Subgroups — D₂₃×#478 #15683150

Order 2944.

448 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₂₃	C₂	1	1	yes	maximal, characteristic
3	#13092720	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092719	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092718	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092717	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092716	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092715	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092714	C₂	1	1	yes	maximal, characteristic, complemented
10	#107×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#106×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄⋊C₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
13	#102×D₂₃	C₂	1	1	yes	maximal, characteristic
14	#102×D₂₃	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#478	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₂₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₂₃	—	2	1	no	—
18	C₂₃×#125	C₂²	1	1	yes	characteristic, complemented
19	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
20	D₄×Dic₂₃	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊D₂₃	—	2	1	no	—
22	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
24	#10625064	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×D₂₃	—	2	1	no	—
26	C₄⋊C₄×D₂₃	—	2	1	no	—
27	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
30	#10625061	C₂²	1	1	yes	characteristic, complemented
31	#10625061	C₂²	1	1	yes	characteristic, complemented
32	#10625060	—	2	1	no	—
33	#10625060	—	2	1	no	—
34	(C₂²⋊D₂₃)⋊C₄	—	2	1	no	—
35	C₂²⋊C₄×D₂₃	—	2	1	no	—
36	C₄×D₉₂	C₂²	1	1	yes	characteristic
37	C₄²×D₂₃	C₂²	1	1	yes	characteristic
38	#10625041	C₂²	1	1	yes	characteristic, complemented
39	C₂₃×#107	C₂²	1	1	yes	characteristic, complemented
40	C₂₃×#106	C₂²	1	1	yes	characteristic, complemented
41	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	C₂₃×#102	C₂²	1	1	yes	characteristic
43	C₂₃×#102	C₂²	1	1	yes	characteristic
44	#10625014	C₂²	1	1	yes	characteristic
45	#10625014	C₂²	1	1	yes	characteristic
46	#10625003	C₂²	1	1	yes	characteristic
47	#10625003	C₂²	1	1	yes	characteristic
48	#10625000	C₂²	1	1	yes	characteristic, complemented
49	#10624999	C₂²	1	1	yes	characteristic, complemented
50	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
51	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
52	#10624991	C₂²	1	1	yes	characteristic
53	#10624991	C₂²	1	1	yes	characteristic
54	#10624990	C₂²	1	1	yes	characteristic, complemented
55	#10624990	C₂²	1	1	yes	characteristic, complemented
56	#10624989	C₂²	1	1	yes	characteristic, complemented
57	#10624989	C₂²	1	1	yes	characteristic, complemented
58	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₂₃	—	2	1	no	—
60	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	—	2	1	no	—
62	C₂×C₂²⋊D₂₃	—	2	1	no	—
63	C₂²×Dic₂₃	—	2	1	no	—
64	D₄×D₂₃	—	2	2	no	—
