Subgroups — C₂₃×#13039 #19160665

Order 5888.

472 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#510	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#444	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#360	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1971	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1375	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
9	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#131	—	2	1	no	—
11	C₂₃×#510	—	2	2	no	—
12	C₂₃×#510	—	2	2	no	—
13	C₂×C₂₃×Q₃₂	C₄	1	2	yes	complemented
14	C₂×C₂₃×SD₃₂	C₄	1	2	yes	complemented
15	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
17	C₂₃×#444	—	2	2	no	—
18	C₂₃×#444	—	2	2	no	—
19	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#104	—	2	1	no	—
22	C₂×C₂₃×#103	—	2	1	no	—
23	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
24	C₂₃×#391	—	2	1	no	—
25	C₂₃×#362	—	2	2	no	—
26	C₂₃×#360	—	2	2	no	—
27	C₂×#41×C₂₃	—	2	1	no	—
28	C₂²×Q₈×C₂₃	—	2	1	no	—
29	C₂₃×#135	—	2	1	no	—
30	C₂₃×#135	—	2	1	no	—
31	C₂₃×#135	D₄	1	2	yes	—
32	C₂₃×#135	D₄	1	2	yes	—
33	C₂×Q₁₆×C₂₃	—	2	1	no	—
34	C₂×Q₁₆×C₂₃	—	2	1	no	—
35	C₂×Q₁₆×C₂₃	—	2	1	no	—
36	C₂×Q₁₆×C₂₃	—	2	1	no	—
37	C₂×Q₁₆×C₂₃	—	2	1	no	—
38	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
39	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂×SD₁₆×C₂₃	—	2	1	no	—
41	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂₃×#131	—	2	4	no	—
43	C₂₃×#131	—	4	2	no	—
44	C₂×M₁₆×C₂₃	—	2	1	no	—
45	C₂×M₁₆×C₂₃	—	4	1	no	—
46	C₂²×C₈×C₂₃	—	4	1	no	—
47	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
50	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
51	C₂₃×Q₃₂	—	2	4	no	—
52	C₂₃×SD₃₂	—	2	4	no	—
53	C₂₃×M₃₂	—	2	1	no	—
54	C₂₃×M₃₂	—	2	1	no	—
55	C₂₃×M₃₂	—	2	1	no	—
56	C₂₃×M₃₂	—	2	1	no	—
57	C₂×C₁₆×C₂₃	—	2	1	no	—
58	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
59	C₂₃×#108	—	2	1	no	—
60	C₂₃×#108	D₄	1	2	yes	—
61	C₂₃×#106	—	2	1	no	—
62	C₂₃×#106	—	2	1	no	—
63	C₂₃×#104	—	2	4	no	—
64	C₂₃×#103	—	4	2	no	—
65	C₂₃×#97	—	2	1	no	—
66	C₂₃×#97	—	2	1	no	—
67	C₂₃×#95	—	4	2	no	—
68	#41×C₂₃	—	2	2	no	—
69	#41×C₂₃	—	2	2	no	—
70	#41×C₂₃	—	4	1	no	—
71	#41×C₂₃	—	4	1	no	—
72	C₂×Q₈×C₂₃	—	2	1	no	—
73	C₂×Q₈×C₂₃	—	2	2	no	—
74	C₂×Q₈×C₂₃	—	4	1	no	—
75	C₂×Q₈×C₂₃	—	4	1	no	—
76	C₂×Q₈×C₂₃	—	4	1	no	—
77	C₂×Q₈×C₂₃	—	4	1	no	—
78	C₂×Q₈×C₂₃	—	4	1	no	—
79	C₂×D₄×C₂₃	—	2	1	no	—
80	C₂×D₄×C₂₃	—	4	1	no	—
81	C₂²×C₄×C₂₃	—	2	1	no	—
82	C₂²×C₄×C₂₃	—	4	1	no	—
83	C₂²×C₄×C₂₃	—	4	1	no	—
84	C₂²×C₄×C₂₃	—	4	1	no	—
85	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
86	Q₁₆×C₂₃	—	2	2	no	—
87	Q₁₆×C₂₃	—	2	1	no	—
88	Q₁₆×C₂₃	—	2	2	no	—
89	Q₁₆×C₂₃	—	4	1	no	—
90	Q₁₆×C₂₃	—	4	1	no	—
91	Q₁₆×C₂₃	C₂²⋊C₄	1	2	yes	—
92	SD₁₆×C₂₃	—	2	2	no	—
93	SD₁₆×C₂₃	—	2	2	no	—
94	D₈×C₂₃	—	2	1	no	—
95	D₈×C₂₃	C₂²⋊C₄	1	2	yes	—
96	M₁₆×C₂₃	—	2	2	no	—
97	M₁₆×C₂₃	—	4	4	no	—
98	M₁₆×C₂₃	—	4	1	no	—
99	C₂×C₈×C₂₃	—	2	1	no	—
100	C₂×C₈×C₂₃	—	2	1	no	—
101	C₂×C₈×C₂₃	—	4	4	no	—
102	C₂×C₈×C₂₃	—	4	1	no	—
103	C₂×C₈×C₂₃	D₈	1	2	yes	—
104	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
109	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
110	C₄⋊C₄×C₂₃	—	4	1	no	—
111	C₄⋊C₄×C₂₃	—	4	1	no	—
112	C₄²×C₂₃	—	2	1	no	—
113	C₄²×C₂₃	—	2	1	no	—
114	C₁₆×C₂₃	—	2	2	no	—
115	C₁₆×C₂₃	—	4	1	no	—
116	#13039	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₂₃	—	4	1	no	—
