Subgroups — D₂₃×#476 #15683144

Order 2944.

480 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	C₄⋊Q₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₂₃	C₂	1	1	yes	maximal, characteristic
3	#13092708	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092707	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092706	C₂	1	2	yes	maximal, complemented
6	#13092705	C₂	1	2	yes	maximal, complemented
7	#13092704	C₂	1	1	yes	maximal, characteristic, complemented
8	#106×D₂₃	C₂	1	2	yes	maximal, complemented
9	C₄⋊C₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#102×D₂₃	C₂	1	2	yes	maximal
11	C₂₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₂₃	—	2	1	no	—
13	C₂×D₄×D₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₂₃	—	2	1	no	—
15	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
16	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
17	#10625115	—	2	1	no	—
18	D₄×Dic₂₃	C₂²	1	1	yes	characteristic
19	C₄×C₂²⋊D₂₃	—	2	1	no	—
20	C₂×C₈×D₂₃	C₂²	1	2	yes	—
21	#10625064	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×D₂₃	—	2	1	no	—
23	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₄×D₂₃	C₂²	1	2	yes	—
25	#10625059	—	2	1	no	—
26	#10625059	C₂²	1	2	yes	complemented
27	(C₂²⋊D₂₃)⋊C₄	—	2	1	no	—
28	C₂²⋊C₄×D₂₃	—	2	1	no	—
29	C₄×D₉₂	C₂²	1	1	yes	characteristic
30	C₄²×D₂₃	C₂²	1	1	yes	characteristic
31	#10625040	C₂²	1	1	yes	characteristic, complemented
32	C₂₃×#106	C₂²	1	2	yes	complemented
33	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
34	C₂₃×#102	C₂²	1	2	yes	—
35	#10625014	C₂²	1	2	yes	—
36	#10625003	C₂²	1	2	yes	—
37	#10624999	C₂²	1	2	yes	complemented
38	Dic₂₃⋊C₈	C₂²	1	2	yes	complemented
39	#10624991	C₂²	1	2	yes	—
40	#10624990	C₂²	1	2	yes	complemented
41	#10624990	C₂²	1	2	yes	complemented
42	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
43	C₂³×D₂₃	—	2	1	no	—
44	C₂×Q₈×C₂₃	—	2	1	no	—
45	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	—	2	1	no	—
47	C₂×C₂²⋊D₂₃	—	2	1	no	—
48	C₂²×Dic₂₃	—	2	1	no	—
49	Q₈×D₂₃	—	4	4	no	—
50	D₄×D₂₃	—	2	2	no	—
51	D₄×D₂₃	Q₈	1	4	yes	complemented
52	C₂×D₉₂	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₂₃	—	2	1	no	—
54	C₂×C₄×D₂₃	—	2	2	no	—
55	C₂×C₄×D₂₃	—	2	1	no	—
56	C₂×C₄×D₂₃	—	2	1	no	—
57	C₂×C₄×D₂₃	—	4	4	no	—
58	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
59	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
60	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
61	C₂×Dic₄₆	—	2	1	no	—
62	C₂×Dic₄₆	—	2	2	no	—
63	C₂×C₈×C₂₃	C₂³	1	2	yes	—
64	C₄⋊C₄×C₂₃	—	2	1	no	—
65	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
66	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²⋊C₄×C₂₃	—	2	1	no	—
68	C₄²×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²⋊Dic₂₃	—	2	1	no	—
70	(C₂×D₂₃)⋊C₄	—	2	1	no	—
71	(C₂×D₂₃)⋊C₄	—	2	1	no	—
72	C₄⋊Dic₂₃	—	2	1	no	—
73	C₄⋊Dic₂₃	C₂³	1	2	yes	—
74	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
75	Dic₂₃⋊C₄	—	2	1	no	—
76	Dic₂₃⋊C₄	—	2	2	no	—
77	Dic₂₃⋊C₄	—	2	2	no	—
78	C₄×Dic₂₃	D₄	1	1	yes	characteristic
79	C₄×Dic₂₃	D₄	1	1	yes	characteristic
80	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
81	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
82	C₈×D₂₃	—	2	4	no	—
83	C₂³×C₂₃	—	2	1	no	—
84	C₂²×D₂₃	—	2	1	no	—
85	C₂²×D₂₃	—	2	4	no	—
86	C₂²×D₂₃	—	4	2	no	—
87	C₂²×D₂₃	—	4	2	no	—
88	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₂₃	—	4	2	no	—
90	D₄×C₂₃	—	2	1	no	—
91	D₄×C₂₃	C₂×Q₈	1	2	yes	complemented
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	4	2	no	—
95	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₂₃	—	2	4	no	—
99	C₂×Dic₂₃	—	2	2	no	—
100	C₂×Dic₂₃	—	2	1	no	—
101	C₂×Dic₂₃	—	4	2	no	—
102	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
105	D₉₂	—	2	1	no	—
106	D₉₂	C₂×Q₈	1	2	yes	complemented
107	C₄×D₂₃	—	2	2	no	—
108	C₄×D₂₃	—	4	2	no	—
109	C₄×D₂₃	—	4	4	no	—
110	C₄×D₂₃	—	4	2	no	—
111	C₄×D₂₃	—	4	2	no	—
112	C₄×D₂₃	SD₁₆	1	4	yes	—
113	C₄×D₂₃	C₂×Q₈	1	2	yes	—
114	C₄×D₂₃	#41	1	2	yes	—
115	Dic₄₆	—	4	4	no	—
116	Dic₄₆	—	4	2	no	—
117	C₈×C₂₃	—	2	2	no	—
118	C₂₃⋊C₈	—	2	2	no	—
119	C₂×#476	—	23	1	no	maximal, Sylow
120	C₂²×C₂₃	—	2	2	no	—
121	C₂²×C₂₃	—	4	1	no	—
122	C₂²×C₂₃	—	4	1	no	—
123	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×D₂₃	—	2	2	no	—
125	C₂×D₂₃	—	4	1	no	—
126	C₂×D₂₃	—	4	1	no	—
127	C₂×D₂₃	—	4	8	no	—
128	C₂×D₂₃	C₂²⋊Q₈	1	2	yes	—
129	C₂×D₂₃	C₂×SD₁₆	1	2	yes	—
130	C₂×D₂₃	AGL(1,ℤ/8)	1	2	yes	—
131	C₄×C₂₃	—	2	1	no	—
132	C₄×C₂₃	—	4	1	no	—
133	C₄×C₂₃	—	4	2	no	—
