Subgroups — #13092614

Order 1472.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×#67273	C₂	1	2	yes	maximal, complemented
3	C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625014	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×C₂₃	—	2	1	no	—
7	C₂×D₄×C₂₃	—	2	1	no	—
8	C₂×D₄×C₂₃	C₂²	1	2	yes	—
9	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
10	C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	#67273	—	2	4	no	—
12	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
13	Dic₂₃⋊C₄	—	2	1	no	—
14	C₄×Dic₂₃	—	2	1	no	—
15	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
16	C₂³×C₂₃	—	2	2	no	—
17	D₄×C₂₃	—	2	2	no	—
18	D₄×C₂₃	—	2	2	no	—
19	D₄×C₂₃	—	4	2	no	—
20	D₄×C₂₃	—	4	2	no	—
21	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₂×Dic₂₃	—	2	1	no	—
25	C₂×Dic₂₃	—	2	1	no	—
26	Dic₄₆	—	2	1	no	—
27	Dic₄₆	D₄	1	2	yes	complemented
28	C₂₃⋊C₈	—	2	2	no	—
29	C₂²×C₂₃	—	4	2	no	—
30	C₂²×C₂₃	—	4	2	no	—
31	C₂²×C₂₃	—	4	2	no	—
32	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
33	C₄×C₂₃	—	2	1	no	—
34	C₄×C₂₃	SD₁₆	1	2	yes	complemented
35	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₄×C₂₃	#41	1	1	yes	characteristic
37	Dic₂₃	—	2	2	no	—
38	Dic₂₃	—	4	1	no	—
39	#460	—	23	1	no	maximal, Sylow
40	C₂×C₂₃	—	8	2	no	—
41	C₂×C₂₃	#121	1	1	yes	characteristic
42	C₂×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
43	C₂×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
44	C₂×SD₁₆	—	23	2	no	—
45	#127	D₂₃	1	1	yes	characteristic
46	C₄×Q₈	—	23	1	no	—
47	C₄⋊C₈	—	23	1	no	—
48	#102	—	23	2	no	—
49	C₂₃	#460	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₈	—	23	1	no	—
51	C₂×D₄	—	2	1	no	—
52	C₂×D₄	—	2	1	no	—
53	C₂×D₄	C₂×D₂₃	1	2	yes	—
54	SD₁₆	—	46	4	no	—
55	C₂×C₈	—	23	2	no	—
56	C₄⋊C₄	—	23	1	no	—
57	C₄⋊C₄	—	46	1	no	—
58	C₄²	—	46	1	no	—
59	C₄²	C₂×D₂₃	1	1	yes	characteristic
60	C₂³	—	2	2	no	—
61	Q₈	—	23	2	no	—
62	Q₈	—	46	1	no	—
63	D₄	—	2	2	no	—
64	D₄	—	2	2	no	—
65	D₄	—	4	2	no	—
66	D₄	—	4	2	no	—
67	C₂×C₄	—	46	1	no	—
68	C₂×C₄	—	46	1	no	—
69	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
70	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
72	C₈	—	46	2	no	—
73	C₂²	—	4	2	no	—
74	C₂²	—	4	2	no	—
75	C₂²	—	4	2	no	—
76	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	46	2	no	—
79	C₄	—	92	1	no	—
80	C₄	#67273	1	2	yes	complemented
81	C₄	D₄×D₂₃	1	1	yes	characteristic
82	C₄	#67290	1	1	yes	characteristic
83	C₂	—	8	2	no	—
84	C₂	#10625103	1	1	yes	characteristic
85	C₂	C₂×#67273	1	1	yes	characteristic
86	C₂	#10625109	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting, prime radical	C₂₃×#127	C₂
socle	C₂²×C₂₃	C₂×D₄
hypercenter	#127	D₂₃
hypocenter	C₂₃	#460
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
2. C₂×#67273	C₂	semidirect
3. C₄×Dic₄₆	C₂	semidirect
4. #10625014	C₂	semidirect
5. C₄⋊(C₂₃⋊C₈)	C₂	semidirect
27. Dic₄₆	D₄	semidirect
34. C₄×C₂₃	SD₁₆	semidirect
49. C₂₃	#460	semidirect
80. C₄	#67273	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#127	2
C₂×#67273	2
C₄×Dic₄₆	2
#10625014	2
C₄⋊(C₂₃⋊C₈)	2
#460	23