65	D₄×D₂₃	Q₈	1	4	yes	—
66	C₂×D₉₂	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₂₃	—	2	1	no	—
68	C₂×C₄×D₂₃	—	2	1	no	—
69	C₂×C₄×D₂₃	—	2	1	no	—
70	C₂×C₄×D₂₃	—	2	1	no	—
71	C₂×C₄×D₂₃	—	2	1	no	—
72	C₂×C₄×D₂₃	—	4	4	no	—
73	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
74	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
75	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₂₃	—	2	1	no	—
79	C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₂₃	—	2	1	no	—
84	C₄²×C₂₃	C₂³	1	1	yes	characteristic
85	C₂²⋊Dic₂₃	—	2	1	no	—
86	(C₂×D₂₃)⋊C₄	—	2	1	no	—
87	(C₂×D₂₃)⋊C₄	—	2	1	no	—
88	C₄⋊Dic₂₃	—	2	1	no	—
89	C₄⋊Dic₂₃	—	2	1	no	—
90	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
93	Dic₂₃⋊C₄	—	2	1	no	—
94	Dic₂₃⋊C₄	—	2	1	no	—
95	Dic₂₃⋊C₄	—	2	1	no	—
96	Dic₂₃⋊C₄	—	2	1	no	—
97	Dic₂₃⋊C₄	—	2	1	no	—
98	Dic₂₃⋊C₄	—	2	1	no	—
99	Dic₂₃⋊C₄	—	2	1	no	—
100	C₄×Dic₂₃	D₄	1	1	yes	characteristic
101	C₄×Dic₂₃	D₄	1	1	yes	characteristic
102	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
105	C₈×D₂₃	—	2	2	no	—
106	C₈×D₂₃	—	2	2	no	—
107	C₂³×C₂₃	—	2	1	no	—
108	C₂²×D₂₃	—	2	1	no	—
109	C₂²×D₂₃	—	2	4	no	—
110	C₂²×D₂₃	—	4	2	no	—
111	C₂²×D₂₃	—	4	2	no	—
112	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
113	D₄×C₂₃	—	2	1	no	—
114	D₄×C₂₃	C₂×Q₈	1	2	yes	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	2	1	no	—
117	C₂×C₄×C₂₃	—	2	1	no	—
118	C₂×C₄×C₂₃	—	4	2	no	—
119	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₂₃	—	2	4	no	—
123	C₂×Dic₂₃	—	2	1	no	—
124	C₂×Dic₂₃	—	2	1	no	—
125	C₂×Dic₂₃	—	2	1	no	—
126	C₂×Dic₂₃	—	4	2	no	—
127	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
130	D₉₂	—	2	1	no	—
131	D₉₂	C₂×Q₈	1	2	yes	—
132	C₄×D₂₃	—	2	2	no	—
133	C₄×D₂₃	—	2	2	no	—
134	C₄×D₂₃	—	4	2	no	—
135	C₄×D₂₃	—	4	2	no	—
136	C₄×D₂₃	—	4	2	no	—
137	C₄×D₂₃	—	4	2	no	—
138	C₄×D₂₃	—	4	2	no	—
139	C₄×D₂₃	C₂×Q₈	1	2	yes	—
140	C₄×D₂₃	#41	1	2	yes	—
141	C₈×C₂₃	—	2	1	no	—
142	C₈×C₂₃	—	2	1	no	—
143	C₂₃⋊C₈	—	2	1	no	—
144	C₂₃⋊C₈	—	2	1	no	—
145	C₂×#478	—	23	1	no	maximal, Sylow
146	C₂²×C₂₃	—	2	2	no	—
147	C₂²×C₂₃	—	4	1	no	—
148	C₂²×C₂₃	—	4	1	no	—
149	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₂₃	—	2	2	no	—
151	C₂×D₂₃	—	4	1	no	—
152	C₂×D₂₃	—	4	1	no	—
153	C₂×D₂₃	—	4	8	no	—
154	C₂×D₂₃	C₂²⋊Q₈	1	2	yes	—
155	C₂×D₂₃	#135	1	2	yes	—
156	C₂×D₂₃	AGL(1,ℤ/8)	1	2	yes	—
157	C₄×C₂₃	—	2	1	no	—
158	C₄×C₂₃	—	2	1	no	—
159	C₄×C₂₃	—	4	1	no	—
160	C₄×C₂₃	—	4	1	no	—
161	C₄×C₂₃	—	4	1	no	—
162	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
163	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
164	Dic₂₃	—	2	1	no	—
165	Dic₂₃	—	2	1	no	—
166	Dic₂₃	—	4	1	no	—
167	Dic₂₃	—	4	1	no	—
168	Dic₂₃	—	4	1	no	—
169	Dic₂₃	C₂²⋊Q₈	1	2	yes	—
170	C₂×#125	—	23	1	no	—
171	C₂×C₄×D₄	—	23	1	no	—
172	#478	—	23	1	no	—
173	#478	—	23	1	no	—
174	#478	—	23	1	no	—
175	#478	—	23	1	no	—
176	#478	—	23	1	no	—
177	#478	—	23	1	no	—
178	#478	—	23	1	no	—
