Subgroups — C₂₃×#17956 #19165582

Order 5888.

750 conjugacy classes of subgroups.

The subgroup lattice has 290 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×#2931	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1589	C₂	1	4	yes	maximal, complemented
4	C₈×D₈×C₂₃	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1492	C₂	1	2	yes	maximal, complemented
7	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#131	—	2	2	no	—
9	C₂₃×#532	C₂²	1	2	yes	—
10	C₂₃×#500	C₄	1	1	yes	characteristic
11	C₂₃×#493	C₄	1	1	yes	characteristic
12	C₂₃×#486	C₄	1	2	yes	—
13	C₂₃×#466	C₄	1	4	yes	—
14	C₄×D₈×C₂₃	C₂²	1	4	yes	—
15	C₂₃×#436	—	2	2	no	—
16	C₂₃×#436	C₂²	1	4	yes	—
17	C₈×D₄×C₂₃	—	2	2	no	—
18	C₈×D₄×C₂₃	C₂²	1	4	yes	—
19	C₂₃×#423	C₂²	1	2	yes	—
20	C₂₃×#408	—	2	2	no	—
21	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#335	C₂²	1	4	yes	—
25	C₂₃×D₄⋊C₈	C₂²	1	8	yes	—
26	C₈²×C₂₃	C₂²	1	2	yes	—
27	C₈²×C₂₃	C₂²	1	2	yes	—
28	C₂×#41×C₂₃	—	2	2	no	—
29	C₂×D₈×C₂₃	C₂×C₄	1	4	yes	—
30	C₂₃×#131	—	4	8	no	—
31	C₂×M₁₆×C₂₃	—	2	2	no	—
32	C₂×M₁₆×C₂₃	—	2	4	no	—
33	C₂²×C₈×C₂₃	—	2	1	no	—
34	C₂²×C₈×C₂₃	—	2	4	no	—
35	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
38	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₂₃×#127	C₂×C₄	1	2	yes	—
40	C₂₃×#124	—	2	2	no	—
41	C₂₃×#121	—	2	2	no	—
42	C₂₃×#121	C₂×C₄	1	4	yes	—
43	C₄×D₄×C₂₃	—	2	2	no	—
44	C₄×D₄×C₂₃	C₂³	1	4	yes	—
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂₃×#107	C₂×C₄	1	4	yes	—
47	C₂₃×C₄⋊C₈	—	2	2	no	—
48	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
49	C₂₃×#102	C₂×C₄	1	8	yes	—
50	C₂₃×C₂²⋊C₈	—	2	4	no	—
51	C₂₃×C₂²⋊C₈	—	2	4	no	—
52	C₄×C₈×C₂₃	—	2	1	no	—
53	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
56	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₂₃	C₂³	1	2	yes	—
58	C₄×C₈×C₂₃	C₂³	1	2	yes	—
59	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
61	#41×C₂₃	—	4	4	no	—
62	#41×C₂₃	—	4	4	no	—
63	C₂×Q₈×C₂₃	—	2	2	no	—
64	C₂×D₄×C₂₃	—	2	4	no	—
65	C₂×D₄×C₂₃	—	2	2	no	—
66	C₂×D₄×C₂₃	—	2	2	no	—
67	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₂₃	—	2	4	no	—
69	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
72	D₈×C₂₃	—	2	8	no	—
73	M₁₆×C₂₃	—	4	4	no	—
74	M₁₆×C₂₃	—	4	8	no	—
75	C₂×C₈×C₂₃	—	2	4	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	—	2	4	no	—
79	C₂×C₈×C₂₃	—	2	4	no	—
80	C₂×C₈×C₂₃	—	4	8	no	—
81	C₂×C₈×C₂₃	D₈	1	4	yes	complemented
82	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
90	C₂²⋊C₄×C₂₃	—	2	4	no	—
91	C₂²⋊C₄×C₂₃	—	2	4	no	—
92	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₂₃	#41	1	1	yes	characteristic
95	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
96	#17956	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₂₃	—	2	4	no	—
98	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
99	Q₈×C₂₃	—	4	4	no	—
100	D₄×C₂₃	—	2	8	no	—
101	D₄×C₂₃	—	4	8	no	—
102	D₄×C₂₃	—	4	4	no	—
103	D₄×C₂₃	—	4	4	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	2	4	no	—
