Subgroups — C₂₃×#10282 #19157908

Order 5888.

678 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3227	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3205	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2858	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1962	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1545	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#532	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#124	—	2	2	no	—
12	C₂₃×#498	—	2	1	no	—
13	C₂₃×#493	—	2	1	no	—
14	C₂₃×#486	—	2	1	no	—
15	C₂₃×#486	—	2	1	no	—
16	C₂₃×#481	—	2	2	no	—
17	C₂₃×#476	—	2	2	no	—
18	C₂₃×#459	—	2	2	no	—
19	C₂₃×#447	—	2	2	no	—
20	C₂₃×#431	C₂²	1	1	yes	characteristic
21	C₂₃×#431	C₂²	1	1	yes	characteristic
22	C₂₃×#423	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#104	—	2	2	no	—
24	C₂₃×#390	—	2	2	no	—
25	C₂₃×#390	—	2	2	no	—
26	C₄³×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#332	—	2	2	no	—
28	C₂₃×#331	—	2	2	no	—
29	C₂×#41×C₂₃	—	2	1	no	—
30	C₂²×Q₈×C₂₃	—	2	1	no	—
31	C₂²×D₄×C₂₃	—	2	1	no	—
32	C₂²×D₄×C₂₃	—	2	1	no	—
33	C₂²×D₄×C₂₃	D₄	1	2	yes	—
34	C₂×D₈×C₂₃	—	2	4	no	—
35	C₂×M₁₆×C₂₃	—	2	1	no	—
36	C₂₃×C₄⋊Q₈	—	2	1	no	—
37	C₂₃×C₄⋊Q₈	—	2	1	no	—
38	C₂₃×C₄⋊Q₈	—	2	1	no	—
39	C₂₃×C₄⋊Q₈	—	2	1	no	—
40	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
41	C₂₃×#127	—	2	1	no	—
42	C₂₃×#127	—	2	1	no	—
43	C₂₃×#127	—	2	1	no	—
44	C₂₃×#127	—	2	1	no	—
45	C₂₃×#127	D₄	1	1	yes	characteristic
46	C₂₃×#124	—	2	1	no	—
47	C₂₃×#124	—	4	8	no	—
48	C₂₃×#121	—	2	1	no	—
49	C₂₃×#121	—	2	1	no	—
50	C₄×D₄×C₂₃	—	2	1	no	—
51	C₄×D₄×C₂₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
54	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
55	C₂×C₄²×C₂₃	—	2	1	no	—
56	C₂×C₄²×C₂₃	—	2	1	no	—
57	C₂×C₄²×C₂₃	D₄	1	2	yes	—
58	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
59	C₂₃×#106	—	2	4	no	—
60	C₂₃×C₄⋊C₈	—	2	1	no	—
61	C₂₃×C₄⋊C₈	—	2	1	no	—
62	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
63	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
64	C₂₃×#104	—	4	4	no	—
65	C₂₃×#102	—	2	4	no	—
66	C₂₃×#102	—	2	4	no	—
67	C₂₃×C₂²⋊C₈	—	2	1	no	—
68	C₂₃×C₂²⋊C₈	—	2	1	no	—
69	C₄×C₈×C₂₃	—	2	1	no	—
70	C₄×C₈×C₂₃	—	2	1	no	—
71	C₂⁴×C₂₃	—	2	2	no	—
72	#41×C₂₃	—	4	2	no	—
73	#41×C₂₃	—	4	2	no	—
74	C₂×Q₈×C₂₃	—	2	1	no	—
75	C₂×Q₈×C₂₃	—	4	4	no	—
76	C₂×Q₈×C₂₃	—	4	1	no	—
77	C₂×Q₈×C₂₃	—	4	1	no	—
78	C₂×D₄×C₂₃	—	2	1	no	—
79	C₂×D₄×C₂₃	—	2	2	no	—
80	C₂×D₄×C₂₃	—	2	1	no	—
81	C₂×D₄×C₂₃	—	2	1	no	—
82	C₂×D₄×C₂₃	—	2	2	no	—
83	C₂×D₄×C₂₃	—	4	4	no	—
84	C₂×D₄×C₂₃	—	4	4	no	—
85	C₂×D₄×C₂₃	—	4	1	no	—
86	C₂×D₄×C₂₃	—	4	1	no	—
87	C₂×D₄×C₂₃	—	4	1	no	—
88	C₂×D₄×C₂₃	—	4	1	no	—
89	C₂×D₄×C₂₃	—	4	1	no	—
90	C₂×D₄×C₂₃	—	4	2	no	—
91	C₂×D₄×C₂₃	—	4	2	no	—
92	C₂²×C₄×C₂₃	—	2	2	no	—
93	C₂²×C₄×C₂₃	—	2	2	no	—
