Subgroups — C₂₃×#33924 #19181550

Order 5888.

482 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3341	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3338	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2237	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2232	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2227	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2226	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2220	C₂	1	2	yes	maximal, complemented
9	C₄×C₂₃×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#2169	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#2159	C₂	1	2	yes	maximal, complemented
12	C₂₃×#555	—	2	1	no	—
13	C₂₃×#554	—	2	1	no	—
14	C₂×C₂₃×SD₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
16	C₂₃×#500	C₂²	1	1	yes	characteristic
17	C₂₃×#500	C₂²	1	1	yes	characteristic
18	C₂₃×#484	—	2	2	no	—
19	C₂₃×#482	—	2	2	no	—
20	C₂₃×#477	—	2	1	no	—
21	C₂₃×#476	—	2	1	no	—
22	C₂₃×#467	C₂²	1	2	yes	—
23	C₂₃×#466	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
25	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#427	C₂²	1	2	yes	—
28	C₂×C₂₃×#107	C₂²	1	2	yes	—
29	C₂₃×#367	C₂²	1	1	yes	characteristic
30	C₂₃×#367	C₂²	1	2	yes	—
31	C₂₃×#366	C₂²	1	2	yes	—
32	C₂₃×#366	C₂²	1	2	yes	—
33	C₂₃×#366	C₂²	1	2	yes	—
34	C₂₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₂₃×#358	C₂²	1	1	yes	characteristic
36	C₂₃×#358	C₂²	1	2	yes	—
37	C₂₃×#357	C₂²	1	1	yes	characteristic
38	C₂₃×#357	C₂²	1	2	yes	—
39	C₂₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₂₃	C₂³	1	1	yes	characteristic
42	C₂×D₈×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
44	C₂₃×C₄⋊Q₈	—	2	1	no	—
45	C₂₃×C₄⋊Q₈	—	2	1	no	—
46	C₂₃×#125	—	4	1	no	—
47	C₂₃×#124	—	4	2	no	—
48	C₂₃×C₂²⋊Q₈	—	2	2	no	—
49	C₂₃×C₂²⋊Q₈	—	4	2	no	—
50	C₂₃×#121	—	2	2	no	—
51	C₂₃×#121	—	4	1	no	—
52	C₄×Q₈×C₂₃	—	2	1	no	—
53	C₄×D₄×C₂₃	—	2	1	no	—
54	C₄×D₄×C₂₃	—	4	1	no	—
55	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
56	C₂₃×#117	—	2	2	no	—
57	C₂₃×#117	—	4	1	no	—
58	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₂₃×SD₃₂	—	2	2	no	—
60	C₂₃×M₃₂	—	2	4	no	—
61	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
63	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
64	C₂₃×#107	—	2	2	no	—
65	C₂₃×#107	—	2	1	no	—
66	C₂₃×#107	—	2	1	no	—
67	C₂₃×#107	C₂³	1	1	yes	characteristic
68	C₂₃×#107	C₂³	1	2	yes	—
69	C₂₃×#107	C₂³	1	2	yes	—
70	C₂₃×#107	C₂³	1	1	yes	characteristic
71	C₂₃×#106	—	2	2	no	—
72	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
73	C₂₃×#103	—	2	1	no	—
74	C₂₃×#103	—	2	2	no	—
75	C₂₃×#102	—	2	1	no	—
76	C₂₃×#102	—	2	2	no	—
77	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
78	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₂₃	—	2	1	no	—
80	C₂×Q₈×C₂₃	—	4	1	no	—
81	C₂×Q₈×C₂₃	—	4	1	no	—
82	C₂×D₄×C₂₃	—	2	1	no	—
83	C₂×D₄×C₂₃	—	4	2	no	—
84	C₂×D₄×C₂₃	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₂₃	—	4	2	no	—
86	C₂²×C₄×C₂₃	—	4	1	no	—
87	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
88	Q₁₆×C₂₃	—	2	1	no	—
89	Q₁₆×C₂₃	—	2	1	no	—
90	D₈×C₂₃	—	2	1	no	—
91	D₈×C₂₃	—	2	1	no	—
92	C₂×C₈×C₂₃	—	2	2	no	—
93	C₂×C₈×C₂₃	—	2	2	no	—
94	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₂₃	—	2	1	no	—
98	C₄⋊C₄×C₂₃	—	2	2	no	—
99	C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₄⋊C₄×C₂₃	—	2	2	no	—
101	C₄⋊C₄×C₂₃	—	4	2	no	—
102	C₄⋊C₄×C₂₃	—	4	1	no	—
103	C₄⋊C₄×C₂₃	—	4	1	no	—
104	C₄⋊C₄×C₂₃	—	4	1	no	—
105	C₄⋊C₄×C₂₃	—	4	2	no	—
106	C₄⋊C₄×C₂₃	D₈	1	1	yes	characteristic
107	C₂²⋊C₄×C₂₃	—	4	2	no	—
108	C₂²⋊C₄×C₂₃	—	4	2	no	—
109	C₂²⋊C₄×C₂₃	—	4	2	no	—
