Subgroups — C₂×#108×C₂₂₃ #28383141

Order 14272.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×C₂₂₃	C₂	1	2	yes	maximal
2	C₂²×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#108×C₂₂₃	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
5	M₁₆×C₂₂₃	—	2	2	no	—
6	M₁₆×C₂₂₃	C₂²	1	4	yes	—
7	C₂×C₈×C₂₂₃	—	2	2	no	—
8	C₂×C₈×C₂₂₃	C₄	1	2	yes	—
9	C₂×C₈×C₂₂₃	C₄	1	2	yes	—
10	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
11	C₂³×C₂₂₃	Q₈	1	1	yes	characteristic
12	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
15	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
16	C₂×C₄×C₂₂₃	Q₈	1	1	yes	characteristic
17	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
18	C₈×C₂₂₃	—	2	4	no	—
19	C₈×C₂₂₃	C₂×C₄	1	4	yes	—
20	C₂²×C₂₂₃	—	2	2	no	—
21	C₂²×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
22	C₂²×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
24	C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
25	C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
26	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
28	C₂×C₂₂₃	—	2	1	no	—
29	C₂×C₂₂₃	—	2	1	no	—
30	C₂×C₂₂₃	#108	1	2	yes	direct product factor
31	C₂×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂₂₃	C₂×#108	1	1	yes	characteristic, direct product factor, Sylow
33	C₂×#108	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×M₁₆	C₂×C₂₂₃	1	2	yes	—
35	C₂²×C₈	C₂×C₂₂₃	1	1	yes	characteristic
36	#108	C₂×C₂₂₃	1	4	yes	direct product factor
37	C₂²×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
38	M₁₆	—	2	2	no	—
39	M₁₆	C₂²×C₂₂₃	1	4	yes	—
40	C₂×C₈	—	2	2	no	—
41	C₂×C₈	C₄×C₂₂₃	1	2	yes	—
42	C₂×C₈	C₄×C₂₂₃	1	2	yes	—
43	C₂×C₈	C₂²×C₂₂₃	1	2	yes	—
44	C₂³	Q₈×C₂₂₃	1	1	yes	characteristic
45	C₂×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
46	C₂×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
47	C₂×C₄	D₄×C₂₂₃	1	1	yes	characteristic
48	C₂×C₄	D₄×C₂₂₃	1	1	yes	characteristic
49	C₂×C₄	Q₈×C₂₂₃	1	1	yes	characteristic
50	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
51	C₈	—	2	4	no	—
52	C₈	C₂×C₄×C₂₂₃	1	4	yes	—
53	C₂²	—	2	2	no	—
54	C₂²	C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
55	C₂²	C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
56	C₂²	C₂×Q₈×C₂₂₃	1	1	yes	characteristic
57	C₄	C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
58	C₄	C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
59	C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
60	C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	—	2	1	no	—
63	C₂	#108×C₂₂₃	1	2	yes	direct product factor
64	C₂	C₂×C₄⋊C₄×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₂₃	D₄
commutator	C₄	C₂²×C₄×C₂₂₃
Frattini, cosocle	C₂×C₄	C₂³×C₂₂₃
socle	C₂²×C₂₂₃	C₄⋊C₄
prime radical	C₂³×C₂₂₃	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₂₃ ◁ C₂²×C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. #108×C₂₂₃	C₂	direct (N × H)
30. C₂×C₂₂₃	#108	direct (N × H)
32. C₂₂₃	C₂×#108	direct (N × H)
33. C₂×#108	C₂₂₃	direct (N × H)
36. #108	C₂×C₂₂₃	direct (N × H)
63. C₂	#108×C₂₂₃	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×M₁₆×C₂₂₃	2
C₂²×C₈×C₂₂₃	2
#108×C₂₂₃	2
C₂×#108	223

Subgroups — C2×#108×C223 #28383141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#108×C₂₂₃ #28383141