Subgroups — C₂₃×#17970 #19165596

Order 5888.

640 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	C₂₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2981	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1594	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1589	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1580	C₂	1	2	yes	maximal, complemented
6	C₈×D₈×C₂₃	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1496	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#532	C₂²	1	2	yes	—
11	C₂₃×#500	C₄	1	1	yes	characteristic
12	C₂₃×#493	C₄	1	1	yes	characteristic
13	C₂₃×#486	C₄	1	2	yes	—
14	C₂₃×#466	C₄	1	2	yes	—
15	C₂₃×#466	C₄	1	2	yes	—
16	C₄×D₈×C₂₃	C₂²	1	2	yes	—
17	C₄×D₈×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#436	C₂²	1	4	yes	—
19	C₂₃×#435	—	2	4	no	—
20	C₈×D₄×C₂₃	C₂²	1	4	yes	—
21	C₂₃×#432	C₂²	1	2	yes	—
22	C₂₃×#431	C₂²	1	1	yes	characteristic
23	C₂₃×#431	C₂²	1	1	yes	characteristic
24	C₂₃×#408	—	2	4	no	—
25	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#335	C₂²	1	2	yes	—
27	C₂₃×#334	C₂²	1	2	yes	—
28	C₂₃×D₄⋊C₈	C₂²	1	4	yes	—
29	C₂₃×D₄⋊C₈	C₂²	1	4	yes	—
30	C₂₃×#322	C₂²	1	1	yes	characteristic
31	C₂₃×#322	C₂²	1	1	yes	characteristic
32	C₈²×C₂₃	C₂²	1	1	yes	characteristic
33	C₈²×C₂₃	C₂²	1	1	yes	characteristic
34	C₂×#41×C₂₃	—	2	2	no	—
35	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
36	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₂₃	—	2	4	no	—
38	C₂²×C₈×C₂₃	—	2	4	no	—
39	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
42	C₂₃×#127	C₂×C₄	1	2	yes	—
43	C₂₃×#124	—	2	2	no	—
44	C₂₃×#121	—	2	2	no	—
45	C₂₃×#121	C₂×C₄	1	4	yes	—
46	C₄×D₄×C₂₃	—	2	2	no	—
47	C₄×D₄×C₂₃	C₂³	1	4	yes	—
48	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
49	C₂₃×#107	C₂×C₄	1	2	yes	—
50	C₂₃×#107	C₂×C₄	1	2	yes	—
51	C₂₃×C₄⋊C₈	—	2	1	no	—
52	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
53	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
54	C₂₃×#102	C₂×C₄	1	4	yes	—
55	C₂₃×#102	C₂×C₄	1	4	yes	—
56	C₂₃×C₂²⋊C₈	—	2	2	no	—
57	C₂₃×C₂²⋊C₈	—	2	1	no	—
58	C₂₃×C₂²⋊C₈	—	2	4	no	—
59	C₂₃×C₂²⋊C₈	—	2	4	no	—
60	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
61	C₂₃×#97	—	2	2	no	—
62	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
70	#41×C₂₃	—	8	4	no	—
71	C₂×Q₈×C₂₃	—	2	2	no	—
72	C₂×D₄×C₂₃	—	2	4	no	—
73	C₂×D₄×C₂₃	—	2	2	no	—
74	C₂×D₄×C₂₃	—	2	2	no	—
75	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₂₃	—	2	4	no	—
77	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
80	D₈×C₂₃	—	2	4	no	—
81	D₈×C₂₃	—	2	4	no	—
82	M₁₆×C₂₃	—	8	4	no	—
83	C₂×C₈×C₂₃	—	2	2	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	—	2	4	no	—
86	C₂×C₈×C₂₃	—	2	1	no	—
87	C₂×C₈×C₂₃	—	2	1	no	—
88	C₂×C₈×C₂₃	—	8	4	no	—
89	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
95	C₄⋊C₄×C₂₃	—	2	2	no	—
96	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
97	C₂²⋊C₄×C₂₃	—	2	4	no	—
98	C₂²⋊C₄×C₂₃	—	2	4	no	—
99	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₂₃	#41	1	1	yes	characteristic
102	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
103	#17970	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₂₃	—	2	4	no	—
105	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
106	Q₈×C₂₃	—	8	2	no	—
107	D₄×C₂₃	—	2	4	no	—
108	D₄×C₂₃	—	2	4	no	—
