Subgroups — D₂₃×#428 #15681978

Order 2944.

536 conjugacy classes of subgroups.

The subgroup lattice has 296 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	1	yes	maximal, characteristic
2	#117×D₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×D₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092460	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092459	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092458	C₂	1	2	yes	maximal, complemented
7	#13092457	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092456	C₂	1	2	yes	maximal, complemented
9	#107×D₂₃	C₂	1	2	yes	maximal, complemented
10	#106×D₂₃	C₂	1	2	yes	maximal, complemented
11	C₂₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₂₃	—	2	1	no	—
13	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#117	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	#10625097	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₃⋊C₄	—	2	1	no	—
20	C₂×#67267	C₂²	1	1	yes	characteristic
21	M₁₆×D₂₃	C₄	1	8	yes	complemented
22	C₂×#67262	C₂²	1	2	yes	—
23	C₂×C₈×D₂₃	C₂²	1	2	yes	—
24	#10625063	C₂²	1	1	yes	characteristic
25	#10625063	C₂²	1	1	yes	characteristic
26	C₄⋊C₄×D₂₃	—	2	1	no	—
27	C₄⋊C₄×D₂₃	—	2	1	no	—
28	C₄⋊C₄×D₂₃	C₂²	1	2	yes	—
29	C₄⋊C₄×D₂₃	C₂²	1	2	yes	—
30	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
32	C₂²⋊C₄×D₂₃	—	2	1	no	—
33	#10625048	—	2	1	no	—
34	#10625043	—	2	1	no	—
35	C₄²×D₂₃	—	2	1	no	—
36	C₂₃×#107	C₂²	1	2	yes	complemented
37	C₂₃×#106	C₂²	1	2	yes	complemented
38	#10625000	C₂²	1	2	yes	complemented
39	#10624999	C₂²	1	2	yes	complemented
40	#10624990	C₂²	1	2	yes	complemented
41	#10624990	C₂²	1	2	yes	complemented
42	#10624989	C₂²	1	2	yes	complemented
43	#10624989	C₂²	1	2	yes	complemented
44	C₂³×D₂₃	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	—	2	1	no	—
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×Dic₂₃	—	2	1	no	—
48	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
49	C₂×C₄×D₂₃	—	2	1	no	—
50	C₂×C₄×D₂₃	—	2	1	no	—
51	C₂×C₄×D₂₃	—	2	2	no	—
52	C₂×C₄×D₂₃	—	2	2	no	—
53	C₂×C₄×D₂₃	—	4	4	no	—
54	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
55	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
56	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
57	C₂×C₄×D₂₃	Q₈	1	4	yes	—
58	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
60	M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
61	C₂×C₈×C₂₃	C₂³	1	2	yes	—
62	C₄⋊C₄×C₂₃	—	2	1	no	—
63	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
65	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂²⋊C₄×C₂₃	—	2	1	no	—
67	C₄²×C₂₃	—	2	1	no	—
68	C₂²⋊Dic₂₃	—	2	1	no	—
69	(C₂×D₂₃)⋊C₄	—	2	1	no	—
70	(C₂×D₂₃)⋊C₄	—	2	1	no	—
71	C₄⋊Dic₂₃	—	2	1	no	—
72	C₄⋊Dic₂₃	C₂³	1	2	yes	—
73	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
75	Dic₂₃⋊C₄	—	2	1	no	—
76	Dic₂₃⋊C₄	—	2	1	no	—
77	Dic₂₃⋊C₄	—	2	2	no	—
78	Dic₂₃⋊C₄	—	2	2	no	—
79	C₄×Dic₂₃	—	2	1	no	—
80	C₄×Dic₂₃	—	2	1	no	—
81	C₄×Dic₂₃	—	2	1	no	—
82	#67267	C₂×C₄	1	4	yes	complemented
83	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
84	#67262	C₂×C₄	1	8	yes	complemented
85	C₈×D₂₃	C₂×C₄	1	8	yes	complemented
86	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×D₂₃	—	2	2	no	—
88	C₂²×D₂₃	—	2	2	no	—
89	C₂²×D₂₃	C₄⋊C₄	1	4	yes	—
90	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	—	2	1	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	4	2	no	—
96	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
