Subgroups — C₂²×#2845 #7633393

Order 512.

5914 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2845	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#398	C₂	1	12	yes	maximal
3	C₂²×#382	C₂	1	3	yes	maximal
4	C₂³×C₄⋊C₄	C₂²	1	6	yes	—
5	C₂³×C₄⋊C₄	C₂²	1	12	yes	—
6	C₂³×C₄²	C₂²	1	1	yes	characteristic
7	#2845	C₂²	1	256	yes	direct product factor
8	C₂×#398	C₂²	1	288	yes	—
9	C₂×#382	C₂²	1	72	yes	—
10	C₂²×#95	C₂²	1	12	yes	—
11	C₂²×#95	C₂²	1	4	yes	—
12	C₂⁴×C₄	C₂³	1	12	yes	—
13	C₂⁴×C₄	C₂³	1	3	yes	—
14	C₂²×C₄⋊C₄	—	4	24	no	—
15	C₂²×C₄⋊C₄	—	4	48	no	—
16	C₂²×C₄⋊C₄	C₂³	1	144	yes	—
17	C₂²×C₄⋊C₄	C₂³	1	72	yes	—
18	C₂²×C₄²	Q₈	1	16	yes	—
19	C₂²×C₄²	C₂³	1	12	yes	—
20	#398	C₂³	1	768	yes	—
21	#382	C₂³	1	192	yes	—
22	C₂×#95	C₂³	1	48	yes	—
23	C₂×#95	C₂³	1	144	yes	—
24	C₂⁵	C₂⁴	1	1	yes	characteristic
25	C₂³×C₄	—	2	12	no	—
26	C₂³×C₄	—	2	12	no	—
27	C₂³×C₄	—	2	48	no	—
28	C₂³×C₄	—	2	48	no	—
29	C₂³×C₄	—	2	48	no	—
30	C₂³×C₄	C₂×Q₈	1	24	yes	—
31	C₂³×C₄	C₂⁴	1	72	yes	—
32	C₂³×C₄	C₂⁴	1	18	yes	—
33	C₂×C₄⋊C₄	—	4	144	no	—
34	C₂×C₄⋊C₄	—	4	288	no	—
35	C₂×C₄⋊C₄	C₂⁴	1	96	yes	—
36	C₂×C₄⋊C₄	C₂⁴	1	192	yes	—
37	C₂×C₄²	C₂×Q₈	1	96	yes	—
38	C₂×C₄²	C₂⁴	1	16	yes	—
39	#95	C₂⁴	1	192	yes	—
40	#95	C₂⁴	1	64	yes	—
41	C₂⁴	C₂²×Q₈	1	4	yes	—
42	C₂⁴	H(5,𝔽₂)	1	12	yes	—
43	C₂⁴	2₋¹⁺⁴	1	12	yes	—
44	C₂⁴	C₂⁵	1	3	yes	—
45	C₂²×C₄	—	2	96	no	—
46	C₂²×C₄	—	2	36	no	—
47	C₂²×C₄	—	2	36	no	—
48	C₂²×C₄	—	2	144	no	—
49	C₂²×C₄	—	2	144	no	—
50	C₂²×C₄	—	2	144	no	—
51	C₂²×C₄	—	4	192	no	—
52	C₂²×C₄	C₂²×Q₈	1	72	yes	—
53	C₂²×C₄	C₂⁵	1	48	yes	—
54	C₂²×C₄	C₂⁵	1	12	yes	—
55	C₄⋊C₄	—	4	96	no	—
56	C₄⋊C₄	—	4	192	no	—
57	C₄²	C₂²×Q₈	1	64	yes	—
58	C₂³	#544	1	48	yes	—
59	C₂³	#562	1	48	yes	—
60	C₂³	#564	1	16	yes	—
61	C₂³	C₂³×Q₈	1	6	yes	—
62	C₂³	C₂×H(5,𝔽₂)	1	18	yes	—
63	C₂³	C₂×2₋¹⁺⁴	1	18	yes	—
64	C₂³	C₂⁶	1	1	yes	characteristic
65	C₂×C₄	—	2	12	no	—
66	C₂×C₄	—	2	12	no	—
67	C₂×C₄	—	2	144	no	—
68	C₂×C₄	—	2	48	no	—
69	C₂×C₄	—	2	48	no	—
70	C₂×C₄	—	2	48	no	—
71	C₂×C₄	—	4	288	no	—
72	C₂×C₄	C₂³×Q₈	1	24	yes	—
73	C₂²	#2845	1	64	yes	direct product factor
74	C₂²	C₂×#544	1	36	yes	—
75	C₂²	C₂×#562	1	36	yes	—
76	C₂²	C₂×#564	1	12	yes	—
77	C₂²	C₂⁴×Q₈	1	1	yes	characteristic
78	C₂²	C₂²×H(5,𝔽₂)	1	3	yes	—
79	C₂²	C₂²×2₋¹⁺⁴	1	3	yes	—
80	C₄	—	2	24	no	—
81	C₄	—	4	48	no	—
82	C₂	C₂×#2845	1	24	yes	direct product factor
83	C₂	C₂²×#544	1	3	yes	—
84	C₂	C₂²×#562	1	3	yes	—
85	C₂	C₂²×#564	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#2845	C₂	direct (N × H)
7. #2845	C₂²	direct (N × H)
73. C₂²	#2845	direct (N × H)
82. C₂	C₂×#2845	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#2845	2
C₂²×#398	2
C₂²×#382	2

Subgroups — C22×#2845 #7633393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#2845 #7633393