Subgroups — C₂×C₂₃×#2258 #19181599

Order 5888.

470 conjugacy classes of subgroups.

The subgroup lattice has 118 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, direct product factor
3	C₂×C₂₃×M₆₄	C₂	1	3	yes	maximal, complemented
4	C₂²×C₂₃×C₃₂	C₂	1	3	yes	maximal, complemented
5	C₂×C₂₃×#131	C₄	1	1	yes	characteristic
6	C₂₃×#504	C₄	1	4	yes	—
7	C₂₃×#504	C₂²	1	4	yes	—
8	C₂×C₂₃×M₃₂	C₄	1	3	yes	—
9	C₂²×C₁₆×C₂₃	C₂²	1	3	yes	—
10	C₂₃×M₆₄	C₂²	1	12	yes	complemented
11	C₂×C₂₃×C₃₂	C₂²	1	3	yes	—
12	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×C₃₂	C₂²	1	12	yes	complemented
14	C₂×#41×C₂₃	C₈	1	1	yes	characteristic
15	C₂₃×#131	C₈	1	4	yes	—
16	C₂₃×#131	C₂×C₄	1	4	yes	—
17	C₂×M₁₆×C₂₃	C₈	1	3	yes	—
18	C₂²×C₈×C₂₃	C₂×C₄	1	3	yes	—
19	C₂₃×M₃₂	C₂×C₄	1	12	yes	—
20	C₂×C₁₆×C₂₃	C₂×C₄	1	3	yes	—
21	C₂×C₁₆×C₂₃	C₂×C₄	1	6	yes	—
22	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
23	C₂×C₁₆×C₂₃	C₂³	1	6	yes	—
24	C₂₃×C₃₂	C₂³	1	6	yes	—
25	C₂₃×C₃₂	C₂³	1	2	yes	—
26	#41×C₂₃	C₁₆	1	4	yes	—
27	#41×C₂₃	C₂×C₈	1	4	yes	—
28	C₂×Q₈×C₂₃	C₁₆	1	1	yes	characteristic
29	C₂×D₄×C₂₃	C₁₆	1	3	yes	—
30	C₂²×C₄×C₂₃	C₂×C₈	1	3	yes	—
31	M₁₆×C₂₃	C₂×C₈	1	12	yes	—
32	C₂×C₈×C₂₃	C₂×C₈	1	3	yes	—
33	C₂×C₈×C₂₃	C₂×C₈	1	6	yes	—
34	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₂₃	C₂²×C₄	1	6	yes	—
36	C₁₆×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₁₆×C₂₃	C₂²×C₄	1	6	yes	—
38	C₁₆×C₂₃	C₂⁴	1	1	yes	characteristic
39	C₂×#2258	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂³×C₂₃	C₂×C₁₆	1	3	yes	—
41	Q₈×C₂₃	C₂×C₁₆	1	4	yes	—
42	D₄×C₂₃	C₂×C₁₆	1	12	yes	—
43	C₂×C₄×C₂₃	C₂×C₁₆	1	3	yes	—
44	C₂×C₄×C₂₃	C₂×C₁₆	1	6	yes	—
45	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₄×C₂₃	C₂²×C₈	1	6	yes	—
47	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
48	C₈×C₂₃	C₂²×C₈	1	6	yes	—
49	C₈×C₂₃	C₂³×C₄	1	1	yes	characteristic
50	C₂×#504	C₂×C₂₃	1	1	yes	characteristic
51	#2258	C₂×C₂₃	1	8	yes	direct product factor
52	C₂×M₆₄	C₂×C₂₃	1	3	yes	complemented
53	C₂²×C₃₂	C₂×C₂₃	1	3	yes	complemented
54	C₂²×C₂₃	—	2	6	no	—
55	C₂²×C₂₃	C₂²×C₁₆	1	6	yes	—
56	C₂²×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
57	C₄×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
58	C₄×C₂₃	C₂²×C₁₆	1	6	yes	—
59	C₄×C₂₃	C₂³×C₈	1	1	yes	characteristic
60	C₂×#131	C₄×C₂₃	1	1	yes	characteristic
61	#504	C₄×C₂₃	1	4	yes	—
62	#504	C₂²×C₂₃	1	4	yes	—
63	C₂×M₃₂	C₄×C₂₃	1	3	yes	—
64	C₂²×C₁₆	C₂²×C₂₃	1	3	yes	—
65	M₆₄	C₂²×C₂₃	1	12	yes	complemented
66	C₂×C₃₂	C₂²×C₂₃	1	3	yes	—
67	C₂×C₃₂	C₂²×C₂₃	1	1	yes	characteristic
68	C₂×C₃₂	C₂²×C₂₃	1	12	yes	complemented
