Subgroups — C₂₃×#8682 #19156308

Order 5888.

430 conjugacy classes of subgroups.

The subgroup lattice has 300 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×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1547	C₂	1	2	yes	maximal
5	C₂₃×#1292	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	—
10	C₈×D₄×C₂₃	—	2	1	no	—
11	C₈×D₄×C₂₃	C₄	1	2	yes	—
12	C₂×C₂₃×#108	—	2	1	no	—
13	C₂×C₂₃×#108	—	2	1	no	—
14	C₂₃×#423	C₂²	1	2	yes	—
15	C₂₃×#423	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂₃×#408	—	2	1	no	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#334	—	2	1	no	—
20	C₂₃×#334	—	2	1	no	—
21	C₂₃×#332	—	2	2	no	—
22	C₂₃×#331	—	2	2	no	—
23	C₂×#41×C₂₃	—	2	1	no	—
24	C₂₃×#131	—	4	2	no	—
25	C₂₃×#131	—	4	2	no	—
26	C₂×M₁₆×C₂₃	—	2	1	no	—
27	C₂×M₁₆×C₂₃	—	2	2	no	—
28	C₂×M₁₆×C₂₃	—	2	2	no	—
29	C₂²×C₈×C₂₃	—	2	2	no	—
30	C₂²×C₈×C₂₃	—	2	2	no	—
31	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
33	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
34	C₂₃×#127	D₄	1	1	yes	characteristic
35	C₂₃×#124	—	2	1	no	—
36	C₂₃×#121	—	2	1	no	—
37	C₂₃×#121	Q₈	1	2	yes	—
38	C₄×D₄×C₂₃	—	2	1	no	—
39	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
41	C₂₃×#108	—	4	2	no	—
42	C₂₃×#108	—	4	2	no	—
43	C₂₃×C₄⋊C₈	—	2	1	no	—
44	C₂₃×C₄⋊C₈	—	2	2	no	—
45	C₂₃×C₄⋊C₈	—	2	2	no	—
46	C₂₃×C₄⋊C₈	—	2	2	no	—
47	C₂₃×C₄⋊C₈	—	2	2	no	—
48	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₂₃×C₂²⋊C₈	—	2	2	no	—
50	C₂₃×C₂²⋊C₈	—	2	2	no	—
51	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
55	#41×C₂₃	—	4	1	no	—
56	#41×C₂₃	—	4	1	no	—
57	#41×C₂₃	—	4	1	no	—
58	#41×C₂₃	—	4	1	no	—
59	C₂×Q₈×C₂₃	—	2	1	no	—
60	C₂×D₄×C₂₃	—	2	2	no	—
61	C₂×D₄×C₂₃	—	2	1	no	—
62	C₂×D₄×C₂₃	—	2	1	no	—
63	C₂×D₄×C₂₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₂₃	—	2	2	no	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×Q₈	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	M₁₆×C₂₃	—	4	2	no	—
72	M₁₆×C₂₃	—	4	2	no	—
73	C₂×C₈×C₂₃	—	2	1	no	—
74	C₂×C₈×C₂₃	—	2	1	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	4	4	no	—
77	C₂×C₈×C₂₃	—	4	2	no	—
78	C₂×C₈×C₂₃	—	4	2	no	—
79	C₂×C₈×C₂₃	—	8	2	no	—
80	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₂₃	D₈	1	2	yes	—
85	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
86	C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
88	C₂²⋊C₄×C₂₃	—	2	2	no	—
89	C₂²⋊C₄×C₂₃	—	2	2	no	—
90	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₂₃	#41	1	1	yes	characteristic
94	#8682	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₂₃	—	2	2	no	—
96	C₂³×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
97	Q₈×C₂₃	—	4	1	no	—
98	Q₈×C₂₃	—	4	1	no	—
99	D₄×C₂₃	—	4	2	no	—
100	D₄×C₂₃	—	4	2	no	—
101	D₄×C₂₃	—	4	2	no	—
102	D₄×C₂₃	—	4	1	no	—
103	D₄×C₂₃	—	4	1	no	—
104	D₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	4	1	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	4	2	no	—
110	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#121	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	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
120	C₈×C₂₃	—	2	1	no	—
121	C₈×C₂₃	—	2	1	no	—
122	C₈×C₂₃	—	4	2	no	—
123	C₈×C₂₃	—	8	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	#1848	C₂×C₂₃	1	1	yes	characteristic, complemented
129	C₂×#334	C₂×C₂₃	1	1	yes	characteristic, complemented
130	#1547	C₂×C₂₃	1	2	yes	—
131	#1292	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	#474	1	1	yes	characteristic
139	C₂²×C₂₃	#476	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	—	4	2	no	—
144	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
145	C₄×C₂₃	#417	1	1	yes	characteristic
146	C₄×C₂₃	#459	1	1	yes	characteristic
147	C₄×C₂₃	#461	1	1	yes	characteristic
148	C₄×C₂₃	#471	1	2	yes	—
149	C₂×#131	—	2	1	no	—
150	#532	C₂²×C₂₃	1	1	yes	characteristic
151	#436	—	2	1	no	—
152	#436	C₄×C₂₃	1	2	yes	—
153	C₈×D₄	—	2	1	no	—
