Subgroups — C₂₃×#22613 #19170239

Order 5888.

778 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	C₂×C₂₃×#427	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2346	C₂	1	2	yes	maximal
3	C₂₃×#1942	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1920	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1825	C₂	1	2	yes	maximal
6	C₂₃×#1808	C₂	1	1	yes	maximal, characteristic
7	C₄×C₂₃×#107	C₂	1	2	yes	maximal, complemented
8	C₈×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#117	C₂²	1	2	yes	—
11	C₂₃×#427	C₄	1	8	yes	complemented
12	C₂×C₂₃×#107	C₂²	1	2	yes	—
13	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#106	C₂²	1	2	yes	—
16	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#389	C₂²	1	4	yes	—
19	C₂₃×#388	—	2	4	no	—
20	C₂₃×#382	—	2	2	no	—
21	C₂₃×#380	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
23	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#376	—	2	2	no	—
25	C₂₃×#340	C₂²	1	2	yes	—
26	C₂₃×#340	C₂²	1	4	yes	—
27	C₂₃×#340	C₂²	1	4	yes	—
28	C₂₃×#336	C₂²	1	2	yes	—
29	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₂₃	—	2	4	no	—
31	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂₃×#117	—	2	4	no	—
36	C₂₃×#117	C₂×C₄	1	8	yes	—
37	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
38	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
41	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
44	C₂×C₄²×C₂₃	—	2	2	no	—
45	C₂×C₄²×C₂₃	—	2	4	no	—
46	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
47	C₂₃×#107	—	2	4	no	—
48	C₂₃×#107	C₂×C₄	1	4	yes	—
49	C₂₃×#107	C₂×C₄	1	4	yes	—
50	C₂₃×#106	C₂×C₄	1	8	yes	—
51	C₂₃×C₂²⋊C₈	—	2	2	no	—
52	C₄×C₈×C₂₃	—	2	4	no	—
53	C₂₃×#95	—	2	2	no	—
54	C₂₃×#95	—	2	4	no	—
55	C₂₃×#95	—	2	4	no	—
56	C₂₃×#95	D₄	1	1	yes	characteristic
57	C₂₃×#95	D₄	1	1	yes	characteristic
58	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	—	2	2	no	—
60	C₂²×C₄×C₂₃	—	2	2	no	—
61	C₂²×C₄×C₂₃	—	2	2	no	—
62	C₂²×C₄×C₂₃	—	2	2	no	—
63	C₂²×C₄×C₂₃	—	2	4	no	—
64	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₂₃	#41	1	2	yes	—
69	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₂₃	—	2	2	no	—
71	C₂×C₈×C₂₃	—	2	8	no	—
72	C₂×C₈×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	—	2	2	no	—
75	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₂₃	#41	1	4	yes	—
78	C₄⋊C₄×C₂₃	—	2	2	no	—
79	C₄⋊C₄×C₂₃	—	2	2	no	—
80	C₄⋊C₄×C₂₃	—	2	4	no	—
81	C₄⋊C₄×C₂₃	—	2	4	no	—
82	C₄⋊C₄×C₂₃	—	8	4	no	—
83	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
84	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
85	C₂²⋊C₄×C₂₃	—	2	8	no	—
86	C₂²⋊C₄×C₂₃	—	2	4	no	—
87	C₂²⋊C₄×C₂₃	—	8	2	no	—
88	C₄²×C₂₃	—	2	8	no	—
89	C₄²×C₂₃	—	2	2	no	—
90	C₄²×C₂₃	—	2	2	no	—
91	C₄²×C₂₃	—	4	8	no	—
92	#22613	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₂₃	—	2	2	no	—
94	C₂³×C₂₃	—	2	2	no	—
95	C₂³×C₂₃	—	2	1	no	—
96	C₂³×C₂₃	—	2	1	no	—
97	C₂³×C₂₃	C₄×D₄	1	2	yes	—
98	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	2	4	no	—
105	C₂×C₄×C₂₃	—	2	4	no	—
106	C₂×C₄×C₂₃	—	2	4	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	4	4	no	—
111	C₂×C₄×C₂₃	—	4	2	no	—
112	C₂×C₄×C₂₃	—	4	2	no	—
113	C₂×C₄×C₂₃	—	4	4	no	—
114	C₂×C₄×C₂₃	—	4	4	no	—
115	C₂×C₄×C₂₃	—	4	4	no	—
116	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₂₃	#135	1	4	yes	—
118	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
120	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
122	C₈×C₂₃	—	2	2	no	—
123	C₈×C₂₃	—	4	2	no	—
124	C₈×C₂₃	#117	1	4	yes	—
125	C₂×#427	C₂×C₂₃	1	1	yes	characteristic
126	#2346	C₂×C₂₃	1	2	yes	—
127	#1942	C₂×C₂₃	1	2	yes	complemented
128	#1920	C₂×C₂₃	1	4	yes	complemented
129	#1825	C₂×C₂₃	1	2	yes	—
130	#1808	C₂×C₂₃	1	1	yes	characteristic
131	C₄×#107	C₂×C₂₃	1	2	yes	complemented
132	C₈×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic
133	C₂²×C₂₃	—	2	2	no	—
134	C₂²×C₂₃	—	2	2	no	—
135	C₂²×C₂₃	—	2	2	no	—
136	C₂²×C₂₃	—	4	2	no	—
137	C₂²×C₂₃	—	4	2	no	—
138	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂²×C₂₃	#547	1	2	yes	—
140	C₂²×C₂₃	C₂×#135	1	2	yes	—
141	C₂²×C₂₃	#576	1	1	yes	characteristic
142	C₂²×C₂₃	#578	1	1	yes	characteristic
143	C₄×C₂₃	—	2	4	no	—
144	C₄×C₂₃	—	2	2	no	—
145	C₄×C₂₃	—	4	2	no	—
146	C₄×C₂₃	—	4	4	no	—
147	C₄×C₂₃	—	4	4	no	—
148	C₄×C₂₃	—	8	4	no	—
149	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
