Subgroups — #478×C₂₂₃ #28383190

Order 14272.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#107×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#106×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#102×C₂₂₃	C₂	1	1	yes	maximal, characteristic
7	#102×C₂₂₃	C₂	1	1	yes	maximal, characteristic
8	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₂₂₃	—	2	1	no	—
10	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₂₃	—	2	1	no	—
13	C₄⋊C₄×C₂₂₃	—	2	1	no	—
14	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
18	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
19	C₂³×C₂₂₃	—	2	1	no	—
20	D₄×C₂₂₃	—	2	1	no	—
21	D₄×C₂₂₃	Q₈	1	2	yes	—
22	C₂×C₄×C₂₂₃	—	2	1	no	—
23	C₂×C₄×C₂₂₃	—	2	1	no	—
24	C₂×C₄×C₂₂₃	—	2	1	no	—
25	C₂×C₄×C₂₂₃	—	4	2	no	—
26	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
28	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
29	C₈×C₂₂₃	—	2	1	no	—
30	C₈×C₂₂₃	—	2	1	no	—
31	C₂²×C₂₂₃	—	2	2	no	—
32	C₂²×C₂₂₃	—	4	1	no	—
33	C₂²×C₂₂₃	—	4	1	no	—
34	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
35	C₄×C₂₂₃	—	2	1	no	—
36	C₄×C₂₂₃	—	2	1	no	—
37	C₄×C₂₂₃	—	4	1	no	—
38	C₄×C₂₂₃	—	4	1	no	—
39	C₄×C₂₂₃	—	4	1	no	—
40	C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
41	C₄×C₂₂₃	#41	1	1	yes	characteristic
42	C₂×C₂₂₃	—	4	2	no	—
43	C₂×C₂₂₃	C₂²⋊Q₈	1	1	yes	characteristic
44	C₂×C₂₂₃	#135	1	1	yes	characteristic
45	C₂×C₂₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
46	C₂₂₃	#478	1	1	yes	characteristic, direct product factor, Sylow
47	#478	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	#125	C₂×C₂₂₃	1	1	yes	characteristic, complemented
49	C₄×D₄	C₂×C₂₂₃	1	1	yes	characteristic
50	#107	C₂×C₂₂₃	1	1	yes	characteristic, complemented
51	#106	C₂×C₂₂₃	1	1	yes	characteristic, complemented
52	C₄⋊C₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
53	#102	C₂×C₂₂₃	1	1	yes	characteristic
54	#102	C₂×C₂₂₃	1	1	yes	characteristic
55	C₂×D₄	C₂²×C₂₂₃	1	1	yes	characteristic
56	C₂²×C₄	—	2	1	no	—
57	C₂×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
58	C₂×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	2	1	no	—
61	C₄⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
62	C₄⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
63	C₄⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
64	C₂²⋊C₄	—	2	1	no	—
65	C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
66	C₂³	—	2	1	no	—
67	D₄	—	2	1	no	—
68	D₄	Q₈×C₂₂₃	1	2	yes	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	D₄×C₂₂₃	1	1	yes	characteristic
74	C₂×C₄	D₄×C₂₂₃	1	1	yes	characteristic
75	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	2	1	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	4	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	C₂×D₄×C₂₂₃	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	4	1	no	—
87	C₄	C₂×Q₈×C₂₂₃	1	1	yes	characteristic
88	C₄	#41×C₂₂₃	1	1	yes	characteristic
89	C₂	—	4	2	no	—
90	C₂	C₂²⋊Q₈×C₂₂₃	1	1	yes	characteristic
91	C₂	#135×C₂₂₃	1	1	yes	characteristic
92	C₂	AGL(1,ℤ/8)×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. #125×C₂₂₃	C₂	semidirect
3. #107×C₂₂₃	C₂	semidirect
4. #106×C₂₂₃	C₂	semidirect
5. C₄⋊C₈×C₂₂₃	C₂	semidirect
46. C₂₂₃	#478	direct (N × H)
47. #478	C₂₂₃	direct (N × H)
48. #125	C₂×C₂₂₃	semidirect
50. #107	C₂×C₂₂₃	semidirect
51. #106	C₂×C₂₂₃	semidirect
52. C₄⋊C₈	C₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#125×C₂₂₃	2
C₄×D₄×C₂₂₃	2
#107×C₂₂₃	2
#106×C₂₂₃	2
C₄⋊C₈×C₂₂₃	2
#102×C₂₂₃	2
#478	223

Subgroups — #478×C223 #28383190

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #478×C₂₂₃ #28383190