94	C₂²×C₄×C₂₃	—	4	1	no	—
95	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
98	D₈×C₂₃	—	8	4	no	—
99	M₁₆×C₂₃	—	4	2	no	—
100	C₂×C₈×C₂₃	—	2	2	no	—
101	C₂×C₈×C₂₃	—	2	2	no	—
102	C₂×C₈×C₂₃	—	4	1	no	—
103	C₄⋊C₄×C₂₃	—	2	1	no	—
104	C₄⋊C₄×C₂₃	—	2	2	no	—
105	C₄⋊C₄×C₂₃	—	2	1	no	—
106	C₄⋊C₄×C₂₃	—	2	2	no	—
107	C₄⋊C₄×C₂₃	—	4	1	no	—
108	C₄⋊C₄×C₂₃	—	4	1	no	—
109	C₂²⋊C₄×C₂₃	—	4	16	no	—
110	C₂²⋊C₄×C₂₃	—	4	1	no	—
111	C₂²⋊C₄×C₂₃	—	4	1	no	—
112	C₄²×C₂₃	—	2	4	no	—
113	C₄²×C₂₃	—	2	2	no	—
114	C₄²×C₂₃	—	2	2	no	—
115	C₄²×C₂₃	—	4	2	no	—
116	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₂₃	#41	1	1	yes	characteristic
120	#10282	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₂₃	—	4	4	no	—
122	C₂³×C₂₃	—	4	4	no	—
123	C₂³×C₂₃	—	4	1	no	—
124	C₂³×C₂₃	—	4	2	no	—
125	C₂³×C₂₃	—	4	4	no	—
126	C₂³×C₂₃	#120	1	1	yes	characteristic
127	Q₈×C₂₃	—	4	2	no	—
128	Q₈×C₂₃	—	8	2	no	—
129	Q₈×C₂₃	—	8	2	no	—
130	D₄×C₂₃	—	4	2	no	—
131	D₄×C₂₃	—	4	4	no	—
132	D₄×C₂₃	—	4	2	no	—
133	D₄×C₂₃	—	4	2	no	—
134	D₄×C₂₃	—	8	2	no	—
135	D₄×C₂₃	—	8	2	no	—
136	D₄×C₂₃	—	8	2	no	—
137	D₄×C₂₃	—	8	2	no	—
138	D₄×C₂₃	—	8	2	no	—
139	D₄×C₂₃	—	8	1	no	—
140	D₄×C₂₃	—	8	1	no	—
141	C₂×C₄×C₂₃	—	2	2	no	—
142	C₂×C₄×C₂₃	—	2	2	no	—
143	C₂×C₄×C₂₃	—	2	4	no	—
144	C₂×C₄×C₂₃	—	4	2	no	—
145	C₂×C₄×C₂₃	—	4	2	no	—
146	C₂×C₄×C₂₃	—	4	4	no	—
147	C₂×C₄×C₂₃	—	4	1	no	—
148	C₂×C₄×C₂₃	—	4	2	no	—
149	C₂×C₄×C₂₃	—	8	1	no	—
150	C₂×C₄×C₂₃	—	8	1	no	—
151	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
152	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
153	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
154	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
155	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
156	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
157	C₂×C₄×C₂₃	#135	1	2	yes	—
158	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
159	C₈×C₂₃	—	4	2	no	—
160	C₈×C₂₃	—	4	2	no	—
161	C₈×C₂₃	—	8	1	no	—
162	#3227	C₂×C₂₃	1	1	yes	characteristic, complemented
163	#3205	C₂×C₂₃	1	1	yes	characteristic
164	#2858	C₂×C₂₃	1	1	yes	characteristic, complemented
165	#1962	C₂×C₂₃	1	1	yes	characteristic, complemented
166	#1549	C₂×C₂₃	1	1	yes	characteristic, complemented
167	#1545	C₂×C₂₃	1	1	yes	characteristic, complemented
168	#1276	C₂×C₂₃	1	1	yes	characteristic, complemented
169	C₂²×C₂₃	—	2	1	no	—
170	C₂²×C₂₃	—	2	1	no	—
171	C₂²×C₂₃	—	8	8	no	—
172	C₂²×C₂₃	—	8	2	no	—
173	C₂²×C₂₃	—	8	1	no	—
174	C₂²×C₂₃	—	8	1	no	—
175	C₂²×C₂₃	—	8	2	no	—
176	C₂²×C₂₃	—	8	1	no	—
177	C₂²×C₂₃	—	8	2	no	—
178	C₂²×C₂₃	#392	1	1	yes	characteristic
179	C₂²×C₂₃	#449	1	2	yes	—
180	C₄×C₂₃	—	2	1	no	—
181	C₄×C₂₃	—	2	1	no	—
182	C₄×C₂₃	—	4	4	no	—
183	C₄×C₂₃	—	8	2	no	—
184	C₄×C₂₃	—	8	1	no	—
185	C₄×C₂₃	#453	1	2	yes	—
186	C₄×C₂₃	#454	1	2	yes	—
