Subgroups — C₂₃×#13142 #19160768

Order 5888.

356 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#361	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2140	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1972	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1773	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1386	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1375	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₂₃	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×#108	C₂²	1	1	yes	characteristic
12	C₂₃×#427	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#102	—	2	1	no	—
15	C₂₃×#405	—	2	1	no	—
16	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
17	C₂₃×#390	—	2	1	no	—
18	C₂₃×#365	—	2	2	no	—
19	C₂₃×#361	—	2	2	no	—
20	C₂₃×#357	C₄	1	4	yes	complemented
21	C₂₃×#339	—	2	1	no	—
22	C₂²×D₄×C₂₃	—	2	1	no	—
23	C₂×D₈×C₂₃	—	2	1	no	—
24	C₂×D₈×C₂₃	—	2	1	no	—
25	C₂×D₈×C₂₃	—	2	1	no	—
26	C₂×D₈×C₂₃	—	2	1	no	—
27	C₂×D₈×C₂₃	—	2	1	no	—
28	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₂₃	—	2	1	no	—
30	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
32	C₂₃×#117	—	2	1	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
34	C₂×C₂²⋊C₄×C₂₃	—	4	2	no	—
35	C₂×C₄²×C₂₃	D₄	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	C₂×C₄	1	2	yes	—
42	C₂₃×#108	—	2	1	no	—
43	C₂₃×#108	D₄	1	2	yes	—
44	C₂₃×#107	C₂×C₄	1	2	yes	—
45	C₂₃×#106	—	2	1	no	—
46	C₂₃×#106	—	2	1	no	—
47	C₂₃×#106	—	2	1	no	—
48	C₂₃×#102	—	4	2	no	—
49	C₂₃×#97	—	2	1	no	—
50	C₂₃×#97	—	2	1	no	—
51	C₂₃×#97	—	4	1	no	—
52	C₄×C₈×C₂₃	—	4	1	no	—
53	C₂⁴×C₂₃	—	4	1	no	—
54	C₂×D₄×C₂₃	—	2	2	no	—
55	C₂×D₄×C₂₃	—	4	1	no	—
56	C₂×D₄×C₂₃	—	4	1	no	—
57	C₂×D₄×C₂₃	—	4	1	no	—
58	C₂×D₄×C₂₃	—	4	1	no	—
59	C₂×D₄×C₂₃	—	4	1	no	—
60	C₂²×C₄×C₂₃	—	2	1	no	—
61	C₂²×C₄×C₂₃	—	4	1	no	—
62	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
63	D₈×C₂₃	—	2	2	no	—
64	D₈×C₂₃	—	2	2	no	—
65	D₈×C₂₃	—	4	1	no	—
66	D₈×C₂₃	—	4	1	no	—
67	M₁₆×C₂₃	—	2	2	no	—
68	M₁₆×C₂₃	—	4	1	no	—
69	C₂×C₈×C₂₃	—	2	1	no	—
70	C₂×C₈×C₂₃	—	2	1	no	—
71	C₂×C₈×C₂₃	—	4	1	no	—
72	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
73	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
76	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
77	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
78	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
79	C₄⋊C₄×C₂₃	—	2	2	no	—
80	C₄⋊C₄×C₂₃	—	4	1	no	—
81	C₄⋊C₄×C₂₃	—	4	1	no	—
82	C₂²⋊C₄×C₂₃	—	4	1	no	—
83	C₂²⋊C₄×C₂₃	—	8	4	no	—
84	C₄²×C₂₃	—	2	1	no	—
85	C₄²×C₂₃	—	2	1	no	—
86	C₄²×C₂₃	—	4	1	no	—
87	C₁₆×C₂₃	—	2	2	no	—
88	C₁₆×C₂₃	—	4	1	no	—
89	#13142	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₃	—	4	2	no	—
91	C₂³×C₂₃	—	8	2	no	—
92	C₂³×C₂₃	—	8	2	no	—
93	C₂³×C₂₃	—	8	1	no	—
94	C₂³×C₂₃	—	8	1	no	—
95	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
96	D₄×C₂₃	—	4	2	no	—
97	D₄×C₂₃	—	4	2	no	—
98	D₄×C₂₃	—	8	1	no	—
99	D₄×C₂₃	—	8	1	no	—
100	C₂×C₄×C₂₃	—	4	2	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	—	4	1	no	—
103	C₂×C₄×C₂₃	—	8	2	no	—
104	C₂×C₄×C₂₃	—	8	1	no	—
105	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
107	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
110	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₈×C₂₃	—	2	2	no	—
112	C₈×C₂₃	—	4	2	no	—
113	C₈×C₂₃	#117	1	1	yes	characteristic
114	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
115	C₈×C₂₃	#124	1	1	yes	characteristic
116	C₈×C₂₃	#126	1	1	yes	characteristic
117	C₂×#365	C₂×C₂₃	1	1	yes	characteristic, complemented
118	C₂×#361	C₂×C₂₃	1	1	yes	characteristic
119	#2140	C₂×C₂₃	1	1	yes	characteristic
120	#1972	C₂×C₂₃	1	1	yes	characteristic
121	#1773	C₂×C₂₃	1	1	yes	characteristic, complemented
122	#1386	C₂×C₂₃	1	1	yes	characteristic, complemented
123	#1375	C₂×C₂₃	1	1	yes	characteristic, complemented
124	C₂²×C₂₃	—	2	2	no	—
125	C₂²×C₂₃	—	8	4	no	—
126	C₂²×C₂₃	—	8	2	no	—
127	C₂²×C₂₃	—	16	2	no	—
128	C₂²×C₂₃	—	16	1	no	—
129	C₂²×C₂₃	—	16	1	no	—
130	C₂²×C₂₃	#442	1	1	yes	characteristic
131	C₂²×C₂₃	#460	1	1	yes	characteristic
132	C₂²×C₂₃	#482	1	1	yes	characteristic
133	C₄×C₂₃	—	8	1	no	—
134	C₄×C₂₃	—	8	2	no	—
135	C₄×C₂₃	—	16	1	no	—
136	C₄×C₂₃	#388	1	1	yes	characteristic
137	C₄×C₂₃	C₄×SD₁₆	1	1	yes	characteristic
138	C₄×C₂₃	#463	1	1	yes	characteristic
139	C₄×C₂₃	#481	1	1	yes	characteristic
140	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
