Subgroups — C₂₃×#8674 #19156300

Order 5888.

434 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2931	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1517	C₂	1	2	yes	maximal
5	C₂₃×#1286	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#131	—	2	1	no	—
7	C₂₃×#532	C₂²	1	1	yes	characteristic
8	C₂₃×#436	—	2	1	no	—
9	C₂₃×#436	C₄	1	2	yes	complemented
10	C₈×D₄×C₂₃	—	2	1	no	—
11	C₈×D₄×C₂₃	C₄	1	2	yes	complemented
12	C₂₃×#431	C₂²	1	2	yes	—
13	C₂×C₂₃×#107	—	2	1	no	—
14	C₂×C₂₃×#106	—	2	1	no	—
15	C₂₃×#423	C₂²	1	1	yes	characteristic
16	C₂₃×#408	—	2	1	no	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#382	C₂²	1	2	yes	—
19	C₂₃×#335	—	2	1	no	—
20	C₂₃×#334	—	2	1	no	—
21	C₂₃×#327	—	2	4	no	—
22	C₂×#41×C₂₃	—	2	1	no	—
23	C₂₃×#131	—	4	2	no	—
24	C₂₃×#131	—	4	2	no	—
25	C₂×M₁₆×C₂₃	—	2	1	no	—
26	C₂×M₁₆×C₂₃	—	2	2	no	—
27	C₂²×C₈×C₂₃	—	2	2	no	—
28	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
30	C₂₃×C₄⋊Q₈	Q₈	1	1	yes	characteristic
31	C₂₃×#127	Q₈	1	1	yes	characteristic
32	C₂₃×#124	—	2	1	no	—
33	C₂₃×#121	—	2	1	no	—
34	C₂₃×#121	D₄	1	2	yes	—
35	C₄×D₄×C₂₃	—	2	1	no	—
36	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
38	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
39	C₂₃×#107	—	4	2	no	—
40	C₂₃×#106	—	4	2	no	—
41	C₂₃×C₄⋊C₈	—	2	1	no	—
42	C₂₃×C₄⋊C₈	—	2	2	no	—
43	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
44	C₂₃×C₂²⋊C₈	—	2	2	no	—
45	C₂₃×C₂²⋊C₈	—	2	2	no	—
46	C₂₃×C₂²⋊C₈	—	2	2	no	—
47	C₄×C₈×C₂₃	—	2	2	no	—
48	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
52	C₂₃×#95	—	2	2	no	—
53	C₂₃×#95	—	2	2	no	—
54	#41×C₂₃	—	4	1	no	—
55	#41×C₂₃	—	4	1	no	—
56	#41×C₂₃	—	4	1	no	—
57	#41×C₂₃	—	4	1	no	—
58	C₂×Q₈×C₂₃	—	2	1	no	—
59	C₂×D₄×C₂₃	—	2	2	no	—
60	C₂×D₄×C₂₃	—	2	1	no	—
61	C₂×D₄×C₂₃	—	2	1	no	—
62	C₂×D₄×C₂₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₂₃	—	2	2	no	—
64	C₂²×C₄×C₂₃	—	4	2	no	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
68	M₁₆×C₂₃	—	4	1	no	—
69	M₁₆×C₂₃	—	4	1	no	—
70	M₁₆×C₂₃	—	4	4	no	—
71	C₂×C₈×C₂₃	—	2	1	no	—
72	C₂×C₈×C₂₃	—	2	1	no	—
73	C₂×C₈×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	—	4	4	no	—
75	C₂×C₈×C₂₃	—	4	2	no	—
76	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₂₃	D₈	1	2	yes	—
81	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
82	C₄⋊C₄×C₂₃	—	2	1	no	—
83	C₄⋊C₄×C₂₃	—	4	2	no	—
84	C₄⋊C₄×C₂₃	—	4	2	no	—
85	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
86	C₂²⋊C₄×C₂₃	—	2	2	no	—
87	C₂²⋊C₄×C₂₃	—	2	2	no	—
88	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
91	C₄²×C₂₃	#41	1	1	yes	characteristic
92	#8674	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₂₃	—	2	2	no	—
94	C₂³×C₂₃	#121	1	1	yes	characteristic
95	Q₈×C₂₃	—	4	1	no	—
96	Q₈×C₂₃	—	4	1	no	—
97	D₄×C₂₃	—	4	2	no	—
98	D₄×C₂₃	—	4	2	no	—
99	D₄×C₂₃	—	4	2	no	—
100	D₄×C₂₃	—	4	2	no	—
101	D₄×C₂₃	—	4	1	no	—
102	D₄×C₂₃	—	4	1	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	4	1	no	—
105	C₂×C₄×C₂₃	—	4	1	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	—	8	2	no	—
109	C₂×C₄×C₂₃	—	8	2	no	—
110	C₂×C₄×C₂₃	—	8	2	no	—
111	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
121	C₈×C₂₃	—	2	1	no	—
122	C₈×C₂₃	—	2	1	no	—
123	C₈×C₂₃	—	4	2	no	—
124	C₈×C₂₃	—	8	2	no	—
125	C₈×C₂₃	#102	1	2	yes	—
126	C₈×C₂₃	#102	1	2	yes	—
127	#2931	C₂×C₂₃	1	1	yes	characteristic
128	#1846	C₂×C₂₃	1	1	yes	characteristic, complemented
129	#1564	C₂×C₂₃	1	1	yes	characteristic, complemented
130	#1517	C₂×C₂₃	1	2	yes	—
131	#1286	C₂×C₂₃	1	2	yes	complemented
132	C₂²×C₂₃	—	2	1	no	—
133	C₂²×C₂₃	—	2	1	no	—
134	C₂²×C₂₃	—	4	2	no	—
135	C₂²×C₂₃	—	4	2	no	—
136	C₂²×C₂₃	—	4	2	no	—
137	C₂²×C₂₃	#380	1	1	yes	characteristic
138	C₂²×C₂₃	#459	1	1	yes	characteristic
139	C₂²×C₂₃	#461	1	1	yes	characteristic
140	C₄×C₂₃	—	2	1	no	—
141	C₄×C₂₃	—	2	1	no	—
142	C₄×C₂₃	—	2	1	no	—
143	C₄×C₂₃	—	2	1	no	—
144	C₄×C₂₃	—	4	2	no	—
145	C₄×C₂₃	—	16	2	no	—
146	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
147	C₄×C₂₃	#417	1	1	yes	characteristic
148	C₄×C₂₃	#474	1	1	yes	characteristic
149	C₄×C₂₃	#476	1	1	yes	characteristic
150	C₂×#131	—	2	1	no	—
151	#532	C₂²×C₂₃	1	1	yes	characteristic
152	#436	—	2	1	no	—
153	#436	C₄×C₂₃	1	2	yes	complemented
154	C₈×D₄	—	2	1	no	—
155	C₈×D₄	C₄×C₂₃	1	2	yes	complemented
156	#431	C₂²×C₂₃	1	2	yes	—
157	C₂×#107	—	2	1	no	—