110	C₂²⋊C₄×C₂₃	—	4	1	no	—
111	C₂²⋊C₄×C₂₃	D₈	1	2	yes	—
112	C₄²×C₂₃	—	4	2	no	—
113	C₄²×C₂₃	—	4	1	no	—
114	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₂₃	—	2	2	no	—
116	C₁₆×C₂₃	—	2	1	no	—
117	C₁₆×C₂₃	#41	1	2	yes	—
118	#33924	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂³×C₂₃	—	4	1	no	—
120	C₂³×C₂₃	C₂×D₈	1	2	yes	—
121	Q₈×C₂₃	—	4	1	no	—
122	Q₈×C₂₃	—	4	1	no	—
123	Q₈×C₂₃	—	16	1	no	—
124	Q₈×C₂₃	—	16	1	no	—
125	D₄×C₂₃	—	2	2	no	—
126	D₄×C₂₃	—	4	1	no	—
127	D₄×C₂₃	—	4	1	no	—
128	D₄×C₂₃	—	16	2	no	—
129	C₂×C₄×C₂₃	—	2	2	no	—
130	C₂×C₄×C₂₃	—	2	2	no	—
131	C₂×C₄×C₂₃	—	4	1	no	—
132	C₂×C₄×C₂₃	—	4	2	no	—
133	C₂×C₄×C₂₃	—	4	1	no	—
134	C₂×C₄×C₂₃	—	4	1	no	—
135	C₂×C₄×C₂₃	—	4	2	no	—
136	C₂×C₄×C₂₃	—	8	2	no	—
137	C₂×C₄×C₂₃	—	16	2	no	—
138	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
142	C₈×C₂₃	—	2	2	no	—
143	C₈×C₂₃	—	2	1	no	—
144	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
145	C₈×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
146	#3341	C₂×C₂₃	1	1	yes	characteristic
147	#3338	C₂×C₂₃	1	1	yes	characteristic, complemented
148	#2252	C₂×C₂₃	1	1	yes	characteristic, complemented
149	#2237	C₂×C₂₃	1	2	yes	complemented
150	#2232	C₂×C₂₃	1	2	yes	complemented
151	#2227	C₂×C₂₃	1	1	yes	characteristic, complemented
152	#2226	C₂×C₂₃	1	1	yes	characteristic, complemented
153	#2220	C₂×C₂₃	1	2	yes	complemented
154	C₄×SD₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
155	#2169	C₂×C₂₃	1	1	yes	characteristic, complemented
156	#2159	C₂×C₂₃	1	2	yes	complemented
157	C₂²×C₂₃	—	2	2	no	—
158	C₂²×C₂₃	—	2	2	no	—
159	C₂²×C₂₃	—	2	2	no	—
160	C₂²×C₂₃	—	8	1	no	—
161	C₂²×C₂₃	—	8	1	no	—
162	C₂²×C₂₃	—	8	1	no	—
163	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
164	C₄×C₂₃	—	2	2	no	—
165	C₄×C₂₃	—	2	1	no	—
166	C₄×C₂₃	—	8	1	no	—
167	C₄×C₂₃	—	8	1	no	—
168	C₄×C₂₃	—	8	1	no	—
169	C₄×C₂₃	—	8	2	no	—
170	C₄×C₂₃	—	8	2	no	—
171	C₄×C₂₃	#510	1	2	yes	—
172	C₄×C₂₃	#547	1	1	yes	characteristic
173	C₄×C₂₃	#578	1	1	yes	characteristic
174	#555	—	2	1	no	—
175	#554	—	2	1	no	—
176	C₂×SD₃₂	C₂²×C₂₃	1	1	yes	characteristic
177	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
178	#500	C₂²×C₂₃	1	1	yes	characteristic
179	#500	C₂²×C₂₃	1	1	yes	characteristic
180	#484	—	2	2	no	—
181	#482	—	2	2	no	—
182	#477	—	2	1	no	—
183	#476	—	2	1	no	—
184	#467	C₂²×C₂₃	1	2	yes	—
185	#466	C₂²×C₂₃	1	2	yes	—
186	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
187	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
188	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
189	#427	C₂²×C₂₃	1	2	yes	—
190	C₂×#107	C₂²×C₂₃	1	2	yes	—
191	#367	C₂²×C₂₃	1	1	yes	characteristic
192	#367	C₂²×C₂₃	1	2	yes	—
193	#366	C₂²×C₂₃	1	2	yes	—
194	#366	C₂²×C₂₃	1	2	yes	—
195	#366	C₂²×C₂₃	1	2	yes	—
196	C₄⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
197	#358	C₂²×C₂₃	1	1	yes	characteristic
198	#358	C₂²×C₂₃	1	2	yes	—
199	#357	C₂²×C₂₃	1	1	yes	characteristic
200	#357	C₂²×C₂₃	1	2	yes	—
201	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	—
202	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
203	C₂×C₂₃	—	4	2	no	—
204	C₂×C₂₃	—	16	1	no	—
205	C₂×C₂₃	#3334	1	1	yes	characteristic
206	C₂×C₂₃	C₂×#510	1	1	yes	characteristic
207	C₂×C₂₃	#3415	1	1	yes	characteristic
208	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
209	C₂×D₈	C₂³×C₂₃	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	#125	—	4	1	no	—
214	#124	—	4	2	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₂²⋊Q₈	—	4	2	no	—
217	#121	—	2	2	no	—
218	#121	—	4	1	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×D₄	—	2	1	no	—
