Subgroups — C₈×D₄×D₂₃ #15682066

Order 2944.

616 conjugacy classes of subgroups.

The subgroup lattice has 284 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	2	yes	maximal, complemented
2	C₄×D₄×D₂₃	C₂	1	1	yes	maximal, characteristic
3	C₈×D₉₂	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092482	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₂²⋊D₂₃	C₂	1	2	yes	maximal, complemented
7	(C₂²⋊D₂₃)⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₄⋊C₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²⋊C₈×D₂₃	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₂₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₃	C₂²	1	2	yes	complemented
15	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
16	D₄×Dic₂₃	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₂₃	C₂²	1	2	yes	—
18	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	complemented
19	C₂×C₈×D₂₃	—	2	8	no	—
20	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₂₃	C₂²	1	2	yes	—
22	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₂₃	C₂²	1	2	yes	complemented
24	C₂×C₈×D₂₃	C₂²	1	2	yes	complemented
25	#10625064	C₄	1	1	yes	characteristic
26	C₄⋊C₄×D₂₃	C₄	1	1	yes	characteristic
27	(C₂²⋊D₂₃)⋊C₄	C₄	1	2	yes	—
28	C₂²⋊C₄×D₂₃	C₄	1	2	yes	—
29	C₄×D₉₂	C₂²	1	1	yes	characteristic
30	C₄²×D₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
32	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
33	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
34	C₂²⋊(C₂₃⋊C₈)	C₂²	1	2	yes	complemented
35	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	complemented
36	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	complemented
37	Dic₂₃⋊C₈	C₂²	1	2	yes	complemented
38	C₈×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
39	C₈×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
40	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
41	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	C₂³×D₂₃	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
45	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
46	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
47	D₄×D₂₃	C₈	1	8	yes	direct product factor
48	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₂₃	—	2	4	no	—
50	C₂×C₄×D₂₃	—	2	4	no	—
51	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
54	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₂₃	C₂³	1	2	yes	—
56	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₂₃	—	2	4	no	—
58	C₂×C₈×C₂₃	C₂³	1	2	yes	—
59	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
62	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
63	C₄²×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
65	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
