Subgroups — #431×C₂₂₃ #28383143

Order 14272.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₈×C₂₂₃	C₂	1	2	yes	maximal, complemented
3	C₂²⋊C₈×C₂₂₃	C₂	1	2	yes	maximal
4	C₄×C₈×C₂₂₃	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₂₂₃	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₂₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₂₂₃	C₂²	1	4	yes	—
9	C₄²×C₂₂₃	C₄	1	1	yes	characteristic
10	C₄²×C₂₂₃	C₄	1	1	yes	characteristic
11	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₂₂₃	—	2	2	no	—
15	C₂×C₄×C₂₂₃	—	2	2	no	—
16	C₂×C₄×C₂₂₃	C₈	1	4	yes	complemented
17	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
23	C₈×C₂₂₃	—	2	4	no	—
24	C₂²×C₂₂₃	—	2	1	no	—
25	C₂²×C₂₂₃	—	2	1	no	—
26	C₂²×C₂₂₃	C₂×C₈	1	2	yes	—
27	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₂₂₃	—	2	1	no	—
29	C₄×C₂₂₃	—	2	1	no	—
30	C₄×C₂₂₃	C₂×C₈	1	2	yes	—
31	C₄×C₂₂₃	C₂×C₈	1	2	yes	—
32	C₄×C₂₂₃	M₁₆	1	2	yes	—
33	C₄×C₂₂₃	#41	1	2	yes	—
34	C₂×C₂₂₃	—	2	2	no	—
35	C₂×C₂₂₃	#117	1	1	yes	characteristic
36	C₂×C₂₂₃	C₂²×C₈	1	1	yes	characteristic
37	C₂×C₂₂₃	C₂×M₁₆	1	1	yes	characteristic
38	C₂₂₃	#431	1	1	yes	characteristic, direct product factor, Sylow
39	#431	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₄²	C₂×C₂₂₃	1	1	yes	characteristic
41	C₄⋊C₈	C₂×C₂₂₃	1	2	yes	complemented
42	C₂²⋊C₈	C₂×C₂₂₃	1	2	yes	—
43	C₄×C₈	C₂×C₂₂₃	1	2	yes	complemented
44	C₂²×C₄	C₄×C₂₂₃	1	1	yes	characteristic
45	C₂²×C₄	C₄×C₂₂₃	1	1	yes	characteristic
46	C₂²×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
47	C₂×C₈	C₂²×C₂₂₃	1	4	yes	—
48	C₄²	C₄×C₂₂₃	1	1	yes	characteristic
49	C₄²	C₄×C₂₂₃	1	1	yes	characteristic
50	C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
51	C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
52	C₂³	C₂×C₄×C₂₂₃	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₈×C₂₂₃	1	4	yes	complemented
56	C₂×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
57	C₂×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
58	C₂×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
59	C₂×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
60	C₂×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
61	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
62	C₈	—	2	4	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	2	1	no	—
65	C₂²	C₂×C₈×C₂₂₃	1	2	yes	—
66	C₂²	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	2	1	no	—
69	C₄	C₂×C₈×C₂₂₃	1	2	yes	—
70	C₄	C₂×C₈×C₂₂₃	1	2	yes	—
71	C₄	M₁₆×C₂₂₃	1	2	yes	—
72	C₄	#41×C₂₂₃	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#117×C₂₂₃	1	1	yes	characteristic
75	C₂	C₂²×C₈×C₂₂₃	1	1	yes	characteristic
76	C₂	C₂×M₁₆×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₂₂₃	C₂²
commutator	C₂	C₂²×C₈×C₂₂₃
Frattini, cosocle	C₂×C₄	C₂³×C₂₂₃
socle	C₂²×C₂₂₃	C₂²×C₄
prime radical	C₂³×C₂₂₃	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₈×C₂₂₃	C₂	semidirect
4. C₄×C₈×C₂₂₃	C₂	semidirect
16. C₂×C₄×C₂₂₃	C₈	semidirect
38. C₂₂₃	#431	direct (N × H)
39. #431	C₂₂₃	direct (N × H)
41. C₄⋊C₈	C₂×C₂₂₃	semidirect
43. C₄×C₈	C₂×C₂₂₃	semidirect
55. C₂×C₄	C₈×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×C₂₂₃	2
C₄⋊C₈×C₂₂₃	2
C₂²⋊C₈×C₂₂₃	2
C₄×C₈×C₂₂₃	2
#431	223

Subgroups — #431×C223 #28383143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #431×C₂₂₃ #28383143