Subgroups — C₂₃×#18058 #19165684

Order 5888.

626 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2984	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1591	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1587	C₂	1	4	yes	maximal, complemented
5	C₈×Q₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1568	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1491	C₂	1	2	yes	maximal
8	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
11	C₂₃×#446	—	2	4	no	—
12	C₂₃×#446	C₂²	1	4	yes	—
13	C₈×Q₈×C₂₃	C₂²	1	4	yes	—
14	C₄×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
15	C₄×Q₁₆×C₂₃	C₄	1	2	yes	—
16	C₄×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
17	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
19	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#432	—	2	2	no	—
21	C₂₃×#432	C₂²	1	2	yes	—
22	C₂₃×#431	—	2	2	no	—
23	C₂₃×#431	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#107	C₄	1	1	yes	characteristic
25	C₂×C₂₃×#103	C₄	1	2	yes	—
26	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#335	C₂²	1	2	yes	—
29	C₂₃×#334	C₂²	1	2	yes	—
30	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	—
32	C₂₃×#322	C₂²	1	2	yes	—
33	C₈²×C₂₃	C₂²	1	2	yes	—
34	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×Q₁₆×C₂₃	—	2	4	no	—
36	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₂₃	—	2	1	no	—
40	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₄×Q₈×C₂₃	—	2	2	no	—
43	C₄×Q₈×C₂₃	—	2	2	no	—
44	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
45	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
46	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
47	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₂₃	—	2	2	no	—
51	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
52	C₂₃×#107	C₂×C₄	1	2	yes	—
53	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
54	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
55	C₂₃×C₄⋊C₈	—	2	4	no	—
56	C₂₃×C₄⋊C₈	—	2	4	no	—
57	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
58	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
59	C₂₃×#103	C₂×C₄	1	2	yes	—
60	C₂₃×#103	C₂×C₄	1	2	yes	—
61	C₂₃×#103	C₂×C₄	1	2	yes	—
62	C₂₃×#103	C₂×C₄	1	2	yes	—
63	C₂₃×C₂²⋊C₈	—	2	2	no	—
64	C₂₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
65	C₂₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
66	C₂₃×#97	—	2	4	no	—
67	C₂₃×#97	—	2	2	no	—
68	C₂₃×#97	—	2	1	no	—
69	C₄×C₈×C₂₃	—	2	4	no	—
70	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₂₃	C₂³	1	2	yes	—
73	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₂₃	C₂³	1	2	yes	—
76	C₂×Q₈×C₂₃	—	2	2	no	—
77	C₂×Q₈×C₂₃	—	2	4	no	—
78	C₂×Q₈×C₂₃	—	4	2	no	—
79	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
80	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₂₃	—	2	2	no	—
82	C₂²×C₄×C₂₃	—	2	2	no	—
83	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
86	Q₁₆×C₂₃	—	2	4	no	—
87	Q₁₆×C₂₃	—	2	2	no	—
88	Q₁₆×C₂₃	M₁₆	1	4	yes	complemented
89	M₁₆×C₂₃	—	2	4	no	—
90	C₂×C₈×C₂₃	—	2	2	no	—
91	C₂×C₈×C₂₃	—	2	2	no	—
92	C₂×C₈×C₂₃	—	2	4	no	—
93	C₂×C₈×C₂₃	—	4	1	no	—
94	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
100	C₄⋊C₄×C₂₃	—	2	2	no	—
101	C₄⋊C₄×C₂₃	—	2	2	no	—
102	C₄⋊C₄×C₂₃	—	2	2	no	—
103	C₄⋊C₄×C₂₃	—	2	2	no	—
104	C₄⋊C₄×C₂₃	—	2	2	no	—
105	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
106	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
107	C₄²×C₂₃	—	2	2	no	—
