Subgroups — C₂₃×#10442 #19158068

Order 5888.

460 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#460	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1801	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1571	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1545	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1545	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1479	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₂₃	—	2	1	no	—
9	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#460	—	2	2	no	—
12	C₂₃×#460	—	2	2	no	—
13	C₂₃×#431	C₂²	1	1	yes	characteristic
14	C₂₃×#431	C₂²	1	1	yes	characteristic
15	C₂₃×#423	C₂²	1	1	yes	characteristic
16	C₂₃×#423	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#102	—	2	1	no	—
19	C₂×C₂₃×#101	—	2	1	no	—
20	C₂₃×#343	—	2	1	no	—
21	C₂₃×#335	—	2	2	no	—
22	C₂₃×#331	—	2	2	no	—
23	C₂₃×#331	—	2	2	no	—
24	C₂₃×Q₈⋊C₈	—	2	2	no	—
25	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
26	C₂²×D₄×C₂₃	—	2	1	no	—
27	C₂²×D₄×C₂₃	—	2	1	no	—
28	C₂²×D₄×C₂₃	—	2	1	no	—
29	C₂×SD₁₆×C₂₃	—	2	4	no	—
30	C₂×SD₁₆×C₂₃	—	4	2	no	—
31	C₂×SD₁₆×C₂₃	—	4	2	no	—
32	C₂×M₁₆×C₂₃	—	2	1	no	—
33	C₂×M₁₆×C₂₃	—	2	1	no	—
34	C₂²×C₈×C₂₃	—	2	1	no	—
35	C₂₃×#127	—	2	1	no	—
36	C₂₃×#127	—	2	1	no	—
37	C₂₃×#127	—	2	1	no	—
38	C₂₃×#127	D₄	1	2	yes	—
39	C₄×Q₈×C₂₃	—	2	1	no	—
40	C₄×Q₈×C₂₃	—	2	1	no	—
41	C₄×Q₈×C₂₃	—	2	1	no	—
42	C₄×Q₈×C₂₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₂₃	—	2	1	no	—
46	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
47	C₂₃×C₄⋊C₈	—	2	1	no	—
48	C₂₃×C₄⋊C₈	—	2	1	no	—
49	C₂₃×C₄⋊C₈	—	2	1	no	—
50	C₂₃×C₄⋊C₈	—	2	1	no	—
51	C₂₃×C₄⋊C₈	—	2	1	no	—
52	C₂₃×C₄⋊C₈	—	2	1	no	—
53	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
54	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
55	C₂₃×#102	—	2	4	no	—
56	C₂₃×#101	—	4	2	no	—
57	C₂₃×C₂²⋊C₈	—	2	1	no	—
58	C₂₃×C₂²⋊C₈	—	2	1	no	—
59	C₄×C₈×C₂₃	—	2	1	no	—
60	C₄×C₈×C₂₃	—	2	1	no	—
61	C₂⁴×C₂₃	—	4	1	no	—
62	C₂×Q₈×C₂₃	—	2	1	no	—
63	C₂×Q₈×C₂₃	—	2	1	no	—
64	C₂×Q₈×C₂₃	—	2	1	no	—
65	C₂×Q₈×C₂₃	—	2	1	no	—
66	C₂×Q₈×C₂₃	—	2	1	no	—
67	C₂×Q₈×C₂₃	D₈	1	2	yes	—
68	C₂×D₄×C₂₃	—	2	2	no	—
69	C₂×D₄×C₂₃	—	2	2	no	—
70	C₂×D₄×C₂₃	—	2	2	no	—
71	C₂×D₄×C₂₃	—	4	1	no	—
72	C₂×D₄×C₂₃	—	4	1	no	—
73	C₂×D₄×C₂₃	—	4	1	no	—
74	C₂×D₄×C₂₃	—	4	1	no	—
75	C₂×D₄×C₂₃	—	4	1	no	—
76	C₂×D₄×C₂₃	—	4	1	no	—
77	C₂×D₄×C₂₃	—	4	1	no	—
78	C₂×D₄×C₂₃	—	8	2	no	—
79	C₂×D₄×C₂₃	—	8	2	no	—
80	C₂²×C₄×C₂₃	—	2	1	no	—
81	C₂²×C₄×C₂₃	—	2	1	no	—
82	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
84	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
85	SD₁₆×C₂₃	—	4	8	no	—
86	M₁₆×C₂₃	—	4	1	no	—
87	M₁₆×C₂₃	—	4	1	no	—
88	M₁₆×C₂₃	—	4	1	no	—
89	M₁₆×C₂₃	—	4	1	no	—
90	C₂×C₈×C₂₃	—	2	2	no	—
91	C₂×C₈×C₂₃	—	2	2	no	—
92	C₂×C₈×C₂₃	—	4	1	no	—
93	C₂×C₈×C₂₃	—	4	1	no	—
94	C₂×C₈×C₂₃	—	4	1	no	—
95	C₂×C₈×C₂₃	—	4	1	no	—
96	C₂×C₈×C₂₃	—	4	1	no	—
97	C₄⋊C₄×C₂₃	—	2	1	no	—
98	C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₄⋊C₄×C₂₃	—	4	1	no	—
100	C₄⋊C₄×C₂₃	—	4	1	no	—
101	C₄²×C₂₃	—	4	1	no	—
102	C₄²×C₂₃	—	4	1	no	—
103	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₂₃	#41	1	1	yes	characteristic
107	#10442	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₂₃	—	4	2	no	—
109	C₂³×C₂₃	—	8	2	no	—
110	C₂³×C₂₃	—	8	2	no	—
111	C₂³×C₂₃	—	8	1	no	—
