Subgroups — C₂₃×#9669 #19157295

Order 5888.

446 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1479	C₂	1	2	yes	maximal
4	C₂₃×#1461	C₂	1	2	yes	maximal
5	C₂₃×#1450	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
8	C₄×Q₁₆×C₂₃	—	2	1	no	—
9	C₄×Q₁₆×C₂₃	—	2	1	no	—
10	C₄×Q₁₆×C₂₃	—	2	1	no	—
11	C₄×Q₁₆×C₂₃	—	2	1	no	—
12	C₂₃×#431	C₂²	1	2	yes	—
13	C₂₃×#431	C₂²	1	1	yes	characteristic
14	C₂₃×#431	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#107	C₄	1	1	yes	characteristic
16	C₂×C₂₃×#103	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#335	—	2	1	no	—
19	C₂₃×#334	—	2	1	no	—
20	C₂₃×Q₈⋊C₈	—	2	4	no	—
21	C₂₃×#324	—	2	4	no	—
22	C₂₃×#322	—	2	1	no	—
23	C₈²×C₂₃	—	2	1	no	—
24	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
25	C₂×Q₁₆×C₂₃	—	2	1	no	—
26	C₂×Q₁₆×C₂₃	—	2	1	no	—
27	C₂×Q₁₆×C₂₃	—	2	2	no	—
28	C₂×Q₁₆×C₂₃	C₈	1	4	yes	complemented
29	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₂₃	—	2	2	no	—
32	C₄×Q₈×C₂₃	—	2	2	no	—
33	C₄×Q₈×C₂₃	—	2	2	no	—
34	C₄×Q₈×C₂₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₂₃	—	2	2	no	—
38	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
39	C₂₃×#107	—	2	1	no	—
40	C₂₃×#107	—	2	1	no	—
41	C₂₃×C₄⋊C₈	—	2	2	no	—
42	C₂₃×C₄⋊C₈	—	2	1	no	—
43	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
44	C₂₃×#103	—	2	2	no	—
45	C₂₃×#103	—	2	2	no	—
46	C₂₃×C₂²⋊C₈	—	2	1	no	—
47	C₂₃×C₂²⋊C₈	—	2	2	no	—
48	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
49	C₄×C₈×C₂₃	—	2	1	no	—
50	C₄×C₈×C₂₃	—	2	1	no	—
51	C₄×C₈×C₂₃	—	2	2	no	—
52	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
56	C₂×Q₈×C₂₃	—	2	2	no	—
57	C₂×Q₈×C₂₃	—	2	2	no	—
58	C₂×Q₈×C₂₃	—	2	2	no	—
59	C₂×Q₈×C₂₃	—	2	2	no	—
60	C₂×Q₈×C₂₃	—	2	2	no	—
61	C₂×Q₈×C₂₃	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₂₃	—	2	2	no	—
63	C₂²×C₄×C₂₃	—	2	2	no	—
64	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
67	Q₁₆×C₂₃	—	2	4	no	—
68	Q₁₆×C₂₃	—	2	4	no	—
69	C₂×C₈×C₂₃	—	2	1	no	—
70	C₂×C₈×C₂₃	—	2	1	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	—	4	2	no	—
74	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
80	C₄⋊C₄×C₂₃	—	2	2	no	—
81	C₄⋊C₄×C₂₃	—	2	2	no	—
82	C₄⋊C₄×C₂₃	—	4	2	no	—
83	C₄⋊C₄×C₂₃	—	4	2	no	—
84	C₄²×C₂₃	—	4	2	no	—
85	C₄²×C₂₃	—	4	2	no	—
86	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₂₃	#41	1	1	yes	characteristic
90	#9669	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
92	Q₈×C₂₃	—	2	4	no	—
93	Q₈×C₂₃	—	2	4	no	—
94	Q₈×C₂₃	—	4	2	no	—
95	Q₈×C₂₃	—	4	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	4	no	—
98	C₂×C₄×C₂₃	—	4	1	no	—
99	C₂×C₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	—	4	2	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	—	8	2	no	—
103	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
106	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
107	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
108	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
112	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
113	C₈×C₂₃	—	2	1	no	—
114	C₈×C₂₃	—	2	1	no	—
115	C₈×C₂₃	—	4	2	no	—
116	C₈×C₂₃	—	4	2	no	—
117	C₈×C₂₃	—	8	2	no	—
118	C₈×C₂₃	C₂²⋊C₈	1	2	yes	—
119	C₈×C₂₃	C₂²⋊C₈	1	2	yes	—
120	C₂×C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic
121	#1564	C₂×C₂₃	1	1	yes	characteristic
122	#1479	C₂×C₂₃	1	2	yes	—
123	#1461	C₂×C₂₃	1	2	yes	—
124	#1450	C₂×C₂₃	1	1	yes	characteristic
125	C₂²×C₂₃	—	2	1	no	—
126	C₂²×C₂₃	—	2	1	no	—
127	C₂²×C₂₃	#390	1	1	yes	characteristic
128	C₂²×C₂₃	C₈×D₄	1	1	yes	characteristic
129	C₂²×C₂₃	#436	1	1	yes	characteristic
130	C₄×C₂₃	—	2	1	no	—
131	C₄×C₂₃	—	2	1	no	—
132	C₄×C₂₃	—	2	1	no	—
133	C₄×C₂₃	—	4	4	no	—
134	C₄×C₂₃	—	8	2	no	—
135	C₄×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
136	C₄×C₂₃	#408	1	1	yes	characteristic
137	C₄×C₂₃	C₈×D₄	1	1	yes	characteristic
138	C₄×C₂₃	#436	1	1	yes	characteristic
139	C₄×C₂₃	#473	1	2	yes	—
140	C₂²×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
141	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
142	C₄×Q₁₆	—	2	1	no	—
143	C₄×Q₁₆	—	2	1	no	—
144	C₄×Q₁₆	—	2	1	no	—
145	C₄×Q₁₆	—	2	1	no	—
146	#431	C₂²×C₂₃	1	2	yes	—
147	#431	C₂²×C₂₃	1	1	yes	characteristic
148	#431	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×#107	C₄×C₂₃	1	1	yes	characteristic
