Subgroups — C₂₃×#8673 #19156299

Order 5888.

386 conjugacy classes of subgroups.

The subgroup lattice has 312 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	C₄×C₂₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1489	C₂	1	2	yes	maximal
5	C₂₃×#1299	C₂	1	2	yes	maximal, complemented
6	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#446	C₄	1	1	yes	characteristic
8	C₂₃×#446	C₄	1	1	yes	characteristic
9	C₂₃×#436	C₄	1	1	yes	characteristic
10	C₂₃×#436	C₄	1	1	yes	characteristic
11	C₂₃×#435	—	2	1	no	—
12	C₂₃×#435	—	2	1	no	—
13	C₂₃×#431	—	2	1	no	—
14	C₂₃×#431	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#108	—	2	1	no	—
16	C₂₃×#423	C₂²	1	2	yes	—
17	C₄×M₁₆×C₂₃	—	2	1	no	—
18	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#341	—	2	1	no	—
21	C₂₃×#335	—	2	1	no	—
22	C₂₃×#335	—	2	1	no	—
23	C₂₃×Q₈⋊C₈	—	2	2	no	—
24	C₂₃×D₄⋊C₈	—	2	2	no	—
25	C₂×#41×C₂₃	Q₈	1	1	yes	characteristic
26	C₂×M₁₆×C₂₃	—	2	1	no	—
27	C₂×M₁₆×C₂₃	—	2	1	no	—
28	C₂×M₁₆×C₂₃	—	2	2	no	—
29	C₂×M₁₆×C₂₃	—	2	2	no	—
30	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
34	C₄×D₄×C₂₃	—	2	1	no	—
35	C₄×D₄×C₂₃	—	2	1	no	—
36	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
38	C₂₃×#117	—	2	1	no	—
39	C₂₃×#117	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	—	2	1	no	—
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂₃×#108	—	2	4	no	—
43	C₂₃×C₄⋊C₈	—	2	1	no	—
44	C₂₃×C₄⋊C₈	—	2	1	no	—
45	C₂₃×C₄⋊C₈	—	2	2	no	—
46	C₂₃×C₄⋊C₈	—	2	2	no	—
47	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₂₃×C₂²⋊C₈	—	2	1	no	—
50	C₂₃×C₂²⋊C₈	—	2	1	no	—
51	C₂₃×C₂²⋊C₈	—	2	1	no	—
52	C₂₃×#97	—	2	1	no	—
53	C₂₃×#97	—	2	1	no	—
54	C₂₃×#97	—	2	2	no	—
55	C₄×C₈×C₂₃	—	2	1	no	—
56	C₄×C₈×C₂₃	—	2	2	no	—
57	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
61	#41×C₂₃	—	4	2	no	—
62	C₂×Q₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂×D₄×C₂₃	—	2	1	no	—
64	C₂×D₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	—	2	1	no	—
66	C₂²×C₄×C₂₃	—	2	1	no	—
67	C₂²×C₄×C₂₃	—	2	1	no	—
68	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
71	M₁₆×C₂₃	—	2	4	no	—
72	M₁₆×C₂₃	—	4	2	no	—
73	M₁₆×C₂₃	—	4	2	no	—
74	M₁₆×C₂₃	—	4	2	no	—
75	M₁₆×C₂₃	—	8	2	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	2	1	no	—
78	C₂×C₈×C₂₃	—	2	1	no	—
79	C₂×C₈×C₂₃	—	4	1	no	—
80	C₂×C₈×C₂₃	—	4	2	no	—
81	C₂×C₈×C₂₃	—	4	2	no	—
82	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₄⋊C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
89	C₄⋊C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂²⋊C₄×C₂₃	—	2	1	no	—
91	C₂²⋊C₄×C₂₃	—	2	1	no	—
92	C₂²⋊C₄×C₂₃	—	2	1	no	—
93	C₄²×C₂₃	—	2	1	no	—
94	C₄²×C₂₃	—	2	1	no	—
95	C₄²×C₂₃	—	2	1	no	—
96	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₂₃	#41	1	1	yes	characteristic
100	#8673	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₂₃	—	2	1	no	—
102	C₂³×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
103	Q₈×C₂₃	—	2	2	no	—
104	D₄×C₂₃	—	2	2	no	—
105	D₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	—	2	1	no	—
107	C₂×C₄×C₂₃	—	2	1	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	2	1	no	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	—	2	1	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	4	2	no	—
115	C₂×C₄×C₂₃	—	4	1	no	—
116	C₂×C₄×C₂₃	—	4	1	no	—
117	C₂×C₄×C₂₃	—	4	1	no	—
118	C₂×C₄×C₂₃	—	4	1	no	—
119	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
122	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
125	C₈×C₂₃	—	2	1	no	—
126	C₈×C₂₃	—	2	1	no	—
127	C₈×C₂₃	—	2	1	no	—
128	C₈×C₂₃	—	4	4	no	—
129	C₈×C₂₃	—	4	1	no	—
130	C₈×C₂₃	—	4	1	no	—
131	C₈×C₂₃	—	8	2	no	—
132	C₈×C₂₃	#104	1	2	yes	—
133	#2965	C₂×C₂₃	1	1	yes	characteristic
134	C₄×#108	C₂×C₂₃	1	1	yes	characteristic, complemented
135	#1567	C₂×C₂₃	1	1	yes	characteristic, complemented
136	#1489	C₂×C₂₃	1	2	yes	—
137	#1299	C₂×C₂₃	1	2	yes	complemented
138	C₂²×C₂₃	—	2	1	no	—
139	C₂²×C₂₃	—	2	1	no	—
140	C₂²×C₂₃	—	2	1	no	—
141	C₂²×C₂₃	—	4	1	no	—
142	C₂²×C₂₃	—	4	1	no	—
143	C₂²×C₂₃	—	4	1	no	—
144	C₂²×C₂₃	#380	1	1	yes	characteristic
145	C₄×C₂₃	—	2	1	no	—
146	C₄×C₂₃	—	2	1	no	—
147	C₄×C₂₃	—	2	1	no	—
148	C₄×C₂₃	—	4	1	no	—
149	C₄×C₂₃	—	4	1	no	—
150	C₄×C₂₃	—	4	1	no	—
151	C₄×C₂₃	C₂×#104	1	1	yes	characteristic
152	C₄×C₂₃	#421	1	1	yes	characteristic
153	C₄×C₂₃	#430	1	2	yes	—
154	C₄×C₂₃	#472	1	1	yes	characteristic
155	C₄×C₂₃	#473	1	1	yes	characteristic
