Subgroups — #13092440

Order 1472.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂²×C₄
Fitting	C₂×M₁₆×C₂₃	C₂
prime radical	C₂³×C₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	C₂×M₁₆	D₂₃
hypocenter	C₂₃	#423
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
4. Dic₂₃⋊C₈	C₂	semidirect
31. Dic₂₃	M₁₆	semidirect
40. C₂₃	#423	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×C₂₃	2
C₂×C₄×Dic₂₃	2
C₂×#67267	2
Dic₂₃⋊C₈	2
#423	23