Subgroups — C₂₃×#11404 #19159030

Order 5888.

298 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	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1313	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×M₃₂	C₄	1	2	yes	—
6	C₂×C₂₃×M₃₂	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#346	C₂²	1	2	yes	—
10	C₂₃×#346	C₂²	1	2	yes	—
11	C₄×C₁₆×C₂₃	C₄	1	2	yes	—
12	C₄×C₁₆×C₂₃	C₄	1	2	yes	—
13	C₂₃×#322	C₄	1	2	yes	—
14	C₂₃×#322	C₂²	1	2	yes	—
15	C₂²×C₈×C₂₃	—	2	2	no	—
16	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂₃×M₃₂	—	2	4	no	—
20	C₂₃×M₃₂	C₈	1	8	yes	complemented
21	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	—
22	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	—
23	C₄×C₈×C₂₃	—	2	2	no	—
24	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
31	C₂×C₈×C₂₃	—	2	4	no	—
32	C₂×C₈×C₂₃	—	4	4	no	—
33	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₂₃	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₂₃	M₁₆	1	4	yes	—
37	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₂₃	C₄²	1	1	yes	characteristic
40	C₄²×C₂₃	M₁₆	1	2	yes	—
41	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
42	C₁₆×C₂₃	—	2	4	no	—
43	C₁₆×C₂₃	C₂×C₈	1	8	yes	complemented
44	#11404	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂³×C₂₃	#97	1	1	yes	characteristic
46	C₂×C₄×C₂₃	—	2	4	no	—
47	C₂×C₄×C₂₃	C₄×C₈	1	2	yes	—
48	C₂×C₄×C₂₃	#97	1	1	yes	characteristic
49	C₂×C₄×C₂₃	#97	1	2	yes	—
50	C₂×C₄×C₂₃	#97	1	2	yes	—
51	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
53	C₈×C₂₃	—	4	4	no	—
54	C₈×C₂₃	C₄×C₈	1	2	yes	—
55	C₈×C₂₃	#97	1	2	yes	—
56	C₈×C₂₃	C₂²×C₈	1	2	yes	—
57	C₈×C₂₃	C₂×M₁₆	1	2	yes	—
58	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic
59	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic
60	C₂×#322	C₂×C₂₃	1	1	yes	characteristic
61	#1313	C₂×C₂₃	1	4	yes	complemented
62	C₂²×C₂₃	—	2	2	no	—
63	C₂²×C₂₃	#322	1	2	yes	—
64	C₂²×C₂₃	C₂×#97	1	1	yes	characteristic
65	C₄×C₂₃	—	2	2	no	—
66	C₄×C₂₃	#322	1	2	yes	—
67	C₄×C₂₃	#347	1	4	yes	—
68	C₄×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
69	C₄×C₂₃	C₂×#97	1	1	yes	characteristic
70	C₂×M₃₂	C₄×C₂₃	1	2	yes	—
71	C₂×M₃₂	C₄×C₂₃	1	2	yes	—
72	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	—
73	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
74	#346	C₂²×C₂₃	1	2	yes	—
75	#346	C₂²×C₂₃	1	2	yes	—
76	C₄×C₁₆	C₄×C₂₃	1	2	yes	—
77	C₄×C₁₆	C₄×C₂₃	1	2	yes	—
78	#322	C₄×C₂₃	1	2	yes	—
79	#322	C₂²×C₂₃	1	2	yes	—
80	C₂×C₂₃	—	2	2	no	—
81	C₂×C₂₃	C₂×#322	1	1	yes	characteristic
82	C₂×C₂₃	C₂×#347	1	2	yes	—
83	C₂²×C₈	—	2	2	no	—
84	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
85	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
86	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
87	M₃₂	—	2	4	no	—
88	M₃₂	C₈×C₂₃	1	8	yes	complemented
89	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	—
90	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	—
91	C₄×C₈	—	2	2	no	—
92	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
93	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
94	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
95	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
96	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
97	C₂₃	#11404	1	1	yes	characteristic, direct product factor, Sylow
98	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
99	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	4	4	no	—
102	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
103	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
104	C₂×C₈	C₂×C₈×C₂₃	1	4	yes	—
105	C₂×C₈	M₁₆×C₂₃	1	4	yes	—
106	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
107	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
108	C₄²	C₄²×C₂₃	1	1	yes	characteristic
109	C₄²	M₁₆×C₂₃	1	2	yes	—
110	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
111	C₁₆	—	2	4	no	—
112	C₁₆	C₂×C₈×C₂₃	1	8	yes	complemented
113	C₂³	C₂₃×#97	1	1	yes	characteristic
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	C₄×C₈×C₂₃	1	2	yes	—
116	C₂×C₄	C₂₃×#97	1	1	yes	characteristic
117	C₂×C₄	C₂₃×#97	1	2	yes	—
118	C₂×C₄	C₂₃×#97	1	2	yes	—
119	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
120	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
121	C₈	—	4	4	no	—
122	C₈	C₄×C₈×C₂₃	1	2	yes	—
123	C₈	C₂₃×#97	1	2	yes	—
124	C₈	C₂²×C₈×C₂₃	1	2	yes	—
125	C₈	C₂×M₁₆×C₂₃	1	2	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	C₂₃×#322	1	2	yes	—
128	C₂²	C₂×C₂₃×#97	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	C₂₃×#322	1	2	yes	—
131	C₄	C₂₃×#347	1	4	yes	—
132	C₄	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
133	C₄	C₂×C₂₃×#97	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	C₂×C₂₃×#322	1	1	yes	characteristic
136	C₂	C₂×C₂₃×#347	1	2	yes	—

	subgroup	G/H
center	C₄²×C₂₃	C₄²
commutator	C₄	C₂×C₄×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#97
prime radical	C₂³×C₂₃	#97
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1313	C₂	semidirect
20. C₂₃×M₃₂	C₈	semidirect
43. C₁₆×C₂₃	C₂×C₈	semidirect
44. #11404	C₂₃	direct (N × H)
61. #1313	C₂×C₂₃	semidirect
88. M₃₂	C₈×C₂₃	semidirect
97. C₂₃	#11404	direct (N × H)
112. C₁₆	C₂×C₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂₃×M₃₂	2
C₂×C₂₃×#322	2
C₂₃×#1313	2
#11404	23

Subgroups — C23×#11404 #19159030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11404 #19159030