Subgroups — C₂₃×#21282 #19168908

Order 5888.

862 conjugacy classes of subgroups.

The subgroup lattice has 308 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	C₂₃×#2319	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1970	C₂	1	2	yes	maximal
4	C₂₃×#1942	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1936	C₂	1	4	yes	maximal, complemented
6	C₄×C₂₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1762	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
11	C₄×D₈×C₂₃	C₄	1	8	yes	complemented
12	C₂×C₂₃×#107	C₂²	1	2	yes	—
13	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#102	C₂²	1	4	yes	—
15	C₂×C₂₃×#102	C₂²	1	2	yes	—
16	C₂×C₂₃×#97	C₂²	1	2	yes	—
17	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#390	C₂²	1	4	yes	—
21	C₂₃×#389	C₂²	1	4	yes	—
22	C₂₃×#388	—	2	4	no	—
23	C₂₃×#381	—	2	2	no	—
24	C₂₃×#380	—	2	2	no	—
25	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₂₃×#376	C₂²	1	1	yes	characteristic
27	C₂₃×#340	C₂²	1	2	yes	—
28	C₂₃×#340	C₂²	1	4	yes	—
29	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
30	C₂³×C₄×C₂₃	—	2	2	no	—
31	C₂×D₈×C₂₃	—	2	4	no	—
32	C₂×D₈×C₂₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
36	C₄×D₄×C₂₃	—	2	4	no	—
37	C₄×D₄×C₂₃	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
41	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
43	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
44	C₂×C₄²×C₂₃	—	2	2	no	—
45	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
47	C₂₃×#107	—	2	4	no	—
48	C₂₃×#107	C₂×C₄	1	4	yes	—
49	C₂₃×#102	—	2	8	no	—
50	C₂₃×#102	C₂×C₄	1	8	yes	—
51	C₂₃×#97	—	2	4	no	—
52	C₂₃×#97	—	2	2	no	—
53	C₄×C₈×C₂₃	—	2	4	no	—
54	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
55	C₂₃×#95	—	2	2	no	—
56	C₂₃×#95	—	2	4	no	—
57	C₂₃×#95	D₄	1	1	yes	characteristic
58	C₂₃×#95	D₄	1	1	yes	characteristic
59	C₂₃×#95	D₄	1	2	yes	—
60	C₂⁴×C₂₃	—	2	2	no	—
61	C₂×D₄×C₂₃	—	2	4	no	—
62	C₂×D₄×C₂₃	—	2	4	no	—
63	C₂×D₄×C₂₃	—	2	2	no	—
64	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₂₃	—	2	2	no	—
66	C₂²×C₄×C₂₃	—	2	8	no	—
67	C₂²×C₄×C₂₃	—	2	4	no	—
68	C₂²×C₄×C₂₃	—	8	4	no	—
69	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
74	D₈×C₂₃	—	2	8	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₂₃	#41	1	4	yes	—
81	C₂×C₈×C₂₃	#41	1	4	yes	—
82	C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₄⋊C₄×C₂₃	—	2	4	no	—
84	C₄⋊C₄×C₂₃	—	2	4	no	—
85	C₄⋊C₄×C₂₃	—	8	4	no	—
86	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
87	C₂²⋊C₄×C₂₃	—	2	8	no	—
88	C₂²⋊C₄×C₂₃	—	8	4	no	—
89	C₂²⋊C₄×C₂₃	—	8	4	no	—
90	C₄²×C₂₃	—	2	2	no	—
91	C₄²×C₂₃	—	2	2	no	—
92	C₄²×C₂₃	—	8	2	no	—
93	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₂₃	#41	1	2	yes	—
95	#21282	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₂₃	—	2	4	no	—
97	C₂³×C₂₃	—	4	4	no	—
98	C₂³×C₂₃	—	4	2	no	—
99	C₂³×C₂₃	—	4	4	no	—
100	C₂³×C₂₃	—	4	2	no	—
101	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
102	D₄×C₂₃	—	2	4	no	—
103	D₄×C₂₃	—	2	4	no	—
104	D₄×C₂₃	—	4	4	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	2	1	no	—
