Subgroups — D₂₃×#559 #15686981

Order 2944.

1240 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093153	C₂	1	1	yes	maximal, characteristic, complemented
2	#13093152	C₂	1	2	yes	maximal, complemented
3	#13093151	C₂	1	4	yes	maximal, complemented
4	#13093150	C₂	1	4	yes	maximal, complemented
5	#13093149	C₂	1	4	yes	maximal, complemented
6	#127×D₂₃	C₂	1	1	yes	maximal, characteristic
7	#124×D₂₃	C₂	1	4	yes	maximal, complemented
8	#123×D₂₃	C₂	1	4	yes	maximal, complemented
9	C₄×Q₈×D₂₃	C₂	1	2	yes	maximal, complemented
10	C₄×D₄×D₂₃	C₂	1	4	yes	maximal, complemented
11	C₂₃×#559	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₂₃	C₂²	1	2	yes	—
13	C₂×D₄×D₂₃	C₂²	1	4	yes	—
14	C₂×D₄×D₂₃	C₂²	1	2	yes	—
15	C₂²×C₄×D₂₃	C₂²	1	4	yes	—
16	C₂₃×#127	C₂²	1	1	yes	characteristic
17	C₂₃×#124	C₂²	1	4	yes	complemented
18	C₂₃×#123	C₂²	1	4	yes	complemented
19	C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
20	C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
21	#10625118	C₂²	1	4	yes	complemented
22	Q₈×Dic₂₃	C₂²	1	2	yes	complemented
23	#10625111	C₂²	1	4	yes	—
24	#10625111	C₂²	1	2	yes	—
25	#10625107	C₂²	1	4	yes	complemented
26	#10625106	C₂²	1	4	yes	complemented
27	D₄×Dic₂₃	C₂²	1	4	yes	complemented
28	#10625100	C₂²	1	4	yes	complemented
29	C₄×C₂²⋊D₂₃	C₂²	1	8	yes	complemented
30	#10625065	C₂²	1	8	yes	complemented
31	#10625064	C₂²	1	4	yes	complemented
32	C₄⋊C₄×D₂₃	C₂²	1	2	yes	—
33	C₄⋊C₄×D₂₃	C₂²	1	4	yes	—
34	#10625058	C₂²	1	2	yes	complemented
35	#10625058	C₂²	1	4	yes	complemented
36	#10625057	C₂²	1	4	yes	complemented
37	#10625056	C₂²	1	8	yes	complemented
38	#10625056	C₂²	1	8	yes	complemented
39	#10625054	C₂²	1	8	yes	complemented
40	(C₂²⋊D₂₃)⋊C₄	C₂²	1	8	yes	complemented
41	C₂²⋊C₄×D₂₃	C₂²	1	8	yes	—
42	C₂²⋊C₄×D₂₃	C₂²	1	4	yes	—
43	#10625046	C₂²	1	4	yes	complemented
44	#10625045	C₂²	1	1	yes	characteristic
45	C₄×D₉₂	C₂²	1	4	yes	complemented
46	C₄²×D₂₃	C₂²	1	4	yes	—
47	C₄²×D₂₃	C₂²	1	1	yes	characteristic
48	C₄×Dic₄₆	C₂²	1	4	yes	complemented
49	C₄×Dic₄₆	C₂²	1	2	yes	complemented
50	C₂³×D₂₃	C₂³	1	4	yes	—
51	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
52	C₂×D₄×C₂₃	C₂³	1	4	yes	—
53	C₂×D₄×C₂₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
55	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	—
56	C₂×C₂²⋊D₂₃	C₂³	1	4	yes	—
57	C₂²×Dic₂₃	C₂³	1	4	yes	—
58	Q₈×D₂₃	—	2	8	no	—
59	D₄×D₂₃	—	2	16	no	—
60	D₄×D₂₃	—	4	4	no	—
61	C₂×D₉₂	C₂³	1	4	yes	—
62	C₂×D₉₂	C₂³	1	2	yes	—
63	C₂×C₄×D₂₃	—	4	8	no	—
64	C₂×C₄×D₂₃	C₂³	1	4	yes	—
65	C₂×C₄×D₂₃	C₂³	1	4	yes	—
66	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₂₃	C₂³	1	8	yes	—
68	C₂×C₄×D₂₃	C₂³	1	2	yes	—
69	C₂×Dic₄₆	C₂³	1	2	yes	—
70	C₂×Dic₄₆	C₂³	1	4	yes	—
71	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
72	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
73	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
74	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
75	C₄²×C₂₃	C₂³	1	1	yes	characteristic
