Subgroups — C₂₃×#8724 #19156350

Order 5888.

630 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#384	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1797	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1797	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1296	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
6	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
7	C₂²×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
12	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
13	C₂₃×#384	—	2	1	no	—
14	C₂₃×#384	—	2	1	no	—
15	C₂₃×#384	C₄	1	2	yes	—
16	C₂₃×#384	C₂²	1	2	yes	—
17	C₂₃×#336	—	2	1	no	—
18	C₂₃×#336	—	2	1	no	—
19	C₂³×C₄×C₂₃	—	2	2	no	—
20	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₂₃	—	2	2	no	—
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
31	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
32	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
33	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
34	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₂₃	—	2	4	no	—
36	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×C₄⋊C₈	—	2	4	no	—
41	C₂₃×C₄⋊C₈	—	2	4	no	—
42	C₂₃×C₂²⋊C₈	—	4	2	no	—
43	C₂₃×C₂²⋊C₈	—	4	2	no	—
44	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	—	2	2	no	—
46	C₂²×C₄×C₂₃	—	2	2	no	—
47	C₂²×C₄×C₂₃	—	2	2	no	—
48	C₂²×C₄×C₂₃	—	2	2	no	—
49	C₂²×C₄×C₂₃	—	2	2	no	—
50	C₂²×C₄×C₂₃	—	2	2	no	—
51	C₂²×C₄×C₂₃	—	2	2	no	—
52	C₂²×C₄×C₂₃	—	2	4	no	—
53	C₂²×C₄×C₂₃	—	4	4	no	—
54	C₂²×C₄×C₂₃	—	4	4	no	—
55	C₂²×C₄×C₂₃	—	4	4	no	—
56	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	D₈	1	2	yes	—
60	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
61	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
62	C₂²×C₄×C₂₃	Q₁₆	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₂₃	—	4	2	no	—
67	C₂×C₈×C₂₃	—	4	2	no	—
68	C₂×C₈×C₂₃	—	4	2	no	—
69	C₂×C₈×C₂₃	—	4	2	no	—
70	C₂×C₈×C₂₃	—	4	2	no	—
71	C₂×C₈×C₂₃	—	4	2	no	—
72	C₄⋊C₄×C₂₃	—	4	4	no	—
73	C₄⋊C₄×C₂₃	—	4	4	no	—
74	C₄⋊C₄×C₂₃	—	4	4	no	—
75	C₄⋊C₄×C₂₃	—	4	4	no	—
76	C₄⋊C₄×C₂₃	—	4	2	no	—
77	C₄⋊C₄×C₂₃	—	4	2	no	—
78	C₄⋊C₄×C₂₃	—	4	4	no	—
79	C₄²×C₂₃	—	2	2	no	—
80	C₄²×C₂₃	—	2	2	no	—
81	C₄²×C₂₃	—	2	2	no	—
82	C₄²×C₂₃	#41	1	4	yes	—
83	#8724	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₂₃	—	2	1	no	—
85	C₂³×C₂₃	—	2	1	no	—
86	C₂³×C₂₃	—	2	1	no	—
87	C₂³×C₂₃	—	2	1	no	—
88	C₂³×C₂₃	#101	1	2	yes	—
89	C₂³×C₂₃	#102	1	2	yes	—
90	C₂³×C₂₃	#103	1	2	yes	—
91	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	—	2	8	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	4	no	—
102	C₂×C₄×C₂₃	—	2	4	no	—
103	C₂×C₄×C₂₃	—	2	4	no	—
104	C₂×C₄×C₂₃	—	2	4	no	—
105	C₂×C₄×C₂₃	—	4	4	no	—
106	C₂×C₄×C₂₃	—	4	4	no	—
107	C₂×C₄×C₂₃	—	4	4	no	—
108	C₂×C₄×C₂₃	—	8	8	no	—
109	C₂×C₄×C₂₃	#100	1	2	yes	—
110	C₂×C₄×C₂₃	#102	1	2	yes	—
111	C₂×C₄×C₂₃	#103	1	2	yes	—
112	C₂×C₄×C₂₃	#117	1	2	yes	—
113	C₂×C₄×C₂₃	#123	1	2	yes	—
114	C₂×C₄×C₂₃	#123	1	2	yes	—
115	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
119	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₂₃	#137	1	2	yes	—
121	C₈×C₂₃	—	8	2	no	—
122	C₈×C₂₃	—	8	2	no	—
123	C₂×#384	C₂×C₂₃	1	1	yes	characteristic
124	#1797	C₂×C₂₃	1	1	yes	characteristic
125	#1797	C₂×C₂₃	1	1	yes	characteristic
126	#1296	C₂×C₂₃	1	4	yes	complemented
127	C₂²×C₂₃	—	2	2	no	—
128	C₂²×C₂₃	—	2	2	no	—
129	C₂²×C₂₃	—	2	2	no	—
130	C₂²×C₂₃	—	2	2	no	—
131	C₂²×C₂₃	—	2	2	no	—
132	C₂²×C₂₃	—	2	2	no	—
133	C₂²×C₂₃	#332	1	4	yes	—
134	C₂²×C₂₃	#381	1	1	yes	characteristic
135	C₂²×C₂₃	C₂×#100	1	1	yes	characteristic
136	C₂²×C₂₃	C₂×#101	1	1	yes	characteristic
137	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
138	C₂²×C₂₃	C₂×#103	1	1	yes	characteristic
139	C₂²×C₂₃	#418	1	1	yes	characteristic
140	C₂²×C₂₃	#419	1	1	yes	characteristic
141	C₄×C₂₃	—	2	2	no	—
142	C₄×C₂₃	—	2	2	no	—
143	C₄×C₂₃	—	2	2	no	—
144	C₄×C₂₃	—	2	2	no	—
145	C₄×C₂₃	—	8	4	no	—
146	C₄×C₂₃	#480	1	2	yes	—
147	C₄×C₂₃	#481	1	2	yes	—
148	C₄×C₂₃	#483	1	2	yes	—
149	C₄×C₂₃	#484	1	2	yes	—
