Subgroups — C₅×#12844 #11935564

Order 1280.

542 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₂²⋊C₁₆	C₂	1	2	yes	maximal
3	C₅×#1374	C₂	1	4	yes	maximal, complemented
4	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
7	C₂²×C₅×C₁₆	C₂²	1	4	yes	—
8	C₂×C₄×C₅×C₈	—	2	4	no	—
9	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
11	C₅×C₂²⋊C₁₆	—	2	4	no	—
12	#12844	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂³×C₄×C₅	C₈	1	1	yes	characteristic
14	C₂³×C₄×C₅	C₈	1	1	yes	characteristic
15	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₅×C₈	—	2	2	no	—
17	C₂²×C₅×C₈	—	2	4	no	—
18	C₂²×C₅×C₈	—	2	4	no	—
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₅×C₈	D₄	1	4	yes	—
22	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₅	—	2	2	no	—
24	C₂×C₄²×C₅	—	2	2	no	—
25	C₂×C₄²×C₅	C₈	1	2	yes	—
26	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
27	C₂×C₅×C₁₆	—	2	4	no	—
28	C₂×C₅×C₁₆	—	2	8	no	—
29	C₄×C₅×C₈	—	2	4	no	—
30	C₄×C₅×C₈	—	2	4	no	—
31	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
32	C₂×C₂²⋊C₁₆	C₂×C₅	1	2	yes	—
33	#1374	C₂×C₅	1	4	yes	complemented
34	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₅	—	2	2	no	—
36	C₂²×C₄×C₅	—	2	2	no	—
37	C₂²×C₄×C₅	—	2	2	no	—
38	C₂²×C₄×C₅	—	2	4	no	—
39	C₂²×C₄×C₅	—	2	2	no	—
40	C₂²×C₄×C₅	—	2	2	no	—
41	C₂²×C₄×C₅	C₁₆	1	4	yes	complemented
42	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₅	M₁₆	1	2	yes	—
47	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₅×C₈	—	2	8	no	—
49	C₂×C₅×C₈	—	2	2	no	—
50	C₂×C₅×C₈	—	2	2	no	—
51	C₂×C₅×C₈	—	2	2	no	—
52	C₂×C₅×C₈	—	2	8	no	—
53	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
54	C₂×C₅×C₈	M₁₆	1	4	yes	—
55	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
56	C₂×C₅×C₈	#41	1	4	yes	—
57	C₄²×C₅	—	2	4	no	—
58	C₄²×C₅	—	2	2	no	—
59	C₄²×C₅	—	2	2	no	—
60	C₅×C₁₆	—	4	4	no	—
61	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
62	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
63	C₂²×C₄²	C₄×C₅	1	1	yes	characteristic
64	C₂²×C₁₆	C₂²×C₅	1	4	yes	—
65	C₂×C₄×C₈	—	2	4	no	—
66	C₂×C₄×C₈	C₄×C₅	1	2	yes	—
67	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
68	C₂²⋊C₁₆	—	2	4	no	—
69	C₂³×C₅	—	2	2	no	—
70	C₂³×C₅	—	2	2	no	—
71	C₂³×C₅	C₂²⋊C₈	1	2	yes	—
72	C₂³×C₅	C₂×C₁₆	1	2	yes	—
73	C₂³×C₅	M₃₂	1	2	yes	—
74	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₅	—	2	8	no	—
76	C₂×C₄×C₅	—	2	8	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	2	4	no	—
85	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
86	C₂×C₄×C₅	C₂×C₁₆	1	4	yes	—
87	C₂×C₄×C₅	M₃₂	1	4	yes	—
88	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅	#117	1	2	yes	—
90	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
91	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
92	C₅×C₈	—	2	2	no	—
93	C₅×C₈	—	2	4	no	—
94	C₅×C₈	#123	1	4	yes	—
95	C₂³×C₄	C₅×C₈	1	1	yes	characteristic
96	C₂³×C₄	C₅×C₈	1	1	yes	characteristic
97	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
98	C₂²×C₈	—	2	2	no	—
99	C₂²×C₈	—	2	4	no	—
100	C₂²×C₈	—	2	4	no	—
101	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
103	C₂²×C₈	C₅×D₄	1	4	yes	—
104	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
105	C₂×C₄²	—	2	2	no	—
106	C₂×C₄²	—	2	2	no	—
107	C₂×C₄²	C₅×C₈	1	2	yes	—
