Subgroups — #1757

Order 128.

142 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	#413	C₂	1	1	yes	maximal, characteristic
3	#410	C₂	1	1	yes	maximal, characteristic
4	#405	C₂	1	2	yes	maximal, complemented
5	#344	C₂	1	2	yes	maximal, complemented
6	C₂×#41	C₂²	1	1	yes	characteristic
7	#131	—	2	2	no	—
8	#131	—	2	2	no	—
9	C₂×M₁₆	C₄	1	2	yes	complemented
10	C₂×M₁₆	C₂²	1	2	yes	—
11	C₂×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₈	C₄	1	2	yes	complemented
13	C₂²×C₈	C₂²	1	1	yes	characteristic
14	#117	C₂²	1	2	yes	—
15	#101	C₄	1	2	yes	complemented
16	#100	C₄	1	2	yes	complemented
17	#99	C₄	1	4	yes	—
18	#97	—	2	2	no	—
19	C₄×C₈	—	2	2	no	—
20	#41	—	2	2	no	—
21	#41	—	2	2	no	—
22	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
23	C₂×D₄	C₂×C₄	1	2	yes	—
24	C₂×D₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄	C₂×C₄	1	2	yes	—
26	C₂²×C₄	C₂³	1	1	yes	characteristic
27	M₁₆	—	2	2	no	—
28	M₁₆	—	2	2	no	—
29	M₁₆	—	2	4	no	—
30	M₁₆	C₂×C₄	1	4	yes	—
31	C₂×C₈	—	2	2	no	—
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	—	2	2	no	—
34	C₂×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈	D₄	1	2	yes	—
38	C₂×C₈	D₄	1	2	yes	—
39	C₄⋊C₄	—	2	2	no	—
40	C₂²⋊C₄	C₂×C₄	1	4	yes	—
41	C₄²	—	2	2	no	—
42	C₂³	C₄²	1	2	yes	—
43	C₂³	C₂²×C₄	1	1	yes	characteristic
44	Q₈	—	2	2	no	—
45	D₄	—	2	4	no	—
46	D₄	—	2	2	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	2	4	no	—
50	C₂×C₄	C₄²	1	2	yes	—
51	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄	C₂×D₄	1	2	yes	—
54	C₂×C₄	#41	1	2	yes	—
55	C₈	—	2	2	no	—
56	C₈	—	2	4	no	—
57	C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₈	C₂²⋊C₄	1	2	yes	—
59	C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²	—	2	2	no	—
61	C₂²	—	4	1	no	—
62	C₂²	—	4	2	no	—
63	C₂²	C₂×C₄²	1	1	yes	characteristic
64	C₂²	C₄×D₄	1	2	yes	—
65	C₄	—	2	2	no	—
66	C₄	—	4	4	no	—
67	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄	#117	1	1	yes	characteristic
69	C₄	C₄×D₄	1	2	yes	—
70	C₂	—	2	2	no	—
71	C₂	—	2	1	no	—
72	C₂	—	4	2	no	—
73	C₂	C₄×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊C₄
commutator	C₂²	C₂×C₄²
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂	C₄×C₂²⋊C₄
prime radical	C₂×#41	C₂²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #405	C₂	semidirect
5. #344	C₂	semidirect
9. C₂×M₁₆	C₄	semidirect
12. C₂²×C₈	C₄	semidirect
15. #101	C₄	semidirect
16. #100	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
#413	2
#410	2
#405	2
#344	2