Subgroups — C₂₃×#19204 #19166830

Order 5888.

646 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×#3229	C₂	1	2	yes	maximal
2	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2960	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1690	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1670	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1546	C₂	1	2	yes	maximal
8	C₂²×Q₁₆×C₂₃	—	2	1	no	—
9	C₂²×Q₁₆×C₂₃	—	2	1	no	—
10	C₂₃×#533	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂₃×#498	—	2	2	no	—
13	C₂₃×#494	—	2	1	no	—
14	C₂₃×#494	—	2	1	no	—
15	C₂₃×#494	C₂²	1	2	yes	—
16	C₂₃×#494	C₂²	1	2	yes	—
17	C₂₃×#487	—	2	2	no	—
18	C₂₃×#483	—	2	4	no	—
19	C₂₃×#475	C₂²	1	8	yes	—
20	C₂₃×#446	—	2	1	no	—
21	C₂₃×#446	C₂²	1	2	yes	—
22	C₈×Q₈×C₂₃	—	2	1	no	—
23	C₈×Q₈×C₂₃	C₂²	1	2	yes	—
24	C₂₃×#431	C₂²	1	2	yes	—
25	C₂₃×#424	—	2	1	no	—
26	C₂₃×#423	C₂²	1	1	yes	characteristic
27	C₂₃×#405	—	2	1	no	—
28	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#335	C₂²	1	2	yes	—
30	C₂₃×#334	C₂²	1	2	yes	—
31	C₂₃×#332	C₂²	1	8	yes	—
32	C₂²×Q₈×C₂₃	—	2	2	no	—
33	C₂×Q₁₆×C₂₃	—	2	2	no	—
34	C₂×Q₁₆×C₂₃	—	2	2	no	—
35	C₂×Q₁₆×C₂₃	—	2	2	no	—
36	C₂×Q₁₆×C₂₃	—	2	2	no	—
37	C₂×Q₁₆×C₂₃	—	4	4	no	—
38	C₂×Q₁₆×C₂₃	—	4	4	no	—
39	C₂×M₁₆×C₂₃	—	2	1	no	—
40	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
42	C₂₃×C₄⋊Q₈	—	2	2	no	—
43	C₂₃×C₄⋊Q₈	—	2	2	no	—
44	C₂₃×C₄⋊Q₈	—	2	2	no	—
45	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
46	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
47	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
48	C₂₃×#125	—	2	1	no	—
49	C₂₃×C₂²⋊Q₈	—	2	1	no	—
50	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
51	C₄×Q₈×C₂₃	—	2	1	no	—
52	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
53	C₂₃×#117	—	2	1	no	—
54	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
57	C₂₃×#106	—	2	8	no	—
58	C₂₃×C₄⋊C₈	—	2	2	no	—
59	C₂₃×C₄⋊C₈	—	2	2	no	—
60	C₂₃×C₄⋊C₈	—	2	1	no	—
61	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
62	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
63	C₂₃×#103	—	2	8	no	—
64	C₂₃×C₂²⋊C₈	—	2	2	no	—
65	C₂₃×#97	—	2	2	no	—
66	C₄×C₈×C₂₃	—	2	2	no	—
67	C₄×C₈×C₂₃	—	2	2	no	—
68	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂×Q₈×C₂₃	—	2	1	no	—
73	C₂×Q₈×C₂₃	—	2	4	no	—
74	C₂×Q₈×C₂₃	—	4	2	no	—
75	C₂×Q₈×C₂₃	—	4	4	no	—
76	C₂×Q₈×C₂₃	—	4	4	no	—
77	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₂₃	—	4	2	no	—
79	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
82	Q₁₆×C₂₃	—	4	4	no	—
83	Q₁₆×C₂₃	—	4	4	no	—
84	Q₁₆×C₂₃	—	4	4	no	—
85	Q₁₆×C₂₃	—	4	4	no	—
86	M₁₆×C₂₃	—	4	2	no	—
87	C₂×C₈×C₂₃	—	2	2	no	—
88	C₂×C₈×C₂₃	—	2	4	no	—
89	C₂×C₈×C₂₃	—	2	2	no	—
90	C₂×C₈×C₂₃	SD₁₆	1	4	yes	—
91	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₄⋊C₄×C₂₃	—	2	1	no	—
96	C₄⋊C₄×C₂₃	—	2	1	no	—
97	C₄⋊C₄×C₂₃	—	2	1	no	—
98	C₄⋊C₄×C₂₃	—	2	2	no	—
99	C₄⋊C₄×C₂₃	—	2	2	no	—
100	C₄⋊C₄×C₂₃	—	2	4	no	—
101	C₄⋊C₄×C₂₃	—	2	4	no	—
102	C₄⋊C₄×C₂₃	—	4	2	no	—
103	C₄⋊C₄×C₂₃	—	4	2	no	—
104	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
105	C₂²⋊C₄×C₂₃	—	2	2	no	—
106	C₄²×C₂₃	—	2	2	no	—
