Subgroups — #15682230

Order 2944.

506 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092900	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092892	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092529	C₂	1	2	yes	maximal, complemented
4	#13092518	C₂	1	2	yes	maximal, complemented
5	SD₁₆×Dic₂₃	C₂	1	2	yes	maximal, complemented
6	D₈×Dic₂₃	C₂	1	2	yes	maximal, complemented
7	#13092459	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092413	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10625014	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092375	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×AGL(1,ℤ/8)	C₄	1	8	yes	complemented
15	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
16	C₂×D₈×C₂₃	C₂²	1	2	yes	—
17	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Dic₂₃	—	2	2	no	—
19	Q₈×Dic₂₃	C₂²	1	2	yes	complemented
20	D₄×Dic₂₃	—	2	2	no	—
21	D₄×Dic₂₃	—	2	1	no	—
22	D₄×Dic₂₃	—	2	1	no	—
23	D₄×Dic₂₃	C₂²	1	2	yes	—
24	D₄×Dic₂₃	C₂²	1	2	yes	—
25	D₄×Dic₂₃	C₂²	1	2	yes	complemented
26	#10625097	—	2	1	no	—
27	#10625097	C₂²	1	1	yes	characteristic
28	#10625097	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Dic₂₃	—	2	1	no	—
30	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
31	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
32	#10625017	C₂²	1	1	yes	characteristic, complemented
33	#10625017	C₂²	1	1	yes	characteristic, complemented
34	#10625014	C₂²	1	2	yes	—
35	#10625014	C₂²	1	2	yes	—
36	#10625014	C₂²	1	1	yes	characteristic, complemented
37	#10625014	C₂²	1	1	yes	characteristic, complemented
38	#10625000	C₂²	1	2	yes	complemented
39	#10624999	C₂²	1	2	yes	complemented
40	#10624997	C₂²	1	2	yes	complemented
41	C₈×Dic₂₃	C₂²	1	2	yes	complemented
42	C₂⁴×C₂₃	—	2	1	no	—
43	#41×C₂₃	—	2	2	no	—
44	#41×C₂₃	C₂×C₄	1	4	yes	complemented
45	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₂₃	—	2	1	no	—
47	C₂×D₄×C₂₃	—	2	1	no	—
48	C₂×D₄×C₂₃	—	2	2	no	—
49	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	complemented
50	C₂×D₄×C₂₃	C₂³	1	2	yes	—
51	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	—	2	1	no	—
53	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×Dic₂₃	—	2	1	no	—
55	C₂²×Dic₂₃	—	2	2	no	—
56	C₂²×Dic₂₃	—	2	2	no	—
57	C₂²×Dic₂₃	—	2	2	no	—
58	SD₁₆×C₂₃	C₂×C₄	1	8	yes	complemented
59	D₈×C₂₃	C₂×C₄	1	8	yes	complemented
60	M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
61	C₂×C₈×C₂₃	C₂³	1	2	yes	—
62	C₂²⋊Dic₂₃	—	2	1	no	—
63	C₂²⋊Dic₂₃	—	2	2	no	—
64	C₂²⋊Dic₂₃	—	2	2	no	—
65	C₂²⋊Dic₂₃	—	2	2	no	—
66	C₂²⋊Dic₂₃	D₄	1	2	yes	—
67	C₄⋊Dic₂₃	—	2	1	no	—
68	C₄⋊Dic₂₃	—	2	2	no	—
69	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
70	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
71	C₄⋊Dic₂₃	C₂³	1	2	yes	—
72	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
74	C₄×Dic₂₃	—	2	1	no	—
75	C₄×Dic₂₃	—	2	2	no	—
76	C₄×Dic₂₃	C₂³	1	2	yes	—
77	C₂×C₂₃⋊C₈	—	2	2	no	—
78	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
80	C₂³×C₂₃	—	2	1	no	—
81	C₂³×C₂₃	—	2	2	no	—
82	C₂³×C₂₃	—	2	4	no	—
83	C₂³×C₂₃	—	8	1	no	—
84	C₂³×C₂₃	—	8	1	no	—
85	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
86	Q₈×C₂₃	—	2	1	no	—
87	Q₈×C₂₃	C₂²×C₄	1	2	yes	—
88	D₄×C₂₃	—	2	4	no	—
89	D₄×C₂₃	—	2	4	no	—
