Subgroups — D₂₃×#427 #15681973

Order 2944.

512 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic
2	#117×D₂₃	C₂	1	2	yes	maximal
3	#13092455	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092454	C₂	1	2	yes	maximal, complemented
5	#13092453	C₂	1	2	yes	maximal, complemented
6	#13092452	C₂	1	2	yes	maximal, complemented
7	#107×D₂₃	C₂	1	2	yes	maximal, complemented
8	#106×D₂₃	C₂	1	2	yes	maximal, complemented
9	C₂₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#117	C₂²	1	2	yes	—
13	#10625097	C₂²	1	2	yes	—
14	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₂₃	C₄	1	8	yes	complemented
16	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
20	#10625063	C₂²	1	4	yes	—
21	C₄⋊C₄×D₂₃	C₂²	1	4	yes	—
22	C₂²⋊C₄×D₂₃	—	2	2	no	—
23	#10625048	—	2	2	no	—
24	#10625043	—	2	2	no	—
25	C₄²×D₂₃	—	2	2	no	—
26	C₂₃×#107	C₂²	1	2	yes	complemented
27	C₂₃×#106	C₂²	1	2	yes	complemented
28	#10625000	C₂²	1	2	yes	complemented
29	#10624999	C₂²	1	2	yes	complemented
30	#10624990	C₂²	1	4	yes	complemented
31	#10624989	C₂²	1	4	yes	complemented
32	C₂³×D₂₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
35	C₂×C₄×D₂₃	—	2	4	no	—
36	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
37	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
38	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
39	C₂×C₄×D₂₃	Q₈	1	4	yes	—
40	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₂₃	C₂×C₄	1	4	yes	complemented
43	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
46	C₂²⋊C₄×C₂₃	—	2	2	no	—
47	C₄²×C₂₃	—	2	2	no	—
48	C₂²⋊Dic₂₃	—	2	2	no	—
49	(C₂×D₂₃)⋊C₄	—	2	4	no	—
50	C₄⋊Dic₂₃	C₂³	1	4	yes	—
51	Dic₂₃⋊C₄	—	2	4	no	—
52	C₄×Dic₂₃	—	2	2	no	—
53	C₄×Dic₂₃	—	2	4	no	—
54	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	complemented
55	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
56	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
57	C₈×D₂₃	C₂×C₄	1	4	yes	—
58	C₈×D₂₃	C₂×C₄	1	4	yes	—
59	C₈×D₂₃	C₂×C₄	1	4	yes	complemented
60	C₈×D₂₃	C₂×C₄	1	4	yes	complemented
61	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×D₂₃	—	2	4	no	—
63	C₂²×D₂₃	C₄⋊C₄	1	4	yes	—
64	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	—	2	4	no	—
67	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
74	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
75	C₄×D₂₃	—	4	8	no	—
76	C₄×D₂₃	C₄⋊C₄	1	4	yes	—
77	C₄×D₂₃	C₄⋊C₄	1	4	yes	—
78	C₄×D₂₃	C₂²×C₄	1	2	yes	—
79	C₄×D₂₃	C₂²×C₄	1	2	yes	—
80	C₄×D₂₃	C₂×Q₈	1	2	yes	—
81	C₄×D₂₃	C₂×Q₈	1	2	yes	—
82	C₈×C₂₃	C₂²×C₄	1	2	yes	—
83	C₈×C₂₃	C₂²×C₄	1	2	yes	—
84	C₂₃⋊C₈	C₂²×C₄	1	2	yes	—
85	C₂₃⋊C₈	C₂²×C₄	1	2	yes	—
86	C₂×#427	—	23	1	no	maximal, Sylow
87	C₂²×C₂₃	—	2	2	no	—
88	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
89	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
90	C₂×D₂₃	—	2	2	no	—
91	C₂×D₂₃	—	2	8	no	—
92	C₂×D₂₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×D₂₃	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×D₂₃	#135	1	4	yes	—
95	C₄×C₂₃	—	4	4	no	—
96	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
98	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
99	Dic₂₃	—	4	4	no	—
100	Dic₂₃	C₂×C₄⋊C₄	1	2	yes	—
101	Dic₂₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂³×C₈	—	23	1	no	—
103	C₂×#117	—	23	2	no	—
104	#427	—	23	1	no	—
105	#427	—	23	2	no	—
106	#427	—	23	2	no	—
107	#427	—	23	2	no	—
108	#427	D₂₃	1	1	yes	characteristic, direct product factor
109	C₂×#107	—	23	2	no	—
110	C₂×#106	—	23	2	no	—
111	C₂×C₂₃	—	2	2	no	—
112	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₂₃	C₂×#135	1	2	yes	—
114	D₂₃	—	2	2	no	—
115	D₂₃	#427	1	4	yes	direct product factor
116	C₂³×C₄	—	23	1	no	—
117	C₂²×C₈	—	23	1	no	—
118	C₂²×C₈	—	23	8	no	—
119	C₂²×C₈	—	23	1	no	—
120	C₂²×C₈	—	23	1	no	—
121	C₂²×C₈	—	23	1	no	—
122	C₂²×C₈	—	23	1	no	—
123	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
124	#117	—	23	2	no	—
125	#117	—	23	4	no	—
126	#117	—	46	2	no	—
127	#117	—	46	2	no	—
128	#117	C₂×D₂₃	1	2	yes	—
129	C₂×C₄⋊C₄	—	23	4	no	—
130	C₂×C₂²⋊C₄	—	46	2	no	—
131	C₂×C₄²	—	46	2	no	—
132	#107	—	23	2	no	—
133	#107	—	23	4	no	—
134	#107	C₂×D₂₃	1	2	yes	complemented
135	#106	—	23	2	no	—
136	#106	—	23	4	no	—
137	#106	C₂×D₂₃	1	2	yes	complemented
138	C₂₃	C₂×#427	1	1	yes	characteristic, complemented, Sylow
139	C₂⁴	—	23	1	no	—
140	C₂²×C₄	—	23	1	no	—
141	C₂²×C₄	—	23	4	no	—
142	C₂²×C₄	—	23	1	no	—
143	C₂²×C₄	—	23	1	no	—
144	C₂²×C₄	—	23	4	no	—
145	C₂²×C₄	—	23	1	no	—
146	C₂²×C₄	—	23	1	no	—
147	C₂²×C₄	—	46	4	no	—
148	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
149	C₂×C₈	—	23	4	no	—
150	C₂×C₈	—	23	1	no	—
151	C₂×C₈	—	23	1	no	—
152	C₂×C₈	—	23	4	no	—
153	C₂×C₈	—	23	4	no	—
154	C₂×C₈	—	23	4	no	—
155	C₂×C₈	—	23	4	no	—
156	C₂×C₈	C₄×D₂₃	1	4	yes	complemented
157	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
158	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
159	C₄⋊C₄	—	23	4	no	—
160	C₄⋊C₄	—	46	4	no	—
161	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	46	4	no	—
164	C₂²⋊C₄	—	46	2	no	—
165	C₄²	—	2	2	no	—
166	C₄²	—	46	4	no	—
167	C₄²	—	46	2	no	—
168	C₂³	—	23	1	no	—
169	C₂³	—	23	4	no	—
170	C₂³	—	23	1	no	—
171	C₂³	—	46	4	no	—
172	C₂³	D₄×D₂₃	1	1	yes	characteristic
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	23	4	no	—
175	C₂×C₄	—	23	1	no	—
176	C₂×C₄	—	23	1	no	—
177	C₂×C₄	—	23	4	no	—
178	C₂×C₄	—	23	4	no	—
179	C₂×C₄	—	23	2	no	—
180	C₂×C₄	—	23	2	no	—
181	C₂×C₄	—	23	2	no	—
182	C₂×C₄	—	23	2	no	—
183	C₂×C₄	—	46	4	no	—
184	C₂×C₄	—	92	8	no	—
185	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
186	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
187	C₂×C₄	Q₈×D₂₃	1	2	yes	—
188	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
189	C₈	—	23	2	no	—
190	C₈	—	23	2	no	—
191	C₈	C₂×C₄×D₂₃	1	2	yes	—
192	C₈	C₂×C₄×D₂₃	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	23	2	no	—
195	C₂²	—	23	4	no	—
196	C₂²	—	23	2	no	—
197	C₂²	—	46	8	no	—
198	C₂²	—	46	2	no	—
199	C₂²	C₄⋊C₄×D₂₃	1	2	yes	—
200	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
201	C₄	—	4	4	no	—
202	C₄	—	23	2	no	—
203	C₄	—	23	2	no	—
204	C₄	—	92	4	no	—
205	C₄	C₄⋊C₄×D₂₃	1	2	yes	—
206	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
207	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	—	23	4	no	—
210	C₂	—	46	2	no	—
211	C₂	C₂×C₄⋊C₄×D₂₃	1	1	yes	characteristic
212	C₂	#135×D₂₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#427	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂³×D₂₃	D₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#427	D₂₃
hypocenter	C₂₃	C₂×#427
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

