Subgroups — C₂²×C₂₃×#489 #19208232

Order 5888.

2486 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#489	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₂₃×#103	C₂	1	2	yes	maximal, complemented
5	C₂²×C₂₃×#102	C₂	1	2	yes	maximal
6	C₂²×C₂₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₂₃	—	2	1	no	—
8	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃×C₄⋊Q₈	—	2	6	no	—
12	C₂×C₂₃×C₄⋊Q₈	C₂²	1	12	yes	complemented
13	C₂×C₂₃×#124	—	2	6	no	—
14	C₂×C₂₃×#124	C₂²	1	12	yes	—
15	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂²×C₂²⋊C₄×C₂₃	—	2	1	no	—
18	C₂²×C₂²⋊C₄×C₂₃	—	2	1	no	—
19	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#489	C₂²	1	64	yes	direct product factor
21	C₂×C₂₃×#103	C₂²	1	24	yes	complemented
22	C₂×C₂₃×#102	C₂²	1	24	yes	—
23	C₂×C₂₃×#97	C₂²	1	12	yes	complemented
24	C₂⁵×C₂₃	—	2	1	no	—
25	C₂²×Q₈×C₂₃	—	2	12	no	—
26	C₂²×Q₈×C₂₃	—	2	3	no	—
27	C₂²×Q₈×C₂₃	—	2	8	no	—
28	C₂²×Q₈×C₂₃	—	4	8	no	—
29	C₂²×Q₈×C₂₃	C₂³	1	6	yes	—
30	C₂²×D₄×C₂₃	—	2	3	no	—
31	C₂²×D₄×C₂₃	—	2	8	no	—
32	C₂²×D₄×C₂₃	C₂³	1	6	yes	—
33	C₂³×C₄×C₂₃	—	2	2	no	—
34	C₂³×C₄×C₂₃	—	2	1	no	—
35	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₂₃	—	2	8	no	—
39	C₂²×C₈×C₂₃	C₂³	1	12	yes	—
40	C₂₃×C₄⋊Q₈	—	2	24	no	—
41	C₂₃×C₄⋊Q₈	C₂³	1	16	yes	complemented
42	C₂₃×#124	—	2	24	no	—
43	C₂₃×#124	C₂³	1	16	yes	—
44	C₂×C₄⋊C₄×C₂₃	—	2	12	no	—
45	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
46	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
47	C₂×C₂²⋊C₄×C₂₃	—	2	12	no	—
48	C₂×C₂²⋊C₄×C₂₃	—	2	12	no	—
49	C₂×C₄²×C₂₃	D₄	1	6	yes	—
50	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
51	C₂₃×#103	C₂³	1	32	yes	complemented
52	C₂₃×#102	C₂³	1	32	yes	—
53	C₂₃×#97	C₂³	1	16	yes	complemented
54	C₂⁴×C₂₃	—	2	6	no	—
55	C₂⁴×C₂₃	—	4	4	no	—
56	C₂⁴×C₂₃	—	4	4	no	—
57	C₂⁴×C₂₃	—	4	4	no	—
58	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂×Q₈×C₂₃	—	2	6	no	—
60	C₂×Q₈×C₂₃	—	2	16	no	—
61	C₂×Q₈×C₂₃	—	2	24	no	—
62	C₂×Q₈×C₂₃	—	4	48	no	—
63	C₂×Q₈×C₂₃	—	4	12	no	—
64	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
65	C₂×D₄×C₂₃	—	2	6	no	—
66	C₂×D₄×C₂₃	—	2	24	no	—
67	C₂×D₄×C₂₃	—	4	12	no	—
68	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
69	C₂²×C₄×C₂₃	—	2	4	no	—
70	C₂²×C₄×C₂₃	—	2	4	no	—
71	C₂²×C₄×C₂₃	—	2	6	no	—
72	C₂²×C₄×C₂₃	—	2	12	no	—
73	C₂²×C₄×C₂₃	—	4	4	no	—
74	C₂²×C₄×C₂₃	—	4	8	no	—
75	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
76	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
77	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
78	C₂²×C₄×C₂₃	#41	1	8	yes	—
79	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
80	C₂×C₈×C₂₃	—	2	24	no	—
81	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
82	C₄⋊C₄×C₂₃	—	2	16	no	—
83	C₄⋊C₄×C₂₃	—	2	12	no	—
84	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
85	C₂²⋊C₄×C₂₃	—	2	16	no	—
86	C₂²⋊C₄×C₂₃	—	2	16	no	—
87	C₄²×C₂₃	C₂×D₄	1	12	yes	—
88	C₄²×C₂₃	C₂⁴	1	4	yes	—
89	C₂²×#489	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₃	—	2	4	no	—
91	C₂³×C₂₃	—	4	12	no	—
92	C₂³×C₂₃	—	4	12	no	—
93	C₂³×C₂₃	—	4	12	no	—
94	C₂³×C₂₃	—	8	16	no	—
95	C₂³×C₂₃	#124	1	4	yes	—
96	C₂³×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
97	C₂³×C₂₃	#137	1	4	yes	—
98	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
99	Q₈×C₂₃	—	2	8	no	—
100	Q₈×C₂₃	—	4	32	no	—
101	Q₈×C₂₃	—	4	12	no	—
102	D₄×C₂₃	—	2	8	no	—
103	D₄×C₂₃	—	4	12	no	—
104	C₂×C₄×C₂₃	—	2	12	no	—
105	C₂×C₄×C₂₃	—	2	12	no	—
106	C₂×C₄×C₂₃	—	2	4	no	—
107	C₂×C₄×C₂₃	—	2	8	no	—
