Subgroups — C₂₃×#13125 #19160751

Order 5888.

324 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×#2151	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#350	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2113	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1764	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1379	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#104	C₄	1	1	yes	characteristic
14	C₂₃×#416	C₄	1	1	yes	characteristic
15	C₂₃×#405	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×C₄⋊C₁₆	—	2	2	no	—
18	C₂₃×#350	—	2	2	no	—
19	C₂₃×C₂²⋊C₁₆	—	2	2	no	—
20	C₂₃×#346	—	2	2	no	—
21	C₂₃×#341	C₄	1	1	yes	characteristic
22	C₂₃×#339	C₄	1	1	yes	characteristic
23	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂₃×#131	—	2	1	no	—
25	C₂₃×#131	—	2	1	no	—
26	C₂₃×#131	—	2	1	no	—
27	C₂₃×#131	—	2	1	no	—
28	C₂×M₁₆×C₂₃	—	2	1	no	—
29	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	—	2	1	no	—
32	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂₃×M₃₂	—	2	1	no	—
37	C₂₃×M₃₂	—	2	1	no	—
38	C₂₃×M₃₂	—	2	1	no	—
39	C₂₃×M₃₂	—	2	1	no	—
40	C₂×C₁₆×C₂₃	—	2	1	no	—
41	C₂×C₁₆×C₂₃	—	2	1	no	—
42	C₂×C₁₆×C₂₃	—	2	1	no	—
43	C₂×C₁₆×C₂₃	—	2	1	no	—
44	C₂×C₁₆×C₂₃	—	2	1	no	—
45	C₂×C₁₆×C₂₃	—	2	1	no	—
46	C₂×C₁₆×C₂₃	D₄	1	2	yes	—
47	C₂×C₁₆×C₂₃	D₄	1	2	yes	—
48	C₂₃×#104	—	2	2	no	—
49	C₂₃×#103	C₈	1	2	yes	—
50	C₂₃×#102	C₈	1	2	yes	—
51	C₂₃×#97	—	2	1	no	—
52	C₄×C₈×C₂₃	—	2	1	no	—
53	C₄×C₈×C₂₃	—	2	1	no	—
54	C₄×C₈×C₂₃	—	2	1	no	—
55	#41×C₂₃	—	2	1	no	—
56	#41×C₂₃	—	2	1	no	—
57	#41×C₂₃	—	2	1	no	—
58	#41×C₂₃	—	2	1	no	—
59	C₂×Q₈×C₂₃	C₂×C₈	1	1	yes	characteristic
60	C₂×D₄×C₂₃	—	2	1	no	—
61	C₂×D₄×C₂₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	—	2	1	no	—
63	C₂²×C₄×C₂₃	—	2	1	no	—
64	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	M₁₆×C₂₃	—	2	1	no	—
66	M₁₆×C₂₃	—	2	1	no	—
67	M₁₆×C₂₃	—	2	1	no	—
68	M₁₆×C₂₃	—	2	1	no	—
69	M₁₆×C₂₃	—	4	1	no	—
70	M₁₆×C₂₃	—	4	1	no	—
71	C₂×C₈×C₂₃	—	2	1	no	—
72	C₂×C₈×C₂₃	—	2	1	no	—
73	C₂×C₈×C₂₃	—	2	1	no	—
74	C₂×C₈×C₂₃	—	2	1	no	—
75	C₂×C₈×C₂₃	—	4	1	no	—
76	C₂×C₈×C₂₃	—	4	1	no	—
77	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
81	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
82	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
84	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
85	C₂²⋊C₄×C₂₃	—	2	1	no	—
86	C₄²×C₂₃	—	2	1	no	—
87	C₄²×C₂₃	—	2	1	no	—
88	C₄²×C₂₃	M₁₆	1	2	yes	—
89	C₁₆×C₂₃	—	2	2	no	—
90	C₁₆×C₂₃	—	2	2	no	—
91	C₁₆×C₂₃	—	4	1	no	—
92	C₁₆×C₂₃	—	4	1	no	—
93	#13125	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₂₃	—	2	1	no	—
95	C₂³×C₂₃	#131	1	1	yes	characteristic
96	Q₈×C₂₃	—	2	1	no	—
97	Q₈×C₂₃	—	2	1	no	—
98	D₄×C₂₃	—	2	1	no	—
99	D₄×C₂₃	—	2	1	no	—
100	D₄×C₂₃	—	4	1	no	—
101	D₄×C₂₃	—	4	1	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	—	4	1	no	—
107	C₂×C₄×C₂₃	—	4	1	no	—
108	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
114	C₈×C₂₃	—	2	2	no	—
115	C₈×C₂₃	—	4	1	no	—
116	C₈×C₂₃	—	4	1	no	—
117	C₈×C₂₃	#121	1	1	yes	characteristic
118	C₈×C₂₃	#123	1	1	yes	characteristic
119	C₈×C₂₃	#124	1	1	yes	characteristic
120	C₈×C₂₃	#126	1	1	yes	characteristic
121	#2151	C₂×C₂₃	1	1	yes	characteristic, complemented
122	C₂×#350	C₂×C₂₃	1	1	yes	characteristic
123	#2113	C₂×C₂₃	1	1	yes	characteristic
124	C₂×#346	C₂×C₂₃	1	1	yes	characteristic, complemented
125	#1764	C₂×C₂₃	1	1	yes	characteristic
126	#1379	C₂×C₂₃	1	1	yes	characteristic, complemented
127	#1376	C₂×C₂₃	1	1	yes	characteristic, complemented
128	C₂²×C₂₃	—	2	2	no	—
129	C₂²×C₂₃	—	4	2	no	—
130	C₂²×C₂₃	—	4	1	no	—
131	C₂²×C₂₃	#433	1	1	yes	characteristic
132	C₂²×C₂₃	C₈×D₄	1	1	yes	characteristic
133	C₂²×C₂₃	#435	1	1	yes	characteristic
134	C₄×C₂₃	—	4	2	no	—
135	C₄×C₂₃	—	4	2	no	—
136	C₄×C₂₃	—	4	1	no	—
137	C₄×C₂₃	#388	1	1	yes	characteristic
138	C₄×C₂₃	#431	1	1	yes	characteristic
139	C₄×C₂₃	C₈×D₄	1	1	yes	characteristic
140	C₄×C₂₃	#436	1	1	yes	characteristic
141	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
143	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
144	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
