Subgroups — C₂×C₄×#97 #20326

Order 256.

895 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#97	C₂	1	4	yes	maximal
2	C₂²×C₄×C₈	C₂	1	2	yes	maximal
3	C₂×C₄³	C₂	1	1	yes	maximal, characteristic
4	C₄×#97	C₂	1	8	yes	maximal, direct product factor
5	C₂³×C₈	C₂²	1	4	yes	—
6	C₂²×C₄²	C₄	1	4	yes	—
7	C₂²×C₄²	C₂²	1	2	yes	—
8	C₂²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×#97	C₄	1	32	yes	direct product factor
10	C₂×#97	C₂²	1	16	yes	—
11	C₂×C₄×C₈	C₄	1	16	yes	complemented
12	C₂×C₄×C₈	C₂²	1	8	yes	—
13	C₄³	C₄	1	4	yes	—
14	C₄³	C₂²	1	4	yes	—
15	C₂³×C₄	C₂×C₄	1	4	yes	—
16	C₂³×C₄	C₂×C₄	1	2	yes	—
17	C₂³×C₄	C₂³	1	1	yes	characteristic
18	C₂²×C₈	C₂×C₄	1	16	yes	—
19	C₂²×C₈	C₂×C₄	1	32	yes	—
20	C₂²×C₈	C₂³	1	8	yes	—
21	C₂×C₄²	—	2	16	no	—
22	C₂×C₄²	C₂×C₄	1	16	yes	—
23	C₂×C₄²	C₂×C₄	1	4	yes	—
24	C₂×C₄²	C₂×C₄	1	2	yes	—
25	C₂×C₄²	C₂×C₄	1	8	yes	—
26	C₂×C₄²	C₂×C₄	1	16	yes	—
27	C₂×C₄²	C₂³	1	2	yes	—
28	C₂×C₄²	C₂³	1	4	yes	—
29	#97	C₂×C₄	1	64	yes	direct product factor
30	C₄×C₈	C₂×C₄	1	32	yes	complemented
31	C₂⁴	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄	—	2	16	no	—
33	C₂²×C₄	C₄²	1	16	yes	—
34	C₂²×C₄	C₄²	1	8	yes	—
35	C₂²×C₄	M₁₆	1	16	yes	—
36	C₂²×C₄	C₂²×C₄	1	8	yes	—
37	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄	C₂²×C₄	1	4	yes	—
39	C₂²×C₄	C₂²×C₄	1	8	yes	—
40	C₂²×C₄	C₂²×C₄	1	4	yes	—
41	C₂²×C₄	C₂⁴	1	1	yes	characteristic
42	C₂×C₈	C₄²	1	64	yes	complemented
43	C₂×C₈	C₂²×C₄	1	16	yes	—
44	C₂×C₈	C₂²×C₄	1	32	yes	—
45	C₄²	—	2	32	no	—
46	C₄²	C₄²	1	16	yes	—
47	C₄²	C₄²	1	8	yes	—
48	C₄²	C₂²×C₄	1	8	yes	—
49	C₄²	C₂²×C₄	1	16	yes	—
50	C₂³	C₂×C₄²	1	2	yes	—
51	C₂³	C₂×C₄²	1	4	yes	—
52	C₂³	C₂×M₁₆	1	8	yes	—
53	C₂³	C₂³×C₄	1	1	yes	characteristic
54	C₂×C₄	—	2	16	no	—
55	C₂×C₄	—	2	32	no	—
56	C₂×C₄	#97	1	32	yes	direct product factor
57	C₂×C₄	C₂×C₄²	1	16	yes	—
58	C₂×C₄	C₂×C₄²	1	2	yes	—
59	C₂×C₄	C₂×C₄²	1	4	yes	—
60	C₂×C₄	C₂×C₄²	1	16	yes	—
61	C₂×C₄	C₂×C₄²	1	8	yes	—
62	C₂×C₄	C₂×M₁₆	1	16	yes	—
63	C₂×C₄	C₂³×C₄	1	2	yes	—
64	C₂×C₄	C₂³×C₄	1	4	yes	—
65	C₈	C₂×C₄²	1	32	yes	complemented
66	C₂²	C₄³	1	4	yes	—
67	C₂²	C₂×#97	1	8	yes	—
68	C₂²	C₄×M₁₆	1	16	yes	—
69	C₂²	C₂²×C₄²	1	1	yes	characteristic
70	C₂²	C₂²×C₄²	1	2	yes	—
71	C₂²	C₂²×M₁₆	1	4	yes	—
72	C₄	—	2	16	no	—
73	C₄	C₄³	1	4	yes	—
74	C₄	C₂×#97	1	16	yes	direct product factor
75	C₄	C₂²×C₄²	1	4	yes	—
76	C₂	C₄×#97	1	8	yes	direct product factor
77	C₂	C₂×C₄³	1	1	yes	characteristic
78	C₂	C₂²×#97	1	2	yes	—
79	C₂	C₂×C₄×M₁₆	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂²
commutator	C₂	C₂×C₄³
Frattini, cosocle	C₂²×C₄	C₂⁴
socle, prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. C₄×#97	C₂	direct (N × H)
9. C₂×#97	C₄	direct (N × H)
11. C₂×C₄×C₈	C₄	semidirect
29. #97	C₂×C₄	direct (N × H)
30. C₄×C₈	C₂×C₄	semidirect
42. C₂×C₈	C₄²	semidirect
56. C₂×C₄	#97	direct (N × H)
65. C₈	C₂×C₄²	semidirect
74. C₄	C₂×#97	direct (N × H)
76. C₂	C₄×#97	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×#97	2
C₂²×C₄×C₈	2
C₂×C₄³	2
C₄×#97	2

Subgroups — C2×C4×#97 #20326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#97 #20326