108	C₄²×C₂₃	—	2	2	no	—
109	C₄²×C₂₃	—	2	2	no	—
110	C₄²×C₂₃	—	2	2	no	—
111	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₂₃	#41	1	1	yes	characteristic
114	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
115	#18058	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
117	Q₈×C₂₃	—	2	2	no	—
118	Q₈×C₂₃	—	2	2	no	—
119	Q₈×C₂₃	—	2	2	no	—
120	Q₈×C₂₃	—	4	4	no	—
121	Q₈×C₂₃	C₂×M₁₆	1	4	yes	—
122	C₂×C₄×C₂₃	—	2	2	no	—
123	C₂×C₄×C₂₃	—	2	2	no	—
124	C₂×C₄×C₂₃	—	2	1	no	—
125	C₂×C₄×C₂₃	—	2	1	no	—
126	C₂×C₄×C₂₃	—	2	2	no	—
127	C₂×C₄×C₂₃	—	2	2	no	—
128	C₂×C₄×C₂₃	—	2	2	no	—
129	C₂×C₄×C₂₃	—	2	2	no	—
130	C₂×C₄×C₂₃	—	4	4	no	—
131	C₂×C₄×C₂₃	—	4	4	no	—
132	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
138	C₈×C₂₃	—	2	4	no	—
139	C₈×C₂₃	—	2	1	no	—
140	C₈×C₂₃	—	2	1	no	—
141	C₈×C₂₃	—	2	1	no	—
142	C₈×C₂₃	—	2	2	no	—
143	C₈×C₂₃	—	2	2	no	—
144	C₈×C₂₃	—	4	4	no	—
145	C₈×C₂₃	C₂×M₁₆	1	2	yes	—
146	#2984	C₂×C₂₃	1	2	yes	—
147	C₂×C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic
148	#1591	C₂×C₂₃	1	2	yes	complemented
149	#1587	C₂×C₂₃	1	4	yes	complemented
150	C₈×Q₁₆	C₂×C₂₃	1	2	yes	complemented
151	#1568	C₂×C₂₃	1	1	yes	characteristic
152	#1491	C₂×C₂₃	1	2	yes	—
153	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic
154	C₂²×C₂₃	—	2	1	no	—
155	C₂²×C₂₃	—	2	1	no	—
156	C₂²×C₂₃	#443	1	2	yes	—
157	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
158	C₄×C₂₃	—	2	1	no	—
159	C₄×C₂₃	—	2	4	no	—
160	C₄×C₂₃	—	2	1	no	—
161	C₄×C₂₃	—	2	1	no	—
162	C₄×C₂₃	—	2	1	no	—
163	C₄×C₂₃	—	4	2	no	—
164	C₄×C₂₃	—	4	2	no	—
165	C₄×C₂₃	—	4	2	no	—
166	C₄×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
167	C₄×C₂₃	#568	1	1	yes	characteristic
168	C₄×C₂₃	#578	1	2	yes	—
169	C₂²×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
170	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
171	#446	—	2	4	no	—
172	#446	C₂²×C₂₃	1	4	yes	—
173	C₈×Q₈	C₂²×C₂₃	1	4	yes	—
174	C₄×Q₁₆	C₄×C₂₃	1	2	yes	—
175	C₄×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
176	C₄×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
177	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
178	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
179	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
180	#432	—	2	2	no	—
181	#432	C₂²×C₂₃	1	2	yes	—
182	#431	—	2	2	no	—
183	#431	C₂²×C₂₃	1	1	yes	characteristic
184	C₂×#107	C₄×C₂₃	1	1	yes	characteristic
185	C₂×#103	C₄×C₂₃	1	2	yes	—
186	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
187	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
188	#335	C₂²×C₂₃	1	2	yes	—
189	#334	C₂²×C₂₃	1	2	yes	—
190	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	—
191	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	—
192	#322	C₂²×C₂₃	1	2	yes	—
193	C₈²	C₂²×C₂₃	1	2	yes	—
194	C₂×C₂₃	—	2	2	no	—
195	C₂×C₂₃	D₄×M₁₆	1	1	yes	characteristic
196	C₂×C₂₃	#2950	1	1	yes	characteristic
197	C₂×C₂₃	C₂×#443	1	1	yes	characteristic
198	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
199	C₂×Q₁₆	—	2	4	no	—
200	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
201	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
202	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
203	C₂×M₁₆	—	2	1	no	—
204	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
206	C₄×Q₈	—	2	2	no	—
207	C₄×Q₈	—	2	2	no	—
208	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
209	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
