Subgroups — #1813

Order 128.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×#108	C₂	1	2	yes	maximal, complemented
3	#413	C₂	1	2	yes	maximal
4	#338	C₂	1	2	yes	maximal, complemented
5	C₂×#41	C₂²	1	1	yes	characteristic
6	#131	—	2	4	no	—
7	C₂×M₁₆	C₄	1	2	yes	—
8	C₂×M₁₆	C₂²	1	4	yes	—
9	C₂×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₈	C₄	1	2	yes	—
11	C₂²×C₈	C₂²	1	1	yes	characteristic
12	#108	—	2	4	no	—
13	#101	—	2	2	no	—
14	#100	—	2	2	no	—
15	#41	—	2	2	no	—
16	#41	—	2	2	no	—
17	C₂×Q₈	D₄	1	1	yes	characteristic
18	C₂×D₄	D₄	1	1	yes	characteristic
19	C₂×D₄	Q₈	1	2	yes	—
20	C₂²×C₄	C₂×C₄	1	2	yes	—
21	C₂²×C₄	C₂³	1	1	yes	characteristic
22	M₁₆	—	2	2	no	—
23	M₁₆	—	2	4	no	—
24	M₁₆	—	2	4	no	—
25	M₁₆	—	2	4	no	—
26	C₂×C₈	—	2	4	no	—
27	C₂×C₈	—	2	4	no	—
28	C₂×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈	D₄	1	4	yes	—
32	C₂³	C₄⋊C₄	1	2	yes	—
33	C₂³	C₂×Q₈	1	1	yes	characteristic
34	Q₈	—	2	2	no	—
35	D₄	—	2	4	no	—
36	D₄	—	2	2	no	—
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	—	2	2	no	—
39	C₂×C₄	C₄⋊C₄	1	2	yes	—
40	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄	C₂×D₄	1	2	yes	—
42	C₂×C₄	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄	#41	1	2	yes	—
44	C₈	—	2	2	no	—
45	C₈	—	4	4	no	—
46	C₈	C₂²⋊C₄	1	2	yes	—
47	C₈	C₂²⋊C₄	1	2	yes	—
48	C₂²	—	2	2	no	—
49	C₂²	—	4	1	no	—
50	C₂²	—	4	2	no	—
51	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²	C₂²⋊Q₈	1	2	yes	—
53	C₄	—	2	2	no	—
54	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄	#117	1	1	yes	characteristic
56	C₄	#121	1	2	yes	—
57	C₂	—	2	1	no	—
58	C₂	—	2	2	no	—
59	C₂	—	4	2	no	—
60	C₂	#380	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂	#380
prime radical	C₂×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ 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×#108	C₂	semidirect
4. #338	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#131	2
C₂×#108	2
#413	2
#338	2