Subgroups — C₂×C₄×C₄⋊C₄ #2269

Order 128.

464 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₄	C₂	1	2	yes	maximal
2	C₂²×C₄²	C₂	1	2	yes	maximal
3	C₂²×C₄²	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊C₄	C₂	1	8	yes	maximal, direct product factor
5	C₂×#95	C₂	1	2	yes	maximal
6	C₂³×C₄	C₂²	1	4	yes	—
7	C₂³×C₄	C₂²	1	1	yes	characteristic
8	C₂³×C₄	C₂²	1	1	yes	characteristic
9	C₂³×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄	C₄	1	16	yes	direct product factor
11	C₂×C₄⋊C₄	C₂²	1	8	yes	—
12	C₂×C₄²	—	2	8	no	—
13	C₂×C₄²	C₄	1	8	yes	complemented
14	C₂×C₄²	C₂²	1	8	yes	—
15	C₂×C₄²	C₂²	1	4	yes	—
16	#95	C₂²	1	8	yes	—
17	C₂⁴	C₂³	1	1	yes	characteristic
18	C₂²×C₄	—	2	8	no	—
19	C₂²×C₄	—	2	4	no	—
20	C₂²×C₄	—	2	8	no	—
21	C₂²×C₄	C₂×C₄	1	4	yes	—
22	C₂²×C₄	C₂×C₄	1	16	yes	—
23	C₂²×C₄	C₂×C₄	1	4	yes	—
24	C₂²×C₄	C₂×C₄	1	8	yes	—
25	C₂²×C₄	C₂×C₄	1	4	yes	—
26	C₂²×C₄	D₄	1	4	yes	—
27	C₂²×C₄	Q₈	1	4	yes	—
28	C₂²×C₄	C₂³	1	2	yes	—
29	C₂²×C₄	C₂³	1	8	yes	—
30	C₂²×C₄	C₂³	1	2	yes	—
31	C₂²×C₄	C₂³	1	2	yes	—
32	C₄⋊C₄	C₂×C₄	1	32	yes	direct product factor
33	C₄²	—	2	16	no	—
34	C₄²	C₂×C₄	1	16	yes	complemented
35	C₂³	C₂²×C₄	1	2	yes	—
36	C₂³	C₂²×C₄	1	4	yes	—
37	C₂³	C₂×D₄	1	2	yes	—
38	C₂³	C₂×Q₈	1	2	yes	—
39	C₂³	#41	1	4	yes	—
40	C₂³	C₂⁴	1	1	yes	characteristic
41	C₂×C₄	—	2	8	no	—
42	C₂×C₄	—	2	32	no	—
43	C₂×C₄	—	2	8	no	—
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	—	2	8	no	—
46	C₂×C₄	C₄²	1	16	yes	complemented
47	C₂×C₄	C₄⋊C₄	1	16	yes	direct product factor
48	C₂×C₄	C₂²×C₄	1	4	yes	—
49	C₂×C₄	C₂²×C₄	1	16	yes	—
50	C₂×C₄	C₂²×C₄	1	4	yes	—
51	C₂×C₄	C₂²×C₄	1	8	yes	—
52	C₂×C₄	C₂²×C₄	1	4	yes	—
53	C₂×C₄	C₂×D₄	1	4	yes	—
54	C₂×C₄	C₂×Q₈	1	4	yes	—
55	C₂²	C₂×C₄²	1	4	yes	—
56	C₂²	C₂×C₄⋊C₄	1	4	yes	—
57	C₂²	#117	1	4	yes	—
58	C₂²	C₄×D₄	1	8	yes	—
59	C₂²	C₄×Q₈	1	8	yes	—
60	C₂²	C₂³×C₄	1	1	yes	characteristic
61	C₂²	C₂³×C₄	1	2	yes	—
62	C₂²	C₂²×D₄	1	1	yes	characteristic
63	C₂²	C₂²×Q₈	1	1	yes	characteristic
64	C₂²	C₂×#41	1	2	yes	—
65	C₄	—	2	4	no	—
66	C₄	—	2	16	no	—
67	C₄	C₂×C₄²	1	8	yes	complemented
68	C₄	C₂×C₄⋊C₄	1	8	yes	direct product factor
69	C₂	C₄×C₄⋊C₄	1	8	yes	direct product factor
70	C₂	C₂²×C₄²	1	1	yes	characteristic
71	C₂	C₂²×C₄⋊C₄	1	1	yes	characteristic
72	C₂	C₂×#117	1	1	yes	characteristic
73	C₂	C₂×C₄×D₄	1	2	yes	—
74	C₂	C₂×C₄×Q₈	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²
commutator	C₂	C₂²×C₄²
Frattini, cosocle	C₂³	C₂⁴
socle, prime radical	C₂⁴	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. C₄×C₄⋊C₄	C₂	direct (N × H)
10. C₂×C₄⋊C₄	C₄	direct (N × H)
13. C₂×C₄²	C₄	semidirect
32. C₄⋊C₄	C₂×C₄	direct (N × H)
34. C₄²	C₂×C₄	semidirect
46. C₂×C₄	C₄²	semidirect
47. C₂×C₄	C₄⋊C₄	direct (N × H)
67. C₄	C₂×C₄²	semidirect
68. C₄	C₂×C₄⋊C₄	direct (N × H)
69. C₂	C₄×C₄⋊C₄	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×C₄⋊C₄	2
C₂²×C₄²	2
C₄×C₄⋊C₄	2
C₂×#95	2

Subgroups — C2×C4×C4⋊C4 #2269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₄⋊C₄ #2269