150	C₂²×C₄⋊C₄	—	2	1	no	—
151	C₂²×C₄⋊C₄	—	2	1	no	—
152	C₂²×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
153	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
154	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
155	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
156	C₂×C₂²⋊C₈	—	2	1	no	—
157	C₂×C₂²⋊C₈	—	2	1	no	—
158	#384	—	2	1	no	—
159	#384	—	2	1	no	—
160	#384	C₄×C₂₃	1	2	yes	—
161	#384	C₂²×C₂₃	1	2	yes	—
162	#336	—	2	1	no	—
163	#336	—	2	1	no	—
164	C₂×C₂₃	—	2	4	no	—
165	C₂×C₂₃	C₂×#332	1	1	yes	characteristic
166	C₂×C₂₃	#1545	1	1	yes	characteristic
167	C₂×C₂₃	#1548	1	1	yes	characteristic
168	C₂×C₂₃	#1556	1	1	yes	characteristic
169	C₂×C₂₃	#1821	1	1	yes	characteristic
170	C₂×C₂₃	#1823	1	1	yes	characteristic
171	C₂×C₂₃	#1824	1	1	yes	characteristic
172	C₂³×C₄	—	2	2	no	—
173	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
174	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
175	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	4	4	no	—
184	C₂×C₄⋊C₄	—	4	2	no	—
185	C₂×C₄⋊C₄	—	4	2	no	—
186	C₂×C₄⋊C₄	—	4	4	no	—
187	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
188	C₂×C₄²	—	2	4	no	—
189	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
190	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
191	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
193	C₄⋊C₈	—	2	4	no	—
194	C₄⋊C₈	—	2	4	no	—
195	C₂²⋊C₈	—	4	2	no	—
196	C₂²⋊C₈	—	4	2	no	—
197	C₂₃	#8724	1	1	yes	characteristic, direct product factor, Sylow
198	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	4	4	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
213	C₂²×C₄	D₈×C₂₃	1	2	yes	—
214	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
215	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
216	C₂²×C₄	Q₁₆×C₂₃	1	2	yes	—
217	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	#41×C₂₃	1	4	yes	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	C₂₃×#101	1	2	yes	—
242	C₂³	C₂₃×#102	1	2	yes	—
243	C₂³	C₂₃×#103	1	2	yes	—
244	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
245	C₂×C₄	—	2	8	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	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	8	8	no	—
262	C₂×C₄	C₂₃×#100	1	2	yes	—
263	C₂×C₄	C₂₃×#102	1	2	yes	—
264	C₂×C₄	C₂₃×#103	1	2	yes	—
265	C₂×C₄	C₂₃×#117	1	2	yes	—
266	C₂×C₄	C₂₃×#123	1	2	yes	—
267	C₂×C₄	C₂₃×#123	1	2	yes	—
268	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
269	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
270	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
271	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
272	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
273	C₂×C₄	C₂₃×#137	1	2	yes	—
274	C₈	—	8	2	no	—
275	C₈	—	8	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	C₂₃×#332	1	4	yes	—
283	C₂²	C₂₃×#381	1	1	yes	characteristic
284	C₂²	C₂×C₂₃×#100	1	1	yes	characteristic
285	C₂²	C₂×C₂₃×#101	1	1	yes	characteristic
286	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
287	C₂²	C₂×C₂₃×#103	1	1	yes	characteristic
288	C₂²	C₂₃×#418	1	1	yes	characteristic
289	C₂²	C₂₃×#419	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	8	4	no	—
295	C₄	C₂₃×#480	1	2	yes	—
296	C₄	C₂₃×#481	1	2	yes	—
297	C₄	C₂₃×#483	1	2	yes	—
298	C₄	C₂₃×#484	1	2	yes	—
299	C₂	—	2	4	no	—
300	C₂	C₂×C₂₃×#332	1	1	yes	characteristic
301	C₂	C₂₃×#1545	1	1	yes	characteristic
302	C₂	C₂₃×#1548	1	1	yes	characteristic
303	C₂	C₂₃×#1556	1	1	yes	characteristic
304	C₂	C₂₃×#1821	1	1	yes	characteristic
305	C₂	C₂₃×#1823	1	1	yes	characteristic
306	C₂	C₂₃×#1824	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂⁴×C₂₃	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1296	C₂	semidirect
83. #8724	C₂₃	direct (N × H)
126. #1296	C₂×C₂₃	semidirect
197. C₂₃	#8724	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#384	2
C₂₃×#1797	2
C₂₃×#1296	2
#8724	23

Subgroups — C23×#8724 #19156350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8724 #19156350