Subgroups — C₂₃×#22275 #19169901

Order 5888.

938 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#492	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2328	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2327	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1961	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1960	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1880	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1877	C₂	1	4	yes	maximal, complemented
8	C₂₃×#1835	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₂₃	C₂²	1	4	yes	—
11	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
15	C₂₃×#492	C₄	1	8	yes	complemented
16	C₂×C₂₃×#103	C₂²	1	4	yes	—
17	C₂×C₂₃×#102	C₂²	1	4	yes	—
18	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#391	C₂²	1	2	yes	—
21	C₂₃×#390	C₂²	1	2	yes	—
22	C₂₃×#387	—	2	4	no	—
23	C₂₃×#386	—	2	4	no	—
24	C₂₃×#382	C₂²	1	1	yes	characteristic
25	C₂₃×#382	C₂²	1	1	yes	characteristic
26	C₂₃×#376	C₂²	1	1	yes	characteristic
27	C₂₃×#340	C₂²	1	4	yes	—
28	C₂₃×#336	C₂²	1	2	yes	—
29	C₂₃×#336	C₂²	1	2	yes	—
30	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
31	C₂²×D₄×C₂₃	—	2	2	no	—
32	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
33	C₂³×C₄×C₂₃	—	2	2	no	—
34	C₂×SD₁₆×C₂₃	—	4	4	no	—
35	C₂×SD₁₆×C₂₃	—	4	4	no	—
36	C₂×SD₁₆×C₂₃	C₂×C₄	1	16	yes	complemented
37	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
39	C₂₃×C₄⋊Q₈	—	2	2	no	—
40	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
41	C₂₃×#127	—	2	2	no	—
42	C₂₃×#127	C₂×C₄	1	4	yes	—
43	C₄×Q₈×C₂₃	—	4	4	no	—
44	C₄×D₄×C₂₃	—	4	4	no	—
45	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
51	C₂×C₄²×C₂₃	—	2	2	no	—
52	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
55	C₂₃×#103	—	4	4	no	—
56	C₂₃×#102	—	4	4	no	—
57	C₂₃×#97	—	2	2	no	—
58	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	complemented
59	C₂₃×#95	—	2	2	no	—
60	C₂₃×#95	—	2	2	no	—
61	C₂₃×#95	—	2	4	no	—
62	C₂₃×#95	D₄	1	4	yes	—
63	C₂⁴×C₂₃	—	2	2	no	—
64	C₂×Q₈×C₂₃	—	2	2	no	—
65	C₂×Q₈×C₂₃	—	2	4	no	—
66	C₂×Q₈×C₂₃	—	2	4	no	—
67	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
68	C₂×D₄×C₂₃	—	2	8	no	—
69	C₂×D₄×C₂₃	—	2	2	no	—
70	C₂×D₄×C₂₃	—	2	4	no	—
71	C₂×D₄×C₂₃	—	2	4	no	—
72	C₂×D₄×C₂₃	—	8	4	no	—
73	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₂₃	—	2	2	no	—
75	C₂²×C₄×C₂₃	—	2	2	no	—
76	C₂²×C₄×C₂₃	—	2	2	no	—
77	C₂²×C₄×C₂₃	—	4	4	no	—
78	C₂²×C₄×C₂₃	—	8	4	no	—
79	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
83	SD₁₆×C₂₃	—	4	16	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	—	2	2	no	—
86	C₂×C₈×C₂₃	—	2	2	no	—
87	C₂×C₈×C₂₃	—	2	2	no	—
88	C₂×C₈×C₂₃	—	2	2	no	—
89	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
90	C₄⋊C₄×C₂₃	—	2	8	no	—
91	C₄⋊C₄×C₂₃	—	2	4	no	—
92	C₄⋊C₄×C₂₃	—	2	4	no	—
93	C₄⋊C₄×C₂₃	—	4	4	no	—
94	C₂²⋊C₄×C₂₃	—	4	4	no	—
95	C₂²⋊C₄×C₂₃	—	8	4	no	—
96	C₄²×C₂₃	—	2	2	no	—
97	C₄²×C₂₃	—	2	2	no	—
98	C₄²×C₂₃	—	4	4	no	—
99	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₂₃	#41	1	2	yes	—
101	#22275	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂³×C₂₃	—	2	4	no	—
103	C₂³×C₂₃	—	4	2	no	—
104	C₂³×C₂₃	—	4	2	no	—
105	C₂³×C₂₃	—	4	4	no	—
