Subgroups — C₂₃×#12937 #19160563

Order 5888.

490 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×C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#357	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#357	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1942	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1380	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₂₃	—	2	1	no	—
10	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
12	C₄×D₈×C₂₃	—	2	2	no	—
13	C₄×D₈×C₂₃	—	2	2	no	—
14	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#102	—	2	1	no	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#389	—	2	1	no	—
20	C₂₃×#367	—	2	2	no	—
21	C₂₃×#357	—	2	2	no	—
22	C₂₃×#357	C₄	1	4	yes	complemented
23	C₂₃×#340	—	2	1	no	—
24	C₂²×D₄×C₂₃	—	2	1	no	—
25	C₂³×C₄×C₂₃	—	4	1	no	—
26	C₂×D₈×C₂₃	—	2	1	no	—
27	C₂×D₈×C₂₃	—	2	1	no	—
28	C₂×D₈×C₂₃	—	2	1	no	—
29	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
30	C₂×D₈×C₂₃	D₄	1	2	yes	—
31	C₂×D₈×C₂₃	Q₈	1	2	yes	—
32	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₄×D₄×C₂₃	—	2	4	no	—
35	C₄×D₄×C₂₃	—	4	2	no	—
36	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
40	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
41	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₂₃	—	2	1	no	—
43	C₂×C₁₆×C₂₃	—	2	1	no	—
44	C₂×C₁₆×C₂₃	—	2	1	no	—
45	C₂×C₁₆×C₂₃	—	2	1	no	—
46	C₂×C₁₆×C₂₃	—	2	1	no	—
47	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	complemented
48	C₂₃×#107	—	2	1	no	—
49	C₂₃×#107	—	2	1	no	—
50	C₂₃×#107	—	2	1	no	—
51	C₂₃×#107	—	2	1	no	—
52	C₂₃×#107	C₂×C₄	1	2	yes	complemented
53	C₂₃×#107	D₄	1	2	yes	—
54	C₂₃×#102	—	2	4	no	—
55	C₄×C₈×C₂₃	—	2	1	no	—
56	C₄×C₈×C₂₃	—	2	1	no	—
57	C₂₃×#95	—	4	1	no	—
58	C₂⁴×C₂₃	—	4	1	no	—
59	C₂×D₄×C₂₃	—	2	2	no	—
60	C₂×D₄×C₂₃	—	2	2	no	—
61	C₂×D₄×C₂₃	—	2	2	no	—
62	C₂×D₄×C₂₃	—	4	1	no	—
63	C₂×D₄×C₂₃	—	4	1	no	—
64	C₂×D₄×C₂₃	—	4	1	no	—
65	C₂²×C₄×C₂₃	—	4	4	no	—
66	C₂²×C₄×C₂₃	—	4	1	no	—
67	C₂²×C₄×C₂₃	—	4	1	no	—
68	C₂²×C₄×C₂₃	—	4	1	no	—
69	C₂²×C₄×C₂₃	—	8	2	no	—
70	C₂²×C₄×C₂₃	—	8	2	no	—
71	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
74	D₈×C₂₃	—	2	2	no	—
75	D₈×C₂₃	—	2	2	no	—
76	D₈×C₂₃	—	2	2	no	—
77	D₈×C₂₃	C₄⋊C₄	1	4	yes	complemented
78	C₂×C₈×C₂₃	—	2	1	no	—
79	C₂×C₈×C₂₃	—	2	1	no	—
80	C₂×C₈×C₂₃	—	2	1	no	—
81	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
86	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
87	C₄⋊C₄×C₂₃	—	2	2	no	—
88	C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₄⋊C₄×C₂₃	—	4	1	no	—
90	C₄⋊C₄×C₂₃	—	4	1	no	—
91	C₄⋊C₄×C₂₃	—	4	1	no	—
92	C₄⋊C₄×C₂₃	—	4	1	no	—
93	C₄⋊C₄×C₂₃	—	8	2	no	—
94	C₂²⋊C₄×C₂₃	—	4	4	no	—
95	C₄²×C₂₃	—	2	1	no	—
96	C₄²×C₂₃	—	2	1	no	—
97	C₁₆×C₂₃	—	2	2	no	—
98	C₁₆×C₂₃	—	4	1	no	—
99	#12937	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₂₃	—	4	2	no	—
101	C₂³×C₂₃	—	4	2	no	—
102	C₂³×C₂₃	—	4	2	no	—
103	C₂³×C₂₃	—	8	1	no	—
104	C₂³×C₂₃	—	8	1	no	—
105	C₂³×C₂₃	—	8	1	no	—
106	C₂³×C₂₃	—	8	1	no	—
107	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
108	D₄×C₂₃	—	2	4	no	—
109	D₄×C₂₃	—	4	2	no	—
110	D₄×C₂₃	—	4	2	no	—
111	D₄×C₂₃	—	4	2	no	—
112	C₂×C₄×C₂₃	—	2	1	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	2	1	no	—
117	C₂×C₄×C₂₃	—	4	2	no	—
118	C₂×C₄×C₂₃	—	4	2	no	—
119	C₂×C₄×C₂₃	—	8	8	no	—
120	C₂×C₄×C₂₃	—	8	1	no	—
121	C₂×C₄×C₂₃	—	8	1	no	—
122	C₂×C₄×C₂₃	—	8	1	no	—
123	C₂×C₄×C₂₃	—	8	1	no	—
124	C₂×C₄×C₂₃	—	8	1	no	—
125	C₂×C₄×C₂₃	—	8	1	no	—
126	C₂×C₄×C₂₃	—	8	1	no	—
