Subgroups — C₂₃×#22689 #19170315

Order 5888.

862 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#443	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2952	C₂	1	2	yes	maximal
3	C₂₃×#2912	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1951	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1859	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1815	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#131	C₂²	1	2	yes	—
11	C₂₃×#520	C₂²	1	4	yes	—
12	C₂₃×#443	C₄	1	8	yes	complemented
13	C₂₃×#442	—	2	2	no	—
14	C₂₃×#441	—	2	2	no	—
15	C₄×SD₁₆×C₂₃	—	2	4	no	—
16	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
18	C₂₃×#428	—	2	4	no	—
19	C₂₃×#428	C₂²	1	4	yes	—
20	C₂₃×#427	—	2	2	no	—
21	C₂₃×#427	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#106	—	2	2	no	—
23	C₂×C₂₃×#106	C₂²	1	2	yes	—
24	C₂×C₂₃×#106	C₂²	1	2	yes	—
25	C₂×C₂₃×#104	C₂²	1	2	yes	—
26	C₂₃×#417	C₂²	1	4	yes	—
27	C₂₃×#405	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#384	C₂²	1	2	yes	—
30	C₂₃×#343	C₂²	1	4	yes	—
31	C₂₃×#339	C₂²	1	2	yes	—
32	C₂×#41×C₂₃	C₂³	1	2	yes	—
33	C₂₃×#135	C₂×C₄	1	4	yes	—
34	C₂₃×#135	Q₈	1	4	yes	—
35	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
36	C₂×SD₁₆×C₂₃	D₄	1	2	yes	—
37	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
38	C₂₃×#131	—	2	4	no	—
39	C₂₃×#131	C₂×C₄	1	8	yes	complemented
40	C₂×M₁₆×C₂₃	—	2	2	no	—
41	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
42	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₂₃	—	2	2	no	—
44	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₄×Q₈×C₂₃	—	2	4	no	—
47	C₄×D₄×C₂₃	—	2	4	no	—
48	C₄×D₄×C₂₃	—	4	4	no	—
49	C₂₃×#117	—	2	4	no	—
50	C₂₃×#117	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
52	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
53	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
54	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
55	C₂₃×#108	—	2	2	no	—
56	C₂₃×#108	C₂×C₄	1	4	yes	complemented
57	C₂₃×#107	—	2	4	no	—
58	C₂₃×#107	D₄	1	2	yes	—
59	C₂₃×#106	—	2	2	no	—
60	C₂₃×#106	—	2	2	no	—
61	C₂₃×#106	—	2	2	no	—
62	C₂₃×#106	—	2	2	no	—
63	C₂₃×#106	—	2	4	no	—
64	C₂₃×#106	D₄	1	2	yes	—
65	C₂₃×#104	C₂×C₄	1	8	yes	complemented
66	C₂₃×#103	—	2	4	no	—
67	C₂₃×#102	—	2	4	no	—
68	C₂₃×#97	—	2	2	no	—
69	C₄×C₈×C₂₃	—	2	2	no	—
70	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	complemented
71	#41×C₂₃	—	2	2	no	—
72	#41×C₂₃	—	2	2	no	—
73	#41×C₂₃	C₂²×C₄	1	4	yes	—
74	#41×C₂₃	C₂×Q₈	1	4	yes	—
75	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
76	C₂×D₄×C₂₃	—	2	2	no	—
77	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₂₃	—	2	1	no	—
79	C₂²×C₄×C₂₃	—	2	2	no	—
80	C₂²×C₄×C₂₃	—	2	4	no	—
81	C₂²×C₄×C₂₃	—	4	4	no	—
82	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
83	Q₁₆×C₂₃	C₄⋊C₄	1	4	yes	complemented
84	SD₁₆×C₂₃	C₄⋊C₄	1	8	yes	complemented
85	D₈×C₂₃	C₄⋊C₄	1	4	yes	complemented
86	M₁₆×C₂₃	—	2	2	no	—
87	M₁₆×C₂₃	—	2	8	no	—
88	M₁₆×C₂₃	—	2	2	no	—
89	M₁₆×C₂₃	—	2	2	no	—
90	M₁₆×C₂₃	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₂₃	—	2	2	no	—
92	C₂×C₈×C₂₃	—	2	8	no	—
93	C₂×C₈×C₂₃	—	2	2	no	—
94	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
101	C₂×C₈×C₂₃	#41	1	2	yes	—
102	C₄⋊C₄×C₂₃	—	2	4	no	—
103	C₄⋊C₄×C₂₃	—	2	4	no	—
104	C₄⋊C₄×C₂₃	—	4	4	no	—
105	C₄⋊C₄×C₂₃	—	8	4	no	—
106	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
107	C₂²⋊C₄×C₂₃	—	2	2	no	—
108	C₂²⋊C₄×C₂₃	—	4	4	no	—
109	C₄²×C₂₃	—	2	2	no	—
110	C₄²×C₂₃	—	2	1	no	—
111	C₄²×C₂₃	—	4	4	no	—
112	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
113	#22689	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂³×C₂₃	—	2	2	no	—
