Subgroups — C₂₃×#13123 #19160749

Order 5888.

312 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2153	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#2117	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#350	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1757	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1379	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1378	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	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#413	C₄	1	1	yes	characteristic
14	C₂₃×#410	C₄	1	1	yes	characteristic
15	C₂₃×#405	C₂²	1	1	yes	characteristic
16	C₂₃×#405	C₂²	1	1	yes	characteristic
17	C₂₃×#364	—	2	2	no	—
18	C₂₃×#350	—	2	2	no	—
19	C₂₃×#350	—	2	1	no	—
20	C₂₃×#350	—	2	1	no	—
21	C₂₃×#346	—	2	1	no	—
22	C₄×C₁₆×C₂₃	—	2	1	no	—
23	C₂₃×#344	C₄	1	1	yes	characteristic
24	C₂₃×#344	C₄	1	1	yes	characteristic
25	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂₃×#131	—	2	1	no	—
27	C₂₃×#131	—	2	1	no	—
28	C₂₃×#131	—	2	1	no	—
29	C₂₃×#131	—	2	1	no	—
30	C₂×M₁₆×C₂₃	—	2	1	no	—
31	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	—	2	1	no	—
35	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
36	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
37	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
38	C₂₃×M₃₂	—	2	1	no	—
39	C₂₃×M₃₂	—	2	1	no	—
40	C₂₃×M₃₂	—	2	1	no	—
41	C₂₃×M₃₂	—	2	1	no	—
42	C₂₃×M₃₂	—	2	1	no	—
43	C₂₃×M₃₂	D₄	1	2	yes	—
44	C₂×C₁₆×C₂₃	—	2	1	no	—
45	C₂×C₁₆×C₂₃	—	2	1	no	—
46	C₂×C₁₆×C₂₃	—	2	1	no	—
47	C₂×C₁₆×C₂₃	—	2	1	no	—
48	C₂×C₁₆×C₂₃	—	2	1	no	—
49	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
50	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
51	C₂₃×#101	C₈	1	2	yes	—
52	C₂₃×#100	C₈	1	2	yes	—
53	C₂₃×#99	—	2	2	no	—
54	C₂₃×#97	—	2	1	no	—
55	C₂₃×#97	—	2	1	no	—
56	C₄×C₈×C₂₃	—	2	1	no	—
57	C₄×C₈×C₂₃	—	2	1	no	—
58	#41×C₂₃	—	2	1	no	—
59	#41×C₂₃	—	2	1	no	—
60	#41×C₂₃	—	2	1	no	—
61	#41×C₂₃	—	2	1	no	—
62	C₂×Q₈×C₂₃	C₂×C₈	1	1	yes	characteristic
63	C₂×D₄×C₂₃	—	2	1	no	—
64	C₂×D₄×C₂₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	—	2	1	no	—
66	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	M₁₆×C₂₃	—	2	1	no	—
68	M₁₆×C₂₃	—	2	1	no	—
69	M₁₆×C₂₃	—	2	1	no	—
70	M₁₆×C₂₃	—	2	1	no	—
71	M₁₆×C₂₃	—	4	1	no	—
72	M₁₆×C₂₃	—	4	1	no	—
73	M₁₆×C₂₃	C₂×C₈	1	2	yes	—
74	M₁₆×C₂₃	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₂₃	—	2	1	no	—
76	C₂×C₈×C₂₃	—	2	1	no	—
77	C₂×C₈×C₂₃	—	2	1	no	—
78	C₂×C₈×C₂₃	—	4	1	no	—
79	C₂×C₈×C₂₃	—	4	1	no	—
80	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
86	C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂²⋊C₄×C₂₃	—	2	1	no	—
89	C₂²⋊C₄×C₂₃	M₁₆	1	2	yes	—
90	C₄²×C₂₃	—	2	1	no	—
91	C₄²×C₂₃	—	2	1	no	—
92	C₁₆×C₂₃	—	2	2	no	—
93	C₁₆×C₂₃	—	2	1	no	—
94	C₁₆×C₂₃	—	2	1	no	—
95	C₁₆×C₂₃	—	4	1	no	—
96	C₁₆×C₂₃	—	4	1	no	—
97	#13123	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₂₃	—	2	1	no	—
