Subgroups — C₂×#349 #2114

Order 128.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×M₃₂	C₂	1	2	yes	maximal, complemented
3	#349	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₄	C₄	1	1	yes	characteristic
5	C₂×M₁₆	—	2	2	no	—
6	C₂×M₁₆	—	2	2	no	—
7	C₂×M₁₆	C₄	1	2	yes	—
8	C₂×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₈	C₄	1	1	yes	characteristic
10	C₂²×C₈	C₂²	1	1	yes	characteristic
11	M₃₂	—	2	2	no	—
12	M₃₂	C₂²	1	4	yes	complemented
13	C₂×C₁₆	—	2	2	no	—
14	C₂⁴	C₈	1	1	yes	characteristic
15	C₂²×C₄	—	2	1	no	—
16	C₂²×C₄	—	2	1	no	—
17	C₂²×C₄	—	2	1	no	—
18	C₂²×C₄	—	2	1	no	—
19	C₂²×C₄	C₈	1	1	yes	characteristic
20	C₂²×C₄	C₈	1	2	yes	—
21	C₂²×C₄	C₂×C₄	1	2	yes	—
22	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
23	M₁₆	—	2	4	no	—
24	M₁₆	—	2	4	no	—
25	C₂×C₈	—	2	1	no	—
26	C₂×C₈	—	2	1	no	—
27	C₂×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈	D₄	1	2	yes	—
30	C₂×C₈	D₄	1	2	yes	—
31	C₂×C₈	C₂³	1	1	yes	characteristic
32	C₁₆	—	2	4	no	—
33	C₂³	—	2	1	no	—
34	C₂³	—	2	1	no	—
35	C₂³	—	4	2	no	—
36	C₂³	C₂×C₈	1	1	yes	characteristic
37	C₂³	C₂×C₈	1	2	yes	—
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄	C₂×C₈	1	2	yes	—
47	C₂×C₄	C₂×C₈	1	1	yes	characteristic
48	C₂×C₄	M₁₆	1	1	yes	characteristic
49	C₂×C₄	M₁₆	1	1	yes	characteristic
50	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	C₂²⋊C₄	1	2	yes	—
53	C₈	C₂×D₄	1	2	yes	—
54	C₂²	—	2	2	no	—
55	C₂²	—	2	2	no	—
56	C₂²	—	4	2	no	—
57	C₂²	—	4	2	no	—
58	C₂²	—	4	2	no	—
59	C₂²	C₂²⋊C₈	1	1	yes	characteristic
60	C₂²	C₂²⋊C₈	1	1	yes	characteristic
61	C₂²	C₂²×C₈	1	1	yes	characteristic
62	C₄	—	2	2	no	—
63	C₄	C₂²⋊C₈	1	1	yes	characteristic
64	C₄	C₂²⋊C₈	1	1	yes	characteristic
65	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₄	C₂×M₁₆	1	1	yes	characteristic
67	C₂	—	2	1	no	—
68	C₂	—	2	1	no	—
69	C₂	—	4	2	no	—
70	C₂	#349	1	2	yes	direct product factor
71	C₂	C₂×C₂²⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄
commutator	C₂²	C₂²×C₈
Frattini, cosocle	C₂×C₈	C₂³
socle	C₂²	C₂²⋊C₈
prime radical	C₂⁴	C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×M₃₂	C₂	semidirect
3. #349	C₂	direct (N × H)
12. M₃₂	C₂²	semidirect
70. C₂	#349	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
C₂×M₃₂	2
#349	2

Subgroups — C2×#349 #2114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#349 #2114