115	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
116	Q₈×C₂₃	—	2	2	no	—
117	Q₈×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
118	D₄×C₂₃	—	2	4	no	—
119	D₄×C₂₃	—	2	4	no	—
120	D₄×C₂₃	—	2	2	no	—
121	D₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
122	C₂×C₄×C₂₃	—	2	4	no	—
123	C₂×C₄×C₂₃	—	2	2	no	—
124	C₂×C₄×C₂₃	—	2	2	no	—
125	C₂×C₄×C₂₃	—	2	4	no	—
126	C₂×C₄×C₂₃	—	2	4	no	—
127	C₂×C₄×C₂₃	—	2	4	no	—
128	C₂×C₄×C₂₃	—	4	4	no	—
129	C₂×C₄×C₂₃	—	8	4	no	—
130	C₂×C₄×C₂₃	—	16	4	no	—
131	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
137	C₈×C₂₃	—	2	4	no	—
138	C₈×C₂₃	—	4	2	no	—
139	C₈×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₈×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
141	C₈×C₂₃	C₄×D₄	1	2	yes	—
142	C₈×C₂₃	C₄×D₄	1	2	yes	—
143	C₂×#443	C₂×C₂₃	1	1	yes	characteristic
144	#2952	C₂×C₂₃	1	2	yes	—
145	#2912	C₂×C₂₃	1	2	yes	complemented
146	#1951	C₂×C₂₃	1	2	yes	complemented
147	#1859	C₂×C₂₃	1	4	yes	complemented
148	#1844	C₂×C₂₃	1	1	yes	characteristic, complemented
149	#1815	C₂×C₂₃	1	2	yes	complemented
150	#1772	C₂×C₂₃	1	1	yes	characteristic, complemented
151	C₂²×C₂₃	—	2	2	no	—
152	C₂²×C₂₃	—	2	4	no	—
153	C₂²×C₂₃	—	4	4	no	—
154	C₂²×C₂₃	#520	1	1	yes	characteristic
155	C₂²×C₂₃	D₄²	1	1	yes	characteristic
156	C₂²×C₂₃	#554	1	1	yes	characteristic
157	C₄×C₂₃	—	2	4	no	—
158	C₄×C₂₃	—	2	4	no	—
159	C₄×C₂₃	—	4	4	no	—
160	C₄×C₂₃	—	8	4	no	—
161	C₄×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
162	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
163	C₄×C₂₃	#547	1	1	yes	characteristic
164	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
165	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
166	C₂×#131	C₂²×C₂₃	1	2	yes	—
167	#520	C₂²×C₂₃	1	4	yes	—
168	#443	C₄×C₂₃	1	8	yes	complemented
169	#442	—	2	2	no	—
170	#441	—	2	2	no	—
171	C₄×SD₁₆	—	2	4	no	—
172	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
173	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
174	#428	—	2	4	no	—
175	#428	C₂²×C₂₃	1	4	yes	—
176	#427	—	2	2	no	—
177	#427	C₂²×C₂₃	1	1	yes	characteristic
178	C₂×#106	—	2	2	no	—
179	C₂×#106	C₂²×C₂₃	1	2	yes	—
180	C₂×#106	C₂²×C₂₃	1	2	yes	—
181	C₂×#104	C₂²×C₂₃	1	2	yes	—
182	#417	C₂²×C₂₃	1	4	yes	—
183	#405	C₂²×C₂₃	1	2	yes	—
184	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
185	#384	C₂²×C₂₃	1	2	yes	—
186	#343	C₂²×C₂₃	1	4	yes	—
187	#339	C₂²×C₂₃	1	2	yes	—
188	C₂×C₂₃	—	2	1	no	—
189	C₂×C₂₃	—	2	1	no	—
190	C₂×C₂₃	—	4	4	no	—
191	C₂×C₂₃	D₄×C₄⋊C₄	1	1	yes	characteristic
192	C₂×C₂₃	#3291	1	2	yes	—
193	C₂×#41	C₂³×C₂₃	1	2	yes	—
194	#135	C₂×C₄×C₂₃	1	4	yes	—
195	#135	Q₈×C₂₃	1	4	yes	—
196	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
197	C₂×SD₁₆	D₄×C₂₃	1	2	yes	—
198	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
199	#131	—	2	4	no	—
200	#131	C₂×C₄×C₂₃	1	8	yes	complemented
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
203	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
207	C₄×Q₈	—	2	4	no	—
208	C₄×D₄	—	2	4	no	—
209	C₄×D₄	—	4	4	no	—
210	#117	—	2	4	no	—
211	#117	C₂³×C₂₃	1	2	yes	—
212	C₂×C₄⋊C₄	—	2	4	no	—
213	C₂×C₄⋊C₄	—	4	2	no	—
214	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
215	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
216	#108	—	2	2	no	—
217	#108	C₂×C₄×C₂₃	1	4	yes	complemented
218	#107	—	2	4	no	—
219	#107	D₄×C₂₃	1	2	yes	—
220	#106	—	2	2	no	—
221	#106	—	2	2	no	—
222	#106	—	2	2	no	—
223	#106	—	2	2	no	—
224	#106	—	2	4	no	—
225	#106	D₄×C₂₃	1	2	yes	—
226	#104	C₂×C₄×C₂₃	1	8	yes	complemented
227	#103	—	2	4	no	—
228	#102	—	2	4	no	—
229	#97	—	2	2	no	—
