Subgroups — C₂₃×#21277 #19168903

Order 5888.

950 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2293	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1972	C₂	1	2	yes	maximal
4	C₂₃×#1943	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1936	C₂	1	4	yes	maximal, complemented
6	C₄×C₂₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂₃×#102	C₂	1	2	yes	maximal, complemented
8	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃×#117	—	2	2	no	—
12	C₄×D₈×C₂₃	C₄	1	8	yes	complemented
13	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#106	C₂²	1	2	yes	—
15	C₂×C₂₃×#102	C₂²	1	4	yes	—
16	C₂×C₂₃×#102	C₂²	1	2	yes	—
17	C₂×C₂₃×#97	C₂²	1	2	yes	—
18	C₂×C₂₃×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#390	C₂²	1	4	yes	—
22	C₂₃×#389	C₂²	1	4	yes	—
23	C₂₃×#388	—	2	4	no	—
24	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
25	C₄×C₂²⋊C₄×C₂₃	—	2	2	no	—
26	C₄³×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#340	C₂²	1	2	yes	—
28	C₂₃×#340	C₂²	1	4	yes	—
29	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
30	C₂³×C₄×C₂₃	—	2	2	no	—
31	C₂×D₈×C₂₃	—	2	4	no	—
32	C₂×D₈×C₂₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
36	C₄×D₄×C₂₃	—	2	4	no	—
37	C₄×D₄×C₂₃	C₂×C₄	1	8	yes	—
38	C₂₃×#117	—	4	8	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
40	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
42	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
44	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
45	C₂×C₄²×C₂₃	—	2	2	no	—
46	C₂×C₄²×C₂₃	D₄	1	2	yes	—
47	C₂×C₄²×C₂₃	D₄	1	2	yes	—
48	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
49	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
50	C₂₃×#107	C₂×C₄	1	4	yes	—
51	C₂₃×#106	—	2	4	no	—
52	C₂₃×#102	—	2	8	no	—
53	C₂₃×#102	C₂×C₄	1	8	yes	—
54	C₂₃×#97	—	2	4	no	—
55	C₂₃×#97	—	2	4	no	—
56	C₄×C₈×C₂₃	—	2	2	no	—
57	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
58	C₂₃×#95	—	2	4	no	—
59	C₂₃×#95	—	2	2	no	—
60	C₂⁴×C₂₃	—	2	2	no	—
61	C₂×D₄×C₂₃	—	2	4	no	—
62	C₂×D₄×C₂₃	—	2	4	no	—
63	C₂×D₄×C₂₃	—	2	2	no	—
64	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₂₃	—	2	2	no	—
66	C₂²×C₄×C₂₃	—	2	8	no	—
67	C₂²×C₄×C₂₃	—	2	4	no	—
68	C₂²×C₄×C₂₃	—	4	4	no	—
69	C₂²×C₄×C₂₃	—	4	4	no	—
70	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
74	D₈×C₂₃	—	2	8	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₂₃	#41	1	4	yes	—
81	C₂×C₈×C₂₃	#41	1	4	yes	—
82	C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₄⋊C₄×C₂₃	—	2	4	no	—
84	C₄⋊C₄×C₂₃	—	2	4	no	—
85	C₄⋊C₄×C₂₃	—	4	8	no	—
86	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
87	C₂²⋊C₄×C₂₃	—	2	8	no	—
88	C₂²⋊C₄×C₂₃	—	4	8	no	—
89	C₂²⋊C₄×C₂₃	—	4	8	no	—
90	C₄²×C₂₃	—	2	4	no	—
91	C₄²×C₂₃	—	2	2	no	—
92	C₄²×C₂₃	—	2	2	no	—
93	C₄²×C₂₃	—	4	2	no	—
94	C₄²×C₂₃	—	4	4	no	—
95	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₂₃	#41	1	2	yes	—
97	#21277	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₂₃	—	2	4	no	—
99	C₂³×C₂₃	—	4	4	no	—
100	C₂³×C₂₃	—	4	2	no	—
101	C₂³×C₂₃	—	4	2	no	—
102	C₂³×C₂₃	—	4	4	no	—
103	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
104	D₄×C₂₃	—	2	4	no	—
105	D₄×C₂₃	—	2	4	no	—
106	D₄×C₂₃	—	4	4	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	2	4	no	—
109	C₂×C₄×C₂₃	—	2	4	no	—
