Subgroups — #13092184

Order 1472.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₂₃	C₂	1	2	yes	maximal
3	C₂⁴×C₂₃	C₂²	1	1	yes	characteristic
4	C₂²×C₄×C₂₃	C₄	1	2	yes	complemented
5	C₂²×Dic₂₃	—	2	2	no	—
6	C₂²⋊C₄×C₂₃	—	2	2	no	—
7	C₂²⋊Dic₂₃	—	2	2	no	—
8	C₂²⋊Dic₂₃	C₄	1	4	yes	complemented
9	C₂³×C₂₃	—	4	2	no	—
10	C₂³×C₂₃	C₂×C₄	1	2	yes	—
11	C₂³×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂³×C₂₃	D₄	1	1	yes	characteristic
13	C₂³×C₂₃	D₄	1	2	yes	—
14	C₂³×C₂₃	Q₈	1	1	yes	characteristic
15	C₂×C₄×C₂₃	—	2	2	no	—
16	C₂×C₄×C₂₃	—	4	2	no	—
17	C₂×Dic₂₃	—	2	4	no	—
18	C₂×Dic₂₃	—	4	4	no	—
19	C₂²×C₂₃	—	2	4	no	—
20	C₂²×C₂₃	—	2	2	no	—
21	C₂²×C₂₃	—	4	2	no	—
22	C₂²×C₂₃	—	4	1	no	—
23	C₂²×C₂₃	—	4	1	no	—
24	C₂²×C₂₃	C₄²	1	1	yes	characteristic
25	C₂²×C₂₃	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₂₃	C₄⋊C₄	1	2	yes	—
28	C₂²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
29	C₄×C₂₃	—	4	2	no	—
30	Dic₂₃	—	4	4	no	—
31	#342	—	23	1	no	maximal, Sylow
32	C₂×C₂₃	—	2	2	no	—
33	C₂×C₂₃	—	2	1	no	—
34	C₂×C₂₃	—	2	2	no	—
35	C₂×C₂₃	—	2	1	no	—
36	C₂×C₂₃	#95	1	1	yes	characteristic
37	C₂×C₂₃	#99	1	2	yes	—
38	C₂×C₂²⋊C₄	—	23	2	no	—
39	C₂×C₂²⋊C₄	D₂₃	1	1	yes	characteristic
40	C₂₃	#342	1	1	yes	characteristic, complemented, Sylow
41	C₂⁴	C₂×D₂₃	1	1	yes	characteristic
42	C₂²×C₄	—	46	2	no	—
43	C₂²×C₄	Dic₂₃	1	2	yes	complemented
44	C₂²⋊C₄	—	2	2	no	—
45	C₂²⋊C₄	—	23	4	no	—
46	C₂²⋊C₄	—	46	2	no	—
47	C₂³	—	4	2	no	—
48	C₂³	Dic₄₆	1	1	yes	characteristic
49	C₂³	C₄×D₂₃	1	2	yes	—
50	C₂³	D₉₂	1	1	yes	characteristic
51	C₂³	C₂×Dic₂₃	1	1	yes	characteristic
52	C₂³	C₂²⋊D₂₃	1	2	yes	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	4	2	no	—
55	C₂×C₄	—	46	4	no	—
56	C₂×C₄	—	92	4	no	—
57	C₂²	—	2	2	no	—
58	C₂²	—	2	4	no	—
59	C₂²	—	4	1	no	—
60	C₂²	—	4	2	no	—
61	C₂²	—	4	1	no	—
62	C₂²	C₄×Dic₂₃	1	1	yes	characteristic
63	C₂²	Dic₂₃⋊C₄	1	2	yes	—
64	C₂²	C₄⋊Dic₂₃	1	1	yes	characteristic
65	C₂²	(C₂×D₂₃)⋊C₄	1	2	yes	—
66	C₂²	C₂²⋊Dic₂₃	1	1	yes	characteristic
67	C₄	—	4	2	no	—
68	C₄	—	92	4	no	—
69	C₂	—	2	2	no	—
70	C₂	—	2	1	no	—
71	C₂	—	2	2	no	—
72	C₂	—	2	1	no	—
73	C₂	C₂₃⋊#95	1	1	yes	characteristic
74	C₂	#10625016	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²⋊Dic₂₃
commutator	C₂²×C₂₃	C₄²
Frattini	C₂⁴	C₂×D₂₃
Fitting	C₂×C₂²⋊C₄×C₂₃	C₂
socle	C₂²×C₂₃	C₂²⋊C₄
prime radical, cosocle	C₂⁴×C₂₃	C₂²
hypercenter	C₂×C₂²⋊C₄	D₂₃
hypocenter	C₂₃	#342
LCS	G ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₂²×C₄×C₂₃	C₄	semidirect
8. C₂²⋊Dic₂₃	C₄	semidirect
40. C₂₃	#342	semidirect
43. C₂²×C₄	Dic₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊C₄×C₂₃	2
C₂×C₂²⋊Dic₂₃	2
#342	23