Subgroups — C₂₃×#8658 #19156284

Order 5888.

482 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1486	C₂	1	2	yes	maximal
5	C₂₃×#1301	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₂₃	—	2	1	no	—
7	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₈×D₄×C₂₃	—	2	1	no	—
9	C₈×D₄×C₂₃	—	2	1	no	—
10	C₈×D₄×C₂₃	C₄	1	2	yes	—
11	C₈×D₄×C₂₃	C₄	1	2	yes	—
12	C₂₃×#431	C₂²	1	2	yes	—
13	C₂×C₂₃×#108	—	2	1	no	—
14	C₂×C₂₃×#108	—	2	1	no	—
15	C₂₃×#423	C₂²	1	2	yes	—
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#335	—	2	1	no	—
20	C₂₃×#334	—	2	1	no	—
21	C₂₃×D₄⋊C₈	—	2	4	no	—
22	C₂²×D₄×C₂₃	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₂₃	—	2	1	no	—
24	C₂×M₁₆×C₂₃	—	2	2	no	—
25	C₂²×C₈×C₂₃	—	2	1	no	—
26	C₂²×C₈×C₂₃	—	2	2	no	—
27	C₂²×C₈×C₂₃	—	2	2	no	—
28	C₂²×C₈×C₂₃	—	4	2	no	—
29	C₂²×C₈×C₂₃	—	4	2	no	—
30	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
32	C₄×D₄×C₂₃	—	2	1	no	—
33	C₄×D₄×C₂₃	—	2	1	no	—
34	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
35	C₄×D₄×C₂₃	D₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₂₃	Q₈	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
38	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
39	C₂₃×#108	—	4	2	no	—
40	C₂₃×#108	—	4	2	no	—
41	C₂₃×C₄⋊C₈	—	2	1	no	—
42	C₂₃×C₄⋊C₈	—	2	2	no	—
43	C₂₃×C₄⋊C₈	—	2	2	no	—
44	C₂₃×C₄⋊C₈	—	2	2	no	—
45	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₂₃×C₂²⋊C₈	—	2	2	no	—
47	C₂₃×C₂²⋊C₈	—	2	2	no	—
48	C₂₃×C₂²⋊C₈	—	2	2	no	—
49	C₄×C₈×C₂₃	—	2	2	no	—
50	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
54	C₂⁴×C₂₃	—	2	1	no	—
55	C₂×D₄×C₂₃	—	2	4	no	—
56	C₂×D₄×C₂₃	—	2	1	no	—
57	C₂×D₄×C₂₃	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₂₃	—	2	1	no	—
59	C₂²×C₄×C₂₃	—	2	2	no	—
60	C₂²×C₄×C₂₃	—	2	2	no	—
61	C₂²×C₄×C₂₃	—	4	1	no	—
62	C₂²×C₄×C₂₃	—	4	1	no	—
63	C₂²×C₄×C₂₃	—	4	1	no	—
64	C₂²×C₄×C₂₃	—	4	1	no	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
68	M₁₆×C₂₃	—	4	2	no	—
69	M₁₆×C₂₃	—	4	2	no	—
70	C₂×C₈×C₂₃	—	2	1	no	—
71	C₂×C₈×C₂₃	—	2	1	no	—
72	C₂×C₈×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	—	4	1	no	—
74	C₂×C₈×C₂₃	—	4	1	no	—
75	C₂×C₈×C₂₃	—	4	4	no	—
76	C₂×C₈×C₂₃	—	4	4	no	—
77	C₂×C₈×C₂₃	—	4	2	no	—
78	C₂×C₈×C₂₃	—	4	2	no	—
79	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₂₃	D₈	1	2	yes	—
84	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
85	C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
87	C₂²⋊C₄×C₂₃	—	2	2	no	—
88	C₂²⋊C₄×C₂₃	—	2	2	no	—
89	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₂₃	#41	1	1	yes	characteristic
93	#8658	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₂₃	—	2	4	no	—
95	C₂³×C₂₃	—	2	2	no	—
96	C₂³×C₂₃	—	4	2	no	—
97	C₂³×C₂₃	—	4	1	no	—
98	C₂³×C₂₃	—	4	1	no	—
99	C₂³×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
100	D₄×C₂₃	—	2	4	no	—
101	D₄×C₂₃	—	2	2	no	—
102	D₄×C₂₃	#108	1	4	yes	complemented
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	4	1	no	—
106	C₂×C₄×C₂₃	—	4	1	no	—
107	C₂×C₄×C₂₃	—	4	1	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	4	2	no	—
110	C₂×C₄×C₂₃	—	4	2	no	—
111	C₂×C₄×C₂₃	—	4	2	no	—
112	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
122	C₈×C₂₃	—	2	1	no	—
123	C₈×C₂₃	—	2	1	no	—
124	C₈×C₂₃	—	4	2	no	—
125	C₈×C₂₃	—	8	2	no	—
126	C₈×C₂₃	—	8	2	no	—
127	C₈×C₂₃	#102	1	2	yes	—
128	C₈×C₂₃	#102	1	2	yes	—
129	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic
130	#1848	C₂×C₂₃	1	1	yes	characteristic, complemented
131	#1564	C₂×C₂₃	1	1	yes	characteristic, complemented
132	#1486	C₂×C₂₃	1	2	yes	—
133	#1301	C₂×C₂₃	1	2	yes	complemented
134	C₂²×C₂₃	—	2	1	no	—
135	C₂²×C₂₃	—	2	1	no	—
136	C₂²×C₂₃	—	2	4	no	—
137	C₂²×C₂₃	—	4	4	no	—
138	C₂²×C₂₃	—	4	4	no	—
139	C₂²×C₂₃	—	4	2	no	—
140	C₂²×C₂₃	—	4	2	no	—
141	C₂²×C₂₃	#380	1	1	yes	characteristic
142	C₂²×C₂₃	#474	1	1	yes	characteristic
143	C₂²×C₂₃	#476	1	1	yes	characteristic
144	C₄×C₂₃	—	2	1	no	—
145	C₄×C₂₃	—	2	1	no	—
146	C₄×C₂₃	—	2	1	no	—
147	C₄×C₂₃	—	4	2	no	—
148	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
149	C₄×C₂₃	#417	1	1	yes	characteristic
150	C₄×C₂₃	C₂×#108	1	2	yes	—
151	C₄×C₂₃	#459	1	1	yes	characteristic
152	C₄×C₂₃	#461	1	1	yes	characteristic
153	C₂³×C₈	—	2	1	no	—
154	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