99	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
100	Q₈×C₂₃	—	2	1	no	—
101	Q₈×C₂₃	—	2	1	no	—
102	D₄×C₂₃	—	2	1	no	—
103	D₄×C₂₃	—	2	1	no	—
104	D₄×C₂₃	—	4	1	no	—
105	D₄×C₂₃	—	4	1	no	—
106	C₂×C₄×C₂₃	—	2	1	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	4	1	no	—
109	C₂×C₄×C₂₃	—	4	1	no	—
110	C₂×C₄×C₂₃	—	4	1	no	—
111	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
116	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
117	C₈×C₂₃	—	2	2	no	—
118	C₈×C₂₃	—	2	2	no	—
119	C₈×C₂₃	—	4	1	no	—
120	C₈×C₂₃	#121	1	1	yes	characteristic
121	C₈×C₂₃	#123	1	1	yes	characteristic
122	C₈×C₂₃	#124	1	1	yes	characteristic
123	C₈×C₂₃	#126	1	1	yes	characteristic
124	#2153	C₂×C₂₃	1	1	yes	characteristic, complemented
125	#2117	C₂×C₂₃	1	1	yes	characteristic
126	C₂×#350	C₂×C₂₃	1	1	yes	characteristic
127	#2108	C₂×C₂₃	1	1	yes	characteristic, complemented
128	#1757	C₂×C₂₃	1	1	yes	characteristic
129	#1379	C₂×C₂₃	1	1	yes	characteristic, complemented
130	#1378	C₂×C₂₃	1	1	yes	characteristic, complemented
131	C₂²×C₂₃	—	4	1	no	—
132	C₂²×C₂₃	—	4	1	no	—
133	C₂²×C₂₃	—	8	1	no	—
134	C₂²×C₂₃	#431	1	1	yes	characteristic
135	C₂²×C₂₃	#435	1	1	yes	characteristic
136	C₂²×C₂₃	#436	1	1	yes	characteristic
137	C₄×C₂₃	—	4	2	no	—
138	C₄×C₂₃	—	4	1	no	—
139	C₄×C₂₃	—	8	1	no	—
140	C₄×C₂₃	#388	1	1	yes	characteristic
141	C₄×C₂₃	#433	1	1	yes	characteristic
142	C₄×C₂₃	C₈×D₄	1	1	yes	characteristic
143	C₄×C₂₃	C₈×D₄	1	1	yes	characteristic
144	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
147	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
148	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
149	#413	C₄×C₂₃	1	1	yes	characteristic
150	#410	C₄×C₂₃	1	1	yes	characteristic
151	#405	C₂²×C₂₃	1	1	yes	characteristic
152	#405	C₂²×C₂₃	1	1	yes	characteristic
153	#364	—	2	2	no	—
154	#350	—	2	2	no	—
155	#350	—	2	1	no	—
156	#350	—	2	1	no	—
157	#346	—	2	1	no	—
158	C₄×C₁₆	—	2	1	no	—
159	#344	C₄×C₂₃	1	1	yes	characteristic
160	#344	C₄×C₂₃	1	1	yes	characteristic
161	C₂×C₂₃	—	2	1	no	—
162	C₂×C₂₃	—	2	1	no	—
163	C₂×C₂₃	—	2	1	no	—
164	C₂×C₂₃	—	8	1	no	—
165	C₂×C₂₃	#1923	1	1	yes	characteristic
166	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
167	#131	—	2	1	no	—
168	#131	—	2	1	no	—
169	#131	—	2	1	no	—
170	#131	—	2	1	no	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
173	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
174	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
177	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
178	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
179	M₃₂	—	2	1	no	—
180	M₃₂	—	2	1	no	—
181	M₃₂	—	2	1	no	—
182	M₃₂	—	2	1	no	—
183	M₃₂	—	2	1	no	—
184	M₃₂	D₄×C₂₃	1	2	yes	—
185	C₂×C₁₆	—	2	1	no	—
186	C₂×C₁₆	—	2	1	no	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
191	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
192	#101	C₈×C₂₃	1	2	yes	—
193	#100	C₈×C₂₃	1	2	yes	—
