Subgroups — #13092707

Order 1472.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	#10625040	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625014	C₂	1	2	yes	maximal
4	#10624990	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₂₃	—	2	1	no	—
8	C₂×Dic₄₆	—	2	1	no	—
9	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²⋊C₄×C₂₃	—	2	1	no	—
11	C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₃	—	2	1	no	—
13	C₄⋊Dic₂₃	C₂²	1	2	yes	—
14	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
15	C₂³×C₂₃	—	2	1	no	—
16	D₄×C₂₃	—	2	1	no	—
17	D₄×C₂₃	Q₈	1	2	yes	complemented
18	C₂×C₄×C₂₃	—	2	1	no	—
19	C₂×C₄×C₂₃	—	4	2	no	—
20	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂×Dic₂₃	—	2	2	no	—
24	Dic₄₆	—	4	2	no	—
25	C₂₃⋊C₈	—	2	2	no	—
26	C₂²×C₂₃	—	2	2	no	—
27	C₂²×C₂₃	—	4	1	no	—
28	C₂²×C₂₃	—	4	1	no	—
29	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
30	C₄×C₂₃	—	2	1	no	—
31	C₄×C₂₃	—	4	1	no	—
32	C₄×C₂₃	SD₁₆	1	2	yes	—
33	C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₂₃	#41	1	1	yes	characteristic
35	Dic₂₃	—	4	2	no	—
36	#476	—	23	1	no	maximal, Sylow
37	C₂×C₂₃	—	4	2	no	—
38	C₂×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
39	C₂×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
40	C₂×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
41	C₄⋊Q₈	—	23	1	no	—
42	C₄×D₄	D₂₃	1	1	yes	characteristic
43	#106	—	23	2	no	—
44	C₄⋊C₈	—	23	1	no	—
45	#102	—	23	2	no	—
46	C₂₃	#476	1	1	yes	characteristic, complemented, Sylow
47	C₂×Q₈	—	46	1	no	—
48	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
49	C₂²×C₄	—	2	1	no	—
50	C₂×C₈	—	23	2	no	—
51	C₄⋊C₄	—	23	2	no	—
52	C₄⋊C₄	—	46	1	no	—
53	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
54	C₂²⋊C₄	—	2	1	no	—
55	C₄²	C₂×D₂₃	1	1	yes	characteristic
56	C₂³	—	2	1	no	—
57	Q₈	—	92	2	no	—
58	D₄	—	2	1	no	—
59	D₄	Dic₄₆	1	2	yes	complemented
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	4	2	no	—
62	C₂×C₄	—	46	2	no	—
63	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
66	C₈	—	46	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	92	2	no	—
74	C₄	#67273	1	2	yes	—
75	C₄	C₂×Dic₄₆	1	1	yes	characteristic
76	C₄	#67288	1	1	yes	characteristic
77	C₂	—	4	2	no	—
78	C₂	#10625095	1	1	yes	characteristic
79	C₂	C₂×#67273	1	1	yes	characteristic
80	C₂	#10625120	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₄×D₄×C₂₃	C₂
socle	C₂²×C₂₃	C₂×D₄
prime radical	C₂×D₄×C₂₃	C₂²
hypercenter	C₄×D₄	D₂₃
hypocenter	C₂₃	#476
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #10625040	C₂	semidirect
4. #10624990	C₂	semidirect
5. C₄⋊(C₂₃⋊C₈)	C₂	semidirect
17. D₄×C₂₃	Q₈	semidirect
46. C₂₃	#476	semidirect
59. D₄	Dic₄₆	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₄×C₂₃	2
#10625040	2
#10625014	2
#10624990	2
C₄⋊(C₂₃⋊C₈)	2
#476	23