Subgroups — C₄×D₈×D₂₃ #15682144

Order 2944.

616 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×D₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄×D₄×D₂₃	C₂	1	2	yes	maximal, complemented
3	C₄×D₁₈₄	C₂	1	1	yes	maximal, characteristic, complemented
4	D₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092499	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#67272	C₂	1	2	yes	maximal, complemented
7	#13092497	C₂	1	2	yes	maximal, complemented
8	#107×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#102×D₂₃	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₂₃	C₂²	1	2	yes	—
13	C₂²×C₄×D₂₃	—	2	2	no	—
14	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
16	D₄×Dic₂₃	C₂²	1	2	yes	complemented
17	C₂×#67272	C₂²	1	2	yes	—
18	C₄×C₂²⋊D₂₃	—	2	2	no	—
19	D₈×D₂₃	C₄	1	8	yes	direct product factor
20	C₂×D₁₈₄	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
23	#10625064	C₂²	1	2	yes	complemented
24	C₄⋊C₄×D₂₃	C₂²	1	2	yes	—
25	(C₂²⋊D₂₃)⋊C₄	—	2	2	no	—
26	C₂²⋊C₄×D₂₃	—	2	2	no	—
27	C₄×D₉₂	C₂²	1	2	yes	complemented
28	C₄²×D₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#107	C₂²	1	1	yes	characteristic, complemented
30	C₂₃×#102	C₂²	1	2	yes	complemented
31	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
32	#10625014	C₂²	1	2	yes	complemented
33	#10625003	C₂²	1	2	yes	complemented
34	#10625000	C₂²	1	1	yes	characteristic, complemented
35	C₈×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
36	C₈×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
37	#10624991	C₂²	1	2	yes	complemented
38	#10624989	C₂²	1	2	yes	complemented
39	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
40	C₂³×D₂₃	—	2	2	no	—
41	C₂×D₄×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	—	2	2	no	—
43	C₂×C₂²⋊D₂₃	—	2	2	no	—
44	C₂²×Dic₂₃	—	2	2	no	—
45	D₄×D₂₃	—	2	4	no	—
46	D₄×D₂₃	C₂×C₄	1	8	yes	complemented
47	C₂×D₉₂	C₂³	1	2	yes	—
48	C₂×C₄×D₂₃	—	2	2	no	—
49	C₂×C₄×D₂₃	—	2	2	no	—
50	C₂×C₄×D₂₃	—	2	2	no	—
51	C₂×C₄×D₂₃	—	4	8	no	—
52	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
53	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
54	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
55	D₈×C₂₃	C₂×C₄	1	4	yes	complemented
56	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
59	C₂²⋊C₄×C₂₃	—	2	2	no	—
60	C₄²×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²⋊Dic₂₃	—	2	2	no	—
62	#67272	C₂×C₄	1	8	yes	complemented
63	(C₂×D₂₃)⋊C₄	—	2	2	no	—
64	(C₂×D₂₃)⋊C₄	—	2	2	no	—
65	C₄⋊Dic₂₃	C₂³	1	2	yes	—
66	Dic₂₃⋊C₄	—	2	2	no	—
