Subgroups — C₂₃×#10294 #19157920

Order 5888.

640 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3229	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3208	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2858	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1545	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#533	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×#492	—	2	1	no	—
14	C₂₃×#487	—	2	1	no	—
15	C₂₃×#486	—	2	1	no	—
16	C₂₃×#483	—	2	2	no	—
17	C₂₃×#475	—	2	2	no	—
18	C₂₃×#460	—	2	2	no	—
19	C₂₃×#450	—	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	C₂₃×#390	—	2	2	no	—
24	C₂₃×#390	—	2	2	no	—
25	C₄³×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#339	—	2	2	no	—
27	C₂₃×#332	—	2	2	no	—
28	C₂₃×#331	—	2	2	no	—
29	C₂²×Q₈×C₂₃	—	2	1	no	—
30	C₂²×D₄×C₂₃	—	2	1	no	—
31	C₂²×D₄×C₂₃	—	2	1	no	—
32	C₂²×D₄×C₂₃	D₄	1	2	yes	—
33	C₂×SD₁₆×C₂₃	—	2	4	no	—
34	C₂×M₁₆×C₂₃	—	2	1	no	—
35	C₂₃×C₄⋊Q₈	—	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	D₄	1	1	yes	characteristic
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×#125	—	2	1	no	—
46	C₂₃×#124	—	4	4	no	—
47	C₂₃×#124	—	4	4	no	—
48	C₂₃×C₂²⋊Q₈	—	2	1	no	—
49	C₂₃×C₂²⋊Q₈	—	2	1	no	—
50	C₄×Q₈×C₂₃	—	2	1	no	—
51	C₄×Q₈×C₂₃	—	2	1	no	—
52	C₂₃×#117	—	2	1	no	—
53	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
56	C₂×C₄²×C₂₃	—	2	1	no	—
57	C₂×C₄²×C₂₃	—	2	1	no	—
58	C₂×C₄²×C₂₃	D₄	1	2	yes	—
59	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
60	C₂₃×#106	—	2	4	no	—
61	C₂₃×C₄⋊C₈	—	2	1	no	—
62	C₂₃×C₄⋊C₈	—	2	1	no	—
63	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
64	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
65	C₂₃×#103	—	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	C₂×Q₈×C₂₃	—	2	1	no	—
73	C₂×Q₈×C₂₃	—	2	1	no	—
74	C₂×Q₈×C₂₃	—	4	1	no	—
75	C₂×Q₈×C₂₃	—	4	1	no	—
76	C₂×Q₈×C₂₃	—	4	2	no	—
77	C₂×Q₈×C₂₃	—	4	2	no	—
78	C₂×D₄×C₂₃	—	2	2	no	—
79	C₂×D₄×C₂₃	—	2	2	no	—
80	C₂×D₄×C₂₃	—	4	4	no	—
81	C₂×D₄×C₂₃	—	4	4	no	—
82	C₂×D₄×C₂₃	—	4	1	no	—
83	C₂×D₄×C₂₃	—	4	1	no	—
84	C₂×D₄×C₂₃	—	4	1	no	—
85	C₂×D₄×C₂₃	—	4	1	no	—
86	C₂×D₄×C₂₃	—	4	2	no	—
87	C₂×D₄×C₂₃	—	4	2	no	—
88	C₂²×C₄×C₂₃	—	2	2	no	—
89	C₂²×C₄×C₂₃	—	2	2	no	—
90	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
93	SD₁₆×C₂₃	—	8	4	no	—
94	M₁₆×C₂₃	—	4	2	no	—
95	C₂×C₈×C₂₃	—	2	2	no	—
96	C₂×C₈×C₂₃	—	2	2	no	—
97	C₂×C₈×C₂₃	—	4	1	no	—
98	C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₄⋊C₄×C₂₃	—	2	2	no	—
100	C₄⋊C₄×C₂₃	—	2	1	no	—
101	C₄⋊C₄×C₂₃	—	2	2	no	—
102	C₄⋊C₄×C₂₃	—	2	1	no	—
103	C₄⋊C₄×C₂₃	—	2	1	no	—
104	C₄⋊C₄×C₂₃	—	4	1	no	—
105	C₄⋊C₄×C₂₃	—	4	1	no	—
106	C₄⋊C₄×C₂₃	—	4	1	no	—
107	C₄⋊C₄×C₂₃	—	4	1	no	—
108	C₂²⋊C₄×C₂₃	—	4	16	no	—
109	C₂²⋊C₄×C₂₃	—	4	1	no	—
110	C₄²×C₂₃	—	2	2	no	—
111	C₄²×C₂₃	—	4	2	no	—
112	C₄²×C₂₃	—	4	2	no	—
113	C₄²×C₂₃	—	4	1	no	—
114	C₄²×C₂₃	SD₁₆	1	4	yes	complemented
115	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₂₃	#41	1	1	yes	characteristic
119	#10294	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₂₃	—	4	4	no	—
121	C₂³×C₂₃	—	4	4	no	—
122	C₂³×C₂₃	—	4	2	no	—
123	C₂³×C₂₃	—	4	4	no	—
124	C₂³×C₂₃	#120	1	1	yes	characteristic
125	Q₈×C₂₃	—	4	2	no	—
126	Q₈×C₂₃	—	4	2	no	—
127	Q₈×C₂₃	—	8	2	no	—
128	Q₈×C₂₃	—	8	1	no	—
129	Q₈×C₂₃	—	8	1	no	—
130	D₄×C₂₃	—	4	4	no	—
131	D₄×C₂₃	—	8	2	no	—
