Subgroups — C₂₃×#8289 #19155915

Order 5888.

410 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	C₄×C₂₃×#106	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1770	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1279	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1279	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₃×#106	C₄	1	2	yes	—
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#97	—	2	1	no	—
12	C₂×C₄×C₈×C₂₃	—	2	1	no	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#340	C₄	1	2	yes	—
19	C₂₃×#336	—	2	1	no	—
20	C₂₃×#336	—	2	1	no	—
21	C₂₃×#335	C₄	1	4	yes	complemented
22	C₂₃×#322	C₄	1	4	yes	complemented
23	C₂²×C₈×C₂₃	—	2	1	no	—
24	C₂²×C₈×C₂₃	—	2	1	no	—
25	C₂²×C₈×C₂₃	—	2	1	no	—
26	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₂₃	—	2	1	no	—
33	C₂×C₄²×C₂₃	—	2	1	no	—
34	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
35	C₂₃×#106	—	2	4	no	—
36	C₂₃×C₄⋊C₈	—	2	1	no	—
37	C₂₃×C₄⋊C₈	—	2	1	no	—
38	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₂₃×#97	—	4	2	no	—
41	C₄×C₈×C₂₃	—	2	1	no	—
42	C₄×C₈×C₂₃	—	4	2	no	—
43	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₂₃	Q₈	1	2	yes	—
48	C₂₃×#95	—	2	1	no	—
49	C₂₃×#95	—	2	1	no	—
50	C₂²×C₄×C₂₃	—	2	2	no	—
51	C₂²×C₄×C₂₃	—	2	2	no	—
52	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
55	C₂×C₈×C₂₃	—	2	4	no	—
56	C₂×C₈×C₂₃	—	2	2	no	—
57	C₂×C₈×C₂₃	—	2	2	no	—
58	C₂×C₈×C₂₃	—	2	2	no	—
59	C₂×C₈×C₂₃	—	2	1	no	—
60	C₂×C₈×C₂₃	—	2	1	no	—
61	C₂×C₈×C₂₃	—	2	1	no	—
62	C₂×C₈×C₂₃	—	2	1	no	—
63	C₂×C₈×C₂₃	—	2	2	no	—
64	C₂×C₈×C₂₃	—	2	2	no	—
65	C₂×C₈×C₂₃	—	4	1	no	—
66	C₂×C₈×C₂₃	—	4	1	no	—
67	C₂×C₈×C₂₃	—	4	1	no	—
68	C₂×C₈×C₂₃	—	4	1	no	—
69	C₂×C₈×C₂₃	C₄²	1	2	yes	—
70	C₂×C₈×C₂₃	C₄²	1	2	yes	—
71	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
72	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
73	C₄⋊C₄×C₂₃	—	2	2	no	—
74	C₄⋊C₄×C₂₃	—	2	2	no	—
75	C₄⋊C₄×C₂₃	—	2	2	no	—
76	C₄⋊C₄×C₂₃	M₁₆	1	4	yes	—
77	C₄²×C₂₃	—	4	1	no	—
78	C₄²×C₂₃	—	4	1	no	—
79	C₄²×C₂₃	—	4	1	no	—
80	C₄²×C₂₃	—	4	1	no	—
81	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
84	C₄²×C₂₃	#41	1	1	yes	characteristic
85	#8289	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	4	no	—
89	C₂×C₄×C₂₃	—	4	2	no	—
90	C₂×C₄×C₂₃	—	4	2	no	—
91	C₂×C₄×C₂₃	—	4	2	no	—
92	C₂×C₄×C₂₃	—	4	2	no	—
93	C₂×C₄×C₂₃	—	4	2	no	—
94	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
101	C₂×C₄×C₂₃	#131	1	2	yes	—
102	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
105	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
106	C₈×C₂₃	—	2	4	no	—
107	C₈×C₂₃	—	2	2	no	—
108	C₈×C₂₃	—	2	2	no	—
109	C₈×C₂₃	—	2	2	no	—
110	C₈×C₂₃	—	2	2	no	—
111	C₈×C₂₃	—	4	2	no	—
112	C₈×C₂₃	—	4	2	no	—
113	C₄×#106	C₂×C₂₃	1	1	yes	characteristic
114	#1770	C₂×C₂₃	1	1	yes	characteristic
115	C₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
116	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
117	C₂×#322	C₂×C₂₃	1	1	yes	characteristic
118	#1279	C₂×C₂₃	1	1	yes	characteristic
119	#1279	C₂×C₂₃	1	1	yes	characteristic
120	C₂²×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₂₃	#435	1	2	yes	—
122	C₂²×C₂₃	#440	1	1	yes	characteristic
123	C₂²×C₂₃	#440	1	1	yes	characteristic
124	C₂²×C₂₃	#443	1	2	yes	—
125	C₄×C₂₃	—	2	1	no	—
126	C₄×C₂₃	—	2	1	no	—
127	C₄×C₂₃	—	4	4	no	—
128	C₄×C₂₃	—	8	2	no	—
129	C₄×C₂₃	C₄×M₁₆	1	1	yes	characteristic
130	C₄×C₂₃	#405	1	1	yes	characteristic
131	C₄×C₂₃	#428	1	1	yes	characteristic
132	C₄×C₂₃	#428	1	1	yes	characteristic
133	C₄×C₂₃	#441	1	1	yes	characteristic
134	C₄×C₂₃	#442	1	1	yes	characteristic
135	C₄×C₂₃	#446	1	2	yes	—
136	C₂×#106	C₄×C₂₃	1	2	yes	—
137	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
138	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
139	C₂×#97	—	2	1	no	—
140	C₂×C₄×C₈	—	2	1	no	—
141	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