76	C₄²×C₂₃	C₂³	1	4	yes	—
77	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
78	C₂²⋊Dic₂₃	C₂³	1	4	yes	—
79	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
80	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
81	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
82	C₄⋊Dic₂₃	C₂³	1	2	yes	—
83	C₄⋊Dic₂₃	C₂³	1	4	yes	—
84	Dic₂₃⋊C₄	C₂³	1	4	yes	—
85	Dic₂₃⋊C₄	C₂³	1	4	yes	—
86	Dic₂₃⋊C₄	C₂³	1	4	yes	—
87	C₄×Dic₂₃	C₂³	1	2	yes	—
88	C₄×Dic₂₃	C₂³	1	4	yes	—
89	C₄×Dic₂₃	C₂³	1	4	yes	—
90	C₄×Dic₂₃	C₂³	1	4	yes	—
91	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
92	C₂³×C₂₃	C₂⁴	1	4	yes	—
93	C₂²×D₂₃	—	2	16	no	—
94	C₂²×D₂₃	—	2	8	no	—
95	C₂²×D₂₃	C₂⁴	1	4	yes	—
96	C₂²×D₂₃	C₂⁴	1	1	yes	characteristic
97	Q₈×C₂₃	—	2	4	no	—
98	D₄×C₂₃	—	2	8	no	—
99	D₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	—	4	4	no	—
101	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
103	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
104	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
105	C₂²⋊D₂₃	—	2	16	no	—
106	C₂²⋊D₂₃	—	4	4	no	—
107	C₂×Dic₂₃	—	4	4	no	—
108	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
109	C₂×Dic₂₃	C₂⁴	1	4	yes	—
110	C₂×Dic₂₃	C₂⁴	1	4	yes	—
111	C₂×Dic₂₃	C₂⁴	1	2	yes	—
112	D₉₂	—	2	8	no	—
113	D₉₂	—	4	2	no	—
114	C₄×D₂₃	—	2	8	no	—
115	C₄×D₂₃	—	2	8	no	—
116	C₄×D₂₃	—	2	2	no	—
117	C₄×D₂₃	—	4	8	no	—
118	C₄×D₂₃	#41	1	8	yes	—
119	Dic₄₆	—	2	4	no	—
120	Dic₄₆	—	2	8	no	—
121	C₂×#559	—	23	1	no	maximal, Sylow
122	C₂²×C₂₃	—	2	8	no	—
123	C₂²×C₂₃	—	2	4	no	—
124	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
125	C₂×D₂₃	—	2	8	no	—
126	C₂×D₂₃	—	2	4	no	—
127	C₂×D₂₃	—	4	16	no	—
128	C₂×D₂₃	C₂×#41	1	4	yes	—
129	C₂×D₂₃	H(5,𝔽₂)	1	2	yes	—
130	C₄×C₂₃	—	2	4	no	—
131	C₄×C₂₃	—	2	4	no	—
132	C₄×C₂₃	—	2	1	no	—
133	C₄×C₂₃	C₂×#41	1	4	yes	—
134	Dic₂₃	—	2	4	no	—
135	Dic₂₃	—	2	4	no	—
136	Dic₂₃	—	2	1	no	—
137	Dic₂₃	C₂×#41	1	4	yes	—
138	#559	—	23	1	no	—
139	#559	—	23	4	no	—
140	#559	—	23	4	no	—
141	#559	—	23	2	no	—
142	#559	—	23	4	no	—
143	#559	D₂₃	1	1	yes	characteristic, direct product factor
144	C₂×#127	—	23	1	no	—
145	C₂×#124	—	23	4	no	—
146	C₂×#123	—	23	4	no	—
147	C₂×C₄×Q₈	—	23	2	no	—
148	C₂×C₄×D₄	—	23	4	no	—
149	C₂×C₂₃	—	4	4	no	—
150	C₂×C₂₃	C₂²×#41	1	2	yes	—
151	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
152	D₂₃	—	4	4	no	—
153	D₂₃	#559	1	4	yes	direct product factor
154	C₂²×Q₈	—	23	2	no	—
155	C₂²×D₄	—	23	4	no	—
156	C₂²×D₄	—	23	2	no	—
157	C₂³×C₄	—	23	4	no	—
158	#127	—	23	2	no	—
159	#127	—	23	4	no	—
160	#127	—	23	1	no	—
161	#127	C₂×D₂₃	1	1	yes	characteristic
162	#124	—	23	4	no	—
163	#124	—	23	4	no	—
164	#124	—	23	8	no	—
165	#124	—	23	8	no	—
166	#124	—	23	4	no	—
167	#124	C₂×D₂₃	1	4	yes	complemented
168	#123	—	23	4	no	—
169	#123	—	23	4	no	—
170	#123	—	23	8	no	—
171	#123	—	23	8	no	—
172	#123	—	23	4	no	—
173	#123	C₂×D₂₃	1	4	yes	complemented
174	C₄×Q₈	—	23	2	no	—
175	C₄×Q₈	—	23	4	no	—
176	C₄×Q₈	—	23	2	no	—
177	C₄×Q₈	—	23	2	no	—