108	C₂×C₄×C₂₃	—	2	4	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	—	2	2	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	2	2	no	—
115	C₂×C₄×C₂₃	—	4	4	no	—
116	C₂×C₄×C₂₃	—	4	4	no	—
117	C₂×C₄×C₂₃	—	4	4	no	—
118	C₂×C₄×C₂₃	—	4	4	no	—
119	C₂×C₄×C₂₃	—	8	8	no	—
120	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
124	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
125	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
126	C₈×C₂₃	—	2	2	no	—
127	C₈×C₂₃	—	2	4	no	—
128	C₈×C₂₃	#117	1	4	yes	—
129	C₂×C₄×D₈	C₂×C₂₃	1	1	yes	characteristic
130	#2319	C₂×C₂₃	1	2	yes	complemented
131	#1970	C₂×C₂₃	1	2	yes	—
132	#1942	C₂×C₂₃	1	2	yes	complemented
133	#1936	C₂×C₂₃	1	4	yes	complemented
134	C₄×#107	C₂×C₂₃	1	1	yes	characteristic, complemented
135	#1762	C₂×C₂₃	1	2	yes	complemented
136	#1726	C₂×C₂₃	1	1	yes	characteristic, complemented
137	C₂²×C₂₃	—	4	4	no	—
138	C₂²×C₂₃	—	4	4	no	—
139	C₂²×C₂₃	—	4	4	no	—
140	C₂²×C₂₃	—	8	4	no	—
141	C₂²×C₂₃	—	8	4	no	—
142	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂²×C₂₃	#532	1	2	yes	—
144	C₂²×C₂₃	#576	1	2	yes	—
145	C₂²×C₂₃	#576	1	1	yes	characteristic
146	C₂²×C₂₃	#578	1	1	yes	characteristic
147	C₄×C₂₃	—	2	2	no	—
148	C₄×C₂₃	—	2	2	no	—
149	C₄×C₂₃	—	4	2	no	—
150	C₄×C₂₃	—	4	2	no	—
151	C₄×C₂₃	—	4	4	no	—
152	C₄×C₂₃	—	8	4	no	—
153	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
154	C₄×C₂₃	C₄×#41	1	1	yes	characteristic
155	C₄×C₂₃	#529	1	2	yes	—
156	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
157	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
158	C₄×D₈	C₄×C₂₃	1	8	yes	complemented
159	C₂×#107	C₂²×C₂₃	1	2	yes	—
160	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
161	C₂×#102	C₂²×C₂₃	1	4	yes	—
162	C₂×#102	C₂²×C₂₃	1	2	yes	—
163	C₂×#97	C₂²×C₂₃	1	2	yes	—
164	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
166	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
167	#390	C₂²×C₂₃	1	4	yes	—
168	#389	C₂²×C₂₃	1	4	yes	—
169	#388	—	2	4	no	—
170	#381	—	2	2	no	—
171	#380	—	2	2	no	—
172	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
173	#376	C₂²×C₂₃	1	1	yes	characteristic
174	#340	C₂²×C₂₃	1	2	yes	—
175	#340	C₂²×C₂₃	1	4	yes	—
176	C₂×C₂₃	—	8	4	no	—
177	C₂×C₂₃	#2293	1	1	yes	characteristic
178	C₂×C₂₃	#2948	1	1	yes	characteristic
179	C₂×C₂₃	#2951	1	1	yes	characteristic
180	C₂×C₂₃	#3175	1	2	yes	—
181	C₂×C₂₃	#3176	1	2	yes	—
182	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
183	C₂³×C₄	—	2	2	no	—
184	C₂×D₈	—	2	4	no	—
185	C₂×D₈	C₂×C₄×C₂₃	1	4	yes	—
186	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
188	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
189	C₄×D₄	—	2	4	no	—
190	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
193	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
194	C₂×C₂²⋊C₄	—	2	2	no	—
195	C₂×C₂²⋊C₄	—	2	4	no	—
196	C₂×C₂²⋊C₄	—	2	4	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
199	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
200	#107	—	2	4	no	—
201	#107	C₂×C₄×C₂₃	1	4	yes	—
202	#102	—	2	8	no	—
203	#102	C₂×C₄×C₂₃	1	8	yes	—
204	#97	—	2	4	no	—
205	#97	—	2	2	no	—
206	C₄×C₈	—	2	4	no	—
207	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
208	#95	—	2	2	no	—
