Subgroups — C₂₃×#21292 #19168918

Order 5888.

770 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×#135	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2922	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1974	C₂	1	2	yes	maximal
4	C₂₃×#1945	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1937	C₂	1	4	yes	maximal, complemented
6	C₄×C₂₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂₃×#104	C₂	1	2	yes	maximal, complemented
8	C₄²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
10	C₄×#41×C₂₃	C₂²	1	2	yes	—
11	C₄×Q₁₆×C₂₃	C₄	1	2	yes	complemented
12	C₄×SD₁₆×C₂₃	C₄	1	4	yes	complemented
13	C₄×D₈×C₂₃	C₄	1	2	yes	complemented
14	C₂₃×#433	—	2	4	no	—
15	C₂₃×#431	—	2	4	no	—
16	C₂×C₂₃×#108	C₂²	1	2	yes	—
17	C₂₃×#427	C₂²	1	1	yes	characteristic
18	C₂₃×#424	C₂²	1	4	yes	—
19	C₂×C₂₃×#104	C₂²	1	4	yes	—
20	C₂₃×#416	C₂²	1	2	yes	—
21	C₂₃×#408	C₂²	1	4	yes	—
22	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
27	C₄³×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#341	C₂²	1	2	yes	—
29	C₂₃×#339	C₂²	1	4	yes	—
30	C₂×#41×C₂₃	C₂³	1	2	yes	—
31	C₂₃×#135	—	2	4	no	—
32	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
34	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
36	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
39	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
40	C₄×D₄×C₂₃	—	2	4	no	—
41	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
42	C₂₃×#117	—	2	2	no	—
43	C₂₃×#117	C₂³	1	2	yes	—
44	C₂×C₄²×C₂₃	—	2	2	no	—
45	C₂×C₄²×C₂₃	—	2	2	no	—
46	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
47	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
48	C₂₃×#108	—	2	4	no	—
49	C₂₃×#107	C₂×C₄	1	2	yes	—
50	C₂₃×#106	C₂×C₄	1	2	yes	—
51	C₂₃×C₄⋊C₈	—	2	8	no	—
52	C₂₃×#104	—	2	8	no	—
53	C₂₃×#103	C₂×C₄	1	4	yes	—
54	C₂₃×#102	C₂×C₄	1	4	yes	—
55	C₂₃×C₂²⋊C₈	—	2	8	no	—
56	C₂₃×#97	—	2	2	no	—
57	C₄×C₈×C₂₃	—	2	4	no	—
58	C₄×C₈×C₂₃	—	2	2	no	—
59	C₄×C₈×C₂₃	—	2	4	no	—
60	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₂₃	D₄	1	2	yes	—
63	C₄×C₈×C₂₃	D₄	1	2	yes	—
64	#41×C₂₃	—	2	4	no	—
65	#41×C₂₃	—	2	4	no	—
66	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
67	C₂×D₄×C₂₃	—	2	2	no	—
68	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₂₃	—	2	2	no	—
70	C₂²×C₄×C₂₃	—	2	4	no	—
71	C₂²×C₄×C₂₃	—	2	2	no	—
72	C₂²×C₄×C₂₃	#41	1	2	yes	—
73	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
74	Q₁₆×C₂₃	—	2	2	no	—
75	SD₁₆×C₂₃	—	2	4	no	—
76	D₈×C₂₃	—	2	2	no	—
77	M₁₆×C₂₃	—	2	4	no	—
78	M₁₆×C₂₃	—	2	4	no	—
79	C₂×C₈×C₂₃	—	2	2	no	—
80	C₂×C₈×C₂₃	—	2	2	no	—
81	C₂×C₈×C₂₃	—	2	2	no	—
82	C₂×C₈×C₂₃	—	2	4	no	—
83	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
87	C₂×C₈×C₂₃	#41	1	4	yes	—
88	C₄⋊C₄×C₂₃	—	2	4	no	—
89	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
90	C₂²⋊C₄×C₂₃	—	2	2	no	—
91	C₂²⋊C₄×C₂₃	—	2	4	no	—
92	C₄²×C₂₃	—	2	8	no	—
93	C₄²×C₂₃	—	2	4	no	—
94	C₄²×C₂₃	—	2	2	no	—
95	C₄²×C₂₃	—	2	2	no	—
96	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
97	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₂₃	#41	1	4	yes	—
100	#21292	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₂₃	—	2	2	no	—
