Subgroups — C₅×#8038 #11930758

Order 1280.

650 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	C₂×C₅×#431	C₂	1	2	yes	maximal
2	C₅×#2285	C₂	1	1	yes	maximal, characteristic
3	C₅×#1296	C₂	1	4	yes	maximal, complemented
4	C₂×C₅×#117	C₄	1	2	yes	—
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₅×#431	—	2	4	no	—
7	C₅×#431	C₄	1	8	yes	complemented
8	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
9	C₂×C₅×C₂²⋊C₈	—	2	2	no	—
10	C₂×C₄×C₅×C₈	—	2	2	no	—
11	C₅×#384	C₂²	1	2	yes	—
12	C₅×#382	C₄	1	2	yes	—
13	C₅×#380	—	2	2	no	—
14	C₄×C₅×C₄⋊C₄	—	2	2	no	—
15	#8038	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₅	D₄	1	2	yes	—
18	C₂²×C₅×C₈	—	2	4	no	—
19	C₅×#117	—	4	4	no	—
20	C₂×C₅×C₄⋊C₄	—	2	4	no	—
21	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
22	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
23	C₂×C₄²×C₅	—	2	2	no	—
24	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₅	D₄	1	2	yes	—
27	C₂×C₄²×C₅	D₄	1	2	yes	—
28	C₂×C₄²×C₅	Q₈	1	2	yes	—
29	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₅×C₄⋊C₈	—	2	4	no	—
31	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
32	C₅×C₂²⋊C₈	—	2	8	no	—
33	C₄×C₅×C₈	—	2	8	no	—
34	C₅×#95	—	2	4	no	—
35	C₂×#431	C₂×C₅	1	2	yes	—
36	#2285	C₂×C₅	1	1	yes	characteristic
37	#1296	C₂×C₅	1	4	yes	complemented
38	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅	—	2	2	no	—
40	C₂²×C₄×C₅	—	2	2	no	—
41	C₂²×C₄×C₅	—	2	8	no	—
42	C₂²×C₄×C₅	—	2	4	no	—
43	C₂²×C₄×C₅	C₄²	1	2	yes	—
44	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
46	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
51	C₂×C₅×C₈	—	2	8	no	—
52	C₂×C₅×C₈	—	4	8	no	—
53	C₅×C₄⋊C₄	—	4	4	no	—
54	C₅×C₄⋊C₄	—	4	8	no	—
55	C₅×C₂²⋊C₄	—	4	4	no	—
56	C₄²×C₅	—	4	4	no	—
57	C₄²×C₅	C₄²	1	2	yes	—
58	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
59	C₄²×C₅	C₄⋊C₄	1	2	yes	—
60	C₄²×C₅	C₄⋊C₄	1	4	yes	—
61	C₄²×C₅	C₂²×C₄	1	2	yes	—
62	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₂×#117	C₄×C₅	1	2	yes	—
65	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
66	#431	—	2	4	no	—
67	#431	C₄×C₅	1	8	yes	complemented
68	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
69	C₂×C₂²⋊C₈	—	2	2	no	—
70	C₂×C₄×C₈	—	2	2	no	—
71	#384	C₂²×C₅	1	2	yes	—
72	#382	C₄×C₅	1	2	yes	—
73	#380	—	2	2	no	—
74	C₄×C₄⋊C₄	—	2	2	no	—
75	C₂³×C₅	—	2	1	no	—
76	C₂³×C₅	—	2	1	no	—
77	C₂³×C₅	—	2	4	no	—
78	C₂³×C₅	#95	1	2	yes	—
79	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅	—	2	1	no	—
81	C₂×C₄×C₅	—	2	1	no	—
82	C₂×C₄×C₅	—	2	4	no	—
83	C₂×C₄×C₅	—	2	4	no	—
84	C₂×C₄×C₅	—	2	4	no	—
85	C₂×C₄×C₅	—	2	8	no	—
86	C₂×C₄×C₅	—	2	8	no	—
87	C₂×C₄×C₅	—	4	4	no	—
88	C₂×C₄×C₅	—	4	8	no	—
89	C₂×C₄×C₅	#95	1	4	yes	—
90	C₂×C₄×C₅	#95	1	2	yes	—
91	C₂×C₄×C₅	#100	1	2	yes	—
92	C₂×C₄×C₅	#101	1	2	yes	—
93	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
95	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₅	#135	1	8	yes	—
98	C₅×C₈	—	4	8	no	—
99	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
100	C₂³×C₄	C₅×D₄	1	2	yes	—
101	C₂²×C₈	—	2	4	no	—
102	#117	—	4	4	no	—
103	C₂×C₄⋊C₄	—	2	4	no	—
104	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
105	C₂×C₂²⋊C₄	—	2	2	no	—
106	C₂×C₄²	—	2	2	no	—
107	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