106	C₂³×C₂₃	—	4	4	no	—
107	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
108	Q₈×C₂₃	—	2	8	no	—
109	Q₈×C₂₃	—	4	4	no	—
110	D₄×C₂₃	—	2	8	no	—
111	D₄×C₂₃	—	4	4	no	—
112	D₄×C₂₃	—	8	8	no	—
113	C₂×C₄×C₂₃	—	2	4	no	—
114	C₂×C₄×C₂₃	—	2	4	no	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	2	2	no	—
117	C₂×C₄×C₂₃	—	2	4	no	—
118	C₂×C₄×C₂₃	—	2	4	no	—
119	C₂×C₄×C₂₃	—	4	2	no	—
120	C₂×C₄×C₂₃	—	4	2	no	—
121	C₂×C₄×C₂₃	—	4	2	no	—
122	C₂×C₄×C₂₃	—	4	2	no	—
123	C₂×C₄×C₂₃	—	4	4	no	—
124	C₂×C₄×C₂₃	—	4	4	no	—
125	C₂×C₄×C₂₃	—	8	8	no	—
126	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
127	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
130	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
131	C₈×C₂₃	—	2	4	no	—
132	C₈×C₂₃	—	4	2	no	—
133	C₂×#492	C₂×C₂₃	1	1	yes	characteristic
134	#2328	C₂×C₂₃	1	1	yes	characteristic
135	#2327	C₂×C₂₃	1	1	yes	characteristic, complemented
136	#1961	C₂×C₂₃	1	1	yes	characteristic
137	#1960	C₂×C₂₃	1	1	yes	characteristic, complemented
138	#1880	C₂×C₂₃	1	4	yes	complemented
139	#1877	C₂×C₂₃	1	4	yes	complemented
140	#1835	C₂×C₂₃	1	1	yes	characteristic, complemented
141	#1746	C₂×C₂₃	1	1	yes	characteristic, complemented
142	C₂²×C₂₃	—	4	4	no	—
143	C₂²×C₂₃	—	4	8	no	—
144	C₂²×C₂₃	—	8	4	no	—
145	C₂²×C₂₃	—	8	2	no	—
146	C₂²×C₂₃	—	8	2	no	—
147	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂²×C₂₃	#535	1	1	yes	characteristic
149	C₂²×C₂₃	#535	1	1	yes	characteristic
150	C₂²×C₂₃	#577	1	4	yes	—
151	C₄×C₂₃	—	2	4	no	—
152	C₄×C₂₃	—	4	4	no	—
153	C₄×C₂₃	—	4	2	no	—
154	C₄×C₂₃	—	4	2	no	—
155	C₄×C₂₃	—	8	2	no	—
156	C₄×C₂₃	—	8	2	no	—
157	C₄×C₂₃	#518	1	2	yes	—
158	C₄×C₂₃	#540	1	2	yes	—
159	C₂²×SD₁₆	C₂²×C₂₃	1	4	yes	—
160	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
161	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
162	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
163	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
164	#492	C₄×C₂₃	1	8	yes	complemented
165	C₂×#103	C₂²×C₂₃	1	4	yes	—
166	C₂×#102	C₂²×C₂₃	1	4	yes	—
167	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
169	#391	C₂²×C₂₃	1	2	yes	—
170	#390	C₂²×C₂₃	1	2	yes	—
171	#387	—	2	4	no	—
172	#386	—	2	4	no	—
173	#382	C₂²×C₂₃	1	1	yes	characteristic
174	#382	C₂²×C₂₃	1	1	yes	characteristic
175	#376	C₂²×C₂₃	1	1	yes	characteristic
176	#340	C₂²×C₂₃	1	4	yes	—
177	#336	C₂²×C₂₃	1	2	yes	—
178	#336	C₂²×C₂₃	1	2	yes	—
179	C₂×C₂₃	—	8	4	no	—
180	C₂×C₂₃	#2328	1	1	yes	characteristic
181	C₂×C₂₃	#2949	1	2	yes	—
182	C₂×C₂₃	#3221	1	2	yes	—
183	C₂×C₂₃	#3222	1	2	yes	—
184	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
185	C₂²×D₄	—	2	2	no	—
186	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
187	C₂³×C₄	—	2	2	no	—
188	C₂×SD₁₆	—	4	4	no	—
189	C₂×SD₁₆	—	4	4	no	—
190	C₂×SD₁₆	C₂×C₄×C₂₃	1	16	yes	complemented
191	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
192	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
195	#127	—	2	2	no	—
196	#127	C₂×C₄×C₂₃	1	4	yes	—
197	C₄×Q₈	—	4	4	no	—
198	C₄×D₄	—	4	4	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
203	C₂×C₂²⋊C₄	—	2	2	no	—
204	C₂×C₂²⋊C₄	—	2	4	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