69	C₂×C₂₃	—	2	6	no	—
70	C₂×C₂₃	#2258	1	2	yes	direct product factor
71	C₂×C₂₃	C₂³×C₁₆	1	1	yes	characteristic
72	C₂×#41	C₈×C₂₃	1	1	yes	characteristic
73	#131	C₈×C₂₃	1	4	yes	—
74	#131	C₂×C₄×C₂₃	1	4	yes	—
75	C₂×M₁₆	C₈×C₂₃	1	3	yes	—
76	C₂²×C₈	C₂×C₄×C₂₃	1	3	yes	—
77	M₃₂	C₂×C₄×C₂₃	1	12	yes	—
78	C₂×C₁₆	C₂×C₄×C₂₃	1	3	yes	—
79	C₂×C₁₆	C₂×C₄×C₂₃	1	6	yes	—
80	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
81	C₂×C₁₆	C₂³×C₂₃	1	6	yes	—
82	C₃₂	C₂³×C₂₃	1	2	yes	—
83	C₃₂	C₂³×C₂₃	1	6	yes	—
84	C₂₃	C₂×#2258	1	1	yes	characteristic, direct product factor, Sylow
85	#41	C₁₆×C₂₃	1	4	yes	—
86	#41	C₂×C₈×C₂₃	1	4	yes	—
87	C₂×Q₈	C₁₆×C₂₃	1	1	yes	characteristic
88	C₂×D₄	C₁₆×C₂₃	1	3	yes	—
89	C₂²×C₄	C₂×C₈×C₂₃	1	3	yes	—
90	M₁₆	C₂×C₈×C₂₃	1	12	yes	—
91	C₂×C₈	C₂×C₈×C₂₃	1	3	yes	—
92	C₂×C₈	C₂×C₈×C₂₃	1	6	yes	—
93	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
94	C₂×C₈	C₂²×C₄×C₂₃	1	6	yes	—
95	C₁₆	C₂²×C₄×C₂₃	1	1	yes	characteristic
96	C₁₆	C₂²×C₄×C₂₃	1	6	yes	—
97	C₁₆	C₂⁴×C₂₃	1	1	yes	characteristic
98	C₂³	C₂×C₁₆×C₂₃	1	3	yes	—
99	Q₈	C₂×C₁₆×C₂₃	1	4	yes	—
100	D₄	C₂×C₁₆×C₂₃	1	12	yes	—
101	C₂×C₄	C₂×C₁₆×C₂₃	1	3	yes	—
102	C₂×C₄	C₂×C₁₆×C₂₃	1	6	yes	—
103	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×C₈×C₂₃	1	6	yes	—
105	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
106	C₈	C₂²×C₈×C₂₃	1	6	yes	—
107	C₈	C₂³×C₄×C₂₃	1	1	yes	characteristic
108	C₂²	—	2	6	no	—
109	C₂²	C₂²×C₁₆×C₂₃	1	6	yes	—
110	C₂²	C₂²×C₁₆×C₂₃	1	1	yes	characteristic
111	C₄	C₂²×C₁₆×C₂₃	1	1	yes	characteristic
112	C₄	C₂²×C₁₆×C₂₃	1	6	yes	—
113	C₄	C₂³×C₈×C₂₃	1	1	yes	characteristic
114	C₂	—	2	6	no	—
115	C₂	C₂₃×#2258	1	2	yes	direct product factor
116	C₂	C₂³×C₁₆×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₂₃×C₃₂	C₂²
commutator	C₂	C₂³×C₁₆×C₂₃
Frattini, cosocle	C₁₆	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₁₆
prime radical	C₂×#41×C₂₃	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃×C₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2258	C₂	direct (N × H)
3. C₂×C₂₃×M₆₄	C₂	semidirect
4. C₂²×C₂₃×C₃₂	C₂	semidirect
10. C₂₃×M₆₄	C₂²	semidirect
13. C₂×C₂₃×C₃₂	C₂²	semidirect
39. C₂×#2258	C₂₃	direct (N × H)
51. #2258	C₂×C₂₃	direct (N × H)
52. C₂×M₆₄	C₂×C₂₃	semidirect
53. C₂²×C₃₂	C₂×C₂₃	semidirect
65. M₆₄	C₂²×C₂₃	semidirect
68. C₂×C₃₂	C₂²×C₂₃	semidirect
70. C₂×C₂₃	#2258	direct (N × H)
84. C₂₃	C₂×#2258	direct (N × H)
115. C₂	C₂₃×#2258	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
C₂²×C₂₃×C₃₂	2
C₂×#2258	23

Subgroups — C2×C23×#2258 #19181599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2258 #19181599