Subgroups — C₂²×#2866 #7633342

Order 512.

7510 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2866	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#388	C₂	1	7	yes	maximal
3	C₂²×#387	C₂	1	7	yes	maximal, complemented
4	C₂²×C₄³	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₄⋊C₄	C₂²	1	7	yes	—
6	C₂³×C₂²⋊C₄	C₂²	1	7	yes	—
7	C₂³×C₄²	C₂²	1	7	yes	—
8	#2866	C₂²	1	256	yes	direct product factor
9	C₂×#388	C₂²	1	168	yes	—
10	C₂×#387	C₂²	1	168	yes	complemented
11	C₂²×#95	C₂²	1	14	yes	—
12	C₂×C₄³	C₂²	1	24	yes	complemented
13	C₂⁶	C₂³	1	1	yes	characteristic
14	C₂⁴×C₄	C₂³	1	7	yes	—
15	C₂⁴×C₄	C₂³	1	7	yes	—
16	C₂²×C₄⋊C₄	—	2	56	no	—
17	C₂²×C₄⋊C₄	C₂³	1	84	yes	—
18	C₂²×C₂²⋊C₄	—	2	56	no	—
19	C₂²×C₂²⋊C₄	C₂³	1	84	yes	—
20	C₂²×C₄²	—	2	56	no	—
21	C₂²×C₄²	C₂³	1	84	yes	—
22	#388	C₂³	1	448	yes	—
23	#387	C₂³	1	448	yes	complemented
24	C₂×#95	C₂³	1	168	yes	—
25	C₄³	C₂³	1	64	yes	complemented
26	C₂⁵	—	2	28	no	—
27	C₂⁵	C₂⁴	1	6	yes	—
28	C₂⁵	C₂⁴	1	1	yes	characteristic
29	C₂³×C₄	—	2	56	no	—
30	C₂³×C₄	—	2	28	no	—
31	C₂³×C₄	—	2	56	no	—
32	C₂³×C₄	#41	1	56	yes	—
33	C₂³×C₄	C₂⁴	1	42	yes	—
34	C₂³×C₄	C₂⁴	1	42	yes	—
35	C₂×C₄⋊C₄	—	2	336	no	—
36	C₂×C₄⋊C₄	C₂⁴	1	112	yes	—
37	C₂×C₂²⋊C₄	—	2	336	no	—
38	C₂×C₂²⋊C₄	C₂⁴	1	112	yes	—
39	C₂×C₄²	—	2	336	no	—
40	C₂×C₄²	C₂⁴	1	112	yes	—
41	#95	C₂⁴	1	224	yes	—
42	C₂⁴	—	2	84	no	—
43	C₂⁴	—	4	112	no	—
44	C₂⁴	C₂×#41	1	28	yes	—
45	C₂⁴	C₂⁵	1	4	yes	—
46	C₂⁴	C₂⁵	1	3	yes	—
47	C₂²×C₄	—	2	224	no	—
48	C₂²×C₄	—	2	168	no	—
49	C₂²×C₄	—	2	84	no	—
50	C₂²×C₄	—	2	168	no	—
51	C₂²×C₄	—	4	112	no	—
52	C₂²×C₄	C₂×#41	1	168	yes	—
53	C₂²×C₄	C₂⁵	1	28	yes	—
54	C₂²×C₄	C₂⁵	1	28	yes	—
55	C₄⋊C₄	—	2	224	no	—
56	C₂²⋊C₄	—	2	224	no	—
57	C₄²	—	2	224	no	—
58	C₂³	—	2	28	no	—
59	C₂³	—	4	168	no	—
60	C₂³	—	8	64	no	—
61	C₂³	#529	1	112	yes	—
62	C₂³	C₂²×#41	1	42	yes	—
63	C₂³	C₂⁶	1	1	yes	characteristic
64	C₂×C₄	—	2	56	no	—
65	C₂×C₄	—	2	336	no	—
66	C₂×C₄	—	2	28	no	—
67	C₂×C₄	—	2	56	no	—
68	C₂×C₄	—	4	168	no	—
69	C₂×C₄	C₂²×#41	1	56	yes	—
70	C₂²	—	4	28	no	—
71	C₂²	—	8	48	no	—
72	C₂²	#2866	1	64	yes	direct product factor
73	C₂²	C₂×#529	1	84	yes	—
74	C₂²	C₂³×#41	1	7	yes	—
75	C₄	—	2	56	no	—
76	C₄	—	4	28	no	—
77	C₂	—	8	4	no	—
78	C₂	C₂×#2866	1	24	yes	direct product factor
79	C₂	C₂²×#529	1	7	yes	—

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁶
prime radical	C₂⁶	C₂³
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×#2866	C₂	direct (N × H)
3. C₂²×#387	C₂	semidirect
4. C₂²×C₄³	C₂	semidirect
8. #2866	C₂²	direct (N × H)
10. C₂×#387	C₂²	semidirect
12. C₂×C₄³	C₂²	semidirect
23. #387	C₂³	semidirect
25. C₄³	C₂³	semidirect
72. C₂²	#2866	direct (N × H)
78. C₂	C₂×#2866	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×#2866	2
C₂²×#388	2
C₂²×#387	2
C₂²×C₄³	2

Subgroups — C22×#2866 #7633342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#2866 #7633342