107	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₂₃	#41	1	1	yes	characteristic
110	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
111	#19204	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂³×C₂₃	#121	1	1	yes	characteristic
113	Q₈×C₂₃	—	2	4	no	—
114	Q₈×C₂₃	—	4	2	no	—
115	Q₈×C₂₃	—	4	4	no	—
116	Q₈×C₂₃	—	4	4	no	—
117	Q₈×C₂₃	—	8	4	no	—
118	C₂×C₄×C₂₃	—	2	2	no	—
119	C₂×C₄×C₂₃	—	2	2	no	—
120	C₂×C₄×C₂₃	—	2	2	no	—
121	C₂×C₄×C₂₃	—	2	2	no	—
122	C₂×C₄×C₂₃	—	4	4	no	—
123	C₂×C₄×C₂₃	—	8	4	no	—
124	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
125	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
126	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
128	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
129	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
132	C₈×C₂₃	—	2	1	no	—
133	C₈×C₂₃	—	2	1	no	—
134	C₈×C₂₃	—	4	2	no	—
135	C₈×C₂₃	—	4	4	no	—
136	C₈×C₂₃	C₂×SD₁₆	1	2	yes	—
137	C₈×C₂₃	C₂×SD₁₆	1	2	yes	—
138	#3229	C₂×C₂₃	1	2	yes	—
139	C₂×#494	C₂×C₂₃	1	1	yes	characteristic
140	#2960	C₂×C₂₃	1	1	yes	characteristic
141	#1690	C₂×C₂₃	1	4	yes	complemented
142	#1670	C₂×C₂₃	1	4	yes	complemented
143	#1564	C₂×C₂₃	1	1	yes	characteristic
144	#1546	C₂×C₂₃	1	2	yes	—
145	C₂²×C₂₃	—	2	1	no	—
146	C₂²×C₂₃	—	2	1	no	—
147	C₂²×C₂₃	#460	1	2	yes	—
148	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
149	C₄×C₂₃	—	2	1	no	—
150	C₄×C₂₃	—	2	1	no	—
151	C₄×C₂₃	—	4	2	no	—
152	C₄×C₂₃	—	4	2	no	—
153	C₄×C₂₃	—	8	4	no	—
154	C₄×C₂₃	#460	1	2	yes	—
155	C₄×C₂₃	#473	1	2	yes	—
156	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
157	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₄×C₂₃	#578	1	2	yes	—
159	C₂²×Q₁₆	—	2	1	no	—
160	C₂²×Q₁₆	—	2	1	no	—
161	#533	C₂²×C₂₃	1	1	yes	characteristic
162	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
163	#498	—	2	2	no	—
164	#494	—	2	1	no	—
165	#494	—	2	1	no	—
166	#494	C₂²×C₂₃	1	2	yes	—
167	#494	C₂²×C₂₃	1	2	yes	—
168	#487	—	2	2	no	—
169	#483	—	2	4	no	—
170	#475	C₂²×C₂₃	1	8	yes	—
171	#446	—	2	1	no	—
172	#446	C₂²×C₂₃	1	2	yes	—
173	C₈×Q₈	—	2	1	no	—
174	C₈×Q₈	C₂²×C₂₃	1	2	yes	—
175	#431	C₂²×C₂₃	1	2	yes	—
176	#424	—	2	1	no	—
177	#423	C₂²×C₂₃	1	1	yes	characteristic
178	#405	—	2	1	no	—
179	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
180	#335	C₂²×C₂₃	1	2	yes	—
181	#334	C₂²×C₂₃	1	2	yes	—
182	#332	C₂²×C₂₃	1	8	yes	—
183	C₂×C₂₃	—	2	2	no	—
184	C₂×C₂₃	C₂×#460	1	1	yes	characteristic
185	C₂×C₂₃	#3059	1	1	yes	characteristic
186	C₂×C₂₃	C₂×#473	1	1	yes	characteristic
187	C₂²×Q₈	—	2	2	no	—
188	C₂×Q₁₆	—	2	2	no	—
189	C₂×Q₁₆	—	2	2	no	—
190	C₂×Q₁₆	—	2	2	no	—
191	C₂×Q₁₆	—	2	2	no	—
192	C₂×Q₁₆	—	4	4	no	—
193	C₂×Q₁₆	—	4	4	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
196	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
197	C₄⋊Q₈	—	2	2	no	—
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
201	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
202	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
203	#125	—	2	1	no	—
204	C₂²⋊Q₈	—	2	1	no	—
205	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
206	C₄×Q₈	—	2	1	no	—
207	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
