Subgroups — C₂₃×#21283 #19168909

Order 5888.

766 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	C₂×C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2320	C₂	1	2	yes	maximal
3	C₂₃×#1971	C₂	1	2	yes	maximal
4	C₂₃×#1943	C₂	1	2	yes	maximal
5	C₂₃×#1933	C₂	1	4	yes	maximal
6	C₄×C₂₃×#106	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1763	C₂	1	2	yes	maximal
8	C₂₃×#1726	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
11	C₄×Q₁₆×C₂₃	C₄	1	8	yes	complemented
12	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#106	C₂²	1	2	yes	—
14	C₂×C₂₃×#103	C₂²	1	4	yes	—
15	C₂×C₂₃×#103	C₂²	1	2	yes	—
16	C₂×C₂₃×#97	C₂²	1	2	yes	—
17	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#391	C₂²	1	4	yes	—
21	C₂₃×#389	C₂²	1	4	yes	—
22	C₂₃×#387	—	2	4	no	—
23	C₂₃×#383	—	2	2	no	—
24	C₂₃×#382	—	2	2	no	—
25	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₂₃×#376	C₂²	1	1	yes	characteristic
27	C₂₃×#340	C₂²	1	2	yes	—
28	C₂₃×#340	C₂²	1	4	yes	—
29	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
30	C₂×Q₁₆×C₂₃	—	2	4	no	—
31	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₂₃	—	2	4	no	—
36	C₄×Q₈×C₂₃	C₂×C₄	1	8	yes	—
37	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
41	C₂×C₄²×C₂₃	—	2	2	no	—
42	C₂×C₄²×C₂₃	—	2	2	no	—
43	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
44	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₂₃×#107	C₂×C₄	1	4	yes	—
46	C₂₃×#106	—	2	4	no	—
47	C₂₃×#103	—	2	8	no	—
48	C₂₃×#103	C₂×C₄	1	8	yes	—
49	C₂₃×#97	—	2	4	no	—
50	C₂₃×#97	—	2	2	no	—
51	C₄×C₈×C₂₃	—	2	4	no	—
52	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
53	C₂₃×#95	—	2	2	no	—
54	C₂₃×#95	—	2	4	no	—
55	C₂₃×#95	—	2	4	no	—
56	C₂₃×#95	—	2	4	no	—
57	C₂₃×#95	D₄	1	2	yes	—
58	C₂₃×#95	D₄	1	2	yes	—
59	C₂×Q₈×C₂₃	—	2	2	no	—
60	C₂×Q₈×C₂₃	—	2	4	no	—
61	C₂×Q₈×C₂₃	—	2	4	no	—
62	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₂₃	—	2	2	no	—
64	C₂²×C₄×C₂₃	—	2	2	no	—
65	C₂²×C₄×C₂₃	—	2	4	no	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
70	Q₁₆×C₂₃	—	2	8	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₂₃	#41	1	4	yes	—
77	C₂×C₈×C₂₃	#41	1	4	yes	—
78	C₄⋊C₄×C₂₃	—	2	8	no	—
79	C₄⋊C₄×C₂₃	—	2	2	no	—
80	C₄⋊C₄×C₂₃	—	2	4	no	—
81	C₄⋊C₄×C₂₃	—	2	4	no	—
82	C₄⋊C₄×C₂₃	—	8	4	no	—
83	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
84	C₄²×C₂₃	—	2	8	no	—
85	C₄²×C₂₃	—	2	2	no	—
86	C₄²×C₂₃	—	2	2	no	—
87	C₄²×C₂₃	—	8	2	no	—
88	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₂₃	#41	1	2	yes	—
90	#21283	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
92	Q₈×C₂₃	—	2	4	no	—
93	Q₈×C₂₃	—	2	4	no	—
94	Q₈×C₂₃	—	4	4	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	4	no	—
98	C₂×C₄×C₂₃	—	2	4	no	—
99	C₂×C₄×C₂₃	—	2	4	no	—
100	C₂×C₄×C₂₃	—	2	4	no	—
101	C₂×C₄×C₂₃	—	2	4	no	—
102	C₂×C₄×C₂₃	—	4	4	no	—
103	C₂×C₄×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	4	4	no	—
106	C₂×C₄×C₂₃	—	4	4	no	—
107	C₂×C₄×C₂₃	—	4	4	no	—
108	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
112	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
113	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
114	C₈×C₂₃	—	2	2	no	—
115	C₈×C₂₃	—	2	4	no	—
116	C₈×C₂₃	#117	1	4	yes	—
117	C₂×C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic
118	#2320	C₂×C₂₃	1	2	yes	—
119	#1971	C₂×C₂₃	1	2	yes	—
120	#1943	C₂×C₂₃	1	2	yes	—
121	#1933	C₂×C₂₃	1	4	yes	—
122	C₄×#106	C₂×C₂₃	1	1	yes	characteristic
123	#1763	C₂×C₂₃	1	2	yes	—
124	#1726	C₂×C₂₃	1	1	yes	characteristic
125	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
126	C₂²×C₂₃	#532	1	2	yes	—
127	C₂²×C₂₃	#577	1	2	yes	—
128	C₂²×C₂₃	#578	1	2	yes	—
129	C₄×C₂₃	—	2	2	no	—
130	C₄×C₂₃	—	2	2	no	—
131	C₄×C₂₃	—	4	4	no	—
132	C₄×C₂₃	—	4	4	no	—
133	C₄×C₂₃	—	4	4	no	—
134	C₄×C₂₃	—	8	4	no	—
135	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
136	C₄×C₂₃	C₄×#41	1	1	yes	characteristic
137	C₄×C₂₃	#529	1	2	yes	—
138	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