67	C₄×Dic₂₃	D₄	1	1	yes	characteristic
68	C₄×Dic₂₃	D₄	1	1	yes	characteristic
69	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
70	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
72	D₁₈₄	C₂×C₄	1	4	yes	complemented
73	C₈×D₂₃	—	2	2	no	—
74	C₈×D₂₃	C₂×C₄	1	4	yes	complemented
75	C₂³×C₂₃	—	2	2	no	—
76	C₂²×D₂₃	—	2	2	no	—
77	C₂²×D₂₃	—	2	8	no	—
78	C₂²×D₂₃	—	4	4	no	—
79	C₂²×D₂₃	—	4	4	no	—
80	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
81	D₄×C₂₃	—	2	2	no	—
82	D₄×C₂₃	C₂²×C₄	1	4	yes	complemented
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	4	4	no	—
85	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
88	C₂²⋊D₂₃	—	2	8	no	—
89	C₂×Dic₂₃	—	2	2	no	—
90	C₂×Dic₂₃	—	4	4	no	—
91	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
93	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
94	D₉₂	—	2	2	no	—
95	D₉₂	C₂²×C₄	1	4	yes	complemented
96	C₄×D₂₃	—	2	2	no	—
97	C₄×D₂₃	—	4	4	no	—
98	C₄×D₂₃	—	4	4	no	—
99	C₄×D₂₃	—	4	4	no	—
100	C₄×D₂₃	D₈	1	4	yes	direct product factor
101	C₄×D₂₃	C₂²×C₄	1	2	yes	—
102	C₄×D₂₃	#41	1	2	yes	—
103	C₈×C₂₃	—	2	1	no	—
104	C₈×C₂₃	C₂²×C₄	1	2	yes	complemented
105	C₂₃⋊C₈	—	2	1	no	—
106	C₂₃⋊C₈	C₂²×C₄	1	2	yes	complemented
107	C₂×C₄×D₈	—	23	1	no	maximal, Sylow
108	C₂²×C₂₃	—	2	4	no	—
109	C₂²×C₂₃	—	4	2	no	—
110	C₂²×C₂₃	—	4	2	no	—
111	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×D₂₃	—	2	4	no	—
113	C₂×D₂₃	—	4	2	no	—
114	C₂×D₂₃	—	4	2	no	—
115	C₂×D₂₃	—	4	16	no	—
116	C₂×D₂₃	C₄×D₄	1	2	yes	—
117	C₂×D₂₃	C₂×D₈	1	2	yes	—
118	C₂×D₂₃	#135	1	2	yes	—
119	C₄×C₂₃	—	2	1	no	—
120	C₄×C₂₃	—	4	2	no	—
121	C₄×C₂₃	C₂×D₈	1	2	yes	complemented
122	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
123	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
124	Dic₂₃	—	2	1	no	—
125	Dic₂₃	—	4	2	no	—
126	Dic₂₃	C₄×D₄	1	2	yes	—
127	Dic₂₃	C₂×D₈	1	2	yes	complemented
128	C₂²×D₈	—	23	1	no	—
129	C₂×C₄×D₄	—	23	2	no	—
130	C₄×D₈	—	23	1	no	—
131	C₄×D₈	—	23	2	no	—
132	C₄×D₈	—	23	1	no	—
133	C₄×D₈	—	23	2	no	—
134	C₄×D₈	—	23	1	no	—
135	C₄×D₈	D₂₃	1	1	yes	characteristic, direct product factor
136	C₂×#107	—	23	1	no	—
137	C₂×#102	—	23	2	no	—
138	C₂×C₄×C₈	—	23	1	no	—
139	C₂×C₂₃	—	4	4	no	—
140	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂×C₂₃	C₂²×D₈	1	1	yes	characteristic
142	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
143	D₂₃	—	4	4	no	—
144	D₂₃	C₄×D₈	1	4	yes	direct product factor
145	C₂²×D₄	—	23	2	no	—
146	C₂³×C₄	—	46	2	no	—
147	C₂×D₈	—	23	2	no	—
148	C₂×D₈	—	23	8	no	—
149	C₂×D₈	—	23	1	no	—
