Subgroups — C₂₃×#9738 #19157364

Order 5888.

538 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#486	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1982	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1981	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#331	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
10	C₂₃×#486	—	2	2	no	—
11	C₂₃×#486	C₄	1	4	yes	complemented
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#103	—	2	2	no	—
16	C₂×C₂₃×#102	—	2	1	no	—
17	C₂×C₂₃×#102	—	2	1	no	—
18	C₂×C₂₃×#102	—	2	2	no	—
19	C₂×C₄×C₈×C₂₃	—	2	1	no	—
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#336	—	2	1	no	—
23	C₂₃×#335	—	2	2	no	—
24	C₂₃×#332	—	2	2	no	—
25	C₂₃×#331	—	2	2	no	—
26	C₂²×Q₈×C₂₃	—	2	1	no	—
27	C₂²×D₄×C₂₃	—	2	1	no	—
28	C₂²×D₄×C₂₃	—	2	1	no	—
29	C₂²×D₄×C₂₃	—	2	1	no	—
30	C₂²×C₈×C₂₃	—	2	1	no	—
31	C₂²×C₈×C₂₃	—	2	1	no	—
32	C₂²×C₈×C₂₃	—	2	1	no	—
33	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
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	C₂×C₄	1	2	yes	—
39	C₂₃×#127	—	2	1	no	—
40	C₂₃×#127	—	2	1	no	—
41	C₂₃×#127	—	2	1	no	—
42	C₂₃×#127	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
46	C₂×C₄²×C₂₃	—	2	1	no	—
47	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
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×#103	—	4	4	no	—
55	C₂₃×#102	—	2	4	no	—
56	C₂₃×#102	—	2	4	no	—
57	C₂₃×#102	—	4	4	no	—
58	C₄×C₈×C₂₃	—	4	2	no	—
59	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₂₃	D₄	1	2	yes	—
61	C₂₃×#95	—	2	1	no	—
62	C₂⁴×C₂₃	—	4	1	no	—
63	C₂×Q₈×C₂₃	—	2	2	no	—
64	C₂×Q₈×C₂₃	—	4	1	no	—
65	C₂×Q₈×C₂₃	—	4	2	no	—
66	C₂×Q₈×C₂₃	—	4	2	no	—
67	C₂×D₄×C₂₃	—	2	2	no	—
68	C₂×D₄×C₂₃	—	2	2	no	—
69	C₂×D₄×C₂₃	—	2	2	no	—
70	C₂×D₄×C₂₃	—	4	1	no	—
71	C₂×D₄×C₂₃	—	4	1	no	—
72	C₂×D₄×C₂₃	—	4	1	no	—
73	C₂×D₄×C₂₃	—	4	2	no	—
74	C₂×D₄×C₂₃	—	4	1	no	—
75	C₂×D₄×C₂₃	—	4	1	no	—
76	C₂×D₄×C₂₃	—	4	2	no	—
77	C₂×D₄×C₂₃	—	4	1	no	—
78	C₂×D₄×C₂₃	—	4	1	no	—
79	C₂×D₄×C₂₃	—	8	2	no	—
80	C₂²×C₄×C₂₃	—	2	2	no	—
81	C₂²×C₄×C₂₃	—	4	1	no	—
82	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₂₃	—	2	1	no	—
86	C₂×C₈×C₂₃	—	2	1	no	—
87	C₂×C₈×C₂₃	—	2	1	no	—
88	C₂×C₈×C₂₃	—	2	1	no	—
89	C₂×C₈×C₂₃	—	2	2	no	—
90	C₂×C₈×C₂₃	—	4	1	no	—
91	C₂×C₈×C₂₃	—	4	1	no	—
92	C₂×C₈×C₂₃	—	4	1	no	—
93	C₂×C₈×C₂₃	—	4	1	no	—
94	C₂×C₈×C₂₃	—	4	2	no	—
95	C₂×C₈×C₂₃	—	4	2	no	—
96	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
97	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
98	C₄⋊C₄×C₂₃	—	2	2	no	—
99	C₄⋊C₄×C₂₃	—	2	2	no	—
100	C₄⋊C₄×C₂₃	—	2	2	no	—
101	C₄⋊C₄×C₂₃	—	4	1	no	—
102	C₄⋊C₄×C₂₃	—	4	1	no	—
103	C₄⋊C₄×C₂₃	D₈	1	4	yes	—
104	C₄²×C₂₃	—	4	1	no	—
105	C₄²×C₂₃	—	4	1	no	—
106	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₂₃	#41	1	1	yes	characteristic
110	#9738	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂³×C₂₃	—	4	2	no	—
112	C₂³×C₂₃	—	8	1	no	—
113	C₂³×C₂₃	—	8	1	no	—
114	C₂³×C₂₃	—	8	2	no	—
115	C₂³×C₂₃	—	8	2	no	—
116	C₂³×C₂₃	#121	1	1	yes	characteristic
117	Q₈×C₂₃	—	4	4	no	—
118	Q₈×C₂₃	—	8	1	no	—
119	Q₈×C₂₃	—	8	1	no	—
120	D₄×C₂₃	—	4	2	no	—
121	D₄×C₂₃	—	4	2	no	—
122	D₄×C₂₃	—	4	4	no	—
123	D₄×C₂₃	—	8	2	no	—
124	D₄×C₂₃	—	8	2	no	—
125	D₄×C₂₃	—	8	1	no	—
126	D₄×C₂₃	—	8	1	no	—
127	D₄×C₂₃	—	8	1	no	—
