Subgroups — C₂₃×#19664 #19167290

Order 5888.

690 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	C₂₃×#3230	C₂	1	2	yes	maximal
2	C₂₃×#3209	C₂	1	2	yes	maximal
3	C₂₃×#3160	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#500	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1723	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1713	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1710	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1450	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#533	C₂²	1	2	yes	—
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂₃×#500	—	2	2	no	—
12	C₂₃×#500	C₂²	1	4	yes	—
13	C₂₃×#500	C₂²	1	1	yes	characteristic
14	C₂₃×#500	C₂²	1	2	yes	—
15	C₂₃×#500	C₂²	1	1	yes	characteristic
16	C₂₃×#500	C₂²	1	2	yes	—
17	C₂₃×#499	—	2	2	no	—
18	C₂₃×#498	C₂²	1	2	yes	—
19	C₂₃×#494	C₂²	1	4	yes	—
20	C₂₃×#487	C₂²	1	4	yes	—
21	C₂₃×#487	C₂²	1	4	yes	—
22	C₂₃×#484	—	2	4	no	—
23	C₂₃×#477	—	2	4	no	—
24	C₂₃×#462	—	2	4	no	—
25	C₂₃×#451	—	2	4	no	—
26	C₂₃×#431	C₂²	1	1	yes	characteristic
27	C₂₃×#431	C₂²	1	1	yes	characteristic
28	C₂₃×#427	—	2	2	no	—
29	C₂×C₂₃×#107	—	2	2	no	—
30	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#322	C₂²	1	2	yes	—
32	C₈²×C₂₃	C₂²	1	2	yes	—
33	C₂²×Q₈×C₂₃	—	2	2	no	—
34	C₂×Q₁₆×C₂₃	—	2	8	no	—
35	C₂²×C₈×C₂₃	D₄	1	2	yes	—
36	C₂₃×C₄⋊Q₈	—	2	2	no	—
37	C₂₃×C₄⋊Q₈	—	2	2	no	—
38	C₂₃×C₄⋊Q₈	—	2	2	no	—
39	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
40	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
41	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
42	C₂₃×#125	—	2	2	no	—
43	C₂₃×C₂²⋊Q₈	—	2	2	no	—
44	C₂₃×C₂²⋊Q₈	—	2	2	no	—
45	C₄×Q₈×C₂₃	—	2	2	no	—
46	C₄×Q₈×C₂₃	—	2	2	no	—
47	C₂₃×#117	—	2	2	no	—
48	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
50	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
51	C₂₃×#107	—	2	4	no	—
52	C₂₃×#107	—	2	4	no	—
53	C₂₃×#107	—	2	2	no	—
54	C₂₃×#107	—	2	2	no	—
55	C₂₃×#107	—	2	8	no	—
56	C₂₃×#106	—	2	4	no	—
57	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
58	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
59	C₂₃×#103	—	2	8	no	—
60	C₂₃×#103	—	2	8	no	—
61	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
62	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
63	C₄×C₈×C₂₃	D₄	1	2	yes	—
64	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₂₃	C₂³	1	2	yes	—
66	C₄×C₈×C₂₃	C₂³	1	2	yes	—
67	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₂×Q₈×C₂₃	—	2	2	no	—
69	C₂×Q₈×C₂₃	—	2	4	no	—
70	C₂×Q₈×C₂₃	—	2	2	no	—
71	C₂×Q₈×C₂₃	—	4	2	no	—
72	C₂×Q₈×C₂₃	—	8	4	no	—
73	C₂²×C₄×C₂₃	—	4	2	no	—
74	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₂₃	—	8	8	no	—
78	C₂×C₈×C₂₃	Q₁₆	1	8	yes	complemented
79	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
81	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
83	C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₄⋊C₄×C₂₃	—	2	4	no	—
85	C₄⋊C₄×C₂₃	—	2	2	no	—
86	C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₄⋊C₄×C₂₃	—	2	4	no	—
88	C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₄⋊C₄×C₂₃	—	4	2	no	—
90	C₄⋊C₄×C₂₃	—	4	2	no	—
91	C₄⋊C₄×C₂₃	—	4	2	no	—
92	C₄⋊C₄×C₂₃	—	4	2	no	—
93	C₂²⋊C₄×C₂₃	—	4	2	no	—
94	C₄²×C₂₃	—	4	2	no	—
95	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
99	#19664	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₂₃	#127	1	1	yes	characteristic
101	Q₈×C₂₃	—	4	4	no	—
102	Q₈×C₂₃	—	4	8	no	—
103	Q₈×C₂₃	—	4	4	no	—
104	Q₈×C₂₃	—	8	4	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	4	4	no	—
110	C₂×C₄×C₂₃	—	8	4	no	—
111	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
112	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂×Q₁₆	1	4	yes	—
115	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
118	C₈×C₂₃	—	2	4	no	—
119	C₈×C₂₃	—	2	4	no	—
120	C₈×C₂₃	C₂×Q₁₆	1	4	yes	—
121	C₈×C₂₃	C₂×Q₁₆	1	4	yes	—
122	#3230	C₂×C₂₃	1	2	yes	—
123	#3209	C₂×C₂₃	1	2	yes	—
124	#3160	C₂×C₂₃	1	1	yes	characteristic
125	C₂×#500	C₂×C₂₃	1	1	yes	characteristic
126	#1723	C₂×C₂₃	1	4	yes	complemented
127	#1713	C₂×C₂₃	1	2	yes	complemented
128	#1710	C₂×C₂₃	1	2	yes	complemented
129	#1450	C₂×C₂₃	1	1	yes	characteristic
130	C₂²×C₂₃	—	2	1	no	—
131	C₂²×C₂₃	—	2	1	no	—
132	C₂²×C₂₃	#494	1	2	yes	—
133	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
134	C₄×C₂₃	—	2	1	no	—
135	C₄×C₂₃	—	2	1	no	—
136	C₄×C₂₃	—	8	4	no	—
137	C₄×C₂₃	—	8	2	no	—
138	C₄×C₂₃	—	8	2	no	—
139	C₄×C₂₃	#494	1	2	yes	—