127	C₂×C₄×C₂₃	SD₃₂	1	2	yes	—
128	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
132	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
134	C₈×C₂₃	—	4	1	no	—
135	C₈×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
137	C₈×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
138	C₈×C₂₃	#123	1	1	yes	characteristic
139	C₂×C₄×D₈	C₂×C₂₃	1	1	yes	characteristic
140	C₂×#367	C₂×C₂₃	1	1	yes	characteristic, complemented
141	C₂×#357	C₂×C₂₃	1	1	yes	characteristic
142	C₂×#357	C₂×C₂₃	1	1	yes	characteristic
143	#1942	C₂×C₂₃	1	1	yes	characteristic, complemented
144	#1380	C₂×C₂₃	1	1	yes	characteristic, complemented
145	#1374	C₂×C₂₃	1	1	yes	characteristic, complemented
146	C₂²×C₂₃	—	4	4	no	—
147	C₂²×C₂₃	—	8	2	no	—
148	C₂²×C₂₃	—	8	2	no	—
149	C₂²×C₂₃	—	8	2	no	—
150	C₂²×C₂₃	—	8	2	no	—
151	C₂²×C₂₃	—	8	2	no	—
152	C₂²×C₂₃	—	8	2	no	—
153	C₂²×C₂₃	C₄×D₈	1	1	yes	characteristic
154	C₂²×C₂₃	#447	1	1	yes	characteristic
155	C₂²×C₂₃	#474	1	1	yes	characteristic
156	C₂²×C₂₃	C₂×SD₃₂	1	1	yes	characteristic
157	C₂²×C₂₃	#508	1	1	yes	characteristic
158	C₂²×C₂₃	#509	1	1	yes	characteristic
159	C₂²×C₂₃	#509	1	1	yes	characteristic
160	C₄×C₂₃	—	2	2	no	—
161	C₄×C₂₃	—	4	1	no	—
162	C₄×C₂₃	—	8	2	no	—
163	C₄×C₂₃	—	8	2	no	—
164	C₄×C₂₃	—	16	1	no	—
165	C₄×C₂₃	#385	1	1	yes	characteristic
166	C₄×C₂₃	#440	1	1	yes	characteristic
167	C₄×C₂₃	#452	1	1	yes	characteristic
168	C₄×C₂₃	#478	1	1	yes	characteristic
169	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
170	C₂×C₄×D₄	—	2	1	no	—
171	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
172	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
173	C₄×D₈	—	2	2	no	—
174	C₄×D₈	—	2	2	no	—
175	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
176	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
177	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
178	C₂×#102	—	2	1	no	—
179	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
180	#389	—	2	1	no	—
181	#367	—	2	2	no	—
182	#357	—	2	2	no	—
183	#357	C₄×C₂₃	1	4	yes	complemented
184	#340	—	2	1	no	—
185	C₂×C₂₃	—	8	4	no	—
186	C₂×C₂₃	#1862	1	1	yes	characteristic
187	C₂×C₂₃	C₄×SD₃₂	1	1	yes	characteristic
188	C₂×C₂₃	#2177	1	1	yes	characteristic
189	C₂×C₂₃	#2186	1	1	yes	characteristic
190	C₂×C₂₃	#2187	1	1	yes	characteristic
191	C₂×C₂₃	#2226	1	1	yes	characteristic
192	C₂×C₂₃	#2228	1	1	yes	characteristic
193	C₂²×D₄	—	2	1	no	—
194	C₂³×C₄	—	4	1	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	—	2	1	no	—
197	C₂×D₈	—	2	1	no	—
198	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
199	C₂×D₈	D₄×C₂₃	1	2	yes	—
200	C₂×D₈	Q₈×C₂₃	1	2	yes	—
201	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
203	C₄×D₄	—	2	4	no	—
204	C₄×D₄	—	4	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	4	1	no	—
209	C₂×C₂²⋊C₄	—	4	1	no	—
210	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
211	C₂×C₁₆	—	2	1	no	—
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	complemented
217	#107	—	2	1	no	—
218	#107	—	2	1	no	—
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#107	C₂×C₄×C₂₃	1	2	yes	complemented
222	#107	D₄×C₂₃	1	2	yes	—
223	#102	—	2	4	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	#95	—	4	1	no	—
227	C₂₃	#12937	1	1	yes	characteristic, direct product factor, Sylow
228	C₂⁴	—	4	1	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	4	1	no	—
233	C₂×D₄	—	4	1	no	—
234	C₂×D₄	—	4	1	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	8	2	no	—