155	C₈×D₄	—	2	1	no	—
156	C₈×D₄	—	2	1	no	—
157	C₈×D₄	C₄×C₂₃	1	2	yes	—
158	C₈×D₄	C₄×C₂₃	1	2	yes	—
159	#431	C₂²×C₂₃	1	2	yes	—
160	C₂×#108	—	2	1	no	—
161	C₂×#108	—	2	1	no	—
162	#423	C₂²×C₂₃	1	2	yes	—
163	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
164	C₂×C₂²⋊C₈	—	2	1	no	—
165	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
166	#335	—	2	1	no	—
167	#334	—	2	1	no	—
168	D₄⋊C₈	—	2	4	no	—
169	C₂×C₂₃	—	2	1	no	—
170	C₂×C₂₃	—	2	1	no	—
171	C₂×C₂₃	—	4	4	no	—
172	C₂×C₂₃	#1802	1	1	yes	characteristic
173	C₂×C₂₃	#1810	1	1	yes	characteristic
174	C₂×C₂₃	#1814	1	1	yes	characteristic
175	C₂²×D₄	Q₈×C₂₃	1	1	yes	characteristic
176	C₂³×C₄	—	2	1	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	4	2	no	—
182	C₂²×C₈	—	4	2	no	—
183	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
184	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
185	C₄×D₄	—	2	1	no	—
186	C₄×D₄	—	2	1	no	—
187	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
188	C₄×D₄	D₄×C₂₃	1	2	yes	—
189	C₂×C₄⋊C₄	Q₈×C₂₃	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	—	2	1	no	—
191	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
192	#108	—	4	2	no	—
193	#108	—	4	2	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
199	C₂²⋊C₈	—	2	2	no	—
200	C₂²⋊C₈	—	2	2	no	—
201	C₂²⋊C₈	—	2	2	no	—
202	C₄×C₈	—	2	2	no	—
203	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
204	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
205	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
206	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
207	C₂₃	#8658	1	1	yes	characteristic, direct product factor, Sylow
208	C₂⁴	—	2	1	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	C₄⋊C₄×C₂₃	1	2	yes	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
221	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	—	4	4	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
234	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
237	C₂×C₈	D₈×C₂₃	1	2	yes	—
238	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
244	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
245	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
246	C₄²	#41×C₂₃	1	1	yes	characteristic
247	C₂³	—	2	4	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
253	D₄	—	2	4	no	—
254	D₄	—	2	2	no	—
255	D₄	C₂₃×#108	1	4	yes	complemented
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
267	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
268	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
269	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
270	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
271	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
272	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
273	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
274	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	4	2	no	—
278	C₈	—	8	2	no	—
279	C₈	—	8	2	no	—
280	C₈	C₂₃×#102	1	2	yes	—
281	C₈	C₂₃×#102	1	2	yes	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	C₂₃×#380	1	1	yes	characteristic
290	C₂²	C₂₃×#474	1	1	yes	characteristic
291	C₂²	C₂₃×#476	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	4	2	no	—
296	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
297	C₄	C₂₃×#417	1	1	yes	characteristic
298	C₄	C₂×C₂₃×#108	1	2	yes	—
299	C₄	C₂₃×#459	1	1	yes	characteristic
300	C₄	C₂₃×#461	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	4	4	no	—
304	C₂	C₂₃×#1802	1	1	yes	characteristic
305	C₂	C₂₃×#1810	1	1	yes	characteristic
306	C₂	C₂₃×#1814	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#380
prime radical	C₂²×D₄×C₂₃	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1848	C₂	semidirect
3. C₂₃×#1564	C₂	semidirect
5. C₂₃×#1301	C₂	semidirect
93. #8658	C₂₃	direct (N × H)
102. D₄×C₂₃	#108	semidirect
130. #1848	C₂×C₂₃	semidirect
131. #1564	C₂×C₂₃	semidirect
133. #1301	C₂×C₂₃	semidirect
207. C₂₃	#8658	direct (N × H)
255. D₄	C₂₃×#108	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₂₃	2
C₂₃×#1848	2
C₂₃×#1564	2
C₂₃×#1486	2
C₂₃×#1301	2
#8658	23

Subgroups — C23×#8658 #19156284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8658 #19156284