156	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
157	#446	C₄×C₂₃	1	1	yes	characteristic
158	#446	C₄×C₂₃	1	1	yes	characteristic
159	#436	C₄×C₂₃	1	1	yes	characteristic
160	#436	C₄×C₂₃	1	1	yes	characteristic
161	#435	—	2	1	no	—
162	#435	—	2	1	no	—
163	#431	—	2	1	no	—
164	#431	C₂²×C₂₃	1	1	yes	characteristic
165	C₂×#108	—	2	1	no	—
166	#423	C₂²×C₂₃	1	2	yes	—
167	C₄×M₁₆	—	2	1	no	—
168	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
169	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
170	#341	—	2	1	no	—
171	#335	—	2	1	no	—
172	#335	—	2	1	no	—
173	Q₈⋊C₈	—	2	2	no	—
174	D₄⋊C₈	—	2	2	no	—
175	C₂×C₂₃	—	4	1	no	—
176	C₂×C₂₃	—	8	1	no	—
177	C₂×C₂₃	#1806	1	1	yes	characteristic
178	C₂×C₂₃	#1811	1	1	yes	characteristic
179	C₂×C₂₃	#1816	1	1	yes	characteristic
180	C₂×#41	Q₈×C₂₃	1	1	yes	characteristic
181	C₂×M₁₆	—	2	1	no	—
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
186	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
187	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
188	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
189	C₄×D₄	—	2	1	no	—
190	C₄×D₄	—	2	1	no	—
191	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
192	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
193	#117	—	2	1	no	—
194	#117	Q₈×C₂₃	1	1	yes	characteristic
195	C₂×C₄²	—	2	1	no	—
196	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
197	#108	—	2	4	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	2	2	no	—
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
203	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
204	C₂²⋊C₈	—	2	1	no	—
205	C₂²⋊C₈	—	2	1	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	#97	—	2	1	no	—
208	#97	—	2	1	no	—
209	#97	—	2	2	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	2	no	—
212	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
213	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
214	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
215	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
216	C₂₃	#8673	1	1	yes	characteristic, direct product factor, Sylow
217	#41	—	4	2	no	—
218	C₂×Q₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
226	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	2	no	—
230	M₁₆	—	4	2	no	—
231	M₁₆	—	8	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
239	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
240	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
241	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
245	C₄⋊C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	2	1	no	—
248	C₂²⋊C₄	—	2	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
253	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
254	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
255	C₄²	#41×C₂₃	1	1	yes	characteristic
256	C₂³	—	2	1	no	—
257	C₂³	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
258	Q₈	—	2	2	no	—
259	D₄	—	2	2	no	—
260	D₄	—	4	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
276	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
277	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
278	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
279	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	4	4	no	—
284	C₈	—	4	1	no	—
285	C₈	—	4	1	no	—
286	C₈	—	8	2	no	—
287	C₈	C₂₃×#104	1	2	yes	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	C₂₃×#380	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	C₂×C₂₃×#104	1	1	yes	characteristic
302	C₄	C₂₃×#421	1	1	yes	characteristic
303	C₄	C₂₃×#430	1	2	yes	—
304	C₄	C₂₃×#472	1	1	yes	characteristic
305	C₄	C₂₃×#473	1	1	yes	characteristic
306	C₂	—	4	1	no	—
307	C₂	—	8	1	no	—
308	C₂	C₂₃×#1806	1	1	yes	characteristic
309	C₂	C₂₃×#1811	1	1	yes	characteristic
310	C₂	C₂₃×#1816	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#380
prime radical	C₂×#41×C₂₃	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₂₃×#108	C₂	semidirect
3. C₂₃×#1567	C₂	semidirect
5. C₂₃×#1299	C₂	semidirect
100. #8673	C₂₃	direct (N × H)
134. C₄×#108	C₂×C₂₃	semidirect
135. #1567	C₂×C₂₃	semidirect
137. #1299	C₂×C₂₃	semidirect
216. C₂₃	#8673	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
C₄×C₂₃×#108	2
C₂₃×#1567	2
C₂₃×#1489	2
C₂₃×#1299	2
#8673	23

Subgroups — C23×#8673 #19156299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8673 #19156299