112	C₂³×C₂₃	—	8	1	no	—
113	C₂³×C₂₃	#123	1	1	yes	characteristic
114	Q₈×C₂₃	—	2	2	no	—
115	Q₈×C₂₃	—	2	2	no	—
116	Q₈×C₂₃	—	4	1	no	—
117	Q₈×C₂₃	—	4	1	no	—
118	D₄×C₂₃	—	4	2	no	—
119	D₄×C₂₃	—	4	2	no	—
120	D₄×C₂₃	—	8	2	no	—
121	D₄×C₂₃	—	8	2	no	—
122	D₄×C₂₃	—	8	2	no	—
123	D₄×C₂₃	—	8	1	no	—
124	D₄×C₂₃	—	8	1	no	—
125	D₄×C₂₃	—	8	2	no	—
126	C₂×C₄×C₂₃	—	2	2	no	—
127	C₂×C₄×C₂₃	—	2	2	no	—
128	C₂×C₄×C₂₃	—	4	1	no	—
129	C₂×C₄×C₂₃	—	4	1	no	—
130	C₂×C₄×C₂₃	—	4	1	no	—
131	C₂×C₄×C₂₃	—	4	1	no	—
132	C₂×C₄×C₂₃	—	8	1	no	—
133	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
134	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
137	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
138	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
139	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
141	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
143	C₈×C₂₃	—	4	2	no	—
144	C₈×C₂₃	—	4	2	no	—
145	C₈×C₂₃	—	8	1	no	—
146	C₈×C₂₃	—	8	1	no	—
147	C₈×C₂₃	—	8	1	no	—
148	C₂×#460	C₂×C₂₃	1	1	yes	characteristic
149	#1801	C₂×C₂₃	1	1	yes	characteristic, complemented
150	#1571	C₂×C₂₃	1	1	yes	characteristic, complemented
151	#1545	C₂×C₂₃	1	1	yes	characteristic
152	#1545	C₂×C₂₃	1	1	yes	characteristic
153	#1479	C₂×C₂₃	1	1	yes	characteristic, complemented
154	#1301	C₂×C₂₃	1	1	yes	characteristic, complemented
155	C₂²×C₂₃	—	2	1	no	—
156	C₂²×C₂₃	—	2	1	no	—
157	C₂²×C₂₃	—	8	2	no	—
158	C₂²×C₂₃	—	8	2	no	—
159	C₂²×C₂₃	—	8	2	no	—
160	C₂²×C₂₃	—	16	2	no	—
161	C₂²×C₂₃	—	16	2	no	—
162	C₂²×C₂₃	#394	1	1	yes	characteristic
163	C₂²×C₂₃	#480	1	1	yes	characteristic
164	C₂²×C₂₃	#482	1	1	yes	characteristic
165	C₄×C₂₃	—	2	1	no	—
166	C₄×C₂₃	—	2	1	no	—
167	C₄×C₂₃	—	2	1	no	—
168	C₄×C₂₃	—	4	2	no	—
169	C₄×C₂₃	—	8	1	no	—
170	C₄×C₂₃	#447	1	1	yes	characteristic
171	C₄×C₂₃	#452	1	1	yes	characteristic
172	C₄×C₂₃	#466	1	1	yes	characteristic
173	C₄×C₂₃	#468	1	1	yes	characteristic
174	C₄×C₂₃	#471	1	2	yes	—
175	C₂²×SD₁₆	—	2	1	no	—
176	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
177	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
178	#460	—	2	2	no	—
179	#460	—	2	2	no	—
180	#431	C₂²×C₂₃	1	1	yes	characteristic
181	#431	C₂²×C₂₃	1	1	yes	characteristic
182	#423	C₂²×C₂₃	1	1	yes	characteristic
183	#423	C₂²×C₂₃	1	1	yes	characteristic
184	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
185	C₂×#102	—	2	1	no	—
186	C₂×#101	—	2	1	no	—
187	#343	—	2	1	no	—
188	#335	—	2	2	no	—
189	#331	—	2	2	no	—
190	#331	—	2	2	no	—
191	Q₈⋊C₈	—	2	2	no	—
192	C₂×C₂₃	—	2	1	no	—
193	C₂×C₂₃	—	2	1	no	—
194	C₂×C₂₃	—	16	2	no	—
195	C₂×C₂₃	#2033	1	1	yes	characteristic
196	C₂×C₂₃	#2046	1	1	yes	characteristic
197	C₂×C₂₃	#2048	1	1	yes	characteristic
198	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
199	C₂²×D₄	—	2	1	no	—
200	C₂²×D₄	—	2	1	no	—
201	C₂²×D₄	—	2	1	no	—
202	C₂×SD₁₆	—	2	4	no	—
203	C₂×SD₁₆	—	4	2	no	—
204	C₂×SD₁₆	—	4	2	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂²×C₈	—	2	1	no	—
208	#127	—	2	1	no	—
209	#127	—	2	1	no	—
210	#127	—	2	1	no	—
211	#127	D₄×C₂₃	1	2	yes	—
212	C₄×Q₈	—	2	1	no	—
213	C₄×Q₈	—	2	1	no	—
214	C₄×Q₈	—	2	1	no	—
215	C₄×Q₈	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
218	C₂×C₄²	—	2	1	no	—
219	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	2	1	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