139	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
140	C₄×Q₁₆	C₄×C₂₃	1	8	yes	complemented
141	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×#106	C₂²×C₂₃	1	2	yes	—
143	C₂×#103	C₂²×C₂₃	1	4	yes	—
144	C₂×#103	C₂²×C₂₃	1	2	yes	—
145	C₂×#97	C₂²×C₂₃	1	2	yes	—
146	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
148	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
149	#391	C₂²×C₂₃	1	4	yes	—
150	#389	C₂²×C₂₃	1	4	yes	—
151	#387	—	2	4	no	—
152	#383	—	2	2	no	—
153	#382	—	2	2	no	—
154	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
155	#376	C₂²×C₂₃	1	1	yes	characteristic
156	#340	C₂²×C₂₃	1	2	yes	—
157	#340	C₂²×C₂₃	1	4	yes	—
158	C₂×C₂₃	#2293	1	1	yes	characteristic
159	C₂×C₂₃	#2950	1	1	yes	characteristic
160	C₂×C₂₃	#2952	1	1	yes	characteristic
161	C₂×C₂₃	#3178	1	4	yes	—
162	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
163	C₂×Q₁₆	—	2	4	no	—
164	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	—
165	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
167	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
168	C₄×Q₈	—	2	4	no	—
169	C₄×Q₈	C₂×C₄×C₂₃	1	8	yes	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	4	no	—
172	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
173	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
177	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
178	#107	C₂×C₄×C₂₃	1	4	yes	—
179	#106	—	2	4	no	—
180	#103	—	2	8	no	—
181	#103	C₂×C₄×C₂₃	1	8	yes	—
182	#97	—	2	4	no	—
183	#97	—	2	2	no	—
184	C₄×C₈	—	2	4	no	—
185	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
186	#95	—	2	2	no	—
187	#95	—	2	4	no	—
188	#95	—	2	4	no	—
189	#95	—	2	4	no	—
190	#95	D₄×C₂₃	1	2	yes	—
191	#95	D₄×C₂₃	1	2	yes	—
192	C₂₃	#21283	1	1	yes	characteristic, direct product factor, Sylow
193	C₂×Q₈	—	2	2	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
201	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
204	Q₁₆	—	2	8	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
209	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
210	C₂×C₈	#41×C₂₃	1	4	yes	—
211	C₂×C₈	#41×C₂₃	1	4	yes	—
212	C₄⋊C₄	—	2	8	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	8	4	no	—
217	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
218	C₄²	—	2	8	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	8	2	no	—
222	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
223	C₄²	#41×C₂₃	1	2	yes	—
224	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
225	Q₈	—	2	4	no	—
226	Q₈	—	2	4	no	—
227	Q₈	—	4	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
242	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
243	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
244	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
245	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
246	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	2	4	no	—
249	C₈	C₂₃×#117	1	4	yes	—
250	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
251	C₂²	C₂₃×#532	1	2	yes	—
252	C₂²	C₂₃×#577	1	2	yes	—
253	C₂²	C₂₃×#578	1	2	yes	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	4	no	—
257	C₄	—	4	4	no	—
258	C₄	—	4	4	no	—
259	C₄	—	8	4	no	—
260	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
261	C₄	C₄×#41×C₂₃	1	1	yes	characteristic
262	C₄	C₂₃×#529	1	2	yes	—
263	C₂	C₂₃×#2293	1	1	yes	characteristic
264	C₂	C₂₃×#2950	1	1	yes	characteristic
265	C₂	C₂₃×#2952	1	1	yes	characteristic
266	C₂	C₂₃×#3178	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
11. C₄×Q₁₆×C₂₃	C₄	semidirect
90. #21283	C₂₃	direct (N × H)
140. C₄×Q₁₆	C₄×C₂₃	semidirect
192. C₂₃	#21283	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×#2320	2
C₂₃×#1971	2
C₂₃×#1943	2
C₂₃×#1933	2
C₄×C₂₃×#106	2
C₂₃×#1763	2
C₂₃×#1726	2
#21283	23

Subgroups — C23×#21283 #19168909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21283 #19168909