109	D₄×C₂₃	—	8	2	no	—
110	D₄×C₂₃	—	8	2	no	—
111	D₄×C₂₃	—	8	4	no	—
112	C₂×C₄×C₂₃	—	2	1	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	2	no	—
115	C₂×C₄×C₂₃	—	2	4	no	—
116	C₂×C₄×C₂₃	—	4	1	no	—
117	C₂×C₄×C₂₃	—	4	4	no	—
118	C₂×C₄×C₂₃	—	4	4	no	—
119	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
125	C₈×C₂₃	—	2	1	no	—
126	C₈×C₂₃	—	2	1	no	—
127	C₈×C₂₃	—	2	2	no	—
128	C₈×C₂₃	—	2	2	no	—
129	C₈×C₂₃	—	4	1	no	—
130	C₈×C₂₃	—	4	1	no	—
131	C₈×C₂₃	—	4	4	no	—
132	C₈×C₂₃	#131	1	2	yes	—
133	C₈×C₂₃	#131	1	2	yes	—
134	#3169	C₂×C₂₃	1	1	yes	characteristic
135	#2981	C₂×C₂₃	1	2	yes	complemented
136	#1594	C₂×C₂₃	1	2	yes	complemented
137	#1589	C₂×C₂₃	1	2	yes	complemented
138	#1580	C₂×C₂₃	1	2	yes	complemented
139	C₈×D₈	C₂×C₂₃	1	2	yes	complemented
140	#1568	C₂×C₂₃	1	1	yes	characteristic, complemented
141	#1496	C₂×C₂₃	1	2	yes	complemented
142	#1450	C₂×C₂₃	1	1	yes	characteristic, complemented
143	C₂²×C₂₃	—	2	1	no	—
144	C₂²×C₂₃	—	2	1	no	—
145	C₂²×C₂₃	—	2	1	no	—
146	C₂²×C₂₃	—	4	4	no	—
147	C₂²×C₂₃	—	4	4	no	—
148	C₂²×C₂₃	—	4	4	no	—
149	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
150	C₄×C₂₃	—	2	1	no	—
151	C₄×C₂₃	—	2	1	no	—
152	C₄×C₂₃	—	2	1	no	—
153	C₄×C₂₃	—	2	1	no	—
154	C₄×C₂₃	—	4	4	no	—
155	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
156	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
157	C₄×C₂₃	#576	1	2	yes	—
158	#532	C₂²×C₂₃	1	2	yes	—
159	#500	C₄×C₂₃	1	1	yes	characteristic
160	#493	C₄×C₂₃	1	1	yes	characteristic
161	#486	C₄×C₂₃	1	2	yes	—
162	#466	C₄×C₂₃	1	2	yes	—
163	#466	C₄×C₂₃	1	2	yes	—
164	C₄×D₈	C₂²×C₂₃	1	2	yes	—
165	C₄×D₈	C₂²×C₂₃	1	2	yes	—
166	#436	C₂²×C₂₃	1	4	yes	—
167	#435	—	2	4	no	—
168	C₈×D₄	C₂²×C₂₃	1	4	yes	—
169	#432	C₂²×C₂₃	1	2	yes	—
170	#431	C₂²×C₂₃	1	1	yes	characteristic
171	#431	C₂²×C₂₃	1	1	yes	characteristic
172	#408	—	2	4	no	—
173	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
174	#335	C₂²×C₂₃	1	2	yes	—
175	#334	C₂²×C₂₃	1	2	yes	—
176	D₄⋊C₈	C₂²×C₂₃	1	4	yes	—
177	D₄⋊C₈	C₂²×C₂₃	1	4	yes	—
178	#322	C₂²×C₂₃	1	1	yes	characteristic
179	#322	C₂²×C₂₃	1	1	yes	characteristic
180	C₈²	C₂²×C₂₃	1	1	yes	characteristic
181	C₈²	C₂²×C₂₃	1	1	yes	characteristic
182	C₂×C₂₃	—	4	1	no	—
183	C₂×C₂₃	—	8	4	no	—
184	C₂×C₂₃	#2931	1	1	yes	characteristic
185	C₂×C₂₃	#2948	1	1	yes	characteristic
186	C₂×C₂₃	#2957	1	1	yes	characteristic
187	C₂×#41	—	2	2	no	—
188	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
189	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
190	C₂×M₁₆	—	2	4	no	—
191	C₂²×C₈	—	2	4	no	—
192	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
193	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
194	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
195	#127	C₂×C₄×C₂₃	1	2	yes	—
196	#124	—	2	2	no	—
197	#121	—	2	2	no	—
198	#121	C₂×C₄×C₂₃	1	4	yes	—
199	C₄×D₄	—	2	2	no	—
200	C₄×D₄	C₂³×C₂₃	1	4	yes	—
201	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
202	#107	C₂×C₄×C₂₃	1	2	yes	—
203	#107	C₂×C₄×C₂₃	1	2	yes	—
204	C₄⋊C₈	—	2	1	no	—
205	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
206	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
207	#102	C₂×C₄×C₂₃	1	4	yes	—
208	#102	C₂×C₄×C₂₃	1	4	yes	—
209	C₂²⋊C₈	—	2	2	no	—
210	C₂²⋊C₈	—	2	1	no	—
211	C₂²⋊C₈	—	2	4	no	—
212	C₂²⋊C₈	—	2	4	no	—
213	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
