Subgroups — D₂₃×#423 #15681949

Order 2944.

576 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₂₃	C₂	1	2	yes	maximal
2	C₂×C₄²×D₂₃	C₂	1	1	yes	maximal, characteristic
3	#13092441	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092440	C₂	1	2	yes	maximal, complemented
5	#13092439	C₂	1	4	yes	maximal, complemented
6	C₄⋊C₈×D₂₃	C₂	1	4	yes	maximal, complemented
7	C₂₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₂₃	C₄	1	2	yes	—
9	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₂₃	C₄	1	2	yes	—
13	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
14	C₂×#67267	C₂²	1	2	yes	—
15	M₁₆×D₂₃	—	2	8	no	—
16	C₂×#67262	C₂²	1	4	yes	—
17	C₂×C₈×D₂₃	C₂²	1	4	yes	—
18	C₄²×D₂₃	C₄	1	1	yes	characteristic
19	C₄²×D₂₃	C₄	1	1	yes	characteristic
20	C₄²×D₂₃	C₄	1	1	yes	characteristic
21	C₄²×D₂₃	C₄	1	1	yes	characteristic
22	C₄²×D₂₃	C₂²	1	2	yes	—
23	C₄²×D₂₃	C₂²	1	2	yes	—
24	C₂₃×C₄⋊C₈	C₂²	1	4	yes	complemented
25	Dic₂₃⋊C₈	C₂²	1	8	yes	complemented
26	C₄⋊(C₂₃⋊C₈)	C₂²	1	4	yes	complemented
27	C₂³×D₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
31	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₂₃	—	2	8	no	—
33	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₂₃	D₄	1	4	yes	—
40	C₂×C₄×D₂₃	Q₈	1	4	yes	—
41	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
43	M₁₆×C₂₃	—	2	4	no	—
44	C₂×C₈×C₂₃	C₂³	1	4	yes	—
45	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₂₃	C₂³	1	2	yes	—
48	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
49	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
50	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
51	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
52	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
53	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
54	C₄×Dic₂₃	C₂³	1	2	yes	—
55	#67267	—	2	4	no	—
56	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
57	#67262	—	2	8	no	—
58	C₈×D₂₃	—	2	8	no	—
59	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×D₂₃	—	2	4	no	—
61	C₂²×D₂₃	C₄⋊C₄	1	4	yes	—
62	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₂₃	—	2	4	no	—
65	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
71	C₂×Dic₂₃	—	2	4	no	—
72	C₂×Dic₂₃	C₄⋊C₄	1	4	yes	—
73	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
77	C₄×D₂₃	—	2	4	no	—
78	C₄×D₂₃	—	2	4	no	—
79	C₄×D₂₃	—	2	4	no	—
80	C₄×D₂₃	C₄⋊C₄	1	4	yes	—
81	C₄×D₂₃	C₄⋊C₄	1	4	yes	—
82	C₄×D₂₃	M₁₆	1	8	yes	complemented
83	C₄×D₂₃	C₂×D₄	1	2	yes	—
84	C₄×D₂₃	C₂×D₄	1	2	yes	—
85	C₄×D₂₃	C₂×Q₈	1	2	yes	—
86	C₄×D₂₃	C₂×Q₈	1	2	yes	—
87	C₈×C₂₃	—	2	4	no	—
88	C₂₃⋊C₈	—	2	4	no	—
89	C₂×#423	—	23	1	no	maximal, Sylow
90	C₂²×C₂₃	—	2	2	no	—
91	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×D₂₃	—	2	2	no	—
94	C₂×D₂₃	—	2	8	no	—
95	C₂×D₂₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×D₂₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×D₂₃	C₂×M₁₆	1	4	yes	—
98	C₄×C₂₃	—	2	2	no	—
99	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₄×C₂₃	C₂×M₁₆	1	4	yes	complemented
101	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
102	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
103	Dic₂₃	—	2	2	no	—
104	Dic₂₃	C₂×C₄⋊C₄	1	2	yes	—
105	Dic₂₃	C₂×C₄⋊C₄	1	2	yes	—
106	Dic₂₃	C₂×M₁₆	1	4	yes	complemented
107	C₂²×M₁₆	—	23	2	no	—
108	C₂²×C₄²	—	23	1	no	—
109	#423	—	23	2	no	—
110	#423	—	23	1	no	—
111	#423	—	23	4	no	—
112	#423	D₂₃	1	1	yes	characteristic, direct product factor
113	C₂×C₄⋊C₈	—	23	4	no	—
114	C₂×C₂₃	—	2	2	no	—
115	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₂₃	C₂²×M₁₆	1	2	yes	—
117	D₂₃	—	2	2	no	—
118	D₂₃	#423	1	4	yes	direct product factor
119	C₂³×C₄	—	23	1	no	—
120	C₂³×C₄	—	23	2	no	—