134	C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
135	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
136	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
137	Dic₂₃	—	2	1	no	—
138	Dic₂₃	—	4	1	no	—
139	Dic₂₃	—	4	2	no	—
140	Dic₂₃	C₂²⋊Q₈	1	2	yes	—
141	Dic₂₃	C₂×SD₁₆	1	2	yes	—
142	C₂×C₄⋊Q₈	—	23	1	no	—
143	C₂×C₄×D₄	—	23	1	no	—
144	#476	—	23	1	no	—
145	#476	—	23	1	no	—
146	#476	—	23	2	no	—
147	#476	—	23	1	no	—
148	#476	—	23	2	no	—
149	#476	D₂₃	1	1	yes	characteristic, direct product factor
150	C₂×#106	—	23	2	no	—
151	C₂×C₄⋊C₈	—	23	1	no	—
152	C₂×#102	—	23	2	no	—
153	C₂×C₂₃	—	4	2	no	—
154	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
155	C₂×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
156	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
157	D₂₃	—	4	2	no	—
158	D₂₃	#476	1	4	yes	direct product factor
159	C₂²×Q₈	—	46	1	no	—
160	C₂²×D₄	—	23	1	no	—
161	C₂³×C₄	—	46	1	no	—
162	C₂²×C₈	—	23	2	no	—
163	C₄⋊Q₈	—	23	2	no	—
164	C₄⋊Q₈	—	23	1	no	—
165	C₄⋊Q₈	—	46	1	no	—
166	C₄⋊Q₈	—	46	1	no	—
167	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
168	C₄×D₄	—	23	1	no	—
169	C₄×D₄	—	23	1	no	—
170	C₄×D₄	—	23	1	no	—
171	C₄×D₄	—	46	1	no	—
172	C₄×D₄	—	46	1	no	—
173	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic
174	C₂×C₄⋊C₄	—	23	1	no	—
175	C₂×C₄⋊C₄	—	23	2	no	—
176	C₂×C₄⋊C₄	—	46	1	no	—
177	C₂×C₂²⋊C₄	—	46	1	no	—
178	C₂×C₄²	—	23	1	no	—
179	#106	—	23	2	no	—
180	#106	—	23	2	no	—
181	#106	—	23	2	no	—
182	#106	C₂×D₂₃	1	2	yes	complemented
183	C₄⋊C₈	—	23	2	no	—
184	C₄⋊C₈	—	23	1	no	—
185	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic, complemented
186	#102	—	23	2	no	—
187	#102	—	23	2	no	—
188	#102	—	23	2	no	—
189	#102	C₂×D₂₃	1	2	yes	—
190	C₂₃	C₂×#476	1	1	yes	characteristic, complemented, Sylow
191	C₂⁴	—	46	1	no	—
192	C₂×Q₈	—	2	1	no	—
193	C₂×Q₈	—	46	2	no	—
194	C₂×Q₈	—	46	1	no	—
195	C₂×Q₈	—	92	4	no	—
196	C₂×D₄	—	23	4	no	—
197	C₂×D₄	—	23	1	no	—
198	C₂×D₄	—	46	2	no	—
199	C₂×D₄	—	46	1	no	—
200	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	23	1	no	—
203	C₂²×C₄	—	23	1	no	—
204	C₂²×C₄	—	23	1	no	—
205	C₂²×C₄	—	46	1	no	—
206	C₂²×C₄	—	46	1	no	—
207	C₂²×C₄	—	46	2	no	—
208	C₂²×C₄	—	46	1	no	—
209	C₂²×C₄	—	46	1	no	—
210	C₂²×C₄	—	92	4	no	—
211	C₂×C₈	—	23	2	no	—
212	C₂×C₈	—	46	4	no	—
213	C₂×C₈	C₂²×D₂₃	1	2	yes	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	—	23	2	no	—
216	C₄⋊C₄	—	23	1	no	—
217	C₄⋊C₄	—	46	1	no	—
218	C₄⋊C₄	—	46	2	no	—
219	C₄⋊C₄	—	46	2	no	—
220	C₄⋊C₄	—	46	1	no	—
221	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
222	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
223	C₂²⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	46	1	no	—
225	C₂²⋊C₄	—	46	1	no	—
226	C₂²⋊C₄	—	46	1	no	—
227	C₄²	—	23	1	no	—
228	C₄²	—	23	1	no	—
229	C₄²	—	23	1	no	—
230	C₄²	C₂²×D₂₃	1	1	yes	characteristic
231	C₂³	—	2	1	no	—
232	C₂³	—	23	1	no	—
233	C₂³	—	46	4	no	—
234	C₂³	—	46	1	no	—
235	C₂³	—	92	2	no	—
236	C₂³	—	92	2	no	—
237	Q₈	—	4	2	no	—
238	Q₈	—	92	2	no	—
239	Q₈	—	92	4	no	—
240	D₄	—	2	1	no	—
241	D₄	—	23	2	no	—
242	D₄	—	46	4	no	—
243	D₄	—	46	1	no	—
244	D₄	Q₈×D₂₃	1	2	yes	complemented
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	23	1	no	—
249	C₂×C₄	—	23	1	no	—
250	C₂×C₄	—	23	4	no	—
251	C₂×C₄	—	23	2	no	—
252	C₂×C₄	—	23	2	no	—
253	C₂×C₄	—	23	1	no	—
254	C₂×C₄	—	46	2	no	—
255	C₂×C₄	—	46	1	no	—
256	C₂×C₄	—	46	2	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	—	92	2	no	—
262	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
263	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
264	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	46	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	23	2	no	—
271	C₂²	—	23	2	no	—
272	C₂²	—	23	2	no	—
273	C₂²	—	46	2	no	—
274	C₂²	—	92	8	no	—
275	C₂²	—	92	1	no	—
276	C₂²	—	92	1	no	—
277	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	2	no	—
281	C₄	—	23	2	no	—
282	C₄	—	23	2	no	—
283	C₄	—	46	1	no	—
284	C₄	—	92	1	no	—
285	C₄	—	92	2	no	—
286	C₄	SD₁₆×D₂₃	1	2	yes	—
287	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
288	C₄	#41×D₂₃	1	1	yes	characteristic
289	C₂	—	4	2	no	—
290	C₂	—	23	4	no	—
291	C₂	—	92	2	no	—
292	C₂	C₂²⋊Q₈×D₂₃	1	1	yes	characteristic
293	C₂	C₂×SD₁₆×D₂₃	1	1	yes	characteristic
294	C₂	AGL(1,ℤ/8)×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#476	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×D₄×D₂₃	C₂²
hypercenter	#476	D₂₃
hypocenter	C₂₃	C₂×#476
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