178	C₄×Q₈	—	23	4	no	—
179	C₄×Q₈	C₂×D₂₃	1	2	yes	complemented
180	C₄×D₄	—	23	4	no	—
181	C₄×D₄	—	23	8	no	—
182	C₄×D₄	—	23	4	no	—
183	C₄×D₄	—	23	8	no	—
184	C₄×D₄	—	23	4	no	—
185	C₄×D₄	C₂×D₂₃	1	4	yes	complemented
186	C₂×C₄⋊C₄	—	23	2	no	—
187	C₂×C₄⋊C₄	—	23	4	no	—
188	C₂×C₂²⋊C₄	—	23	8	no	—
189	C₂×C₂²⋊C₄	—	23	4	no	—
190	C₂×C₄²	—	23	4	no	—
191	C₂×C₄²	—	23	1	no	—
192	C₂₃	C₂×#559	1	1	yes	characteristic, complemented, Sylow
193	C₂⁴	—	23	4	no	—
194	C₂×Q₈	—	23	4	no	—
195	C₂×Q₈	—	23	2	no	—
196	C₂×Q₈	—	46	8	no	—
197	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
198	C₂×D₄	—	23	8	no	—
199	C₂×D₄	—	23	4	no	—
200	C₂×D₄	—	23	4	no	—
201	C₂×D₄	—	23	2	no	—
202	C₂×D₄	—	46	16	no	—
203	C₂×D₄	—	92	4	no	—
204	C₂×D₄	C₂²×D₂₃	1	4	yes	—
205	C₂×D₄	C₂²×D₂₃	1	2	yes	—
206	C₂²×C₄	—	23	2	no	—
207	C₂²×C₄	—	23	4	no	—
208	C₂²×C₄	—	23	4	no	—
209	C₂²×C₄	—	23	4	no	—
210	C₂²×C₄	—	23	1	no	—
211	C₂²×C₄	—	23	8	no	—
212	C₂²×C₄	—	92	8	no	—
213	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
214	C₄⋊C₄	—	23	4	no	—
215	C₄⋊C₄	—	23	2	no	—
216	C₄⋊C₄	—	23	4	no	—
217	C₄⋊C₄	—	23	4	no	—
218	C₄⋊C₄	—	23	4	no	—
219	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
220	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
221	C₂²⋊C₄	—	23	8	no	—
222	C₂²⋊C₄	—	23	4	no	—
223	C₂²⋊C₄	—	23	8	no	—
224	C₂²⋊C₄	—	23	8	no	—
225	C₂²⋊C₄	—	23	8	no	—
226	C₂²⋊C₄	C₂²×D₂₃	1	8	yes	—
227	C₂²⋊C₄	C₂²×D₂₃	1	4	yes	—
228	C₄²	—	23	4	no	—
229	C₄²	—	23	4	no	—
230	C₄²	—	23	4	no	—
231	C₄²	—	23	1	no	—
232	C₄²	—	23	2	no	—
233	C₄²	C₂²×D₂₃	1	4	yes	—
234	C₄²	C₂²×D₂₃	1	1	yes	characteristic
235	C₂³	—	23	1	no	—
236	C₂³	—	23	4	no	—
237	C₂³	—	46	8	no	—
238	C₂³	—	46	16	no	—
239	C₂³	C₂³×D₂₃	1	4	yes	—
240	Q₈	—	2	4	no	—
241	Q₈	—	46	8	no	—
242	Q₈	—	46	4	no	—
243	D₄	—	2	8	no	—
244	D₄	—	4	2	no	—
245	D₄	—	46	16	no	—
246	D₄	—	46	8	no	—
247	D₄	—	92	2	no	—
248	D₄	—	92	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	23	2	no	—
251	C₂×C₄	—	23	8	no	—
252	C₂×C₄	—	23	4	no	—
253	C₂×C₄	—	23	4	no	—
254	C₂×C₄	—	23	1	no	—
255	C₂×C₄	—	46	8	no	—
256	C₂×C₄	—	46	2	no	—
257	C₂×C₄	—	46	8	no	—
258	C₂×C₄	—	92	8	no	—
259	C₂×C₄	—	92	4	no	—
260	C₂×C₄	C₂³×D₂₃	1	4	yes	—
261	C₂×C₄	C₂³×D₂₃	1	4	yes	—
262	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
263	C₂×C₄	C₂³×D₂₃	1	2	yes	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	23	4	no	—
267	C₂²	—	23	2	no	—
268	C₂²	—	46	8	no	—
269	C₂²	—	46	4	no	—
270	C₂²	—	92	16	no	—
271	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
272	C₄	—	2	4	no	—
273	C₄	—	2	4	no	—
274	C₄	—	2	1	no	—
275	C₄	—	23	4	no	—
276	C₄	—	46	4	no	—
277	C₄	—	46	4	no	—
278	C₄	—	46	1	no	—
279	C₄	#41×D₂₃	1	4	yes	—
280	C₂	—	4	4	no	—
281	C₂	—	23	4	no	—
282	C₂	—	92	4	no	—
283	C₂	C₂×#41×D₂₃	1	2	yes	—
284	C₂	H(5,𝔽₂)×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Fitting	C₂₃×#559	C₂
prime radical	#127×D₂₃	C₂
hypercenter	#559	D₂₃
hypocenter	C₂₃	C₂×#559
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #559