102	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
103	Q₈×C₂₃	—	2	2	no	—
104	Q₈×C₂₃	—	2	2	no	—
105	D₄×C₂₃	—	2	2	no	—
106	D₄×C₂₃	—	2	2	no	—
107	D₄×C₂₃	—	4	4	no	—
108	C₂×C₄×C₂₃	—	2	8	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	—	2	2	no	—
113	C₂×C₄×C₂₃	—	4	4	no	—
114	C₂×C₄×C₂₃	—	4	8	no	—
115	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
120	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
121	C₈×C₂₃	—	2	2	no	—
122	C₈×C₂₃	—	2	4	no	—
123	C₈×C₂₃	—	4	4	no	—
124	C₈×C₂₃	#117	1	2	yes	—
125	C₈×C₂₃	#117	1	2	yes	—
126	C₄×#135	C₂×C₂₃	1	1	yes	characteristic
127	#2922	C₂×C₂₃	1	2	yes	complemented
128	#1974	C₂×C₂₃	1	2	yes	—
129	#1945	C₂×C₂₃	1	2	yes	complemented
130	#1937	C₂×C₂₃	1	4	yes	complemented
131	C₄×#108	C₂×C₂₃	1	1	yes	characteristic, complemented
132	C₄×#104	C₂×C₂₃	1	2	yes	complemented
133	C₄²×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
134	C₂²×C₂₃	—	2	2	no	—
135	C₂²×C₂₃	—	4	4	no	—
136	C₂²×C₂₃	—	4	2	no	—
137	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂²×C₂₃	#529	1	2	yes	—
139	C₄×C₂₃	—	2	8	no	—
140	C₄×C₂₃	—	2	2	no	—
141	C₄×C₂₃	—	4	4	no	—
142	C₄×C₂₃	—	4	2	no	—
143	C₄×C₂₃	#443	1	4	yes	—
144	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
145	C₄×C₂₃	C₄×#41	1	1	yes	characteristic
146	C₄×C₂₃	#532	1	2	yes	—
147	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
148	C₄×#41	C₂²×C₂₃	1	2	yes	—
149	C₄×Q₁₆	C₄×C₂₃	1	2	yes	complemented
150	C₄×SD₁₆	C₄×C₂₃	1	4	yes	complemented
151	C₄×D₈	C₄×C₂₃	1	2	yes	complemented
152	#433	—	2	4	no	—
153	#431	—	2	4	no	—
154	C₂×#108	C₂²×C₂₃	1	2	yes	—
155	#427	C₂²×C₂₃	1	1	yes	characteristic
156	#424	C₂²×C₂₃	1	4	yes	—
157	C₂×#104	C₂²×C₂₃	1	4	yes	—
158	#416	C₂²×C₂₃	1	2	yes	—
159	#408	C₂²×C₂₃	1	4	yes	—
160	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
161	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
162	C₂×C₄×C₈	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	C₄³	C₂²×C₂₃	1	1	yes	characteristic
166	#341	C₂²×C₂₃	1	2	yes	—
167	#339	C₂²×C₂₃	1	4	yes	—
168	C₂×C₂₃	—	2	2	no	—
169	C₂×C₂₃	—	8	2	no	—
170	C₂×C₂₃	#2293	1	1	yes	characteristic
171	C₂×C₂₃	C₂×#443	1	2	yes	—
172	C₂×#41	C₂³×C₂₃	1	2	yes	—
173	#135	—	2	4	no	—
174	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
175	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
176	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
177	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
178	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
180	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
181	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
182	C₄×D₄	—	2	4	no	—
183	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
184	#117	—	2	2	no	—
185	#117	C₂³×C₂₃	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
189	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
190	#108	—	2	4	no	—
191	#107	C₂×C₄×C₂₃	1	2	yes	—
192	#106	C₂×C₄×C₂₃	1	2	yes	—
193	C₄⋊C₈	—	2	8	no	—
194	#104	—	2	8	no	—
195	#103	C₂×C₄×C₂₃	1	4	yes	—
196	#102	C₂×C₄×C₂₃	1	4	yes	—
197	C₂²⋊C₈	—	2	8	no	—
198	#97	—	2	2	no	—
199	C₄×C₈	—	2	4	no	—
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	—	2	4	no	—