99	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
100	C₂×Dic₂₃	—	2	1	no	—
101	C₂×Dic₂₃	—	2	1	no	—
102	C₂×Dic₂₃	—	2	2	no	—
103	C₂×Dic₂₃	—	4	2	no	—
104	C₂×Dic₂₃	C₄⋊C₄	1	4	yes	—
105	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
106	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
107	C₄×D₂₃	—	4	4	no	—
108	C₄×D₂₃	—	4	2	no	—
109	C₄×D₂₃	—	4	4	no	—
110	C₄×D₂₃	—	4	2	no	—
111	C₄×D₂₃	C₄⋊C₄	1	4	yes	—
112	C₄×D₂₃	C₄⋊C₄	1	4	yes	—
113	C₄×D₂₃	C₂²×C₄	1	2	yes	—
114	C₄×D₂₃	C₂²×C₄	1	2	yes	—
115	C₄×D₂₃	C₂×Q₈	1	2	yes	—
116	C₄×D₂₃	C₂×Q₈	1	2	yes	—
117	C₈×C₂₃	C₂²×C₄	1	4	yes	complemented
118	C₂₃⋊C₈	C₂²×C₄	1	4	yes	complemented
119	C₂×#428	—	23	1	no	maximal, Sylow
120	C₂²×C₂₃	—	2	1	no	—
121	C₂²×C₂₃	—	2	1	no	—
122	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×D₂₃	—	2	1	no	—
125	C₂×D₂₃	—	2	1	no	—
126	C₂×D₂₃	—	2	8	no	—
127	C₂×D₂₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×D₂₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×D₂₃	AGL(1,ℤ/8)	1	2	yes	—
130	C₂×D₂₃	#137	1	2	yes	—
131	C₄×C₂₃	—	4	2	no	—
132	C₄×C₂₃	—	4	1	no	—
133	C₄×C₂₃	—	4	1	no	—
134	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
135	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
136	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
137	Dic₂₃	—	4	2	no	—
138	Dic₂₃	—	4	1	no	—
139	Dic₂₃	—	4	1	no	—
140	Dic₂₃	C₂×C₄⋊C₄	1	2	yes	—
141	Dic₂₃	C₂×C₄⋊C₄	1	2	yes	—
142	C₂²×M₁₆	—	23	1	no	—
143	C₂×#117	—	23	1	no	—
144	C₂²×C₄⋊C₄	—	23	1	no	—
145	#428	—	23	1	no	—
146	#428	—	23	1	no	—
147	#428	—	23	1	no	—
148	#428	—	23	2	no	—
149	#428	—	23	2	no	—
150	#428	D₂₃	1	1	yes	characteristic, direct product factor
151	C₂×#107	—	23	2	no	—
152	C₂×#106	—	23	2	no	—
153	C₂×C₂₃	—	2	2	no	—
154	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
157	D₂₃	—	2	2	no	—
158	D₂₃	#428	1	4	yes	direct product factor
159	C₂³×C₄	—	23	1	no	—
160	C₂³×C₄	—	46	1	no	—
161	C₂×M₁₆	—	23	1	no	—
162	C₂×M₁₆	—	23	8	no	—
163	C₂×M₁₆	—	23	2	no	—
164	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
165	C₂²×C₈	—	23	2	no	—
166	#117	—	23	1	no	—
167	#117	—	23	1	no	—
168	#117	—	23	1	no	—
169	#117	—	46	1	no	—
170	#117	—	46	1	no	—
171	#117	C₂×D₂₃	1	1	yes	characteristic
172	C₂×C₄⋊C₄	—	23	1	no	—
173	C₂×C₄⋊C₄	—	23	2	no	—
174	C₂×C₄⋊C₄	—	23	2	no	—
175	C₂×C₄⋊C₄	—	23	1	no	—
176	C₂×C₄⋊C₄	—	23	1	no	—
177	C₂×C₄⋊C₄	—	46	1	no	—
178	C₂×C₄⋊C₄	—	46	1	no	—
179	C₂×C₄⋊C₄	—	46	1	no	—
180	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
181	C₂×C₂²⋊C₄	—	46	1	no	—
182	C₂×C₄²	—	46	1	no	—
183	#107	—	23	2	no	—
184	#107	—	23	2	no	—
185	#107	—	23	2	no	—
186	#107	C₂×D₂₃	1	2	yes	complemented
187	#106	—	23	2	no	—
188	#106	—	23	2	no	—
189	#106	—	23	2	no	—
190	#106	C₂×D₂₃	1	2	yes	complemented
191	C₂₃	C₂×#428	1	1	yes	characteristic, complemented, Sylow
192	C₂⁴	—	23	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	23	1	no	—
195	C₂²×C₄	—	23	4	no	—
196	C₂²×C₄	—	23	1	no	—
197	C₂²×C₄	—	23	1	no	—
198	C₂²×C₄	—	23	4	no	—
199	C₂²×C₄	—	23	1	no	—
200	C₂²×C₄	—	23	1	no	—
201	C₂²×C₄	—	46	1	no	—
202	C₂²×C₄	—	46	1	no	—
203	C₂²×C₄	—	46	2	no	—
204	C₂²×C₄	—	46	2	no	—
205	C₂²×C₄	—	46	1	no	—
206	C₂²×C₄	—	92	4	no	—
207	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
208	M₁₆	—	23	4	no	—
209	M₁₆	—	23	8	no	—
210	M₁₆	C₄×D₂₃	1	4	yes	complemented
211	C₂×C₈	—	23	2	no	—
212	C₂×C₈	—	23	8	no	—
213	C₂×C₈	C₂²×D₂₃	1	2	yes	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	—	23	1	no	—
216	C₄⋊C₄	—	23	2	no	—
217	C₄⋊C₄	—	23	1	no	—
