Subgroups — C₂₃×#9743 #19157369

Order 5888.

642 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#493	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1982	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#331	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₂₃	—	2	1	no	—
7	C₂²×D₈×C₂₃	—	2	1	no	—
8	C₂×C₂₃×#127	C₂²	1	2	yes	—
9	C₂₃×#493	—	2	2	no	—
10	C₂₃×#493	C₄	1	4	yes	complemented
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#102	—	2	4	no	—
14	C₂×C₂₃×#97	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#336	—	2	1	no	—
18	C₂₃×#334	—	2	2	no	—
19	C₂₃×#331	—	2	4	no	—
20	C₂²×D₄×C₂₃	—	2	2	no	—
21	C₂²×D₄×C₂₃	—	2	2	no	—
22	C₂²×D₄×C₂₃	—	2	2	no	—
23	C₂×D₈×C₂₃	—	2	4	no	—
24	C₂×D₈×C₂₃	—	2	4	no	—
25	C₂×D₈×C₂₃	—	4	2	no	—
26	C₂×D₈×C₂₃	—	4	2	no	—
27	C₂×D₈×C₂₃	—	4	2	no	—
28	C₂×D₈×C₂₃	—	4	2	no	—
29	C₂²×C₈×C₂₃	—	2	1	no	—
30	C₂²×C₈×C₂₃	—	2	2	no	—
31	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
33	C₂₃×#127	—	2	2	no	—
34	C₂₃×#127	—	2	2	no	—
35	C₂₃×#127	—	2	2	no	—
36	C₂₃×#127	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
38	C₂×C₄²×C₂₃	—	2	1	no	—
39	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×C₄⋊C₈	—	2	2	no	—
41	C₂₃×C₄⋊C₈	—	2	2	no	—
42	C₂₃×C₄⋊C₈	—	2	1	no	—
43	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
44	C₂₃×#102	—	4	8	no	—
45	C₂₃×#97	—	4	2	no	—
46	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₂₃	D₄	1	2	yes	—
48	C₂₃×#95	—	2	1	no	—
49	C₂⁴×C₂₃	—	4	2	no	—
50	C₂×D₄×C₂₃	—	2	4	no	—
51	C₂×D₄×C₂₃	—	2	4	no	—
52	C₂×D₄×C₂₃	—	2	4	no	—
53	C₂×D₄×C₂₃	—	4	2	no	—
54	C₂×D₄×C₂₃	—	4	2	no	—
55	C₂×D₄×C₂₃	—	4	2	no	—
56	C₂×D₄×C₂₃	—	4	2	no	—
57	C₂×D₄×C₂₃	—	4	2	no	—
58	C₂×D₄×C₂₃	—	4	4	no	—
59	C₂×D₄×C₂₃	—	4	2	no	—
60	C₂×D₄×C₂₃	—	4	2	no	—
61	C₂×D₄×C₂₃	—	4	4	no	—
62	C₂×D₄×C₂₃	—	8	4	no	—
63	C₂²×C₄×C₂₃	—	2	2	no	—
64	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
67	D₈×C₂₃	—	4	8	no	—
68	D₈×C₂₃	—	4	8	no	—
69	C₂×C₈×C₂₃	—	2	1	no	—
70	C₂×C₈×C₂₃	—	2	1	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	4	1	no	—
73	C₂×C₈×C₂₃	—	4	1	no	—
74	C₂×C₈×C₂₃	—	4	2	no	—
75	C₂×C₈×C₂₃	—	4	4	no	—
76	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₂₃	D₈	1	2	yes	—
79	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
80	C₄⋊C₄×C₂₃	—	2	4	no	—
81	C₄²×C₂₃	—	4	1	no	—
82	C₄²×C₂₃	—	4	1	no	—
83	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₂₃	#41	1	1	yes	characteristic
87	#9743	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₂₃	—	4	4	no	—
89	C₂³×C₂₃	—	8	2	no	—
90	C₂³×C₂₃	—	8	2	no	—
91	C₂³×C₂₃	—	8	4	no	—
92	C₂³×C₂₃	—	8	4	no	—
93	C₂³×C₂₃	#121	1	1	yes	characteristic
94	D₄×C₂₃	—	4	4	no	—
95	D₄×C₂₃	—	4	4	no	—
96	D₄×C₂₃	—	4	8	no	—
97	D₄×C₂₃	—	8	4	no	—
98	D₄×C₂₃	—	8	4	no	—
99	D₄×C₂₃	—	8	2	no	—
100	D₄×C₂₃	—	8	2	no	—
101	D₄×C₂₃	—	8	2	no	—
102	D₄×C₂₃	—	8	2	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