214	#97	—	2	2	no	—
215	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
216	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
217	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
218	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
219	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
220	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
221	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
222	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
223	C₂₃	#17970	1	1	yes	characteristic, direct product factor, Sylow
224	#41	—	8	4	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
233	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
234	D₈	—	2	4	no	—
235	D₈	—	2	4	no	—
236	M₁₆	—	8	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	8	4	no	—
243	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
244	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
245	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
246	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
247	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
248	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
254	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
255	C₄²	#41×C₂₃	1	1	yes	characteristic
256	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
257	C₂³	—	2	4	no	—
258	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
259	Q₈	—	8	2	no	—
260	D₄	—	2	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	8	2	no	—
263	D₄	—	8	2	no	—
264	D₄	—	8	4	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
273	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
274	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
275	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
277	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	2	no	—
281	C₈	—	2	2	no	—
282	C₈	—	4	1	no	—
283	C₈	—	4	1	no	—
284	C₈	—	4	4	no	—
285	C₈	C₂₃×#131	1	2	yes	—
286	C₈	C₂₃×#131	1	2	yes	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	4	no	—
292	C₂²	—	4	4	no	—
293	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	4	no	—
299	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
300	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
301	C₄	C₂₃×#576	1	2	yes	—
302	C₂	—	4	1	no	—
303	C₂	—	8	4	no	—
304	C₂	C₂₃×#2931	1	1	yes	characteristic
305	C₂	C₂₃×#2948	1	1	yes	characteristic
306	C₂	C₂₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂₃×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2981	C₂	semidirect
3. C₂₃×#1594	C₂	semidirect
4. C₂₃×#1589	C₂	semidirect
5. C₂₃×#1580	C₂	semidirect
6. C₈×D₈×C₂₃	C₂	semidirect
7. C₂₃×#1568	C₂	semidirect
8. C₂₃×#1496	C₂	semidirect
9. C₂₃×#1450	C₂	semidirect
103. #17970	C₂₃	direct (N × H)
135. #2981	C₂×C₂₃	semidirect
136. #1594	C₂×C₂₃	semidirect
137. #1589	C₂×C₂₃	semidirect
138. #1580	C₂×C₂₃	semidirect
139. C₈×D₈	C₂×C₂₃	semidirect
140. #1568	C₂×C₂₃	semidirect
141. #1496	C₂×C₂₃	semidirect
142. #1450	C₂×C₂₃	semidirect
223. C₂₃	#17970	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×#3169	2
C₂₃×#2981	2
C₂₃×#1594	2
C₂₃×#1589	2
C₂₃×#1580	2
C₈×D₈×C₂₃	2
C₂₃×#1568	2
C₂₃×#1496	2
C₂₃×#1450	2
#17970	23

Subgroups — C23×#17970 #19165596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17970 #19165596