Subgroups — C₂₃×#12597 #19160223

Order 5888.

258 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2110	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2108	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1747	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1379	C₂	1	2	yes	maximal
5	C₂₃×#1378	C₂	1	2	yes	maximal
6	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
7	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
8	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#405	C₂²	1	2	yes	—
10	C₂₃×#405	C₂²	1	1	yes	characteristic
11	C₂₃×#347	C₄	1	4	yes	complemented
12	C₂₃×#346	C₄	1	2	yes	complemented
13	C₄×C₁₆×C₂₃	C₄	1	2	yes	complemented
14	C₂₃×#338	C₄	1	1	yes	characteristic
15	C₂₃×H(3,ℤ/4)	C₄	1	1	yes	characteristic
16	C₂₃×H(3,ℤ/4)	C₄	1	2	yes	—
17	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
18	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
20	C₂₃×#117	C₂×C₄	1	2	yes	—
21	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
22	C₂₃×M₃₂	—	2	2	no	—
23	C₂₃×M₃₂	—	2	2	no	—
24	C₂₃×M₃₂	C₂×C₄	1	4	yes	—
25	C₂×C₁₆×C₂₃	—	2	2	no	—
26	C₂×C₁₆×C₂₃	—	2	2	no	—
27	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂₃×#97	—	2	2	no	—
31	C₂₃×#97	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₂₃	—	2	2	no	—
33	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
35	M₁₆×C₂₃	—	2	2	no	—
36	M₁₆×C₂₃	—	2	2	no	—
37	M₁₆×C₂₃	—	2	2	no	—
38	C₂×C₈×C₂₃	—	2	2	no	—
39	C₂×C₈×C₂₃	C₄²	1	2	yes	—
40	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₂₃	#41	1	2	yes	—
43	C₂×C₈×C₂₃	#41	1	2	yes	—
44	C₄⋊C₄×C₂₃	—	2	1	no	—
45	C₄⋊C₄×C₂₃	—	2	2	no	—
46	C₂²⋊C₄×C₂₃	—	2	1	no	—
47	C₂²⋊C₄×C₂₃	—	2	2	no	—
48	C₄²×C₂₃	—	2	2	no	—
49	C₄²×C₂₃	C₄²	1	2	yes	—
50	C₁₆×C₂₃	—	2	1	no	—
51	C₁₆×C₂₃	—	2	1	no	—
52	C₁₆×C₂₃	—	2	4	no	—
53	C₁₆×C₂₃	—	2	2	no	—
54	#12597	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₂₃	#131	1	1	yes	characteristic
56	C₂×C₄×C₂₃	—	2	2	no	—
57	C₂×C₄×C₂₃	—	4	2	no	—
58	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₄×C₂₃	#117	1	2	yes	—
60	C₂×C₄×C₂₃	#131	1	2	yes	—
61	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
62	C₈×C₂₃	—	2	2	no	—
63	C₈×C₂₃	—	4	2	no	—
64	C₈×C₂₃	#117	1	1	yes	characteristic
65	C₈×C₂₃	#117	1	2	yes	—
66	C₈×C₂₃	#117	1	1	yes	characteristic
67	#2110	C₂×C₂₃	1	1	yes	characteristic
68	#2108	C₂×C₂₃	1	1	yes	characteristic
69	#1747	C₂×C₂₃	1	1	yes	characteristic
70	#1379	C₂×C₂₃	1	2	yes	—
71	#1378	C₂×C₂₃	1	2	yes	—
72	C₂²×C₂₃	—	4	1	no	—
73	C₂²×C₂₃	#405	1	1	yes	characteristic
74	C₂²×C₂₃	#433	1	2	yes	—
75	C₄×C₂₃	—	4	2	no	—
76	C₄×C₂₃	—	8	2	no	—
77	C₄×C₂₃	#376	1	1	yes	characteristic
78	C₄×C₂₃	#405	1	1	yes	characteristic
79	C₄×C₂₃	#433	1	2	yes	—
80	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
81	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
82	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
83	#405	C₂²×C₂₃	1	2	yes	—
84	#405	C₂²×C₂₃	1	1	yes	characteristic
85	#347	C₄×C₂₃	1	4	yes	complemented
86	#346	C₄×C₂₃	1	2	yes	complemented
87	C₄×C₁₆	C₄×C₂₃	1	2	yes	complemented
88	#338	C₄×C₂₃	1	1	yes	characteristic
89	H(3,ℤ/4)	C₄×C₂₃	1	1	yes	characteristic
90	H(3,ℤ/4)	C₄×C₂₃	1	2	yes	—
91	C₂×C₂₃	—	2	1	no	—
92	C₂×C₂₃	—	2	2	no	—
93	C₂×C₂₃	#1746	1	1	yes	characteristic
94	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
95	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
96	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
97	#117	C₂×C₄×C₂₃	1	2	yes	—
98	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
99	M₃₂	—	2	2	no	—
100	M₃₂	—	2	2	no	—
101	M₃₂	C₂×C₄×C₂₃	1	4	yes	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
105	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
106	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
107	#97	—	2	2	no	—
108	#97	C₂×C₄×C₂₃	1	2	yes	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
111	C₂₃	#12597	1	1	yes	characteristic, direct product factor, Sylow
112	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	C₄²×C₂₃	1	2	yes	—
118	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
119	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
120	C₂×C₈	#41×C₂₃	1	2	yes	—
121	C₂×C₈	#41×C₂₃	1	2	yes	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	—	2	1	no	—
125	C₂²⋊C₄	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	C₄²×C₂₃	1	2	yes	—
128	C₁₆	—	2	1	no	—
129	C₁₆	—	2	1	no	—
130	C₁₆	—	2	4	no	—
131	C₁₆	—	2	2	no	—
132	C₂³	C₂₃×#131	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
136	C₂×C₄	C₂₃×#117	1	2	yes	—
137	C₂×C₄	C₂₃×#131	1	2	yes	—
138	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	4	2	no	—
141	C₈	C₂₃×#117	1	1	yes	characteristic
142	C₈	C₂₃×#117	1	2	yes	—
143	C₈	C₂₃×#117	1	1	yes	characteristic
144	C₂²	—	4	1	no	—
145	C₂²	C₂₃×#405	1	1	yes	characteristic
146	C₂²	C₂₃×#433	1	2	yes	—
147	C₄	—	4	2	no	—
148	C₄	—	8	2	no	—
149	C₄	C₂₃×#376	1	1	yes	characteristic
150	C₄	C₂₃×#405	1	1	yes	characteristic
151	C₄	C₂₃×#433	1	2	yes	—
152	C₂	—	2	1	no	—
153	C₂	—	2	2	no	—
154	C₂	C₂₃×#1746	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#117
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#1746
prime radical	C₂³×C₂₃	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂₃×#347	C₄	semidirect
12. C₂₃×#346	C₄	semidirect
13. C₄×C₁₆×C₂₃	C₄	semidirect
54. #12597	C₂₃	direct (N × H)
85. #347	C₄×C₂₃	semidirect
86. #346	C₄×C₂₃	semidirect
87. C₄×C₁₆	C₄×C₂₃	semidirect
111. C₂₃	#12597	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×#2110	2
C₂₃×#2108	2
C₂₃×#1747	2
C₂₃×#1379	2
C₂₃×#1378	2
#12597	23

Subgroups — C23×#12597 #19160223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12597 #19160223