Complements

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

N	H ≅ G/N	type
1. #13093153	C₂	semidirect
2. #13093152	C₂	semidirect
3. #13093151	C₂	semidirect
4. #13093150	C₂	semidirect
5. #13093149	C₂	semidirect
7. #124×D₂₃	C₂	semidirect
8. #123×D₂₃	C₂	semidirect
9. C₄×Q₈×D₂₃	C₂	semidirect
10. C₄×D₄×D₂₃	C₂	semidirect
11. C₂₃×#559	C₂	semidirect
17. C₂₃×#124	C₂²	semidirect
18. C₂₃×#123	C₂²	semidirect
19. C₄×Q₈×C₂₃	C₂²	semidirect
20. C₄×D₄×C₂₃	C₂²	semidirect
21. #10625118	C₂²	semidirect
22. Q₈×Dic₂₃	C₂²	semidirect
25. #10625107	C₂²	semidirect
26. #10625106	C₂²	semidirect
27. D₄×Dic₂₃	C₂²	semidirect
28. #10625100	C₂²	semidirect
29. C₄×C₂²⋊D₂₃	C₂²	semidirect
30. #10625065	C₂²	semidirect
31. #10625064	C₂²	semidirect
34. #10625058	C₂²	semidirect
35. #10625058	C₂²	semidirect
36. #10625057	C₂²	semidirect
37. #10625056	C₂²	semidirect
38. #10625056	C₂²	semidirect
39. #10625054	C₂²	semidirect
40. (C₂²⋊D₂₃)⋊C₄	C₂²	semidirect
43. #10625046	C₂²	semidirect
45. C₄×D₉₂	C₂²	semidirect
48. C₄×Dic₄₆	C₂²	semidirect
49. C₄×Dic₄₆	C₂²	semidirect
143. #559	D₂₃	direct (N × H)
153. D₂₃	#559	direct (N × H)
167. #124	C₂×D₂₃	semidirect
173. #123	C₂×D₂₃	semidirect
179. C₄×Q₈	C₂×D₂₃	semidirect
185. C₄×D₄	C₂×D₂₃	semidirect
192. C₂₃	C₂×#559	semidirect

Maximal subgroups

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

subgroup	index
#13093153	2
#13093152	2
#13093151	2
#13093150	2
#13093149	2
#127×D₂₃	2
#124×D₂₃	2
#123×D₂₃	2
C₄×Q₈×D₂₃	2
C₄×D₄×D₂₃	2
C₂₃×#559	2
C₂×#559	23

Subgroups — D23×#559 #15686981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₃×#559 #15686981