Subgroups — C₂²×C₅×C₄⋊C₁₆ #11956114

Order 1280.

862 conjugacy classes of subgroups.

The subgroup lattice has 156 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	2	yes	maximal
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₄⋊C₁₆	C₂	1	12	yes	maximal, direct product factor
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	—	2	8	no	—
8	C₂²×C₅×C₁₆	C₂²	1	12	yes	—
9	C₂×C₄×C₅×C₈	C₄	1	6	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	6	yes	—
11	C₅×C₄⋊C₁₆	C₂²	1	16	yes	direct product factor
12	C₂²×C₄⋊C₁₆	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	4	no	—
17	C₂²×C₅×C₈	C₂×C₄	1	6	yes	—
18	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
19	C₂²×C₅×C₈	D₄	1	4	yes	—
20	C₂²×C₅×C₈	Q₈	1	4	yes	—
21	C₂²×C₅×C₈	C₂³	1	3	yes	—
22	C₂×C₄²×C₅	C₈	1	6	yes	—
23	C₂×C₄²×C₅	C₂×C₄	1	6	yes	—
24	C₂×C₅×C₁₆	—	2	24	no	—
25	C₂×C₅×C₁₆	C₂³	1	8	yes	—
26	C₄×C₅×C₈	C₂×C₄	1	12	yes	—
27	C₄×C₅×C₈	C₂³	1	4	yes	—
28	C₂³×C₁₆	C₂×C₅	1	2	yes	—
29	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
30	C₂×C₄⋊C₁₆	C₂×C₅	1	12	yes	direct product factor
31	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₅	—	2	4	no	—
33	C₂²×C₄×C₅	C₁₆	1	8	yes	complemented
34	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
35	C₂²×C₄×C₅	C₂×C₈	1	6	yes	—
36	C₂²×C₄×C₅	C₂×C₈	1	3	yes	—
37	C₂²×C₄×C₅	C₂×C₈	1	6	yes	—
38	C₂²×C₄×C₅	M₁₆	1	4	yes	—
39	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
40	C₂×C₅×C₈	—	2	12	no	—
41	C₂×C₅×C₈	C₄⋊C₄	1	12	yes	—
42	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
43	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
44	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
45	C₂×C₅×C₈	C₂×Q₈	1	6	yes	—
46	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
47	C₄²×C₅	C₂×C₈	1	12	yes	—
48	C₄²×C₅	C₂²×C₄	1	4	yes	—
49	C₅×C₁₆	—	2	8	no	—
50	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
51	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
52	C₂²×C₄²	C₄×C₅	1	1	yes	characteristic
53	C₂²×C₁₆	—	2	8	no	—
54	C₂²×C₁₆	C₂²×C₅	1	12	yes	—
55	C₂×C₄×C₈	C₄×C₅	1	6	yes	—
56	C₂×C₄×C₈	C₂²×C₅	1	6	yes	—
57	C₄⋊C₁₆	C₂²×C₅	1	16	yes	direct product factor
58	C₂³×C₅	C₄⋊C₈	1	4	yes	—
59	C₂³×C₅	C₂×C₁₆	1	4	yes	—
60	C₂³×C₅	M₃₂	1	4	yes	—
61	C₂³×C₅	C₂²×C₈	1	3	yes	—
62	C₂×C₄×C₅	—	2	12	no	—
63	C₂×C₄×C₅	C₄⋊C₈	1	12	yes	—
64	C₂×C₄×C₅	C₂×C₁₆	1	24	yes	complemented
65	C₂×C₄×C₅	C₂×C₄⋊C₄	1	6	yes	—
66	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
67	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
69	C₂×C₄×C₅	C₂×M₁₆	1	6	yes	—
70	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
71	C₅×C₈	—	2	4	no	—
72	C₅×C₈	C₂×C₄⋊C₄	1	6	yes	—
73	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
74	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
75	C₂³×C₄	C₅×C₈	1	1	yes	characteristic
76	C₂³×C₄	C₅×C₈	1	1	yes	characteristic
77	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
78	C₂²×C₈	—	2	4	no	—
79	C₂²×C₈	C₂×C₄×C₅	1	6	yes	—
80	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
81	C₂²×C₈	C₅×D₄	1	4	yes	—
82	C₂²×C₈	C₅×Q₈	1	4	yes	—
83	C₂²×C₈	C₂³×C₅	1	3	yes	—
84	C₂×C₄²	C₅×C₈	1	6	yes	—
85	C₂×C₄²	C₂×C₄×C₅	1	6	yes	—
86	C₂×C₁₆	—	2	24	no	—
87	C₂×C₁₆	C₂³×C₅	1	8	yes	—
88	C₄×C₈	C₂×C₄×C₅	1	12	yes	—
89	C₄×C₈	C₂³×C₅	1	4	yes	—
90	C₂²×C₅	C₄⋊C₁₆	1	16	yes	direct product factor
91	C₂²×C₅	C₂×C₄⋊C₈	1	6	yes	—
