Subgroups — #72234396

Order 1024.

267 conjugacy classes of subgroups.

The subgroup lattice has 111 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₁₂₈	C₂	1	2	yes	maximal
2	C₂×#7329	C₂	1	1	yes	maximal, characteristic
3	#94505	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₂₈	C₄	1	4	yes	—
5	C₂×C₄×C₃₂	C₄	1	1	yes	characteristic
6	C₂×C₄×C₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₁₆	C₄	1	1	yes	characteristic
8	#7510	C₂²	1	4	yes	—
9	C₄×C₆₄	C₄	1	4	yes	—
10	#7329	C₄	1	2	yes	—
11	#7329	C₂²	1	2	yes	—
12	C₂²×C₃₂	C₈	1	2	yes	—
13	C₂²×C₃₂	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₁₆	C₈	1	1	yes	characteristic
15	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈²	C₈	1	1	yes	characteristic
17	M₁₂₈	C₈	1	16	yes	complemented
18	C₂×C₆₄	C₂×C₄	1	8	yes	—
19	C₄×C₃₂	C₈	1	2	yes	—
20	C₄×C₃₂	C₂×C₄	1	2	yes	—
21	C₄×C₃₂	C₂×C₄	1	1	yes	characteristic
22	C₄×C₃₂	C₂×C₄	1	2	yes	—
23	C₄×C₃₂	C₂³	1	1	yes	characteristic
24	C₈×C₁₆	C₈	1	2	yes	—
25	C₈×C₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₁₆	C₁₆	1	2	yes	—
27	C₂²×C₁₆	C₄²	1	1	yes	characteristic
28	C₂²×C₁₆	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₈	C₁₆	1	2	yes	—
30	C₂×C₄×C₈	C₂×C₈	1	1	yes	characteristic
31	C₂×C₃₂	C₁₆	1	8	yes	—
32	C₂×C₃₂	C₄²	1	2	yes	—
33	C₂×C₃₂	C₂×C₈	1	8	yes	—
34	C₂×C₃₂	C₂×C₈	1	4	yes	—
35	C₂×C₃₂	C₂²×C₄	1	2	yes	—
36	C₄×C₁₆	C₁₆	1	2	yes	—
37	C₄×C₁₆	C₄²	1	1	yes	characteristic
38	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
39	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
40	C₄×C₁₆	C₂×C₈	1	2	yes	—
41	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
42	C₈²	C₁₆	1	2	yes	—
43	C₈²	C₂×C₈	1	2	yes	—
44	C₆₄	C₂×C₈	1	16	yes	complemented
45	C₂²×C₈	—	2	2	no	—
46	C₂²×C₈	C₄×C₈	1	1	yes	characteristic
47	C₂²×C₈	C₂×C₁₆	1	1	yes	characteristic
48	C₂×C₄²	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₁₆	—	2	4	no	—
50	C₂×C₁₆	C₄×C₈	1	1	yes	characteristic
51	C₂×C₁₆	C₄×C₈	1	4	yes	—
52	C₂×C₁₆	C₄×C₈	1	1	yes	characteristic
53	C₂×C₁₆	C₂×C₁₆	1	4	yes	—
54	C₂×C₁₆	C₂×C₁₆	1	4	yes	—
55	C₂×C₁₆	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
57	C₄×C₈	—	2	2	no	—
58	C₄×C₈	C₄×C₈	1	1	yes	characteristic
59	C₄×C₈	C₂×C₁₆	1	1	yes	characteristic
60	C₄×C₈	C₂×C₁₆	1	4	yes	—
61	C₄×C₈	C₂×C₁₆	1	1	yes	characteristic
62	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
63	C₃₂	C₄×C₈	1	4	yes	—
64	C₃₂	C₂×C₁₆	1	8	yes	—
65	C₃₂	C₂²×C₈	1	4	yes	—
66	C₂²×C₄	C₄×C₁₆	1	1	yes	characteristic
67	C₂²×C₄	M₆₄	1	2	yes	—
68	C₂×C₈	—	2	4	no	—
69	C₂×C₈	—	4	4	no	—
70	C₂×C₈	C₈²	1	2	yes	—
71	C₂×C₈	C₄×C₁₆	1	1	yes	characteristic
72	C₂×C₈	C₄×C₁₆	1	2	yes	—
73	C₂×C₈	C₄×C₁₆	1	1	yes	characteristic
74	C₂×C₈	M₆₄	1	4	yes	—
75	C₂×C₈	C₂×C₄×C₈	1	1	yes	characteristic
76	C₂×C₈	C₂²×C₁₆	1	1	yes	characteristic
77	C₄²	C₄×C₁₆	1	1	yes	characteristic
78	C₄²	M₆₄	1	2	yes	—
79	C₄²	C₂²×C₁₆	1	1	yes	characteristic
80	C₁₆	—	2	4	no	—
81	C₁₆	C₈²	1	2	yes	—
82	C₁₆	C₄×C₁₆	1	2	yes	—
83	C₁₆	C₂×C₄×C₈	1	2	yes	—
84	C₁₆	C₂²×C₁₆	1	2	yes	—
85	C₂³	#1397	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	C₈×C₁₆	1	2	yes	—
88	C₂×C₄	#1397	1	2	yes	—
89	C₂×C₄	#1397	1	1	yes	characteristic
90	C₂×C₄	#1397	1	2	yes	—
91	C₂×C₄	C₂×C₄×C₁₆	1	1	yes	characteristic
92	C₂×C₄	C₂×M₆₄	1	2	yes	—
93	C₈	—	4	4	no	—
94	C₈	C₈×C₁₆	1	2	yes	—
95	C₈	#1397	1	2	yes	—
96	C₈	C₂×C₈²	1	1	yes	characteristic
97	C₈	C₂×C₄×C₁₆	1	1	yes	characteristic
98	C₈	C₂×M₆₄	1	2	yes	—
99	C₂²	—	2	2	no	—
100	C₂²	#7329	1	2	yes	—
101	C₂²	C₂×#1397	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	#7329	1	2	yes	—
104	C₄	#7511	1	4	yes	—
105	C₄	C₂×C₈×C₁₆	1	1	yes	characteristic
106	C₄	C₂×#1397	1	1	yes	characteristic
107	C₂	—	2	2	no	—
108	C₂	C₂×#7329	1	1	yes	characteristic
109	C₂	C₂×#7511	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆	C₄²
commutator	C₄	C₂×C₈×C₁₆
Frattini, cosocle	C₄×C₃₂	C₂³
socle	C₂²	C₂×#1397
prime radical	C₂³	#1397
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #94505	C₂	semidirect
17. M₁₂₈	C₈	semidirect
44. C₆₄	C₂×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₁₂₈	2
C₂×#7329	2
#94505	2