Subgroups — C₅×#12868 #11935588

Order 1280.

426 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#1808	C₂	1	1	yes	maximal, characteristic
4	C₅×#1380	C₂	1	2	yes	maximal, complemented
5	C₅×#1380	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×#117	—	2	1	no	—
8	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
9	C₅×#427	—	2	2	no	—
10	C₅×#427	C₄	1	4	yes	complemented
11	C₂×C₅×#107	C₂²	1	2	yes	—
12	C₂×C₅×#107	C₂²	1	1	yes	characteristic
13	C₂×C₅×#107	C₂²	1	1	yes	characteristic
14	C₂×C₅×#106	—	2	1	no	—
15	C₅×#380	—	2	1	no	—
16	C₅×C₂²⋊C₁₆	—	2	2	no	—
17	C₅×#340	—	2	1	no	—
18	#12868	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	—	2	4	no	—
21	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	D₄	1	2	yes	—
24	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
25	C₅×#117	—	2	4	no	—
26	C₅×#117	—	4	2	no	—
27	C₂×C₅×C₄⋊C₄	—	2	1	no	—
28	C₂×C₅×C₄⋊C₄	—	2	1	no	—
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
31	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
32	C₂×C₄²×C₅	—	4	1	no	—
33	C₂×C₅×C₁₆	—	2	2	no	—
34	C₂×C₅×C₁₆	—	2	4	no	—
35	C₅×#107	—	2	1	no	—
36	C₅×#107	—	2	1	no	—
37	C₅×#107	—	2	4	no	—
38	C₅×#107	C₂×C₄	1	2	yes	—
39	C₅×#107	C₂×C₄	1	2	yes	—
40	C₅×#106	—	2	4	no	—
41	C₅×#95	—	4	1	no	—
42	C₂×#427	C₂×C₅	1	1	yes	characteristic
43	C₂×C₂²⋊C₁₆	C₂×C₅	1	1	yes	characteristic
44	#1808	C₂×C₅	1	1	yes	characteristic
45	#1380	C₂×C₅	1	2	yes	complemented
46	#1380	C₂×C₅	1	2	yes	complemented
47	C₂⁴×C₅	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₅	—	2	2	no	—
49	C₂²×C₄×C₅	—	2	2	no	—
50	C₂²×C₄×C₅	—	4	1	no	—
51	C₂²×C₄×C₅	—	4	1	no	—
52	C₂²×C₄×C₅	—	4	2	no	—
53	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
56	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	2	4	no	—
59	C₂×C₅×C₈	—	2	4	no	—
60	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
61	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₅×C₈	#41	1	4	yes	—
64	C₅×C₄⋊C₄	—	2	2	no	—
65	C₅×C₄⋊C₄	—	2	2	no	—
66	C₅×C₄⋊C₄	—	4	1	no	—
67	C₅×C₄⋊C₄	—	4	1	no	—
68	C₅×C₄⋊C₄	—	4	2	no	—
69	C₅×C₄⋊C₄	—	4	2	no	—
70	C₅×C₂²⋊C₄	—	4	4	no	—
71	C₅×C₂²⋊C₄	—	8	2	no	—
72	C₄²×C₅	—	4	4	no	—
73	C₅×C₁₆	—	4	2	no	—
74	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
75	C₂×#117	—	2	1	no	—
76	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
77	#427	—	2	2	no	—
78	#427	C₄×C₅	1	4	yes	complemented
79	C₂×#107	C₂²×C₅	1	2	yes	—
80	C₂×#107	C₂²×C₅	1	1	yes	characteristic
81	C₂×#107	C₂²×C₅	1	1	yes	characteristic
82	C₂×#106	—	2	1	no	—
83	#380	—	2	1	no	—
84	C₂²⋊C₁₆	—	2	2	no	—
85	#340	—	2	1	no	—
86	C₂³×C₅	—	2	2	no	—
87	C₂³×C₅	—	2	2	no	—
88	C₂³×C₅	—	2	1	no	—
89	C₂³×C₅	—	2	1	no	—
90	C₂³×C₅	#102	1	2	yes	—
91	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₅	—	2	2	no	—
93	C₂×C₄×C₅	—	2	2	no	—
94	C₂×C₄×C₅	—	2	2	no	—
95	C₂×C₄×C₅	—	2	4	no	—
96	C₂×C₄×C₅	—	4	2	no	—
97	C₂×C₄×C₅	—	4	2	no	—
98	C₂×C₄×C₅	—	8	2	no	—
99	C₂×C₄×C₅	—	8	1	no	—
100	C₂×C₄×C₅	—	8	1	no	—
101	C₂×C₄×C₅	—	8	2	no	—
102	C₂×C₄×C₅	—	8	2	no	—
103	C₂×C₄×C₅	—	8	2	no	—
104	C₂×C₄×C₅	#102	1	2	yes	—
105	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅	#123	1	2	yes	—
107	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅	#137	1	2	yes	—
109	C₅×C₈	—	2	2	no	—
110	C₅×C₈	#117	1	2	yes	—
111	C₅×C₈	#123	1	2	yes	—
112	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
113	C₂²×C₈	—	2	4	no	—
114	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
115	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
116	C₂²×C₈	C₅×D₄	1	2	yes	—
117	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
