Subgroups — #28383603

Order 14272.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813154	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
3	Dic₂₂₃⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₂₃	C₄	1	2	yes	—
6	#16813154	—	2	2	no	—
7	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
8	C₄×Dic₂₂₃	C₄	1	1	yes	characteristic
9	C₄×Dic₂₂₃	C₄	1	1	yes	characteristic
10	C₄×Dic₂₂₃	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
12	M₁₆×C₂₂₃	—	2	2	no	—
13	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
14	C₂×Dic₂₂₃	—	2	4	no	—
15	C₂×Dic₂₂₃	C₂×C₄	1	2	yes	—
16	C₂×Dic₂₂₃	C₂×C₄	1	2	yes	—
17	C₂₂₃⋊C₈	—	2	2	no	—
18	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
21	C₂×C₄×C₂₂₃	Q₈	1	2	yes	—
22	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
23	C₈×C₂₂₃	—	2	2	no	—
24	C₂²×C₂₂₃	—	2	2	no	—
25	C₂²×C₂₂₃	C₄⋊C₄	1	2	yes	—
26	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
28	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
29	C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
30	Dic₂₂₃	—	2	2	no	—
31	Dic₂₂₃	M₁₆	1	4	yes	complemented
32	C₂×C₂₂₃	—	2	2	no	—
33	C₂×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₂₂₃	C₂×M₁₆	1	2	yes	—
35	C₂₂₃	#423	1	1	yes	characteristic, complemented, Sylow
36	#423	—	223	1	no	maximal, Sylow
37	C₂×M₁₆	—	223	1	no	—
38	C₂×M₁₆	D₂₂₃	1	1	yes	characteristic
39	C₂×C₄²	—	223	1	no	—
40	C₄⋊C₈	—	223	4	no	—
41	C₂²×C₄	—	223	2	no	—
42	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
43	M₁₆	—	2	2	no	—
44	M₁₆	—	446	2	no	—
45	C₂×C₈	—	223	2	no	—
46	C₂×C₈	C₂×D₂₂₃	1	2	yes	—
47	C₄²	—	223	1	no	—
48	C₄²	—	223	1	no	—
49	C₄²	—	223	2	no	—
50	C₂³	C₄×D₂₂₃	1	1	yes	characteristic
51	C₂×C₄	—	446	4	no	—
52	C₂×C₄	—	223	2	no	—
53	C₂×C₄	—	223	2	no	—
54	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
56	C₂×C₄	Dic₄₄₆	1	2	yes	—
57	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	446	2	no	—
60	C₂²	—	2	2	no	—
61	C₂²	Dic₂₂₃⋊C₄	1	2	yes	—
62	C₂²	C₂×C₄×D₂₂₃	1	1	yes	characteristic
63	C₄	—	446	2	no	—
64	C₄	—	223	4	no	—
65	C₄	Dic₂₂₃⋊C₄	1	2	yes	—
66	C₄	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
67	C₄	C₂×Dic₄₄₆	1	1	yes	characteristic
68	C₂	—	2	2	no	—
69	C₂	C₂×Dic₂₂₃⋊C₄	1	1	yes	characteristic
70	C₂	M₁₆×D₂₂₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₂₃
commutator, socle	C₂²×C₂₂₃	C₂²×C₄
Fitting	C₂×M₁₆×C₂₂₃	C₂
prime radical	C₂³×C₂₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂×M₁₆	D₂₂₃
hypocenter	C₂₂₃	#423
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. Dic₂₂₃⋊C₈	C₂	semidirect
31. Dic₂₂₃	M₁₆	semidirect
35. C₂₂₃	#423	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813154	2
C₂×C₄×Dic₂₂₃	2
Dic₂₂₃⋊C₈	2
C₂×M₁₆×C₂₂₃	2
#423	223