Subgroups — C₂×#1747 #20369

Order 256.

465 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#405	C₂	1	3	yes	maximal
2	#1747	C₂	1	8	yes	maximal, direct product factor
3	C₂×#338	C₂	1	1	yes	maximal, characteristic
4	C₂×H(3,ℤ/4)	C₂	1	3	yes	maximal
5	C₂²×M₁₆	C₂²	1	3	yes	—
6	C₂³×C₈	C₂²	1	1	yes	characteristic
7	C₂×#117	C₂²	1	3	yes	—
8	#405	C₄	1	12	yes	complemented
9	#405	C₂²	1	12	yes	—
10	C₂×#97	C₄	1	6	yes	complemented
11	C₂×C₄×C₈	C₄	1	6	yes	complemented
12	#338	C₂²	1	4	yes	—
13	H(3,ℤ/4)	C₂²	1	12	yes	—
14	C₂³×C₄	C₂³	1	1	yes	characteristic
15	C₂×M₁₆	—	2	6	no	—
16	C₂×M₁₆	—	2	6	no	—
17	C₂×M₁₆	C₂×C₄	1	6	yes	—
18	C₂×M₁₆	C₂³	1	6	yes	—
19	C₂²×C₈	C₂×C₄	1	6	yes	—
20	C₂²×C₈	C₂×C₄	1	3	yes	—
21	C₂²×C₈	C₂×C₄	1	6	yes	—
22	C₂²×C₈	C₂×C₄	1	3	yes	—
23	C₂²×C₈	C₂³	1	2	yes	—
24	#117	—	2	3	no	—
25	#117	—	2	3	no	—
26	#117	C₂×C₄	1	6	yes	—
27	#117	C₂³	1	6	yes	—
28	C₂×C₄⋊C₄	—	2	3	no	—
29	C₂×C₂²⋊C₄	—	2	3	no	—
30	C₂×C₄²	C₂×C₄	1	6	yes	—
31	#97	C₂×C₄	1	24	yes	complemented
32	C₄×C₈	C₂×C₄	1	24	yes	complemented
33	C₂⁴	#41	1	1	yes	characteristic
34	C₂²×C₄	—	2	6	no	—
35	C₂²×C₄	C₂²×C₄	1	3	yes	—
36	C₂²×C₄	C₂²×C₄	1	3	yes	—
37	C₂²×C₄	#41	1	1	yes	characteristic
38	C₂²×C₄	#41	1	3	yes	—
39	C₂²×C₄	#41	1	3	yes	—
40	C₂²×C₄	C₂⁴	1	1	yes	characteristic
41	M₁₆	—	2	12	no	—
42	M₁₆	—	2	12	no	—
43	C₂×C₈	—	2	6	no	—
44	C₂×C₈	—	2	6	no	—
45	C₂×C₈	C₄²	1	6	yes	—
46	C₂×C₈	C₄²	1	3	yes	—
47	C₂×C₈	C₄²	1	6	yes	—
48	C₂×C₈	C₄²	1	1	yes	characteristic
49	C₂×C₈	C₂²×C₄	1	12	yes	—
50	C₂×C₈	C₂²×C₄	1	6	yes	—
51	C₂×C₈	C₂²×C₄	1	6	yes	—
52	C₄⋊C₄	—	2	6	no	—
53	C₄⋊C₄	—	2	6	no	—
54	C₂²⋊C₄	—	2	6	no	—
55	C₂²⋊C₄	—	2	6	no	—
56	C₄²	—	2	6	no	—
57	C₄²	C₂²×C₄	1	12	yes	—
58	C₂³	—	4	2	no	—
59	C₂³	#117	1	3	yes	—
60	C₂³	#117	1	3	yes	—
61	C₂³	C₂×#41	1	1	yes	characteristic
62	C₂×C₄	—	2	12	no	—
63	C₂×C₄	—	4	12	no	—
64	C₂×C₄	C₂×C₄²	1	3	yes	—
65	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄	#117	1	3	yes	—
67	C₂×C₄	#117	1	3	yes	—
68	C₂×C₄	#117	1	6	yes	—
69	C₂×C₄	#117	1	3	yes	—
70	C₂×C₄	#117	1	3	yes	—
71	C₂×C₄	C₂³×C₄	1	3	yes	—
72	C₂×C₄	C₂×#41	1	3	yes	—
73	C₈	—	2	12	no	—
74	C₈	C₂×C₄²	1	6	yes	—
75	C₈	C₂×C₄²	1	2	yes	—
76	C₂²	—	2	6	no	—
77	C₂²	—	4	3	no	—
78	C₂²	—	4	1	no	—
79	C₂²	#376	1	3	yes	—
80	C₂²	#376	1	1	yes	characteristic
81	C₂²	C₂×#117	1	3	yes	—
82	C₄	—	4	12	no	—
83	C₄	#376	1	3	yes	—
84	C₄	#376	1	1	yes	characteristic
85	C₄	C₂²×C₄²	1	1	yes	characteristic
86	C₄	C₂×#117	1	3	yes	—
87	C₂	—	2	3	no	—
88	C₂	—	2	3	no	—
89	C₂	#1747	1	2	yes	direct product factor
90	C₂	C₂×#376	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄²
commutator	C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	#376
prime radical	C₂⁴	#41
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #1747	C₂	direct (N × H)
8. #405	C₄	semidirect
10. C₂×#97	C₄	semidirect
11. C₂×C₄×C₈	C₄	semidirect
31. #97	C₂×C₄	semidirect
32. C₄×C₈	C₂×C₄	semidirect
89. C₂	#1747	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×#405	2
#1747	2
C₂×#338	2
C₂×H(3,ℤ/4)	2

Subgroups — C2×#1747 #20369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1747 #20369