Subgroups — C₂₃×#11626 #19159252

Order 5888.

328 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×#2965	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1338	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1330	C₂	1	2	yes	maximal, complemented
6	C₄×#41×C₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₂₃	—	2	1	no	—
8	C₂₃×#446	C₄	1	1	yes	characteristic
9	C₂₃×#446	C₂²	1	1	yes	characteristic
10	C₂₃×#436	C₄	1	1	yes	characteristic
11	C₂₃×#436	C₂²	1	1	yes	characteristic
12	C₂₃×#435	—	2	1	no	—
13	C₂₃×#435	—	2	1	no	—
14	C₂₃×#431	—	2	1	no	—
15	C₂₃×#431	C₄	1	1	yes	characteristic
16	C₄×M₁₆×C₂₃	—	2	1	no	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×C₄⋊C₁₆	—	2	1	no	—
19	C₂₃×C₄⋊C₁₆	—	2	1	no	—
20	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
21	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
22	C₂₃×#346	C₂²	1	2	yes	—
23	C₂×#41×C₂₃	C₈	1	1	yes	characteristic
24	C₂×M₁₆×C₂₃	—	2	1	no	—
25	C₂×M₁₆×C₂₃	—	2	1	no	—
26	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
28	C₄×Q₈×C₂₃	C₈	1	1	yes	characteristic
29	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₄×D₄×C₂₃	—	2	1	no	—
31	C₄×D₄×C₂₃	—	2	1	no	—
32	C₄×D₄×C₂₃	C₈	1	1	yes	characteristic
33	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂₃×#117	—	2	1	no	—
35	C₂₃×#117	C₈	1	1	yes	characteristic
36	C₂×C₄²×C₂₃	—	2	1	no	—
37	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₂₃	—	2	2	no	—
39	C₂×C₁₆×C₂₃	—	2	2	no	—
40	C₂×C₁₆×C₂₃	—	4	2	no	—
41	C₂₃×C₄⋊C₈	—	2	1	no	—
42	C₂₃×C₄⋊C₈	—	2	1	no	—
43	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₂₃×C₂²⋊C₈	—	2	1	no	—
46	C₂₃×C₂²⋊C₈	—	2	1	no	—
47	C₂₃×C₂²⋊C₈	—	2	1	no	—
48	C₂₃×#97	—	2	1	no	—
49	C₂₃×#97	—	2	1	no	—
50	C₄×C₈×C₂₃	—	2	1	no	—
51	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
55	#41×C₂₃	—	4	2	no	—
56	C₂×Q₈×C₂₃	C₂×C₈	1	1	yes	characteristic
57	C₂×D₄×C₂₃	—	2	1	no	—
58	C₂×D₄×C₂₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	—	2	1	no	—
60	C₂²×C₄×C₂₃	—	2	1	no	—
61	C₂²×C₄×C₂₃	—	2	1	no	—
62	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
65	M₁₆×C₂₃	—	4	2	no	—
66	M₁₆×C₂₃	—	4	1	no	—
67	M₁₆×C₂₃	—	4	1	no	—
68	C₂×C₈×C₂₃	—	2	1	no	—
69	C₂×C₈×C₂₃	—	2	1	no	—
70	C₂×C₈×C₂₃	—	2	1	no	—
71	C₂×C₈×C₂₃	—	2	1	no	—
72	C₂×C₈×C₂₃	—	4	1	no	—
73	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₂₃	—	2	1	no	—
78	C₄⋊C₄×C₂₃	—	2	1	no	—
79	C₄⋊C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
80	C₄⋊C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
81	C₂²⋊C₄×C₂₃	—	2	1	no	—
82	C₂²⋊C₄×C₂₃	—	2	1	no	—
83	C₂²⋊C₄×C₂₃	—	2	1	no	—
84	C₄²×C₂₃	—	2	1	no	—
85	C₄²×C₂₃	—	2	1	no	—
86	C₄²×C₂₃	—	2	1	no	—
87	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
89	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
90	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₁₆×C₂₃	—	4	2	no	—
92	C₁₆×C₂₃	—	4	2	no	—
93	#11626	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₂₃	—	2	1	no	—
95	C₂³×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
96	Q₈×C₂₃	—	4	1	no	—
97	D₄×C₂₃	—	4	1	no	—
98	D₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	—	4	1	no	—
109	C₂×C₄×C₂₃	—	4	1	no	—
110	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
116	C₈×C₂₃	—	2	1	no	—
117	C₈×C₂₃	—	2	1	no	—
118	C₈×C₂₃	—	2	1	no	—
119	C₈×C₂₃	—	4	1	no	—
120	C₈×C₂₃	—	4	1	no	—
121	C₈×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₈×C₂₃	#137	1	1	yes	characteristic
123	#2965	C₂×C₂₃	1	1	yes	characteristic
124	#2163	C₂×C₂₃	1	1	yes	characteristic, complemented
125	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
126	#1338	C₂×C₂₃	1	2	yes	complemented
127	#1330	C₂×C₂₃	1	2	yes	complemented
128	C₂²×C₂₃	—	2	1	no	—
129	C₂²×C₂₃	—	2	1	no	—
130	C₂²×C₂₃	—	4	1	no	—
131	C₂²×C₂₃	—	4	1	no	—
132	C₂²×C₂₃	—	4	1	no	—
133	C₂²×C₂₃	#349	1	2	yes	—
134	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
135	C₄×C₂₃	—	2	1	no	—
136	C₄×C₂₃	—	2	1	no	—
137	C₄×C₂₃	—	2	1	no	—
138	C₄×C₂₃	—	2	1	no	—
139	C₄×C₂₃	—	4	1	no	—
140	C₄×C₂₃	—	4	1	no	—
141	C₄×C₂₃	—	4	1	no	—
142	C₄×C₂₃	#350	1	2	yes	—
143	C₄×C₂₃	#417	1	1	yes	characteristic
144	C₄×C₂₃	#421	1	1	yes	characteristic
145	C₄×#41	C₄×C₂₃	1	1	yes	characteristic
146	C₂²×C₁₆	—	2	1	no	—
