Subgroups — C₅×#8784 #11931504

Order 1280.

422 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	C₅×#1842	C₂	1	2	yes	maximal
2	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#1277	C₂	1	4	yes	maximal
4	C₂×C₅×C₄⋊C₈	—	2	2	no	—
5	C₂×C₅×C₄⋊C₈	—	2	2	no	—
6	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
7	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
11	C₄³×C₅	C₂²	1	1	yes	characteristic
12	#8784	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₅×C₈	—	2	4	no	—
14	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
15	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
23	C₅×C₄⋊C₈	—	2	8	no	—
24	C₅×C₄⋊C₈	—	4	4	no	—
25	C₅×C₄⋊C₈	—	4	4	no	—
26	C₄×C₅×C₈	—	2	4	no	—
27	C₄×C₅×C₈	—	2	2	no	—
28	C₄×C₅×C₈	—	2	4	no	—
29	C₄×C₅×C₈	C₈	1	4	yes	complemented
30	#1842	C₂×C₅	1	2	yes	—
31	C₄²×C₈	C₂×C₅	1	1	yes	characteristic
32	#1277	C₂×C₅	1	4	yes	—
33	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₅×C₈	—	2	2	no	—
41	C₂×C₅×C₈	—	2	2	no	—
42	C₂×C₅×C₈	—	2	2	no	—
43	C₂×C₅×C₈	—	2	2	no	—
44	C₂×C₅×C₈	—	4	4	no	—
45	C₂×C₅×C₈	—	4	4	no	—
46	C₂×C₅×C₈	—	4	4	no	—
47	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
48	C₂×C₅×C₈	M₁₆	1	4	yes	—
49	C₄²×C₅	—	2	2	no	—
50	C₄²×C₅	—	2	2	no	—
51	C₄²×C₅	—	2	2	no	—
52	C₄²×C₅	C₂×C₈	1	2	yes	—
53	C₄²×C₅	M₁₆	1	2	yes	—
54	C₄²×C₅	D₈	1	2	yes	—
55	C₄²×C₅	SD₁₆	1	4	yes	—
56	C₄²×C₅	Q₁₆	1	2	yes	—
57	C₄²×C₅	#41	1	4	yes	—
58	C₂×C₄⋊C₈	—	2	2	no	—
59	C₂×C₄⋊C₈	—	2	2	no	—
60	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
61	C₂×C₄×C₈	C₄×C₅	1	2	yes	—
62	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
63	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
65	C₄³	C₂²×C₅	1	1	yes	characteristic
66	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅	—	2	1	no	—
68	C₂×C₄×C₅	—	2	1	no	—
69	C₂×C₄×C₅	—	2	1	no	—
70	C₂×C₄×C₅	—	2	1	no	—
71	C₂×C₄×C₅	—	2	1	no	—
72	C₂×C₄×C₅	—	2	1	no	—
73	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
75	C₂×C₄×C₅	#106	1	2	yes	—
76	C₂×C₄×C₅	#106	1	2	yes	—
77	C₂×C₄×C₅	#107	1	2	yes	—
78	C₂×C₄×C₅	#107	1	2	yes	—
79	C₂×C₄×C₅	#108	1	2	yes	—
80	C₂×C₄×C₅	#108	1	2	yes	—
81	C₂×C₄×C₅	#117	1	2	yes	—
82	C₂×C₄×C₅	#124	1	2	yes	—
83	C₂×C₄×C₅	#125	1	2	yes	—
84	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
90	C₅×C₈	—	2	4	no	—
91	C₅×C₈	—	2	4	no	—
92	C₅×C₈	—	8	4	no	—
93	C₂²×C₈	—	2	4	no	—
94	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
95	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
96	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
97	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
98	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
99	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
100	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
101	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
102	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
103	C₄⋊C₈	—	2	8	no	—
104	C₄⋊C₈	—	4	4	no	—
105	C₄⋊C₈	—	4	4	no	—
106	C₄×C₈	—	2	4	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	—	2	4	no	—
109	C₄×C₈	C₅×C₈	1	4	yes	complemented
110	C₂²×C₅	#382	1	1	yes	characteristic
