Subgroups — D₂₃×#550 #15686873

Order 2944.

1736 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₂₃	C₂	1	2	yes	maximal, complemented
2	#13093089	C₂	1	1	yes	maximal, characteristic, complemented
3	#13093088	C₂	1	3	yes	maximal, complemented
4	#13093087	C₂	1	3	yes	maximal, complemented
5	#13093086	C₂	1	2	yes	maximal, complemented
6	#13093085	C₂	1	6	yes	maximal, complemented
7	#127×D₂₃	C₂	1	3	yes	maximal, complemented
8	#121×D₂₃	C₂	1	6	yes	maximal, complemented
9	C₄×Q₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×D₂₃	C₂	1	3	yes	maximal, complemented
11	C₂₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	2	yes	complemented
13	#41×D₂₃	—	2	16	no	—
14	C₂×#67292	C₂²	1	2	yes	complemented
15	C₂×Q₈×D₂₃	C₂²	1	1	yes	characteristic
16	C₂×#67290	C₂²	1	6	yes	complemented
17	C₂×D₄×D₂₃	C₂²	1	3	yes	—
18	C₂×D₄×D₂₃	C₂²	1	6	yes	—
19	C₂×D₄×D₂₃	C₂²	1	6	yes	—
20	C₂×#67288	C₂²	1	6	yes	complemented
21	C₂²×C₄×D₂₃	C₂²	1	6	yes	—
22	C₂₃×#127	C₂²	1	3	yes	complemented
23	C₂₃×#121	C₂²	1	6	yes	complemented
24	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
25	C₄×D₄×C₂₃	C₂²	1	3	yes	complemented
26	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
27	#10625111	C₂²	1	6	yes	complemented
28	#10625111	C₂²	1	6	yes	complemented
29	#10625111	C₂²	1	6	yes	complemented
30	#10625110	C₂²	1	12	yes	complemented
31	#10625109	C₂²	1	6	yes	complemented
32	D₄×Dic₂₃	C₂²	1	3	yes	complemented
33	#10625101	C₂²	1	6	yes	complemented
34	C₄×C₂²⋊D₂₃	C₂²	1	6	yes	complemented
35	#10625066	C₂²	1	6	yes	complemented
36	#10625064	C₂²	1	3	yes	complemented
37	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₂₃	C₂²	1	3	yes	—
39	#10625058	C₂²	1	3	yes	complemented
40	#10625055	C₂²	1	12	yes	complemented
41	(C₂²⋊D₂₃)⋊C₄	C₂²	1	6	yes	complemented
42	C₂²⋊C₄×D₂₃	C₂²	1	6	yes	—
43	#10625045	C₂²	1	3	yes	complemented
44	C₄×D₉₂	C₂²	1	3	yes	complemented
45	C₄²×D₂₃	C₂²	1	3	yes	—
46	C₄×Dic₄₆	C₂²	1	3	yes	complemented
47	C₂³×D₂₃	C₂³	1	6	yes	—
48	#41×C₂₃	—	2	8	no	—
49	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₂₃	C₂³	1	3	yes	—
51	C₂×D₄×C₂₃	C₂³	1	6	yes	—
52	C₂×D₄×C₂₃	C₂³	1	6	yes	—
53	C₂²×C₄×C₂₃	C₂³	1	6	yes	—
54	C₂×C₂²⋊D₂₃	C₂³	1	6	yes	—
55	C₂×C₂²⋊D₂₃	C₂³	1	12	yes	—
56	C₂×C₂²⋊D₂₃	C₂³	1	12	yes	—
57	C₂²×Dic₂₃	C₂³	1	6	yes	—
58	#67292	—	2	8	no	—
59	Q₈×D₂₃	D₄	1	8	yes	complemented
60	#67290	—	2	24	no	—
61	D₄×D₂₃	—	2	24	no	—
62	D₄×D₂₃	—	2	12	no	—
63	D₄×D₂₃	—	4	12	no	—
64	#67288	—	2	24	no	—
65	C₂×D₉₂	C₂³	1	3	yes	—
66	C₂×D₉₂	C₂³	1	6	yes	—
67	C₂×D₉₂	C₂³	1	6	yes	—
68	C₂×C₄×D₂₃	—	2	24	no	—
69	C₂×C₄×D₂₃	C₂³	1	3	yes	—
70	C₂×C₄×D₂₃	C₂³	1	6	yes	—
71	C₂×C₄×D₂₃	C₂³	1	2	yes	—
72	C₂×C₄×D₂₃	C₂³	1	6	yes	—
73	C₂×C₄×D₂₃	C₂³	1	3	yes	—
74	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