147	#446	C₄×C₂₃	1	1	yes	characteristic
148	#446	C₂²×C₂₃	1	1	yes	characteristic
149	#436	C₄×C₂₃	1	1	yes	characteristic
150	#436	C₂²×C₂₃	1	1	yes	characteristic
151	#435	—	2	1	no	—
152	#435	—	2	1	no	—
153	#431	—	2	1	no	—
154	#431	C₄×C₂₃	1	1	yes	characteristic
155	C₄×M₁₆	—	2	1	no	—
156	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
157	C₄⋊C₁₆	—	2	1	no	—
158	C₄⋊C₁₆	—	2	1	no	—
159	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
160	C₂²⋊C₁₆	—	2	1	no	—
161	#346	C₂²×C₂₃	1	2	yes	—
162	C₂×C₂₃	—	2	2	no	—
163	C₂×C₂₃	—	8	1	no	—
164	C₂×C₂₃	#1477	1	1	yes	characteristic
165	C₂×C₂₃	C₂×#349	1	1	yes	characteristic
166	C₂×C₂₃	C₂×#350	1	1	yes	characteristic
167	C₂×#41	C₈×C₂₃	1	1	yes	characteristic
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
172	C₄×Q₈	C₈×C₂₃	1	1	yes	characteristic
173	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
174	C₄×D₄	—	2	1	no	—
175	C₄×D₄	—	2	1	no	—
176	C₄×D₄	C₈×C₂₃	1	1	yes	characteristic
177	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
178	#117	—	2	1	no	—
179	#117	C₈×C₂₃	1	1	yes	characteristic
180	C₂×C₄²	—	2	1	no	—
181	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
182	C₂×C₁₆	—	2	2	no	—
183	C₂×C₁₆	—	2	2	no	—
184	C₂×C₁₆	—	4	2	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₄⋊C₈	—	2	1	no	—
187	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
188	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
189	C₂²⋊C₈	—	2	1	no	—
190	C₂²⋊C₈	—	2	1	no	—
191	C₂²⋊C₈	—	2	1	no	—
192	#97	—	2	1	no	—
193	#97	—	2	1	no	—
194	C₄×C₈	—	2	1	no	—
195	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
196	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
197	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
198	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
199	C₂₃	#11626	1	1	yes	characteristic, direct product factor, Sylow
200	#41	—	4	2	no	—
201	C₂×Q₈	C₂×C₈×C₂₃	1	1	yes	characteristic
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	C₂×C₈×C₂₃	1	1	yes	characteristic
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
220	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
221	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
225	C₄⋊C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
226	C₂²⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₄²	—	2	1	no	—
230	C₄²	—	2	1	no	—
231	C₄²	—	2	1	no	—
232	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
233	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
234	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
235	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
236	C₁₆	—	4	2	no	—
237	C₁₆	—	4	2	no	—
238	C₂³	—	2	1	no	—
239	C₂³	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
240	Q₈	—	4	1	no	—
241	D₄	—	4	1	no	—
242	D₄	—	4	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
255	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
256	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
258	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	4	1	no	—
264	C₈	—	4	1	no	—
265	C₈	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
266	C₈	C₂₃×#137	1	1	yes	characteristic
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	1	no	—
272	C₂²	C₂₃×#349	1	2	yes	—
273	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	1	no	—
281	C₄	C₂₃×#350	1	2	yes	—
282	C₄	C₂₃×#417	1	1	yes	characteristic
283	C₄	C₂₃×#421	1	1	yes	characteristic
284	C₂	—	2	2	no	—
285	C₂	—	8	1	no	—
286	C₂	C₂₃×#1477	1	1	yes	characteristic
287	C₂	C₂×C₂₃×#349	1	1	yes	characteristic
288	C₂	C₂×C₂₃×#350	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₂₃	C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2163	C₂	semidirect
3. C₂×C₂₃×C₄⋊C₁₆	C₂	semidirect
4. C₂₃×#1338	C₂	semidirect
5. C₂₃×#1330	C₂	semidirect
93. #11626	C₂₃	direct (N × H)
124. #2163	C₂×C₂₃	semidirect
125. C₂×C₄⋊C₁₆	C₂×C₂₃	semidirect
126. #1338	C₂×C₂₃	semidirect
127. #1330	C₂×C₂₃	semidirect
199. C₂₃	#11626	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×#2965	2
C₂₃×#2163	2
C₂×C₂₃×C₄⋊C₁₆	2
C₂₃×#1338	2
C₂₃×#1330	2
#11626	23

Subgroups — C23×#11626 #19159252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11626 #19159252