Subgroups — C₄×C₂₃×M₆₄ #19161174

Order 5888.

250 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×M₆₄	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1397	C₂	1	2	yes	maximal, complemented
4	C₄×C₂₃×C₃₂	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×C₂₃	C₄	1	1	yes	characteristic
6	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
8	C₂₃×M₆₄	C₄	1	8	yes	direct product factor
9	C₂×C₂₃×C₃₂	C₂²	1	4	yes	—
10	C₄×C₁₆×C₂₃	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₂₃	C₄	1	1	yes	characteristic
12	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₃	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₂₃	C₈	1	1	yes	characteristic
17	C₂×C₁₆×C₂₃	C₈	1	4	yes	—
18	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	—
20	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₂₃	C₈	1	1	yes	characteristic
24	C₄×C₈×C₂₃	C₈	1	1	yes	characteristic
25	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂₃×C₃₂	C₂×C₄	1	8	yes	complemented
28	C₂²×C₄×C₂₃	C₁₆	1	2	yes	—
29	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₂₃	C₁₆	1	4	yes	—
31	C₂×C₈×C₂₃	C₄²	1	2	yes	—
32	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₂₃	C₁₆	1	2	yes	—
38	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
39	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
40	C₁₆×C₂₃	C₄²	1	2	yes	—
41	C₁₆×C₂₃	C₂×C₈	1	2	yes	—
42	C₁₆×C₂₃	C₂×C₈	1	2	yes	—
43	C₁₆×C₂₃	C₂²×C₄	1	2	yes	—
44	C₄×M₆₄	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂³×C₂₃	C₂×C₁₆	1	1	yes	characteristic
46	C₂×C₄×C₂₃	—	2	4	no	—
47	C₂×C₄×C₂₃	C₄×C₈	1	2	yes	—
48	C₂×C₄×C₂₃	C₂×C₁₆	1	2	yes	—
49	C₂×C₄×C₂₃	C₂×C₁₆	1	1	yes	characteristic
50	C₂×C₄×C₂₃	C₂×C₁₆	1	2	yes	—
51	C₂×C₄×C₂₃	C₂×C₁₆	1	2	yes	—
52	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
53	C₈×C₂₃	C₄×C₈	1	2	yes	—
54	C₈×C₂₃	C₂×C₁₆	1	2	yes	—
55	C₈×C₂₃	C₂×C₁₆	1	2	yes	—
56	C₈×C₂₃	C₂×C₄²	1	1	yes	characteristic
57	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
58	C₂×M₆₄	C₂×C₂₃	1	2	yes	—
59	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic
60	#1397	C₂×C₂₃	1	2	yes	complemented
61	C₄×C₃₂	C₂×C₂₃	1	2	yes	complemented
62	C₂²×C₂₃	—	2	2	no	—
63	C₂²×C₂₃	C₄×C₁₆	1	2	yes	—
64	C₂²×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
65	C₄×C₂₃	—	2	2	no	—
66	C₄×C₂₃	C₄×C₁₆	1	2	yes	—
67	C₄×C₂₃	M₆₄	1	4	yes	direct product factor
68	C₄×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
69	C₄×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
70	C₂²×C₁₆	C₄×C₂₃	1	1	yes	characteristic
71	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
72	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
73	M₆₄	C₄×C₂₃	1	8	yes	direct product factor
74	C₂×C₃₂	C₂²×C₂₃	1	4	yes	—
75	C₄×C₁₆	C₄×C₂₃	1	1	yes	characteristic
76	C₄×C₁₆	C₄×C₂₃	1	1	yes	characteristic
77	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
78	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
79	C₂×C₂₃	—	2	2	no	—
80	C₂×C₂₃	C₂×C₄×C₁₆	1	1	yes	characteristic
81	C₂×C₂₃	C₂×M₆₄	1	2	yes	—
82	C₂²×C₈	C₈×C₂₃	1	1	yes	characteristic
83	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
