Subgroups — C₁₆×#97 #126525

Order 512.

426 conjugacy classes of subgroups.

The subgroup lattice has 121 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12590	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1312	C₂	1	2	yes	maximal
4	C₂×C₈×C₁₆	C₂	1	2	yes	maximal
5	C₈×#97	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₁₆	C₄	1	4	yes	—
7	C₂×C₄×C₁₆	C₂²	1	2	yes	—
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	#1744	C₄	1	1	yes	characteristic
11	C₄×#97	C₄	1	1	yes	characteristic
12	C₄²×C₈	C₂²	1	1	yes	characteristic
13	C₂×#322	C₄	1	2	yes	—
14	C₂×C₈²	C₂²	1	2	yes	—
15	#1374	C₄	1	4	yes	—
16	#1312	C₄	1	8	yes	complemented
17	C₈×C₁₆	C₄	1	8	yes	complemented
18	C₂²×C₁₆	C₂×C₄	1	2	yes	—
19	C₂²×C₁₆	C₂×C₄	1	4	yes	—
20	C₂²×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×#97	C₈	1	4	yes	—
22	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	C₄³	C₂×C₄	1	1	yes	characteristic
29	C₄×C₁₆	—	2	8	no	—
30	C₄×C₁₆	C₂×C₄	1	4	yes	—
31	C₄×C₁₆	C₂×C₄	1	4	yes	—
32	C₄×C₁₆	C₂×C₄	1	4	yes	—
33	C₄×C₁₆	C₂×C₄	1	4	yes	—
34	#336	C₈	1	4	yes	—
35	#322	C₈	1	8	yes	—
36	C₈²	C₂×C₄	1	4	yes	—
37	C₈²	C₂×C₄	1	4	yes	—
38	C₂²×C₈	C₄²	1	2	yes	—
39	C₂²×C₈	C₂×C₈	1	4	yes	—
40	C₂²×C₈	C₂×C₈	1	4	yes	—
41	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄²	C₄²	1	2	yes	—
44	C₂×C₄²	C₂×C₈	1	2	yes	—
45	C₂×C₄²	C₂×C₈	1	2	yes	—
46	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₁₆	—	2	4	no	—
48	C₂×C₁₆	—	2	8	no	—
49	C₂×C₁₆	C₄²	1	4	yes	—
50	C₂×C₁₆	C₄²	1	8	yes	—
51	C₂×C₁₆	C₄²	1	4	yes	—
52	C₂×C₁₆	M₁₆	1	8	yes	—
53	#97	C₁₆	1	16	yes	direct product factor
54	C₄×C₈	—	2	4	no	—
55	C₄×C₈	—	2	4	no	—
56	C₄×C₈	C₄²	1	4	yes	—
57	C₄×C₈	C₄²	1	4	yes	—
58	C₄×C₈	C₂×C₈	1	4	yes	—
59	C₄×C₈	C₂×C₈	1	4	yes	—
60	C₄×C₈	C₂×C₈	1	4	yes	—
61	C₄×C₈	C₂²×C₄	1	2	yes	—
62	C₄×C₈	C₂²×C₄	1	2	yes	—
63	C₂²×C₄	C₄×C₈	1	4	yes	—
64	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
65	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
66	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	8	no	—
70	C₂×C₈	C₄×C₈	1	8	yes	—
71	C₂×C₈	C₄×C₈	1	4	yes	—
72	C₂×C₈	C₄×C₈	1	8	yes	—
73	C₂×C₈	C₂×C₁₆	1	16	yes	—
74	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₈	C₂×C₄²	1	2	yes	—
77	C₂×C₈	C₂×M₁₆	1	4	yes	—
78	C₂×C₈	#131	1	4	yes	—
79	C₄²	—	2	2	no	—
80	C₄²	—	2	2	no	—
81	C₄²	—	2	4	no	—
82	C₄²	C₂×C₁₆	1	8	yes	—
83	C₄²	C₂×C₄²	1	2	yes	—
84	C₄²	C₂²×C₈	1	2	yes	—
85	C₁₆	—	2	4	no	—
86	C₁₆	—	2	8	no	—
87	C₁₆	#97	1	8	yes	direct product factor
88	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	C₄×C₁₆	1	8	yes	—
92	C₂×C₄	C₄³	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
94	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
95	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
96	C₂×C₄	C₄×M₁₆	1	2	yes	—
97	C₂×C₄	#405	1	2	yes	—
98	C₂×C₄	C₂²×C₁₆	1	4	yes	—
99	C₂×C₄	#504	1	4	yes	—
100	C₈	—	2	8	no	—
101	C₈	—	2	4	no	—
102	C₈	C₄×C₁₆	1	16	yes	complemented
103	C₈	C₂×#97	1	2	yes	—
104	C₈	C₄×M₁₆	1	2	yes	—
105	C₈	#405	1	4	yes	—
106	C₂²	C₈×M₁₆	1	2	yes	—
107	C₂²	C₄²×C₈	1	1	yes	characteristic
108	C₂²	C₂×C₄×C₁₆	1	2	yes	—
109	C₂²	#2108	1	2	yes	—
110	C₄	—	2	8	no	—
111	C₄	C₈×M₁₆	1	2	yes	—
112	C₄	C₄×#97	1	1	yes	characteristic
113	C₄	#1726	1	1	yes	characteristic
114	C₄	C₂×C₄×C₁₆	1	4	yes	—
115	C₄	#2108	1	4	yes	—
116	C₂	C₈×#97	1	1	yes	characteristic
117	C₂	C₁₆×M₁₆	1	4	yes	—
118	C₂	C₄²×C₁₆	1	1	yes	characteristic
119	C₂	#12539	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₆	C₂²
commutator	C₂	C₄²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×C₄×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
16. #1312	C₄	semidirect
17. C₈×C₁₆	C₄	semidirect
53. #97	C₁₆	direct (N × H)
87. C₁₆	#97	direct (N × H)
102. C₈	C₄×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#12590	2
C₄²×C₁₆	2
C₂×#1312	2
C₂×C₈×C₁₆	2
C₈×#97	2

Subgroups — C16×#97 #126525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×#97 #126525