150	C₂×#103	C₄×C₂₃	1	2	yes	—
151	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
152	#335	—	2	1	no	—
153	#334	—	2	1	no	—
154	Q₈⋊C₈	—	2	4	no	—
155	#324	—	2	4	no	—
156	#322	—	2	1	no	—
157	C₈²	—	2	1	no	—
158	C₂×C₂₃	—	2	1	no	—
159	C₂×C₂₃	—	2	1	no	—
160	C₂×C₂₃	#1954	1	1	yes	characteristic
161	C₂×C₂₃	#1976	1	1	yes	characteristic
162	C₂×C₂₃	#1979	1	1	yes	characteristic
163	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
164	C₂×Q₁₆	—	2	1	no	—
165	C₂×Q₁₆	—	2	1	no	—
166	C₂×Q₁₆	—	2	2	no	—
167	C₂×Q₁₆	C₈×C₂₃	1	4	yes	complemented
168	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
169	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
170	C₄×Q₈	—	2	2	no	—
171	C₄×Q₈	—	2	2	no	—
172	C₄×Q₈	—	2	2	no	—
173	C₄×Q₈	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
176	C₂×C₄²	—	2	2	no	—
177	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
178	#107	—	2	1	no	—
179	#107	—	2	1	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	2	1	no	—
182	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
183	#103	—	2	2	no	—
184	#103	—	2	2	no	—
185	C₂²⋊C₈	—	2	1	no	—
186	C₂²⋊C₈	—	2	2	no	—
187	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
188	C₄×C₈	—	2	1	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	2	2	no	—
191	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
192	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
193	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
194	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
195	C₂₃	#9669	1	1	yes	characteristic, direct product factor, Sylow
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	C₂×C₈×C₂₃	1	4	yes	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
207	Q₁₆	—	2	4	no	—
208	Q₁₆	—	2	4	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
215	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
216	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
217	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
218	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
219	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄²	—	4	2	no	—
225	C₄²	—	4	2	no	—
226	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
227	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
228	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
229	C₄²	#41×C₂₃	1	1	yes	characteristic
230	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
231	Q₈	—	2	4	no	—
232	Q₈	—	2	4	no	—
233	Q₈	—	4	2	no	—
234	Q₈	—	4	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
243	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
244	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
245	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
246	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
247	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
249	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
250	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
251	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	4	2	no	—
255	C₈	—	4	2	no	—
256	C₈	—	8	2	no	—
257	C₈	C₂₃×C₂²⋊C₈	1	2	yes	—
258	C₈	C₂₃×C₂²⋊C₈	1	2	yes	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	C₂₃×#390	1	1	yes	characteristic
262	C₂²	C₈×D₄×C₂₃	1	1	yes	characteristic
263	C₂²	C₂₃×#436	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	4	no	—
268	C₄	—	8	2	no	—
269	C₄	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
270	C₄	C₂₃×#408	1	1	yes	characteristic
271	C₄	C₈×D₄×C₂₃	1	1	yes	characteristic
272	C₄	C₂₃×#436	1	1	yes	characteristic
273	C₄	C₂₃×#473	1	2	yes	—
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	C₂₃×#1954	1	1	yes	characteristic
277	C₂	C₂₃×#1976	1	1	yes	characteristic
278	C₂	C₂₃×#1979	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#390
prime radical	C₂³×C₂₃	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
28. C₂×Q₁₆×C₂₃	C₈	semidirect
90. #9669	C₂₃	direct (N × H)
167. C₂×Q₁₆	C₈×C₂₃	semidirect
195. C₂₃	#9669	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×C₄×Q₁₆×C₂₃	2
C₂₃×#1564	2
C₂₃×#1479	2
C₂₃×#1461	2
C₂₃×#1450	2
#9669	23

Subgroups — C23×#9669 #19157295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9669 #19157295