Subgroups — C₂²×#1840 #6371061

Order 512.

2650 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#376	C₂	1	1	yes	maximal, characteristic
2	C₂×#1840	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#336	C₂	1	6	yes	maximal
4	C₂⁴×C₈	C₂²	1	4	yes	—
5	C₂³×C₄²	C₂²	1	3	yes	—
6	C₂×#376	C₄	1	12	yes	—
7	C₂×#376	C₂²	1	12	yes	—
8	C₂²×#95	C₄	1	4	yes	—
9	#1840	C₂²	1	64	yes	direct product factor
10	C₂×#336	C₂²	1	72	yes	—
11	C₂⁴×C₄	C₂×C₄	1	6	yes	—
12	C₂⁴×C₄	C₂³	1	1	yes	characteristic
13	C₂³×C₈	—	2	48	no	—
14	C₂³×C₈	C₂³	1	24	yes	—
15	C₂²×C₄²	—	2	24	no	—
16	C₂²×C₄²	C₂×C₄	1	18	yes	—
17	C₂²×C₄²	C₂³	1	18	yes	—
18	#376	C₂×C₄	1	48	yes	—
19	#376	C₂³	1	16	yes	—
20	C₂×#95	C₂×C₄	1	48	yes	—
21	#336	C₂³	1	96	yes	—
22	C₂⁵	C₂²×C₄	1	1	yes	characteristic
23	C₂³×C₄	—	2	24	no	—
24	C₂³×C₄	—	2	48	no	—
25	C₂³×C₄	C₂²×C₄	1	36	yes	—
26	C₂³×C₄	C₂²×C₄	1	3	yes	—
27	C₂³×C₄	#41	1	24	yes	—
28	C₂³×C₄	C₂⁴	1	3	yes	—
29	C₂²×C₈	—	2	144	no	—
30	C₂²×C₈	—	4	64	no	—
31	C₂²×C₈	C₂⁴	1	16	yes	—
32	C₂×C₄²	—	2	72	no	—
33	C₂×C₄²	—	2	72	no	—
34	C₂×C₄²	C₂²×C₄	1	36	yes	—
35	C₂×C₄²	C₂⁴	1	12	yes	—
36	#95	C₂²×C₄	1	64	yes	—
37	C₂⁴	#117	1	12	yes	—
38	C₂⁴	#131	1	16	yes	—
39	C₂⁴	C₂³×C₄	1	3	yes	—
40	C₂²×C₄	—	2	72	no	—
41	C₂²×C₄	—	2	144	no	—
42	C₂²×C₄	—	4	96	no	—
43	C₂²×C₄	#117	1	36	yes	—
44	C₂²×C₄	#126	1	64	yes	—
45	C₂²×C₄	C₂³×C₄	1	24	yes	—
46	C₂²×C₄	C₂³×C₄	1	3	yes	—
47	C₂²×C₄	C₂×#41	1	36	yes	—
48	C₂²×C₄	C₂⁵	1	1	yes	characteristic
49	C₂×C₈	—	2	48	no	—
50	C₂×C₈	—	4	96	no	—
51	C₄²	—	2	72	no	—
52	C₄²	—	2	24	no	—
53	C₂³	#383	1	16	yes	—
54	C₂³	#433	1	96	yes	—
55	C₂³	C₂×#117	1	18	yes	—
56	C₂³	C₂×#131	1	24	yes	—
57	C₂³	C₂⁴×C₄	1	1	yes	characteristic
58	C₂×C₄	—	2	24	no	—
59	C₂×C₄	—	2	48	no	—
60	C₂×C₄	—	4	144	no	—
61	C₂×C₄	#383	1	48	yes	—
62	C₂×C₄	C₂×#117	1	18	yes	—
63	C₂×C₄	C₂×#126	1	48	yes	—
64	C₂×C₄	C₂²×#41	1	6	yes	—
65	C₈	—	4	16	no	—
66	C₂²	#1840	1	64	yes	direct product factor
67	C₂²	C₂×#383	1	12	yes	—
68	C₂²	C₂×#433	1	72	yes	—
69	C₂²	C₂²×#117	1	3	yes	—
70	C₂²	C₂²×#131	1	4	yes	—
71	C₄	—	4	24	no	—
72	C₄	C₂×#383	1	12	yes	—
73	C₄	C₂²×#126	1	4	yes	—
74	C₂	C₂×#1840	1	24	yes	direct product factor
75	C₂	C₂²×#383	1	1	yes	characteristic
76	C₂	C₂²×#433	1	6	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂³
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle, prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#1840	C₂	direct (N × H)
9. #1840	C₂²	direct (N × H)
66. C₂²	#1840	direct (N × H)
74. C₂	C₂×#1840	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#376	2
C₂×#1840	2
C₂²×#336	2

Subgroups — C22×#1840 #6371061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#1840 #6371061