221	C₄×D₄	—	4	1	no	—
222	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
223	#117	—	2	2	no	—
224	#117	—	4	1	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	SD₃₂	—	2	2	no	—
227	M₃₂	—	2	4	no	—
228	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
229	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
230	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
231	#107	—	2	2	no	—
232	#107	—	2	1	no	—
233	#107	—	2	1	no	—
234	#107	C₂³×C₂₃	1	1	yes	characteristic
235	#107	C₂³×C₂₃	1	2	yes	—
236	#107	C₂³×C₂₃	1	2	yes	—
237	#107	C₂³×C₂₃	1	1	yes	characteristic
238	#106	—	2	2	no	—
239	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
240	#103	—	2	1	no	—
241	#103	—	2	2	no	—
242	#102	—	2	1	no	—
243	#102	—	2	2	no	—
244	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
245	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
246	C₂₃	#33924	1	1	yes	characteristic, direct product factor, Sylow
247	C₂×Q₈	—	2	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	D₈×C₂₃	1	1	yes	characteristic
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
256	Q₁₆	—	2	1	no	—
257	Q₁₆	—	2	1	no	—
258	D₈	—	2	1	no	—
259	D₈	—	2	1	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
264	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	D₈×C₂₃	1	1	yes	characteristic
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	D₈×C₂₃	1	2	yes	—
280	C₄²	—	4	2	no	—
281	C₄²	—	4	1	no	—
282	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
283	C₁₆	—	2	2	no	—
284	C₁₆	—	2	1	no	—
285	C₁₆	#41×C₂₃	1	2	yes	—
286	C₂³	—	4	1	no	—
287	C₂³	C₂×D₈×C₂₃	1	2	yes	—
288	Q₈	—	4	1	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	16	1	no	—
291	Q₈	—	16	1	no	—
292	D₄	—	2	2	no	—
293	D₄	—	4	1	no	—
294	D₄	—	4	1	no	—
295	D₄	—	16	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	—	16	2	no	—
305	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
306	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
307	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	2	1	no	—
311	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
312	C₈	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	—	8	1	no	—
319	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	2	no	—
327	C₄	C₂₃×#510	1	2	yes	—
328	C₄	C₂₃×#547	1	1	yes	characteristic
329	C₄	C₂₃×#578	1	1	yes	characteristic
330	C₂	—	4	2	no	—
331	C₂	—	16	1	no	—
332	C₂	C₂₃×#3334	1	1	yes	characteristic
333	C₂	C₂×C₂₃×#510	1	1	yes	characteristic
334	C₂	C₂₃×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂₃×#3341	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×D₄×C₂₃ ◁ C₈×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#3338	C₂	semidirect
3. C₂₃×#2252	C₂	semidirect
4. C₂₃×#2237	C₂	semidirect
5. C₂₃×#2232	C₂	semidirect
6. C₂₃×#2227	C₂	semidirect
7. C₂₃×#2226	C₂	semidirect
8. C₂₃×#2220	C₂	semidirect
9. C₄×C₂₃×SD₃₂	C₂	semidirect
10. C₂₃×#2169	C₂	semidirect
11. C₂₃×#2159	C₂	semidirect
118. #33924	C₂₃	direct (N × H)
147. #3338	C₂×C₂₃	semidirect
148. #2252	C₂×C₂₃	semidirect
149. #2237	C₂×C₂₃	semidirect
150. #2232	C₂×C₂₃	semidirect
151. #2227	C₂×C₂₃	semidirect
152. #2226	C₂×C₂₃	semidirect
153. #2220	C₂×C₂₃	semidirect
154. C₄×SD₃₂	C₂×C₂₃	semidirect
155. #2169	C₂×C₂₃	semidirect
156. #2159	C₂×C₂₃	semidirect
246. C₂₃	#33924	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×#3341	2
C₂₃×#3338	2
C₂₃×#2252	2
C₂₃×#2237	2
C₂₃×#2232	2
C₂₃×#2227	2
C₂₃×#2226	2
C₂₃×#2220	2
C₄×C₂₃×SD₃₂	2
C₂₃×#2169	2
C₂₃×#2159	2
#33924	23

Subgroups — C23×#33924 #19181550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33924 #19181550