209	#103	—	4	4	no	—
210	#102	—	4	4	no	—
211	#97	—	2	2	no	—
212	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	complemented
213	#95	—	2	2	no	—
214	#95	—	2	2	no	—
215	#95	—	2	4	no	—
216	#95	D₄×C₂₃	1	4	yes	—
217	C₂₃	#22275	1	1	yes	characteristic, direct product factor, Sylow
218	C₂⁴	—	2	2	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
223	C₂×D₄	—	2	8	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	8	4	no	—
228	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	4	4	no	—
233	C₂²×C₄	—	8	4	no	—
234	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
237	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
238	SD₁₆	—	4	16	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₂²⋊C₄	—	4	4	no	—
250	C₂²⋊C₄	—	8	4	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	4	4	no	—
254	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
255	C₄²	#41×C₂₃	1	2	yes	—
256	C₂³	—	2	4	no	—
257	C₂³	—	4	4	no	—
258	C₂³	—	4	4	no	—
259	C₂³	—	4	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
262	Q₈	—	2	8	no	—
263	Q₈	—	4	4	no	—
264	D₄	—	2	8	no	—
265	D₄	—	4	4	no	—
266	D₄	—	8	8	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	8	8	no	—
280	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
281	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
282	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
283	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
284	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
285	C₈	—	2	4	no	—
286	C₈	—	4	2	no	—
287	C₂²	—	4	8	no	—
288	C₂²	—	4	4	no	—
289	C₂²	—	8	4	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	8	2	no	—
292	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
293	C₂²	C₂₃×#535	1	1	yes	characteristic
294	C₂²	C₂₃×#535	1	1	yes	characteristic
295	C₂²	C₂₃×#577	1	4	yes	—
296	C₄	—	2	4	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	4	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	2	no	—
302	C₄	C₂₃×#518	1	2	yes	—
303	C₄	C₂₃×#540	1	2	yes	—
304	C₂	—	8	4	no	—
305	C₂	C₂₃×#2328	1	1	yes	characteristic
306	C₂	C₂₃×#2949	1	2	yes	—
307	C₂	C₂₃×#3221	1	2	yes	—
308	C₂	C₂₃×#3222	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2327	C₂	semidirect
5. C₂₃×#1960	C₂	semidirect
6. C₂₃×#1880	C₂	semidirect
7. C₂₃×#1877	C₂	semidirect
8. C₂₃×#1835	C₂	semidirect
9. C₂₃×#1746	C₂	semidirect
15. C₂₃×#492	C₄	semidirect
36. C₂×SD₁₆×C₂₃	C₂×C₄	semidirect
40. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
58. C₄×C₈×C₂₃	C₂×C₄	semidirect
101. #22275	C₂₃	direct (N × H)
135. #2327	C₂×C₂₃	semidirect
137. #1960	C₂×C₂₃	semidirect
138. #1880	C₂×C₂₃	semidirect
139. #1877	C₂×C₂₃	semidirect
140. #1835	C₂×C₂₃	semidirect
141. #1746	C₂×C₂₃	semidirect
164. #492	C₄×C₂₃	semidirect
190. C₂×SD₁₆	C₂×C₄×C₂₃	semidirect
194. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
212. C₄×C₈	C₂×C₄×C₂₃	semidirect
217. C₂₃	#22275	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×#492	2
C₂₃×#2328	2
C₂₃×#2327	2
C₂₃×#1961	2
C₂₃×#1960	2
C₂₃×#1880	2
C₂₃×#1877	2
C₂₃×#1835	2
C₂₃×#1746	2
#22275	23

Subgroups — C23×#22275 #19169901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22275 #19169901