145	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×#104	C₄×C₂₃	1	1	yes	characteristic
147	#416	C₄×C₂₃	1	1	yes	characteristic
148	#405	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
150	C₄⋊C₁₆	—	2	2	no	—
151	#350	—	2	2	no	—
152	C₂²⋊C₁₆	—	2	2	no	—
153	#346	—	2	2	no	—
154	#341	C₄×C₂₃	1	1	yes	characteristic
155	#339	C₄×C₂₃	1	1	yes	characteristic
156	C₂×C₂₃	—	2	1	no	—
157	C₂×C₂₃	—	2	1	no	—
158	C₂×C₂₃	—	8	1	no	—
159	C₂×C₂₃	#1923	1	1	yes	characteristic
160	C₂×C₂₃	#2171	1	2	yes	—
161	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
162	#131	—	2	1	no	—
163	#131	—	2	1	no	—
164	#131	—	2	1	no	—
165	#131	—	2	1	no	—
166	C₂×M₁₆	—	2	1	no	—
167	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
168	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
172	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
173	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
174	M₃₂	—	2	1	no	—
175	M₃₂	—	2	1	no	—
176	M₃₂	—	2	1	no	—
177	M₃₂	—	2	1	no	—
178	C₂×C₁₆	—	2	1	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	D₄×C₂₃	1	2	yes	—
185	C₂×C₁₆	D₄×C₂₃	1	2	yes	—
186	#104	—	2	2	no	—
187	#103	C₈×C₂₃	1	2	yes	—
188	#102	C₈×C₂₃	1	2	yes	—
189	#97	—	2	1	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₂₃	#13125	1	1	yes	characteristic, direct product factor, Sylow
194	#41	—	2	1	no	—
195	#41	—	2	1	no	—
196	#41	—	2	1	no	—
197	#41	—	2	1	no	—
198	C₂×Q₈	C₂×C₈×C₂₃	1	1	yes	characteristic
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	C₂×C₈×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
204	M₁₆	—	2	1	no	—
205	M₁₆	—	2	1	no	—
206	M₁₆	—	2	1	no	—
207	M₁₆	—	2	1	no	—
208	M₁₆	—	4	1	no	—
209	M₁₆	—	4	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
217	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
218	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
219	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
220	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
221	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
222	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
223	C₄⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	1	no	—
227	C₄²	M₁₆×C₂₃	1	2	yes	—
228	C₁₆	—	2	2	no	—
229	C₁₆	—	2	2	no	—
230	C₁₆	—	4	1	no	—
231	C₁₆	—	4	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	C₂₃×#131	1	1	yes	characteristic
234	Q₈	—	2	1	no	—
235	Q₈	—	2	1	no	—
236	D₄	—	2	1	no	—
237	D₄	—	2	1	no	—
238	D₄	—	4	1	no	—
239	D₄	—	4	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
247	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
248	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
249	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
250	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	4	1	no	—
254	C₈	—	4	1	no	—
255	C₈	C₂₃×#121	1	1	yes	characteristic
256	C₈	C₂₃×#123	1	1	yes	characteristic
257	C₈	C₂₃×#124	1	1	yes	characteristic
258	C₈	C₂₃×#126	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	C₂₃×#433	1	1	yes	characteristic
263	C₂²	C₈×D₄×C₂₃	1	1	yes	characteristic
264	C₂²	C₂₃×#435	1	1	yes	characteristic
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	1	no	—
268	C₄	C₂₃×#388	1	1	yes	characteristic
269	C₄	C₂₃×#431	1	1	yes	characteristic
270	C₄	C₈×D₄×C₂₃	1	1	yes	characteristic
271	C₄	C₂₃×#436	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	8	1	no	—
275	C₂	C₂₃×#1923	1	1	yes	characteristic
276	C₂	C₂₃×#2171	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₈×D₄
prime radical	C₂×#41×C₂₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂²×C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2151	C₂	semidirect
4. C₂×C₂₃×#346	C₂	semidirect
6. C₂₃×#1379	C₂	semidirect
7. C₂₃×#1376	C₂	semidirect
93. #13125	C₂₃	direct (N × H)
121. #2151	C₂×C₂₃	semidirect
124. C₂×#346	C₂×C₂₃	semidirect
126. #1379	C₂×C₂₃	semidirect
127. #1376	C₂×C₂₃	semidirect
193. C₂₃	#13125	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2151	2
C₂×C₂₃×#350	2
C₂₃×#2113	2
C₂×C₂₃×#346	2
C₂₃×#1764	2
C₂₃×#1379	2
C₂₃×#1376	2
#13125	23

Subgroups — C23×#13125 #19160751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13125 #19160751