107	C₂×C₄×C₂₃	—	4	4	no	—
108	C₂×C₄×C₂₃	—	4	4	no	—
109	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₂₃	#135	1	2	yes	—
114	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
117	C₈×C₂₃	—	2	4	no	—
118	C₈×C₂₃	—	2	2	no	—
119	C₈×C₂₃	—	2	1	no	—
120	C₈×C₂₃	—	2	1	no	—
121	C₈×C₂₃	—	4	4	no	—
122	C₈×C₂₃	#131	1	4	yes	complemented
123	C₈×C₂₃	C₂×D₈	1	2	yes	—
124	C₈×C₂₃	C₂×D₈	1	2	yes	—
125	#3169	C₂×C₂₃	1	1	yes	characteristic
126	#2931	C₂×C₂₃	1	2	yes	complemented
127	#1589	C₂×C₂₃	1	4	yes	complemented
128	C₈×D₈	C₂×C₂₃	1	4	yes	complemented
129	#1567	C₂×C₂₃	1	1	yes	characteristic, complemented
130	#1492	C₂×C₂₃	1	2	yes	complemented
131	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic, complemented
132	C₂²×C₂₃	—	2	1	no	—
133	C₂²×C₂₃	—	2	1	no	—
134	C₂²×C₂₃	—	4	4	no	—
135	C₂²×C₂₃	—	4	4	no	—
136	C₂²×C₂₃	—	4	4	no	—
137	C₂²×C₂₃	C₄×D₈	1	2	yes	—
138	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
139	C₄×C₂₃	—	2	1	no	—
140	C₄×C₂₃	—	2	1	no	—
141	C₄×C₂₃	—	4	4	no	—
142	C₄×C₂₃	C₄×D₈	1	2	yes	—
143	C₄×C₂₃	#443	1	2	yes	—
144	C₄×C₂₃	C₂×#131	1	2	yes	—
145	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
146	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
147	C₂×#131	—	2	2	no	—
148	#532	C₂²×C₂₃	1	2	yes	—
149	#500	C₄×C₂₃	1	1	yes	characteristic
150	#493	C₄×C₂₃	1	1	yes	characteristic
151	#486	C₄×C₂₃	1	2	yes	—
152	#466	C₄×C₂₃	1	4	yes	—
153	C₄×D₈	C₂²×C₂₃	1	4	yes	—
154	#436	—	2	2	no	—
155	#436	C₂²×C₂₃	1	4	yes	—
156	C₈×D₄	—	2	2	no	—
157	C₈×D₄	C₂²×C₂₃	1	4	yes	—
158	#423	C₂²×C₂₃	1	2	yes	—
159	#408	—	2	2	no	—
160	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
163	#335	C₂²×C₂₃	1	4	yes	—
164	D₄⋊C₈	C₂²×C₂₃	1	8	yes	—
165	C₈²	C₂²×C₂₃	1	2	yes	—
166	C₈²	C₂²×C₂₃	1	2	yes	—
167	C₂×C₂₃	—	2	2	no	—
168	C₂×C₂₃	—	8	4	no	—
169	C₂×C₂₃	#2931	1	1	yes	characteristic
170	C₂×C₂₃	C₂×C₄×D₈	1	1	yes	characteristic
171	C₂×C₂₃	C₂×#443	1	1	yes	characteristic
172	C₂×#41	—	2	2	no	—
173	C₂×D₈	C₂×C₄×C₂₃	1	4	yes	—
174	#131	—	4	8	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	4	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	2	4	no	—
179	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
180	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
181	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
182	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
183	#127	C₂×C₄×C₂₃	1	2	yes	—
184	#124	—	2	2	no	—
185	#121	—	2	2	no	—
186	#121	C₂×C₄×C₂₃	1	4	yes	—
187	C₄×D₄	—	2	2	no	—
188	C₄×D₄	C₂³×C₂₃	1	4	yes	—
189	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
190	#107	C₂×C₄×C₂₃	1	4	yes	—
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
193	#102	C₂×C₄×C₂₃	1	8	yes	—
194	C₂²⋊C₈	—	2	4	no	—
195	C₂²⋊C₈	—	2	4	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
198	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
199	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
200	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
201	C₄×C₈	C₂³×C₂₃	1	2	yes	—
202	C₄×C₈	C₂³×C₂₃	1	2	yes	—