66	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
67	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
68	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
69	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
72	C₂×C₂₃⋊C₈	—	2	4	no	—
73	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
74	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₈×D₂₃	—	2	4	no	—
77	C₈×D₂₃	—	2	2	no	—
78	C₈×D₂₃	—	2	4	no	—
79	C₈×D₂₃	—	2	4	no	—
80	C₈×D₂₃	D₄	1	4	yes	direct product factor
81	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
82	C₂²×D₂₃	—	2	4	no	—
83	C₂²×D₂₃	—	2	4	no	—
84	C₂²×D₂₃	C₂×C₈	1	8	yes	complemented
85	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
86	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
87	D₄×C₂₃	C₂×C₈	1	4	yes	complemented
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₂₃	C₂×C₈	1	8	yes	complemented
95	C₂×Dic₂₃	—	2	4	no	—
96	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
98	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
100	D₉₂	C₂×C₈	1	4	yes	complemented
101	C₄×D₂₃	—	2	4	no	—
102	C₄×D₂₃	—	2	2	no	—
103	C₄×D₂₃	—	2	2	no	—
104	C₄×D₂₃	—	2	2	no	—
105	C₄×D₂₃	—	2	4	no	—
106	C₄×D₂₃	C₂×C₈	1	4	yes	complemented
107	C₄×D₂₃	C₂×D₄	1	2	yes	—
108	C₄×D₂₃	#41	1	2	yes	—
109	C₈×C₂₃	—	2	2	no	—
110	C₈×C₂₃	—	2	1	no	—
111	C₈×C₂₃	C₂×D₄	1	2	yes	complemented
112	C₂₃⋊C₈	—	2	2	no	—
113	C₂₃⋊C₈	—	2	1	no	—
114	C₂₃⋊C₈	C₂×D₄	1	2	yes	complemented
115	C₂×C₈×D₄	—	23	1	no	maximal, Sylow
116	C₂²×C₂₃	—	2	2	no	—
117	C₂²×C₂₃	—	2	2	no	—
118	C₂²×C₂₃	C₂²×C₈	1	4	yes	complemented
119	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×D₂₃	—	2	2	no	—
121	C₂×D₂₃	—	2	2	no	—
122	C₂×D₂₃	—	2	16	no	—
123	C₂×D₂₃	C₄×D₄	1	2	yes	—
124	C₂×D₂₃	C₂²×C₈	1	2	yes	—
125	C₂×D₂₃	C₂²×C₈	1	4	yes	complemented
126	C₂×D₂₃	#131	1	2	yes	—
127	C₄×C₂₃	—	2	2	no	—
128	C₄×C₂₃	—	2	1	no	—
129	C₄×C₂₃	C₂²×C₈	1	2	yes	complemented
130	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
132	Dic₂₃	—	2	2	no	—
133	Dic₂₃	—	2	1	no	—
134	Dic₂₃	C₄×D₄	1	2	yes	—
135	Dic₂₃	C₂²×C₈	1	2	yes	complemented
136	C₂³×C₈	—	23	2	no	—
137	C₂×C₄×D₄	—	23	1	no	—
138	C₈×D₄	—	23	2	no	—
139	C₈×D₄	—	23	1	no	—
140	C₈×D₄	—	23	1	no	—
141	C₈×D₄	—	23	2	no	—
142	C₈×D₄	—	23	1	no	—
143	C₈×D₄	D₂₃	1	1	yes	characteristic, direct product factor
144	C₂×C₄⋊C₈	—	23	1	no	—
145	C₂×C₂²⋊C₈	—	23	2	no	—
146	C₂×C₄×C₈	—	23	1	no	—
147	C₂×C₂₃	—	2	4	no	—
148	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×C₂₃	C₂³×C₈	1	1	yes	characteristic
150	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
151	D₂₃	—	2	4	no	—
152	D₂₃	C₈×D₄	1	4	yes	direct product factor
153	C₂²×D₄	—	23	1	no	—
154	C₂³×C₄	—	23	2	no	—
155	C₂²×C₈	—	23	1	no	—
156	C₂²×C₈	—	23	2	no	—
