Subgroups — C₂₃×#7532 #19155158

Order 5888.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1713	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1411	C₂	1	2	yes	maximal
3	C₂₃×#500	—	2	2	no	—
4	C₂₃×#494	C₄	1	1	yes	characteristic
5	C₂₃×#494	C₂²	1	1	yes	characteristic
6	C₂₃×#356	—	2	2	no	—
7	C₈²×C₂₃	C₄	1	1	yes	characteristic
8	C₂×Q₁₆×C₂₃	—	4	1	no	—
9	C₂×Q₁₆×C₂₃	—	4	1	no	—
10	C₂₃×C₄⋊Q₈	—	2	1	no	—
11	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
12	C₂₃×#107	—	4	4	no	—
13	C₂₃×#101	—	4	2	no	—
14	C₄×C₈×C₂₃	—	2	1	no	—
15	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂×Q₈×C₂₃	—	2	2	no	—
17	C₂×Q₈×C₂₃	—	4	1	no	—
18	Q₁₆×C₂₃	—	8	2	no	—
19	Q₁₆×C₂₃	—	8	2	no	—
20	M₁₆×C₂₃	—	8	2	no	—
21	C₂×C₈×C₂₃	—	2	2	no	—
22	C₂×C₈×C₂₃	—	2	1	no	—
23	C₄⋊C₄×C₂₃	—	4	2	no	—
24	C₄⋊C₄×C₂₃	—	4	2	no	—
25	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	#7532	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	Q₈×C₂₃	—	8	2	no	—
28	Q₈×C₂₃	—	8	2	no	—
29	C₂×C₄×C₂₃	—	2	1	no	—
30	C₂×C₄×C₂₃	—	4	2	no	—
31	C₂×C₄×C₂₃	—	8	1	no	—
32	C₂×C₄×C₂₃	#99	1	1	yes	characteristic
33	C₈×C₂₃	—	2	4	no	—
34	C₈×C₂₃	—	2	2	no	—
35	C₈×C₂₃	—	16	2	no	—
36	#1713	C₂×C₂₃	1	1	yes	characteristic
37	#1411	C₂×C₂₃	1	2	yes	—
38	C₂²×C₂₃	#353	1	1	yes	characteristic
39	C₄×C₂₃	—	2	2	no	—
40	C₄×C₂₃	—	2	1	no	—
41	C₄×C₂₃	—	8	2	no	—
42	C₄×C₂₃	—	16	1	no	—
43	#500	—	2	2	no	—
44	#494	C₄×C₂₃	1	1	yes	characteristic
45	#494	C₂²×C₂₃	1	1	yes	characteristic
46	#356	—	2	2	no	—
47	C₈²	C₄×C₂₃	1	1	yes	characteristic
48	C₂×C₂₃	—	2	1	no	—
49	C₂×C₂₃	#1408	1	1	yes	characteristic
50	C₂×Q₁₆	—	4	1	no	—
51	C₂×Q₁₆	—	4	1	no	—
52	C₄⋊Q₈	—	2	1	no	—
53	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
54	#107	—	4	4	no	—
55	#101	—	4	2	no	—
56	C₄×C₈	—	2	1	no	—
57	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
58	C₂₃	#7532	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	4	1	no	—
61	Q₁₆	—	8	2	no	—
62	Q₁₆	—	8	2	no	—
63	M₁₆	—	8	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	1	no	—
66	C₄⋊C₄	—	4	2	no	—
67	C₄⋊C₄	—	4	2	no	—
68	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
69	Q₈	—	8	2	no	—
70	Q₈	—	8	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	8	1	no	—
74	C₂×C₄	C₂₃×#99	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₈	—	2	4	no	—
77	C₈	—	16	2	no	—
78	C₂²	C₂₃×#353	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	1	no	—
81	C₄	—	8	2	no	—
82	C₄	—	16	1	no	—
83	C₂	—	2	1	no	—
84	C₂	C₂₃×#1408	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#1408
commutator	C₄×C₈	C₂×C₄×C₂₃
Frattini, cosocle	#494	C₂²×C₂₃
prime radical	C₂²×C₂₃	#353
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂²×C₂₃ ◁ C₂×C₄×C₂₃ ◁ C₄²×C₂₃ ◁ C₂₃×#494 ◁ G

Complements

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

N	H ≅ G/N	type
26. #7532	C₂₃	direct (N × H)
58. C₂₃	#7532	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#1713	2
C₂₃×#1411	2
#7532	23

Subgroups — C23×#7532 #19155158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7532 #19155158