Subgroups — C₅×#8807 #11931527

Order 1280.

422 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1831	C₂	1	2	yes	maximal
2	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#335	C₂	1	2	yes	maximal
4	C₂×C₅×#334	C₂	1	2	yes	maximal
5	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
6	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
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	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₄³×C₅	C₂²	1	1	yes	characteristic
13	C₅×#336	—	2	4	no	—
14	C₅×#335	—	2	4	no	—
15	C₅×#334	—	2	4	no	—
16	#8807	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₅×C₈	—	2	4	no	—
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	D₄	1	2	yes	—
25	C₂×C₄²×C₅	Q₈	1	2	yes	—
26	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
27	C₅×C₄⋊C₈	—	2	4	no	—
28	C₅×C₄⋊C₈	—	2	4	no	—
29	C₄×C₅×C₈	—	2	2	no	—
30	C₄×C₅×C₈	—	2	2	no	—
31	C₄×C₅×C₈	—	2	2	no	—
32	C₄×C₅×C₈	C₈	1	4	yes	complemented
33	C₄×C₅×C₈	D₄	1	2	yes	—
34	C₄×C₅×C₈	D₄	1	2	yes	—
35	C₄×C₅×C₈	Q₈	1	2	yes	—
36	C₄×C₅×C₈	Q₈	1	2	yes	—
37	#1831	C₂×C₅	1	2	yes	—
38	C₄²×C₈	C₂×C₅	1	1	yes	characteristic
39	C₂×#335	C₂×C₅	1	2	yes	—
40	C₂×#334	C₂×C₅	1	2	yes	—
41	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₅×C₈	—	2	1	no	—
47	C₂×C₅×C₈	—	2	1	no	—
48	C₂×C₅×C₈	—	2	1	no	—
49	C₂×C₅×C₈	—	2	1	no	—
50	C₂×C₅×C₈	—	4	4	no	—
51	C₂×C₅×C₈	—	4	8	no	—
52	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
53	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
54	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
55	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
56	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
57	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
58	C₂×C₅×C₈	M₁₆	1	2	yes	—
59	C₂×C₅×C₈	M₁₆	1	2	yes	—
60	C₄²×C₅	—	2	4	no	—
61	C₄²×C₅	—	2	1	no	—
62	C₄²×C₅	—	2	1	no	—
63	C₄²×C₅	—	2	4	no	—
64	C₄²×C₅	C₂×C₈	1	2	yes	—
65	C₄²×C₅	M₁₆	1	2	yes	—
66	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
69	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
71	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
72	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
73	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
74	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
77	C₄³	C₂²×C₅	1	1	yes	characteristic
78	#336	—	2	4	no	—
79	#335	—	2	4	no	—
80	#334	—	2	4	no	—
81	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	—	2	2	no	—
87	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
88	C₂×C₄×C₅	#108	1	4	yes	—
89	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅	#127	1	1	yes	characteristic
92	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅	#135	1	4	yes	—
100	C₅×C₈	—	2	2	no	—
101	C₅×C₈	—	2	2	no	—
102	C₅×C₈	—	8	4	no	—
103	C₅×C₈	C₄⋊C₈	1	4	yes	—
104	C₅×C₈	C₄⋊C₈	1	4	yes	—
105	C₂²×C₈	—	2	4	no	—
106	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
107	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
108	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
109	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
110	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
111	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
112	C₂×C₄²	C₅×D₄	1	2	yes	—
113	C₂×C₄²	C₅×Q₈	1	2	yes	—
114	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
115	C₄⋊C₈	—	2	4	no	—
116	C₄⋊C₈	—	2	4	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	C₅×C₈	1	4	yes	complemented
121	C₄×C₈	C₅×D₄	1	2	yes	—
122	C₄×C₈	C₅×D₄	1	2	yes	—
123	C₄×C₈	C₅×Q₈	1	2	yes	—
124	C₄×C₈	C₅×Q₈	1	2	yes	—
125	C₂²×C₅	#384	1	1	yes	characteristic
126	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
127	C₂²×C₅	#423	1	1	yes	characteristic
128	C₂²×C₅	#427	1	2	yes	—
129	C₂²×C₅	C₂×#108	1	2	yes	—
130	C₄×C₅	—	2	4	no	—
131	C₄×C₅	—	2	4	no	—
132	C₄×C₅	—	2	1	no	—
133	C₄×C₅	—	2	1	no	—
134	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
135	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
136	C₄×C₅	#423	1	1	yes	characteristic
137	C₄×C₅	#423	1	1	yes	characteristic
138	C₄×C₅	#495	1	2	yes	—
139	C₄×C₅	#499	1	2	yes	—
140	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
141	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
142	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	4	4	no	—
150	C₂×C₈	—	4	8	no	—
151	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
152	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
153	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
154	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
155	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
156	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
157	C₂×C₈	C₅×M₁₆	1	2	yes	—
158	C₂×C₈	C₅×M₁₆	1	2	yes	—
159	C₄²	—	2	1	no	—
160	C₄²	—	2	1	no	—
161	C₄²	—	2	4	no	—
162	C₄²	—	2	4	no	—
163	C₄²	C₂×C₅×C₈	1	2	yes	—
164	C₄²	C₅×M₁₆	1	2	yes	—
165	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
166	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
167	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
168	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
169	C₂×C₅	#1564	1	2	yes	—
170	C₂×C₅	#1568	1	2	yes	—
171	C₂×C₅	#1842	1	1	yes	characteristic
172	C₂×C₅	#1846	1	1	yes	characteristic
173	C₂×C₅	#1848	1	1	yes	characteristic
174	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	C₅×C₄⋊C₈	1	4	yes	—
181	C₂×C₄	C₅×#108	1	4	yes	—
182	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
184	C₂×C₄	C₅×#127	1	1	yes	characteristic
185	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
186	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
187	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
188	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
189	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
190	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
191	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
192	C₂×C₄	C₅×#135	1	4	yes	—
193	C₈	—	2	2	no	—
194	C₈	—	2	2	no	—
195	C₈	—	8	4	no	—
196	C₈	C₅×C₄⋊C₈	1	4	yes	—
197	C₈	C₅×C₄⋊C₈	1	4	yes	—
198	C₅	#8807	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²	C₅×#384	1	1	yes	characteristic
200	C₂²	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
201	C₂²	C₅×#423	1	1	yes	characteristic
202	C₂²	C₅×#427	1	2	yes	—
203	C₂²	C₂×C₅×#108	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	2	4	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
209	C₄	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
210	C₄	C₅×#423	1	1	yes	characteristic
211	C₄	C₅×#423	1	1	yes	characteristic
212	C₄	C₅×#495	1	2	yes	—
213	C₄	C₅×#499	1	2	yes	—
214	C₂	C₅×#1564	1	2	yes	—
215	C₂	C₅×#1568	1	2	yes	—
216	C₂	C₅×#1842	1	1	yes	characteristic
217	C₂	C₅×#1846	1	1	yes	characteristic
218	C₂	C₅×#1848	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
16. #8807	C₅	direct (N × H)
32. C₄×C₅×C₈	C₈	semidirect
120. C₄×C₈	C₅×C₈	semidirect
198. C₅	#8807	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×#1831	2
C₄²×C₅×C₈	2
C₂×C₅×#335	2
C₂×C₅×#334	2
#8807	5

Subgroups — C5×#8807 #11931527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8807 #11931527