Subgroups — #15673537

Order 2944.

180 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×#10624985	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	#10624985	C₄	1	4	yes	—
12	C₂²×C₄×C₂₃	Q₈	1	2	yes	—
13	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
14	M₁₆×C₂₃	—	2	4	no	—
15	C₂×C₈×C₂₃	—	2	2	no	—
16	C₄²×C₂₃	C₂×C₄	1	2	yes	—
17	C₄²×C₂₃	D₄	1	2	yes	—
18	#67267	—	2	4	no	—
19	C₂×C₂₃⋊C₈	—	2	2	no	—
20	C₂×C₂₃⋊C₈	—	2	2	no	—
21	C₂×C₂₃⋊C₈	—	2	2	no	—
22	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
23	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₄×C₂₃	—	4	2	no	—
25	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
26	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₂₃	#41	1	2	yes	—
29	C₂×C₄×C₂₃	#41	1	2	yes	—
30	C₈×C₂₃	—	4	2	no	—
31	C₂₃⋊C₈	—	2	4	no	—
32	C₂₃⋊C₈	—	4	2	no	—
33	#1838	—	23	1	no	maximal, Sylow
34	C₂²×C₂₃	—	2	2	no	—
35	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₂₃	#125	1	2	yes	—
37	C₄×C₂₃	—	2	4	no	—
38	C₄×C₂₃	#117	1	2	yes	—
39	C₄×C₂₃	#124	1	1	yes	characteristic
40	C₄×C₂₃	#124	1	1	yes	characteristic
41	#423	—	23	1	no	—
42	#423	D₂₃	1	1	yes	characteristic
43	C₂×#97	—	23	1	no	—
44	#341	—	23	4	no	—
45	C₂×C₂₃	—	2	2	no	—
46	C₂×C₂₃	#382	1	1	yes	characteristic
47	C₂×C₂₃	#430	1	2	yes	—
48	C₂×M₁₆	—	23	2	no	—
49	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
50	C₂²×C₈	—	23	2	no	—
51	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
52	C₄⋊C₈	—	2	2	no	—
53	C₄⋊C₈	—	46	2	no	—
54	#97	—	23	4	no	—
55	C₂₃	#1838	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₄	Dic₄₆	1	2	yes	—
57	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
58	M₁₆	—	2	4	no	—
59	M₁₆	—	46	4	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	23	4	no	—
62	C₂×C₈	—	46	2	no	—
63	C₂×C₈	—	46	2	no	—
64	C₂×C₈	—	46	2	no	—
65	C₄²	C₄×D₂₃	1	2	yes	—
66	C₄²	C₂²⋊D₂₃	1	2	yes	—
67	C₂³	C₂×Dic₄₆	1	1	yes	characteristic
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	Dic₂₃⋊C₄	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
71	C₂×C₄	#67290	1	2	yes	—
72	C₂×C₄	#67292	1	2	yes	—
73	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
74	C₈	—	4	2	no	—
75	C₈	—	46	4	no	—
76	C₈	—	92	2	no	—
77	C₂²	—	2	2	no	—
78	C₂²	#10625061	1	2	yes	—
79	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
80	C₄	—	2	4	no	—
81	C₄	#10625063	1	2	yes	—
82	C₄	#10625107	1	1	yes	characteristic
83	C₄	#10625118	1	1	yes	characteristic
84	C₂	—	2	2	no	—
85	C₂	#13092282	1	1	yes	characteristic
86	C₂	#13092465	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	#1838
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
55. C₂₃	#1838	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×#10624985	2
#13092181	2
C₂₃×#423	2
#1838	23