Subgroups — C₂₃×#23540 #19171166

Order 5888.

1682 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#418	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2309	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2054	C₂	1	8	yes	maximal, complemented
6	C₂₃×#1735	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₂₃	C₂²	1	2	yes	—
8	C₂²×D₈×C₂₃	—	2	8	no	—
9	C₂²×D₈×C₂₃	—	2	4	no	—
10	C₂²×D₈×C₂₃	C₂²	1	4	yes	—
11	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#117	C₂²	1	2	yes	—
14	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
15	C₂₃×#424	—	2	4	no	—
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	complemented
17	C₂₃×#418	—	2	8	no	—
18	C₂×C₂₃×#102	C₂²	1	8	yes	complemented
19	C₂₃×#390	—	2	4	no	—
20	C₂₃×#390	C₂²	1	8	yes	complemented
21	C₂₃×#379	—	2	4	no	—
22	C₂₃×#340	C₂²	1	4	yes	complemented
23	C₂⁵×C₂₃	—	2	2	no	—
24	C₂²×D₄×C₂₃	—	2	4	no	—
25	C₂²×D₄×C₂₃	—	2	4	no	—
26	C₂²×D₄×C₂₃	—	2	2	no	—
27	C₂²×D₄×C₂₃	—	2	4	no	—
28	C₂²×D₄×C₂₃	—	2	4	no	—
29	C₂²×D₄×C₂₃	—	2	4	no	—
30	C₂²×D₄×C₂₃	D₄	1	8	yes	—
31	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
32	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₂×D₈×C₂₃	—	2	32	no	—
34	C₂×D₈×C₂₃	—	2	16	no	—
35	C₂×D₈×C₂₃	—	2	8	no	—
36	C₂×M₁₆×C₂₃	—	2	4	no	—
37	C₂×M₁₆×C₂₃	—	2	2	no	—
38	C₂²×C₈×C₂₃	—	2	4	no	—
39	C₂²×C₈×C₂₃	D₄	1	4	yes	—
40	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
42	C₂₃×#117	—	2	2	no	—
43	C₂₃×#117	—	2	2	no	—
44	C₂₃×#117	—	2	4	no	—
45	C₂₃×#117	—	2	4	no	—
46	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
48	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
50	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
51	C₂×C₄²×C₂₃	—	2	2	no	—
52	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
53	C₂₃×C₄⋊C₈	—	2	8	no	—
54	C₂₃×#102	—	2	16	no	—
55	C₂⁴×C₂₃	—	2	4	no	—
56	C₂⁴×C₂₃	—	2	8	no	—
57	C₂⁴×C₂₃	—	4	4	no	—
58	C₂⁴×C₂₃	—	4	4	no	—
59	C₂⁴×C₂₃	—	8	8	no	—
60	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₂×D₄×C₂₃	—	2	16	no	—
62	C₂×D₄×C₂₃	—	2	8	no	—
63	C₂×D₄×C₂₃	—	2	16	no	—
64	C₂×D₄×C₂₃	—	2	4	no	—
65	C₂×D₄×C₂₃	—	2	4	no	—
66	C₂×D₄×C₂₃	—	2	4	no	—
67	C₂×D₄×C₂₃	—	2	4	no	—
68	C₂×D₄×C₂₃	—	2	4	no	—
69	C₂×D₄×C₂₃	—	4	4	no	—
70	C₂×D₄×C₂₃	—	4	4	no	—
71	C₂×D₄×C₂₃	—	4	8	no	—
72	C₂×D₄×C₂₃	—	4	8	no	—
73	C₂×D₄×C₂₃	C₂×D₄	1	8	yes	—
74	C₂²×C₄×C₂₃	—	2	4	no	—
75	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₃	#41	1	2	yes	—
80	C₂²×C₄×C₂₃	#41	1	4	yes	—
81	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
82	D₈×C₂₃	—	2	32	no	—
83	M₁₆×C₂₃	—	4	4	no	—
84	C₂×C₈×C₂₃	—	2	8	no	—
85	C₂×C₈×C₂₃	—	2	4	no	—
86	C₂×C₈×C₂₃	—	2	4	no	—
87	C₂×C₈×C₂₃	—	2	2	no	—
88	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
89	C₄⋊C₄×C₂₃	—	2	4	no	—
90	C₄⋊C₄×C₂₃	—	2	4	no	—
91	C₄⋊C₄×C₂₃	—	2	4	no	—
92	C₂²⋊C₄×C₂₃	—	2	4	no	—
93	C₂²⋊C₄×C₂₃	—	4	4	no	—
94	C₂²⋊C₄×C₂₃	—	8	16	no	—
95	C₄²×C₂₃	—	2	2	no	—
96	C₄²×C₂₃	—	2	2	no	—
97	C₄²×C₂₃	—	4	4	no	—
98	#23540	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₂₃	—	2	4	no	—
100	C₂³×C₂₃	—	2	8	no	—
101	C₂³×C₂₃	—	4	4	no	—
102	C₂³×C₂₃	—	4	4	no	—
103	C₂³×C₂₃	—	4	16	no	—
104	C₂³×C₂₃	—	4	16	no	—
105	C₂³×C₂₃	—	8	16	no	—
106	C₂³×C₂₃	—	8	16	no	—
107	C₂³×C₂₃	—	8	16	no	—
108	C₂³×C₂₃	#121	1	2	yes	—
109	C₂³×C₂₃	#121	1	4	yes	—
