Subgroups — C₂×C₄×C₂²⋊C₄ #2268

Order 128.

680 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊C₄	C₂	1	2	yes	maximal
3	C₂²×C₄²	C₂	1	2	yes	maximal, complemented
4	C₄×C₂²⋊C₄	C₂	1	8	yes	maximal, direct product factor
5	C₂×#95	C₂	1	2	yes	maximal, complemented
6	C₂⁵	C₂²	1	1	yes	characteristic
7	C₂³×C₄	—	2	8	no	—
8	C₂³×C₄	C₄	1	8	yes	complemented
9	C₂³×C₄	C₂²	1	1	yes	characteristic
10	C₂³×C₄	C₂²	1	4	yes	—
11	C₂³×C₄	C₂²	1	4	yes	—
12	C₂³×C₄	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₄	C₄	1	16	yes	direct product factor
14	C₂×C₂²⋊C₄	C₂²	1	8	yes	—
15	C₂×C₄²	—	2	8	no	—
16	C₂×C₄²	C₂²	1	8	yes	complemented
17	#95	C₂²	1	8	yes	complemented
18	C₂⁴	—	2	4	no	—
19	C₂⁴	—	2	4	no	—
20	C₂⁴	C₂×C₄	1	8	yes	—
21	C₂⁴	C₂×C₄	1	4	yes	—
22	C₂⁴	C₂³	1	2	yes	—
23	C₂⁴	C₂³	1	1	yes	characteristic
24	C₂²×C₄	—	2	8	no	—
25	C₂²×C₄	—	2	4	no	—
26	C₂²×C₄	—	2	8	no	—
27	C₂²×C₄	—	2	32	no	—
28	C₂²×C₄	—	2	16	no	—
29	C₂²×C₄	C₂×C₄	1	4	yes	—
30	C₂²×C₄	C₂×C₄	1	4	yes	—
31	C₂²×C₄	C₂×C₄	1	16	yes	complemented
32	C₂²×C₄	C₂×C₄	1	16	yes	complemented
33	C₂²×C₄	D₄	1	8	yes	—
34	C₂²×C₄	C₂³	1	2	yes	—
35	C₂²×C₄	C₂³	1	8	yes	—
36	C₂²×C₄	C₂³	1	2	yes	—
37	C₂²⋊C₄	C₂×C₄	1	32	yes	direct product factor
38	C₄²	—	2	16	no	—
39	C₂³	—	2	8	no	—
40	C₂³	—	2	8	no	—
41	C₂³	—	2	32	no	—
42	C₂³	—	2	4	no	—
43	C₂³	—	2	4	no	—
44	C₂³	C₄²	1	16	yes	complemented
45	C₂³	C₂²×C₄	1	8	yes	—
46	C₂³	C₂²×C₄	1	4	yes	—
47	C₂³	C₂²×C₄	1	2	yes	—
48	C₂³	C₂²×C₄	1	4	yes	—
49	C₂³	C₂×D₄	1	4	yes	—
50	C₂³	#41	1	4	yes	—
51	C₂³	C₂⁴	1	1	yes	characteristic
52	C₂×C₄	—	2	8	no	—
53	C₂×C₄	—	2	32	no	—
54	C₂×C₄	—	2	8	no	—
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	8	no	—
57	C₂×C₄	—	2	32	no	—
58	C₂×C₄	C₂²⋊C₄	1	16	yes	direct product factor
59	C₂×C₄	C₂²×C₄	1	4	yes	—
60	C₂×C₄	C₂²×C₄	1	4	yes	—
61	C₂×C₄	C₂²×C₄	1	16	yes	complemented
62	C₂×C₄	C₂×D₄	1	8	yes	—
63	C₂²	—	2	4	no	—
64	C₂²	—	2	4	no	—
65	C₂²	—	2	16	no	—
66	C₂²	—	2	32	no	—
67	C₂²	C₂×C₄²	1	4	yes	—
68	C₂²	C₂×C₄²	1	8	yes	complemented
69	C₂²	C₂×C₂²⋊C₄	1	4	yes	—
70	C₂²	#117	1	4	yes	—
71	C₂²	C₄×D₄	1	16	yes	—
72	C₂²	C₂³×C₄	1	1	yes	characteristic
73	C₂²	C₂³×C₄	1	2	yes	—
74	C₂²	C₂²×D₄	1	2	yes	—
75	C₂²	C₂×#41	1	2	yes	—
76	C₄	—	2	4	no	—
77	C₄	—	2	16	no	—
78	C₄	C₂×C₂²⋊C₄	1	8	yes	direct product factor
79	C₂	—	2	8	no	—
80	C₂	C₄×C₂²⋊C₄	1	8	yes	direct product factor
81	C₂	C₂²×C₄²	1	1	yes	characteristic
82	C₂	C₂²×C₂²⋊C₄	1	1	yes	characteristic
83	C₂	C₂×#117	1	1	yes	characteristic
84	C₂	C₂×C₄×D₄	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²
commutator	C₂	C₂²×C₄²
Frattini, cosocle	C₂³	C₂⁴
socle	C₂⁴	C₂³
prime radical	C₂⁵	C₂²
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
3. C₂²×C₄²	C₂	semidirect
4. C₄×C₂²⋊C₄	C₂	direct (N × H)
5. C₂×#95	C₂	semidirect
8. C₂³×C₄	C₄	semidirect
13. C₂×C₂²⋊C₄	C₄	direct (N × H)
16. C₂×C₄²	C₂²	semidirect
17. #95	C₂²	semidirect
31. C₂²×C₄	C₂×C₄	semidirect
32. C₂²×C₄	C₂×C₄	semidirect
37. C₂²⋊C₄	C₂×C₄	direct (N × H)
44. C₂³	C₄²	semidirect
58. C₂×C₄	C₂²⋊C₄	direct (N × H)
61. C₂×C₄	C₂²×C₄	semidirect
68. C₂²	C₂×C₄²	semidirect
78. C₄	C₂×C₂²⋊C₄	direct (N × H)
80. C₂	C₄×C₂²⋊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
C₂⁴×C₄	2
C₂²×C₂²⋊C₄	2
C₂²×C₄²	2
C₄×C₂²⋊C₄	2
C₂×#95	2

Subgroups — C2×C4×C22⋊C4 #2268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₂²⋊C₄ #2268