144	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
145	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
146	#340	C₄×C₂₃	1	2	yes	—
147	#336	—	2	1	no	—
148	#336	—	2	1	no	—
149	#335	C₄×C₂₃	1	4	yes	complemented
150	#322	C₄×C₂₃	1	4	yes	complemented
151	C₂×C₂₃	#1585	1	2	yes	—
152	C₂×C₂₃	#1588	1	2	yes	—
153	C₂×C₂₃	#1771	1	1	yes	characteristic
154	C₂×C₂₃	#1772	1	1	yes	characteristic
155	C₂×C₂₃	#1776	1	1	yes	characteristic
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
160	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
161	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
162	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
163	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
164	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
168	#106	—	2	4	no	—
169	C₄⋊C₈	—	2	1	no	—
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
172	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
173	#97	—	4	2	no	—
174	C₄×C₈	—	2	1	no	—
175	C₄×C₈	—	4	2	no	—
176	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
177	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
178	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
179	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
180	C₄×C₈	Q₈×C₂₃	1	2	yes	—
181	#95	—	2	1	no	—
182	#95	—	2	1	no	—
183	C₂₃	#8289	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
188	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	C₄²×C₂₃	1	2	yes	—
204	C₂×C₈	C₄²×C₂₃	1	2	yes	—
205	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
206	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	M₁₆×C₂₃	1	4	yes	—
211	C₄²	—	4	1	no	—
212	C₄²	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	—	4	1	no	—
215	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
216	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
217	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
218	C₄²	#41×C₂₃	1	1	yes	characteristic
219	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
229	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
230	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
231	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
232	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
233	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
234	C₂×C₄	C₂₃×#131	1	2	yes	—
235	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
236	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
237	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
238	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
239	C₈	—	2	4	no	—
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	—	4	2	no	—
245	C₈	—	4	2	no	—
246	C₂²	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
247	C₂²	C₂₃×#435	1	2	yes	—
248	C₂²	C₂₃×#440	1	1	yes	characteristic
249	C₂²	C₂₃×#440	1	1	yes	characteristic
250	C₂²	C₂₃×#443	1	2	yes	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	2	no	—
255	C₄	C₄×M₁₆×C₂₃	1	1	yes	characteristic
256	C₄	C₂₃×#405	1	1	yes	characteristic
257	C₄	C₂₃×#428	1	1	yes	characteristic
258	C₄	C₂₃×#428	1	1	yes	characteristic
259	C₄	C₂₃×#441	1	1	yes	characteristic
260	C₄	C₂₃×#442	1	1	yes	characteristic
261	C₄	C₂₃×#446	1	2	yes	—
262	C₂	C₂₃×#1585	1	2	yes	—
263	C₂	C₂₃×#1588	1	2	yes	—
264	C₂	C₂₃×#1771	1	1	yes	characteristic
265	C₂	C₂₃×#1772	1	1	yes	characteristic
266	C₂	C₂₃×#1776	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
21. C₂₃×#335	C₄	semidirect
22. C₂₃×#322	C₄	semidirect
85. #8289	C₂₃	direct (N × H)
149. #335	C₄×C₂₃	semidirect
150. #322	C₄×C₂₃	semidirect
183. C₂₃	#8289	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×#106	2
C₂₃×#1770	2
C₈×C₄⋊C₄×C₂₃	2
C₂×C₂₃×#335	2
C₂×C₂₃×#322	2
C₂₃×#1279	2
#8289	23

Subgroups — C23×#8289 #19155915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8289 #19155915