Subgroups — C₂₃×#33974 #19181600

Order 5888.

462 conjugacy classes of subgroups.

The subgroup lattice has 112 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#504	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2258	C₂	1	8	yes	maximal, complemented
3	C₂×C₂₃×M₆₄	C₂	1	6	yes	maximal, complemented
4	C₂×C₂₃×#131	C₄	1	1	yes	characteristic
5	C₂₃×#504	C₄	1	4	yes	—
6	C₂₃×#504	C₂²	1	4	yes	—
7	C₂×C₂₃×M₃₂	C₄	1	3	yes	—
8	C₂²×C₁₆×C₂₃	C₂²	1	3	yes	—
9	C₂₃×M₆₄	C₂²	1	4	yes	—
10	C₂₃×M₆₄	C₂²	1	12	yes	complemented
11	C₂×C₂₃×C₃₂	C₂²	1	12	yes	complemented
12	C₂×#41×C₂₃	C₈	1	1	yes	characteristic
13	C₂₃×#131	C₈	1	4	yes	—
14	C₂₃×#131	C₂×C₄	1	4	yes	—
15	C₂×M₁₆×C₂₃	C₈	1	3	yes	—
16	C₂²×C₈×C₂₃	C₂×C₄	1	3	yes	—
17	C₂₃×M₃₂	C₂×C₄	1	12	yes	—
18	C₂×C₁₆×C₂₃	C₂×C₄	1	3	yes	—
19	C₂×C₁₆×C₂₃	C₂×C₄	1	6	yes	—
20	C₂×C₁₆×C₂₃	C₂³	1	6	yes	—
21	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
22	C₂₃×C₃₂	C₂³	1	8	yes	—
23	#41×C₂₃	C₁₆	1	4	yes	—
24	#41×C₂₃	C₂×C₈	1	4	yes	—
25	C₂×Q₈×C₂₃	C₁₆	1	1	yes	characteristic
26	C₂×D₄×C₂₃	C₁₆	1	3	yes	—
27	C₂²×C₄×C₂₃	C₂×C₈	1	3	yes	—
28	M₁₆×C₂₃	C₂×C₈	1	12	yes	—
29	C₂×C₈×C₂₃	C₂×C₈	1	3	yes	—
30	C₂×C₈×C₂₃	C₂×C₈	1	6	yes	—
31	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₂₃	C₂²×C₄	1	6	yes	—
33	C₁₆×C₂₃	C₂²×C₄	1	1	yes	characteristic
34	C₁₆×C₂₃	C₂²×C₄	1	6	yes	—
35	C₁₆×C₂₃	C₂⁴	1	1	yes	characteristic
36	#33974	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂³×C₂₃	C₂×C₁₆	1	3	yes	—
38	Q₈×C₂₃	C₂×C₁₆	1	4	yes	—
39	D₄×C₂₃	C₂×C₁₆	1	12	yes	—
40	C₂×C₄×C₂₃	C₂×C₁₆	1	3	yes	—
41	C₂×C₄×C₂₃	C₂×C₁₆	1	6	yes	—
42	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
43	C₂×C₄×C₂₃	C₂²×C₈	1	6	yes	—
44	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
45	C₈×C₂₃	C₂²×C₈	1	6	yes	—
46	C₈×C₂₃	C₂³×C₄	1	1	yes	characteristic
47	C₂×#504	C₂×C₂₃	1	1	yes	characteristic
48	#2258	C₂×C₂₃	1	8	yes	complemented
49	C₂×M₆₄	C₂×C₂₃	1	6	yes	complemented
50	C₂²×C₂₃	—	4	3	no	—
51	C₂²×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
52	C₂²×C₂₃	C₂²×C₁₆	1	6	yes	—
53	C₄×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
54	C₄×C₂₃	C₂²×C₁₆	1	6	yes	—
55	C₄×C₂₃	C₂³×C₈	1	1	yes	characteristic
56	C₂×#131	C₄×C₂₃	1	1	yes	characteristic
57	#504	C₄×C₂₃	1	4	yes	—
58	#504	C₂²×C₂₃	1	4	yes	—
59	C₂×M₃₂	C₄×C₂₃	1	3	yes	—
60	C₂²×C₁₆	C₂²×C₂₃	1	3	yes	—
61	M₆₄	C₂²×C₂₃	1	4	yes	—
