Subgroups — C₂×C₂₃×#518 #15671044

Order 2944.

1920 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#518	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×D₄×C₂₃	C₂	1	8	yes	maximal, complemented
4	C₂×C₂₃×#117	C₂	1	2	yes	maximal, complemented
5	C₂²×C₂²⋊C₄×C₂₃	C₂	1	4	yes	maximal, complemented
6	C₂⁵×C₂₃	C₂²	1	2	yes	—
7	C₂²×D₄×C₂₃	C₄	1	16	yes	complemented
8	C₂²×D₄×C₂₃	C₂²	1	4	yes	—
9	C₂²×D₄×C₂₃	C₂²	1	8	yes	—
10	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃	C₂²	1	8	yes	complemented
12	C₄×D₄×C₂₃	C₂²	1	64	yes	complemented
13	C₂₃×#117	C₂²	1	16	yes	complemented
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	complemented
15	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
16	C₂×C₂²⋊C₄×C₂₃	C₂²	1	32	yes	complemented
17	C₂×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	complemented
18	C₂×C₄²×C₂₃	C₂²	1	4	yes	complemented
19	C₂⁴×C₂₃	—	4	8	no	—
20	C₂⁴×C₂₃	C₂×C₄	1	16	yes	complemented
21	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
22	C₂⁴×C₂₃	C₂³	1	4	yes	—
23	C₂⁴×C₂₃	C₂³	1	8	yes	—
24	C₂×D₄×C₂₃	C₂×C₄	1	32	yes	complemented
25	C₂×D₄×C₂₃	C₂×C₄	1	64	yes	complemented
26	C₂×D₄×C₂₃	C₂³	1	16	yes	—
27	C₂²×C₄×C₂₃	—	4	16	no	—
28	C₂²×C₄×C₂₃	C₂×C₄	1	8	yes	complemented
29	C₂²×C₄×C₂₃	C₂³	1	8	yes	—
30	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	32	yes	complemented
33	C₄⋊C₄×C₂₃	C₂³	1	16	yes	complemented
34	C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
35	C₂²⋊C₄×C₂₃	C₂³	1	32	yes	complemented
36	C₄²×C₂₃	C₂³	1	16	yes	complemented
37	C₂³×C₂₃	—	2	16	no	—
38	C₂³×C₂₃	—	2	4	no	—
39	C₂³×C₂₃	—	4	32	no	—
40	C₂³×C₂₃	—	4	16	no	—
41	C₂³×C₂₃	C₂²×C₄	1	4	yes	—
42	C₂³×C₂₃	C₂²×C₄	1	16	yes	—
43	C₂³×C₂₃	C₂²×C₄	1	32	yes	complemented
44	C₂³×C₂₃	C₂⁴	1	2	yes	—
45	C₂³×C₂₃	C₂⁴	1	8	yes	—
46	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
47	D₄×C₂₃	C₂²×C₄	1	64	yes	complemented
48	C₂×C₄×C₂₃	—	2	16	no	—
49	C₂×C₄×C₂₃	—	4	32	no	—
50	C₂×C₄×C₂₃	C₂²×C₄	1	8	yes	—
51	C₂×C₄×C₂₃	C₂²×C₄	1	16	yes	complemented
52	C₂×C₄×C₂₃	C₂⁴	1	16	yes	—
53	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
54	C₂×#518	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂²×C₂₃	—	2	8	no	—
56	C₂²×C₂₃	—	2	32	no	—
57	C₂²×C₂₃	—	2	16	no	—
58	C₂²×C₂₃	—	2	4	no	—
59	C₂²×C₂₃	—	4	32	no	—
60	C₂²×C₂₃	C₂³×C₄	1	4	yes	—
61	C₂²×C₂₃	C₂³×C₄	1	16	yes	—
62	C₂²×C₂₃	C₂³×C₄	1	2	yes	—
63	C₂²×C₂₃	H(5,𝔽₂)	1	4	yes	—
64	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
65	C₄×C₂₃	—	2	16	no	—