90	D₄×C₂₃	—	2	2	no	—
91	D₄×C₂₃	—	2	1	no	—
92	D₄×C₂₃	C₂²×C₄	1	2	yes	—
93	D₄×C₂₃	C₂²×C₄	1	4	yes	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	4	no	—
96	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	#41	1	2	yes	—
99	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
100	C₂×Dic₂₃	—	2	2	no	—
101	C₂×Dic₂₃	—	2	1	no	—
102	C₂×Dic₂₃	—	2	1	no	—
103	C₂×Dic₂₃	—	8	2	no	—
104	C₂×Dic₂₃	—	8	2	no	—
105	C₂×Dic₂₃	—	8	2	no	—
106	C₂×Dic₂₃	—	8	1	no	—
107	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
108	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
109	C₈×C₂₃	C₂²×C₄	1	4	yes	—
110	C₂₃⋊C₈	—	2	1	no	—
111	C₂₃⋊C₈	—	2	1	no	—
112	#2946	—	23	1	no	maximal, Sylow
113	C₂²×C₂₃	—	2	1	no	—
114	C₂²×C₂₃	—	2	1	no	—
115	C₂²×C₂₃	—	2	2	no	—
116	C₂²×C₂₃	—	2	4	no	—
117	C₂²×C₂₃	—	4	1	no	—
118	C₂²×C₂₃	—	4	2	no	—
119	C₂²×C₂₃	—	4	2	no	—
120	C₂²×C₂₃	—	4	1	no	—
121	C₂²×C₂₃	—	8	4	no	—
122	C₂²×C₂₃	C₄×D₄	1	2	yes	—
123	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
124	C₄×C₂₃	—	2	2	no	—
125	C₄×C₂₃	C₄×D₄	1	2	yes	—
126	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
127	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
128	Dic₂₃	—	2	2	no	—
129	Dic₂₃	—	2	2	no	—
130	Dic₂₃	—	4	2	no	—
131	Dic₂₃	—	4	1	no	—
132	Dic₂₃	—	4	1	no	—
133	C₂×AGL(1,ℤ/8)	D₂₃	1	1	yes	characteristic
134	#520	—	23	1	no	—
135	#518	—	23	1	no	—
136	#442	—	23	2	no	—
137	#440	—	23	2	no	—
138	C₄×SD₁₆	—	23	2	no	—
139	C₄×D₈	—	23	2	no	—
140	#428	—	23	1	no	—
141	#416	—	23	1	no	—
142	C₂×#102	—	23	1	no	—
143	#405	—	23	1	no	—
144	C₂×C₂₃	—	2	2	no	—
145	C₂×C₂₃	—	4	2	no	—
146	C₂×C₂₃	—	4	4	no	—
147	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂×C₂₃	#576	1	1	yes	characteristic
149	C₂×C₂₃	#577	1	1	yes	characteristic
150	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
151	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
152	AGL(1,ℤ/8)	Dic₂₃	1	8	yes	complemented
153	C₂×SD₁₆	C₂×D₂₃	1	2	yes	—
154	C₂×D₈	C₂×D₂₃	1	2	yes	—
155	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
156	C₂²×C₈	—	23	1	no	—
157	C₄×Q₈	—	23	2	no	—
158	C₄×D₄	—	23	2	no	—
159	C₄×D₄	—	23	2	no	—
160	C₄×D₄	—	23	2	no	—
161	C₄×D₄	—	46	2	no	—
162	C₄×D₄	—	46	1	no	—
163	C₄×D₄	—	46	1	no	—
164	#117	—	23	1	no	—
165	#117	—	23	1	no	—
166	#117	—	46	1	no	—
167	C₂×C₄⋊C₄	—	23	1	no	—
168	C₂×C₄⋊C₄	—	46	1	no	—
169	C₂×C₂²⋊C₄	—	46	2	no	—
170	#107	—	23	2	no	—
171	#106	—	23	2	no	—
172	#103	—	23	1	no	—
173	#103	—	23	1	no	—
174	#102	—	23	2	no	—
175	#102	—	23	2	no	—
176	#102	—	23	1	no	—
177	#102	—	23	1	no	—
178	#97	—	23	2	no	—
179	C₄×C₈	—	23	2	no	—
180	C₂₃	#2946	1	1	yes	characteristic, complemented, Sylow
181	C₂⁴	—	2	1	no	—
182	#41	—	2	2	no	—
183	#41	C₂×Dic₂₃	1	4	yes	complemented
184	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	C₂×Dic₂₃	1	4	yes	complemented
189	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
190	C₂×D₄	C₂²×D₂₃	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	46	2	no	—
193	C₂²×C₄	—	46	2	no	—
194	C₂²×C₄	—	46	2	no	—
195	C₂²×C₄	—	46	1	no	—
196	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
197	SD₁₆	C₂×Dic₂₃	1	8	yes	complemented
198	D₈	C₂×Dic₂₃	1	8	yes	complemented
199	M₁₆	C₂×Dic₂₃	1	4	yes	complemented
200	C₂×C₈	—	23	1	no	—
201	C₂×C₈	—	23	1	no	—
