Subgroups — C₂×C₅×#1381 #11935278

Order 1280.

574 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	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#108	C₂	1	2	yes	maximal
3	C₅×#1381	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₅×M₁₆	—	2	2	no	—
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₅×C₁₆	—	2	4	no	—
7	C₂²×C₅×C₁₆	C₄	1	2	yes	—
8	C₂²×C₅×C₁₆	C₄	1	2	yes	—
9	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
10	C₂×C₅×#108	—	2	2	no	—
11	C₂×C₅×#108	—	2	4	no	—
12	C₂×C₅×#108	C₄	1	8	yes	—
13	C₂×C₅×#108	C₂²	1	4	yes	—
14	C₂×#1381	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
16	C₂×C₅×M₁₆	—	2	8	no	—
17	C₂×C₅×M₁₆	—	2	4	no	—
18	C₂×C₅×M₁₆	—	4	2	no	—
19	C₂×C₅×M₁₆	—	4	4	no	—
20	C₂²×C₅×C₈	—	4	2	no	—
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	D₄	1	4	yes	—
26	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
27	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₂×C₅×C₁₆	—	2	4	no	—
30	C₂×C₅×C₁₆	—	2	4	no	—
31	C₂×C₅×C₁₆	—	2	4	no	—
32	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
33	C₅×#108	—	2	8	no	—
34	C₅×#108	—	2	4	no	—
35	C₅×#108	C₂×C₄	1	8	yes	—
36	C₂³×C₁₆	C₂×C₅	1	1	yes	characteristic
37	C₂²×#108	C₂×C₅	1	2	yes	—
38	#1381	C₂×C₅	1	4	yes	direct product factor
39	C₂⁴×C₅	Q₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
41	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₅	SD₁₆	1	4	yes	—
46	C₂²×C₄×C₅	Q₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₅×M₁₆	—	2	8	no	—
49	C₅×M₁₆	—	4	8	no	—
50	C₅×M₁₆	—	4	4	no	—
51	C₂×C₅×C₈	—	4	8	no	—
52	C₂×C₅×C₈	—	4	4	no	—
53	C₂×C₅×C₈	C₄²	1	2	yes	—
54	C₂×C₅×C₈	C₄²	1	2	yes	—
55	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
56	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
57	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
58	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
61	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
62	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
64	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
65	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
66	C₅×C₁₆	—	2	4	no	—
67	C₂²×M₁₆	—	2	2	no	—
68	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
69	C₂²×C₁₆	—	2	4	no	—
70	C₂²×C₁₆	C₄×C₅	1	2	yes	—
71	C₂²×C₁₆	C₄×C₅	1	2	yes	—
72	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
73	C₂×#108	—	2	2	no	—
74	C₂×#108	—	2	4	no	—
75	C₂×#108	C₄×C₅	1	8	yes	—
76	C₂×#108	C₂²×C₅	1	4	yes	—
77	C₂³×C₅	—	2	4	no	—
78	C₂³×C₅	#103	1	4	yes	—
79	C₂³×C₅	#107	1	1	yes	characteristic
80	C₂³×C₅	#107	1	1	yes	characteristic
81	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅	#95	1	2	yes	—
83	C₂×C₄×C₅	#95	1	2	yes	—
84	C₂×C₄×C₅	#102	1	4	yes	—
85	C₂×C₄×C₅	#102	1	4	yes	—
86	C₂×C₄×C₅	#103	1	4	yes	—
87	C₂×C₄×C₅	#106	1	2	yes	—
88	C₂×C₄×C₅	#106	1	2	yes	—
89	C₂×C₄×C₅	#107	1	1	yes	characteristic
90	C₂×C₄×C₅	#107	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
92	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
95	C₅×C₈	—	4	8	no	—
96	C₅×C₈	#95	1	2	yes	—
97	C₅×C₈	#95	1	2	yes	—
98	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
99	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
101	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
102	C₂×M₁₆	—	2	8	no	—
103	C₂×M₁₆	—	2	4	no	—
104	C₂×M₁₆	—	4	2	no	—
105	C₂×M₁₆	—	4	4	no	—
106	C₂²×C₈	—	4	2	no	—
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	4	yes	—
110	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
111	C₂²×C₈	C₅×D₄	1	4	yes	—
112	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
113	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
114	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
115	C₂×C₁₆	—	2	4	no	—
116	C₂×C₁₆	—	2	4	no	—
117	C₂×C₁₆	—	2	4	no	—
118	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
