Subgroups — C₄×#126 #2304

Order 128.

258 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	#388	C₂	1	3	yes	maximal
3	#387	C₂	1	3	yes	maximal, complemented
4	#383	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
6	C₄×C₂²⋊C₄	C₂	1	3	yes	maximal
7	C₄³	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄	C₂²	1	1	yes	characteristic
9	#126	C₄	1	8	yes	direct product factor
10	C₂×C₄⋊C₄	C₂²	1	3	yes	—
11	C₂×C₄⋊C₄	C₂²	1	3	yes	—
12	C₂×C₂²⋊C₄	C₂²	1	3	yes	—
13	C₂×C₂²⋊C₄	C₂²	1	3	yes	—
14	C₂×C₄²	C₂²	1	3	yes	—
15	C₂×C₄²	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	—	2	6	no	—
24	C₂²×C₄	C₂³	1	1	yes	characteristic
25	C₂²×C₄	C₂³	1	3	yes	—
26	C₂²×C₄	C₂³	1	3	yes	—
27	C₂²×C₄	C₂³	1	3	yes	—
28	C₂²×C₄	C₂³	1	3	yes	—
29	C₂²×C₄	C₂³	1	1	yes	characteristic
30	C₄⋊C₄	—	2	6	no	—
31	C₄⋊C₄	C₂×C₄	1	12	yes	complemented
32	C₂²⋊C₄	—	2	6	no	—
33	C₂²⋊C₄	C₂×C₄	1	12	yes	—
34	C₄²	—	2	6	no	—
35	C₄²	—	2	6	no	—
36	C₄²	—	2	6	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	—	4	4	no	—
53	C₂×C₄	C₂²×C₄	1	6	yes	—
54	C₂×C₄	C₂²×C₄	1	6	yes	—
55	C₂×C₄	#41	1	6	yes	—
56	C₂×C₄	#41	1	6	yes	—
57	C₂²	—	2	6	no	—
58	C₂²	—	4	1	no	—
59	C₂²	—	4	3	no	—
60	C₂²	C₂³×C₄	1	1	yes	characteristic
61	C₂²	C₂×#41	1	3	yes	—
62	C₂²	C₂×#41	1	3	yes	—
63	C₄	—	2	6	no	—
64	C₄	—	2	6	no	—
65	C₄	—	2	6	no	—
66	C₄	—	4	1	no	—
67	C₄	—	4	3	no	—
68	C₄	#126	1	4	yes	direct product factor
69	C₂	—	4	2	no	—
70	C₂	C₄×#41	1	3	yes	—
71	C₂	C₂×#126	1	1	yes	characteristic
72	C₂	#529	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #387	C₂	semidirect
4. #383	C₂	semidirect
5. C₄×C₄⋊C₄	C₂	semidirect
7. C₄³	C₂	semidirect
9. #126	C₄	direct (N × H)
31. C₄⋊C₄	C₂×C₄	semidirect
37. C₄²	C₂×C₄	semidirect
68. C₄	#126	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#126	2
#388	2
#387	2
#383	2
C₄×C₄⋊C₄	2
C₄×C₂²⋊C₄	2
C₄³	2

Subgroups — C4×#126 #2304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#126 #2304