Subgroups — #72234310

Order 1024.

351 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×C₃₂	C₂	1	2	yes	maximal
2	C₂×C₁₆²	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃₂	C₄	1	4	yes	—
4	C₂×C₈×C₁₆	C₄	1	2	yes	—
5	C₂×C₈×C₁₆	C₂²	1	1	yes	characteristic
6	C₈×C₃₂	—	2	2	no	—
7	C₈×C₃₂	—	2	2	no	—
8	C₁₆²	—	2	2	no	—
9	C₂²×C₃₂	C₈	1	8	yes	—
10	C₂×C₄×C₁₆	C₈	1	4	yes	—
11	C₂×C₄×C₁₆	C₂×C₄	1	2	yes	—
12	C₂×C₈²	C₂×C₄	1	1	yes	characteristic
13	C₄×C₃₂	—	2	4	no	—
14	C₄×C₃₂	—	2	4	no	—
15	C₈×C₁₆	—	2	2	no	—
16	C₈×C₁₆	—	2	2	no	—
17	C₈×C₁₆	D₄	1	2	yes	—
18	C₈×C₁₆	D₄	1	1	yes	characteristic
19	C₈×C₁₆	Q₈	1	1	yes	characteristic
20	C₂²×C₁₆	C₁₆	1	8	yes	—
21	C₂²×C₁₆	C₂×C₈	1	4	yes	—
22	C₂×C₄×C₈	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂×C₈	1	2	yes	—
24	C₂×C₃₂	—	2	8	no	—
25	C₂×C₃₂	—	2	8	no	—
26	C₂×C₃₂	C₁₆	1	16	yes	complemented
27	C₄×C₁₆	—	2	4	no	—
28	C₄×C₁₆	—	2	4	no	—
29	C₄×C₁₆	C₂²⋊C₄	1	2	yes	—
30	C₄×C₁₆	C₄⋊C₄	1	2	yes	—
31	C₄×C₁₆	M₁₆	1	4	yes	—
32	C₈²	C₂²⋊C₄	1	1	yes	characteristic
33	C₈²	C₄⋊C₄	1	1	yes	characteristic
34	C₈²	M₁₆	1	2	yes	—
35	C₂²×C₈	C₄×C₈	1	2	yes	—
36	C₂²×C₈	C₂×C₁₆	1	4	yes	—
37	C₂×C₄²	C₄×C₈	1	1	yes	characteristic
38	C₂×C₁₆	—	2	8	no	—
39	C₂×C₁₆	—	2	8	no	—
40	C₂×C₁₆	C₃₂	1	16	yes	complemented
41	C₂×C₁₆	C₂²⋊C₈	1	4	yes	—
42	C₂×C₁₆	C₄⋊C₈	1	4	yes	—
43	C₂×C₁₆	C₂×C₁₆	1	8	yes	—
44	C₂×C₁₆	M₃₂	1	8	yes	—
45	C₄×C₈	#95	1	1	yes	characteristic
46	C₄×C₈	#97	1	2	yes	—
47	C₄×C₈	#97	1	1	yes	characteristic
48	C₄×C₈	C₂²⋊C₈	1	2	yes	—
49	C₄×C₈	C₄⋊C₈	1	2	yes	—
50	C₄×C₈	M₃₂	1	4	yes	—
51	C₃₂	—	2	16	no	—
52	C₂²×C₄	C₈²	1	1	yes	characteristic
53	C₂²×C₄	C₄×C₁₆	1	2	yes	—
54	C₂×C₈	#322	1	2	yes	—
55	C₂×C₈	#336	1	2	yes	—
56	C₂×C₈	C₄×C₁₆	1	4	yes	—
57	C₂×C₈	#346	1	4	yes	—
58	C₂×C₈	C₂²⋊C₁₆	1	4	yes	—
59	C₂×C₈	C₄⋊C₁₆	1	4	yes	—
60	C₂×C₈	C₂×C₃₂	1	8	yes	—
61	C₂×C₈	M₆₄	1	8	yes	—
62	C₄²	#322	1	1	yes	characteristic
63	C₄²	#336	1	1	yes	characteristic
64	C₄²	#346	1	2	yes	—
65	C₁₆	—	2	16	no	—
66	C₁₆	C₂²⋊C₁₆	1	8	yes	—
67	C₁₆	C₄⋊C₁₆	1	8	yes	—
68	C₂³	C₈×C₁₆	1	1	yes	characteristic
69	C₂×C₄	#1273	1	1	yes	characteristic
70	C₂×C₄	C₈×C₁₆	1	2	yes	—
71	C₂×C₄	#1311	1	1	yes	characteristic
72	C₂×C₄	#1311	1	2	yes	—
73	C₂×C₄	#1312	1	2	yes	—
74	C₂×C₄	#1374	1	2	yes	—
75	C₂×C₄	C₄×C₃₂	1	4	yes	—
76	C₂×C₄	#1397	1	4	yes	—
77	C₈	#1312	1	4	yes	—
78	C₈	#1374	1	4	yes	—
79	C₈	C₂²⋊C₃₂	1	8	yes	—
80	C₈	C₄⋊C₃₂	1	8	yes	—
81	C₂²	C₁₆²	1	1	yes	characteristic
82	C₂²	#7052	1	1	yes	characteristic
83	C₂²	#7136	1	1	yes	characteristic
84	C₂²	C₈×C₃₂	1	2	yes	—
85	C₂²	#7329	1	2	yes	—
86	C₄	#7052	1	2	yes	—
87	C₄	#7136	1	2	yes	—
88	C₄	#7331	1	4	yes	—
89	C₄	#7465	1	4	yes	—
90	C₂	#92895	1	1	yes	characteristic
91	C₂	C₁₆×C₃₂	1	1	yes	characteristic
92	C₂	#93664	1	1	yes	characteristic
93	C₂	#93665	1	2	yes	—
94	C₂	#93989	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
26. C₂×C₃₂	C₁₆	semidirect
40. C₂×C₁₆	C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×C₃₂	2
C₂×C₁₆²	2