128	D₄×C₂₃	—	8	1	no	—
129	C₂×C₄×C₂₃	—	2	2	no	—
130	C₂×C₄×C₂₃	—	2	4	no	—
131	C₂×C₄×C₂₃	—	4	2	no	—
132	C₂×C₄×C₂₃	—	4	2	no	—
133	C₂×C₄×C₂₃	—	4	2	no	—
134	C₂×C₄×C₂₃	—	8	2	no	—
135	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
140	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
141	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
143	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
144	C₂×C₄×C₂₃	#135	1	2	yes	—
145	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
147	C₈×C₂₃	—	2	2	no	—
148	C₈×C₂₃	—	2	2	no	—
149	C₈×C₂₃	—	4	2	no	—
150	C₈×C₂₃	—	8	1	no	—
151	C₈×C₂₃	—	8	1	no	—
152	C₂×#486	C₂×C₂₃	1	1	yes	characteristic
153	#1982	C₂×C₂₃	1	1	yes	characteristic
154	#1981	C₂×C₂₃	1	1	yes	characteristic, complemented
155	C₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
156	C₂×#335	C₂×C₂₃	1	1	yes	characteristic, complemented
157	C₂×#332	C₂×C₂₃	1	1	yes	characteristic, complemented
158	C₂×#331	C₂×C₂₃	1	1	yes	characteristic
159	C₂²×C₂₃	—	8	2	no	—
160	C₂²×C₂₃	—	8	2	no	—
161	C₂²×C₂₃	—	8	2	no	—
162	C₂²×C₂₃	—	16	2	no	—
163	C₂²×C₂₃	—	16	2	no	—
164	C₂²×C₂₃	#390	1	1	yes	characteristic
165	C₂²×C₂₃	#463	1	1	yes	characteristic
166	C₂²×C₂₃	#464	1	1	yes	characteristic
167	C₂²×C₂₃	#466	1	2	yes	—
168	C₂²×C₂₃	#471	1	2	yes	—
169	C₄×C₂₃	—	2	1	no	—
170	C₄×C₂₃	—	2	1	no	—
171	C₄×C₂₃	—	4	4	no	—
172	C₄×C₂₃	—	8	2	no	—
173	C₄×C₂₃	#416	1	1	yes	characteristic
174	C₄×C₂₃	#417	1	1	yes	characteristic
175	C₄×C₂₃	C₄×D₈	1	2	yes	—
176	C₄×C₂₃	#463	1	1	yes	characteristic
177	C₄×C₂₃	#464	1	1	yes	characteristic
178	C₄×C₂₃	#493	1	1	yes	characteristic
179	C₄×C₂₃	#495	1	1	yes	characteristic
180	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
181	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
182	#486	—	2	2	no	—
183	#486	C₄×C₂₃	1	4	yes	complemented
184	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
185	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
186	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
187	C₂×#103	—	2	2	no	—
188	C₂×#102	—	2	1	no	—
189	C₂×#102	—	2	1	no	—
190	C₂×#102	—	2	2	no	—
191	C₂×C₄×C₈	—	2	1	no	—
192	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
193	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
194	#336	—	2	1	no	—
195	#335	—	2	2	no	—
196	#332	—	2	2	no	—
197	#331	—	2	2	no	—
198	C₂×C₂₃	—	16	2	no	—
199	C₂×C₂₃	#1681	1	2	yes	—
200	C₂×C₂₃	#1682	1	2	yes	—
201	C₂×C₂₃	#1970	1	1	yes	characteristic
202	C₂×C₂₃	#1977	1	1	yes	characteristic
203	C₂×C₂₃	#1983	1	1	yes	characteristic
204	C₂²×Q₈	—	2	1	no	—
205	C₂²×D₄	—	2	1	no	—
206	C₂²×D₄	—	2	1	no	—
207	C₂²×D₄	—	2	1	no	—
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
212	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
217	#127	—	2	1	no	—
218	#127	—	2	1	no	—
219	#127	—	2	1	no	—
220	#127	C₂×C₄×C₂₃	1	2	yes	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
224	C₂×C₄²	—	2	1	no	—
225	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
232	#103	—	4	4	no	—
233	#102	—	2	4	no	—
234	#102	—	2	4	no	—
235	#102	—	4	4	no	—
236	C₄×C₈	—	4	2	no	—
237	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
238	C₄×C₈	D₄×C₂₃	1	2	yes	—
239	#95	—	2	1	no	—
240	C₂₃	#9738	1	1	yes	characteristic, direct product factor, Sylow
