Subgroups — #489×C₂₂₃ #28383201

Order 14272.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#124×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#103×C₂₂₃	C₂	1	2	yes	maximal, complemented
4	#102×C₂₂₃	C₂	1	2	yes	maximal
5	#97×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₂₂₃	—	2	1	no	—
7	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
10	C₄⋊C₄×C₂₂₃	—	2	1	no	—
11	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
12	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
13	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
14	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₂³×C₂₂₃	—	2	1	no	—
16	Q₈×C₂₂₃	—	2	2	no	—
17	Q₈×C₂₂₃	—	4	2	no	—
18	D₄×C₂₂₃	—	2	2	no	—
19	C₂×C₄×C₂₂₃	—	2	1	no	—
20	C₂×C₄×C₂₂₃	—	2	2	no	—
21	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
24	C₈×C₂₂₃	—	2	2	no	—
25	C₂²×C₂₂₃	—	4	1	no	—
26	C₂²×C₂₂₃	—	4	1	no	—
27	C₂²×C₂₂₃	—	4	1	no	—
28	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
29	C₄×C₂₂₃	—	2	1	no	—
30	C₄×C₂₂₃	—	2	1	no	—
31	C₄×C₂₂₃	—	4	1	no	—
32	C₄×C₂₂₃	—	4	2	no	—
33	C₄×C₂₂₃	#41	1	2	yes	—
34	C₂×C₂₂₃	—	8	1	no	—
35	C₂×C₂₂₃	#124	1	1	yes	characteristic
36	C₂×C₂₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
37	C₂×C₂₂₃	#137	1	1	yes	characteristic
38	C₂₂₃	#489	1	1	yes	characteristic, direct product factor, Sylow
39	#489	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₄⋊Q₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
41	#124	C₂×C₂₂₃	1	1	yes	characteristic
42	#103	C₂×C₂₂₃	1	2	yes	complemented
43	#102	C₂×C₂₂₃	1	2	yes	—
44	#97	C₂×C₂₂₃	1	1	yes	characteristic, complemented
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
47	C₂×D₄	C₂²×C₂₂₃	1	1	yes	characteristic
48	C₂×C₈	C₂²×C₂₂₃	1	2	yes	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
51	C₂²⋊C₄	—	2	1	no	—
52	C₂²⋊C₄	—	2	1	no	—
53	C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
54	C₂³	—	2	1	no	—
55	Q₈	—	2	2	no	—
56	Q₈	—	4	2	no	—
57	D₄	—	2	2	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	D₄×C₂₂₃	1	1	yes	characteristic
61	C₂×C₄	D₄×C₂₂₃	1	1	yes	characteristic
62	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₂²	—	4	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	C₂×D₄×C₂₂₃	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	1	no	—
71	C₄	—	4	2	no	—
72	C₄	#41×C₂₂₃	1	2	yes	—
73	C₂	—	8	1	no	—
74	C₂	#124×C₂₂₃	1	1	yes	characteristic
75	C₂	AGL(1,ℤ/8)×C₂₂₃	1	1	yes	characteristic
76	C₂	#137×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₂₂₃
prime radical	C₂×D₄×C₂₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ C₄²×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄⋊Q₈×C₂₂₃	C₂	semidirect
3. #103×C₂₂₃	C₂	semidirect
5. #97×C₂₂₃	C₂	semidirect
38. C₂₂₃	#489	direct (N × H)
39. #489	C₂₂₃	direct (N × H)
40. C₄⋊Q₈	C₂×C₂₂₃	semidirect
42. #103	C₂×C₂₂₃	semidirect
44. #97	C₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×C₂₂₃	2
#124×C₂₂₃	2
#103×C₂₂₃	2
#102×C₂₂₃	2
#97×C₂₂₃	2
#489	223

Subgroups — #489×C223 #28383201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #489×C₂₂₃ #28383201