Subgroups — #13092391

Order 1472.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
2	#10625097	C₂	1	2	yes	maximal, complemented
3	#10625016	C₂	1	4	yes	maximal, complemented
4	#41×C₂₃	—	2	2	no	—
5	#41×C₂₃	—	2	2	no	—
6	C₂×Q₈×C₂₃	C₄	1	1	yes	characteristic, complemented
7	C₂×D₄×C₂₃	C₄	1	1	yes	characteristic, complemented
8	C₂×D₄×C₂₃	C₂²	1	2	yes	—
9	C₂²×C₄×C₂₃	C₄	1	2	yes	complemented
10	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₂₃	C₂²	1	4	yes	—
12	C₄⋊Dic₂₃	—	2	2	no	—
13	C₄×Dic₂₃	—	2	2	no	—
14	C₂³×C₂₃	C₂×C₄	1	2	yes	—
15	C₂³×C₂₃	C₂³	1	1	yes	characteristic
16	Q₈×C₂₃	—	2	2	no	—
17	D₄×C₂₃	—	2	4	no	—
18	D₄×C₂₃	—	2	2	no	—
19	C₂×C₄×C₂₃	—	2	2	no	—
20	C₂×C₄×C₂₃	—	2	2	no	—
21	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₃	D₄	1	2	yes	—
25	C₂×C₄×C₂₃	D₄	1	2	yes	—
26	C₂×Dic₂₃	—	2	4	no	—
27	C₂²×C₂₃	—	2	2	no	—
28	C₂²×C₂₃	—	4	1	no	—
29	C₂²×C₂₃	—	4	2	no	—
30	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₂₃	C₂×D₄	1	2	yes	—
32	C₄×C₂₃	—	2	2	no	—
33	C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
35	C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₂₃	—	4	4	no	—
37	#410	—	23	1	no	maximal, Sylow
38	C₂×C₂₃	—	2	2	no	—
39	C₂×C₂₃	—	2	1	no	—
40	C₂×C₂₃	—	4	2	no	—
41	C₂×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
42	C₂×#41	D₂₃	1	1	yes	characteristic
43	#117	—	23	2	no	—
44	#99	—	23	4	no	—
45	C₂₃	#410	1	1	yes	characteristic, complemented, Sylow
46	#41	—	2	2	no	—
47	#41	—	2	2	no	—
48	C₂×Q₈	Dic₂₃	1	1	yes	characteristic, complemented
49	C₂×D₄	Dic₂₃	1	1	yes	characteristic, complemented
50	C₂×D₄	C₂×D₂₃	1	2	yes	—
51	C₂²×C₄	Dic₂₃	1	2	yes	complemented
52	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
53	C₄⋊C₄	—	46	2	no	—
54	C₂²⋊C₄	—	23	4	no	—
55	C₄²	—	46	2	no	—
56	C₂³	C₂×Dic₂₃	1	2	yes	—
57	C₂³	C₂²×D₂₃	1	1	yes	characteristic
58	Q₈	—	2	2	no	—
59	D₄	—	2	4	no	—
60	D₄	—	2	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	46	4	no	—
64	C₂×C₄	C₂×Dic₂₃	1	2	yes	—
65	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
66	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
67	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
68	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
69	C₂²	—	2	2	no	—
70	C₂²	—	4	1	no	—
71	C₂²	—	4	2	no	—
72	C₂²	C₂²×Dic₂₃	1	1	yes	characteristic
73	C₂²	C₂×C₂²⋊D₂₃	1	2	yes	—
74	C₄	—	2	2	no	—
75	C₄	—	92	4	no	—
76	C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
77	C₄	C₂²⋊Dic₂₃	1	2	yes	—
78	C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
79	C₂	—	2	2	no	—
80	C₂	—	2	1	no	—
81	C₂	—	4	2	no	—
82	C₂	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊Dic₂₃
commutator	C₂²×C₂₃	C₂²×C₄
Frattini	C₂³	C₂²×D₂₃
Fitting, prime radical	C₂×#41×C₂₃	C₂
socle	C₂×C₂₃	C₂×C₂²⋊C₄
cosocle	C₂³×C₂₃	C₂³
hypercenter	C₂×#41	D₂₃
hypocenter	C₂₃	#410
LCS	G ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. #10625097	C₂	semidirect
3. #10625016	C₂	semidirect
6. C₂×Q₈×C₂₃	C₄	semidirect
7. C₂×D₄×C₂₃	C₄	semidirect
9. C₂²×C₄×C₂₃	C₄	semidirect
45. C₂₃	#410	semidirect
48. C₂×Q₈	Dic₂₃	semidirect
49. C₂×D₄	Dic₂₃	semidirect
51. C₂²×C₄	Dic₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×C₂₃	2
#10625097	2
#10625016	2
#410	23