Subgroups — #13092495

Order 1472.

122 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	C₂×#67267	C₂	1	2	yes	maximal, complemented
3	C₂²⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×C₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₂₃	C₂²	1	2	yes	—
8	C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
9	C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
10	C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	#67267	—	2	4	no	—
12	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
13	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂³×C₂₃	C₂×C₄	1	2	yes	—
16	D₄×C₂₃	—	2	2	no	—
17	C₂×C₄×C₂₃	—	2	2	no	—
18	C₂×C₄×C₂₃	—	2	2	no	—
19	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂₃⋊C₈	—	2	2	no	—
24	C₂₃⋊C₈	—	2	1	no	—
25	C₂₃⋊C₈	D₄	1	2	yes	complemented
26	C₂²×C₂₃	—	2	2	no	—
27	C₂²×C₂₃	—	2	2	no	—
28	C₂²×C₂₃	—	2	2	no	—
29	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₂₃	—	2	2	no	—
31	C₄×C₂₃	—	2	1	no	—
32	C₄×C₂₃	M₁₆	1	2	yes	complemented
33	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₄×C₂₃	#41	1	1	yes	characteristic
35	#436	—	23	1	no	maximal, Sylow
36	C₂×C₂₃	—	4	2	no	—
37	C₂×C₂₃	C₄×D₄	1	1	yes	characteristic
38	C₂×C₂₃	C₂×M₁₆	1	1	yes	characteristic
39	C₂×C₂₃	#131	1	1	yes	characteristic
40	C₂×M₁₆	—	23	2	no	—
41	C₄×D₄	D₂₃	1	1	yes	characteristic
42	C₄⋊C₈	—	23	1	no	—
43	C₂²⋊C₈	—	23	2	no	—
44	C₄×C₈	—	23	1	no	—
45	C₂₃	#436	1	1	yes	characteristic, complemented, Sylow
46	C₂×D₄	Dic₂₃	1	1	yes	characteristic
47	C₂²×C₄	C₂×D₂₃	1	2	yes	—
48	M₁₆	—	46	4	no	—
49	C₂×C₈	—	23	1	no	—
50	C₂×C₈	—	23	1	no	—
51	C₂×C₈	—	23	2	no	—
52	C₄⋊C₄	Dic₂₃	1	1	yes	characteristic
53	C₂²⋊C₄	Dic₂₃	1	2	yes	—
54	C₄²	C₂×D₂₃	1	1	yes	characteristic
55	C₂³	C₂×Dic₂₃	1	2	yes	—
56	D₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₂₃	1	2	yes	—
61	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
63	C₈	—	23	2	no	—
64	C₈	—	46	2	no	—
65	C₈	—	46	1	no	—
66	C₂²	—	2	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	C₂²×Dic₂₃	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	#67267	1	2	yes	complemented
73	C₄	D₄×D₂₃	1	1	yes	characteristic
74	C₄	#67290	1	1	yes	characteristic
75	C₂	—	4	2	no	—
76	C₂	C₂×#67267	1	1	yes	characteristic
77	C₂	D₄×Dic₂₃	1	1	yes	characteristic
78	C₂	#10625119	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂²×C₄
Fitting	C₄×D₄×C₂₃	C₂
prime radical	C₂×D₄×C₂₃	C₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	C₄×D₄	D₂₃
hypocenter	C₂₃	#436
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×#67267	C₂	semidirect
3. C₂²⋊(C₂₃⋊C₈)	C₂	semidirect
4. C₄⋊(C₂₃⋊C₈)	C₂	semidirect
5. C₄×C₂₃⋊C₈	C₂	semidirect
25. C₂₃⋊C₈	D₄	semidirect
32. C₄×C₂₃	M₁₆	semidirect
45. C₂₃	#436	semidirect
72. C₄	#67267	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
C₂×#67267	2
C₂²⋊(C₂₃⋊C₈)	2
C₄⋊(C₂₃⋊C₈)	2
C₄×C₂₃⋊C₈	2
#436	23