210	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
211	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
216	#107	C₂×C₄×C₂₃	1	2	yes	—
217	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
218	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
219	C₄⋊C₈	—	2	4	no	—
220	C₄⋊C₈	—	2	4	no	—
221	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
222	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
223	#103	C₂×C₄×C₂₃	1	2	yes	—
224	#103	C₂×C₄×C₂₃	1	2	yes	—
225	#103	C₂×C₄×C₂₃	1	2	yes	—
226	#103	C₂×C₄×C₂₃	1	2	yes	—
227	C₂²⋊C₈	—	2	2	no	—
228	C₂²⋊C₈	D₄×C₂₃	1	1	yes	characteristic
229	C₂²⋊C₈	D₄×C₂₃	1	1	yes	characteristic
230	#97	—	2	4	no	—
231	#97	—	2	2	no	—
232	#97	—	2	1	no	—
233	C₄×C₈	—	2	4	no	—
234	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
235	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₂₃	1	2	yes	—
237	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₂₃	1	2	yes	—
240	C₂₃	#18058	1	1	yes	characteristic, direct product factor, Sylow
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
245	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
250	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
251	Q₁₆	—	2	4	no	—
252	Q₁₆	—	2	2	no	—
253	Q₁₆	M₁₆×C₂₃	1	4	yes	complemented
254	M₁₆	—	2	4	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
260	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
261	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
262	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
264	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
271	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
278	C₄²	#41×C₂₃	1	1	yes	characteristic
279	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
280	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
281	Q₈	—	2	2	no	—
282	Q₈	—	2	2	no	—
283	Q₈	—	2	2	no	—
284	Q₈	—	4	4	no	—
285	Q₈	C₂×M₁₆×C₂₃	1	4	yes	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
297	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
298	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
299	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
301	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
302	C₈	—	2	4	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	2	no	—
307	C₈	—	2	2	no	—
308	C₈	—	4	4	no	—
309	C₈	C₂×M₁₆×C₂₃	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	C₂₃×#443	1	2	yes	—
313	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	4	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	4	2	no	—
322	C₄	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
323	C₄	C₂₃×#568	1	1	yes	characteristic
324	C₄	C₂₃×#578	1	2	yes	—
325	C₂	—	2	2	no	—
326	C₂	D₄×M₁₆×C₂₃	1	1	yes	characteristic
327	C₂	C₂₃×#2950	1	1	yes	characteristic
328	C₂	C₂×C₂₃×#443	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 ◁ C₂₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1591	C₂	semidirect
4. C₂₃×#1587	C₂	semidirect
5. C₈×Q₁₆×C₂₃	C₂	semidirect
88. Q₁₆×C₂₃	M₁₆	semidirect
115. #18058	C₂₃	direct (N × H)
148. #1591	C₂×C₂₃	semidirect
149. #1587	C₂×C₂₃	semidirect
150. C₈×Q₁₆	C₂×C₂₃	semidirect
240. C₂₃	#18058	direct (N × H)
253. Q₁₆	M₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2984	2
C₂×C₄×Q₁₆×C₂₃	2
C₂₃×#1591	2
C₂₃×#1587	2
C₈×Q₁₆×C₂₃	2
C₂₃×#1568	2
C₂₃×#1491	2
C₈×M₁₆×C₂₃	2
#18058	23

Subgroups — C23×#18058 #19165684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18058 #19165684