110	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
111	D₄×C₂₃	—	2	8	no	—
112	D₄×C₂₃	—	2	8	no	—
113	D₄×C₂₃	—	4	4	no	—
114	D₄×C₂₃	—	4	4	no	—
115	D₄×C₂₃	—	4	16	no	—
116	C₂×C₄×C₂₃	—	2	4	no	—
117	C₂×C₄×C₂₃	—	2	8	no	—
118	C₂×C₄×C₂₃	—	4	4	no	—
119	C₂×C₄×C₂₃	—	4	8	no	—
120	C₂×C₄×C₂₃	#120	1	4	yes	—
121	C₂×C₄×C₂₃	#121	1	4	yes	—
122	C₂×C₄×C₂₃	#121	1	2	yes	—
123	C₂×C₄×C₂₃	#123	1	8	yes	—
124	C₂×C₄×C₂₃	#124	1	4	yes	—
125	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
126	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
129	C₈×C₂₃	—	2	4	no	—
130	C₈×C₂₃	—	4	2	no	—
131	C₂³×D₈	C₂×C₂₃	1	1	yes	characteristic
132	C₂×#424	C₂×C₂₃	1	1	yes	characteristic, complemented
133	C₂×#418	C₂×C₂₃	1	2	yes	complemented
134	#2309	C₂×C₂₃	1	2	yes	complemented
135	#2054	C₂×C₂₃	1	8	yes	complemented
136	#1735	C₂×C₂₃	1	1	yes	characteristic, complemented
137	C₂²×C₂₃	—	2	8	no	—
138	C₂²×C₂₃	—	2	4	no	—
139	C₂²×C₂₃	—	4	8	no	—
140	C₂²×C₂₃	—	4	16	no	—
141	C₂²×C₂₃	—	8	32	no	—
142	C₂²×C₂₃	—	8	8	no	—
143	C₂²×C₂₃	—	8	8	no	—
144	C₂²×C₂₃	#394	1	4	yes	—
145	C₂²×C₂₃	C₂×#121	1	2	yes	—
146	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
147	C₂²×C₂₃	#576	1	4	yes	—
148	C₄×C₂₃	—	4	4	no	—
149	C₄×C₂₃	—	8	4	no	—
150	C₄×C₂₃	#394	1	4	yes	—
151	C₄×C₂₃	C₂×#120	1	1	yes	characteristic
152	C₄×C₂₃	C₂×#123	1	2	yes	—
153	C₄×C₂₃	C₂×#124	1	1	yes	characteristic
154	C₂³×D₄	C₂²×C₂₃	1	2	yes	—
155	C₂²×D₈	—	2	8	no	—
156	C₂²×D₈	—	2	4	no	—
157	C₂²×D₈	C₂²×C₂₃	1	4	yes	—
158	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
159	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
160	C₂×#117	C₂²×C₂₃	1	2	yes	—
161	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
162	#424	—	2	4	no	—
163	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	complemented
164	#418	—	2	8	no	—
165	C₂×#102	C₂²×C₂₃	1	8	yes	complemented
166	#390	—	2	4	no	—
167	#390	C₂²×C₂₃	1	8	yes	complemented
168	#379	—	2	4	no	—
169	#340	C₂²×C₂₃	1	4	yes	complemented
170	C₂×C₂₃	—	2	4	no	—
171	C₂×C₂₃	—	8	8	no	—
172	C₂×C₂₃	C₂×#394	1	1	yes	characteristic
173	C₂×C₂₃	#3041	1	4	yes	—
174	C₂×C₂₃	#3058	1	2	yes	—
175	C₂⁵	—	2	2	no	—
176	C₂²×D₄	—	2	4	no	—
177	C₂²×D₄	—	2	4	no	—
178	C₂²×D₄	—	2	2	no	—
179	C₂²×D₄	—	2	4	no	—
180	C₂²×D₄	—	2	4	no	—
181	C₂²×D₄	—	2	4	no	—
182	C₂²×D₄	D₄×C₂₃	1	8	yes	—
183	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
184	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
185	C₂×D₈	—	2	32	no	—
186	C₂×D₈	—	2	16	no	—
187	C₂×D₈	—	2	8	no	—
188	C₂×M₁₆	—	2	4	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂²×C₈	—	2	4	no	—
191	C₂²×C₈	D₄×C₂₃	1	4	yes	—
192	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
193	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
194	#117	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	—	2	4	no	—
197	#117	—	2	4	no	—
198	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
199	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₂²⋊C₄	—	2	16	no	—
202	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
205	C₄⋊C₈	—	2	8	no	—
206	#102	—	2	16	no	—
207	C₂₃	#23540	1	1	yes	characteristic, direct product factor, Sylow
208	C₂⁴	—	2	4	no	—
209	C₂⁴	—	2	8	no	—
210	C₂⁴	—	4	4	no	—
211	C₂⁴	—	4	4	no	—
212	C₂⁴	—	8	8	no	—
