Subgroups — C₂×#341 #1737

Order 128.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆	C₂	1	2	yes	maximal
2	C₂³×C₈	C₂	1	1	yes	maximal, characteristic
3	#341	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₄	C₂²	1	1	yes	characteristic
5	C₂×M₁₆	—	2	4	no	—
6	C₂×M₁₆	—	2	2	no	—
7	C₂×M₁₆	C₄	1	8	yes	—
8	C₂×M₁₆	C₂²	1	4	yes	—
9	C₂²×C₈	—	2	4	no	—
10	C₂²×C₈	—	2	2	no	—
11	C₂²×C₈	C₄	1	2	yes	—
12	C₂²×C₈	C₄	1	2	yes	—
13	C₂²×C₈	C₂²	1	2	yes	—
14	C₂⁴	Q₈	1	1	yes	characteristic
15	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄	C₂×C₄	1	4	yes	—
18	C₂²×C₄	D₄	1	4	yes	—
19	C₂²×C₄	D₄	1	1	yes	characteristic
20	C₂²×C₄	D₄	1	1	yes	characteristic
21	C₂²×C₄	Q₈	1	1	yes	characteristic
22	C₂²×C₄	C₂³	1	1	yes	characteristic
23	M₁₆	—	2	8	no	—
24	M₁₆	—	2	4	no	—
25	M₁₆	C₂×C₄	1	8	yes	—
26	C₂×C₈	—	2	4	no	—
27	C₂×C₈	—	2	4	no	—
28	C₂×C₈	—	2	8	no	—
29	C₂×C₈	—	2	4	no	—
30	C₂×C₈	—	2	4	no	—
31	C₂×C₈	C₂×C₄	1	4	yes	—
32	C₂³	—	2	4	no	—
33	C₂³	C₄⋊C₄	1	4	yes	—
34	C₂³	C₄⋊C₄	1	1	yes	characteristic
35	C₂³	C₄⋊C₄	1	1	yes	characteristic
36	C₂³	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄	C₄²	1	2	yes	—
38	C₂×C₄	C₄²	1	2	yes	—
39	C₂×C₄	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄	C₂²⋊C₄	1	4	yes	—
42	C₂×C₄	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄	C₄⋊C₄	1	4	yes	—
44	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄	C₂²×C₄	1	2	yes	—
48	C₂×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄	C₂×D₄	1	2	yes	—
50	C₈	—	2	4	no	—
51	C₈	—	2	8	no	—
52	C₂²	—	2	8	no	—
53	C₂²	—	2	2	no	—
54	C₂²	—	2	2	no	—
55	C₂²	#95	1	2	yes	—
56	C₂²	#95	1	2	yes	—
57	C₂²	#108	1	4	yes	—
58	C₂²	C₂×C₄⋊C₄	1	2	yes	—
59	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄	#95	1	2	yes	—
61	C₄	#95	1	2	yes	—
62	C₄	C₂×C₄²	1	1	yes	characteristic
63	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₄	C₂×C₂²⋊C₄	1	2	yes	—
65	C₂	—	2	2	no	—
66	C₂	—	2	2	no	—
67	C₂	#341	1	4	yes	direct product factor
68	C₂	C₂×#95	1	1	yes	characteristic
69	C₂	C₂×#108	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄
commutator	C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂³	C₄⋊C₄
prime radical	C₂⁴	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ 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
3. #341	C₂	direct (N × H)
67. C₂	#341	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₁₆	2
C₂³×C₈	2
#341	2

Subgroups — C2×#341 #1737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#341 #1737