Subgroups — #15671557

Order 2944.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092214	C₂	1	1	yes	maximal, characteristic
2	#13092212	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#497	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆×C₂₃	—	2	1	no	—
5	C₂×SD₁₆×C₂₃	—	2	1	no	—
6	C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
7	C₂₃×#124	C₂²	1	1	yes	characteristic
8	C₂₃×#97	C₄	1	1	yes	characteristic, complemented
9	#10625018	—	2	1	no	—
10	#10625016	—	2	1	no	—
11	C₂×Q₈×C₂₃	—	2	1	no	—
12	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
13	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
14	Q₁₆×C₂₃	—	4	2	no	—
15	SD₁₆×C₂₃	—	4	2	no	—
16	C₂×C₈×C₂₃	—	2	1	no	—
17	C₄⋊C₄×C₂₃	—	4	1	no	—
18	C₂²⋊C₄×C₂₃	—	4	1	no	—
19	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²⋊Dic₂₃	—	4	1	no	—
21	#67267	—	4	1	no	—
22	C₂³×C₂₃	—	2	1	no	—
23	Q₈×C₂₃	—	4	1	no	—
24	Q₈×C₂₃	—	4	1	no	—
25	Q₈×C₂₃	—	4	1	no	—
26	D₄×C₂₃	—	4	1	no	—
27	C₂×C₄×C₂₃	—	2	1	no	—
28	C₂×C₄×C₂₃	—	2	1	no	—
29	C₂×C₄×C₂₃	—	4	1	no	—
30	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×Dic₂₃	—	8	1	no	—
32	C₈×C₂₃	—	2	2	no	—
33	C₂₃⋊C₈	—	8	1	no	—
34	#1407	—	23	1	no	maximal, Sylow
35	C₂²×C₂₃	—	4	1	no	—
36	C₂²×C₂₃	—	8	1	no	—
37	C₂²×C₂₃	#99	1	1	yes	characteristic
38	C₄×C₂₃	—	2	1	no	—
39	C₄×C₂₃	—	4	1	no	—
40	C₄×C₂₃	—	4	1	no	—
41	C₄×C₂₃	—	8	1	no	—
42	Dic₂₃	—	16	1	no	—
43	#497	D₂₃	1	1	yes	characteristic
44	#355	—	23	1	no	—
45	#354	—	23	1	no	—
46	C₂×C₂₃	—	2	1	no	—
47	C₂×C₂₃	—	8	1	no	—
48	C₂×C₂₃	#353	1	1	yes	characteristic
49	C₂×Q₁₆	—	2	1	no	—
50	C₂×SD₁₆	—	2	1	no	—
51	C₄⋊Q₈	Dic₂₃	1	1	yes	characteristic, complemented
52	#124	C₂×D₂₃	1	1	yes	characteristic
53	#101	—	46	1	no	—
54	#99	—	46	1	no	—
55	#97	Dic₂₃	1	1	yes	characteristic, complemented
56	C₂₃	#1407	1	1	yes	characteristic, complemented, Sylow
57	C₂×Q₈	—	2	1	no	—
58	C₂×Q₈	C₂²⋊D₂₃	1	1	yes	characteristic
59	C₂×D₄	C₂²⋊D₂₃	1	1	yes	characteristic
60	Q₁₆	—	4	2	no	—
61	SD₁₆	—	4	2	no	—
62	M₁₆	—	92	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₄⋊C₄	—	4	1	no	—
65	C₂²⋊C₄	—	4	1	no	—
66	C₂²⋊C₄	—	92	1	no	—
67	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
68	C₂³	—	2	1	no	—
69	Q₈	—	4	1	no	—
70	Q₈	—	4	1	no	—
71	Q₈	—	4	1	no	—
72	D₄	—	4	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	184	1	no	—
77	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
78	C₈	—	2	2	no	—
79	C₈	—	184	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	—	8	1	no	—
82	C₂²	#10625016	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	8	1	no	—
87	C₄	—	368	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	8	1	no	—
90	C₂	#13092210	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13092210
commutator	C₄²×C₂₃	C₂×C₄
Frattini	#124	C₂×D₂₃
Fitting	C₂₃×#497	C₂
socle	C₂×C₂₃	#353
prime radical	C₂×D₄×C₂₃	D₄
cosocle	C₂₃×#124	C₂²
hypercenter	#497	D₂₃
hypocenter	C₂₃	#1407
LCS	G ▷ C₄²×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #497

Complements

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

N	H ≅ G/N	type
6. C₂₃×C₄⋊Q₈	C₄	semidirect
8. C₂₃×#97	C₄	semidirect
51. C₄⋊Q₈	Dic₂₃	semidirect
55. #97	Dic₂₃	semidirect
56. C₂₃	#1407	semidirect

Maximal subgroups

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

subgroup	index
#13092214	2
#13092212	2
C₂₃×#497	2
#1407	23