241	C₂⁴	—	4	1	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	4	2	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	8	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
263	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	2	no	—
274	C₂×C₈	—	4	2	no	—
275	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
276	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	D₈×C₂₃	1	4	yes	—
283	C₄²	—	4	1	no	—
284	C₄²	—	4	1	no	—
285	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
287	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
288	C₄²	#41×C₂₃	1	1	yes	characteristic
289	C₂³	—	4	2	no	—
290	C₂³	—	8	1	no	—
291	C₂³	—	8	1	no	—
292	C₂³	—	8	2	no	—
293	C₂³	—	8	2	no	—
294	C₂³	C₂₃×#121	1	1	yes	characteristic
295	Q₈	—	4	4	no	—
296	Q₈	—	8	1	no	—
297	Q₈	—	8	1	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	4	no	—
301	D₄	—	8	2	no	—
302	D₄	—	8	2	no	—
303	D₄	—	8	1	no	—
304	D₄	—	8	1	no	—
305	D₄	—	8	1	no	—
306	D₄	—	8	1	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
314	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
315	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
316	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
317	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
318	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
319	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
320	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
321	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
322	C₂×C₄	C₂₃×#135	1	2	yes	—
323	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
324	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	—	4	2	no	—
328	C₈	—	8	1	no	—
329	C₈	—	8	1	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	16	2	no	—
334	C₂²	—	16	2	no	—
335	C₂²	C₂₃×#390	1	1	yes	characteristic
336	C₂²	C₂₃×#463	1	1	yes	characteristic
337	C₂²	C₂₃×#464	1	1	yes	characteristic
338	C₂²	C₂₃×#466	1	2	yes	—
339	C₂²	C₂₃×#471	1	2	yes	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	4	no	—
343	C₄	—	8	2	no	—
344	C₄	C₂₃×#416	1	1	yes	characteristic
345	C₄	C₂₃×#417	1	1	yes	characteristic
346	C₄	C₄×D₈×C₂₃	1	2	yes	—
347	C₄	C₂₃×#463	1	1	yes	characteristic
348	C₄	C₂₃×#464	1	1	yes	characteristic
349	C₄	C₂₃×#493	1	1	yes	characteristic
350	C₄	C₂₃×#495	1	1	yes	characteristic
351	C₂	—	16	2	no	—
352	C₂	C₂₃×#1681	1	2	yes	—
353	C₂	C₂₃×#1682	1	2	yes	—
354	C₂	C₂₃×#1970	1	1	yes	characteristic
355	C₂	C₂₃×#1977	1	1	yes	characteristic
356	C₂	C₂₃×#1983	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#121
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1981	C₂	semidirect
4. C₈×C₄⋊C₄×C₂₃	C₂	semidirect
5. C₂×C₂₃×#335	C₂	semidirect
6. C₂×C₂₃×#332	C₂	semidirect
11. C₂₃×#486	C₄	semidirect
110. #9738	C₂₃	direct (N × H)
154. #1981	C₂×C₂₃	semidirect
155. C₈×C₄⋊C₄	C₂×C₂₃	semidirect
156. C₂×#335	C₂×C₂₃	semidirect
157. C₂×#332	C₂×C₂₃	semidirect
183. #486	C₄×C₂₃	semidirect
240. C₂₃	#9738	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×#486	2
C₂₃×#1982	2
C₂₃×#1981	2
C₈×C₄⋊C₄×C₂₃	2
C₂×C₂₃×#335	2
C₂×C₂₃×#332	2
C₂×C₂₃×#331	2
#9738	23

Subgroups — C23×#9738 #19157364

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9738 #19157364