141	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
143	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
144	#427	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×#102	—	2	1	no	—
147	#405	—	2	1	no	—
148	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
149	#390	—	2	1	no	—
150	#365	—	2	2	no	—
151	#361	—	2	2	no	—
152	#357	C₄×C₂₃	1	4	yes	complemented
153	#339	—	2	1	no	—
154	C₂×C₂₃	—	2	1	no	—
155	C₂×C₂₃	—	2	1	no	—
156	C₂×C₂₃	—	16	2	no	—
157	C₂×C₂₃	#1926	1	1	yes	characteristic
158	C₂×C₂₃	AGL(1,ℤ/16)	1	2	yes	—
159	C₂²×D₄	—	2	1	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	—	2	1	no	—
162	C₂×D₈	—	2	1	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
166	C₂×M₁₆	—	2	1	no	—
167	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
169	#117	—	2	1	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₂²⋊C₄	—	4	2	no	—
172	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
173	M₃₂	—	2	1	no	—
174	M₃₂	—	2	1	no	—
175	M₃₂	—	2	1	no	—
176	M₃₂	—	2	1	no	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
179	#108	—	2	1	no	—
180	#108	D₄×C₂₃	1	2	yes	—
181	#107	C₂×C₄×C₂₃	1	2	yes	—
182	#106	—	2	1	no	—
183	#106	—	2	1	no	—
184	#106	—	2	1	no	—
185	#102	—	4	2	no	—
186	#97	—	2	1	no	—
187	#97	—	2	1	no	—
188	#97	—	4	1	no	—
189	C₄×C₈	—	4	1	no	—
190	C₂₃	#13142	1	1	yes	characteristic, direct product factor, Sylow
191	C₂⁴	—	4	1	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
201	D₈	—	2	2	no	—
202	D₈	—	2	2	no	—
203	D₈	—	4	1	no	—
204	D₈	—	4	1	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	4	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
211	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
212	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
213	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
214	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
215	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
216	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₂²⋊C₄	—	4	1	no	—
221	C₂²⋊C₄	—	8	4	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	4	1	no	—
225	C₁₆	—	2	2	no	—
226	C₁₆	—	4	1	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	8	2	no	—
229	C₂³	—	8	2	no	—
230	C₂³	—	8	1	no	—
231	C₂³	—	8	1	no	—
232	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
233	D₄	—	4	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	8	1	no	—
236	D₄	—	8	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	1	no	—
242	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
243	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
244	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
245	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
246	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
247	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	4	2	no	—
250	C₈	C₂₃×#117	1	1	yes	characteristic
251	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
252	C₈	C₂₃×#124	1	1	yes	characteristic
253	C₈	C₂₃×#126	1	1	yes	characteristic
254	C₂²	—	2	2	no	—
255	C₂²	—	8	4	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	16	2	no	—
258	C₂²	—	16	1	no	—
259	C₂²	—	16	1	no	—
260	C₂²	C₂₃×#442	1	1	yes	characteristic
261	C₂²	C₂₃×#460	1	1	yes	characteristic
262	C₂²	C₂₃×#482	1	1	yes	characteristic
263	C₄	—	8	1	no	—
264	C₄	—	8	2	no	—
265	C₄	—	16	1	no	—
266	C₄	C₂₃×#388	1	1	yes	characteristic
267	C₄	C₄×SD₁₆×C₂₃	1	1	yes	characteristic
268	C₄	C₂₃×#463	1	1	yes	characteristic
269	C₄	C₂₃×#481	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	16	2	no	—
273	C₂	C₂₃×#1926	1	1	yes	characteristic
274	C₂	C₂₃×AGL(1,ℤ/16)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	#442
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₂₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#365	C₂	semidirect
5. C₂₃×#1773	C₂	semidirect
6. C₂₃×#1386	C₂	semidirect
7. C₂₃×#1375	C₂	semidirect
20. C₂₃×#357	C₄	semidirect
89. #13142	C₂₃	direct (N × H)
117. C₂×#365	C₂×C₂₃	semidirect
121. #1773	C₂×C₂₃	semidirect
122. #1386	C₂×C₂₃	semidirect
123. #1375	C₂×C₂₃	semidirect
152. #357	C₄×C₂₃	semidirect
190. C₂₃	#13142	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×C₂₃×#365	2
C₂×C₂₃×#361	2
C₂₃×#2140	2
C₂₃×#1972	2
C₂₃×#1773	2
C₂₃×#1386	2
C₂₃×#1375	2
#13142	23

Subgroups — C23×#13142 #19160768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13142 #19160768