Subgroups — #15672024

Order 2944.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092474	C₂	1	2	yes	maximal
2	#13092159	C₂	1	4	yes	maximal, complemented
3	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#125	—	2	2	no	—
5	C₂₃×#125	C₄	1	2	yes	—
6	C₂₃×#125	C₂²	1	2	yes	—
7	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
8	C₂×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
9	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²⋊(C₂₃⋊C₈)	—	2	2	no	—
11	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
12	#10624985	C₂²	1	4	yes	—
13	C₂²×C₄×C₂₃	—	2	2	no	—
14	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₃	D₄	1	2	yes	—
16	C₄⋊C₄×C₂₃	—	2	4	no	—
17	C₄⋊C₄×C₂₃	—	2	2	no	—
18	C₄⋊C₄×C₂₃	—	2	4	no	—
19	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
20	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₂₃	D₄	1	2	yes	—
22	C₄²×C₂₃	C₂³	1	1	yes	characteristic
23	C₂×C₂₃⋊C₈	—	2	4	no	—
24	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₃	—	2	2	no	—
26	C₂×C₄×C₂₃	—	2	4	no	—
27	C₂×C₄×C₂₃	—	4	2	no	—
28	C₂×C₄×C₂₃	—	4	4	no	—
29	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
30	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
33	C₂₃⋊C₈	—	4	4	no	—
34	#1534	—	23	1	no	maximal, Sylow
35	C₂²×C₂₃	—	2	2	no	—
36	C₂²×C₂₃	#101	1	2	yes	—
37	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₄×C₂₃	—	2	1	no	—
39	C₄×C₂₃	—	2	1	no	—
40	C₄×C₂₃	—	2	2	no	—
41	C₄×C₂₃	—	2	2	no	—
42	C₄×C₂₃	—	4	4	no	—
43	C₂×#125	D₂₃	1	1	yes	characteristic
44	#432	—	23	2	no	—
45	#330	—	23	4	no	—
46	C₂×C₂₃	—	2	2	no	—
47	C₂×C₂₃	C₂×#101	1	1	yes	characteristic
48	C₂×C₂₃	#421	1	2	yes	—
49	#125	—	2	2	no	—
50	#125	Dic₂₃	1	2	yes	—
51	#125	C₂×D₂₃	1	2	yes	—
52	C₂×C₄⋊C₄	—	2	2	no	—
53	C₂×C₄⋊C₄	Dic₂₃	1	2	yes	—
54	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
55	C₄⋊C₈	—	46	2	no	—
56	C₂²⋊C₈	—	46	2	no	—
57	#97	—	23	4	no	—
58	C₂₃	#1534	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₄	—	2	2	no	—
60	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
61	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
62	C₂×C₈	—	46	4	no	—
63	C₄⋊C₄	—	2	4	no	—
64	C₄⋊C₄	—	2	2	no	—
65	C₄⋊C₄	—	2	4	no	—
66	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	—
67	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
68	C₄²	C₂²⋊D₂₃	1	2	yes	—
69	C₄²	C₂²×D₂₃	1	1	yes	characteristic
70	C₂³	C₂²⋊Dic₂₃	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	4	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	4	4	no	—
75	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
76	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
77	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
79	C₈	—	92	4	no	—
80	C₂²	—	2	2	no	—
81	C₂²	#10625018	1	2	yes	—
82	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	2	2	no	—
86	C₄	—	2	2	no	—
87	C₄	—	4	4	no	—
88	C₂	—	2	2	no	—
89	C₂	C₂×#10625018	1	1	yes	characteristic
90	C₂	#13092434	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂×C₂₃×#125	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₃	C₂²⋊C₄
cosocle	C₄²×C₂₃	C₂³
hypercenter	C₂×#125	D₂₃
hypocenter	C₂₃	#1534
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
2. #13092159	C₂	semidirect
58. C₂₃	#1534	semidirect

Maximal subgroups

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

subgroup	index
#13092474	2
#13092159	2
C₂×C₂₃×#125	2
#1534	23