108	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
109	C₂×C₄²	C₅×D₄	1	2	yes	—
110	C₂×C₄²	C₅×D₄	1	2	yes	—
111	C₂×C₄²	C₅×Q₈	1	2	yes	—
112	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	—
115	C₂²⋊C₈	—	2	8	no	—
116	C₄×C₈	—	2	8	no	—
117	#95	—	2	4	no	—
118	C₂²×C₅	—	2	4	no	—
119	C₂²×C₅	—	2	2	no	—
120	C₂²×C₅	—	4	2	no	—
121	C₂²×C₅	—	4	2	no	—
122	C₂²×C₅	C₂×#95	1	1	yes	characteristic
123	C₂²×C₅	C₂×#100	1	1	yes	characteristic
124	C₂²×C₅	C₂×#101	1	1	yes	characteristic
125	C₂²×C₅	#416	1	4	yes	—
126	C₄×C₅	—	2	4	no	—
127	C₄×C₅	—	2	1	no	—
128	C₄×C₅	—	2	1	no	—
129	C₄×C₅	—	8	4	no	—
130	C₄×C₅	#343	1	4	yes	—
131	C₄×C₅	#416	1	4	yes	—
132	C₄×C₅	#427	1	4	yes	—
133	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₄	—	2	8	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	C₄²×C₅	1	2	yes	—
139	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
140	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
141	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
143	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
144	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
146	C₂×C₈	—	2	8	no	—
147	C₂×C₈	—	4	8	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	8	no	—
150	C₂²⋊C₄	—	4	4	no	—
151	C₄²	—	4	4	no	—
152	C₄²	C₄²×C₅	1	2	yes	—
153	C₄²	C₅×C₂²⋊C₄	1	4	yes	—
154	C₄²	C₅×C₄⋊C₄	1	2	yes	—
155	C₄²	C₅×C₄⋊C₄	1	4	yes	—
156	C₄²	C₂²×C₄×C₅	1	2	yes	—
157	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
158	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
159	C₂×C₅	—	4	2	no	—
160	C₂×C₅	#1543	1	2	yes	—
161	C₂×C₅	#1544	1	2	yes	—
162	C₂×C₅	#1735	1	2	yes	—
163	C₂×C₅	C₂×#343	1	1	yes	characteristic
164	C₂³	—	2	1	no	—
165	C₂³	—	2	1	no	—
166	C₂³	—	2	4	no	—
167	C₂³	C₅×#95	1	2	yes	—
168	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	8	no	—
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	—	4	8	no	—
178	C₂×C₄	C₅×#95	1	4	yes	—
179	C₂×C₄	C₅×#95	1	2	yes	—
180	C₂×C₄	C₅×#100	1	2	yes	—
181	C₂×C₄	C₅×#101	1	2	yes	—
182	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
183	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
184	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
185	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
186	C₂×C₄	C₅×#135	1	8	yes	—
187	C₈	—	4	8	no	—
188	C₅	#8038	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²	—	2	4	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	C₂×C₅×#95	1	1	yes	characteristic
194	C₂²	C₂×C₅×#100	1	1	yes	characteristic
195	C₂²	C₂×C₅×#101	1	1	yes	characteristic
196	C₂²	C₅×#416	1	4	yes	—
197	C₄	—	2	4	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	8	4	no	—
201	C₄	C₅×#343	1	4	yes	—
202	C₄	C₅×#416	1	4	yes	—
203	C₄	C₅×#427	1	4	yes	—
204	C₂	—	4	2	no	—
205	C₂	C₅×#1543	1	2	yes	—
206	C₂	C₅×#1544	1	2	yes	—
207	C₂	C₅×#1735	1	2	yes	—
208	C₂	C₂×C₅×#343	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₅×#1296	C₂	semidirect
7. C₅×#431	C₄	semidirect
15. #8038	C₅	direct (N × H)
37. #1296	C₂×C₅	semidirect
67. #431	C₄×C₅	semidirect
188. C₅	#8038	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×#431	2
C₅×#2285	2
C₅×#1296	2
#8038	5

Subgroups — C5×#8038 #11930758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8038 #11930758