Complements

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

N	H ≅ G/N	type
1. C₄⋊Q₈×D₂₃	C₂	semidirect
3. #13092708	C₂	semidirect
4. #13092707	C₂	semidirect
5. #13092706	C₂	semidirect
6. #13092705	C₂	semidirect
7. #13092704	C₂	semidirect
8. #106×D₂₃	C₂	semidirect
9. C₄⋊C₈×D₂₃	C₂	semidirect
11. C₂₃×#476	C₂	semidirect
15. C₂₃×C₄⋊Q₈	C₂²	semidirect
26. #10625059	C₂²	semidirect
31. #10625040	C₂²	semidirect
32. C₂₃×#106	C₂²	semidirect
33. C₂₃×C₄⋊C₈	C₂²	semidirect
37. #10624999	C₂²	semidirect
38. Dic₂₃⋊C₈	C₂²	semidirect
40. #10624990	C₂²	semidirect
41. #10624990	C₂²	semidirect
42. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
51. D₄×D₂₃	Q₈	semidirect
91. D₄×C₂₃	C₂×Q₈	semidirect
106. D₉₂	C₂×Q₈	semidirect
149. #476	D₂₃	direct (N × H)
158. D₂₃	#476	direct (N × H)
167. C₄⋊Q₈	C₂×D₂₃	semidirect
182. #106	C₂×D₂₃	semidirect
185. C₄⋊C₈	C₂×D₂₃	semidirect
190. C₂₃	C₂×#476	semidirect
244. D₄	Q₈×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×D₂₃	2
C₄×D₄×D₂₃	2
#13092708	2
#13092707	2
#13092706	2
#13092705	2
#13092704	2
#106×D₂₃	2
C₄⋊C₈×D₂₃	2
#102×D₂₃	2
C₂₃×#476	2
C₂×#476	23

Subgroups — D23×#476 #15683144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₃×#476 #15683144