66	C₄×C₂₃	C₂³×C₄	1	8	yes	—
67	C₂³×D₄	C₂×C₂₃	1	1	yes	characteristic
68	#518	C₂×C₂₃	1	16	yes	direct product factor
69	C₂×C₄×D₄	C₂×C₂₃	1	8	yes	complemented
70	C₂×#117	C₂×C₂₃	1	2	yes	complemented
71	C₂²×C₂²⋊C₄	C₂×C₂₃	1	4	yes	complemented
72	C₂×C₂₃	—	2	4	no	—
73	C₂×C₂₃	—	2	16	no	—
74	C₂×C₂₃	#518	1	4	yes	direct product factor
75	C₂×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
76	C₂×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
77	C₂⁵	C₂²×C₂₃	1	2	yes	—
78	C₂²×D₄	C₄×C₂₃	1	16	yes	complemented
79	C₂²×D₄	C₂²×C₂₃	1	4	yes	—
80	C₂²×D₄	C₂²×C₂₃	1	8	yes	—
81	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
82	C₂³×C₄	C₂²×C₂₃	1	8	yes	complemented
83	C₄×D₄	C₂²×C₂₃	1	64	yes	complemented
84	#117	C₂²×C₂₃	1	16	yes	complemented
85	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	complemented
86	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
87	C₂×C₂²⋊C₄	C₂²×C₂₃	1	32	yes	complemented
88	C₂×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	complemented
89	C₂×C₄²	C₂²×C₂₃	1	4	yes	complemented
90	C₂₃	C₂×#518	1	1	yes	characteristic, direct product factor, Sylow
91	C₂⁴	—	4	8	no	—
92	C₂⁴	C₂×C₄×C₂₃	1	16	yes	complemented
93	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
94	C₂⁴	C₂³×C₂₃	1	4	yes	—
95	C₂⁴	C₂³×C₂₃	1	8	yes	—
96	C₂×D₄	C₂×C₄×C₂₃	1	32	yes	complemented
97	C₂×D₄	C₂×C₄×C₂₃	1	64	yes	complemented
98	C₂×D₄	C₂³×C₂₃	1	16	yes	—
99	C₂²×C₄	—	4	16	no	—
100	C₂²×C₄	C₂×C₄×C₂₃	1	8	yes	complemented
101	C₂²×C₄	C₂³×C₂₃	1	8	yes	—
102	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
103	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
104	C₂²×C₄	C₂³×C₂₃	1	32	yes	complemented
105	C₄⋊C₄	C₂³×C₂₃	1	16	yes	complemented
106	C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
107	C₂²⋊C₄	C₂³×C₂₃	1	32	yes	complemented
108	C₄²	C₂³×C₂₃	1	16	yes	complemented
109	C₂³	—	2	16	no	—
110	C₂³	—	2	4	no	—
111	C₂³	—	4	32	no	—
112	C₂³	—	4	16	no	—
113	C₂³	C₂²×C₄×C₂₃	1	4	yes	—
114	C₂³	C₂²×C₄×C₂₃	1	16	yes	—
115	C₂³	C₂²×C₄×C₂₃	1	32	yes	complemented
116	C₂³	C₂⁴×C₂₃	1	2	yes	—
117	C₂³	C₂⁴×C₂₃	1	8	yes	—
118	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
119	D₄	C₂²×C₄×C₂₃	1	64	yes	complemented
120	C₂×C₄	—	2	16	no	—
121	C₂×C₄	—	4	32	no	—
122	C₂×C₄	C₂²×C₄×C₂₃	1	8	yes	—
123	C₂×C₄	C₂²×C₄×C₂₃	1	16	yes	complemented
124	C₂×C₄	C₂⁴×C₂₃	1	16	yes	—
125	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
126	C₂²	—	2	16	no	—
127	C₂²	—	2	4	no	—
128	C₂²	—	2	8	no	—