187	C₄×C₂₃	#463	1	2	yes	—
188	C₄×C₂₃	#495	1	1	yes	characteristic
189	C₄×C₂₃	#496	1	1	yes	characteristic
190	#532	C₂²×C₂₃	1	1	yes	characteristic
191	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
192	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
193	C₂×#124	—	2	2	no	—
194	#498	—	2	1	no	—
195	#493	—	2	1	no	—
196	#486	—	2	1	no	—
197	#486	—	2	1	no	—
198	#481	—	2	2	no	—
199	#476	—	2	2	no	—
200	#459	—	2	2	no	—
201	#447	—	2	2	no	—
202	#431	C₂²×C₂₃	1	1	yes	characteristic
203	#431	C₂²×C₂₃	1	1	yes	characteristic
204	#423	C₂²×C₂₃	1	1	yes	characteristic
205	C₂×#104	—	2	2	no	—
206	#390	—	2	2	no	—
207	#390	—	2	2	no	—
208	C₄³	C₂²×C₂₃	1	1	yes	characteristic
209	#332	—	2	2	no	—
210	#331	—	2	2	no	—
211	C₂×C₂₃	—	2	2	no	—
212	C₂×C₂₃	—	16	1	no	—
213	C₂×C₂₃	—	16	2	no	—
214	C₂×C₂₃	#2002	1	1	yes	characteristic
215	C₂×C₂₃	#2003	1	1	yes	characteristic
216	C₂×C₂₃	#2012	1	1	yes	characteristic
217	C₂×#41	—	2	1	no	—
218	C₂²×Q₈	—	2	1	no	—
219	C₂²×D₄	—	2	1	no	—
220	C₂²×D₄	—	2	1	no	—
221	C₂²×D₄	D₄×C₂₃	1	2	yes	—
222	C₂×D₈	—	2	4	no	—
223	C₂×M₁₆	—	2	1	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	2	1	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
229	#127	—	2	1	no	—
230	#127	—	2	1	no	—
231	#127	—	2	1	no	—
232	#127	—	2	1	no	—
233	#127	D₄×C₂₃	1	1	yes	characteristic
234	#124	—	2	1	no	—
235	#124	—	4	8	no	—
236	#121	—	2	1	no	—
237	#121	—	2	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	—	2	1	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
242	C₂×C₂²⋊C₄	—	2	8	no	—
243	C₂×C₄²	—	2	1	no	—
244	C₂×C₄²	—	2	1	no	—
245	C₂×C₄²	D₄×C₂₃	1	2	yes	—
246	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
247	#106	—	2	4	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
251	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
252	#104	—	4	4	no	—
253	#102	—	2	4	no	—
254	#102	—	2	4	no	—
255	C₂²⋊C₈	—	2	1	no	—
256	C₂²⋊C₈	—	2	1	no	—
257	C₄×C₈	—	2	1	no	—
258	C₄×C₈	—	2	1	no	—
259	C₂₃	#10282	1	1	yes	characteristic, direct product factor, Sylow
260	C₂⁴	—	2	2	no	—
261	#41	—	4	2	no	—
262	#41	—	4	2	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	4	4	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	2	1	no	—
270	C₂×D₄	—	2	1	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	4	4	no	—
273	C₂×D₄	—	4	4	no	—
274	C₂×D₄	—	4	1	no	—
275	C₂×D₄	—	4	1	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	—	4	1	no	—
279	C₂×D₄	—	4	2	no	—
280	C₂×D₄	—	4	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
287	D₈	—	8	4	no	—
288	M₁₆	—	4	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	4	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	—	4	16	no	—
299	C₂²⋊C₄	—	4	1	no	—
300	C₂²⋊C₄	—	4	1	no	—
301	C₄²	—	2	4	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	4	2	no	—
