Subgroups — #2913

Order 128.

238 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	#520	C₂	1	2	yes	maximal
3	#428	C₂	1	6	yes	maximal, complemented
4	#427	C₂	1	3	yes	maximal, complemented
5	C₂×#107	C₂	1	3	yes	maximal, complemented
6	C₂×#41	C₂²	1	1	yes	characteristic
7	#131	C₄	1	8	yes	complemented
8	C₂×M₁₆	C₂²	1	3	yes	—
9	C₂²×C₈	C₂²	1	3	yes	—
10	C₄×Q₈	—	2	2	no	—
11	C₄×D₄	—	2	6	no	—
12	#117	C₂²	1	6	yes	—
13	C₂×C₄⋊C₄	C₂²	1	6	yes	—
14	#107	C₂²	1	3	yes	—
15	#107	C₂²	1	6	yes	—
16	#107	C₂²	1	1	yes	characteristic
17	#106	C₂²	1	6	yes	—
18	#41	C₂×C₄	1	4	yes	—
19	#41	Q₈	1	4	yes	—
20	C₂×Q₈	D₄	1	1	yes	characteristic
21	C₂×D₄	D₄	1	3	yes	—
22	C₂²×C₄	—	2	6	no	—
23	C₂²×C₄	C₂³	1	3	yes	—
24	M₁₆	C₂×C₄	1	12	yes	complemented
25	C₂×C₈	C₂×C₄	1	12	yes	complemented
26	C₂×C₈	C₂³	1	3	yes	—
27	C₂×C₈	C₂³	1	1	yes	characteristic
28	C₄⋊C₄	—	2	6	no	—
29	C₄⋊C₄	C₂³	1	2	yes	—
30	C₄⋊C₄	C₂³	1	6	yes	—
31	C₂²⋊C₄	—	2	6	no	—
32	C₄²	—	2	6	no	—
33	C₂³	C₂×D₄	1	3	yes	—
34	Q₈	C₄⋊C₄	1	4	yes	—
35	D₄	C₄⋊C₄	1	12	yes	—
36	C₂×C₄	—	2	2	no	—
37	C₂×C₄	—	2	6	no	—
38	C₂×C₄	—	8	6	no	—
39	C₂×C₄	C₂²×C₄	1	6	yes	—
40	C₂×C₄	C₂×D₄	1	3	yes	—
41	C₂×C₄	C₂×Q₈	1	6	yes	—
42	C₂×C₄	C₂⁴	1	1	yes	characteristic
43	C₈	C₂²×C₄	1	2	yes	—
44	C₈	C₂²×C₄	1	6	yes	—
45	C₂²	—	2	3	no	—
46	C₂²	—	2	3	no	—
47	C₂²	C₂×C₄⋊C₄	1	6	yes	—
48	C₂²	C₂²×D₄	1	1	yes	characteristic
49	C₄	—	4	6	no	—
50	C₄	—	4	2	no	—
51	C₄	C₂×C₄⋊C₄	1	6	yes	—
52	C₄	C₂³×C₄	1	1	yes	characteristic
53	C₄	C₂²×Q₈	1	1	yes	characteristic
54	C₂	—	2	6	no	—
55	C₂	C₂²×C₄⋊C₄	1	1	yes	characteristic
56	C₂	#576	1	1	yes	characteristic
57	C₂	#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator	C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂×#41	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #428	C₂	semidirect
4. #427	C₂	semidirect
5. C₂×#107	C₂	semidirect
7. #131	C₄	semidirect
24. M₁₆	C₂×C₄	semidirect
25. C₂×C₈	C₂×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
#520	2
#428	2
#427	2
C₂×#107	2