Subgroups — #28383639

Order 14272.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297261	C₂	1	1	yes	maximal, characteristic
2	Dic₂₂₃⋊C₈	C₂	1	2	yes	maximal, complemented
3	(C₂×D₂₂₃)⋊C₈	C₂	1	2	yes	maximal
4	#20297215	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₂₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
8	Dic₂₂₃⋊C₄	C₄	1	2	yes	—
9	(C₂×D₂₂₃)⋊C₄	C₄	1	2	yes	—
10	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
12	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₂²×D₂₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₂₂₃	C₂×C₄	1	2	yes	—
15	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₄×D₂₂₃	—	2	2	no	—
17	C₂₂₃⋊C₈	—	2	2	no	—
18	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
20	C₈×C₂₂₃	—	2	2	no	—
21	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
22	C₂×D₂₂₃	—	2	1	no	—
23	C₂×D₂₂₃	—	2	2	no	—
24	C₄×C₂₂₃	—	2	2	no	—
25	C₄×C₂₂₃	#41	1	2	yes	—
26	Dic₂₂₃	—	2	1	no	—
27	Dic₂₂₃	—	2	2	no	—
28	C₂×C₂₂₃	#117	1	1	yes	characteristic
29	C₂×C₂₂₃	#131	1	2	yes	—
30	D₂₂₃	—	4	1	no	—
31	C₂₂₃	#433	1	1	yes	characteristic, complemented, Sylow
32	#433	—	223	1	no	maximal, Sylow
33	#117	—	223	1	no	—
34	C₄⋊C₈	—	223	2	no	—
35	C₂²⋊C₈	—	223	2	no	—
36	#97	—	223	1	no	—
37	C₄×C₈	D₂₂₃	1	1	yes	characteristic, complemented
38	C₂²×C₄	—	223	1	no	—
39	C₂×C₈	—	223	2	no	—
40	C₂×C₈	C₂×D₂₂₃	1	2	yes	—
41	C₄⋊C₄	—	223	2	no	—
42	C₂²⋊C₄	—	223	2	no	—
43	C₄²	—	223	1	no	—
44	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
45	C₂³	—	223	1	no	—
46	C₂×C₄	—	446	2	no	—
47	C₂×C₄	—	223	2	no	—
48	C₂×C₄	—	223	1	no	—
49	C₂×C₄	C₄×D₂₂₃	1	2	yes	—
50	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	446	2	no	—
53	C₂²	—	446	1	no	—
54	C₂²	—	446	2	no	—
55	C₂²	C₂×C₄×D₂₂₃	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	446	1	no	—
58	C₄	—	446	2	no	—
59	C₄	#16813170	1	2	yes	—
60	C₂	—	892	1	no	—
61	C₂	#20297261	1	1	yes	characteristic
62	C₂	#20297310	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₂₃
commutator, socle	C₂²×C₂₂₃	C₂²×C₄
Fitting	C₄×C₈×C₂₂₃	C₂
prime radical	C₂²×D₂₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₄×C₈	D₂₂₃
hypocenter	C₂₂₃	#433
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. Dic₂₂₃⋊C₈	C₂	semidirect
4. #20297215	C₂	semidirect
5. C₄×C₈×C₂₂₃	C₂	semidirect
31. C₂₂₃	#433	semidirect
37. C₄×C₈	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297261	2
Dic₂₂₃⋊C₈	2
(C₂×D₂₂₃)⋊C₈	2
#20297215	2
C₄×C₈×C₂₂₃	2
#433	223