150	C₂×D₈	C₂×D₂₃	1	1	yes	characteristic
151	C₂²×C₈	—	23	1	no	—
152	C₂²×C₈	—	23	1	no	—
153	C₄×D₄	—	23	2	no	—
154	C₄×D₄	—	23	2	no	—
155	C₄×D₄	—	23	2	no	—
156	C₄×D₄	—	46	2	no	—
157	C₄×D₄	—	46	2	no	—
158	C₄×D₄	C₂×D₂₃	1	2	yes	complemented
159	C₂×C₄⋊C₄	—	23	2	no	—
160	C₂×C₂²⋊C₄	—	46	2	no	—
161	C₂×C₄²	—	23	1	no	—
162	#107	—	23	1	no	—
163	#107	—	23	2	no	—
164	#107	C₂×D₂₃	1	1	yes	characteristic, complemented
165	#102	—	23	2	no	—
166	#102	—	23	2	no	—
167	#102	—	23	2	no	—
168	#102	C₂×D₂₃	1	2	yes	complemented
169	C₄×C₈	—	23	1	no	—
170	C₄×C₈	—	23	1	no	—
171	C₄×C₈	—	23	1	no	—
172	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic, complemented
173	C₂₃	C₂×C₄×D₈	1	1	yes	characteristic, complemented, Sylow
174	C₂⁴	—	46	2	no	—
175	C₂×D₄	—	23	8	no	—
176	C₂×D₄	—	23	2	no	—
177	C₂×D₄	—	46	4	no	—
178	C₂×D₄	—	46	2	no	—
179	C₂×D₄	C₂²×D₂₃	1	2	yes	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	23	1	no	—
182	C₂²×C₄	—	23	1	no	—
183	C₂²×C₄	—	23	1	no	—
184	C₂²×C₄	—	46	2	no	—
185	C₂²×C₄	—	46	2	no	—
186	C₂²×C₄	—	46	2	no	—
187	C₂²×C₄	—	46	2	no	—
188	C₂²×C₄	—	92	8	no	—
189	D₈	—	23	8	no	—
190	D₈	—	23	4	no	—
191	D₈	C₄×D₂₃	1	4	yes	direct product factor
192	C₂×C₈	—	23	1	no	—
193	C₂×C₈	—	23	4	no	—
194	C₂×C₈	—	23	1	no	—
195	C₂×C₈	—	46	2	no	—
196	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
197	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
198	C₄⋊C₄	—	23	2	no	—
199	C₄⋊C₄	—	46	2	no	—
200	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	46	2	no	—
203	C₂²⋊C₄	—	46	2	no	—
204	C₂²⋊C₄	—	46	2	no	—
205	C₄²	—	23	1	no	—
206	C₄²	—	23	1	no	—
207	C₄²	—	23	1	no	—
208	C₄²	C₂²×D₂₃	1	1	yes	characteristic
209	C₂³	—	2	2	no	—
210	C₂³	—	23	1	no	—
211	C₂³	—	46	8	no	—
212	C₂³	—	46	2	no	—
213	C₂³	—	92	4	no	—
214	C₂³	—	92	4	no	—
215	D₄	—	2	2	no	—
216	D₄	—	23	4	no	—
217	D₄	—	46	8	no	—
218	D₄	—	46	2	no	—
219	D₄	C₂×C₄×D₂₃	1	4	yes	complemented
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	23	1	no	—
223	C₂×C₄	—	23	1	no	—
224	C₂×C₄	—	23	4	no	—
225	C₂×C₄	—	23	2	no	—
226	C₂×C₄	—	23	2	no	—
227	C₂×C₄	—	23	1	no	—
228	C₂×C₄	—	46	2	no	—
229	C₂×C₄	—	46	2	no	—
230	C₂×C₄	—	92	4	no	—
231	C₂×C₄	—	92	4	no	—
232	C₂×C₄	—	92	4	no	—
233	C₂×C₄	—	92	4	no	—
234	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
235	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
236	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	23	2	no	—
239	C₈	—	46	1	no	—
240	C₈	C₂×C₄×D₂₃	1	2	yes	complemented
241	C₂²	—	2	4	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	23	2	no	—