240	C₂²×C₄	—	8	2	no	—
241	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
242	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
244	D₈	—	2	2	no	—
245	D₈	—	2	2	no	—
246	D₈	—	2	2	no	—
247	D₈	C₄⋊C₄×C₂₃	1	4	yes	complemented
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
252	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
253	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
254	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
255	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
256	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	8	2	no	—
264	C₂²⋊C₄	—	4	4	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₁₆	—	2	2	no	—
268	C₁₆	—	4	1	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	8	1	no	—
273	C₂³	—	8	1	no	—
274	C₂³	—	8	1	no	—
275	C₂³	—	8	1	no	—
276	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
277	D₄	—	2	4	no	—
278	D₄	—	4	2	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	8	8	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	8	1	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	C₂₃×SD₃₂	1	2	yes	—
297	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
298	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
299	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
300	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
301	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
302	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
303	C₈	—	4	1	no	—
304	C₈	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
305	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
306	C₈	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
307	C₈	C₂₃×#123	1	1	yes	characteristic
308	C₂²	—	4	4	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₄×D₈×C₂₃	1	1	yes	characteristic
316	C₂²	C₂₃×#447	1	1	yes	characteristic
317	C₂²	C₂₃×#474	1	1	yes	characteristic
318	C₂²	C₂×C₂₃×SD₃₂	1	1	yes	characteristic
319	C₂²	C₂₃×#508	1	1	yes	characteristic
320	C₂²	C₂₃×#509	1	1	yes	characteristic
321	C₂²	C₂₃×#509	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	4	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	16	1	no	—
327	C₄	C₂₃×#385	1	1	yes	characteristic
328	C₄	C₂₃×#440	1	1	yes	characteristic
329	C₄	C₂₃×#452	1	1	yes	characteristic
330	C₄	C₂₃×#478	1	1	yes	characteristic
331	C₂	—	8	4	no	—
332	C₂	C₂₃×#1862	1	1	yes	characteristic
333	C₂	C₄×C₂₃×SD₃₂	1	1	yes	characteristic
334	C₂	C₂₃×#2177	1	1	yes	characteristic
335	C₂	C₂₃×#2186	1	1	yes	characteristic
336	C₂	C₂₃×#2187	1	1	yes	characteristic
337	C₂	C₂₃×#2226	1	1	yes	characteristic
338	C₂	C₂₃×#2228	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#367	C₂	semidirect
5. C₂₃×#1942	C₂	semidirect
6. C₂₃×#1380	C₂	semidirect
7. C₂₃×#1374	C₂	semidirect
22. C₂₃×#357	C₄	semidirect
47. C₂×C₁₆×C₂₃	C₂×C₄	semidirect
52. C₂₃×#107	C₂×C₄	semidirect
77. D₈×C₂₃	C₄⋊C₄	semidirect
99. #12937	C₂₃	direct (N × H)
140. C₂×#367	C₂×C₂₃	semidirect
143. #1942	C₂×C₂₃	semidirect
144. #1380	C₂×C₂₃	semidirect
145. #1374	C₂×C₂₃	semidirect
183. #357	C₄×C₂₃	semidirect
216. C₂×C₁₆	C₂×C₄×C₂₃	semidirect
221. #107	C₂×C₄×C₂₃	semidirect
227. C₂₃	#12937	direct (N × H)
247. D₈	C₄⋊C₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₂₃	2
C₂×C₂₃×#367	2
C₂×C₂₃×#357	2
C₂₃×#1942	2
C₂₃×#1380	2
C₂₃×#1374	2
#12937	23

Subgroups — C23×#12937 #19160563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12937 #19160563