140	C₄×C₂₃	C₂²×Q₁₆	1	2	yes	—
141	C₄×C₂₃	#576	1	2	yes	—
142	C₄×C₂₃	#578	1	2	yes	—
143	#533	C₂²×C₂₃	1	2	yes	—
144	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
145	#500	—	2	2	no	—
146	#500	C₂²×C₂₃	1	2	yes	—
147	#500	C₂²×C₂₃	1	2	yes	—
148	#500	C₂²×C₂₃	1	1	yes	characteristic
149	#500	C₂²×C₂₃	1	1	yes	characteristic
150	#500	C₂²×C₂₃	1	4	yes	—
151	#499	—	2	2	no	—
152	#498	C₂²×C₂₃	1	2	yes	—
153	#494	C₂²×C₂₃	1	4	yes	—
154	#487	C₂²×C₂₃	1	4	yes	—
155	#487	C₂²×C₂₃	1	4	yes	—
156	#484	—	2	4	no	—
157	#477	—	2	4	no	—
158	#462	—	2	4	no	—
159	#451	—	2	4	no	—
160	#431	C₂²×C₂₃	1	1	yes	characteristic
161	#431	C₂²×C₂₃	1	1	yes	characteristic
162	#427	—	2	2	no	—
163	C₂×#107	—	2	2	no	—
164	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
165	#322	C₂²×C₂₃	1	2	yes	—
166	C₈²	C₂²×C₂₃	1	2	yes	—
167	C₂×C₂₃	—	2	2	no	—
168	C₂×C₂₃	C₂×#494	1	1	yes	characteristic
169	C₂×C₂₃	#3155	1	1	yes	characteristic
170	C₂×C₂₃	#3156	1	1	yes	characteristic
171	C₂²×Q₈	—	2	2	no	—
172	C₂×Q₁₆	—	2	8	no	—
173	C₂²×C₈	D₄×C₂₃	1	2	yes	—
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	—	2	2	no	—
176	C₄⋊Q₈	—	2	2	no	—
177	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
178	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
179	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
180	#125	—	2	2	no	—
181	C₂²⋊Q₈	—	2	2	no	—
182	C₂²⋊Q₈	—	2	2	no	—
183	C₄×Q₈	—	2	2	no	—
184	C₄×Q₈	—	2	2	no	—
185	#117	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
189	#107	—	2	4	no	—
190	#107	—	2	4	no	—
191	#107	—	2	2	no	—
192	#107	—	2	2	no	—
193	#107	—	2	8	no	—
194	#106	—	2	4	no	—
195	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
196	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
197	#103	—	2	8	no	—
198	#103	—	2	8	no	—
199	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
200	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
201	C₄×C₈	D₄×C₂₃	1	2	yes	—
202	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
203	C₄×C₈	C₂³×C₂₃	1	2	yes	—
204	C₄×C₈	C₂³×C₂₃	1	2	yes	—
205	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
206	C₂₃	#19664	1	1	yes	characteristic, direct product factor, Sylow
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	4	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	—	8	4	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
216	Q₁₆	—	8	8	no	—
217	C₂×C₈	Q₁₆×C₂₃	1	8	yes	complemented
218	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
219	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
220	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
221	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₂²⋊C₄	—	4	2	no	—
233	C₄²	—	4	2	no	—
234	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
235	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
236	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
237	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
238	C₂³	C₂₃×#127	1	1	yes	characteristic
239	Q₈	—	4	4	no	—
240	Q₈	—	4	8	no	—
241	Q₈	—	4	4	no	—
242	Q₈	—	8	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
250	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
251	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
252	C₂×C₄	C₂×Q₁₆×C₂₃	1	4	yes	—
253	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
256	C₈	—	2	4	no	—
257	C₈	—	2	4	no	—
258	C₈	C₂×Q₁₆×C₂₃	1	4	yes	—
259	C₈	C₂×Q₁₆×C₂₃	1	4	yes	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	C₂₃×#494	1	2	yes	—
263	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	8	4	no	—
267	C₄	—	8	2	no	—
268	C₄	—	8	2	no	—
269	C₄	C₂₃×#494	1	2	yes	—
270	C₄	C₂²×Q₁₆×C₂₃	1	2	yes	—
271	C₄	C₂₃×#576	1	2	yes	—
272	C₄	C₂₃×#578	1	2	yes	—
273	C₂	—	2	2	no	—
274	C₂	C₂×C₂₃×#494	1	1	yes	characteristic
275	C₂	C₂₃×#3155	1	1	yes	characteristic
276	C₂	C₂₃×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂³×C₂₃	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₂₃×#1723	C₂	semidirect
6. C₂₃×#1713	C₂	semidirect
7. C₂₃×#1710	C₂	semidirect
78. C₂×C₈×C₂₃	Q₁₆	semidirect
99. #19664	C₂₃	direct (N × H)
126. #1723	C₂×C₂₃	semidirect
127. #1713	C₂×C₂₃	semidirect
128. #1710	C₂×C₂₃	semidirect
206. C₂₃	#19664	direct (N × H)
217. C₂×C₈	Q₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3230	2
C₂₃×#3209	2
C₂₃×#3160	2
C₂×C₂₃×#500	2
C₂₃×#1723	2
C₂₃×#1713	2
C₂₃×#1710	2
C₂₃×#1450	2
#19664	23

Subgroups — C23×#19664 #19167290

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19664 #19167290