Subgroups — C₂₃×#7197 #19154823

Order 5888.

184 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2163	C₂	1	2	yes	maximal
2	C₂₃×#1452	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#431	C₄	1	2	yes	—
4	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
5	C₂₃×C₄⋊C₁₆	—	2	2	no	—
6	C₂₃×C₂²⋊C₁₆	C₄	1	4	yes	—
7	C₂₃×#346	—	2	2	no	—
8	C₂₃×#322	—	2	2	no	—
9	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
10	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
11	C₂×C₁₆×C₂₃	—	2	4	no	—
12	C₂₃×C₄⋊C₈	—	2	2	no	—
13	C₂₃×C₂²⋊C₈	C₈	1	4	yes	—
14	C₄×C₈×C₂₃	—	2	2	no	—
15	C₄×C₈×C₂₃	D₄	1	2	yes	—
16	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
17	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	C₂×C₈	1	2	yes	—
20	C₂×C₈×C₂₃	—	2	4	no	—
21	C₂×C₈×C₂₃	—	4	2	no	—
22	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
23	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
24	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
26	C₄²×C₂₃	M₁₆	1	2	yes	—
27	C₁₆×C₂₃	—	4	4	no	—
28	#7197	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂³×C₂₃	C₄×C₈	1	1	yes	characteristic
30	C₂×C₄×C₂₃	—	2	2	no	—
31	C₂×C₄×C₂₃	—	4	2	no	—
32	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
33	C₂×C₄×C₂₃	#97	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
35	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
36	C₈×C₂₃	—	2	2	no	—
37	C₈×C₂₃	—	2	2	no	—
38	C₈×C₂₃	—	4	4	no	—
39	#2163	C₂×C₂₃	1	2	yes	—
40	#1452	C₂×C₂₃	1	1	yes	characteristic
41	C₂²×C₂₃	—	2	1	no	—
42	C₂²×C₂₃	—	2	2	no	—
43	C₂²×C₂₃	#336	1	1	yes	characteristic
44	C₄×C₂₃	—	2	1	no	—
45	C₄×C₂₃	—	2	2	no	—
46	C₄×C₂₃	—	2	2	no	—
47	C₄×C₂₃	#344	1	2	yes	—
48	#431	C₄×C₂₃	1	2	yes	—
49	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
50	C₄⋊C₁₆	—	2	2	no	—
51	C₂²⋊C₁₆	C₄×C₂₃	1	4	yes	—
52	#346	—	2	2	no	—
53	#322	—	2	2	no	—
54	C₂×C₂₃	—	4	1	no	—
55	C₂×C₂₃	#1281	1	1	yes	characteristic
56	C₂×C₂₃	#1378	1	2	yes	—
57	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
58	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
59	C₂×C₁₆	—	2	4	no	—
60	C₄⋊C₈	—	2	2	no	—
61	C₂²⋊C₈	C₈×C₂₃	1	4	yes	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	D₄×C₂₃	1	2	yes	—
64	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
65	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
66	C₂₃	#7197	1	1	yes	characteristic, direct product factor, Sylow
67	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
68	C₂²×C₄	C₂×C₈×C₂₃	1	2	yes	—
69	C₂×C₈	—	2	4	no	—
70	C₂×C₈	—	4	2	no	—
71	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
72	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
73	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
74	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
75	C₄²	M₁₆×C₂₃	1	2	yes	—
76	C₁₆	—	4	4	no	—
77	C₂³	C₄×C₈×C₂₃	1	1	yes	characteristic
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
81	C₂×C₄	C₂₃×#97	1	1	yes	characteristic
82	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
83	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
84	C₈	—	2	2	no	—
85	C₈	—	2	2	no	—
86	C₈	—	4	4	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	2	1	no	—
89	C₂²	C₂₃×#336	1	1	yes	characteristic
90	C₄	—	2	2	no	—
91	C₄	—	2	2	no	—
92	C₄	—	2	1	no	—
93	C₄	C₂₃×#344	1	2	yes	—
94	C₂	—	4	1	no	—
95	C₂	C₂₃×#1281	1	1	yes	characteristic
96	C₂	C₂₃×#1378	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#95
commutator	C₂³	C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
socle	C₂²×C₂₃	#336
prime radical	C₂³×C₂₃	C₄×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
28. #7197	C₂₃	direct (N × H)
66. C₂₃	#7197	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×#2163	2
C₂₃×#1452	2
#7197	23

Subgroups — C23×#7197 #19154823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7197 #19154823