203	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
204	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
205	C₂₃	#17956	1	1	yes	characteristic, direct product factor, Sylow
206	#41	—	4	4	no	—
207	#41	—	4	4	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
217	D₈	—	2	8	no	—
218	M₁₆	—	4	4	no	—
219	M₁₆	—	4	8	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	4	8	no	—
226	C₂×C₈	D₈×C₂₃	1	4	yes	complemented
227	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
228	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
229	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
230	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
232	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
238	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
239	C₄²	#41×C₂₃	1	1	yes	characteristic
240	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
241	C₂³	—	2	4	no	—
242	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
243	Q₈	—	4	4	no	—
244	D₄	—	2	8	no	—
245	D₄	—	4	8	no	—
246	D₄	—	4	4	no	—
247	D₄	—	4	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
256	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
257	C₂×C₄	C₂₃×#135	1	2	yes	—
258	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
260	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
261	C₈	—	2	4	no	—
262	C₈	—	2	2	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	4	4	no	—
266	C₈	C₂₃×#131	1	4	yes	complemented
267	C₈	C₂×D₈×C₂₃	1	2	yes	—
268	C₈	C₂×D₈×C₂₃	1	2	yes	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	4	4	no	—
273	C₂²	—	4	4	no	—
274	C₂²	C₄×D₈×C₂₃	1	2	yes	—
275	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	4	4	no	—
279	C₄	C₄×D₈×C₂₃	1	2	yes	—
280	C₄	C₂₃×#443	1	2	yes	—
281	C₄	C₂×C₂₃×#131	1	2	yes	—
282	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
283	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
284	C₂	—	2	2	no	—
285	C₂	—	8	4	no	—
286	C₂	C₂₃×#2931	1	1	yes	characteristic
287	C₂	C₂×C₄×D₈×C₂₃	1	1	yes	characteristic
288	C₂	C₂×C₂₃×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂×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×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2931	C₂	semidirect
3. C₂₃×#1589	C₂	semidirect
4. C₈×D₈×C₂₃	C₂	semidirect
5. C₂₃×#1567	C₂	semidirect
6. C₂₃×#1492	C₂	semidirect
7. C₂×C₈²×C₂₃	C₂	semidirect
81. C₂×C₈×C₂₃	D₈	semidirect
96. #17956	C₂₃	direct (N × H)
122. C₈×C₂₃	#131	semidirect
126. #2931	C₂×C₂₃	semidirect
127. #1589	C₂×C₂₃	semidirect
128. C₈×D₈	C₂×C₂₃	semidirect
129. #1567	C₂×C₂₃	semidirect
130. #1492	C₂×C₂₃	semidirect
131. C₂×C₈²	C₂×C₂₃	semidirect
205. C₂₃	#17956	direct (N × H)
226. C₂×C₈	D₈×C₂₃	semidirect
266. C₈	C₂₃×#131	semidirect

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×#2931	2
C₂₃×#1589	2
C₈×D₈×C₂₃	2
C₂₃×#1567	2
C₂₃×#1492	2
C₂×C₈²×C₂₃	2
#17956	23

Subgroups — C23×#17956 #19165582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17956 #19165582