218	C₄⋊C₄	—	46	2	no	—
219	C₄⋊C₄	—	46	2	no	—
220	C₄⋊C₄	—	46	1	no	—
221	C₄⋊C₄	—	46	1	no	—
222	C₄⋊C₄	—	46	1	no	—
223	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
224	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
225	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
226	C₂²⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	46	1	no	—
228	C₂²⋊C₄	—	46	1	no	—
229	C₂²⋊C₄	—	46	1	no	—
230	C₄²	—	2	1	no	—
231	C₄²	—	46	1	no	—
232	C₄²	—	46	1	no	—
233	C₄²	—	46	1	no	—
234	C₂³	—	23	1	no	—
235	C₂³	—	23	4	no	—
236	C₂³	—	23	1	no	—
237	C₂³	—	46	2	no	—
238	C₂³	—	46	2	no	—
239	C₂³	D₄×D₂₃	1	1	yes	characteristic
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	23	4	no	—
245	C₂×C₄	—	23	1	no	—
246	C₂×C₄	—	23	1	no	—
247	C₂×C₄	—	23	4	no	—
248	C₂×C₄	—	23	4	no	—
249	C₂×C₄	—	23	2	no	—
250	C₂×C₄	—	23	2	no	—
251	C₂×C₄	—	23	2	no	—
252	C₂×C₄	—	23	2	no	—
253	C₂×C₄	—	46	1	no	—
254	C₂×C₄	—	46	1	no	—
255	C₂×C₄	—	46	2	no	—
256	C₂×C₄	—	92	4	no	—
257	C₂×C₄	—	92	2	no	—
258	C₂×C₄	—	92	2	no	—
259	C₂×C₄	—	92	4	no	—
260	C₂×C₄	—	92	2	no	—
261	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
262	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
263	C₂×C₄	Q₈×D₂₃	1	2	yes	—
264	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
265	C₈	—	23	4	no	—
266	C₈	C₂×C₄×D₂₃	1	4	yes	complemented
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	23	2	no	—
270	C₂²	—	23	2	no	—
271	C₂²	—	23	2	no	—
272	C₂²	—	23	2	no	—
273	C₂²	—	46	8	no	—
274	C₂²	—	46	1	no	—
275	C₂²	—	46	1	no	—
276	C₂²	C₄⋊C₄×D₂₃	1	2	yes	—
277	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
278	C₄	—	4	2	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	1	no	—
281	C₄	—	23	2	no	—
282	C₄	—	23	2	no	—
283	C₄	—	92	2	no	—
284	C₄	—	92	1	no	—
285	C₄	—	92	1	no	—
286	C₄	C₄⋊C₄×D₂₃	1	2	yes	—
287	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
288	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	—	23	4	no	—
291	C₂	—	46	2	no	—
292	C₂	C₂×C₄⋊C₄×D₂₃	1	1	yes	characteristic
293	C₂	AGL(1,ℤ/8)×D₂₃	1	1	yes	characteristic
294	C₂	#137×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#428	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂³×D₂₃	D₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#428	D₂₃
hypocenter	C₂₃	C₂×#428
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
4. #13092460	C₂	semidirect
5. #13092459	C₂	semidirect
6. #13092458	C₂	semidirect
7. #13092457	C₂	semidirect
8. #13092456	C₂	semidirect
9. #107×D₂₃	C₂	semidirect
10. #106×D₂₃	C₂	semidirect
11. C₂₃×#428	C₂	semidirect
21. M₁₆×D₂₃	C₄	semidirect
36. C₂₃×#107	C₂²	semidirect
37. C₂₃×#106	C₂²	semidirect
38. #10625000	C₂²	semidirect
39. #10624999	C₂²	semidirect
40. #10624990	C₂²	semidirect
41. #10624990	C₂²	semidirect
42. #10624989	C₂²	semidirect
43. #10624989	C₂²	semidirect
60. M₁₆×C₂₃	C₂×C₄	semidirect
82. #67267	C₂×C₄	semidirect
84. #67262	C₂×C₄	semidirect
85. C₈×D₂₃	C₂×C₄	semidirect
117. C₈×C₂₃	C₂²×C₄	semidirect
118. C₂₃⋊C₈	C₂²×C₄	semidirect
150. #428	D₂₃	direct (N × H)
158. D₂₃	#428	direct (N × H)
186. #107	C₂×D₂₃	semidirect
190. #106	C₂×D₂₃	semidirect
191. C₂₃	C₂×#428	semidirect
210. M₁₆	C₄×D₂₃	semidirect
266. C₈	C₂×C₄×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
#117×D₂₃	2
C₂×C₄⋊C₄×D₂₃	2
#13092460	2
#13092459	2
#13092458	2
#13092457	2
#13092456	2
#107×D₂₃	2
#106×D₂₃	2
C₂₃×#428	2
C₂×#428	23

Subgroups — D23×#428 #15681978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₃×#428 #15681978