245	C₂²	—	23	2	no	—
246	C₂²	—	23	2	no	—
247	C₂²	—	46	4	no	—
248	C₂²	—	92	16	no	—
249	C₂²	—	92	2	no	—
250	C₂²	—	92	2	no	—
251	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	23	2	no	—
255	C₄	—	23	2	no	—
256	C₄	—	46	1	no	—
257	C₄	—	92	2	no	—
258	C₄	D₈×D₂₃	1	2	yes	direct product factor
259	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
260	C₄	#41×D₂₃	1	1	yes	characteristic
261	C₂	—	4	4	no	—
262	C₂	—	23	4	no	—
263	C₂	—	92	4	no	—
264	C₂	C₄×D₄×D₂₃	1	1	yes	characteristic
265	C₂	C₂×D₈×D₂₃	1	1	yes	characteristic
266	C₂	#135×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₄×D₈×C₂₃	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×D₈×D₂₃	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₄×D₈	D₂₃
hypocenter	C₂₃	C₂×C₄×D₈
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₄×D₄×D₂₃	C₂	semidirect
3. C₄×D₁₈₄	C₂	semidirect
4. D₈×Dic₂₃	C₂	semidirect
5. #13092499	C₂	semidirect
6. C₄×#67272	C₂	semidirect
7. #13092497	C₂	semidirect
8. #107×D₂₃	C₂	semidirect
9. #102×D₂₃	C₂	semidirect
10. C₄×C₈×D₂₃	C₂	semidirect
11. C₄×D₈×C₂₃	C₂	semidirect
15. C₄×D₄×C₂₃	C₂²	semidirect
16. D₄×Dic₂₃	C₂²	semidirect
19. D₈×D₂₃	C₄	direct (N × H)
23. #10625064	C₂²	semidirect
27. C₄×D₉₂	C₂²	semidirect
29. C₂₃×#107	C₂²	semidirect
30. C₂₃×#102	C₂²	semidirect
31. C₄×C₈×C₂₃	C₂²	semidirect
32. #10625014	C₂²	semidirect
33. #10625003	C₂²	semidirect
34. #10625000	C₂²	semidirect
35. C₈×Dic₂₃	C₂²	semidirect
36. C₈×Dic₂₃	C₂²	semidirect
37. #10624991	C₂²	semidirect
38. #10624989	C₂²	semidirect
39. C₄×C₂₃⋊C₈	C₂²	semidirect
46. D₄×D₂₃	C₂×C₄	semidirect
55. D₈×C₂₃	C₂×C₄	semidirect
62. #67272	C₂×C₄	semidirect
72. D₁₈₄	C₂×C₄	semidirect
74. C₈×D₂₃	C₂×C₄	semidirect
82. D₄×C₂₃	C₂²×C₄	semidirect
95. D₉₂	C₂²×C₄	semidirect
100. C₄×D₂₃	D₈	direct (N × H)
104. C₈×C₂₃	C₂²×C₄	semidirect
106. C₂₃⋊C₈	C₂²×C₄	semidirect
121. C₄×C₂₃	C₂×D₈	semidirect
127. Dic₂₃	C₂×D₈	semidirect
135. C₄×D₈	D₂₃	direct (N × H)
144. D₂₃	C₄×D₈	direct (N × H)
158. C₄×D₄	C₂×D₂₃	semidirect
164. #107	C₂×D₂₃	semidirect
168. #102	C₂×D₂₃	semidirect
172. C₄×C₈	C₂×D₂₃	semidirect
173. C₂₃	C₂×C₄×D₈	semidirect
191. D₈	C₄×D₂₃	direct (N × H)
219. D₄	C₂×C₄×D₂₃	semidirect
240. C₈	C₂×C₄×D₂₃	semidirect
258. C₄	D₈×D₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₈×D₂₃	2
C₄×D₄×D₂₃	2
C₄×D₁₈₄	2
D₈×Dic₂₃	2
#13092499	2
C₄×#67272	2
#13092497	2
#107×D₂₃	2
#102×D₂₃	2
C₄×C₈×D₂₃	2
C₄×D₈×C₂₃	2
C₂×C₄×D₈	23

Subgroups — C4×D8×D23 #15682144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₈×D₂₃ #15682144