158	C₂×#106	—	2	1	no	—
159	#423	C₂²×C₂₃	1	1	yes	characteristic
160	#408	—	2	1	no	—
161	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
162	#382	C₂²×C₂₃	1	2	yes	—
163	#335	—	2	1	no	—
164	#334	—	2	1	no	—
165	#327	—	2	4	no	—
166	C₂×C₂₃	—	2	1	no	—
167	C₂×C₂₃	—	2	1	no	—
168	C₂×C₂₃	—	8	2	no	—
169	C₂×C₂₃	#1802	1	1	yes	characteristic
170	C₂×C₂₃	#1812	1	1	yes	characteristic
171	C₂×C₂₃	#1814	1	1	yes	characteristic
172	C₂×#41	—	2	1	no	—
173	#131	—	4	2	no	—
174	#131	—	4	2	no	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
179	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
180	C₄⋊Q₈	Q₈×C₂₃	1	1	yes	characteristic
181	#127	Q₈×C₂₃	1	1	yes	characteristic
182	#124	—	2	1	no	—
183	#121	—	2	1	no	—
184	#121	D₄×C₂₃	1	2	yes	—
185	C₄×D₄	—	2	1	no	—
186	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
189	#107	—	4	2	no	—
190	#106	—	4	2	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	2	2	no	—
196	C₂²⋊C₈	—	2	2	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
199	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
200	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
201	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
202	#95	—	2	2	no	—
203	#95	—	2	2	no	—
204	C₂₃	#8674	1	1	yes	characteristic, direct product factor, Sylow
205	#41	—	4	1	no	—
206	#41	—	4	1	no	—
207	#41	—	4	1	no	—
208	#41	—	4	1	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	C₄⋊C₄×C₂₃	1	2	yes	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
218	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
219	M₁₆	—	4	1	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	4	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	4	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
228	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
229	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
231	C₂×C₈	D₈×C₂₃	1	2	yes	—
232	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
237	C₂²⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
240	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
241	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
242	C₄²	#41×C₂₃	1	1	yes	characteristic
243	C₂³	—	2	2	no	—
244	C₂³	C₂₃×#121	1	1	yes	characteristic
245	Q₈	—	4	1	no	—
246	Q₈	—	4	1	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	1	no	—
252	D₄	—	4	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	8	2	no	—
260	C₂×C₄	—	8	2	no	—
261	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
263	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
264	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
265	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
266	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
267	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
268	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
269	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
270	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	4	2	no	—
274	C₈	—	8	2	no	—
275	C₈	C₂₃×#102	1	2	yes	—
276	C₈	C₂₃×#102	1	2	yes	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	C₂₃×#380	1	1	yes	characteristic
283	C₂²	C₂₃×#459	1	1	yes	characteristic
284	C₂²	C₂₃×#461	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	4	2	no	—
290	C₄	—	16	2	no	—
291	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
292	C₄	C₂₃×#417	1	1	yes	characteristic
293	C₄	C₂₃×#474	1	1	yes	characteristic
294	C₄	C₂₃×#476	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	2	1	no	—
297	C₂	—	8	2	no	—
298	C₂	C₂₃×#1802	1	1	yes	characteristic
299	C₂	C₂₃×#1812	1	1	yes	characteristic
300	C₂	C₂₃×#1814	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#380
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1846	C₂	semidirect
3. C₂₃×#1564	C₂	semidirect
5. C₂₃×#1286	C₂	semidirect
9. C₂₃×#436	C₄	semidirect
11. C₈×D₄×C₂₃	C₄	semidirect
92. #8674	C₂₃	direct (N × H)
128. #1846	C₂×C₂₃	semidirect
129. #1564	C₂×C₂₃	semidirect
131. #1286	C₂×C₂₃	semidirect
153. #436	C₄×C₂₃	semidirect
155. C₈×D₄	C₄×C₂₃	semidirect
204. C₂₃	#8674	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×#2931	2
C₂₃×#1846	2
C₂₃×#1564	2
C₂₃×#1517	2
C₂₃×#1286	2
#8674	23

Subgroups — C23×#8674 #19156300

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8674 #19156300