179	#478	D₂₃	1	1	yes	characteristic, direct product factor
180	C₂×#107	—	23	1	no	—
181	C₂×#106	—	23	1	no	—
182	C₂×C₄⋊C₈	—	23	1	no	—
183	C₂×#102	—	23	1	no	—
184	C₂×#102	—	23	1	no	—
185	C₂×C₂₃	—	4	2	no	—
186	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
187	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
188	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
189	D₂₃	—	4	2	no	—
190	D₂₃	#478	1	4	yes	direct product factor
191	C₂²×D₄	—	23	1	no	—
192	C₂³×C₄	—	46	1	no	—
193	C₂²×C₈	—	23	1	no	—
194	C₂²×C₈	—	23	1	no	—
195	#125	—	23	1	no	—
196	#125	—	23	1	no	—
197	#125	—	23	1	no	—
198	#125	—	46	1	no	—
199	#125	—	46	1	no	—
200	#125	C₂×D₂₃	1	1	yes	characteristic, complemented
201	C₄×D₄	—	23	1	no	—
202	C₄×D₄	—	23	1	no	—
203	C₄×D₄	—	23	1	no	—
204	C₄×D₄	—	46	1	no	—
205	C₄×D₄	—	46	1	no	—
206	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic
207	C₂×C₄⋊C₄	—	23	1	no	—
208	C₂×C₄⋊C₄	—	23	1	no	—
209	C₂×C₄⋊C₄	—	23	1	no	—
210	C₂×C₄⋊C₄	—	46	1	no	—
211	C₂×C₄⋊C₄	—	46	1	no	—
212	C₂×C₂²⋊C₄	—	46	1	no	—
213	C₂×C₄²	—	23	1	no	—
214	#107	—	23	1	no	—
215	#107	—	23	1	no	—
216	#107	—	23	1	no	—
217	#107	C₂×D₂₃	1	1	yes	characteristic, complemented
218	#106	—	23	1	no	—
219	#106	—	23	1	no	—
220	#106	—	23	1	no	—
221	#106	C₂×D₂₃	1	1	yes	characteristic, complemented
222	C₄⋊C₈	—	23	1	no	—
223	C₄⋊C₈	—	23	1	no	—
224	C₄⋊C₈	—	23	1	no	—
225	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic, complemented
226	#102	—	23	1	no	—
227	#102	—	23	1	no	—
228	#102	—	23	1	no	—
229	#102	—	23	1	no	—
230	#102	—	23	1	no	—
231	#102	—	23	1	no	—
232	#102	C₂×D₂₃	1	1	yes	characteristic
233	#102	C₂×D₂₃	1	1	yes	characteristic
234	C₂₃	C₂×#478	1	1	yes	characteristic, complemented, Sylow
235	C₂⁴	—	46	1	no	—
236	C₂×D₄	—	23	4	no	—
237	C₂×D₄	—	23	1	no	—
238	C₂×D₄	—	46	2	no	—
239	C₂×D₄	—	46	1	no	—
240	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	23	1	no	—
243	C₂²×C₄	—	23	1	no	—
244	C₂²×C₄	—	23	1	no	—
245	C₂²×C₄	—	46	1	no	—
246	C₂²×C₄	—	46	1	no	—
247	C₂²×C₄	—	46	1	no	—
248	C₂²×C₄	—	46	1	no	—
249	C₂²×C₄	—	46	1	no	—
250	C₂²×C₄	—	46	1	no	—
251	C₂²×C₄	—	92	4	no	—
252	C₂×C₈	—	23	1	no	—
253	C₂×C₈	—	23	1	no	—
254	C₂×C₈	—	46	2	no	—
255	C₂×C₈	—	46	2	no	—
256	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
257	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	23	1	no	—
261	C₄⋊C₄	—	23	1	no	—
262	C₄⋊C₄	—	23	1	no	—
263	C₄⋊C₄	—	46	1	no	—
264	C₄⋊C₄	—	46	1	no	—
265	C₄⋊C₄	—	46	1	no	—
266	C₄⋊C₄	—	46	1	no	—
267	C₄⋊C₄	—	46	1	no	—
268	C₄⋊C₄	—	46	1	no	—
269	C₄⋊C₄	—	46	1	no	—
270	C₄⋊C₄	—	46	1	no	—
271	C₄⋊C₄	—	46	1	no	—
272	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
273	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	46	1	no	—
277	C₂²⋊C₄	—	46	1	no	—
278	C₂²⋊C₄	—	46	1	no	—
279	C₄²	—	23	1	no	—
280	C₄²	—	23	1	no	—
281	C₄²	—	23	1	no	—
282	C₄²	C₂²×D₂₃	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	23	1	no	—
285	C₂³	—	46	4	no	—
286	C₂³	—	46	1	no	—
287	C₂³	—	92	2	no	—
