Subgroups — C₂₃×#10812 #19158438

Order 5888.

612 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2873	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#344	C₂	1	2	yes	maximal
3	C₂₃×#1738	C₂	1	2	yes	maximal
4	C₂×C₂₃×#338	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×#338	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
11	C₂₃×#405	—	2	2	no	—
12	C₂₃×#405	C₄	1	4	yes	—
13	C₄×M₁₆×C₂₃	—	2	2	no	—
14	C₂₃×#344	—	2	4	no	—
15	C₂₃×#341	C₄	1	8	yes	complemented
16	C₂₃×#338	—	2	2	no	—
17	C₂₃×#338	—	2	2	no	—
18	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₂₃	—	2	1	no	—
20	C₂×M₁₆×C₂₃	—	2	2	no	—
21	C₂×M₁₆×C₂₃	—	2	2	no	—
22	C₂×M₁₆×C₂₃	—	2	2	no	—
23	C₂×M₁₆×C₂₃	—	2	2	no	—
24	C₂×M₁₆×C₂₃	—	2	2	no	—
25	C₂×M₁₆×C₂₃	—	2	2	no	—
26	C₂×M₁₆×C₂₃	—	2	2	no	—
27	C₂×M₁₆×C₂₃	—	2	2	no	—
28	C₂×M₁₆×C₂₃	—	2	2	no	—
29	C₂×M₁₆×C₂₃	—	2	2	no	—
30	C₂×M₁₆×C₂₃	—	2	2	no	—
31	C₂×M₁₆×C₂₃	—	2	2	no	—
32	C₂×M₁₆×C₂₃	—	2	2	no	—
33	C₂×M₁₆×C₂₃	—	2	2	no	—
34	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
35	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
36	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
38	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
39	C₂²×C₈×C₂₃	—	2	1	no	—
40	C₂²×C₈×C₂₃	—	2	2	no	—
41	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
43	C₂₃×#117	—	2	1	no	—
44	C₂₃×#117	—	2	1	no	—
45	C₂₃×#117	C₂×C₄	1	2	yes	—
46	C₂₃×#117	D₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₂₃	Q₈	1	2	yes	—
49	C₂×C₄²×C₂₃	—	2	1	no	—
50	C₂₃×#97	—	2	4	no	—
51	C₄×C₈×C₂₃	—	2	4	no	—
52	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	—	2	2	no	—
54	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	#41	1	2	yes	—
62	C₂²×C₄×C₂₃	#41	1	2	yes	—
63	C₂²×C₄×C₂₃	#41	1	2	yes	—
64	M₁₆×C₂₃	—	2	2	no	—
65	M₁₆×C₂₃	—	2	4	no	—
66	M₁₆×C₂₃	—	2	4	no	—
67	M₁₆×C₂₃	—	2	4	no	—
68	M₁₆×C₂₃	—	2	4	no	—
69	M₁₆×C₂₃	—	2	4	no	—
70	M₁₆×C₂₃	—	4	2	no	—
71	M₁₆×C₂₃	—	4	2	no	—
72	M₁₆×C₂₃	—	4	2	no	—
73	M₁₆×C₂₃	—	4	2	no	—
74	M₁₆×C₂₃	—	4	2	no	—
75	M₁₆×C₂₃	—	4	2	no	—
76	M₁₆×C₂₃	C₄²	1	4	yes	—
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	—	2	2	no	—
80	C₂×C₈×C₂₃	—	2	4	no	—
81	C₂×C₈×C₂₃	—	2	4	no	—
82	C₂×C₈×C₂₃	—	2	4	no	—
83	C₂×C₈×C₂₃	—	2	4	no	—
84	C₂×C₈×C₂₃	—	2	4	no	—
85	C₂×C₈×C₂₃	—	4	2	no	—
86	C₂×C₈×C₂₃	—	4	2	no	—
87	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
88	C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₄⋊C₄×C₂₃	—	2	2	no	—
90	C₂²⋊C₄×C₂₃	—	2	2	no	—
91	C₂²⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
92	C₄²×C₂₃	—	2	2	no	—
93	C₄²×C₂₃	—	2	2	no	—
94	#10812	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₂₃	—	8	1	no	—
96	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₂₃	C₄×Q₈	1	2	yes	—
98	C₂³×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂³×C₂₃	#125	1	2	yes	—
101	C₂×C₄×C₂₃	—	2	4	no	—
102	C₂×C₄×C₂₃	—	8	2	no	—
103	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₂₃	#117	1	2	yes	—
108	C₂×C₄×C₂₃	#117	1	2	yes	—
109	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
110	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₂₃	#123	1	2	yes	—
117	C₂×C₄×C₂₃	#123	1	2	yes	—
118	C₂×C₄×C₂₃	#124	1	2	yes	—
119	C₂×C₄×C₂₃	#126	1	2	yes	—
120	C₂×C₄×C₂₃	#126	1	2	yes	—
121	C₈×C₂₃	—	2	4	no	—
122	C₈×C₂₃	—	2	4	no	—
123	C₈×C₂₃	—	4	4	no	—
124	C₈×C₂₃	—	4	4	no	—
125	#2873	C₂×C₂₃	1	1	yes	characteristic
126	C₂×#344	C₂×C₂₃	1	2	yes	—
127	#1738	C₂×C₂₃	1	2	yes	—
128	C₂×#338	C₂×C₂₃	1	1	yes	characteristic
129	C₂×#338	C₂×C₂₃	1	1	yes	characteristic
130	C₂²×C₂₃	—	4	1	no	—
131	C₂²×C₂₃	—	4	1	no	—
132	C₂²×C₂₃	—	4	2	no	—
133	C₂²×C₂₃	—	4	1	no	—
134	C₂²×C₂₃	—	4	2	no	—
135	C₂²×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₂₃	#380	1	1	yes	characteristic
137	C₂²×C₂₃	#382	1	1	yes	characteristic
138	C₂²×C₂₃	#387	1	2	yes	—
139	C₂²×C₂₃	#387	1	2	yes	—
140	C₄×C₂₃	—	4	4	no	—
141	C₄×C₂₃	#376	1	1	yes	characteristic
142	C₄×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₄×C₂₃	#381	1	1	yes	characteristic