Complements

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

N	H ≅ G/N	type
3. #13092455	C₂	semidirect
4. #13092454	C₂	semidirect
5. #13092453	C₂	semidirect
6. #13092452	C₂	semidirect
7. #107×D₂₃	C₂	semidirect
8. #106×D₂₃	C₂	semidirect
9. C₂₃×#427	C₂	semidirect
15. C₂×C₈×D₂₃	C₄	semidirect
26. C₂₃×#107	C₂²	semidirect
27. C₂₃×#106	C₂²	semidirect
28. #10625000	C₂²	semidirect
29. #10624999	C₂²	semidirect
30. #10624990	C₂²	semidirect
31. #10624989	C₂²	semidirect
42. C₂×C₈×C₂₃	C₂×C₄	semidirect
54. C₂×C₂₃⋊C₈	C₂×C₄	semidirect
59. C₈×D₂₃	C₂×C₄	semidirect
60. C₈×D₂₃	C₂×C₄	semidirect
108. #427	D₂₃	direct (N × H)
115. D₂₃	#427	direct (N × H)
134. #107	C₂×D₂₃	semidirect
137. #106	C₂×D₂₃	semidirect
138. C₂₃	C₂×#427	semidirect
156. C₂×C₈	C₄×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×D₂₃	2
#117×D₂₃	2
#13092455	2
#13092454	2
#13092453	2
#13092452	2
#107×D₂₃	2
#106×D₂₃	2
C₂₃×#427	2
C₂×#427	23

Subgroups — D23×#427 #15681973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₃×#427 #15681973