202	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
203	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
204	C₄×C₈	D₄×C₂₃	1	2	yes	—
205	C₄×C₈	D₄×C₂₃	1	2	yes	—
206	C₂₃	#21292	1	1	yes	characteristic, direct product factor, Sylow
207	#41	—	2	4	no	—
208	#41	—	2	4	no	—
209	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	#41×C₂₃	1	2	yes	—
216	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
217	Q₁₆	—	2	2	no	—
218	SD₁₆	—	2	4	no	—
219	D₈	—	2	2	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
227	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
228	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
229	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
230	C₂×C₈	#41×C₂₃	1	4	yes	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₄²	—	2	8	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
240	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
241	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
242	C₄²	#41×C₂₃	1	4	yes	—
243	C₂³	—	2	2	no	—
244	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
245	Q₈	—	2	2	no	—
246	Q₈	—	2	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	2	2	no	—
249	D₄	—	4	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	8	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
258	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
259	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
261	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
262	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
263	C₈	—	2	2	no	—
264	C₈	—	2	4	no	—
265	C₈	—	4	4	no	—
266	C₈	C₂₃×#117	1	2	yes	—
267	C₈	C₂₃×#117	1	2	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	2	no	—
271	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
272	C₂²	C₂₃×#529	1	2	yes	—
273	C₄	—	2	8	no	—
274	C₄	—	2	2	no	—
275	C₄	—	4	4	no	—
276	C₄	—	4	2	no	—
277	C₄	C₂₃×#443	1	4	yes	—
278	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
279	C₄	C₄×#41×C₂₃	1	1	yes	characteristic
280	C₄	C₂₃×#532	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	8	2	no	—
283	C₂	C₂₃×#2293	1	1	yes	characteristic
284	C₂	C₂×C₂₃×#443	1	2	yes	—

	subgroup	G/H
center	C₄²×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂×C₂₃×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2922	C₂	semidirect
4. C₂₃×#1945	C₂	semidirect
5. C₂₃×#1937	C₂	semidirect
6. C₄×C₂₃×#108	C₂	semidirect
7. C₄×C₂₃×#104	C₂	semidirect
8. C₄²×C₈×C₂₃	C₂	semidirect
11. C₄×Q₁₆×C₂₃	C₄	semidirect
12. C₄×SD₁₆×C₂₃	C₄	semidirect
13. C₄×D₈×C₂₃	C₄	semidirect
100. #21292	C₂₃	direct (N × H)
127. #2922	C₂×C₂₃	semidirect
129. #1945	C₂×C₂₃	semidirect
130. #1937	C₂×C₂₃	semidirect
131. C₄×#108	C₂×C₂₃	semidirect
132. C₄×#104	C₂×C₂₃	semidirect
133. C₄²×C₈	C₂×C₂₃	semidirect
149. C₄×Q₁₆	C₄×C₂₃	semidirect
150. C₄×SD₁₆	C₄×C₂₃	semidirect
151. C₄×D₈	C₄×C₂₃	semidirect
206. C₂₃	#21292	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×#135	2
C₂₃×#2922	2
C₂₃×#1974	2
C₂₃×#1945	2
C₂₃×#1937	2
C₄×C₂₃×#108	2
C₄×C₂₃×#104	2
C₄²×C₈×C₂₃	2
#21292	23

Subgroups — C23×#21292 #19168918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21292 #19168918