Subgroups — #1808

Order 128.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#107	C₂	1	2	yes	maximal, complemented
3	#380	C₂	1	2	yes	maximal
4	#340	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄	C₂²	1	1	yes	characteristic
6	C₂²×C₈	—	2	4	no	—
7	C₂²×C₈	C₄	1	4	yes	complemented
8	C₂²×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₂²⋊C₄	—	2	2	no	—
12	#107	—	2	4	no	—
13	#95	—	2	2	no	—
14	C₂⁴	D₄	1	1	yes	characteristic
15	C₂²×C₄	—	2	2	no	—
16	C₂²×C₄	—	2	2	no	—
17	C₂²×C₄	—	2	4	no	—
18	C₂²×C₄	C₂×C₄	1	2	yes	—
19	C₂²×C₄	D₄	1	1	yes	characteristic
20	C₂²×C₄	Q₈	1	2	yes	—
21	C₂²×C₄	C₂³	1	1	yes	characteristic
22	C₂×C₈	—	2	2	no	—
23	C₂×C₈	—	2	8	no	—
24	C₂×C₈	C₂×C₄	1	2	yes	—
25	C₂×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₈	D₄	1	4	yes	—
27	C₄⋊C₄	—	2	4	no	—
28	C₄⋊C₄	—	2	4	no	—
29	C₂²⋊C₄	—	4	4	no	—
30	C₂³	—	2	2	no	—
31	C₂³	—	2	2	no	—
32	C₂³	C₄⋊C₄	1	2	yes	—
33	C₂³	D₈	1	2	yes	—
34	C₂³	Q₁₆	1	2	yes	—
35	C₂³	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄	—	2	2	no	—
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	—	2	2	no	—
39	C₂×C₄	—	2	4	no	—
40	C₂×C₄	—	4	4	no	—
41	C₂×C₄	—	4	4	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	4	yes	complemented
50	C₂²	—	2	2	no	—
51	C₂²	—	2	2	no	—
52	C₂²	—	2	4	no	—
53	C₂²	—	2	4	no	—
54	C₂²	#107	1	4	yes	complemented
55	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²	#121	1	2	yes	—
57	C₂²	C₂×D₈	1	1	yes	characteristic
58	C₂²	C₂×Q₁₆	1	1	yes	characteristic
59	C₂²	#135	1	2	yes	—
60	C₄	—	2	2	no	—
61	C₄	—	8	4	no	—
62	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄	#117	1	1	yes	characteristic
64	C₄	C₂²⋊Q₈	1	2	yes	—
65	C₂	—	2	4	no	—
66	C₂	#380	1	1	yes	characteristic
67	C₂	C₂×#107	1	1	yes	characteristic
68	C₂	#427	1	1	yes	characteristic
69	C₂	#466	1	2	yes	—
70	C₂	#467	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×#107	C₂	semidirect
4. #340	C₂	semidirect
7. C₂²×C₈	C₄	semidirect
49. C₈	C₂²⋊C₄	semidirect
54. C₂²	#107	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₈	2
C₂×#107	2
#380	2
#340	2