108	C₂×C₄×C₂₃	—	4	12	no	—
109	C₂×C₄×C₂₃	—	4	24	no	—
110	C₂×C₄×C₂₃	#124	1	12	yes	—
111	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
112	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
113	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
114	C₂×C₄×C₂₃	C₂×#41	1	12	yes	—
115	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
116	C₈×C₂₃	—	2	8	no	—
117	C₂²×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
118	C₂²×#124	C₂×C₂₃	1	1	yes	characteristic
119	C₂×#489	C₂×C₂₃	1	24	yes	direct product factor
120	C₂²×#103	C₂×C₂₃	1	2	yes	complemented
121	C₂²×#102	C₂×C₂₃	1	2	yes	—
122	C₂²×#97	C₂×C₂₃	1	1	yes	characteristic, complemented
123	C₂²×C₂₃	—	4	4	no	—
124	C₂²×C₂₃	—	4	4	no	—
125	C₂²×C₂₃	—	4	4	no	—
126	C₂²×C₂₃	—	8	24	no	—
127	C₂²×C₂₃	#489	1	16	yes	direct product factor
128	C₂²×C₂₃	C₂×#124	1	6	yes	—
129	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	6	yes	—
130	C₂²×C₂₃	C₂×#137	1	6	yes	—
131	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
132	C₄×C₂₃	—	2	4	no	—
133	C₄×C₂₃	—	2	4	no	—
134	C₄×C₂₃	—	4	4	no	—
135	C₄×C₂₃	—	4	8	no	—
136	C₄×C₂₃	C₂×#124	1	6	yes	—
137	C₄×C₂₃	C₂²×#41	1	2	yes	—
138	C₂³×Q₈	—	2	1	no	—
139	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
140	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
141	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
142	C₂×C₄⋊Q₈	—	2	6	no	—
143	C₂×C₄⋊Q₈	C₂²×C₂₃	1	12	yes	complemented
144	C₂×#124	—	2	6	no	—
145	C₂×#124	C₂²×C₂₃	1	12	yes	—
146	C₂²×C₄⋊C₄	—	2	1	no	—
147	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
148	C₂²×C₂²⋊C₄	—	2	1	no	—
149	C₂²×C₂²⋊C₄	—	2	1	no	—
150	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
151	#489	C₂²×C₂₃	1	64	yes	direct product factor
152	C₂×#103	C₂²×C₂₃	1	24	yes	complemented
153	C₂×#102	C₂²×C₂₃	1	24	yes	—
154	C₂×#97	C₂²×C₂₃	1	12	yes	complemented
155	C₂×C₂₃	—	8	4	no	—
156	C₂×C₂₃	C₂×#489	1	12	yes	direct product factor
157	C₂×C₂₃	C₂²×#124	1	1	yes	characteristic
158	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
159	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
160	C₂⁵	—	2	1	no	—
161	C₂²×Q₈	—	2	3	no	—
162	C₂²×Q₈	—	2	12	no	—
163	C₂²×Q₈	—	2	8	no	—
164	C₂²×Q₈	—	4	8	no	—
165	C₂²×Q₈	C₂³×C₂₃	1	6	yes	—
166	C₂²×D₄	—	2	3	no	—
167	C₂²×D₄	—	2	8	no	—
168	C₂²×D₄	C₂³×C₂₃	1	6	yes	—
169	C₂³×C₄	—	2	1	no	—
170	C₂³×C₄	—	2	2	no	—
171	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
172	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
173	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
174	C₂²×C₈	—	2	8	no	—
175	C₂²×C₈	C₂³×C₂₃	1	12	yes	—
176	C₄⋊Q₈	—	2	24	no	—
177	C₄⋊Q₈	C₂³×C₂₃	1	16	yes	complemented
178	#124	—	2	24	no	—
179	#124	C₂³×C₂₃	1	16	yes	—
180	C₂×C₄⋊C₄	—	2	12	no	—
181	C₂×C₄⋊C₄	—	2	6	no	—
182	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
183	C₂×C₂²⋊C₄	—	2	12	no	—
184	C₂×C₂²⋊C₄	—	2	12	no	—
185	C₂×C₄²	D₄×C₂₃	1	6	yes	—
186	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
187	#103	C₂³×C₂₃	1	32	yes	complemented
188	#102	C₂³×C₂₃	1	32	yes	—
189	#97	C₂³×C₂₃	1	16	yes	complemented
190	C₂₃	C₂²×#489	1	1	yes	characteristic, direct product factor, Sylow
191	C₂⁴	—	2	6	no	—
192	C₂⁴	—	4	4	no	—
193	C₂⁴	—	4	4	no	—
194	C₂⁴	—	4	4	no	—
195	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
196	C₂×Q₈	—	2	6	no	—
197	C₂×Q₈	—	2	16	no	—
198	C₂×Q₈	—	2	24	no	—
199	C₂×Q₈	—	4	48	no	—
200	C₂×Q₈	—	4	12	no	—
201	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
202	C₂×D₄	—	2	6	no	—
203	C₂×D₄	—	2	24	no	—