157	C₂²×C₈	—	23	2	no	—
158	C₂²×C₈	—	23	1	no	—
159	C₂²×C₈	—	23	2	no	—
160	C₂²×C₈	—	23	2	no	—
161	C₂²×C₈	—	46	8	no	—
162	C₂²×C₈	C₂×D₂₃	1	2	yes	complemented
163	C₄×D₄	—	23	1	no	—
164	C₄×D₄	—	23	2	no	—
165	C₄×D₄	—	23	1	no	—
166	C₄×D₄	—	23	2	no	—
167	C₄×D₄	—	23	1	no	—
168	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic
169	C₂×C₄⋊C₄	—	23	1	no	—
170	C₂×C₂²⋊C₄	—	23	2	no	—
171	C₂×C₄²	—	23	1	no	—
172	C₄⋊C₈	—	23	2	no	—
173	C₄⋊C₈	—	23	1	no	—
174	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic, complemented
175	C₂²⋊C₈	—	23	2	no	—
176	C₂²⋊C₈	—	23	2	no	—
177	C₂²⋊C₈	—	23	2	no	—
178	C₂²⋊C₈	C₂×D₂₃	1	2	yes	complemented
179	C₄×C₈	—	23	1	no	—
180	C₄×C₈	—	23	1	no	—
181	C₄×C₈	—	23	1	no	—
182	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic, complemented
183	C₂₃	C₂×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
184	C₂⁴	—	23	2	no	—
185	C₂×D₄	—	23	2	no	—
186	C₂×D₄	—	23	8	no	—
187	C₂×D₄	—	23	1	no	—
188	C₂×D₄	C₄×D₂₃	1	1	yes	characteristic
189	C₂²×C₄	—	23	1	no	—
190	C₂²×C₄	—	23	1	no	—
191	C₂²×C₄	—	23	2	no	—
192	C₂²×C₄	—	23	1	no	—
193	C₂²×C₄	—	23	2	no	—
194	C₂²×C₄	—	23	2	no	—
195	C₂²×C₄	—	23	2	no	—
196	C₂²×C₄	—	46	4	no	—
197	C₂²×C₄	—	46	4	no	—
198	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	23	1	no	—
201	C₂×C₈	—	23	4	no	—
202	C₂×C₈	—	23	1	no	—
203	C₂×C₈	—	23	2	no	—
204	C₂×C₈	—	46	4	no	—
205	C₂×C₈	—	46	4	no	—
206	C₂×C₈	—	46	2	no	—
207	C₂×C₈	—	46	4	no	—
208	C₂×C₈	—	46	4	no	—
209	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₂₃	1	2	yes	—
211	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
212	C₄⋊C₄	—	23	2	no	—
213	C₄⋊C₄	—	23	1	no	—
214	C₄⋊C₄	C₄×D₂₃	1	1	yes	characteristic
215	C₂²⋊C₄	—	23	2	no	—
216	C₂²⋊C₄	—	23	2	no	—
217	C₂²⋊C₄	—	23	2	no	—
218	C₂²⋊C₄	C₄×D₂₃	1	2	yes	—
219	C₄²	—	23	1	no	—
220	C₄²	—	23	1	no	—
221	C₄²	—	23	1	no	—
222	C₄²	C₂²×D₂₃	1	1	yes	characteristic
223	C₂³	—	23	1	no	—
224	C₂³	—	23	8	no	—
225	C₂³	—	23	2	no	—
226	C₂³	—	46	4	no	—
227	C₂³	—	46	4	no	—
228	C₂³	C₂×C₄×D₂₃	1	2	yes	—
229	D₄	—	23	8	no	—
230	D₄	—	23	4	no	—
231	D₄	C₈×D₂₃	1	4	yes	direct product factor
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	23	1	no	—
235	C₂×C₄	—	23	1	no	—
236	C₂×C₄	—	23	4	no	—
237	C₂×C₄	—	23	2	no	—
238	C₂×C₄	—	23	2	no	—
239	C₂×C₄	—	23	1	no	—
240	C₂×C₄	—	23	2	no	—
241	C₂×C₄	—	46	4	no	—
242	C₂×C₄	—	46	4	no	—
243	C₂×C₄	—	46	2	no	—
244	C₂×C₄	—	46	4	no	—
245	C₂×C₄	—	46	2	no	—
246	C₂×C₄	—	46	2	no	—
247	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
249	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
250	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	1	no	—
253	C₈	—	23	2	no	—
254	C₈	—	46	2	no	—
255	C₈	—	46	1	no	—
256	C₈	D₄×D₂₃	1	2	yes	direct product factor