144	C₄×C₂₃	#383	1	1	yes	characteristic
145	C₄×C₂₃	#388	1	2	yes	—
146	C₄×C₂₃	#388	1	2	yes	—
147	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
148	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
149	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
150	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
152	#405	—	2	2	no	—
153	#405	C₄×C₂₃	1	4	yes	—
154	C₄×M₁₆	—	2	2	no	—
155	#344	—	2	4	no	—
156	#341	C₄×C₂₃	1	8	yes	complemented
157	#338	—	2	2	no	—
158	#338	—	2	2	no	—
159	C₂×C₂₃	—	2	1	no	—
160	C₂×C₂₃	—	2	1	no	—
161	C₂×C₂₃	—	2	1	no	—
162	C₂×C₂₃	—	2	2	no	—
163	C₂×C₂₃	—	2	2	no	—
164	C₂×C₂₃	#1434	1	1	yes	characteristic
165	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
166	C₂×M₁₆	—	2	1	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
182	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
183	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
184	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
185	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
189	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
190	#117	—	2	1	no	—
191	#117	—	2	1	no	—
192	#117	C₂×C₄×C₂₃	1	2	yes	—
193	#117	D₄×C₂₃	1	2	yes	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₂²⋊C₄	Q₈×C₂₃	1	2	yes	—
196	C₂×C₄²	—	2	1	no	—
197	#97	—	2	4	no	—
198	C₄×C₈	—	2	4	no	—
199	C₂₃	#10812	1	1	yes	characteristic, direct product factor, Sylow
200	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
208	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	#41×C₂₃	1	2	yes	—
210	C₂²×C₄	#41×C₂₃	1	2	yes	—
211	C₂²×C₄	#41×C₂₃	1	2	yes	—
212	M₁₆	—	2	2	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	C₄²×C₂₃	1	4	yes	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₂³	—	8	1	no	—
243	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
244	C₂³	C₄×Q₈×C₂₃	1	2	yes	—
245	C₂³	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
246	C₂³	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
247	C₂³	C₂₃×#125	1	2	yes	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	8	2	no	—
250	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
253	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
254	C₂×C₄	C₂₃×#117	1	2	yes	—
255	C₂×C₄	C₂₃×#117	1	2	yes	—
256	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
257	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
258	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
259	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
260	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
261	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
262	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
263	C₂×C₄	C₂₃×#123	1	2	yes	—
264	C₂×C₄	C₂₃×#123	1	2	yes	—
265	C₂×C₄	C₂₃×#124	1	2	yes	—
266	C₂×C₄	C₂₃×#126	1	2	yes	—
267	C₂×C₄	C₂₃×#126	1	2	yes	—
268	C₈	—	2	4	no	—
269	C₈	—	2	4	no	—
270	C₈	—	4	4	no	—
271	C₈	—	4	4	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
278	C₂²	C₂₃×#380	1	1	yes	characteristic
279	C₂²	C₂₃×#382	1	1	yes	characteristic
280	C₂²	C₂₃×#387	1	2	yes	—
281	C₂²	C₂₃×#387	1	2	yes	—
282	C₄	—	4	4	no	—
283	C₄	C₂₃×#376	1	1	yes	characteristic
284	C₄	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
285	C₄	C₂₃×#381	1	1	yes	characteristic
286	C₄	C₂₃×#383	1	1	yes	characteristic
287	C₄	C₂₃×#388	1	2	yes	—
288	C₄	C₂₃×#388	1	2	yes	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	C₂₃×#1434	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂×C₂₃	#1434
prime radical	C₂⁴×C₂₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×M₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₂₃×#341	C₄	semidirect
94. #10812	C₂₃	direct (N × H)
156. #341	C₄×C₂₃	semidirect
199. C₂₃	#10812	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×#2873	2
C₂×C₂₃×#344	2
C₂₃×#1738	2
C₂×C₂₃×#338	2
#10812	23

Subgroups — C23×#10812 #19158438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10812 #19158438