132	D₄×C₂₃	—	8	2	no	—
133	D₄×C₂₃	—	8	2	no	—
134	D₄×C₂₃	—	8	2	no	—
135	D₄×C₂₃	—	8	1	no	—
136	D₄×C₂₃	—	8	1	no	—
137	C₂×C₄×C₂₃	—	2	2	no	—
138	C₂×C₄×C₂₃	—	2	2	no	—
139	C₂×C₄×C₂₃	—	2	4	no	—
140	C₂×C₄×C₂₃	—	4	2	no	—
141	C₂×C₄×C₂₃	—	4	2	no	—
142	C₂×C₄×C₂₃	—	4	1	no	—
143	C₂×C₄×C₂₃	—	4	1	no	—
144	C₂×C₄×C₂₃	—	4	2	no	—
145	C₂×C₄×C₂₃	—	4	4	no	—
146	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
147	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
148	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
149	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
150	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
151	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
152	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
153	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
154	C₈×C₂₃	—	4	2	no	—
155	C₈×C₂₃	—	4	2	no	—
156	C₈×C₂₃	—	8	1	no	—
157	#3229	C₂×C₂₃	1	1	yes	characteristic, complemented
158	#3208	C₂×C₂₃	1	1	yes	characteristic
159	#2858	C₂×C₂₃	1	1	yes	characteristic
160	#1994	C₂×C₂₃	1	1	yes	characteristic, complemented
161	#1546	C₂×C₂₃	1	1	yes	characteristic, complemented
162	#1545	C₂×C₂₃	1	1	yes	characteristic
163	#1276	C₂×C₂₃	1	1	yes	characteristic, complemented
164	C₂²×C₂₃	—	2	1	no	—
165	C₂²×C₂₃	—	2	1	no	—
166	C₂²×C₂₃	—	8	8	no	—
167	C₂²×C₂₃	—	8	2	no	—
168	C₂²×C₂₃	—	8	2	no	—
169	C₂²×C₂₃	—	8	2	no	—
170	C₂²×C₂₃	#392	1	1	yes	characteristic
171	C₂²×C₂₃	#450	1	2	yes	—
172	C₄×C₂₃	—	2	1	no	—
173	C₄×C₂₃	—	2	1	no	—
174	C₄×C₂₃	—	4	4	no	—
175	C₄×C₂₃	—	8	1	no	—
176	C₄×C₂₃	—	8	1	no	—
177	C₄×C₂₃	—	8	2	no	—
178	C₄×C₂₃	#454	1	2	yes	—
179	C₄×C₂₃	#455	1	2	yes	—
180	C₄×C₂₃	#460	1	2	yes	—
181	C₄×C₂₃	#492	1	1	yes	characteristic
182	C₄×C₂₃	#496	1	1	yes	characteristic
183	#533	C₂²×C₂₃	1	1	yes	characteristic
184	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
185	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
186	C₂×#124	—	2	2	no	—
187	#498	—	2	1	no	—
188	#492	—	2	1	no	—
189	#487	—	2	1	no	—
190	#486	—	2	1	no	—
191	#483	—	2	2	no	—
192	#475	—	2	2	no	—
193	#460	—	2	2	no	—
194	#450	—	2	2	no	—
195	#431	C₂²×C₂₃	1	1	yes	characteristic
196	#431	C₂²×C₂₃	1	1	yes	characteristic
197	#423	C₂²×C₂₃	1	1	yes	characteristic
198	#390	—	2	2	no	—
199	#390	—	2	2	no	—
200	C₄³	C₂²×C₂₃	1	1	yes	characteristic
201	#339	—	2	2	no	—
202	#332	—	2	2	no	—
203	#331	—	2	2	no	—
204	C₂×C₂₃	—	2	2	no	—
205	C₂×C₂₃	—	16	2	no	—
206	C₂×C₂₃	#2003	1	1	yes	characteristic
207	C₂×C₂₃	#2004	1	1	yes	characteristic
208	C₂×C₂₃	#2014	1	1	yes	characteristic
209	C₂²×Q₈	—	2	1	no	—
210	C₂²×D₄	—	2	1	no	—
211	C₂²×D₄	—	2	1	no	—
212	C₂²×D₄	D₄×C₂₃	1	2	yes	—
213	C₂×SD₁₆	—	2	4	no	—
214	C₂×M₁₆	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
220	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
221	#127	—	2	1	no	—
222	#127	—	2	1	no	—
223	#127	—	2	1	no	—
224	#127	—	2	1	no	—
225	#125	—	2	1	no	—
226	#124	—	4	4	no	—
227	#124	—	4	4	no	—
228	C₂²⋊Q₈	—	2	1	no	—
229	C₂²⋊Q₈	—	2	1	no	—
230	C₄×Q₈	—	2	1	no	—
231	C₄×Q₈	—	2	1	no	—
232	#117	—	2	1	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
235	C₂×C₂²⋊C₄	—	2	8	no	—
236	C₂×C₄²	—	2	1	no	—
237	C₂×C₄²	—	2	1	no	—
238	C₂×C₄²	D₄×C₂₃	1	2	yes	—
239	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
240	#106	—	2	4	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