118	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
119	Q₈×C₂₃	—	2	2	no	—
120	Q₈×C₂₃	—	4	1	no	—
121	Q₈×C₂₃	—	4	2	no	—
122	Q₈×C₂₃	—	4	2	no	—
123	Q₈×C₂₃	—	8	1	no	—
124	Q₈×C₂₃	—	8	1	no	—
125	D₄×C₂₃	—	2	2	no	—
126	D₄×C₂₃	—	4	2	no	—
127	D₄×C₂₃	—	4	2	no	—
128	D₄×C₂₃	—	4	1	no	—
129	C₂×C₄×C₂₃	—	4	1	no	—
130	C₂×C₄×C₂₃	—	4	2	no	—
131	C₂×C₄×C₂₃	—	4	2	no	—
132	C₂×C₄×C₂₃	—	4	2	no	—
133	C₂×C₄×C₂₃	—	4	2	no	—
134	C₂×C₄×C₂₃	—	4	1	no	—
135	C₂×C₄×C₂₃	—	8	1	no	—
136	C₂×C₄×C₂₃	—	8	1	no	—
137	C₂×C₄×C₂₃	—	8	2	no	—
138	C₂×C₄×C₂₃	—	8	1	no	—
139	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
141	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
142	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
144	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₈×C₂₃	—	2	2	no	—
146	C₈×C₂₃	—	4	2	no	—
147	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
149	C₈×C₂₃	#121	1	1	yes	characteristic
150	C₈×C₂₃	#123	1	1	yes	characteristic
151	C₂×#510	C₂×C₂₃	1	1	yes	characteristic
152	C₂×#444	C₂×C₂₃	1	1	yes	characteristic
153	C₂×#362	C₂×C₂₃	1	1	yes	characteristic, complemented
154	C₂×#360	C₂×C₂₃	1	1	yes	characteristic
155	#1971	C₂×C₂₃	1	1	yes	characteristic, complemented
156	#1383	C₂×C₂₃	1	1	yes	characteristic, complemented
157	#1375	C₂×C₂₃	1	1	yes	characteristic, complemented
158	C₂²×C₂₃	—	2	2	no	—
159	C₂²×C₂₃	—	4	2	no	—
160	C₂²×C₂₃	—	8	2	no	—
161	C₂²×C₂₃	#442	1	1	yes	characteristic
162	C₂²×C₂₃	#449	1	1	yes	characteristic
163	C₂²×C₂₃	#459	1	1	yes	characteristic
164	C₄×C₂₃	—	4	2	no	—
165	C₄×C₂₃	—	8	1	no	—
166	C₄×C₂₃	—	8	2	no	—
167	C₄×C₂₃	—	16	1	no	—
168	C₄×C₂₃	#386	1	1	yes	characteristic
169	C₄×C₂₃	C₄×D₈	1	1	yes	characteristic
170	C₄×C₂₃	#447	1	1	yes	characteristic
171	C₄×C₂₃	#463	1	1	yes	characteristic
172	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
173	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
174	C₂×#131	—	2	1	no	—
175	#510	—	2	2	no	—
176	#510	—	2	2	no	—
177	C₂×Q₃₂	C₄×C₂₃	1	2	yes	complemented
178	C₂×SD₃₂	C₄×C₂₃	1	2	yes	complemented
179	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
180	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
181	#444	—	2	2	no	—
182	#444	—	2	2	no	—
183	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
184	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
185	C₂×#104	—	2	1	no	—
186	C₂×#103	—	2	1	no	—
187	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
188	#391	—	2	1	no	—
189	#362	—	2	2	no	—
190	#360	—	2	2	no	—
191	C₂×C₂₃	—	2	1	no	—
192	C₂×C₂₃	—	2	1	no	—
193	C₂×C₂₃	—	8	2	no	—
194	C₂×C₂₃	#1879	1	1	yes	characteristic
195	C₂×C₂₃	#2180	1	2	yes	—
196	C₂×#41	—	2	1	no	—
197	C₂²×Q₈	—	2	1	no	—
198	#135	—	2	1	no	—
199	#135	—	2	1	no	—
200	#135	D₄×C₂₃	1	2	yes	—
201	#135	D₄×C₂₃	1	2	yes	—
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	—	2	1	no	—
204	C₂×Q₁₆	—	2	1	no	—
205	C₂×Q₁₆	—	2	1	no	—
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
208	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
209	C₂×SD₁₆	—	2	1	no	—
210	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
211	#131	—	2	4	no	—
212	#131	—	4	2	no	—
213	C₂×M₁₆	—	2	1	no	—
214	C₂×M₁₆	—	4	1	no	—
215	C₂²×C₈	—	4	1	no	—
216	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
217	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
