Subgroups — #13092186

Order 1472.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#67267	C₂	1	2	yes	maximal
3	C₂²×C₄×C₂₃	C₄	1	2	yes	—
4	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
5	C₄⋊C₄×C₂₃	—	2	2	no	—
6	#67267	—	2	2	no	—
7	#67267	C₄	1	4	yes	complemented
8	C₂×C₂₃⋊C₈	—	2	2	no	—
9	C₂³×C₂₃	C₂×C₄	1	1	yes	characteristic
10	C₂×C₄×C₂₃	—	2	2	no	—
11	C₂×C₄×C₂₃	—	4	2	no	—
12	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
13	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
14	C₂×C₄×C₂₃	D₄	1	2	yes	—
15	C₂×C₄×C₂₃	Q₈	1	1	yes	characteristic
16	C₂₃⋊C₈	—	2	4	no	—
17	C₂²×C₂₃	—	2	1	no	—
18	C₂²×C₂₃	—	2	1	no	—
19	C₂²×C₂₃	C₄²	1	1	yes	characteristic
20	C₂²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₂²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
22	C₄×C₂₃	—	4	2	no	—
23	C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
24	C₄×C₂₃	C₄⋊C₄	1	2	yes	—
25	#343	—	23	1	no	maximal, Sylow
26	C₂×C₂₃	—	2	1	no	—
27	C₂×C₂₃	—	2	1	no	—
28	C₂×C₂₃	#95	1	1	yes	characteristic
29	C₂×C₂₃	#100	1	1	yes	characteristic
30	C₂×C₂₃	#101	1	1	yes	characteristic
31	C₂×M₁₆	—	23	2	no	—
32	C₂×C₄⋊C₄	D₂₃	1	1	yes	characteristic
33	C₂₃	#343	1	1	yes	characteristic, complemented, Sylow
34	C₂²×C₄	Dic₂₃	1	2	yes	—
35	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
36	M₁₆	—	23	4	no	—
37	M₁₆	—	46	2	no	—
38	C₂×C₈	—	46	2	no	—
39	C₄⋊C₄	—	2	2	no	—
40	C₂³	C₂×Dic₂₃	1	1	yes	characteristic
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	4	2	no	—
43	C₂×C₄	Dic₄₆	1	1	yes	characteristic
44	C₂×C₄	C₄×D₂₃	1	2	yes	—
45	C₂×C₄	D₉₂	1	1	yes	characteristic
46	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
47	C₈	—	46	4	no	—
48	C₂²	—	2	1	no	—
49	C₂²	—	2	1	no	—
50	C₂²	C₄×Dic₂₃	1	1	yes	characteristic
51	C₂²	C₄⋊Dic₂₃	1	1	yes	characteristic
52	C₂²	C₂²⋊Dic₂₃	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	Dic₂₃⋊C₄	1	2	yes	—
55	C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	C₂₃⋊#95	1	1	yes	characteristic
59	C₂	#10625015	1	1	yes	characteristic
60	C₂	#10625018	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₂₃
commutator	C₂²×C₂₃	C₄²
Frattini	C₂²×C₄	C₂×D₂₃
Fitting	C₂×C₄⋊C₄×C₂₃	C₂
socle	C₂²×C₂₃	C₂²⋊C₄
prime radical	C₂³×C₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₃	C₂²
hypercenter	C₂×C₄⋊C₄	D₂₃
hypocenter	C₂₃	#343
LCS	G ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. #67267	C₄	semidirect
33. C₂₃	#343	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×C₂₃	2
C₂×#67267	2
#343	23