213	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
214	C₂×D₄	—	2	16	no	—
215	C₂×D₄	—	2	8	no	—
216	C₂×D₄	—	2	16	no	—
217	C₂×D₄	—	2	4	no	—
218	C₂×D₄	—	2	4	no	—
219	C₂×D₄	—	2	4	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	2	4	no	—
222	C₂×D₄	—	4	4	no	—
223	C₂×D₄	—	4	4	no	—
224	C₂×D₄	—	4	8	no	—
225	C₂×D₄	—	4	8	no	—
226	C₂×D₄	C₂×D₄×C₂₃	1	8	yes	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
229	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
230	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
232	C₂²×C₄	#41×C₂₃	1	2	yes	—
233	C₂²×C₄	#41×C₂₃	1	4	yes	—
234	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
235	D₈	—	2	32	no	—
236	M₁₆	—	4	4	no	—
237	C₂×C₈	—	2	8	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	—	4	4	no	—
247	C₂²⋊C₄	—	8	16	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	4	4	no	—
251	C₂³	—	2	4	no	—
252	C₂³	—	2	8	no	—
253	C₂³	—	4	16	no	—
254	C₂³	—	4	16	no	—
255	C₂³	—	4	4	no	—
256	C₂³	—	4	4	no	—
257	C₂³	—	8	16	no	—
258	C₂³	—	8	16	no	—
259	C₂³	—	8	16	no	—
260	C₂³	C₂₃×#121	1	2	yes	—
261	C₂³	C₂₃×#121	1	4	yes	—
262	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
263	D₄	—	2	8	no	—
264	D₄	—	2	8	no	—
265	D₄	—	4	16	no	—
266	D₄	—	4	4	no	—
267	D₄	—	4	4	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	C₂₃×#120	1	4	yes	—
273	C₂×C₄	C₂₃×#121	1	4	yes	—
274	C₂×C₄	C₂₃×#121	1	2	yes	—
275	C₂×C₄	C₂₃×#123	1	8	yes	—
276	C₂×C₄	C₂₃×#124	1	4	yes	—
277	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
278	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
280	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	4	2	no	—
283	C₂²	—	2	8	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	4	16	no	—
286	C₂²	—	4	8	no	—
287	C₂²	—	8	32	no	—
288	C₂²	—	8	8	no	—
289	C₂²	—	8	8	no	—
290	C₂²	C₂₃×#394	1	4	yes	—
291	C₂²	C₂×C₂₃×#121	1	2	yes	—
292	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
293	C₂²	C₂₃×#576	1	4	yes	—
294	C₄	—	4	4	no	—
295	C₄	—	8	4	no	—
296	C₄	C₂₃×#394	1	4	yes	—
297	C₄	C₂×C₂₃×#120	1	1	yes	characteristic
298	C₄	C₂×C₂₃×#123	1	2	yes	—
299	C₄	C₂×C₂₃×#124	1	1	yes	characteristic
300	C₂	—	2	4	no	—
301	C₂	—	8	8	no	—
302	C₂	C₂×C₂₃×#394	1	1	yes	characteristic
303	C₂	C₂₃×#3041	1	4	yes	—
304	C₂	C₂₃×#3058	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂³×D₈×C₂₃	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#424	C₂	semidirect
3. C₂×C₂₃×#418	C₂	semidirect
4. C₂₃×#2309	C₂	semidirect
5. C₂₃×#2054	C₂	semidirect
6. C₂₃×#1735	C₂	semidirect
16. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
18. C₂×C₂₃×#102	C₂²	semidirect
20. C₂₃×#390	C₂²	semidirect
22. C₂₃×#340	C₂²	semidirect
98. #23540	C₂₃	direct (N × H)
132. C₂×#424	C₂×C₂₃	semidirect
133. C₂×#418	C₂×C₂₃	semidirect
134. #2309	C₂×C₂₃	semidirect
135. #2054	C₂×C₂₃	semidirect
136. #1735	C₂×C₂₃	semidirect
163. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
165. C₂×#102	C₂²×C₂₃	semidirect
167. #390	C₂²×C₂₃	semidirect
169. #340	C₂²×C₂₃	semidirect
207. C₂₃	#23540	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×D₈×C₂₃	2
C₂×C₂₃×#424	2
C₂×C₂₃×#418	2
C₂₃×#2309	2
C₂₃×#2054	2
C₂₃×#1735	2
#23540	23

Subgroups — C23×#23540 #19171166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23540 #19171166