75	C₂×Dic₄₆	C₂³	1	3	yes	—
76	C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
77	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
79	C₄²×C₂₃	C₂³	1	3	yes	—
80	C₂²⋊Dic₂₃	C₂³	1	6	yes	—
81	(C₂×D₂₃)⋊C₄	C₂³	1	6	yes	—
82	(C₂×D₂₃)⋊C₄	C₂³	1	6	yes	—
83	C₄⋊Dic₂₃	C₂³	1	3	yes	—
84	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
85	Dic₂₃⋊C₄	C₂³	1	6	yes	—
86	Dic₂₃⋊C₄	C₂³	1	2	yes	—
87	C₄×Dic₂₃	C₂³	1	3	yes	—
88	C₄×Dic₂₃	C₂³	1	3	yes	—
89	C₄×Dic₂₃	C₂³	1	3	yes	—
90	C₂³×C₂₃	C₂⁴	1	6	yes	—
91	C₂²×D₂₃	—	2	12	no	—
92	C₂²×D₂₃	—	2	12	no	—
93	C₂²×D₂₃	—	2	12	no	—
94	C₂²×D₂₃	C₂⁴	1	6	yes	—
95	C₂²×D₂₃	C₂⁴	1	1	yes	characteristic
96	Q₈×C₂₃	C₂×D₄	1	4	yes	complemented
97	D₄×C₂₃	—	2	12	no	—
98	D₄×C₂₃	—	2	6	no	—
99	D₄×C₂₃	—	4	6	no	—
100	C₂×C₄×C₂₃	—	2	12	no	—
101	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
102	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
103	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
104	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
105	C₂²⋊D₂₃	—	2	24	no	—
106	C₂²⋊D₂₃	—	2	12	no	—
107	C₂²⋊D₂₃	—	4	12	no	—
108	C₂×Dic₂₃	—	2	12	no	—
109	C₂×Dic₂₃	C₂⁴	1	3	yes	—
110	C₂×Dic₂₃	C₂⁴	1	2	yes	—
111	C₂×Dic₂₃	C₂⁴	1	3	yes	—
112	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
113	D₉₂	—	2	12	no	—
114	D₉₂	—	2	6	no	—
115	D₉₂	—	4	6	no	—
116	C₄×D₂₃	—	2	4	no	—
117	C₄×D₂₃	—	2	6	no	—
118	C₄×D₂₃	—	2	12	no	—
119	C₄×D₂₃	—	2	12	no	—
120	C₄×D₂₃	C₂×D₄	1	12	yes	—
121	C₄×D₂₃	#41	1	4	yes	complemented
122	Dic₄₆	C₂×D₄	1	12	yes	complemented
123	C₂×#550	—	23	1	no	maximal, Sylow
124	C₂²×C₂₃	—	2	6	no	—
125	C₂²×C₂₃	—	2	6	no	—
126	C₂²×C₂₃	—	2	6	no	—
127	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
128	C₂×D₂₃	—	2	6	no	—
129	C₂×D₂₃	—	2	6	no	—
130	C₂×D₂₃	—	2	6	no	—
131	C₂×D₂₃	—	4	24	no	—
132	C₂×D₂₃	C₂²×D₄	1	2	yes	—
133	C₂×D₂₃	C₂×#41	1	2	yes	—
134	C₂×D₂₃	H(5,𝔽₂)	1	2	yes	—
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	—	2	3	no	—
137	C₄×C₂₃	C₂²×D₄	1	6	yes	—
138	C₄×C₂₃	C₂×#41	1	2	yes	complemented
139	Dic₂₃	—	2	2	no	—
140	Dic₂₃	—	2	3	no	—
141	Dic₂₃	C₂²×D₄	1	6	yes	—
142	Dic₂₃	C₂×#41	1	2	yes	complemented
143	C₂²×#41	—	23	2	no	—
144	#550	—	23	2	no	—
145	#550	—	23	3	no	—
146	#550	—	23	6	no	—
147	#550	—	23	1	no	—
148	#550	—	23	3	no	—
149	#550	D₂₃	1	1	yes	characteristic, direct product factor
150	C₂×#127	—	23	3	no	—
151	C₂×#121	—	23	6	no	—
152	C₂×C₄×Q₈	—	23	1	no	—
153	C₂×C₄×D₄	—	23	3	no	—
154	C₂×C₂₃	—	4	6	no	—
155	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
156	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
157	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
158	D₂₃	—	4	6	no	—
159	D₂₃	#550	1	4	yes	direct product factor
160	C₂×#41	—	23	2	no	—