84	C₂×C₄²	C₈×C₂₃	1	1	yes	characteristic
85	C₂×C₁₆	C₈×C₂₃	1	4	yes	—
86	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
87	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	—
88	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
89	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
90	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
91	C₄×C₈	C₈×C₂₃	1	1	yes	characteristic
92	C₄×C₈	C₈×C₂₃	1	1	yes	characteristic
93	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
94	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
95	C₃₂	C₂×C₄×C₂₃	1	8	yes	complemented
96	C₂₃	C₄×M₆₄	1	1	yes	characteristic, direct product factor, Sylow
97	C₂²×C₄	C₁₆×C₂₃	1	2	yes	—
98	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
99	C₂×C₈	C₁₆×C₂₃	1	4	yes	—
100	C₂×C₈	C₄²×C₂₃	1	2	yes	—
101	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
102	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
103	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
104	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
105	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
106	C₄²	C₁₆×C₂₃	1	2	yes	—
107	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
108	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
109	C₁₆	C₄²×C₂₃	1	2	yes	—
110	C₁₆	C₂×C₈×C₂₃	1	2	yes	—
111	C₁₆	C₂×C₈×C₂₃	1	2	yes	—
112	C₁₆	C₂²×C₄×C₂₃	1	2	yes	—
113	C₂³	C₂×C₁₆×C₂₃	1	1	yes	characteristic
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	C₄×C₈×C₂₃	1	2	yes	—
116	C₂×C₄	C₂×C₁₆×C₂₃	1	2	yes	—
117	C₂×C₄	C₂×C₁₆×C₂₃	1	1	yes	characteristic
118	C₂×C₄	C₂×C₁₆×C₂₃	1	2	yes	—
119	C₂×C₄	C₂×C₁₆×C₂₃	1	2	yes	—
120	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
121	C₈	C₄×C₈×C₂₃	1	2	yes	—
122	C₈	C₂×C₁₆×C₂₃	1	2	yes	—
123	C₈	C₂×C₁₆×C₂₃	1	2	yes	—
124	C₈	C₂×C₄²×C₂₃	1	1	yes	characteristic
125	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
126	C₂²	—	2	2	no	—
127	C₂²	C₄×C₁₆×C₂₃	1	2	yes	—
128	C₂²	C₂²×C₁₆×C₂₃	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	C₄×C₁₆×C₂₃	1	2	yes	—
131	C₄	C₂₃×M₆₄	1	4	yes	direct product factor
132	C₄	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
133	C₄	C₂²×C₁₆×C₂₃	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	C₂×C₄×C₁₆×C₂₃	1	1	yes	characteristic
136	C₂	C₂×C₂₃×M₆₄	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆×C₂₃	C₂²
commutator	C₂	C₂×C₄×C₁₆×C₂₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₂₃
socle	C₂²×C₂₃	C₂²×C₁₆
prime radical	C₂³×C₂₃	C₂×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1397	C₂	semidirect
4. C₄×C₂₃×C₃₂	C₂	semidirect
8. C₂₃×M₆₄	C₄	direct (N × H)
27. C₂₃×C₃₂	C₂×C₄	semidirect
44. C₄×M₆₄	C₂₃	direct (N × H)
60. #1397	C₂×C₂₃	semidirect
61. C₄×C₃₂	C₂×C₂₃	semidirect
67. C₄×C₂₃	M₆₄	direct (N × H)
73. M₆₄	C₄×C₂₃	direct (N × H)
95. C₃₂	C₂×C₄×C₂₃	semidirect
96. C₂₃	C₄×M₆₄	direct (N × H)
131. C₄	C₂₃×M₆₄	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×M₆₄	2
C₂×C₄×C₁₆×C₂₃	2
C₂₃×#1397	2
C₄×C₂₃×C₃₂	2
C₄×M₆₄	23

Subgroups — C4×C23×M64 #19161174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₃×M₆₄ #19161174