Subgroups — C₂₃×#18068 #19165694

Order 5888.

586 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3170	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2962	C₂	1	2	yes	maximal
3	C₂₃×#1591	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1584	C₂	1	4	yes	maximal, complemented
5	C₈×Q₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1481	C₂	1	2	yes	maximal
8	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#533	C₂²	1	2	yes	—
10	C₂₃×#500	C₄	1	1	yes	characteristic
11	C₂₃×#494	C₄	1	1	yes	characteristic
12	C₂₃×#487	C₄	1	2	yes	—
13	C₂₃×#467	C₄	1	2	yes	—
14	C₂₃×#467	C₄	1	2	yes	—
15	C₂₃×#446	—	2	4	no	—
16	C₂₃×#446	C₂²	1	4	yes	—
17	C₈×Q₈×C₂₃	C₂²	1	4	yes	—
18	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
19	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
20	C₂₃×#431	C₂²	1	1	yes	characteristic
21	C₂₃×#424	—	2	2	no	—
22	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂₃×#405	—	2	2	no	—
24	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#335	C₂²	1	1	yes	characteristic
27	C₂₃×#335	C₂²	1	2	yes	—
28	C₂₃×#335	C₂²	1	1	yes	characteristic
29	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	—
30	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₂₃×#322	C₂²	1	2	yes	—
32	C₈²×C₂₃	C₂²	1	2	yes	—
33	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
34	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
36	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₂₃	—	2	2	no	—
38	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
41	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
42	C₂₃×#125	—	2	2	no	—
43	C₂₃×C₂²⋊Q₈	—	2	2	no	—
44	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
45	C₄×Q₈×C₂₃	—	2	2	no	—
46	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
47	C₂₃×#117	—	2	2	no	—
48	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
49	C₂₃×#107	C₂×C₄	1	2	yes	—
50	C₂₃×#107	C₂×C₄	1	2	yes	—
51	C₂₃×C₄⋊C₈	—	2	4	no	—
52	C₂₃×C₄⋊C₈	—	2	4	no	—
53	C₂₃×C₄⋊C₈	—	2	2	no	—
54	C₂₃×C₄⋊C₈	—	2	1	no	—
55	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
56	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
57	C₂₃×#103	C₂×C₄	1	4	yes	—
58	C₂₃×#103	C₂×C₄	1	4	yes	—
59	C₂₃×C₂²⋊C₈	—	2	1	no	—
60	C₂₃×#97	—	2	4	no	—
61	C₂₃×#97	D₄	1	2	yes	—
62	C₄×C₈×C₂₃	—	2	4	no	—
63	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₂₃	C₂³	1	2	yes	—
66	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₂₃	C₂³	1	2	yes	—
69	C₂×Q₈×C₂₃	—	2	2	no	—
70	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
74	Q₁₆×C₂₃	—	2	4	no	—
75	Q₁₆×C₂₃	—	4	2	no	—
76	M₁₆×C₂₃	—	2	2	no	—
77	M₁₆×C₂₃	Q₁₆	1	4	yes	complemented
78	C₂×C₈×C₂₃	—	2	2	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	—	4	1	no	—
83	C₂×C₈×C₂₃	—	4	4	no	—
84	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
90	C₄⋊C₄×C₂₃	—	2	2	no	—
91	C₄⋊C₄×C₂₃	—	2	2	no	—
92	C₄⋊C₄×C₂₃	—	2	2	no	—
93	C₄⋊C₄×C₂₃	—	2	4	no	—
94	C₄⋊C₄×C₂₃	—	2	4	no	—
95	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
96	C₂²⋊C₄×C₂₃	—	2	4	no	—
97	C₄²×C₂₃	—	2	4	no	—
98	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₂₃	#41	1	1	yes	characteristic
101	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
102	#18068	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
104	Q₈×C₂₃	—	2	4	no	—
105	Q₈×C₂₃	—	2	4	no	—
106	Q₈×C₂₃	—	8	2	no	—
107	C₂×C₄×C₂₃	—	2	1	no	—
108	C₂×C₄×C₂₃	—	2	1	no	—
109	C₂×C₄×C₂₃	—	2	4	no	—
110	C₂×C₄×C₂₃	—	2	4	no	—
111	C₂×C₄×C₂₃	—	4	1	no	—
112	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
116	C₂×C₄×C₂₃	#135	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	1	yes	characteristic
120	C₈×C₂₃	—	2	1	no	—
121	C₈×C₂₃	—	2	1	no	—
122	C₈×C₂₃	—	2	1	no	—
123	C₈×C₂₃	—	2	2	no	—
124	C₈×C₂₃	—	4	2	no	—
125	C₈×C₂₃	—	4	2	no	—
126	C₈×C₂₃	—	4	2	no	—
127	C₈×C₂₃	#131	1	2	yes	—
128	C₈×C₂₃	C₂×Q₁₆	1	4	yes	—
129	#3170	C₂×C₂₃	1	1	yes	characteristic
130	#2962	C₂×C₂₃	1	2	yes	—
131	#1591	C₂×C₂₃	1	2	yes	complemented
132	#1584	C₂×C₂₃	1	4	yes	complemented
133	C₈×Q₁₆	C₂×C₂₃	1	2	yes	complemented
134	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
135	#1481	C₂×C₂₃	1	2	yes	—
136	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic
137	C₂²×C₂₃	—	2	1	no	—
138	C₂²×C₂₃	—	2	1	no	—
139	C₂²×C₂₃	C₄×Q₁₆	1	2	yes	—
140	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₄×C₂₃	—	2	1	no	—
142	C₄×C₂₃	—	2	1	no	—
143	C₄×C₂₃	—	2	1	no	—
144	C₄×C₂₃	—	4	4	no	—
145	C₄×C₂₃	—	4	4	no	—
146	C₄×C₂₃	C₄×Q₁₆	1	2	yes	—
147	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
148	C₄×C₂₃	#568	1	1	yes	characteristic
149	C₄×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
150	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
151	#533	C₂²×C₂₃	1	2	yes	—
152	#500	C₄×C₂₃	1	1	yes	characteristic
153	#494	C₄×C₂₃	1	1	yes	characteristic
154	#487	C₄×C₂₃	1	2	yes	—
155	#467	C₄×C₂₃	1	2	yes	—
156	#467	C₄×C₂₃	1	2	yes	—
157	#446	—	2	4	no	—
158	#446	C₂²×C₂₃	1	4	yes	—
159	C₈×Q₈	C₂²×C₂₃	1	4	yes	—