202	C₂×C₈	—	46	2	no	—
203	C₂×C₈	C₂²×D₂₃	1	2	yes	—
204	C₄⋊C₄	—	23	1	no	—
205	C₄⋊C₄	—	23	1	no	—
206	C₄⋊C₄	—	23	2	no	—
207	C₄⋊C₄	—	23	1	no	—
208	C₄⋊C₄	—	23	1	no	—
209	C₄⋊C₄	—	46	2	no	—
210	C₄⋊C₄	—	46	1	no	—
211	C₂²⋊C₄	—	23	2	no	—
212	C₂²⋊C₄	—	46	2	no	—
213	C₂²⋊C₄	—	46	2	no	—
214	C₂²⋊C₄	—	46	2	no	—
215	C₂²⋊C₄	—	46	1	no	—
216	C₄²	—	23	2	no	—
217	C₄²	—	46	2	no	—
218	C₄²	—	46	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	4	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	8	1	no	—
224	C₂³	D₄×D₂₃	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	C₂²×Dic₂₃	1	2	yes	—
227	D₄	—	2	4	no	—
228	D₄	—	2	4	no	—
229	D₄	—	2	2	no	—
230	D₄	—	2	1	no	—
231	D₄	C₂²×Dic₂₃	1	2	yes	—
232	D₄	C₂²×Dic₂₃	1	4	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	23	2	no	—
236	C₂×C₄	—	23	2	no	—
237	C₂×C₄	—	46	1	no	—
238	C₂×C₄	—	46	1	no	—
239	C₂×C₄	—	46	2	no	—
240	C₂×C₄	—	184	2	no	—
241	C₂×C₄	—	184	2	no	—
242	C₂×C₄	—	184	2	no	—
243	C₂×C₄	—	184	1	no	—
244	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
245	C₂×C₄	#67290	1	2	yes	—
246	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
247	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
248	C₈	—	46	1	no	—
249	C₈	—	46	1	no	—
250	C₈	C₂²×Dic₂₃	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	8	4	no	—
260	C₂²	D₄×Dic₂₃	1	2	yes	—
261	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	46	2	no	—
264	C₄	—	46	2	no	—
265	C₄	—	92	1	no	—
266	C₄	—	92	2	no	—
267	C₄	—	92	1	no	—
268	C₄	D₄×Dic₂₃	1	2	yes	—
269	C₄	C₂×#67290	1	1	yes	characteristic
270	C₄	C₂³×Dic₂₃	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	4	no	—
273	C₂	—	4	2	no	—
274	C₂	C₂×D₄×Dic₂₃	1	1	yes	characteristic
275	C₂	#13093221	1	1	yes	characteristic
276	C₂	#13093227	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting, prime radical	C₂×C₂₃×AGL(1,ℤ/8)	C₂
socle	C₂²×C₂₃	C₂²×D₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂×AGL(1,ℤ/8)	D₂₃
hypocenter	C₂₃	#2946
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #13092900	C₂	semidirect
2. #13092892	C₂	semidirect
3. #13092529	C₂	semidirect
4. #13092518	C₂	semidirect
5. SD₁₆×Dic₂₃	C₂	semidirect
6. D₈×Dic₂₃	C₂	semidirect
7. #13092459	C₂	semidirect
8. #13092413	C₂	semidirect
9. C₂×#10625014	C₂	semidirect
10. #13092375	C₂	semidirect
14. C₂₃×AGL(1,ℤ/8)	C₄	semidirect
19. Q₈×Dic₂₃	C₂²	semidirect
25. D₄×Dic₂₃	C₂²	semidirect
32. #10625017	C₂²	semidirect
33. #10625017	C₂²	semidirect
36. #10625014	C₂²	semidirect
37. #10625014	C₂²	semidirect
38. #10625000	C₂²	semidirect
39. #10624999	C₂²	semidirect
40. #10624997	C₂²	semidirect
41. C₈×Dic₂₃	C₂²	semidirect
44. #41×C₂₃	C₂×C₄	semidirect
49. C₂×D₄×C₂₃	C₂×C₄	semidirect
58. SD₁₆×C₂₃	C₂×C₄	semidirect
59. D₈×C₂₃	C₂×C₄	semidirect
60. M₁₆×C₂₃	C₂×C₄	semidirect
152. AGL(1,ℤ/8)	Dic₂₃	semidirect
180. C₂₃	#2946	semidirect
183. #41	C₂×Dic₂₃	semidirect
188. C₂×D₄	C₂×Dic₂₃	semidirect
197. SD₁₆	C₂×Dic₂₃	semidirect
198. D₈	C₂×Dic₂₃	semidirect
199. M₁₆	C₂×Dic₂₃	semidirect

Maximal subgroups

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

subgroup	index
#13092900	2
#13092892	2
#13092529	2
#13092518	2
SD₁₆×Dic₂₃	2
D₈×Dic₂₃	2
#13092459	2
#13092413	2
C₂×#10625014	2
#13092375	2
C₂×C₂₃×AGL(1,ℤ/8)	2
#2946	23