Subgroups — C₂³×C₂₃×#106 #19207727

Order 5888.

3614 conjugacy classes of subgroups.

The subgroup lattice has 122 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal
3	C₂²×C₂₃×#106	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₄×C₂₃	—	2	2	no	—
5	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₂₃	C₄	1	16	yes	complemented
7	C₂³×C₈×C₂₃	C₂²	1	14	yes	—
8	C₂²×C₄⋊C₄×C₂₃	—	2	14	no	—
9	C₂²×C₄⋊C₄×C₂₃	C₂²	1	28	yes	—
10	C₂×C₂₃×#106	C₂²	1	112	yes	direct product factor
11	C₂⁵×C₂₃	D₄	1	1	yes	characteristic
12	C₂³×C₄×C₂₃	—	2	28	no	—
13	C₂³×C₄×C₂₃	—	4	16	no	—
14	C₂³×C₄×C₂₃	C₂×C₄	1	8	yes	—
15	C₂³×C₄×C₂₃	D₄	1	7	yes	—
16	C₂³×C₄×C₂₃	Q₈	1	8	yes	—
17	C₂³×C₄×C₂₃	C₂³	1	7	yes	—
18	C₂²×C₈×C₂₃	C₂×C₄	1	112	yes	complemented
19	C₂²×C₈×C₂₃	C₂³	1	28	yes	—
20	C₂×C₄⋊C₄×C₂₃	—	2	84	no	—
21	C₂×C₄⋊C₄×C₂₃	C₂³	1	56	yes	—
22	C₂₃×#106	C₂³	1	64	yes	direct product factor
23	C₂⁴×C₂₃	C₄⋊C₄	1	8	yes	—
24	C₂⁴×C₂₃	SD₁₆	1	16	yes	—
25	C₂⁴×C₂₃	C₂×D₄	1	7	yes	—
26	C₂²×C₄×C₂₃	—	2	56	no	—
27	C₂²×C₄×C₂₃	—	4	112	no	—
28	C₂²×C₄×C₂₃	C₄⋊C₄	1	56	yes	—
29	C₂²×C₄×C₂₃	C₂²×C₄	1	28	yes	—
30	C₂²×C₄×C₂₃	C₂×D₄	1	21	yes	—
31	C₂²×C₄×C₂₃	C₂×Q₈	1	28	yes	—
32	C₂²×C₄×C₂₃	C₂⁴	1	7	yes	—
33	C₂×C₈×C₂₃	C₂²×C₄	1	112	yes	complemented
34	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
35	C₄⋊C₄×C₂₃	—	2	56	no	—
36	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
37	C₂³×#106	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂³×C₂₃	#106	1	64	yes	direct product factor
39	C₂³×C₂₃	C₂×C₄⋊C₄	1	28	yes	—
40	C₂³×C₂₃	C₂×SD₁₆	1	56	yes	—
41	C₂³×C₂₃	C₂²×D₄	1	7	yes	—
42	C₂×C₄×C₂₃	—	2	16	no	—
43	C₂×C₄×C₂₃	—	4	112	no	—
44	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	84	yes	—
45	C₂×C₄×C₂₃	C₂³×C₄	1	14	yes	—
46	C₂×C₄×C₂₃	C₂²×D₄	1	7	yes	—
47	C₂×C₄×C₂₃	C₂²×Q₈	1	14	yes	—
48	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
49	C₈×C₂₃	C₂³×C₄	1	16	yes	complemented
50	C₂⁴×C₈	C₂×C₂₃	1	1	yes	characteristic
51	C₂³×C₄⋊C₄	C₂×C₂₃	1	2	yes	—
52	C₂²×#106	C₂×C₂₃	1	28	yes	direct product factor
53	C₂²×C₂₃	C₂×#106	1	112	yes	direct product factor
54	C₂²×C₂₃	C₂²×C₄⋊C₄	1	14	yes	—
55	C₂²×C₂₃	C₂²×SD₁₆	1	28	yes	—
56	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
57	C₄×C₂₃	—	4	16	no	—
58	C₄×C₂₃	C₂²×C₄⋊C₄	1	14	yes	—
59	C₄×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
60	C₄×C₂₃	C₂³×Q₈	1	1	yes	characteristic
61	C₂⁴×C₄	—	2	2	no	—
62	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
63	C₂³×C₈	C₄×C₂₃	1	16	yes	complemented
64	C₂³×C₈	C₂²×C₂₃	1	14	yes	—
65	C₂²×C₄⋊C₄	—	2	14	no	—
66	C₂²×C₄⋊C₄	C₂²×C₂₃	1	28	yes	—
67	C₂×#106	C₂²×C₂₃	1	112	yes	direct product factor
68	C₂×C₂₃	C₂²×#106	1	28	yes	direct product factor
69	C₂×C₂₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₂₃	C₂³×SD₁₆	1	2	yes	—
71	C₂⁵	D₄×C₂₃	1	1	yes	characteristic
72	C₂³×C₄	—	2	28	no	—
73	C₂³×C₄	—	4	16	no	—