161	C₂×#41	—	23	6	no	—
162	C₂×#41	—	23	6	no	—
163	C₂×#41	—	46	16	no	—
164	C₂×#41	C₂×D₂₃	1	2	yes	complemented
165	C₂²×Q₈	—	23	1	no	—
166	C₂²×D₄	—	23	3	no	—
167	C₂²×D₄	—	23	6	no	—
168	C₂²×D₄	—	23	6	no	—
169	C₂³×C₄	—	23	6	no	—
170	#127	—	23	6	no	—
171	#127	—	23	6	no	—
172	#127	—	23	6	no	—
173	#127	—	23	3	no	—
174	#127	C₂×D₂₃	1	3	yes	complemented
175	#121	—	23	6	no	—
176	#121	—	23	12	no	—
177	#121	—	23	6	no	—
178	#121	—	23	6	no	—
179	#121	—	23	12	no	—
180	#121	C₂×D₂₃	1	6	yes	complemented
181	C₄×Q₈	—	23	1	no	—
182	C₄×Q₈	—	23	3	no	—
183	C₄×Q₈	—	23	3	no	—
184	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
185	C₄×D₄	—	23	3	no	—
186	C₄×D₄	—	23	6	no	—
187	C₄×D₄	—	23	3	no	—
188	C₄×D₄	—	23	6	no	—
189	C₄×D₄	—	23	3	no	—
190	C₄×D₄	C₂×D₂₃	1	3	yes	complemented
191	C₂×C₄⋊C₄	—	23	1	no	—
192	C₂×C₄⋊C₄	—	23	3	no	—
193	C₂×C₂²⋊C₄	—	23	6	no	—
194	C₂×C₄²	—	23	3	no	—
195	C₂₃	C₂×#550	1	1	yes	characteristic, complemented, Sylow
196	C₂⁴	—	23	6	no	—
197	#41	—	2	8	no	—
198	#41	—	46	24	no	—
199	#41	—	46	24	no	—
200	#41	—	46	8	no	—
201	C₂×Q₈	—	23	8	no	—
202	C₂×Q₈	—	23	3	no	—
203	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
204	C₂×D₄	—	23	6	no	—
205	C₂×D₄	—	23	12	no	—
206	C₂×D₄	—	23	12	no	—
207	C₂×D₄	—	23	3	no	—
208	C₂×D₄	—	23	6	no	—
209	C₂×D₄	—	23	6	no	—
210	C₂×D₄	—	46	24	no	—
211	C₂×D₄	—	46	12	no	—
212	C₂×D₄	—	92	12	no	—
213	C₂×D₄	C₂²×D₂₃	1	3	yes	—
214	C₂×D₄	C₂²×D₂₃	1	6	yes	—
215	C₂×D₄	C₂²×D₂₃	1	6	yes	—
216	C₂²×C₄	—	23	1	no	—
217	C₂²×C₄	—	23	3	no	—
218	C₂²×C₄	—	23	6	no	—
219	C₂²×C₄	—	23	3	no	—
220	C₂²×C₄	—	23	2	no	—
221	C₂²×C₄	—	23	6	no	—
222	C₂²×C₄	—	23	6	no	—
223	C₂²×C₄	—	46	24	no	—
224	C₂²×C₄	C₂²×D₂₃	1	6	yes	—
225	C₄⋊C₄	—	23	2	no	—
226	C₄⋊C₄	—	23	1	no	—
227	C₄⋊C₄	—	23	6	no	—
228	C₄⋊C₄	—	23	3	no	—
229	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
230	C₄⋊C₄	C₂²×D₂₃	1	3	yes	—
231	C₂²⋊C₄	—	23	6	no	—
232	C₂²⋊C₄	—	23	6	no	—
233	C₂²⋊C₄	—	23	6	no	—
234	C₂²⋊C₄	C₂²×D₂₃	1	6	yes	—
235	C₄²	—	23	3	no	—
236	C₄²	—	23	3	no	—
237	C₄²	—	23	3	no	—
238	C₄²	C₂²×D₂₃	1	3	yes	—
239	C₂³	—	23	1	no	—
240	C₂³	—	23	6	no	—
241	C₂³	—	46	12	no	—
242	C₂³	—	46	12	no	—
243	C₂³	—	46	12	no	—
244	C₂³	C₂³×D₂₃	1	6	yes	—
245	Q₈	—	23	12	no	—
246	Q₈	D₄×D₂₃	1	4	yes	complemented
247	D₄	—	2	12	no	—
248	D₄	—	2	6	no	—
249	D₄	—	4	6	no	—
250	D₄	—	46	24	no	—
251	D₄	—	46	12	no	—
252	D₄	—	46	12	no	—
253	D₄	—	46	6	no	—
254	D₄	—	92	6	no	—
255	D₄	—	92	12	no	—
256	C₂×C₄	—	2	12	no	—
257	C₂×C₄	—	23	1	no	—
258	C₂×C₄	—	23	3	no	—
259	C₂×C₄	—	23	12	no	—
260	C₂×C₄	—	23	4	no	—
261	C₂×C₄	—	23	3	no	—
262	C₂×C₄	—	23	2	no	—
263	C₂×C₄	—	46	12	no	—
264	C₂×C₄	—	46	4	no	—
265	C₂×C₄	—	46	6	no	—
266	C₂×C₄	—	46	12	no	—
267	C₂×C₄	—	46	12	no	—
268	C₂×C₄	C₂³×D₂₃	1	3	yes	—