62	M₆₄	C₂²×C₂₃	1	12	yes	complemented
63	C₂×C₃₂	C₂²×C₂₃	1	12	yes	complemented
64	C₂×C₂₃	—	2	1	no	—
65	C₂×C₂₃	—	2	6	no	—
66	C₂×C₂₃	C₂³×C₁₆	1	1	yes	characteristic
67	C₂×#41	C₈×C₂₃	1	1	yes	characteristic
68	#131	C₈×C₂₃	1	4	yes	—
69	#131	C₂×C₄×C₂₃	1	4	yes	—
70	C₂×M₁₆	C₈×C₂₃	1	3	yes	—
71	C₂²×C₈	C₂×C₄×C₂₃	1	3	yes	—
72	M₃₂	C₂×C₄×C₂₃	1	12	yes	—
73	C₂×C₁₆	C₂×C₄×C₂₃	1	3	yes	—
74	C₂×C₁₆	C₂×C₄×C₂₃	1	6	yes	—
75	C₂×C₁₆	C₂³×C₂₃	1	6	yes	—
76	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
77	C₃₂	C₂³×C₂₃	1	8	yes	—
78	C₂₃	#33974	1	1	yes	characteristic, direct product factor, Sylow
79	#41	C₁₆×C₂₃	1	4	yes	—
80	#41	C₂×C₈×C₂₃	1	4	yes	—
81	C₂×Q₈	C₁₆×C₂₃	1	1	yes	characteristic
82	C₂×D₄	C₁₆×C₂₃	1	3	yes	—
83	C₂²×C₄	C₂×C₈×C₂₃	1	3	yes	—
84	M₁₆	C₂×C₈×C₂₃	1	12	yes	—
85	C₂×C₈	C₂×C₈×C₂₃	1	3	yes	—
86	C₂×C₈	C₂×C₈×C₂₃	1	6	yes	—
87	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
88	C₂×C₈	C₂²×C₄×C₂₃	1	6	yes	—
89	C₁₆	C₂²×C₄×C₂₃	1	1	yes	characteristic
90	C₁₆	C₂²×C₄×C₂₃	1	6	yes	—
91	C₁₆	C₂⁴×C₂₃	1	1	yes	characteristic
92	C₂³	C₂×C₁₆×C₂₃	1	3	yes	—
93	Q₈	C₂×C₁₆×C₂₃	1	4	yes	—
94	D₄	C₂×C₁₆×C₂₃	1	12	yes	—
95	C₂×C₄	C₂×C₁₆×C₂₃	1	3	yes	—
96	C₂×C₄	C₂×C₁₆×C₂₃	1	6	yes	—
97	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
98	C₂×C₄	C₂²×C₈×C₂₃	1	6	yes	—
99	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
100	C₈	C₂²×C₈×C₂₃	1	6	yes	—
101	C₈	C₂³×C₄×C₂₃	1	1	yes	characteristic
102	C₂²	—	4	3	no	—
103	C₂²	C₂²×C₁₆×C₂₃	1	1	yes	characteristic
104	C₂²	C₂²×C₁₆×C₂₃	1	6	yes	—
105	C₄	C₂²×C₁₆×C₂₃	1	1	yes	characteristic
106	C₄	C₂²×C₁₆×C₂₃	1	6	yes	—
107	C₄	C₂³×C₈×C₂₃	1	1	yes	characteristic
108	C₂	—	2	1	no	—
109	C₂	—	2	6	no	—
110	C₂	C₂³×C₁₆×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₂₃	C₂⁴
commutator	C₂	C₂³×C₁₆×C₂₃
Frattini, cosocle	C₁₆	C₂⁴×C₂₃
socle	C₂×C₂₃	C₂³×C₁₆
prime radical	C₂×#41×C₂₃	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2258	C₂	semidirect
3. C₂×C₂₃×M₆₄	C₂	semidirect
10. C₂₃×M₆₄	C₂²	semidirect
11. C₂×C₂₃×C₃₂	C₂²	semidirect
36. #33974	C₂₃	direct (N × H)
48. #2258	C₂×C₂₃	semidirect
49. C₂×M₆₄	C₂×C₂₃	semidirect
62. M₆₄	C₂²×C₂₃	semidirect
63. C₂×C₃₂	C₂²×C₂₃	semidirect
78. C₂₃	#33974	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#504	2
C₂₃×#2258	2
C₂×C₂₃×M₆₄	2
#33974	23

Subgroups — C23×#33974 #19181600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33974 #19181600