Subgroups — C₂₃×#20815 #19168441

Order 5888.

862 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	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1842	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1831	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1798	C₂	1	4	yes	maximal
7	C₂₃×#1768	C₂	1	4	yes	maximal, complemented
8	C₂×C₂₃×#117	C₂²	1	2	yes	—
9	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×#432	—	2	4	no	—
11	C₂₃×#431	—	2	4	no	—
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	8	yes	—
15	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
16	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₂₃×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
19	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
20	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
21	C₄×C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
22	C₄×C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
23	C₂₃×#376	C₂²	1	2	yes	—
24	C₄³×C₂₃	C₂²	1	1	yes	characteristic
25	C₄³×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#336	C₂²	1	8	yes	—
27	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
31	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
32	C₂₃×#117	—	2	8	no	—
33	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₂₃	D₄	1	4	yes	—
42	C₂×C₄²×C₂₃	Q₈	1	4	yes	—
43	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
44	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
47	C₂₃×C₄⋊C₈	—	2	4	no	—
48	C₂₃×C₄⋊C₈	—	2	4	no	—
49	C₂₃×C₄⋊C₈	—	2	16	no	—
50	C₂₃×C₂²⋊C₈	—	2	4	no	—
51	C₂₃×C₂²⋊C₈	—	2	4	no	—
52	C₂₃×#97	—	2	4	no	—
53	C₄×C₈×C₂₃	—	2	4	no	—
54	C₂₃×#95	C₂×C₄	1	4	yes	—
55	C₂₃×#95	C₂×C₄	1	4	yes	—
56	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	—	2	4	no	—
58	C₂²×C₄×C₂₃	—	2	4	no	—
59	C₂²×C₄×C₂₃	—	4	2	no	—
60	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₂₃	—	2	4	no	—
73	C₂×C₈×C₂₃	—	2	4	no	—
74	C₂×C₈×C₂₃	—	2	8	no	—
75	C₂×C₈×C₂₃	—	2	8	no	—
76	C₄⋊C₄×C₂₃	—	2	8	no	—
77	C₂²⋊C₄×C₂₃	—	2	8	no	—
78	C₄²×C₂₃	—	2	4	no	—
79	C₄²×C₂₃	—	2	4	no	—
80	C₄²×C₂₃	—	2	4	no	—
81	C₄²×C₂₃	—	4	2	no	—
82	C₄²×C₂₃	—	4	2	no	—
83	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
84	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
85	C₄²×C₂₃	C₂×D₄	1	2	yes	—
86	C₄²×C₂₃	C₂×D₄	1	2	yes	—
87	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
88	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
89	C₄²×C₂₃	#41	1	8	yes	—
90	#20815	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₃	—	2	2	no	—
92	C₂³×C₂₃	—	2	2	no	—
93	C₂³×C₂₃	—	2	2	no	—
94	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	4	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	4	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	4	no	—
103	C₂×C₄×C₂₃	—	2	4	no	—
104	C₂×C₄×C₂₃	—	2	8	no	—
105	C₂×C₄×C₂₃	—	2	4	no	—
106	C₂×C₄×C₂₃	—	2	4	no	—
107	C₂×C₄×C₂₃	—	2	4	no	—
108	C₂×C₄×C₂₃	—	4	4	no	—
109	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₂₃	#117	1	8	yes	—
115	C₂×C₄×C₂₃	#131	1	4	yes	—
116	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
119	C₈×C₂₃	—	4	4	no	—
120	C₈×C₂₃	—	4	4	no	—
121	C₂×#432	C₂×C₂₃	1	1	yes	characteristic
122	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
123	C₄×#117	C₂×C₂₃	1	1	yes	characteristic
124	#1842	C₂×C₂₃	1	2	yes	complemented
125	#1831	C₂×C₂₃	1	2	yes	complemented
126	#1798	C₂×C₂₃	1	4	yes	—
127	#1768	C₂×C₂₃	1	4	yes	complemented
128	C₂²×C₂₃	—	2	2	no	—
129	C₂²×C₂₃	—	2	2	no	—
130	C₂²×C₂₃	—	2	2	no	—
131	C₂²×C₂₃	—	4	4	no	—
132	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₂₃	C₂×#117	1	2	yes	—
134	C₂²×C₂₃	C₂×#131	1	2	yes	—
135	C₂²×C₂₃	#568	1	2	yes	—
136	C₄×C₂₃	—	2	2	no	—
137	C₄×C₂₃	—	2	2	no	—
138	C₄×C₂₃	—	2	2	no	—
139	C₄×C₂₃	—	2	8	no	—
140	C₄×C₂₃	—	2	2	no	—
141	C₄×C₂₃	—	4	4	no	—
142	C₄×C₂₃	#424	1	4	yes	—
143	C₄×C₂₃	#532	1	2	yes	—
144	C₄×C₂₃	#533	1	2	yes	—
145	C₂×#117	C₂²×C₂₃	1	2	yes	—
146	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
147	#432	—	2	4	no	—
148	#431	—	2	4	no	—
149	C₂×C₄⋊C₈	C₂²×C₂₃	1	8	yes	—
150	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
151	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
152	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
153	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
154	C₂×#97	C₂²×C₂₃	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