194	#99	—	2	2	no	—
195	#97	—	2	1	no	—
196	#97	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	—	2	1	no	—
199	C₂₃	#13123	1	1	yes	characteristic, direct product factor, Sylow
200	#41	—	2	1	no	—
201	#41	—	2	1	no	—
202	#41	—	2	1	no	—
203	#41	—	2	1	no	—
204	C₂×Q₈	C₂×C₈×C₂₃	1	1	yes	characteristic
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	C₂×C₈×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
209	M₁₆	—	2	1	no	—
210	M₁₆	—	2	1	no	—
211	M₁₆	—	2	1	no	—
212	M₁₆	—	2	1	no	—
213	M₁₆	—	4	1	no	—
214	M₁₆	—	4	1	no	—
215	M₁₆	C₂×C₈×C₂₃	1	2	yes	—
216	M₁₆	C₂×C₈×C₂₃	1	2	yes	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
223	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
224	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
225	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
226	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
227	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	2	1	no	—
231	C₂²⋊C₄	M₁₆×C₂₃	1	2	yes	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₁₆	—	2	2	no	—
235	C₁₆	—	2	1	no	—
236	C₁₆	—	2	1	no	—
237	C₁₆	—	4	1	no	—
238	C₁₆	—	4	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
241	Q₈	—	2	1	no	—
242	Q₈	—	2	1	no	—
243	D₄	—	2	1	no	—
244	D₄	—	2	1	no	—
245	D₄	—	4	1	no	—
246	D₄	—	4	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
253	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
256	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
257	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₈	—	4	1	no	—
261	C₈	C₂₃×#121	1	1	yes	characteristic
262	C₈	C₂₃×#123	1	1	yes	characteristic
263	C₈	C₂₃×#124	1	1	yes	characteristic
264	C₈	C₂₃×#126	1	1	yes	characteristic
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	8	1	no	—
268	C₂²	C₂₃×#431	1	1	yes	characteristic
269	C₂²	C₂₃×#435	1	1	yes	characteristic
270	C₂²	C₂₃×#436	1	1	yes	characteristic
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	8	1	no	—
274	C₄	C₂₃×#388	1	1	yes	characteristic
275	C₄	C₂₃×#433	1	1	yes	characteristic
276	C₄	C₈×D₄×C₂₃	1	1	yes	characteristic
277	C₄	C₈×D₄×C₂₃	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	8	1	no	—
282	C₂	C₂₃×#1923	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#121
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#1923
prime radical	C₂×#41×C₂₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2153	C₂	semidirect
4. C₂₃×#2108	C₂	semidirect
6. C₂₃×#1379	C₂	semidirect
7. C₂₃×#1378	C₂	semidirect
97. #13123	C₂₃	direct (N × H)
124. #2153	C₂×C₂₃	semidirect
127. #2108	C₂×C₂₃	semidirect
129. #1379	C₂×C₂₃	semidirect
130. #1378	C₂×C₂₃	semidirect
199. C₂₃	#13123	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×#2153	2
C₂₃×#2117	2
C₂×C₂₃×#350	2
C₂₃×#2108	2
C₂₃×#1757	2
C₂₃×#1379	2
C₂₃×#1378	2
#13123	23

Subgroups — C23×#13123 #19160749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13123 #19160749