112	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
118	C₈×C₂₃	—	2	2	no	—
119	C₈×C₂₃	—	4	2	no	—
120	C₈×C₂₃	—	8	2	no	—
121	C₈×C₂₃	#102	1	4	yes	complemented
122	C₂×#493	C₂×C₂₃	1	1	yes	characteristic
123	#1982	C₂×C₂₃	1	2	yes	complemented
124	#1770	C₂×C₂₃	1	1	yes	characteristic, complemented
125	C₂×#334	C₂×C₂₃	1	1	yes	characteristic, complemented
126	C₂×#331	C₂×C₂₃	1	2	yes	complemented
127	C₂²×C₂₃	—	8	4	no	—
128	C₂²×C₂₃	—	8	4	no	—
129	C₂²×C₂₃	—	8	4	no	—
130	C₂²×C₂₃	—	16	4	no	—
131	C₂²×C₂₃	—	16	4	no	—
132	C₂²×C₂₃	#390	1	1	yes	characteristic
133	C₂²×C₂₃	#459	1	1	yes	characteristic
134	C₂²×C₂₃	#460	1	1	yes	characteristic
135	C₂²×C₂₃	#469	1	2	yes	—
136	C₂²×C₂₃	#472	1	2	yes	—
137	C₄×C₂₃	—	2	1	no	—
138	C₄×C₂₃	—	2	1	no	—
139	C₄×C₂₃	—	8	2	no	—
140	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
141	C₄×C₂₃	#418	1	1	yes	characteristic
142	C₄×C₂₃	#442	1	2	yes	—
143	C₄×C₂₃	#459	1	1	yes	characteristic
144	C₄×C₂₃	#460	1	1	yes	characteristic
145	C₄×C₂₃	#496	1	2	yes	—
146	C₂²×D₈	—	2	1	no	—
147	C₂²×D₈	—	2	1	no	—
148	C₂×#127	C₂²×C₂₃	1	2	yes	—
149	#493	—	2	2	no	—
150	#493	C₄×C₂₃	1	4	yes	complemented
151	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
152	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×#102	—	2	4	no	—
154	C₂×#97	—	2	1	no	—
155	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
156	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
157	#336	—	2	1	no	—
158	#334	—	2	2	no	—
159	#331	—	2	4	no	—
160	C₂×C₂₃	—	16	4	no	—
161	C₂×C₂₃	#1687	1	2	yes	—
162	C₂×C₂₃	#1688	1	2	yes	—
163	C₂×C₂₃	#1972	1	1	yes	characteristic
164	C₂×C₂₃	#1978	1	1	yes	characteristic
165	C₂×C₂₃	#1982	1	1	yes	characteristic
166	C₂²×D₄	—	2	2	no	—
167	C₂²×D₄	—	2	2	no	—
168	C₂²×D₄	—	2	2	no	—
169	C₂×D₈	—	2	4	no	—
170	C₂×D₈	—	2	4	no	—
171	C₂×D₈	—	4	2	no	—
172	C₂×D₈	—	4	2	no	—
173	C₂×D₈	—	4	2	no	—
174	C₂×D₈	—	4	2	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
178	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
179	#127	—	2	2	no	—
180	#127	—	2	2	no	—
181	#127	—	2	2	no	—
182	#127	C₂×C₄×C₂₃	1	4	yes	—
183	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
184	C₂×C₄²	—	2	1	no	—
185	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
190	#102	—	4	8	no	—
191	#97	—	4	2	no	—
192	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
193	C₄×C₈	D₄×C₂₃	1	2	yes	—
194	#95	—	2	1	no	—
195	C₂₃	#9743	1	1	yes	characteristic, direct product factor, Sylow
196	C₂⁴	—	4	2	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	4	2	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	4	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	4	no	—
209	C₂×D₄	—	8	4	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
213	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
214	D₈	—	4	8	no	—
215	D₈	—	4	8	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	4	no	—