208	#117	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
212	#106	—	2	8	no	—
213	C₄⋊C₈	—	2	2	no	—
214	C₄⋊C₈	—	2	2	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
217	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
218	#103	—	2	8	no	—
219	C₂²⋊C₈	—	2	2	no	—
220	#97	—	2	2	no	—
221	C₄×C₈	—	2	2	no	—
222	C₄×C₈	—	2	2	no	—
223	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
224	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
225	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
226	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
227	C₂₃	#19204	1	1	yes	characteristic, direct product factor, Sylow
228	C₂×Q₈	—	2	1	no	—
229	C₂×Q₈	—	2	4	no	—
230	C₂×Q₈	—	4	2	no	—
231	C₂×Q₈	—	4	4	no	—
232	C₂×Q₈	—	4	4	no	—
233	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
237	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
238	Q₁₆	—	4	4	no	—
239	Q₁₆	—	4	4	no	—
240	Q₁₆	—	4	4	no	—
241	Q₁₆	—	4	4	no	—
242	M₁₆	—	4	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	SD₁₆×C₂₃	1	4	yes	—
247	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
248	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
249	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
264	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
265	C₄²	#41×C₂₃	1	1	yes	characteristic
266	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
267	C₂³	C₂₃×#121	1	1	yes	characteristic
268	Q₈	—	2	4	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	4	4	no	—
271	Q₈	—	4	4	no	—
272	Q₈	—	8	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	8	4	no	—
279	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
280	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
281	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
282	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
283	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
284	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
285	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
286	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	—	4	2	no	—
290	C₈	—	4	4	no	—
291	C₈	C₂×SD₁₆×C₂₃	1	2	yes	—
292	C₈	C₂×SD₁₆×C₂₃	1	2	yes	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	C₂₃×#460	1	2	yes	—
296	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	8	4	no	—
302	C₄	C₂₃×#460	1	2	yes	—
303	C₄	C₂₃×#473	1	2	yes	—
304	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
305	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₄	C₂₃×#578	1	2	yes	—
307	C₂	—	2	2	no	—
308	C₂	C₂×C₂₃×#460	1	1	yes	characteristic
309	C₂	C₂₃×#3059	1	1	yes	characteristic
310	C₂	C₂×C₂₃×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂³×C₂₃	#121
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
4. C₂₃×#1690	C₂	semidirect
5. C₂₃×#1670	C₂	semidirect
111. #19204	C₂₃	direct (N × H)
141. #1690	C₂×C₂₃	semidirect
142. #1670	C₂×C₂₃	semidirect
227. C₂₃	#19204	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×#3229	2
C₂×C₂₃×#494	2
C₂₃×#2960	2
C₂₃×#1690	2
C₂₃×#1670	2
C₂₃×#1564	2
C₂₃×#1546	2
#19204	23

Subgroups — C23×#19204 #19166830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19204 #19166830