156	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
157	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
158	C₄×C₂²⋊C₄	C₄×C₂₃	1	2	yes	—
159	C₄×C₂²⋊C₄	C₄×C₂₃	1	2	yes	—
160	#376	C₂²×C₂₃	1	2	yes	—
161	C₄³	C₂²×C₂₃	1	1	yes	characteristic
162	C₄³	C₂²×C₂₃	1	1	yes	characteristic
163	#336	C₂²×C₂₃	1	8	yes	—
164	C₂×C₂₃	—	4	2	no	—
165	C₂×C₂₃	#2285	1	1	yes	characteristic
166	C₂×C₂₃	C₂×#424	1	1	yes	characteristic
167	C₂×C₂₃	#2905	1	1	yes	characteristic
168	C₂×C₂₃	#2925	1	4	yes	—
169	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
170	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
171	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
173	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
174	#117	—	2	8	no	—
175	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
176	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
177	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
178	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
179	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
180	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
181	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
182	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄²	D₄×C₂₃	1	4	yes	—
184	C₂×C₄²	Q₈×C₂₃	1	4	yes	—
185	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
186	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
187	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
188	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
189	C₄⋊C₈	—	2	4	no	—
190	C₄⋊C₈	—	2	4	no	—
191	C₄⋊C₈	—	2	16	no	—
192	C₂²⋊C₈	—	2	4	no	—
193	C₂²⋊C₈	—	2	4	no	—
194	#97	—	2	4	no	—
195	C₄×C₈	—	2	4	no	—
196	#95	C₂×C₄×C₂₃	1	4	yes	—
197	#95	C₂×C₄×C₂₃	1	4	yes	—
198	C₂₃	#20815	1	1	yes	characteristic, direct product factor, Sylow
199	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
204	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
205	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
206	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
211	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
213	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
214	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	8	no	—
218	C₂×C₈	—	2	8	no	—
219	C₄⋊C₄	—	2	8	no	—
220	C₂²⋊C₄	—	2	8	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	4	no	—
223	C₄²	—	2	4	no	—
224	C₄²	—	4	2	no	—
225	C₄²	—	4	2	no	—
226	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
227	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
228	C₄²	C₂×D₄×C₂₃	1	2	yes	—
229	C₄²	C₂×D₄×C₂₃	1	2	yes	—
230	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
231	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
232	C₄²	#41×C₂₃	1	8	yes	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
237	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
252	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
253	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
254	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
255	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
256	C₂×C₄	C₂₃×#117	1	8	yes	—
257	C₂×C₄	C₂₃×#131	1	4	yes	—
258	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
260	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
261	C₈	—	4	4	no	—
262	C₈	—	4	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	4	no	—
267	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
268	C₂²	C₂×C₂₃×#117	1	2	yes	—
269	C₂²	C₂×C₂₃×#131	1	2	yes	—
270	C₂²	C₂₃×#568	1	2	yes	—
271	C₄	—	2	8	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	C₂₃×#424	1	4	yes	—
278	C₄	C₂₃×#532	1	2	yes	—
279	C₄	C₂₃×#533	1	2	yes	—
280	C₂	—	4	2	no	—
281	C₂	C₂₃×#2285	1	1	yes	characteristic
282	C₂	C₂×C₂₃×#424	1	1	yes	characteristic
283	C₂	C₂₃×#2905	1	1	yes	characteristic
284	C₂	C₂₃×#2925	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1842	C₂	semidirect
5. C₂₃×#1831	C₂	semidirect
7. C₂₃×#1768	C₂	semidirect
90. #20815	C₂₃	direct (N × H)
124. #1842	C₂×C₂₃	semidirect
125. #1831	C₂×C₂₃	semidirect
127. #1768	C₂×C₂₃	semidirect
198. C₂₃	#20815	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#432	2
C₂×C₂₃×#431	2
C₄×C₂₃×#117	2
C₂₃×#1842	2
C₂₃×#1831	2
C₂₃×#1798	2
C₂₃×#1768	2
#20815	23

Subgroups — C23×#20815 #19168441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20815 #19168441