92	C₂²×C₅	C₂²×C₁₆	1	6	yes	—
93	C₂²×C₅	C₂×M₃₂	1	6	yes	—
94	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
95	C₄×C₅	—	2	4	no	—
96	C₄×C₅	C₂×C₄⋊C₈	1	6	yes	—
97	C₄×C₅	C₂²×C₁₆	1	8	yes	complemented
98	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
99	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
100	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	C₅×C₁₆	1	8	yes	complemented
103	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
104	C₂²×C₄	C₂×C₅×C₈	1	6	yes	—
105	C₂²×C₄	C₂×C₅×C₈	1	3	yes	—
106	C₂²×C₄	C₂×C₅×C₈	1	6	yes	—
107	C₂²×C₄	C₅×M₁₆	1	4	yes	—
108	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
109	C₂×C₈	—	2	12	no	—
110	C₂×C₈	C₅×C₄⋊C₄	1	12	yes	—
111	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
112	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
113	C₂×C₈	C₂×C₅×D₄	1	6	yes	—
114	C₂×C₈	C₂×C₅×Q₈	1	6	yes	—
115	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
116	C₄²	C₂×C₅×C₈	1	12	yes	—
117	C₄²	C₂²×C₄×C₅	1	4	yes	—
118	C₁₆	—	2	8	no	—
119	C₂×C₅	C₂×C₄⋊C₁₆	1	12	yes	direct product factor
120	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
121	C₂×C₅	C₂³×C₁₆	1	1	yes	characteristic
122	C₂×C₅	C₂²×M₃₂	1	1	yes	characteristic
123	C₂³	C₅×C₄⋊C₈	1	4	yes	—
124	C₂³	C₂×C₅×C₁₆	1	4	yes	—
125	C₂³	C₅×M₃₂	1	4	yes	—
126	C₂³	C₂²×C₅×C₈	1	3	yes	—
127	C₂×C₄	—	2	12	no	—
128	C₂×C₄	C₅×C₄⋊C₈	1	12	yes	—
129	C₂×C₄	C₂×C₅×C₁₆	1	24	yes	complemented
130	C₂×C₄	C₂×C₅×C₄⋊C₄	1	6	yes	—
131	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
132	C₂×C₄	C₂²×C₅×C₈	1	3	yes	—
133	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
134	C₂×C₄	C₂×C₅×M₁₆	1	6	yes	—
135	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
136	C₈	—	2	4	no	—
137	C₈	C₂×C₅×C₄⋊C₄	1	6	yes	—
138	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
139	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
140	C₅	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, direct product factor, Sylow
141	C₂²	C₅×C₄⋊C₁₆	1	16	yes	direct product factor
142	C₂²	C₂×C₅×C₄⋊C₈	1	6	yes	—
143	C₂²	C₂²×C₅×C₁₆	1	6	yes	—
144	C₂²	C₂×C₅×M₃₂	1	6	yes	—
145	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
146	C₄	—	2	4	no	—
147	C₄	C₂×C₅×C₄⋊C₈	1	6	yes	—
148	C₄	C₂²×C₅×C₁₆	1	8	yes	complemented
149	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
150	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
151	C₂	C₂×C₅×C₄⋊C₁₆	1	12	yes	direct product factor
152	C₂	C₂²×C₅×C₄⋊C₈	1	1	yes	characteristic
153	C₂	C₂³×C₅×C₁₆	1	1	yes	characteristic
154	C₂	C₂²×C₅×M₃₂	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×C₄⋊C₁₆	C₂	direct (N × H)
11. C₅×C₄⋊C₁₆	C₂²	direct (N × H)
12. C₂²×C₄⋊C₁₆	C₅	direct (N × H)
30. C₂×C₄⋊C₁₆	C₂×C₅	direct (N × H)
33. C₂²×C₄×C₅	C₁₆	semidirect
57. C₄⋊C₁₆	C₂²×C₅	direct (N × H)
64. C₂×C₄×C₅	C₂×C₁₆	semidirect
90. C₂²×C₅	C₄⋊C₁₆	direct (N × H)
97. C₄×C₅	C₂²×C₁₆	semidirect
102. C₂²×C₄	C₅×C₁₆	semidirect
119. C₂×C₅	C₂×C₄⋊C₁₆	direct (N × H)
129. C₂×C₄	C₂×C₅×C₁₆	semidirect
140. C₅	C₂²×C₄⋊C₁₆	direct (N × H)
141. C₂²	C₅×C₄⋊C₁₆	direct (N × H)
148. C₄	C₂²×C₅×C₁₆	semidirect
151. C₂	C₂×C₅×C₄⋊C₁₆	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×C₄×C₅×C₈	2
C₂×C₅×C₄⋊C₁₆	2
C₂²×C₄⋊C₁₆	5

Subgroups — C22×C5×C4⋊C16 #11956114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×C₄⋊C₁₆ #11956114