160	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
161	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
162	#431	C₂²×C₂₃	1	1	yes	characteristic
163	#424	—	2	2	no	—
164	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
165	#405	—	2	2	no	—
166	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
168	#335	C₂²×C₂₃	1	1	yes	characteristic
169	#335	C₂²×C₂₃	1	2	yes	—
170	#335	C₂²×C₂₃	1	1	yes	characteristic
171	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	—
172	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	—
173	#322	C₂²×C₂₃	1	2	yes	—
174	C₈²	C₂²×C₂₃	1	2	yes	—
175	C₂×C₂₃	—	2	2	no	—
176	C₂×C₂₃	#2935	1	1	yes	characteristic
177	C₂×C₂₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
178	C₂×C₂₃	#2957	1	1	yes	characteristic
179	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
180	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
181	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
182	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
185	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
186	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
187	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
188	#125	—	2	2	no	—
189	C₂²⋊Q₈	—	2	2	no	—
190	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
191	C₄×Q₈	—	2	2	no	—
192	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
193	#117	—	2	2	no	—
194	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
195	#107	C₂×C₄×C₂₃	1	2	yes	—
196	#107	C₂×C₄×C₂₃	1	2	yes	—
197	C₄⋊C₈	—	2	4	no	—
198	C₄⋊C₈	—	2	4	no	—
199	C₄⋊C₈	—	2	2	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
202	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
203	#103	C₂×C₄×C₂₃	1	4	yes	—
204	#103	C₂×C₄×C₂₃	1	4	yes	—
205	C₂²⋊C₈	—	2	1	no	—
206	#97	—	2	4	no	—
207	#97	D₄×C₂₃	1	2	yes	—
208	C₄×C₈	—	2	4	no	—
209	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
210	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
211	C₄×C₈	C₂³×C₂₃	1	2	yes	—
212	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
213	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
214	C₄×C₈	C₂³×C₂₃	1	2	yes	—
215	C₂₃	#18068	1	1	yes	characteristic, direct product factor, Sylow
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
218	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
220	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
221	Q₁₆	—	2	4	no	—
222	Q₁₆	—	4	2	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	Q₁₆×C₂₃	1	4	yes	complemented
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	4	no	—
231	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
232	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
233	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
234	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
235	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
236	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
243	C₂²⋊C₄	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
246	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
247	C₄²	#41×C₂₃	1	1	yes	characteristic
248	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
249	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
250	Q₈	—	2	4	no	—
251	Q₈	—	2	4	no	—
252	Q₈	—	8	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
259	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
260	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
261	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
262	C₂×C₄	C₂₃×#135	1	2	yes	—
263	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
265	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	4	2	no	—
272	C₈	—	4	2	no	—
273	C₈	C₂₃×#131	1	2	yes	—
274	C₈	C₂×Q₁₆×C₂₃	1	4	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	C₄×Q₁₆×C₂₃	1	2	yes	—
278	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	4	no	—
284	C₄	C₄×Q₁₆×C₂₃	1	2	yes	—
285	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
286	C₄	C₂₃×#568	1	1	yes	characteristic
287	C₄	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
288	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	C₂₃×#2935	1	1	yes	characteristic
291	C₂	C₂×C₄×Q₁₆×C₂₃	1	1	yes	characteristic
292	C₂	C₂₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂³×C₂₃	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₄×M₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1591	C₂	semidirect
4. C₂₃×#1584	C₂	semidirect
5. C₈×Q₁₆×C₂₃	C₂	semidirect
77. M₁₆×C₂₃	Q₁₆	semidirect
102. #18068	C₂₃	direct (N × H)
131. #1591	C₂×C₂₃	semidirect
132. #1584	C₂×C₂₃	semidirect
133. C₈×Q₁₆	C₂×C₂₃	semidirect
215. C₂₃	#18068	direct (N × H)
224. M₁₆	Q₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3170	2
C₂₃×#2962	2
C₂₃×#1591	2
C₂₃×#1584	2
C₈×Q₁₆×C₂₃	2
C₂×C₂₃×#335	2
C₂₃×#1481	2
C₈×M₁₆×C₂₃	2
#18068	23

Subgroups — C23×#18068 #19165694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18068 #19165694