Subgroups — #3138

Order 128.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#127	C₂	1	1	yes	maximal, characteristic
3	#490	C₂	1	4	yes	maximal, complemented
4	#486	C₂	1	4	yes	maximal, complemented
5	#418	C₂	1	4	yes	maximal, complemented
6	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄	—	2	2	no	—
8	C₂²×D₄	C₂²	1	2	yes	—
9	C₂×M₁₆	C₂²	1	2	yes	—
10	C₄⋊Q₈	C₂²	1	2	yes	complemented
11	#127	—	2	2	no	—
12	#127	C₂²	1	2	yes	—
13	#127	C₂²	1	2	yes	—
14	#125	C₂²	1	2	yes	complemented
15	C₂²⋊Q₈	—	2	2	no	—
16	C₄×Q₈	—	2	2	no	—
17	#117	C₂²	1	2	yes	—
18	C₂×C₄²	C₂²	1	1	yes	characteristic
19	#102	C₂²	1	16	yes	complemented
20	#97	C₂²	1	2	yes	complemented
21	C₄×C₈	C₂²	1	2	yes	complemented
22	C₂⁴	—	2	2	no	—
23	C₂×Q₈	—	2	2	no	—
24	C₂×D₄	—	2	2	no	—
25	C₂×D₄	—	2	4	no	—
26	C₂×D₄	—	2	4	no	—
27	C₂×D₄	—	2	8	no	—
28	C₂×D₄	—	4	8	no	—
29	C₂×D₄	C₂³	1	4	yes	—
30	C₂²×C₄	D₄	1	2	yes	—
31	C₂²×C₄	C₂³	1	1	yes	characteristic
32	M₁₆	—	2	4	no	—
33	C₂×C₈	C₂³	1	4	yes	—
34	C₄⋊C₄	—	2	4	no	—
35	C₄⋊C₄	—	2	2	no	—
36	C₄⋊C₄	C₂³	1	4	yes	—
37	C₂²⋊C₄	—	2	2	no	—
38	C₄²	—	2	2	no	—
39	C₄²	D₄	1	2	yes	—
40	C₄²	C₂³	1	1	yes	characteristic
41	C₄²	C₂³	1	1	yes	characteristic
42	C₂³	—	2	4	no	—
43	C₂³	—	4	8	no	—
44	C₂³	—	4	4	no	—
45	C₂³	C₂×D₄	1	1	yes	characteristic
46	Q₈	—	4	4	no	—
47	D₄	—	2	8	no	—
48	D₄	—	4	8	no	—
49	D₄	—	4	4	no	—
50	D₄	—	4	8	no	—
51	C₂×C₄	—	2	4	no	—
52	C₂×C₄	—	2	4	no	—
53	C₂×C₄	C₂×D₄	1	2	yes	—
54	C₂×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄	C₂×D₄	1	2	yes	—
56	C₂×C₄	#41	1	4	yes	—
57	C₂×C₄	C₂⁴	1	1	yes	characteristic
58	C₈	—	2	4	no	—
59	C₂²	—	2	2	no	—
60	C₂²	—	4	4	no	—
61	C₂²	—	4	8	no	—
62	C₂²	—	8	8	no	—
63	C₂²	#124	1	2	yes	—
64	C₂²	C₂²×D₄	1	1	yes	characteristic
65	C₄	—	2	2	no	—
66	C₄	—	4	4	no	—
67	C₄	#124	1	2	yes	—
68	C₄	AGL(1,ℤ/8)	1	4	yes	—
69	C₄	C₂×#41	1	2	yes	—
70	C₂	—	2	2	no	—
71	C₂	—	8	4	no	—
72	C₂	C₂×#124	1	1	yes	characteristic
73	C₂	C₂×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #533	C₂	semidirect
3. #490	C₂	semidirect
4. #486	C₂	semidirect
5. #418	C₂	semidirect
6. C₄×M₁₆	C₂	semidirect
10. C₄⋊Q₈	C₂²	semidirect
14. #125	C₂²	semidirect
19. #102	C₂²	semidirect
20. #97	C₂²	semidirect
21. C₄×C₈	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#533	2
C₂×#127	2
#490	2
#486	2
#418	2
C₄×M₁₆	2