288	C₂³	—	92	2	no	—
289	D₄	—	2	1	no	—
290	D₄	—	23	2	no	—
291	D₄	—	46	4	no	—
292	D₄	—	46	1	no	—
293	D₄	Q₈×D₂₃	1	2	yes	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	23	1	no	—
299	C₂×C₄	—	23	2	no	—
300	C₂×C₄	—	23	2	no	—
301	C₂×C₄	—	23	1	no	—
302	C₂×C₄	—	23	1	no	—
303	C₂×C₄	—	46	2	no	—
304	C₂×C₄	—	46	2	no	—
305	C₂×C₄	—	46	1	no	—
306	C₂×C₄	—	46	1	no	—
307	C₂×C₄	—	46	1	no	—
308	C₂×C₄	—	92	2	no	—
309	C₂×C₄	—	92	2	no	—
310	C₂×C₄	—	92	2	no	—
311	C₂×C₄	—	92	2	no	—
312	C₂×C₄	—	92	2	no	—
313	C₂×C₄	—	92	2	no	—
314	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
315	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	46	1	no	—
320	C₈	—	46	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	23	2	no	—
325	C₂²	—	23	2	no	—
326	C₂²	—	23	2	no	—
327	C₂²	—	46	2	no	—
328	C₂²	—	92	8	no	—
329	C₂²	—	92	1	no	—
330	C₂²	—	92	1	no	—
331	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	23	2	no	—
338	C₄	—	46	1	no	—
339	C₄	—	46	1	no	—
340	C₄	—	92	1	no	—
341	C₄	—	92	1	no	—
342	C₄	—	92	1	no	—
343	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
344	C₄	#41×D₂₃	1	1	yes	characteristic
345	C₂	—	4	2	no	—
346	C₂	—	23	4	no	—
347	C₂	—	92	2	no	—
348	C₂	C₂²⋊Q₈×D₂₃	1	1	yes	characteristic
349	C₂	#135×D₂₃	1	1	yes	characteristic
350	C₂	AGL(1,ℤ/8)×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#478	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×D₄×D₂₃	C₂²
hypercenter	#478	D₂₃
hypocenter	C₂₃	C₂×#478
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #125×D₂₃	C₂	semidirect
3. #13092720	C₂	semidirect
4. #13092719	C₂	semidirect
5. #13092718	C₂	semidirect
6. #13092717	C₂	semidirect
7. #13092716	C₂	semidirect
8. #13092715	C₂	semidirect
9. #13092714	C₂	semidirect
10. #107×D₂₃	C₂	semidirect
11. #106×D₂₃	C₂	semidirect
12. C₄⋊C₈×D₂₃	C₂	semidirect
15. C₂₃×#478	C₂	semidirect
18. C₂₃×#125	C₂²	semidirect
30. #10625061	C₂²	semidirect
31. #10625061	C₂²	semidirect
38. #10625041	C₂²	semidirect
39. C₂₃×#107	C₂²	semidirect
40. C₂₃×#106	C₂²	semidirect
41. C₂₃×C₄⋊C₈	C₂²	semidirect
48. #10625000	C₂²	semidirect
49. #10624999	C₂²	semidirect
50. Dic₂₃⋊C₈	C₂²	semidirect
51. Dic₂₃⋊C₈	C₂²	semidirect
54. #10624990	C₂²	semidirect
55. #10624990	C₂²	semidirect
56. #10624989	C₂²	semidirect
57. #10624989	C₂²	semidirect
58. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
179. #478	D₂₃	direct (N × H)
190. D₂₃	#478	direct (N × H)
200. #125	C₂×D₂₃	semidirect
217. #107	C₂×D₂₃	semidirect
221. #106	C₂×D₂₃	semidirect
225. C₄⋊C₈	C₂×D₂₃	semidirect
234. C₂₃	C₂×#478	semidirect

Maximal subgroups

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

subgroup	index
#125×D₂₃	2
C₄×D₄×D₂₃	2
#13092720	2
#13092719	2
#13092718	2
#13092717	2
#13092716	2
#13092715	2
#13092714	2
#107×D₂₃	2
#106×D₂₃	2
C₄⋊C₈×D₂₃	2
#102×D₂₃	2
C₂₃×#478	2
C₂×#478	23

Subgroups — D23×#478 #15683150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₃×#478 #15683150