110	C₂×C₄×C₂₃	—	2	4	no	—
111	C₂×C₄×C₂₃	—	2	4	no	—
112	C₂×C₄×C₂₃	—	4	2	no	—
113	C₂×C₄×C₂₃	—	4	2	no	—
114	C₂×C₄×C₂₃	—	4	16	no	—
115	C₂×C₄×C₂₃	—	4	4	no	—
116	C₂×C₄×C₂₃	—	4	4	no	—
117	C₂×C₄×C₂₃	—	4	4	no	—
118	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
121	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
123	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
124	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
125	C₈×C₂₃	—	2	2	no	—
126	C₈×C₂₃	—	2	4	no	—
127	C₈×C₂₃	#117	1	4	yes	complemented
128	C₂×C₄×D₈	C₂×C₂₃	1	1	yes	characteristic
129	#2293	C₂×C₂₃	1	2	yes	complemented
130	#1972	C₂×C₂₃	1	2	yes	—
131	#1943	C₂×C₂₃	1	2	yes	complemented
132	#1936	C₂×C₂₃	1	4	yes	complemented
133	C₄×#106	C₂×C₂₃	1	1	yes	characteristic, complemented
134	C₄×#102	C₂×C₂₃	1	2	yes	complemented
135	C₄×#97	C₂×C₂₃	1	1	yes	characteristic, complemented
136	C₂²×C₂₃	—	4	4	no	—
137	C₂²×C₂₃	—	4	4	no	—
138	C₂²×C₂₃	—	4	4	no	—
139	C₂²×C₂₃	—	8	4	no	—
140	C₂²×C₂₃	—	8	4	no	—
141	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²×C₂₃	#532	1	2	yes	—
143	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
144	C₂²×C₂₃	#575	1	2	yes	—
145	C₄×C₂₃	—	2	2	no	—
146	C₄×C₂₃	—	4	4	no	—
147	C₄×C₂₃	—	4	4	no	—
148	C₄×C₂₃	—	8	4	no	—
149	C₄×C₂₃	#442	1	4	yes	—
150	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
151	C₄×C₂₃	C₄×#41	1	1	yes	characteristic
152	C₄×C₂₃	#529	1	2	yes	—
153	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
154	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
155	C₂×#117	—	2	2	no	—
156	C₄×D₈	C₄×C₂₃	1	8	yes	complemented
157	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
158	C₂×#106	C₂²×C₂₃	1	2	yes	—
159	C₂×#102	C₂²×C₂₃	1	4	yes	—
160	C₂×#102	C₂²×C₂₃	1	2	yes	—
161	C₂×#97	C₂²×C₂₃	1	2	yes	—
162	C₂×#97	C₂²×C₂₃	1	2	yes	—
163	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
165	#390	C₂²×C₂₃	1	4	yes	—
166	#389	C₂²×C₂₃	1	4	yes	—
167	#388	—	2	4	no	—
168	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
169	C₄×C₂²⋊C₄	—	2	2	no	—
170	C₄³	C₂²×C₂₃	1	1	yes	characteristic
171	#340	C₂²×C₂₃	1	2	yes	—
172	#340	C₂²×C₂₃	1	4	yes	—
173	C₂×C₂₃	—	8	4	no	—
174	C₂×C₂₃	#2293	1	1	yes	characteristic
175	C₂×C₂₃	C₂×#442	1	1	yes	characteristic
176	C₂×C₂₃	#2943	1	1	yes	characteristic
177	C₂×C₂₃	#3171	1	4	yes	—
178	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
179	C₂³×C₄	—	2	2	no	—
180	C₂×D₈	—	2	4	no	—
181	C₂×D₈	C₂×C₄×C₂₃	1	4	yes	—
182	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
184	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
185	C₄×D₄	—	2	4	no	—
186	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	—
187	#117	—	4	8	no	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
190	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₂²⋊C₄	—	2	4	no	—
193	C₂×C₂²⋊C₄	—	2	4	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	D₄×C₂₃	1	2	yes	—
196	C₂×C₄²	D₄×C₂₃	1	2	yes	—
197	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
198	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
199	#107	C₂×C₄×C₂₃	1	4	yes	—
200	#106	—	2	4	no	—
201	#102	—	2	8	no	—
202	#102	C₂×C₄×C₂₃	1	8	yes	—
203	#97	—	2	4	no	—
204	#97	—	2	4	no	—
205	C₄×C₈	—	2	2	no	—
206	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
