Subgroups — #2383

Order 128.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#126	C₂	1	1	yes	maximal, characteristic
2	#389	C₂	1	3	yes	maximal, complemented
3	#388	C₂	1	3	yes	maximal
4	#387	C₂	1	3	yes	maximal, complemented
5	#385	C₂	1	3	yes	maximal
6	#384	C₂	1	1	yes	maximal, characteristic, complemented
7	#383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄	C₂²	1	1	yes	characteristic
9	#126	C₄	1	8	yes	complemented
10	C₂×C₄⋊C₄	C₂²	1	3	yes	—
11	C₂×C₄⋊C₄	C₂²	1	3	yes	—
12	C₂×C₄⋊C₄	C₂²	1	3	yes	—
13	C₂×C₄⋊C₄	C₂²	1	3	yes	—
14	C₂×C₂²⋊C₄	C₂²	1	3	yes	—
15	C₂×C₂²⋊C₄	C₂²	1	3	yes	—
16	C₂×C₄²	C₂²	1	3	yes	—
17	C₂×C₄²	C₂²	1	1	yes	characteristic
18	#95	C₂²	1	3	yes	—
19	#95	C₂²	1	3	yes	—
20	#95	C₂²	1	3	yes	—
21	#95	C₂²	1	3	yes	—
22	C₂⁴	C₂³	1	1	yes	characteristic
23	C₂²×C₄	—	4	3	no	—
24	C₂²×C₄	C₂³	1	3	yes	—
25	C₂²×C₄	C₂³	1	1	yes	characteristic
26	C₂²×C₄	C₂³	1	1	yes	characteristic
27	C₂²×C₄	C₂³	1	3	yes	—
28	C₂²×C₄	C₂³	1	3	yes	—
29	C₂²×C₄	C₂³	1	3	yes	—
30	C₄⋊C₄	—	4	3	no	—
31	C₄⋊C₄	—	4	3	no	—
32	C₄⋊C₄	—	4	3	no	—
33	C₄⋊C₄	C₂×C₄	1	12	yes	complemented
34	C₂²⋊C₄	—	4	3	no	—
35	C₂²⋊C₄	C₂×C₄	1	12	yes	—
36	C₄²	—	4	3	no	—
37	C₄²	C₂×C₄	1	4	yes	complemented
38	C₂³	—	2	3	no	—
39	C₂³	—	2	3	no	—
40	C₂³	C₂²×C₄	1	2	yes	—
41	C₂³	C₂⁴	1	1	yes	characteristic
42	C₂×C₄	—	2	3	no	—
43	C₂×C₄	—	2	3	no	—
44	C₂×C₄	—	2	3	no	—
45	C₂×C₄	—	2	3	no	—
46	C₂×C₄	—	2	3	no	—
47	C₂×C₄	—	2	3	no	—
48	C₂×C₄	—	2	3	no	—
49	C₂×C₄	—	2	3	no	—
50	C₂×C₄	—	2	3	no	—
51	C₂×C₄	—	2	3	no	—
52	C₂×C₄	—	2	6	no	—
53	C₂×C₄	—	4	4	no	—
54	C₂×C₄	C₂²×C₄	1	6	yes	—
55	C₂×C₄	C₂²×C₄	1	6	yes	—
56	C₂²	—	2	6	no	—
57	C₂²	—	4	1	no	—
58	C₂²	—	4	3	no	—
59	C₂²	C₂³×C₄	1	1	yes	characteristic
60	C₂²	H(5,𝔽₂)	1	3	yes	—
61	C₂²	2₋¹⁺⁴	1	3	yes	—
62	C₄	—	2	6	no	—
63	C₄	—	2	6	no	—
64	C₄	—	4	3	no	—
65	C₄	—	4	1	no	—
66	C₄	—	4	1	no	—
67	C₄	—	4	3	no	—
68	C₂	—	4	2	no	—
69	C₂	#520	1	3	yes	—
70	C₂	#560	1	1	yes	characteristic
71	C₂	#563	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
2. #389	C₂	semidirect
4. #387	C₂	semidirect
6. #384	C₂	semidirect
7. #383	C₂	semidirect
9. #126	C₄	semidirect
33. C₄⋊C₄	C₂×C₄	semidirect
37. C₄²	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×#126	2
#389	2
#388	2
#387	2
#385	2
#384	2
#383	2