223	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
224	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
225	C₂×C₈	D₈×C₂₃	1	2	yes	—
226	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄²	—	4	1	no	—
229	C₄²	—	4	1	no	—
230	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
231	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
232	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
233	C₄²	#41×C₂₃	1	1	yes	characteristic
234	C₂³	—	4	4	no	—
235	C₂³	—	8	2	no	—
236	C₂³	—	8	2	no	—
237	C₂³	—	8	4	no	—
238	C₂³	—	8	4	no	—
239	C₂³	C₂₃×#121	1	1	yes	characteristic
240	D₄	—	4	4	no	—
241	D₄	—	4	4	no	—
242	D₄	—	4	8	no	—
243	D₄	—	8	4	no	—
244	D₄	—	8	4	no	—
245	D₄	—	8	2	no	—
246	D₄	—	8	2	no	—
247	D₄	—	8	2	no	—
248	D₄	—	8	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
256	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
257	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
258	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
259	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
260	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
261	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
262	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
263	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	—	8	2	no	—
267	C₈	C₂₃×#102	1	4	yes	complemented
268	C₂²	—	8	4	no	—
269	C₂²	—	8	4	no	—
270	C₂²	—	8	4	no	—
271	C₂²	—	16	4	no	—
272	C₂²	—	16	4	no	—
273	C₂²	C₂₃×#390	1	1	yes	characteristic
274	C₂²	C₂₃×#459	1	1	yes	characteristic
275	C₂²	C₂₃×#460	1	1	yes	characteristic
276	C₂²	C₂₃×#469	1	2	yes	—
277	C₂²	C₂₃×#472	1	2	yes	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	8	2	no	—
281	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
282	C₄	C₂₃×#418	1	1	yes	characteristic
283	C₄	C₂₃×#442	1	2	yes	—
284	C₄	C₂₃×#459	1	1	yes	characteristic
285	C₄	C₂₃×#460	1	1	yes	characteristic
286	C₄	C₂₃×#496	1	2	yes	—
287	C₂	—	16	4	no	—
288	C₂	C₂₃×#1687	1	2	yes	—
289	C₂	C₂₃×#1688	1	2	yes	—
290	C₂	C₂₃×#1972	1	1	yes	characteristic
291	C₂	C₂₃×#1978	1	1	yes	characteristic
292	C₂	C₂₃×#1982	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#121
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂×C₂₃×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1982	C₂	semidirect
3. C₂₃×#1770	C₂	semidirect
4. C₂×C₂₃×#334	C₂	semidirect
5. C₂×C₂₃×#331	C₂	semidirect
10. C₂₃×#493	C₄	semidirect
87. #9743	C₂₃	direct (N × H)
121. C₈×C₂₃	#102	semidirect
123. #1982	C₂×C₂₃	semidirect
124. #1770	C₂×C₂₃	semidirect
125. C₂×#334	C₂×C₂₃	semidirect
126. C₂×#331	C₂×C₂₃	semidirect
150. #493	C₄×C₂₃	semidirect
195. C₂₃	#9743	direct (N × H)
267. C₈	C₂₃×#102	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#493	2
C₂₃×#1982	2
C₂₃×#1770	2
C₂×C₂₃×#334	2
C₂×C₂₃×#331	2
#9743	23

Subgroups — C23×#9743 #19157369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9743 #19157369