Subgroups — #13092441

Order 1472.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#67267	C₂	1	2	yes	maximal
3	C₄⋊(C₂₃⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×C₂₃	C₄	1	2	yes	—
5	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₄²×C₂₃	C₄	1	1	yes	characteristic
7	C₄²×C₂₃	C₄	1	1	yes	characteristic
8	C₄²×C₂₃	C₂²	1	2	yes	—
9	#67267	—	2	4	no	—
10	C₂×C₂₃⋊C₈	C₂²	1	4	yes	—
11	C₂³×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₂₃	—	2	4	no	—
13	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₂₃	D₄	1	2	yes	—
17	C₂×C₄×C₂₃	Q₈	1	2	yes	—
18	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂₃⋊C₈	—	2	4	no	—
20	C₂²×C₂₃	—	2	2	no	—
21	C₂²×C₂₃	C₄⋊C₄	1	2	yes	—
22	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
23	C₄×C₂₃	—	2	2	no	—
24	C₄×C₂₃	C₄⋊C₄	1	2	yes	—
25	C₄×C₂₃	M₁₆	1	4	yes	complemented
26	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
27	C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
28	#423	—	23	1	no	maximal, Sylow
29	C₂×C₂₃	—	2	2	no	—
30	C₂×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₂₃	C₂×M₁₆	1	2	yes	—
32	C₂×M₁₆	—	23	2	no	—
33	C₂×C₄²	D₂₃	1	1	yes	characteristic
34	C₄⋊C₈	—	23	4	no	—
35	C₂₃	#423	1	1	yes	characteristic, complemented, Sylow
36	C₂²×C₄	Dic₂₃	1	2	yes	—
37	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
38	M₁₆	—	46	4	no	—
39	C₂×C₈	—	23	4	no	—
40	C₄²	Dic₂₃	1	1	yes	characteristic
41	C₄²	Dic₂₃	1	1	yes	characteristic
42	C₄²	C₂×D₂₃	1	2	yes	—
43	C₂³	C₂×Dic₂₃	1	1	yes	characteristic
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	Dic₄₆	1	2	yes	—
46	C₂×C₄	D₉₂	1	2	yes	—
47	C₂×C₄	C₂×Dic₂₃	1	2	yes	—
48	C₂×C₄	C₂×Dic₂₃	1	2	yes	—
49	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
51	C₈	—	46	4	no	—
52	C₂²	—	2	2	no	—
53	C₂²	C₄⋊Dic₂₃	1	2	yes	—
54	C₂²	C₂²×Dic₂₃	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	#67267	1	4	yes	complemented
57	C₄	C₄⋊Dic₂₃	1	2	yes	—
58	C₄	C₂×Dic₄₆	1	1	yes	characteristic
59	C₄	C₂×D₉₂	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×#67267	1	2	yes	—
62	C₂	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂²×C₄
Fitting	C₂×C₄²×C₂₃	C₂
prime radical	C₂³×C₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	C₂×C₄²	D₂₃
hypocenter	C₂₃	#423
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₂₃⋊C₈)	C₂	semidirect
25. C₄×C₂₃	M₁₆	semidirect
35. C₂₃	#423	semidirect
56. C₄	#67267	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×C₂₃	2
C₂×#67267	2
C₄⋊(C₂₃⋊C₈)	2
#423	23