74	C₂³×C₄	C₂×C₄×C₂₃	1	8	yes	—
75	C₂³×C₄	D₄×C₂₃	1	7	yes	—
76	C₂³×C₄	Q₈×C₂₃	1	8	yes	—
77	C₂³×C₄	C₂³×C₂₃	1	7	yes	—
78	C₂²×C₈	C₂×C₄×C₂₃	1	112	yes	complemented
79	C₂²×C₈	C₂³×C₂₃	1	28	yes	—
80	C₂×C₄⋊C₄	—	2	84	no	—
81	C₂×C₄⋊C₄	C₂³×C₂₃	1	56	yes	—
82	#106	C₂³×C₂₃	1	64	yes	direct product factor
83	C₂₃	C₂³×#106	1	1	yes	characteristic, direct product factor, Sylow
84	C₂⁴	C₄⋊C₄×C₂₃	1	8	yes	—
85	C₂⁴	SD₁₆×C₂₃	1	16	yes	—
86	C₂⁴	C₂×D₄×C₂₃	1	7	yes	—
87	C₂²×C₄	—	2	56	no	—
88	C₂²×C₄	—	4	112	no	—
89	C₂²×C₄	C₄⋊C₄×C₂₃	1	56	yes	—
90	C₂²×C₄	C₂²×C₄×C₂₃	1	28	yes	—
91	C₂²×C₄	C₂×D₄×C₂₃	1	21	yes	—
92	C₂²×C₄	C₂×Q₈×C₂₃	1	28	yes	—
93	C₂²×C₄	C₂⁴×C₂₃	1	7	yes	—
94	C₂×C₈	C₂²×C₄×C₂₃	1	112	yes	complemented
95	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
96	C₄⋊C₄	—	2	56	no	—
97	C₄⋊C₄	C₂⁴×C₂₃	1	16	yes	—
98	C₂³	C₂₃×#106	1	64	yes	direct product factor
99	C₂³	C₂×C₄⋊C₄×C₂₃	1	28	yes	—
100	C₂³	C₂×SD₁₆×C₂₃	1	56	yes	—
101	C₂³	C₂²×D₄×C₂₃	1	7	yes	—
102	C₂×C₄	—	2	16	no	—
103	C₂×C₄	—	4	112	no	—
104	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	84	yes	—
105	C₂×C₄	C₂³×C₄×C₂₃	1	14	yes	—
106	C₂×C₄	C₂²×D₄×C₂₃	1	7	yes	—
107	C₂×C₄	C₂²×Q₈×C₂₃	1	14	yes	—
108	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
109	C₈	C₂³×C₄×C₂₃	1	16	yes	complemented
110	C₂²	C₂×C₂₃×#106	1	112	yes	direct product factor
111	C₂²	C₂²×C₄⋊C₄×C₂₃	1	14	yes	—
112	C₂²	C₂²×SD₁₆×C₂₃	1	28	yes	—
113	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
114	C₄	—	4	16	no	—
115	C₄	C₂²×C₄⋊C₄×C₂₃	1	14	yes	—
116	C₄	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
117	C₄	C₂³×Q₈×C₂₃	1	1	yes	characteristic
118	C₂	C₂²×C₂₃×#106	1	28	yes	direct product factor
119	C₂	C₂³×C₄⋊C₄×C₂₃	1	1	yes	characteristic
120	C₂	C₂³×SD₁₆×C₂₃	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₂₃	D₄
commutator	C₄	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₂₃ ◁ C₂⁴×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₂₃×#106	C₂	direct (N × H)
6. C₂³×C₈×C₂₃	C₄	semidirect
10. C₂×C₂₃×#106	C₂²	direct (N × H)
18. C₂²×C₈×C₂₃	C₂×C₄	semidirect
22. C₂₃×#106	C₂³	direct (N × H)
33. C₂×C₈×C₂₃	C₂²×C₄	semidirect
37. C₂³×#106	C₂₃	direct (N × H)
38. C₂³×C₂₃	#106	direct (N × H)
49. C₈×C₂₃	C₂³×C₄	semidirect
52. C₂²×#106	C₂×C₂₃	direct (N × H)
53. C₂²×C₂₃	C₂×#106	direct (N × H)
63. C₂³×C₈	C₄×C₂₃	semidirect
67. C₂×#106	C₂²×C₂₃	direct (N × H)
68. C₂×C₂₃	C₂²×#106	direct (N × H)
78. C₂²×C₈	C₂×C₄×C₂₃	semidirect
82. #106	C₂³×C₂₃	direct (N × H)
83. C₂₃	C₂³×#106	direct (N × H)
94. C₂×C₈	C₂²×C₄×C₂₃	semidirect
98. C₂³	C₂₃×#106	direct (N × H)
109. C₈	C₂³×C₄×C₂₃	semidirect
110. C₂²	C₂×C₂₃×#106	direct (N × H)
118. C₂	C₂²×C₂₃×#106	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₂₃	2
C₂³×C₄⋊C₄×C₂₃	2
C₂²×C₂₃×#106	2
C₂³×#106	23

Subgroups — C23×C23×#106 #19207727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂₃×#106 #19207727