Subgroups — C₂×C₂₃×#1398 #19161176

Order 5888.

222 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×C₂₃×#346	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1398	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₂₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
7	C₂₃×M₆₄	C₄	1	4	yes	complemented
8	C₂₃×M₆₄	C₂²	1	4	yes	—
9	C₂×C₂₃×C₃₂	C₄	1	4	yes	complemented
10	C₂₃×#346	C₄	1	2	yes	—
11	C₂₃×#346	C₂²	1	2	yes	—
12	C₂²×C₈×C₂₃	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₂₃	C₈	1	1	yes	characteristic
15	C₂×C₁₆×C₂₃	C₈	1	2	yes	—
16	C₂×C₁₆×C₂₃	C₈	1	2	yes	—
17	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₂₃	C₈	1	2	yes	—
23	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
24	C₂₃×C₃₂	C₂×C₄	1	8	yes	complemented
25	C₂²×C₄×C₂₃	—	2	1	no	—
26	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₂₃	—	2	1	no	—
28	C₂×C₈×C₂₃	—	2	1	no	—
29	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
30	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
31	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₂₃	—	2	1	no	—
37	C₄²×C₂₃	C₂×C₈	1	2	yes	—
38	C₁₆×C₂₃	C₄²	1	2	yes	—
39	C₁₆×C₂₃	C₂×C₈	1	4	yes	—
40	C₁₆×C₂₃	C₂²×C₄	1	2	yes	—
41	C₂×#1398	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂³×C₂₃	M₃₂	1	1	yes	characteristic
43	C₂×C₄×C₂₃	—	2	2	no	—
44	C₂×C₄×C₂₃	—	4	2	no	—
45	C₂×C₄×C₂₃	C₄×C₈	1	1	yes	characteristic
46	C₂×C₄×C₂₃	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₂₃	M₃₂	1	1	yes	characteristic
48	C₂×C₄×C₂₃	M₃₂	1	1	yes	characteristic
49	C₂×C₄×C₂₃	M₃₂	1	1	yes	characteristic
50	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
51	C₈×C₂₃	—	2	2	no	—
52	C₈×C₂₃	C₄×C₈	1	1	yes	characteristic
53	C₈×C₂₃	C₄×C₈	1	1	yes	characteristic
54	C₈×C₂₃	C₂×C₄²	1	1	yes	characteristic
55	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
56	C₂×M₆₄	C₂×C₂₃	1	2	yes	—
57	C₂×#346	C₂×C₂₃	1	1	yes	characteristic
58	#1398	C₂×C₂₃	1	4	yes	direct product factor
59	C₂²×C₂₃	—	2	2	no	—
60	C₂²×C₂₃	#346	1	1	yes	characteristic
61	C₂²×C₂₃	#346	1	1	yes	characteristic
62	C₂²×C₂₃	C₂×M₃₂	1	1	yes	characteristic
63	C₄×C₂₃	—	4	2	no	—
64	C₄×C₂₃	#346	1	1	yes	characteristic
65	C₄×C₂₃	#346	1	1	yes	characteristic
66	C₄×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
67	C₄×C₂₃	C₂×M₃₂	1	1	yes	characteristic
68	C₂²×C₁₆	C₄×C₂₃	1	1	yes	characteristic
69	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
70	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
71	M₆₄	C₄×C₂₃	1	4	yes	complemented
72	M₆₄	C₂²×C₂₃	1	4	yes	—
73	C₂×C₃₂	C₄×C₂₃	1	4	yes	complemented
74	#346	C₄×C₂₃	1	2	yes	—
75	#346	C₂²×C₂₃	1	2	yes	—
76	C₂×C₂₃	—	2	1	no	—
77	C₂×C₂₃	—	2	1	no	—
78	C₂×C₂₃	#1398	1	2	yes	direct product factor
79	C₂×C₂₃	C₂×#346	1	1	yes	characteristic
80	C₂²×C₈	C₈×C₂₃	1	1	yes	characteristic
81	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
82	C₂×C₄²	C₈×C₂₃	1	1	yes	characteristic
83	C₂×C₁₆	C₈×C₂₃	1	2	yes	—
84	C₂×C₁₆	C₈×C₂₃	1	2	yes	—
85	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
86	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
87	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
88	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
89	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
90	C₄×C₈	C₈×C₂₃	1	2	yes	—
91	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
92	C₃₂	C₂×C₄×C₂₃	1	8	yes	complemented
93	C₂₃	C₂×#1398	1	1	yes	characteristic, direct product factor, Sylow
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
99	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
100	C₂×C₈	C₂×C₈×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	1	yes	characteristic
104	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
105	C₄²	—	2	1	no	—
106	C₄²	C₂×C₈×C₂₃	1	2	yes	—
107	C₁₆	C₄²×C₂₃	1	2	yes	—
108	C₁₆	C₂×C₈×C₂₃	1	4	yes	—
109	C₁₆	C₂²×C₄×C₂₃	1	2	yes	—
110	C₂³	C₂₃×M₃₂	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₄×C₈×C₂₃	1	1	yes	characteristic
114	C₂×C₄	C₄×C₈×C₂₃	1	1	yes	characteristic
115	C₂×C₄	C₂₃×M₃₂	1	1	yes	characteristic
116	C₂×C₄	C₂₃×M₃₂	1	1	yes	characteristic
117	C₂×C₄	C₂₃×M₃₂	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	C₄×C₈×C₂₃	1	1	yes	characteristic
121	C₈	C₄×C₈×C₂₃	1	1	yes	characteristic
122	C₈	C₂×C₄²×C₂₃	1	1	yes	characteristic
123	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
124	C₂²	—	2	2	no	—
125	C₂²	C₂₃×#346	1	1	yes	characteristic
126	C₂²	C₂₃×#346	1	1	yes	characteristic
127	C₂²	C₂×C₂₃×M₃₂	1	1	yes	characteristic
128	C₄	—	4	2	no	—
129	C₄	C₂₃×#346	1	1	yes	characteristic
130	C₄	C₂₃×#346	1	1	yes	characteristic
131	C₄	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
132	C₄	C₂×C₂₃×M₃₂	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	2	1	no	—
135	C₂	C₂₃×#1398	1	2	yes	direct product factor
136	C₂	C₂×C₂₃×#346	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₄²
commutator	C₄	C₂×C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₂₃
socle	C₂²×C₂₃	#346
prime radical	C₂³×C₂₃	M₃₂
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1398	C₂	direct (N × H)
7. C₂₃×M₆₄	C₄	semidirect
9. C₂×C₂₃×C₃₂	C₄	semidirect
24. C₂₃×C₃₂	C₂×C₄	semidirect
41. C₂×#1398	C₂₃	direct (N × H)
58. #1398	C₂×C₂₃	direct (N × H)
71. M₆₄	C₄×C₂₃	semidirect
73. C₂×C₃₂	C₄×C₂₃	semidirect
78. C₂×C₂₃	#1398	direct (N × H)
92. C₃₂	C₂×C₄×C₂₃	semidirect
93. C₂₃	C₂×#1398	direct (N × H)
135. C₂	C₂₃×#1398	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×C₂₃×#346	2
C₂₃×#1398	2
C₂×#1398	23

Subgroups — C2×C23×#1398 #19161176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1398 #19161176