227	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
228	#102	—	2	4	no	—
229	#101	—	4	2	no	—
230	C₂²⋊C₈	—	2	1	no	—
231	C₂²⋊C₈	—	2	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₂₃	#10442	1	1	yes	characteristic, direct product factor, Sylow
235	C₂⁴	—	4	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	2	1	no	—
238	C₂×Q₈	—	2	1	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	D₈×C₂₃	1	2	yes	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	8	2	no	—
253	C₂×D₄	—	8	2	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
257	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
258	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
259	SD₁₆	—	4	8	no	—
260	M₁₆	—	4	1	no	—
261	M₁₆	—	4	1	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	4	1	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
278	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
279	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
280	C₄²	#41×C₂₃	1	1	yes	characteristic
281	C₂³	—	4	2	no	—
282	C₂³	—	8	2	no	—
283	C₂³	—	8	2	no	—
284	C₂³	—	8	1	no	—
285	C₂³	—	8	1	no	—
286	C₂³	C₂₃×#123	1	1	yes	characteristic
287	Q₈	—	2	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	4	1	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	8	2	no	—
294	D₄	—	8	2	no	—
295	D₄	—	8	2	no	—
296	D₄	—	8	1	no	—
297	D₄	—	8	1	no	—
298	D₄	—	8	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
307	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
308	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
309	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
310	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
311	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
312	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
313	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
314	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
315	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
316	C₈	—	4	2	no	—
317	C₈	—	4	2	no	—
318	C₈	—	8	1	no	—
319	C₈	—	8	1	no	—
320	C₈	—	8	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	16	2	no	—
327	C₂²	—	16	2	no	—
328	C₂²	C₂₃×#394	1	1	yes	characteristic
329	C₂²	C₂₃×#480	1	1	yes	characteristic
330	C₂²	C₂₃×#482	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	2	no	—
335	C₄	—	8	1	no	—
336	C₄	C₂₃×#447	1	1	yes	characteristic
337	C₄	C₂₃×#452	1	1	yes	characteristic
338	C₄	C₂₃×#466	1	1	yes	characteristic
339	C₄	C₂₃×#468	1	1	yes	characteristic
340	C₄	C₂₃×#471	1	2	yes	—
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	16	2	no	—
344	C₂	C₂₃×#2033	1	1	yes	characteristic
345	C₂	C₂₃×#2046	1	1	yes	characteristic
346	C₂	C₂₃×#2048	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1801	C₂	semidirect
3. C₂₃×#1571	C₂	semidirect
6. C₂₃×#1479	C₂	semidirect
7. C₂₃×#1301	C₂	semidirect
107. #10442	C₂₃	direct (N × H)
149. #1801	C₂×C₂₃	semidirect
150. #1571	C₂×C₂₃	semidirect
153. #1479	C₂×C₂₃	semidirect
154. #1301	C₂×C₂₃	semidirect
234. C₂₃	#10442	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×#460	2
C₂₃×#1801	2
C₂₃×#1571	2
C₂₃×#1545	2
C₂₃×#1479	2
C₂₃×#1301	2
#10442	23

Subgroups — C23×#10442 #19158068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10442 #19158068