Subgroups — #1812

Order 128.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131	C₂	1	1	yes	maximal, characteristic
2	#427	C₂	1	2	yes	maximal, complemented
3	#410	C₂	1	2	yes	maximal
4	H(3,ℤ/4)	C₂	1	2	yes	maximal, complemented
5	C₂×#41	C₂²	1	1	yes	characteristic
6	#131	—	2	2	no	—
7	#131	—	2	2	no	—
8	C₂×M₁₆	C₄	1	2	yes	complemented
9	C₂×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₈	C₄	1	2	yes	complemented
11	C₂²×C₈	C₂²	1	1	yes	characteristic
12	#117	C₂²	1	4	yes	—
13	#107	—	2	2	no	—
14	#106	—	2	2	no	—
15	#99	—	2	4	no	—
16	#41	—	2	2	no	—
17	#41	—	2	2	no	—
18	C₂×Q₈	Q₈	1	1	yes	characteristic
19	C₂×D₄	D₄	1	2	yes	—
20	C₂×D₄	Q₈	1	1	yes	characteristic
21	C₂²×C₄	C₂×C₄	1	2	yes	—
22	C₂²×C₄	C₂³	1	1	yes	characteristic
23	M₁₆	—	2	2	no	—
24	M₁₆	—	2	4	no	—
25	C₂×C₈	—	2	2	no	—
26	C₂×C₈	—	2	2	no	—
27	C₂×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈	D₄	1	2	yes	—
31	C₂×C₈	D₄	1	2	yes	—
32	C₄⋊C₄	—	2	4	no	—
33	C₂²⋊C₄	—	2	4	no	—
34	C₄²	—	2	4	no	—
35	C₂³	C₄⋊C₄	1	2	yes	—
36	C₂³	C₂×D₄	1	1	yes	characteristic
37	Q₈	—	2	2	no	—
38	D₄	—	2	4	no	—
39	D₄	—	2	2	no	—
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	4	4	no	—
43	C₂×C₄	C₄⋊C₄	1	2	yes	—
44	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄	C₂×D₄	1	2	yes	—
46	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄	#41	1	2	yes	—
48	C₈	—	2	2	no	—
49	C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₈	C₂²⋊C₄	1	1	yes	characteristic
51	C₈	C₂²⋊C₄	1	2	yes	—
52	C₂²	—	2	2	no	—
53	C₂²	—	4	1	no	—
54	C₂²	—	4	2	no	—
55	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²	#121	1	2	yes	—
57	C₄	—	2	2	no	—
58	C₄	—	8	4	no	—
59	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄	#117	1	1	yes	characteristic
61	C₄	C₂²⋊Q₈	1	2	yes	—
62	C₂	—	2	2	no	—
63	C₂	—	2	1	no	—
64	C₂	—	4	2	no	—
65	C₂	#380	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂	#380
prime radical	C₂×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #427	C₂	semidirect
4. H(3,ℤ/4)	C₂	semidirect
8. C₂×M₁₆	C₄	semidirect
10. C₂²×C₈	C₄	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#131	2
#427	2
#410	2
H(3,ℤ/4)	2