118	#117	—	2	4	no	—
119	#117	—	4	2	no	—
120	C₂×C₄⋊C₄	—	2	1	no	—
121	C₂×C₄⋊C₄	—	2	1	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₂²⋊C₄	—	4	1	no	—
124	C₂×C₂²⋊C₄	—	4	1	no	—
125	C₂×C₄²	—	4	1	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	4	no	—
128	#107	—	2	1	no	—
129	#107	—	2	1	no	—
130	#107	—	2	4	no	—
131	#107	C₂×C₄×C₅	1	2	yes	—
132	#107	C₂×C₄×C₅	1	2	yes	—
133	#106	—	2	4	no	—
134	#95	—	4	1	no	—
135	C₂²×C₅	—	2	2	no	—
136	C₂²×C₅	—	2	2	no	—
137	C₂²×C₅	—	2	2	no	—
138	C₂²×C₅	—	4	2	no	—
139	C₂²×C₅	—	4	2	no	—
140	C₂²×C₅	C₂×#102	1	1	yes	characteristic
141	C₂²×C₅	#480	1	2	yes	—
142	C₂²×C₅	#508	1	2	yes	—
143	C₂²×C₅	#509	1	1	yes	characteristic
144	C₂²×C₅	#510	1	1	yes	characteristic
145	C₄×C₅	—	2	2	no	—
146	C₄×C₅	—	8	2	no	—
147	C₄×C₅	—	8	2	no	—
148	C₄×C₅	—	16	2	no	—
149	C₄×C₅	#381	1	1	yes	characteristic
150	C₄×C₅	#419	1	1	yes	characteristic
151	C₄×C₅	#483	1	2	yes	—
152	C₂⁴	C₅×D₈	1	1	yes	characteristic
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	4	1	no	—
156	C₂²×C₄	—	4	1	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
159	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
160	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
161	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
166	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
167	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
168	C₂×C₈	C₅×#41	1	4	yes	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	4	1	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₂²⋊C₄	—	4	4	no	—
176	C₂²⋊C₄	—	8	2	no	—
177	C₄²	—	4	4	no	—
178	C₁₆	—	4	2	no	—
179	C₂×C₅	—	4	2	no	—
180	C₂×C₅	#1823	1	1	yes	characteristic
181	C₂×C₅	#2138	1	1	yes	characteristic
182	C₂×C₅	#2139	1	1	yes	characteristic
183	C₂×C₅	#2234	1	2	yes	—
184	C₂×C₅	#2237	1	2	yes	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	1	no	—
188	C₂³	—	2	1	no	—
189	C₂³	C₅×#102	1	2	yes	—
190	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	—	8	1	no	—
199	C₂×C₄	—	8	1	no	—
200	C₂×C₄	—	8	2	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	C₅×#102	1	2	yes	—
204	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
205	C₂×C₄	C₅×#123	1	2	yes	—
206	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
207	C₂×C₄	C₅×#137	1	2	yes	—
208	C₈	—	2	2	no	—
209	C₈	C₅×#117	1	2	yes	—
210	C₈	C₅×#123	1	2	yes	—
211	C₅	#12868	1	1	yes	characteristic, direct product factor, Sylow
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	C₂×C₅×#102	1	1	yes	characteristic
218	C₂²	C₅×#480	1	2	yes	—
219	C₂²	C₅×#508	1	2	yes	—
220	C₂²	C₅×#509	1	1	yes	characteristic
221	C₂²	C₅×#510	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	8	2	no	—
224	C₄	—	8	2	no	—
225	C₄	—	16	2	no	—
226	C₄	C₅×#381	1	1	yes	characteristic
227	C₄	C₅×#419	1	1	yes	characteristic
228	C₄	C₅×#483	1	2	yes	—
229	C₂	—	4	2	no	—
230	C₂	C₅×#1823	1	1	yes	characteristic
231	C₂	C₅×#2138	1	1	yes	characteristic
232	C₂	C₅×#2139	1	1	yes	characteristic
233	C₂	C₅×#2234	1	2	yes	—
234	C₂	C₅×#2237	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂⁴×C₅	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1380	C₂	semidirect
5. C₅×#1380	C₂	semidirect
10. C₅×#427	C₄	semidirect
18. #12868	C₅	direct (N × H)
45. #1380	C₂×C₅	semidirect
46. #1380	C₂×C₅	semidirect
78. #427	C₄×C₅	semidirect
211. C₅	#12868	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×#427	2
C₂×C₅×C₂²⋊C₁₆	2
C₅×#1808	2
C₅×#1380	2
#12868	5

Subgroups — C5×#12868 #11935588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12868 #11935588