129	C₂²	—	2	32	no	—
130	C₂²	—	4	32	no	—
131	C₂²	C₂³×C₄×C₂₃	1	4	yes	—
132	C₂²	C₂³×C₄×C₂₃	1	16	yes	—
133	C₂²	C₂³×C₄×C₂₃	1	2	yes	—
134	C₂²	C₂₃×H(5,𝔽₂)	1	4	yes	—
135	C₂²	C₂⁵×C₂₃	1	1	yes	characteristic
136	C₄	—	2	16	no	—
137	C₄	C₂³×C₄×C₂₃	1	8	yes	—
138	C₂	—	2	4	no	—
139	C₂	—	2	16	no	—
140	C₂	C₂₃×#518	1	4	yes	direct product factor
141	C₂	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
142	C₂	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator	C₂	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂²	C₂⁵×C₂₃
prime radical	C₂³×D₄×C₂₃	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#518	C₂	direct (N × H)
3. C₂×C₄×D₄×C₂₃	C₂	semidirect
4. C₂×C₂₃×#117	C₂	semidirect
5. C₂²×C₂²⋊C₄×C₂₃	C₂	semidirect
7. C₂²×D₄×C₂₃	C₄	semidirect
11. C₂³×C₄×C₂₃	C₂²	semidirect
12. C₄×D₄×C₂₃	C₂²	semidirect
13. C₂₃×#117	C₂²	semidirect
14. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
16. C₂×C₂²⋊C₄×C₂₃	C₂²	semidirect
17. C₂×C₂²⋊C₄×C₂₃	C₂²	semidirect
18. C₂×C₄²×C₂₃	C₂²	semidirect
20. C₂⁴×C₂₃	C₂×C₄	semidirect
24. C₂×D₄×C₂₃	C₂×C₄	semidirect
25. C₂×D₄×C₂₃	C₂×C₄	semidirect
28. C₂²×C₄×C₂₃	C₂×C₄	semidirect
32. C₂²×C₄×C₂₃	C₂³	semidirect
33. C₄⋊C₄×C₂₃	C₂³	semidirect
35. C₂²⋊C₄×C₂₃	C₂³	semidirect
36. C₄²×C₂₃	C₂³	semidirect
43. C₂³×C₂₃	C₂²×C₄	semidirect
47. D₄×C₂₃	C₂²×C₄	semidirect
51. C₂×C₄×C₂₃	C₂²×C₄	semidirect
54. C₂×#518	C₂₃	direct (N × H)
68. #518	C₂×C₂₃	direct (N × H)
69. C₂×C₄×D₄	C₂×C₂₃	semidirect
70. C₂×#117	C₂×C₂₃	semidirect
71. C₂²×C₂²⋊C₄	C₂×C₂₃	semidirect
74. C₂×C₂₃	#518	direct (N × H)
78. C₂²×D₄	C₄×C₂₃	semidirect
82. C₂³×C₄	C₂²×C₂₃	semidirect
83. C₄×D₄	C₂²×C₂₃	semidirect
84. #117	C₂²×C₂₃	semidirect
85. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
87. C₂×C₂²⋊C₄	C₂²×C₂₃	semidirect
88. C₂×C₂²⋊C₄	C₂²×C₂₃	semidirect
89. C₂×C₄²	C₂²×C₂₃	semidirect
90. C₂₃	C₂×#518	direct (N × H)
92. C₂⁴	C₂×C₄×C₂₃	semidirect
96. C₂×D₄	C₂×C₄×C₂₃	semidirect
97. C₂×D₄	C₂×C₄×C₂₃	semidirect
100. C₂²×C₄	C₂×C₄×C₂₃	semidirect
104. C₂²×C₄	C₂³×C₂₃	semidirect
105. C₄⋊C₄	C₂³×C₂₃	semidirect
107. C₂²⋊C₄	C₂³×C₂₃	semidirect
108. C₄²	C₂³×C₂₃	semidirect
115. C₂³	C₂²×C₄×C₂₃	semidirect
119. D₄	C₂²×C₄×C₂₃	semidirect
123. C₂×C₄	C₂²×C₄×C₂₃	semidirect
140. C₂	C₂₃×#518	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×D₄×C₂₃	2
C₂₃×#518	2
C₂×C₄×D₄×C₂₃	2
C₂×C₂₃×#117	2
C₂²×C₂²⋊C₄×C₂₃	2
C₂×#518	23

Subgroups — C2×C23×#518 #15671044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#518 #15671044