154	C₈×D₄	C₄×C₂₃	1	2	yes	—
155	C₂×#108	—	2	1	no	—
156	C₂×#108	—	2	1	no	—
157	#423	C₂²×C₂₃	1	2	yes	—
158	#423	C₂²×C₂₃	1	1	yes	characteristic
159	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
160	#408	—	2	1	no	—
161	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
162	#334	—	2	1	no	—
163	#334	—	2	1	no	—
164	#332	—	2	2	no	—
165	#331	—	2	2	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	#1813	1	1	yes	characteristic
171	C₂×C₂₃	#1815	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	C₂×M₁₆	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
181	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
182	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
183	#127	D₄×C₂₃	1	1	yes	characteristic
184	#124	—	2	1	no	—
185	#121	—	2	1	no	—
186	#121	Q₈×C₂₃	1	2	yes	—
187	C₄×D₄	—	2	1	no	—
188	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
189	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
190	#108	—	4	2	no	—
191	#108	—	4	2	no	—
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
198	C₂²⋊C₈	—	2	2	no	—
199	C₂²⋊C₈	—	2	2	no	—
200	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
201	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
202	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
203	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
204	C₂₃	#8682	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	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	1	no	—
220	M₁₆	—	4	4	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	4	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	8	2	no	—
230	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
231	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
233	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
234	C₂×C₈	D₈×C₂₃	1	2	yes	—
235	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
241	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
243	C₄²	#41×C₂₃	1	1	yes	characteristic
244	C₂³	—	2	2	no	—
245	C₂³	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
246	Q₈	—	4	1	no	—
247	Q₈	—	4	1	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	D₄	—	4	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
261	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
262	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
263	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
264	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
265	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
266	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
267	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
268	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
269	C₈	—	2	1	no	—
270	C₈	—	2	1	no	—
271	C₈	—	4	2	no	—
272	C₈	—	8	2	no	—
273	C₈	—	8	2	no	—
274	C₈	C₂₃×#102	1	2	yes	—
275	C₈	C₂₃×#102	1	2	yes	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₂₃×#380	1	1	yes	characteristic
282	C₂²	C₂₃×#474	1	1	yes	characteristic
283	C₂²	C₂₃×#476	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	2	no	—
288	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
289	C₄	C₂₃×#417	1	1	yes	characteristic
290	C₄	C₂₃×#459	1	1	yes	characteristic
291	C₄	C₂₃×#461	1	1	yes	characteristic
292	C₄	C₂₃×#471	1	2	yes	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	8	2	no	—
296	C₂	C₂₃×#1802	1	1	yes	characteristic
297	C₂	C₂₃×#1813	1	1	yes	characteristic
298	C₂	C₂₃×#1815	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×#1848	C₂	semidirect
3. C₂×C₂₃×#334	C₂	semidirect
5. C₂₃×#1292	C₂	semidirect
94. #8682	C₂₃	direct (N × H)
128. #1848	C₂×C₂₃	semidirect
129. C₂×#334	C₂×C₂₃	semidirect
131. #1292	C₂×C₂₃	semidirect
204. C₂₃	#8682	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×#1848	2
C₂×C₂₃×#334	2
C₂₃×#1547	2
C₂₃×#1292	2
#8682	23

Subgroups — C23×#8682 #19156308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8682 #19156308