305	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
306	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
307	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
308	C₄²	#41×C₂₃	1	1	yes	characteristic
309	C₂³	—	4	4	no	—
310	C₂³	—	4	4	no	—
311	C₂³	—	4	1	no	—
312	C₂³	—	4	2	no	—
313	C₂³	—	4	4	no	—
314	C₂³	C₂₃×#120	1	1	yes	characteristic
315	Q₈	—	4	2	no	—
316	Q₈	—	8	2	no	—
317	Q₈	—	8	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	4	no	—
320	D₄	—	4	2	no	—
321	D₄	—	4	2	no	—
322	D₄	—	8	2	no	—
323	D₄	—	8	2	no	—
324	D₄	—	8	2	no	—
325	D₄	—	8	2	no	—
326	D₄	—	8	2	no	—
327	D₄	—	8	1	no	—
328	D₄	—	8	1	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	4	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	8	1	no	—
338	C₂×C₄	—	8	1	no	—
339	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
340	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
341	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
342	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
343	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
344	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
345	C₂×C₄	C₂₃×#135	1	2	yes	—
346	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
347	C₈	—	4	2	no	—
348	C₈	—	4	2	no	—
349	C₈	—	8	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	8	8	no	—
353	C₂²	—	8	2	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	8	1	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	1	no	—
358	C₂²	—	8	2	no	—
359	C₂²	C₂₃×#392	1	1	yes	characteristic
360	C₂²	C₂₃×#449	1	2	yes	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	4	4	no	—
364	C₄	—	8	2	no	—
365	C₄	—	8	1	no	—
366	C₄	C₂₃×#453	1	2	yes	—
367	C₄	C₂₃×#454	1	2	yes	—
368	C₄	C₂₃×#463	1	2	yes	—
369	C₄	C₂₃×#495	1	1	yes	characteristic
370	C₄	C₂₃×#496	1	1	yes	characteristic
371	C₂	—	2	2	no	—
372	C₂	—	16	2	no	—
373	C₂	—	16	1	no	—
374	C₂	C₂₃×#2002	1	1	yes	characteristic
375	C₂	C₂₃×#2003	1	1	yes	characteristic
376	C₂	C₂₃×#2012	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂₃×#3205	C₂
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
1. C₂₃×#3227	C₂	semidirect
3. C₂₃×#2858	C₂	semidirect
4. C₂₃×#1962	C₂	semidirect
5. C₂₃×#1549	C₂	semidirect
6. C₂₃×#1545	C₂	semidirect
7. C₂₃×#1276	C₂	semidirect
120. #10282	C₂₃	direct (N × H)
162. #3227	C₂×C₂₃	semidirect
164. #2858	C₂×C₂₃	semidirect
165. #1962	C₂×C₂₃	semidirect
166. #1549	C₂×C₂₃	semidirect
167. #1545	C₂×C₂₃	semidirect
168. #1276	C₂×C₂₃	semidirect
259. C₂₃	#10282	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3227	2
C₂₃×#3205	2
C₂₃×#2858	2
C₂₃×#1962	2
C₂₃×#1549	2
C₂₃×#1545	2
C₂₃×#1276	2
#10282	23

Subgroups — C23×#10282 #19157908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10282 #19157908