207	#95	—	2	4	no	—
208	#95	—	2	2	no	—
209	C₂₃	#21277	1	1	yes	characteristic, direct product factor, Sylow
210	C₂⁴	—	2	2	no	—
211	C₂×D₄	—	2	4	no	—
212	C₂×D₄	—	2	4	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	4	4	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
221	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
223	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
224	D₈	—	2	8	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
229	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
230	C₂×C₈	#41×C₂₃	1	4	yes	—
231	C₂×C₈	#41×C₂₃	1	4	yes	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	4	8	no	—
236	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
237	C₂²⋊C₄	—	2	8	no	—
238	C₂²⋊C₄	—	4	8	no	—
239	C₂²⋊C₄	—	4	8	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	4	2	no	—
244	C₄²	—	4	4	no	—
245	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
246	C₄²	#41×C₂₃	1	2	yes	—
247	C₂³	—	2	4	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	4	no	—
251	C₂³	—	4	4	no	—
252	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
253	D₄	—	2	4	no	—
254	D₄	—	2	4	no	—
255	D₄	—	4	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	16	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
268	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
269	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
270	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
271	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
272	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
273	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	2	4	no	—
276	C₈	C₂₃×#117	1	4	yes	complemented
277	C₂²	—	4	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	4	4	no	—
280	C₂²	—	8	4	no	—
281	C₂²	—	8	4	no	—
282	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
283	C₂²	C₂₃×#532	1	2	yes	—
284	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
285	C₂²	C₂₃×#575	1	2	yes	—
286	C₄	—	2	2	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	4	no	—
289	C₄	—	8	4	no	—
290	C₄	C₂₃×#442	1	4	yes	—
291	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
292	C₄	C₄×#41×C₂₃	1	1	yes	characteristic
293	C₄	C₂₃×#529	1	2	yes	—
294	C₂	—	8	4	no	—
295	C₂	C₂₃×#2293	1	1	yes	characteristic
296	C₂	C₂×C₂₃×#442	1	1	yes	characteristic
297	C₂	C₂₃×#2943	1	1	yes	characteristic
298	C₂	C₂₃×#3171	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂²×D₄
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2293	C₂	semidirect
4. C₂₃×#1943	C₂	semidirect
5. C₂₃×#1936	C₂	semidirect
6. C₄×C₂₃×#106	C₂	semidirect
7. C₄×C₂₃×#102	C₂	semidirect
8. C₄×C₂₃×#97	C₂	semidirect
12. C₄×D₈×C₂₃	C₄	semidirect
97. #21277	C₂₃	direct (N × H)
127. C₈×C₂₃	#117	semidirect
129. #2293	C₂×C₂₃	semidirect
131. #1943	C₂×C₂₃	semidirect
132. #1936	C₂×C₂₃	semidirect
133. C₄×#106	C₂×C₂₃	semidirect
134. C₄×#102	C₂×C₂₃	semidirect
135. C₄×#97	C₂×C₂₃	semidirect
156. C₄×D₈	C₄×C₂₃	semidirect
209. C₂₃	#21277	direct (N × H)
276. C₈	C₂₃×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₂₃	2
C₂₃×#2293	2
C₂₃×#1972	2
C₂₃×#1943	2
C₂₃×#1936	2
C₄×C₂₃×#106	2
C₄×C₂₃×#102	2
C₄×C₂₃×#97	2
#21277	23

Subgroups — C23×#21277 #19168903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21277 #19168903