150	C₄×C₂₃	#519	1	1	yes	characteristic
151	C₄×C₂₃	#552	1	2	yes	—
152	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×#117	C₂²×C₂₃	1	2	yes	—
154	#427	C₄×C₂₃	1	8	yes	complemented
155	C₂×#107	C₂²×C₂₃	1	2	yes	—
156	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
157	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
158	C₂×#106	C₂²×C₂₃	1	2	yes	—
159	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
161	#389	C₂²×C₂₃	1	4	yes	—
162	#388	—	2	4	no	—
163	#382	—	2	2	no	—
164	#380	C₂²×C₂₃	1	2	yes	—
165	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
166	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
167	#376	—	2	2	no	—
168	#340	C₂²×C₂₃	1	2	yes	—
169	#340	C₂²×C₂₃	1	4	yes	—
170	#340	C₂²×C₂₃	1	4	yes	—
171	#336	C₂²×C₂₃	1	2	yes	—
172	C₂×C₂₃	—	4	2	no	—
173	C₂×C₂₃	#2344	1	1	yes	characteristic
174	C₂×C₂₃	C₄×#135	1	1	yes	characteristic
175	C₂×C₂₃	#2951	1	1	yes	characteristic
176	C₂×C₂₃	#3336	1	2	yes	—
177	C₂×C₂₃	#3341	1	2	yes	—
178	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
179	C₂²×C₈	—	2	4	no	—
180	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
181	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
182	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
184	#117	—	2	4	no	—
185	#117	C₂×C₄×C₂₃	1	8	yes	—
186	C₂×C₄⋊C₄	—	2	4	no	—
187	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
188	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
189	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	4	no	—
195	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
196	#107	—	2	4	no	—
197	#107	C₂×C₄×C₂₃	1	4	yes	—
198	#107	C₂×C₄×C₂₃	1	4	yes	—
199	#106	C₂×C₄×C₂₃	1	8	yes	—
200	C₂²⋊C₈	—	2	2	no	—
201	C₄×C₈	—	2	4	no	—
202	#95	—	2	2	no	—
203	#95	—	2	4	no	—
204	#95	—	2	4	no	—
205	#95	D₄×C₂₃	1	1	yes	characteristic
206	#95	D₄×C₂₃	1	1	yes	characteristic
207	C₂₃	#22613	1	1	yes	characteristic, direct product factor, Sylow
208	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
218	C₂²×C₄	#41×C₂₃	1	2	yes	—
219	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	8	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
226	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
227	C₂×C₈	#41×C₂₃	1	4	yes	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	8	4	no	—
233	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
234	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
235	C₂²⋊C₄	—	2	8	no	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₂²⋊C₄	—	8	2	no	—
238	C₄²	—	2	8	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	4	8	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	C₄×D₄×C₂₃	1	2	yes	—
247	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
266	C₂×C₄	C₂₃×#135	1	4	yes	—
267	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
268	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
269	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
270	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
271	C₈	—	2	2	no	—
272	C₈	—	4	2	no	—
273	C₈	C₂₃×#117	1	4	yes	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
280	C₂²	C₂₃×#547	1	2	yes	—
281	C₂²	C₂×C₂₃×#135	1	2	yes	—
282	C₂²	C₂₃×#576	1	1	yes	characteristic
283	C₂²	C₂₃×#578	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	2	2	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	2	no	—
289	C₄	—	8	4	no	—
290	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
291	C₄	C₂₃×#519	1	1	yes	characteristic
292	C₄	C₂₃×#552	1	2	yes	—
293	C₂	—	4	2	no	—
294	C₂	C₂₃×#2344	1	1	yes	characteristic
295	C₂	C₄×C₂₃×#135	1	1	yes	characteristic
296	C₂	C₂₃×#2951	1	1	yes	characteristic
297	C₂	C₂₃×#3336	1	2	yes	—
298	C₂	C₂₃×#3341	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₂²⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1942	C₂	semidirect
4. C₂₃×#1920	C₂	semidirect
7. C₄×C₂₃×#107	C₂	semidirect
11. C₂₃×#427	C₄	semidirect
92. #22613	C₂₃	direct (N × H)
127. #1942	C₂×C₂₃	semidirect
128. #1920	C₂×C₂₃	semidirect
131. C₄×#107	C₂×C₂₃	semidirect
154. #427	C₄×C₂₃	semidirect
207. C₂₃	#22613	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×#427	2
C₂₃×#2346	2
C₂₃×#1942	2
C₂₃×#1920	2
C₂₃×#1825	2
C₂₃×#1808	2
C₄×C₂₃×#107	2
C₈×C₂²⋊C₄×C₂₃	2
#22613	23

Subgroups — C23×#22613 #19170239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22613 #19170239