269	C₂×C₄	C₂³×D₂₃	1	2	yes	—
270	C₂×C₄	C₂³×D₂₃	1	3	yes	—
271	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
272	C₂²	—	2	6	no	—
273	C₂²	—	2	6	no	—
274	C₂²	—	2	6	no	—
275	C₂²	—	23	2	no	—
276	C₂²	—	23	2	no	—
277	C₂²	—	23	2	no	—
278	C₂²	—	46	6	no	—
279	C₂²	—	46	6	no	—
280	C₂²	—	46	6	no	—
281	C₂²	—	92	24	no	—
282	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	3	no	—
285	C₄	—	23	6	no	—
286	C₄	—	23	2	no	—
287	C₄	—	46	2	no	—
288	C₄	—	46	3	no	—
289	C₄	C₂×D₄×D₂₃	1	6	yes	—
290	C₄	#41×D₂₃	1	2	yes	complemented
291	C₂	—	4	6	no	—
292	C₂	—	23	4	no	—
293	C₂	—	92	6	no	—
294	C₂	C₂²×D₄×D₂₃	1	1	yes	characteristic
295	C₂	C₂×#41×D₂₃	1	1	yes	characteristic
296	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×#550	C₂
hypercenter	#550	D₂₃
hypocenter	C₂₃	C₂×#550
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #550

Complements

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

N	H ≅ G/N	type
1. C₂×#41×D₂₃	C₂	semidirect
2. #13093089	C₂	semidirect
3. #13093088	C₂	semidirect
4. #13093087	C₂	semidirect
5. #13093086	C₂	semidirect
6. #13093085	C₂	semidirect
7. #127×D₂₃	C₂	semidirect
8. #121×D₂₃	C₂	semidirect
9. C₄×Q₈×D₂₃	C₂	semidirect
10. C₄×D₄×D₂₃	C₂	semidirect
11. C₂₃×#550	C₂	semidirect
12. C₂×#41×C₂₃	C₂²	semidirect
14. C₂×#67292	C₂²	semidirect
16. C₂×#67290	C₂²	semidirect
20. C₂×#67288	C₂²	semidirect
22. C₂₃×#127	C₂²	semidirect
23. C₂₃×#121	C₂²	semidirect
24. C₄×Q₈×C₂₃	C₂²	semidirect
25. C₄×D₄×C₂₃	C₂²	semidirect
26. Q₈×Dic₂₃	C₂²	semidirect
27. #10625111	C₂²	semidirect
28. #10625111	C₂²	semidirect
29. #10625111	C₂²	semidirect
30. #10625110	C₂²	semidirect
31. #10625109	C₂²	semidirect
32. D₄×Dic₂₃	C₂²	semidirect
33. #10625101	C₂²	semidirect
34. C₄×C₂²⋊D₂₃	C₂²	semidirect
35. #10625066	C₂²	semidirect
36. #10625064	C₂²	semidirect
39. #10625058	C₂²	semidirect
40. #10625055	C₂²	semidirect
41. (C₂²⋊D₂₃)⋊C₄	C₂²	semidirect
43. #10625045	C₂²	semidirect
44. C₄×D₉₂	C₂²	semidirect
46. C₄×Dic₄₆	C₂²	semidirect
59. Q₈×D₂₃	D₄	semidirect
96. Q₈×C₂₃	C₂×D₄	semidirect
121. C₄×D₂₃	#41	semidirect
122. Dic₄₆	C₂×D₄	semidirect
138. C₄×C₂₃	C₂×#41	semidirect
142. Dic₂₃	C₂×#41	semidirect
149. #550	D₂₃	direct (N × H)
159. D₂₃	#550	direct (N × H)
164. C₂×#41	C₂×D₂₃	semidirect
174. #127	C₂×D₂₃	semidirect
180. #121	C₂×D₂₃	semidirect
184. C₄×Q₈	C₂×D₂₃	semidirect
190. C₄×D₄	C₂×D₂₃	semidirect
195. C₂₃	C₂×#550	semidirect
246. Q₈	D₄×D₂₃	semidirect
290. C₄	#41×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×D₂₃	2
#13093089	2
#13093088	2
#13093087	2
#13093086	2
#13093085	2
#127×D₂₃	2
#121×D₂₃	2
C₄×Q₈×D₂₃	2
C₄×D₄×D₂₃	2
C₂₃×#550	2
C₂×#550	23

Subgroups — D23×#550 #15686873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₃×#550 #15686873