119	#108	—	2	8	no	—
120	#108	—	2	4	no	—
121	#108	C₂×C₄×C₅	1	8	yes	—
122	C₂²×C₅	—	2	8	no	—
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	—	2	2	no	—
125	C₂²×C₅	#340	1	2	yes	—
126	C₂²×C₅	#340	1	2	yes	—
127	C₂²×C₅	#368	1	4	yes	—
128	C₂²×C₅	C₂×#103	1	2	yes	—
129	C₂²×C₅	C₂×#107	1	1	yes	characteristic
130	C₄×C₅	#340	1	2	yes	—
131	C₄×C₅	#340	1	2	yes	—
132	C₄×C₅	C₂×#95	1	1	yes	characteristic
133	C₄×C₅	C₂×#102	1	2	yes	—
134	C₄×C₅	C₂×#106	1	1	yes	characteristic
135	C₂⁴	C₅×Q₁₆	1	1	yes	characteristic
136	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
137	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
140	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
141	C₂²×C₄	C₅×SD₁₆	1	4	yes	—
142	C₂²×C₄	C₅×Q₁₆	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
144	M₁₆	—	2	8	no	—
145	M₁₆	—	4	4	no	—
146	M₁₆	—	4	8	no	—
147	C₂×C₈	—	4	4	no	—
148	C₂×C₈	—	4	8	no	—
149	C₂×C₈	C₄²×C₅	1	2	yes	—
150	C₂×C₈	C₄²×C₅	1	2	yes	—
151	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
152	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
153	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
154	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
155	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
156	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
159	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
160	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
161	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
162	C₁₆	—	2	4	no	—
163	C₂×C₅	—	2	2	no	—
164	C₂×C₅	—	2	2	no	—
165	C₂×C₅	#1381	1	4	yes	direct product factor
166	C₂×C₅	C₂×#340	1	1	yes	characteristic
167	C₂×C₅	C₂×#368	1	2	yes	—
168	C₂³	—	2	4	no	—
169	C₂³	C₅×#103	1	4	yes	—
170	C₂³	C₅×#107	1	1	yes	characteristic
171	C₂³	C₅×#107	1	1	yes	characteristic
172	C₂³	C₂×C₅×Q₁₆	1	1	yes	characteristic
173	C₂×C₄	C₅×#95	1	2	yes	—
174	C₂×C₄	C₅×#95	1	2	yes	—
175	C₂×C₄	C₅×#102	1	4	yes	—
176	C₂×C₄	C₅×#102	1	4	yes	—
177	C₂×C₄	C₅×#103	1	4	yes	—
178	C₂×C₄	C₅×#106	1	2	yes	—
179	C₂×C₄	C₅×#106	1	2	yes	—
180	C₂×C₄	C₅×#107	1	1	yes	characteristic
181	C₂×C₄	C₅×#107	1	1	yes	characteristic
182	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
183	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
184	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
185	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
186	C₈	—	4	8	no	—
187	C₈	C₅×#95	1	2	yes	—
188	C₈	C₅×#95	1	2	yes	—
189	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
190	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
191	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
192	C₅	C₂×#1381	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²	—	2	8	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	C₅×#340	1	2	yes	—
197	C₂²	C₅×#340	1	2	yes	—
198	C₂²	C₅×#368	1	4	yes	—
199	C₂²	C₂×C₅×#103	1	2	yes	—
200	C₂²	C₂×C₅×#107	1	1	yes	characteristic
201	C₄	C₅×#340	1	2	yes	—
202	C₄	C₅×#340	1	2	yes	—
203	C₄	C₂×C₅×#95	1	1	yes	characteristic
204	C₄	C₂×C₅×#102	1	2	yes	—
205	C₄	C₂×C₅×#106	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	—	2	2	no	—
208	C₂	C₅×#1381	1	4	yes	direct product factor
209	C₂	C₂×C₅×#340	1	1	yes	characteristic
210	C₂	C₂×C₅×#368	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	D₈
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂³×C₅	#107
prime radical	C₂⁴×C₅	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1381	C₂	direct (N × H)
14. C₂×#1381	C₅	direct (N × H)
38. #1381	C₂×C₅	direct (N × H)
165. C₂×C₅	#1381	direct (N × H)
192. C₅	C₂×#1381	direct (N × H)
208. C₂	C₅×#1381	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×C₁₆	2
C₂²×C₅×#108	2
C₅×#1381	2
C₂×#1381	5

Subgroups — C2×C5×#1381 #11935278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1381 #11935278