230	C₄×C₈	—	2	2	no	—
231	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	complemented
232	C₂₃	#22689	1	1	yes	characteristic, direct product factor, Sylow
233	#41	—	2	2	no	—
234	#41	—	2	2	no	—
235	#41	C₂²×C₄×C₂₃	1	4	yes	—
236	#41	C₂×Q₈×C₂₃	1	4	yes	—
237	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
245	Q₁₆	C₄⋊C₄×C₂₃	1	4	yes	complemented
246	SD₁₆	C₄⋊C₄×C₂₃	1	8	yes	complemented
247	D₈	C₄⋊C₄×C₂₃	1	4	yes	complemented
248	M₁₆	—	2	2	no	—
249	M₁₆	—	2	8	no	—
250	M₁₆	—	2	2	no	—
251	M₁₆	—	2	2	no	—
252	M₁₆	C₂²×C₄×C₂₃	1	4	yes	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	8	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
257	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
262	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
263	C₂×C₈	#41×C₂₃	1	2	yes	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₄⋊C₄	—	8	4	no	—
268	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₂²⋊C₄	—	4	4	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	4	4	no	—
274	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
275	C₂³	—	2	2	no	—
276	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
277	Q₈	—	2	2	no	—
278	Q₈	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
279	D₄	—	2	4	no	—
280	D₄	—	2	4	no	—
281	D₄	—	2	2	no	—
282	D₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	8	4	no	—
291	C₂×C₄	—	16	4	no	—
292	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
295	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
296	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
297	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
298	C₈	—	2	4	no	—
299	C₈	—	4	2	no	—
300	C₈	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
301	C₈	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
302	C₈	C₄×D₄×C₂₃	1	2	yes	—
303	C₈	C₄×D₄×C₂₃	1	2	yes	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	4	4	no	—
307	C₂²	C₂₃×#520	1	1	yes	characteristic
308	C₂²	D₄²×C₂₃	1	1	yes	characteristic
309	C₂²	C₂₃×#554	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	2	4	no	—
312	C₄	—	4	4	no	—
313	C₄	—	8	4	no	—
314	C₄	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
315	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
316	C₄	C₂₃×#547	1	1	yes	characteristic
317	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	4	4	no	—
321	C₂	D₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
322	C₂	C₂₃×#3291	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2912	C₂	semidirect
4. C₂₃×#1951	C₂	semidirect
5. C₂₃×#1859	C₂	semidirect
6. C₂₃×#1844	C₂	semidirect
7. C₂₃×#1815	C₂	semidirect
8. C₂₃×#1772	C₂	semidirect
12. C₂₃×#443	C₄	semidirect
39. C₂₃×#131	C₂×C₄	semidirect
56. C₂₃×#108	C₂×C₄	semidirect
65. C₂₃×#104	C₂×C₄	semidirect
70. C₄×C₈×C₂₃	C₂×C₄	semidirect
83. Q₁₆×C₂₃	C₄⋊C₄	semidirect
84. SD₁₆×C₂₃	C₄⋊C₄	semidirect
85. D₈×C₂₃	C₄⋊C₄	semidirect
113. #22689	C₂₃	direct (N × H)
145. #2912	C₂×C₂₃	semidirect
146. #1951	C₂×C₂₃	semidirect
147. #1859	C₂×C₂₃	semidirect
148. #1844	C₂×C₂₃	semidirect
149. #1815	C₂×C₂₃	semidirect
150. #1772	C₂×C₂₃	semidirect
168. #443	C₄×C₂₃	semidirect
200. #131	C₂×C₄×C₂₃	semidirect
217. #108	C₂×C₄×C₂₃	semidirect
226. #104	C₂×C₄×C₂₃	semidirect
231. C₄×C₈	C₂×C₄×C₂₃	semidirect
232. C₂₃	#22689	direct (N × H)
245. Q₁₆	C₄⋊C₄×C₂₃	semidirect
246. SD₁₆	C₄⋊C₄×C₂₃	semidirect
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×C₂₃×#443	2
C₂₃×#2952	2
C₂₃×#2912	2
C₂₃×#1951	2
C₂₃×#1859	2
C₂₃×#1844	2
C₂₃×#1815	2
C₂₃×#1772	2
#22689	23

Subgroups — C23×#22689 #19170315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22689 #19170315