Subgroups — #15671343

Order 2944.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092143	C₂	1	2	yes	maximal
2	C₂₃×#384	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
4	C₂×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₂₃	C₄	1	1	yes	characteristic
7	C₂²⋊(C₂₃⋊C₈)	—	2	2	no	—
8	C₂²×C₄×C₂₃	—	2	1	no	—
9	C₂²×C₄×C₂₃	—	2	1	no	—
10	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₄×C₂₃	D₄	1	2	yes	—
12	C₄⋊C₄×C₂₃	—	4	1	no	—
13	C₄⋊C₄×C₂₃	—	4	1	no	—
14	C₄⋊C₄×C₂₃	—	4	4	no	—
15	C₄²×C₂₃	—	2	1	no	—
16	C₄²×C₂₃	C₈	1	2	yes	complemented
17	C₂×C₂₃⋊C₈	—	4	2	no	—
18	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₃	—	2	2	no	—
20	C₂×C₄×C₂₃	—	2	2	no	—
21	C₂×C₄×C₂₃	—	2	2	no	—
22	C₂×C₄×C₂₃	—	2	1	no	—
23	C₂×C₄×C₂₃	—	2	1	no	—
24	C₂×C₄×C₂₃	—	2	2	no	—
25	C₂×C₄×C₂₃	—	4	2	no	—
26	C₂×C₄×C₂₃	—	4	2	no	—
27	C₂×C₄×C₂₃	—	4	1	no	—
28	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₂₃	M₁₆	1	1	yes	characteristic
30	C₂₃⋊C₈	—	8	2	no	—
31	#1325	—	23	1	no	maximal, Sylow
32	C₂²×C₂₃	—	2	1	no	—
33	C₂²×C₂₃	—	2	1	no	—
34	C₂²×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
35	C₂²×C₂₃	#99	1	1	yes	characteristic
36	C₂²×C₂₃	#101	1	1	yes	characteristic
37	C₄×C₂₃	—	2	4	no	—
38	C₄×C₂₃	—	2	1	no	—
39	C₄×C₂₃	—	2	1	no	—
40	C₄×C₂₃	—	4	2	no	—
41	C₄×C₂₃	—	8	1	no	—
42	#384	D₂₃	1	1	yes	characteristic
43	#324	—	23	2	no	—
44	C₂×C₂₃	—	2	1	no	—
45	C₂×C₂₃	—	2	1	no	—
46	C₂×C₂₃	#324	1	1	yes	characteristic
47	C₂×C₂₃	#353	1	1	yes	characteristic
48	C₂×C₂₃	#356	1	1	yes	characteristic
49	C₂×C₄⋊C₄	—	2	2	no	—
50	C₂×C₄⋊C₄	Dic₂₃	1	1	yes	characteristic
51	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
52	C₂×C₄²	Dic₂₃	1	1	yes	characteristic
53	C₂²⋊C₈	—	46	2	no	—
54	C₂₃	#1325	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₄	—	2	1	no	—
56	C₂²×C₄	—	2	1	no	—
57	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
58	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
59	C₂×C₈	—	92	2	no	—
60	C₄⋊C₄	—	4	1	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	4	4	no	—
63	C₄²	—	2	1	no	—
64	C₄²	C₂₃⋊C₈	1	2	yes	complemented
65	C₂³	C₂²⋊Dic₂₃	1	1	yes	characteristic
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	C₂×C₂₃⋊C₈	1	1	yes	characteristic
76	C₂×C₄	#67267	1	1	yes	characteristic
77	C₈	—	184	2	no	—
78	C₂²	—	2	1	no	—
79	C₂²	—	2	1	no	—
80	C₂²	C₂²⋊(C₂₃⋊C₈)	1	1	yes	characteristic
81	C₂²	#10625016	1	1	yes	characteristic
82	C₂²	#10625018	1	1	yes	characteristic
83	C₄	—	2	4	no	—
84	C₄	—	2	1	no	—
85	C₄	—	2	1	no	—
86	C₄	—	4	2	no	—
87	C₄	—	8	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	2	1	no	—
90	C₂	#13092143	1	1	yes	characteristic
91	C₂	#13092210	1	1	yes	characteristic
92	C₂	#13092216	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10625016
commutator	C₂×C₄×C₂₃	C₂×C₈
Frattini	C₂×C₄⋊C₄	C₂×D₂₃
Fitting	C₂₃×#384	C₂
socle	C₂²×C₂₃	#99
prime radical	C₂³×C₂₃	C₂²⋊C₄
cosocle	C₂×C₄⋊C₄×C₂₃	C₂²
hypercenter	#384	D₂₃
hypocenter	C₂₃	#1325
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #384

Complements

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

N	H ≅ G/N	type
16. C₄²×C₂₃	C₈	semidirect
54. C₂₃	#1325	semidirect
64. C₄²	C₂₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#13092143	2
C₂₃×#384	2
#1325	23