	no	—
105	C₂²×C₈	C₂×C₄×C₅	1	8	yes	—
106	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
107	C₂×C₄⋊C₄	—	2	4	no	—
108	C₂×C₄⋊C₄	—	2	8	no	—
109	C₂×C₄⋊C₄	C₂×C₄×C₅	1	16	yes	—
110	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
111	C₂×C₄²	—	2	8	no	—
112	C₂×C₄²	—	4	8	no	—
113	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
114	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
115	C₂×C₄²	C₅×D₄	1	4	yes	—
116	C₂×C₄²	C₅×Q₈	1	2	yes	—
117	#97	—	2	8	no	—
118	#95	—	2	8	no	—
119	C₂²×C₅	C₂×#95	1	1	yes	characteristic
120	C₂²×C₅	C₄×C₂²⋊C₄	1	4	yes	—
121	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
122	C₂²×C₅	#417	1	4	yes	—
123	C₂²×C₅	#418	1	2	yes	—
124	C₂²×C₅	#419	1	2	yes	—
125	C₂²×C₅	#428	1	4	yes	—
126	C₂²×C₅	#440	1	8	yes	—
127	C₂²×C₅	#441	1	4	yes	—
128	C₂²×C₅	#442	1	4	yes	—
129	C₄×C₅	—	2	8	no	—
130	C₄×C₅	—	4	16	no	—
131	C₄×C₅	C₄³	1	1	yes	characteristic
132	C₄×C₅	#376	1	1	yes	characteristic
133	C₄×C₅	C₄×C₂²⋊C₄	1	2	yes	—
134	C₄×C₅	C₄×C₄⋊C₄	1	4	yes	—
135	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	—	2	8	no	—
139	C₂²×C₄	—	2	16	no	—
140	C₂²×C₄	—	4	8	no	—
141	C₂²×C₄	—	4	4	no	—
142	C₂²×C₄	—	4	4	no	—
143	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
144	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
145	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
146	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
147	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
148	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
149	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
150	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
151	C₂²×C₄	C₅×#41	1	4	yes	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	16	no	—
155	C₂×C₈	C₄²×C₅	1	8	yes	complemented
156	C₄⋊C₄	—	2	8	no	—
157	C₄⋊C₄	—	2	16	no	—
158	C₄⋊C₄	C₄²×C₅	1	16	yes	complemented
159	C₄²	—	4	16	no	—
160	C₄²	C₄²×C₅	1	4	yes	—
161	C₄²	C₅×C₂²⋊C₄	1	4	yes	—
162	C₄²	C₅×C₄⋊C₄	1	8	yes	—
163	C₂×C₅	C₄×#95	1	1	yes	characteristic
164	C₂×C₅	#1736	1	2	yes	—
165	C₂×C₅	#1762	1	4	yes	—
166	C₂×C₅	#1763	1	4	yes	—
167	C₂×C₅	#1776	1	4	yes	—
168	C₂³	C₂×C₅×C₂²⋊C₄	1	2	yes	—
169	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
170	C₂³	C₄×C₅×D₄	1	4	yes	—
171	C₂³	C₅×AGL(1,ℤ/8)	1	4	yes	—
172	C₂³	C₅×#137	1	4	yes	—
173	C₂×C₄	—	2	8	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	16	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	8	no	—
180	C₂×C₄	—	4	16	no	—
181	C₂×C₄	—	4	16	no	—
182	C₂×C₄	C₅×#95	1	8	yes	—
183	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
185	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
186	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
187	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
188	C₂×C₄	C₅×#117	1	2	yes	—
189	C₂×C₄	C₅×#117	1	4	yes	—
190	C₂×C₄	C₄×C₅×D₄	1	8	yes	—
191	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
192	C₈	—	2	8	no	—
193	C₅	#10703	1	1	yes	characteristic, direct product factor, Sylow
194	C₂²	C₂×C₅×#95	1	1	yes	characteristic
195	C₂²	C₄×C₅×C₂²⋊C₄	1	4	yes	—
196	C₂²	C₄×C₅×C₄⋊C₄	1	2	yes	—
197	C₂²	C₅×#417	1	4	yes	—
198	C₂²	C₅×#418	1	2	yes	—
199	C₂²	C₅×#419	1	2	yes	—
200	C₂²	C₅×#428	1	4	yes	—
201	C₂²	C₅×#440	1	8	yes	—
202	C₂²	C₅×#441	1	4	yes	—
203	C₂²	C₅×#442	1	4	yes	—
204	C₄	—	2	8	no	—
205	C₄	—	4	16	no	—
206	C₄	C₄³×C₅	1	1	yes	characteristic
207	C₄	C₅×#376	1	1	yes	characteristic
208	C₄	C₄×C₅×C₂²⋊C₄	1	2	yes	—
209	C₄	C₄×C₅×C₄⋊C₄	1	4	yes	—
210	C₂	C₄×C₅×#95	1	1	yes	characteristic
211	C₂	C₅×#1736	1	2	yes	—
212	C₂	C₅×#1762	1	4	yes	—
213	C₂	C₅×#1763	1	4	yes	—
214	C₂	C₅×#1776	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂×D₄
commutator	C₄	C₄³×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₂×C₅×#97	C₄	semidirect
11. C₄×C₅×C₄⋊C₄	C₄	semidirect
13. C₅×#340	C₄	semidirect
14. #10703	C₅	direct (N × H)
57. C₂×C₅×C₈	C₄²	semidirect
60. C₅×C₄⋊C₄	C₄²	semidirect
70. C₂×#97	C₄×C₅	semidirect
72. C₄×C₄⋊C₄	C₄×C₅	semidirect
74. #340	C₄×C₅	semidirect
155. C₂×C₈	C₄²×C₅	semidirect
158. C₄⋊C₄	C₄²×C₅	semidirect
193. C₅	#10703	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#97	2
C₂×C₄×C₅×C₄⋊C₄	2
C₂×C₅×#340	2
#10703	5

Subgroups — C5×#10703 #11933423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10703 #11933423