209	#95	—	2	4	no	—
210	#95	D₄×C₂₃	1	1	yes	characteristic
211	#95	D₄×C₂₃	1	1	yes	characteristic
212	#95	D₄×C₂₃	1	2	yes	—
213	C₂₃	#21282	1	1	yes	characteristic, direct product factor, Sylow
214	C₂⁴	—	2	2	no	—
215	C₂×D₄	—	2	4	no	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	8	4	no	—
223	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
227	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
228	D₈	—	2	8	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
233	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
234	C₂×C₈	#41×C₂₃	1	4	yes	—
235	C₂×C₈	#41×C₂₃	1	4	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	8	4	no	—
240	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₂²⋊C₄	—	8	4	no	—
243	C₂²⋊C₄	—	8	4	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	8	2	no	—
247	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
248	C₄²	#41×C₂₃	1	2	yes	—
249	C₂³	—	2	4	no	—
250	C₂³	—	4	2	no	—
251	C₂³	—	4	4	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	4	no	—
254	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
255	D₄	—	2	4	no	—
256	D₄	—	2	4	no	—
257	D₄	—	4	4	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	8	8	no	—
273	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
274	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
275	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
276	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
277	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
278	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	2	4	no	—
281	C₈	C₂₃×#117	1	4	yes	—
282	C₂²	—	4	4	no	—
283	C₂²	—	4	4	no	—
284	C₂²	—	4	4	no	—
285	C₂²	—	8	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
288	C₂²	C₂₃×#532	1	2	yes	—
289	C₂²	C₂₃×#576	1	2	yes	—
290	C₂²	C₂₃×#576	1	1	yes	characteristic
291	C₂²	C₂₃×#578	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	2	no	—
295	C₄	—	4	2	no	—
296	C₄	—	4	4	no	—
297	C₄	—	8	4	no	—
298	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
299	C₄	C₄×#41×C₂₃	1	1	yes	characteristic
300	C₄	C₂₃×#529	1	2	yes	—
301	C₂	—	8	4	no	—
302	C₂	C₂₃×#2293	1	1	yes	characteristic
303	C₂	C₂₃×#2948	1	1	yes	characteristic
304	C₂	C₂₃×#2951	1	1	yes	characteristic
305	C₂	C₂₃×#3175	1	2	yes	—
306	C₂	C₂₃×#3176	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	C₂²×D₈×C₂₃	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
2. C₂₃×#2319	C₂	semidirect
4. C₂₃×#1942	C₂	semidirect
5. C₂₃×#1936	C₂	semidirect
6. C₄×C₂₃×#107	C₂	semidirect
7. C₂₃×#1762	C₂	semidirect
8. C₂₃×#1726	C₂	semidirect
11. C₄×D₈×C₂₃	C₄	semidirect
95. #21282	C₂₃	direct (N × H)
130. #2319	C₂×C₂₃	semidirect
132. #1942	C₂×C₂₃	semidirect
133. #1936	C₂×C₂₃	semidirect
134. C₄×#107	C₂×C₂₃	semidirect
135. #1762	C₂×C₂₃	semidirect
136. #1726	C₂×C₂₃	semidirect
158. C₄×D₈	C₄×C₂₃	semidirect
213. C₂₃	#21282	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×C₄×D₈×C₂₃	2
C₂₃×#2319	2
C₂₃×#1970	2
C₂₃×#1942	2
C₂₃×#1936	2
C₄×C₂₃×#107	2
C₂₃×#1762	2
C₂₃×#1726	2
#21282	23

Subgroups — C23×#21282 #19168908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21282 #19168908