220	Q₃₂	—	2	4	no	—
221	SD₃₂	—	2	4	no	—
222	M₃₂	—	2	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	2	1	no	—
225	M₃₂	—	2	1	no	—
226	C₂×C₁₆	—	2	1	no	—
227	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
228	#108	—	2	1	no	—
229	#108	D₄×C₂₃	1	2	yes	—
230	#106	—	2	1	no	—
231	#106	—	2	1	no	—
232	#104	—	2	4	no	—
233	#103	—	4	2	no	—
234	#97	—	2	1	no	—
235	#97	—	2	1	no	—
236	#95	—	4	2	no	—
237	C₂₃	#13039	1	1	yes	characteristic, direct product factor, Sylow
238	#41	—	2	2	no	—
239	#41	—	2	2	no	—
240	#41	—	4	1	no	—
241	#41	—	4	1	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	2	1	no	—
258	Q₁₆	—	2	2	no	—
259	Q₁₆	—	4	1	no	—
260	Q₁₆	—	4	1	no	—
261	Q₁₆	C₂²⋊C₄×C₂₃	1	2	yes	—
262	SD₁₆	—	2	2	no	—
263	SD₁₆	—	2	2	no	—
264	D₈	—	2	1	no	—
265	D₈	C₂²⋊C₄×C₂₃	1	2	yes	—
266	M₁₆	—	2	2	no	—
267	M₁₆	—	4	4	no	—
268	M₁₆	—	4	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	4	4	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	D₈×C₂₃	1	2	yes	—
274	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
277	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
278	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
279	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	4	1	no	—
286	C₂³	—	4	1	no	—
287	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	8	1	no	—
293	Q₈	—	8	1	no	—
294	D₄	—	2	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
309	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
310	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
311	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
312	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
313	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	4	2	no	—
316	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
317	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
318	C₈	C₂₃×#121	1	1	yes	characteristic
319	C₈	C₂₃×#123	1	1	yes	characteristic
320	C₂²	—	2	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	C₂₃×#442	1	1	yes	characteristic
324	C₂²	C₂₃×#449	1	1	yes	characteristic
325	C₂²	C₂₃×#459	1	1	yes	characteristic
326	C₄	—	4	2	no	—
327	C₄	—	8	1	no	—
328	C₄	—	8	2	no	—
329	C₄	—	16	1	no	—
330	C₄	C₂₃×#386	1	1	yes	characteristic
331	C₄	C₄×D₈×C₂₃	1	1	yes	characteristic
332	C₄	C₂₃×#447	1	1	yes	characteristic
333	C₄	C₂₃×#463	1	1	yes	characteristic
334	C₂	—	2	1	no	—
335	C₂	—	2	1	no	—
336	C₂	—	8	2	no	—
337	C₂	C₂₃×#1879	1	1	yes	characteristic
338	C₂	C₂₃×#2180	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	#442
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂×C₂₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₂₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₂₃×#362	C₂	semidirect
5. C₂₃×#1971	C₂	semidirect
6. C₂₃×#1383	C₂	semidirect
7. C₂₃×#1375	C₂	semidirect
13. C₂×C₂₃×Q₃₂	C₄	semidirect
14. C₂×C₂₃×SD₃₂	C₄	semidirect
116. #13039	C₂₃	direct (N × H)
153. C₂×#362	C₂×C₂₃	semidirect
155. #1971	C₂×C₂₃	semidirect
156. #1383	C₂×C₂₃	semidirect
157. #1375	C₂×C₂₃	semidirect
177. C₂×Q₃₂	C₄×C₂₃	semidirect
178. C₂×SD₃₂	C₄×C₂₃	semidirect
237. C₂₃	#13039	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×#510	2
C₂×C₂₃×#444	2
C₂×C₂₃×#362	2
C₂×C₂₃×#360	2
C₂₃×#1971	2
C₂₃×#1383	2
C₂₃×#1375	2
#13039	23

Subgroups — C23×#13039 #19160665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13039 #19160665