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	23	2	no	—
260	C₂²	—	23	2	no	—
261	C₂²	—	23	2	no	—
262	C₂²	—	23	4	no	—
263	C₂²	—	46	16	no	—
264	C₂²	—	46	2	no	—
265	C₂²	—	46	2	no	—
266	C₂²	C₂×C₈×D₂₃	1	4	yes	complemented
267	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	23	2	no	—
271	C₄	—	23	2	no	—
272	C₄	—	46	2	no	—
273	C₄	—	46	1	no	—
274	C₄	C₂×C₈×D₂₃	1	2	yes	complemented
275	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
276	C₄	#41×D₂₃	1	1	yes	characteristic
277	C₂	—	2	4	no	—
278	C₂	—	23	4	no	—
279	C₂	—	46	4	no	—
280	C₂	C₄×D₄×D₂₃	1	1	yes	characteristic
281	C₂	C₂²×C₈×D₂₃	1	1	yes	characteristic
282	C₂	#131×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₂₃
commutator	C₂×C₂₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₈×D₄×C₂₃	C₂
socle	C₂²×C₂₃	C₂³×C₄
prime radical	C₂×D₄×D₂₃	C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₈×D₄	D₂₃
hypocenter	C₂₃	C₂×C₈×D₄
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₈×D₂₃	C₂	semidirect
3. C₈×D₉₂	C₂	semidirect
4. D₄×C₂₃⋊C₈	C₂	semidirect
5. #13092482	C₂	semidirect
6. C₈×C₂²⋊D₂₃	C₂	semidirect
7. (C₂²⋊D₂₃)⋊C₈	C₂	semidirect
8. C₄⋊C₈×D₂₃	C₂	semidirect
9. C₂²⋊C₈×D₂₃	C₂	semidirect
10. C₄×C₈×D₂₃	C₂	semidirect
11. C₈×D₄×C₂₃	C₂	semidirect
14. C₂²×C₈×C₂₃	C₂²	semidirect
18. C₂²×C₂₃⋊C₈	C₂²	semidirect
23. C₂×C₈×D₂₃	C₂²	semidirect
24. C₂×C₈×D₂₃	C₂²	semidirect
31. C₂₃×C₄⋊C₈	C₂²	semidirect
32. C₂₃×C₂²⋊C₈	C₂²	semidirect
33. C₄×C₈×C₂₃	C₂²	semidirect
34. C₂²⋊(C₂₃⋊C₈)	C₂²	semidirect
35. (C₂×D₂₃)⋊C₈	C₂²	semidirect
36. (C₂×D₂₃)⋊C₈	C₂²	semidirect
37. Dic₂₃⋊C₈	C₂²	semidirect
38. C₈×Dic₂₃	C₂²	semidirect
39. C₈×Dic₂₃	C₂²	semidirect
40. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
41. C₄×C₂₃⋊C₈	C₂²	semidirect
47. D₄×D₂₃	C₈	direct (N × H)
80. C₈×D₂₃	D₄	direct (N × H)
84. C₂²×D₂₃	C₂×C₈	semidirect
87. D₄×C₂₃	C₂×C₈	semidirect
94. C₂²⋊D₂₃	C₂×C₈	semidirect
100. D₉₂	C₂×C₈	semidirect
106. C₄×D₂₃	C₂×C₈	semidirect
111. C₈×C₂₃	C₂×D₄	semidirect
114. C₂₃⋊C₈	C₂×D₄	semidirect
118. C₂²×C₂₃	C₂²×C₈	semidirect
125. C₂×D₂₃	C₂²×C₈	semidirect
129. C₄×C₂₃	C₂²×C₈	semidirect
135. Dic₂₃	C₂²×C₈	semidirect
143. C₈×D₄	D₂₃	direct (N × H)
152. D₂₃	C₈×D₄	direct (N × H)
162. C₂²×C₈	C₂×D₂₃	semidirect
174. C₄⋊C₈	C₂×D₂₃	semidirect
178. C₂²⋊C₈	C₂×D₂₃	semidirect
182. C₄×C₈	C₂×D₂₃	semidirect
183. C₂₃	C₂×C₈×D₄	semidirect
231. D₄	C₈×D₂₃	direct (N × H)
256. C₈	D₄×D₂₃	direct (N × H)
266. C₂²	C₂×C₈×D₂₃	semidirect
274. C₄	C₂×C₈×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
C₄×D₄×D₂₃	2
C₈×D₉₂	2
D₄×C₂₃⋊C₈	2
#13092482	2
C₈×C₂²⋊D₂₃	2
(C₂²⋊D₂₃)⋊C₈	2
C₄⋊C₈×D₂₃	2
C₂²⋊C₈×D₂₃	2
C₄×C₈×D₂₃	2
C₈×D₄×C₂₃	2
C₂×C₈×D₄	23

Subgroups — C8×D4×D23 #15682066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×D₄×D₂₃ #15682066