111	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂²×C₅	#423	1	1	yes	characteristic
113	C₂²×C₅	C₂×#106	1	1	yes	characteristic
114	C₂²×C₅	C₂×#107	1	1	yes	characteristic
115	C₂²×C₅	C₂×#108	1	1	yes	characteristic
116	C₂²×C₅	C₂×#108	1	1	yes	characteristic
117	C₄×C₅	—	2	2	no	—
118	C₄×C₅	—	2	2	no	—
119	C₄×C₅	—	2	2	no	—
120	C₄×C₅	#334	1	4	yes	—
121	C₄×C₅	#335	1	4	yes	—
122	C₄×C₅	#431	1	2	yes	—
123	C₄×C₅	#432	1	2	yes	—
124	C₄×C₅	#486	1	2	yes	—
125	C₄×C₅	#487	1	2	yes	—
126	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	4	4	no	—
138	C₂×C₈	—	4	4	no	—
139	C₂×C₈	—	4	4	no	—
140	C₂×C₈	C₂×C₅×C₈	1	4	yes	—
141	C₂×C₈	C₅×M₁₆	1	4	yes	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	C₂×C₅×C₈	1	2	yes	—
146	C₄²	C₅×M₁₆	1	2	yes	—
147	C₄²	C₅×D₈	1	2	yes	—
148	C₄²	C₅×SD₁₆	1	4	yes	—
149	C₄²	C₅×Q₁₆	1	2	yes	—
150	C₄²	C₅×#41	1	4	yes	—
151	C₂×C₅	C₂×#334	1	1	yes	characteristic
152	C₂×C₅	C₂×#335	1	1	yes	characteristic
153	C₂×C₅	#1566	1	1	yes	characteristic
154	C₂×C₅	#1567	1	1	yes	characteristic
155	C₂×C₅	#1831	1	1	yes	characteristic
156	C₂×C₅	#1834	1	1	yes	characteristic
157	C₂×C₅	#1837	1	1	yes	characteristic
158	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	C₅×C₄⋊C₈	1	2	yes	—
166	C₂×C₄	C₅×C₄⋊C₈	1	2	yes	—
167	C₂×C₄	C₅×#106	1	2	yes	—
168	C₂×C₄	C₅×#106	1	2	yes	—
169	C₂×C₄	C₅×#107	1	2	yes	—
170	C₂×C₄	C₅×#107	1	2	yes	—
171	C₂×C₄	C₅×#108	1	2	yes	—
172	C₂×C₄	C₅×#108	1	2	yes	—
173	C₂×C₄	C₅×#117	1	2	yes	—
174	C₂×C₄	C₅×#124	1	2	yes	—
175	C₂×C₄	C₅×#125	1	2	yes	—
176	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
177	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
178	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
179	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
180	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
181	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
182	C₈	—	2	4	no	—
183	C₈	—	2	4	no	—
184	C₈	—	8	4	no	—
185	C₅	#8784	1	1	yes	characteristic, direct product factor, Sylow
186	C₂²	C₅×#382	1	1	yes	characteristic
187	C₂²	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
188	C₂²	C₅×#423	1	1	yes	characteristic
189	C₂²	C₂×C₅×#106	1	1	yes	characteristic
190	C₂²	C₂×C₅×#107	1	1	yes	characteristic
191	C₂²	C₂×C₅×#108	1	1	yes	characteristic
192	C₂²	C₂×C₅×#108	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	C₅×#334	1	4	yes	—
197	C₄	C₅×#335	1	4	yes	—
198	C₄	C₅×#431	1	2	yes	—
199	C₄	C₅×#432	1	2	yes	—
200	C₄	C₅×#486	1	2	yes	—
201	C₄	C₅×#487	1	2	yes	—
202	C₂	C₂×C₅×#334	1	1	yes	characteristic
203	C₂	C₂×C₅×#335	1	1	yes	characteristic
204	C₂	C₅×#1566	1	1	yes	characteristic
205	C₂	C₅×#1567	1	1	yes	characteristic
206	C₂	C₅×#1831	1	1	yes	characteristic
207	C₂	C₅×#1834	1	1	yes	characteristic
208	C₂	C₅×#1837	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
12. #8784	C₅	direct (N × H)
29. C₄×C₅×C₈	C₈	semidirect
109. C₄×C₈	C₅×C₈	semidirect
185. C₅	#8784	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#1842	2
C₄²×C₅×C₈	2
C₅×#1277	2
#8784	5

Subgroups — C5×#8784 #11931504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8784 #11931504