244	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
245	#103	—	2	4	no	—
246	#102	—	2	4	no	—
247	C₂²⋊C₈	—	2	1	no	—
248	C₂²⋊C₈	—	2	1	no	—
249	C₄×C₈	—	2	1	no	—
250	C₄×C₈	—	2	1	no	—
251	C₂₃	#10294	1	1	yes	characteristic, direct product factor, Sylow
252	C₂⁴	—	2	2	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×Q₈	—	4	2	no	—
258	C₂×Q₈	—	4	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	4	4	no	—
262	C₂×D₄	—	4	4	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂×D₄	—	4	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
274	SD₁₆	—	8	4	no	—
275	M₁₆	—	4	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	4	1	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	4	16	no	—
290	C₂²⋊C₄	—	4	1	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	4	2	no	—
293	C₄²	—	4	2	no	—
294	C₄²	—	4	1	no	—
295	C₄²	SD₁₆×C₂₃	1	4	yes	complemented
296	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
297	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
298	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
299	C₄²	#41×C₂₃	1	1	yes	characteristic
300	C₂³	—	4	4	no	—
301	C₂³	—	4	4	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	4	4	no	—
304	C₂³	C₂₃×#120	1	1	yes	characteristic
305	Q₈	—	4	2	no	—
306	Q₈	—	4	2	no	—
307	Q₈	—	8	2	no	—
308	Q₈	—	8	1	no	—
309	Q₈	—	8	1	no	—
310	D₄	—	4	4	no	—
311	D₄	—	8	2	no	—
312	D₄	—	8	2	no	—
313	D₄	—	8	2	no	—
314	D₄	—	8	2	no	—
315	D₄	—	8	1	no	—
316	D₄	—	8	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
327	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
328	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
329	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
330	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
331	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
332	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
333	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
334	C₈	—	4	2	no	—
335	C₈	—	4	2	no	—
336	C₈	—	8	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	8	8	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	C₂₃×#392	1	1	yes	characteristic
344	C₂²	C₂₃×#450	1	2	yes	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	4	4	no	—
348	C₄	—	8	1	no	—
349	C₄	—	8	1	no	—
350	C₄	—	8	2	no	—
351	C₄	C₂₃×#454	1	2	yes	—
352	C₄	C₂₃×#455	1	2	yes	—
353	C₄	C₂₃×#460	1	2	yes	—
354	C₄	C₂₃×#492	1	1	yes	characteristic
355	C₄	C₂₃×#496	1	1	yes	characteristic
356	C₂	—	2	2	no	—
357	C₂	—	16	2	no	—
358	C₂	C₂₃×#2003	1	1	yes	characteristic
359	C₂	C₂₃×#2004	1	1	yes	characteristic
360	C₂	C₂₃×#2014	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#392
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
1. C₂₃×#3229	C₂	semidirect
4. C₂₃×#1994	C₂	semidirect
5. C₂₃×#1546	C₂	semidirect
7. C₂₃×#1276	C₂	semidirect
114. C₄²×C₂₃	SD₁₆	semidirect
119. #10294	C₂₃	direct (N × H)
157. #3229	C₂×C₂₃	semidirect
160. #1994	C₂×C₂₃	semidirect
161. #1546	C₂×C₂₃	semidirect
163. #1276	C₂×C₂₃	semidirect
251. C₂₃	#10294	direct (N × H)
295. C₄²	SD₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3229	2
C₂₃×#3208	2
C₂₃×#2858	2
C₂₃×#1994	2
C₂₃×#1546	2
C₂₃×#1545	2
C₂₃×#1276	2
#10294	23

Subgroups — C23×#10294 #19157920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10294 #19157920