108	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
109	C₂×C₁₆	—	2	4	no	—
110	C₂×C₁₆	—	2	8	no	—
111	C₄×C₈	—	2	4	no	—
112	C₄×C₈	—	2	4	no	—
113	C₂²×C₅	—	2	2	no	—
114	C₂²×C₅	—	2	2	no	—
115	C₂²×C₅	—	2	4	no	—
116	C₂²×C₅	—	2	4	no	—
117	C₂²×C₅	C₂²⋊C₁₆	1	4	yes	—
118	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₅	#431	1	2	yes	—
120	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
121	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
122	C₂²×C₅	C₂×M₃₂	1	2	yes	—
123	C₄×C₅	—	2	2	no	—
124	C₄×C₅	—	2	4	no	—
125	C₄×C₅	—	2	4	no	—
126	C₄×C₅	#381	1	1	yes	characteristic
127	C₄×C₅	#407	1	1	yes	characteristic
128	C₄×C₅	#432	1	2	yes	—
129	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	C₅×C₁₆	1	4	yes	complemented
137	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
138	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
140	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
141	C₂²×C₄	C₅×M₁₆	1	2	yes	—
142	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
143	C₂×C₈	—	2	8	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	8	no	—
148	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
149	C₂×C₈	C₅×M₁₆	1	4	yes	—
150	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
151	C₂×C₈	C₅×#41	1	4	yes	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	2	no	—
154	C₄²	—	2	4	no	—
155	C₁₆	—	4	4	no	—
156	C₂×C₅	—	2	4	no	—
157	C₂×C₅	#1817	1	1	yes	characteristic
158	C₂×C₅	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
159	C₂×C₅	#2112	1	1	yes	characteristic
160	C₂×C₅	#2162	1	2	yes	—
161	C₂×C₅	#2163	1	2	yes	—
162	C₂³	—	2	2	no	—
163	C₂³	—	2	2	no	—
164	C₂³	C₅×C₂²⋊C₈	1	2	yes	—
165	C₂³	C₂×C₅×C₁₆	1	2	yes	—
166	C₂³	C₅×M₃₂	1	2	yes	—
167	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
179	C₂×C₄	C₂×C₅×C₁₆	1	4	yes	—
180	C₂×C₄	C₅×M₃₂	1	4	yes	—
181	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
182	C₂×C₄	C₅×#117	1	2	yes	—
183	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
184	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	2	4	no	—
187	C₈	C₅×#123	1	4	yes	—
188	C₅	#12844	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	C₅×C₂²⋊C₁₆	1	4	yes	—
194	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
195	C₂²	C₅×#431	1	2	yes	—
196	C₂²	C₂²×C₅×C₁₆	1	1	yes	characteristic
197	C₂²	C₂×C₅×M₃₂	1	1	yes	characteristic
198	C₂²	C₂×C₅×M₃₂	1	2	yes	—
199	C₄	—	2	2	no	—
200	C₄	—	2	4	no	—
201	C₄	—	2	4	no	—
202	C₄	C₅×#381	1	1	yes	characteristic
203	C₄	C₅×#407	1	1	yes	characteristic
204	C₄	C₅×#432	1	2	yes	—
205	C₂	—	2	4	no	—
206	C₂	C₅×#1817	1	1	yes	characteristic
207	C₂	C₂×C₅×C₂²⋊C₁₆	1	1	yes	characteristic
208	C₂	C₅×#2112	1	1	yes	characteristic
209	C₂	C₅×#2162	1	2	yes	—
210	C₂	C₅×#2163	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅×C₈	C₂³
commutator	C₂²	C₂²×C₅×C₁₆
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂⁴×C₅	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1374	C₂	semidirect
12. #12844	C₅	direct (N × H)
33. #1374	C₂×C₅	semidirect
41. C₂²×C₄×C₅	C₁₆	semidirect
136. C₂²×C₄	C₅×C₁₆	semidirect
188. C₅	#12844	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×C₄×C₅×C₈	2
C₂×C₅×C₂²⋊C₁₆	2
C₅×#1374	2
#12844	5

Subgroups — C5×#12844 #11935564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12844 #11935564