121	C₂×M₁₆	—	23	2	no	—
122	C₂×M₁₆	—	23	4	no	—
123	C₂×M₁₆	—	46	8	no	—
124	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
125	C₂²×C₈	—	23	4	no	—
126	C₂×C₄²	—	23	2	no	—
127	C₂×C₄²	—	23	1	no	—
128	C₂×C₄²	—	23	1	no	—
129	C₂×C₄²	—	23	1	no	—
130	C₂×C₄²	—	23	1	no	—
131	C₂×C₄²	—	23	1	no	—
132	C₂×C₄²	—	23	2	no	—
133	C₂×C₄²	—	23	2	no	—
134	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
135	C₄⋊C₈	—	23	8	no	—
136	C₄⋊C₈	—	23	4	no	—
137	C₄⋊C₈	C₂×D₂₃	1	4	yes	complemented
138	C₂₃	C₂×#423	1	1	yes	characteristic, complemented, Sylow
139	C₂⁴	—	23	1	no	—
140	C₂²×C₄	—	23	1	no	—
141	C₂²×C₄	—	23	1	no	—
142	C₂²×C₄	—	23	1	no	—
143	C₂²×C₄	—	23	2	no	—
144	C₂²×C₄	—	23	4	no	—
145	C₂²×C₄	—	23	4	no	—
146	C₂²×C₄	—	23	1	no	—
147	C₂²×C₄	—	23	1	no	—
148	C₂²×C₄	—	23	2	no	—
149	C₂²×C₄	—	23	2	no	—
150	C₂²×C₄	—	23	2	no	—
151	C₂²×C₄	—	23	2	no	—
152	C₂²×C₄	—	46	8	no	—
153	C₂²×C₄	C₄×D₂₃	1	2	yes	—
154	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
155	M₁₆	—	2	4	no	—
156	M₁₆	—	46	8	no	—
157	M₁₆	—	46	4	no	—
158	C₂×C₈	—	23	4	no	—
159	C₂×C₈	—	46	8	no	—
160	C₂×C₈	C₂²×D₂₃	1	4	yes	—
161	C₄²	—	23	2	no	—
162	C₄²	—	23	1	no	—
163	C₄²	—	23	2	no	—
164	C₄²	—	23	2	no	—
165	C₄²	—	23	2	no	—
166	C₄²	—	23	1	no	—
167	C₄²	—	23	2	no	—
168	C₄²	C₄×D₂₃	1	1	yes	characteristic
169	C₄²	C₄×D₂₃	1	1	yes	characteristic
170	C₄²	C₂²×D₂₃	1	2	yes	—
171	C₂³	—	23	1	no	—
172	C₂³	—	23	4	no	—
173	C₂³	—	23	1	no	—
174	C₂³	—	46	4	no	—
175	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	23	4	no	—
178	C₂×C₄	—	23	1	no	—
179	C₂×C₄	—	23	1	no	—
180	C₂×C₄	—	23	2	no	—
181	C₂×C₄	—	23	4	no	—
182	C₂×C₄	—	23	4	no	—
183	C₂×C₄	—	23	8	no	—
184	C₂×C₄	—	23	2	no	—
185	C₂×C₄	—	23	2	no	—
186	C₂×C₄	—	23	2	no	—
187	C₂×C₄	—	23	2	no	—
188	C₂×C₄	—	23	2	no	—
189	C₂×C₄	—	46	4	no	—
190	C₂×C₄	—	46	4	no	—
191	C₂×C₄	—	46	4	no	—
192	C₂×C₄	—	46	4	no	—
193	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
194	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
196	C₂×C₄	D₄×D₂₃	1	2	yes	—
197	C₂×C₄	Q₈×D₂₃	1	2	yes	—
198	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
199	C₈	—	2	4	no	—
200	C₈	—	46	4	no	—
201	C₂²	—	2	2	no	—
202	C₂²	—	23	2	no	—
203	C₂²	—	23	4	no	—
204	C₂²	—	23	2	no	—
205	C₂²	—	46	8	no	—
206	C₂²	—	46	2	no	—
207	C₂²	C₄⋊C₄×D₂₃	1	2	yes	—
208	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	23	2	no	—
211	C₄	—	23	2	no	—
212	C₄	—	23	4	no	—
213	C₄	—	46	2	no	—
214	C₄	C₄⋊C₄×D₂₃	1	2	yes	—
215	C₄	M₁₆×D₂₃	1	4	yes	complemented
216	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
217	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
218	C₂	—	2	2	no	—
219	C₂	—	23	4	no	—
220	C₂	—	46	2	no	—
221	C₂	C₂×C₄⋊C₄×D₂₃	1	1	yes	characteristic
222	C₂	C₂×M₁₆×D₂₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#423	C₂
prime radical	C₂³×D₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#423	D₂₃
hypocenter	C₂₃	C₂×#423
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
3. #13092441	C₂	semidirect
4. #13092440	C₂	semidirect
5. #13092439	C₂	semidirect
6. C₄⋊C₈×D₂₃	C₂	semidirect
7. C₂₃×#423	C₂	semidirect
24. C₂₃×C₄⋊C₈	C₂²	semidirect
25. Dic₂₃⋊C₈	C₂²	semidirect
26. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
82. C₄×D₂₃	M₁₆	semidirect
100. C₄×C₂₃	C₂×M₁₆	semidirect
106. Dic₂₃	C₂×M₁₆	semidirect
112. #423	D₂₃	direct (N × H)
118. D₂₃	#423	direct (N × H)
137. C₄⋊C₈	C₂×D₂₃	semidirect
138. C₂₃	C₂×#423	semidirect
215. C₄	M₁₆×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₂₃	2
C₂×C₄²×D₂₃	2
#13092441	2
#13092440	2
#13092439	2
C₄⋊C₈×D₂₃	2
C₂₃×#423	2
C₂×#423	23

Subgroups — D23×#423 #15681949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₃×#423 #15681949