204	C₂×D₄	—	4	12	no	—
205	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	6	no	—
209	C₂²×C₄	—	2	12	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	4	8	no	—
212	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
213	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
214	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
215	C₂²×C₄	#41×C₂₃	1	8	yes	—
216	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
217	C₂×C₈	—	2	24	no	—
218	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
219	C₄⋊C₄	—	2	16	no	—
220	C₄⋊C₄	—	2	12	no	—
221	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
222	C₂²⋊C₄	—	2	16	no	—
223	C₂²⋊C₄	—	2	16	no	—
224	C₄²	C₂×D₄×C₂₃	1	12	yes	—
225	C₄²	C₂⁴×C₂₃	1	4	yes	—
226	C₂³	—	2	4	no	—
227	C₂³	—	4	12	no	—
228	C₂³	—	4	12	no	—
229	C₂³	—	4	12	no	—
230	C₂³	—	8	16	no	—
231	C₂³	C₂₃×#124	1	4	yes	—
232	C₂³	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
233	C₂³	C₂₃×#137	1	4	yes	—
234	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
235	Q₈	—	2	8	no	—
236	Q₈	—	4	32	no	—
237	Q₈	—	4	12	no	—
238	D₄	—	2	8	no	—
239	D₄	—	4	12	no	—
240	C₂×C₄	—	2	12	no	—
241	C₂×C₄	—	2	12	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	4	24	no	—
245	C₂×C₄	—	4	12	no	—
246	C₂×C₄	C₂₃×#124	1	12	yes	—
247	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
248	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
249	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
250	C₂×C₄	C₂×#41×C₂₃	1	12	yes	—
251	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
252	C₈	—	2	8	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	8	24	no	—
257	C₂²	C₂₃×#489	1	16	yes	direct product factor
258	C₂²	C₂×C₂₃×#124	1	6	yes	—
259	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	6	yes	—
260	C₂²	C₂×C₂₃×#137	1	6	yes	—
261	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
262	C₄	—	2	4	no	—
263	C₄	—	2	4	no	—
264	C₄	—	4	4	no	—
265	C₄	—	4	8	no	—
266	C₄	C₂×C₂₃×#124	1	6	yes	—
267	C₄	C₂²×#41×C₂₃	1	2	yes	—
268	C₂	—	8	4	no	—
269	C₂	C₂×C₂₃×#489	1	12	yes	direct product factor
270	C₂	C₂²×C₂₃×#124	1	1	yes	characteristic
271	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
272	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂³×D₄×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂₃×C₄⋊Q₈	C₂	semidirect
3. C₂×C₂₃×#489	C₂	direct (N × H)
4. C₂²×C₂₃×#103	C₂	semidirect
6. C₂²×C₂₃×#97	C₂	semidirect
12. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
20. C₂₃×#489	C₂²	direct (N × H)
21. C₂×C₂₃×#103	C₂²	semidirect
23. C₂×C₂₃×#97	C₂²	semidirect
41. C₂₃×C₄⋊Q₈	C₂³	semidirect
51. C₂₃×#103	C₂³	semidirect
53. C₂₃×#97	C₂³	semidirect
89. C₂²×#489	C₂₃	direct (N × H)
117. C₂²×C₄⋊Q₈	C₂×C₂₃	semidirect
119. C₂×#489	C₂×C₂₃	direct (N × H)
120. C₂²×#103	C₂×C₂₃	semidirect
122. C₂²×#97	C₂×C₂₃	semidirect
127. C₂²×C₂₃	#489	direct (N × H)
143. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
151. #489	C₂²×C₂₃	direct (N × H)
152. C₂×#103	C₂²×C₂₃	semidirect
154. C₂×#97	C₂²×C₂₃	semidirect
156. C₂×C₂₃	C₂×#489	direct (N × H)
177. C₄⋊Q₈	C₂³×C₂₃	semidirect
187. #103	C₂³×C₂₃	semidirect
189. #97	C₂³×C₂₃	semidirect
190. C₂₃	C₂²×#489	direct (N × H)
257. C₂²	C₂₃×#489	direct (N × H)
269. C₂	C₂×C₂₃×#489	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×C₄⋊Q₈	2
C₂²×C₂₃×#124	2
C₂×C₂₃×#489	2
C₂²×C₂₃×#103	2
C₂²×C₂₃×#102	2
C₂²×C₂₃×#97	2
C₂²×#489	23

Subgroups — C22×C23×#489 #19208232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#489 #19208232