Subgroups — #15673534

Order 2944.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092441	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#13092181	C₂	1	4	yes	maximal
4	C₂₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
6	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×#67267	C₂²	1	2	yes	—
8	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
9	C₂₃×C₄⋊C₈	—	2	2	no	—
10	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
11	C₄×C₂₃⋊C₈	C₄	1	2	yes	—
12	C₄×C₂₃⋊C₈	C₄	1	2	yes	—
13	C₂²×C₄×C₂₃	Q₈	1	2	yes	—
14	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
15	M₁₆×C₂₃	—	2	4	no	—
16	C₂×C₈×C₂₃	—	2	2	no	—
17	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₂₃	D₄	1	2	yes	—
20	#67267	—	2	4	no	—
21	C₂×C₂₃⋊C₈	—	2	2	no	—
22	C₂×C₂₃⋊C₈	—	2	2	no	—
23	C₂×C₂₃⋊C₈	—	2	2	no	—
24	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
25	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₄×C₂₃	—	2	4	no	—
27	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₂₃	#41	1	2	yes	—
32	C₂×C₄×C₂₃	#41	1	2	yes	—
33	C₈×C₂₃	—	4	2	no	—
34	C₂₃⋊C₈	—	2	4	no	—
35	C₂₃⋊C₈	—	4	2	no	—
36	#1837	—	23	1	no	maximal, Sylow
37	C₂²×C₂₃	—	2	2	no	—
38	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₂₃	#125	1	2	yes	—
40	C₄×C₂₃	—	2	2	no	—
41	C₄×C₂₃	#108	1	4	yes	—
42	C₄×C₂₃	#117	1	2	yes	—
43	C₄×C₂₃	#124	1	1	yes	characteristic
44	C₄×C₂₃	#124	1	1	yes	characteristic
45	#423	—	23	1	no	—
46	#423	D₂₃	1	1	yes	characteristic
47	C₂×C₄×C₈	—	23	1	no	—
48	#341	—	23	4	no	—
49	C₂×C₂₃	—	2	2	no	—
50	C₂×C₂₃	#382	1	1	yes	characteristic
51	C₂×C₂₃	C₂×#108	1	2	yes	—
52	C₂×M₁₆	—	23	2	no	—
53	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
54	C₂²×C₈	—	23	2	no	—
55	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
56	C₄⋊C₈	—	2	2	no	—
57	C₄⋊C₈	—	46	2	no	—
58	C₄×C₈	—	23	2	no	—
59	C₄×C₈	—	23	2	no	—
60	C₂₃	#1837	1	1	yes	characteristic, complemented, Sylow
61	C₂²×C₄	Dic₄₆	1	2	yes	—
62	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
63	M₁₆	—	2	4	no	—
64	M₁₆	—	46	4	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	23	4	no	—
67	C₂×C₈	—	46	2	no	—
68	C₂×C₈	—	46	2	no	—
69	C₂×C₈	—	46	2	no	—
70	C₄²	C₄×D₂₃	1	1	yes	characteristic
71	C₄²	C₄×D₂₃	1	1	yes	characteristic
72	C₄²	C₂²⋊D₂₃	1	2	yes	—
73	C₂³	C₂×Dic₄₆	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
76	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
77	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
78	C₂×C₄	#67290	1	2	yes	—
79	C₂×C₄	#67292	1	2	yes	—
80	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
81	C₈	—	4	2	no	—
82	C₈	—	46	4	no	—
83	C₈	—	92	2	no	—
84	C₂²	—	2	2	no	—
85	C₂²	#10625061	1	2	yes	—
86	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	#10625004	1	4	yes	—
89	C₄	#10625063	1	2	yes	—
90	C₄	#10625107	1	1	yes	characteristic
91	C₄	#10625118	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	#13092282	1	1	yes	characteristic
94	C₂	C₂×#10625004	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#423	C₂
socle	C₂²×C₂₃	C₂×C₄⋊C₄
prime radical	C₂³×C₂₃	C₂×Q₈
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#423	D₂₃
hypocenter	C₂₃	#1837
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
60. C₂₃	#1837	semidirect

Maximal subgroups

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

subgroup	index
#13092441	2
C₂×C₄×C₂₃⋊C₈	2
#13092181	2
C₂₃×#423	2
#1837	23