Subgroups — C₂×#20297311 #28384343

Order 14272.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297311	C₂	1	8	yes	maximal, direct product factor
2	C₂×#16813154	C₂	1	3	yes	maximal, complemented
3	C₂²×C₂₂₃⋊C₈	C₂	1	3	yes	maximal, complemented
4	C₂×#41×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	#16813154	C₂²	1	12	yes	complemented
6	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₂₂₃⋊C₈	C₂²	1	3	yes	—
8	C₂×C₂₂₃⋊C₈	C₂²	1	12	yes	complemented
9	#41×C₂₂₃	C₄	1	4	yes	—
10	#41×C₂₂₃	C₂²	1	4	yes	—
11	C₂×Q₈×C₂₂₃	C₄	1	1	yes	characteristic
12	C₂×D₄×C₂₂₃	C₄	1	3	yes	—
13	C₂²×C₄×C₂₂₃	C₂²	1	3	yes	—
14	C₂₂₃⋊C₈	C₂³	1	2	yes	—
15	C₂₂₃⋊C₈	C₂³	1	6	yes	—
16	C₂³×C₂₂₃	C₂×C₄	1	3	yes	—
17	Q₈×C₂₂₃	C₂×C₄	1	4	yes	—
18	D₄×C₂₂₃	C₂×C₄	1	12	yes	—
19	C₂×C₄×C₂₂₃	C₂×C₄	1	3	yes	—
20	C₂×C₄×C₂₂₃	C₂×C₄	1	6	yes	—
21	C₂×C₄×C₂₂₃	C₂³	1	6	yes	—
22	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
23	C₂²×C₂₂₃	—	2	6	no	—
24	C₂²×C₂₂₃	C₂²×C₄	1	6	yes	—
25	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
26	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₂₂₃	C₂²×C₄	1	6	yes	—
28	C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
29	C₂×C₂₂₃	—	2	6	no	—
30	C₂×C₂₂₃	#131	1	2	yes	complemented
31	C₂×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
32	C₂₂₃	C₂×#131	1	1	yes	characteristic, complemented, Sylow
33	C₂×#131	—	223	1	no	maximal, Sylow
34	C₂×#41	D₂₂₃	1	1	yes	characteristic
35	#131	—	223	8	no	—
36	C₂×M₁₆	—	223	3	no	—
37	C₂²×C₈	—	223	3	no	—
38	#41	Dic₂₂₃	1	4	yes	—
39	#41	C₂×D₂₂₃	1	4	yes	—
40	C₂×Q₈	Dic₂₂₃	1	1	yes	characteristic
41	C₂×D₄	Dic₂₂₃	1	3	yes	—
42	C₂²×C₄	C₂×D₂₂₃	1	3	yes	—
43	M₁₆	—	223	12	no	—
44	C₂×C₈	—	223	12	no	—
45	C₂×C₈	—	223	1	no	—
46	C₂×C₈	—	223	3	no	—
47	C₂³	C₂×Dic₂₂₃	1	3	yes	—
48	Q₈	C₂×Dic₂₂₃	1	4	yes	—
49	D₄	C₂×Dic₂₂₃	1	12	yes	—
50	C₂×C₄	C₂×Dic₂₂₃	1	3	yes	—
51	C₂×C₄	C₂×Dic₂₂₃	1	6	yes	—
52	C₂×C₄	C₂²×D₂₂₃	1	6	yes	—
53	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
54	C₈	—	223	2	no	—
55	C₈	—	223	6	no	—
56	C₂²	—	2	6	no	—
57	C₂²	C₂²×Dic₂₂₃	1	6	yes	—
58	C₂²	C₂²×Dic₂₂₃	1	1	yes	characteristic
59	C₄	C₂²×Dic₂₂₃	1	1	yes	characteristic
60	C₄	C₂²×Dic₂₂₃	1	6	yes	—
61	C₄	C₂³×D₂₂₃	1	1	yes	characteristic
62	C₂	—	2	6	no	—
63	C₂	#20297311	1	2	yes	direct product factor
64	C₂	C₂³×Dic₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₂₃
commutator	C₂×C₂₂₃	C₂³×C₄
Frattini	C₄	C₂³×D₂₂₃
Fitting, prime radical	C₂×#41×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²×C₄
cosocle	C₄×C₂₂₃	C₂⁴
hypercenter	C₂×#41	D₂₂₃
hypocenter	C₂₂₃	C₂×#131
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #20297311	C₂	direct (N × H)
2. C₂×#16813154	C₂	semidirect
3. C₂²×C₂₂₃⋊C₈	C₂	semidirect
5. #16813154	C₂²	semidirect
8. C₂×C₂₂₃⋊C₈	C₂²	semidirect
30. C₂×C₂₂₃	#131	semidirect
32. C₂₂₃	C₂×#131	semidirect
63. C₂	#20297311	direct (N × H)

Maximal subgroups

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

subgroup	index
#20297311	2
C₂×#16813154	2
C₂²×C₂₂₃⋊C₈	2
C₂×#41×C₂₂₃	2
C₂×#131	223

Subgroups — C2×#20297311 #28384343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20297311 #28384343