Subgroups — C₂₃×#21484 #19169110

Order 5888.

1306 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3573	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3025	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2930	C₂	1	3	yes	maximal, complemented
4	C₂₃×#1891	C₂	1	3	yes	maximal, complemented
5	C₂₃×#1868	C₂	1	3	yes	maximal, complemented
6	C₂₃×#1791	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1757	C₂	1	3	yes	maximal, complemented
8	C₂₃×#585	C₂²	1	1	yes	characteristic
9	C₂₃×#568	—	2	6	no	—
10	C₂₃×#568	C₂²	1	3	yes	—
11	C₂×C₂₃×#131	—	2	6	no	—
12	C₂×C₂₃×#131	C₂²	1	3	yes	—
13	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
14	C₂₃×#525	C₂²	1	3	yes	—
15	C₂₃×#458	C₄	1	4	yes	—
16	C₂₃×B(4,𝔽₂)	C₄	1	4	yes	—
17	C₂₃×#436	—	2	6	no	—
18	C₂₃×#435	—	2	6	no	—
19	C₂₃×#424	C₂²	1	3	yes	complemented
20	C₂₃×#413	C₂²	1	3	yes	—
21	C₂₃×#410	C₂²	1	3	yes	—
22	C₂₃×#408	C₂²	1	3	yes	—
23	C₂₃×#407	C₂²	1	3	yes	complemented
24	C₂₃×#405	C₂²	1	3	yes	complemented
25	C₂₃×#344	C₂²	1	6	yes	complemented
26	C₂₃×2₋¹⁺⁴	—	2	3	no	—
27	C₂₃×H(5,𝔽₂)	—	2	3	no	—
28	C₂₃×H(5,𝔽₂)	C₂×C₄	1	4	yes	—
29	C₂×#41×C₂₃	—	2	6	no	—
30	C₂×#41×C₂₃	C₂³	1	3	yes	—
31	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	C₂₃×#131	—	2	4	no	—
33	C₂₃×#131	—	2	12	no	—
34	C₂₃×#131	—	2	6	no	—
35	C₂₃×#131	—	2	12	no	—
36	C₂₃×#131	—	2	6	no	—
37	C₂×M₁₆×C₂₃	—	2	12	no	—
38	C₂×M₁₆×C₂₃	—	2	6	no	—
39	C₂×M₁₆×C₂₃	—	2	3	no	—
40	C₂×M₁₆×C₂₃	—	2	3	no	—
41	C₂×M₁₆×C₂₃	D₄	1	6	yes	complemented
42	C₂×M₁₆×C₂₃	C₂³	1	3	yes	—
43	C₂×M₁₆×C₂₃	C₂³	1	3	yes	—
44	C₂²×C₈×C₂₃	—	2	4	no	—
45	C₂²×C₈×C₂₃	D₄	1	6	yes	complemented
46	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂₃×#123	C₂×C₄	1	6	yes	—
49	C₂₃×C₂²⋊Q₈	—	2	3	no	—
50	C₂₃×#121	—	2	3	no	—
51	C₂₃×#120	C₂×C₄	1	6	yes	—
52	C₄×D₄×C₂₃	—	2	6	no	—
53	C₂₃×#117	C₂³	1	3	yes	—
54	C₂₃×C₄⋊C₈	—	2	6	no	—
55	C₂₃×#101	—	2	3	no	—
56	C₂₃×#100	—	2	3	no	—
57	C₂₃×#99	C₂×C₄	1	12	yes	—
58	C₂₃×C₂²⋊C₈	—	2	6	no	—
59	C₂₃×C₂²⋊C₈	—	2	6	no	—
60	C₂₃×#97	—	2	3	no	—
61	C₄×C₈×C₂₃	—	2	3	no	—
62	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
63	#41×C₂₃	—	2	4	no	—
64	#41×C₂₃	—	2	12	no	—
65	#41×C₂₃	—	2	12	no	—
66	#41×C₂₃	—	2	6	no	—
67	#41×C₂₃	—	2	6	no	—
68	C₂×Q₈×C₂₃	—	2	6	no	—
69	C₂×Q₈×C₂₃	#41	1	3	yes	—
70	C₂×D₄×C₂₃	—	2	6	no	—
71	C₂×D₄×C₂₃	—	2	12	no	—
72	C₂×D₄×C₂₃	—	2	6	no	—
73	C₂×D₄×C₂₃	C₂²×C₄	1	6	yes	—
74	C₂×D₄×C₂₃	#41	1	3	yes	—
75	C₂²×C₄×C₂₃	—	2	4	no	—
76	C₂²×C₄×C₂₃	—	2	3	no	—
77	C₂²×C₄×C₂₃	—	2	3	no	—
78	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
80	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
81	M₁₆×C₂₃	—	2	12	no	—
82	M₁₆×C₂₃	—	2	6	no	—
83	M₁₆×C₂₃	—	2	12	no	—
84	M₁₆×C₂₃	—	2	6	no	—
85	M₁₆×C₂₃	—	4	1	no	—
86	M₁₆×C₂₃	—	4	3	no	—
87	C₂×C₈×C₂₃	—	2	12	no	—
88	C₂×C₈×C₂₃	—	2	6	no	—
89	C₂×C₈×C₂₃	—	2	6	no	—
90	C₂×C₈×C₂₃	—	2	3	no	—
91	C₂×C₈×C₂₃	—	2	3	no	—
92	C₂×C₈×C₂₃	C₂×D₄	1	6	yes	—
93	C₂×C₈×C₂₃	C₂×D₄	1	3	yes	—
94	C₂×C₈×C₂₃	C₂×D₄	1	3	yes	—
95	C₄⋊C₄×C₂₃	—	2	6	no	—
96	C₄⋊C₄×C₂₃	—	2	3	no	—
97	C₂²⋊C₄×C₂₃	—	2	6	no	—
98	C₂²⋊C₄×C₂₃	—	2	6	no	—
99	C₂²⋊C₄×C₂₃	C₂²×C₄	1	6	yes	—
100	C₄²×C₂₃	—	2	3	no	—
101	#21484	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂³×C₂₃	—	2	4	no	—
103	C₂³×C₂₃	—	2	3	no	—
104	C₂³×C₂₃	—	8	1	no	—
105	C₂³×C₂₃	C₄×D₄	1	6	yes	—
106	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
107	Q₈×C₂₃	—	2	4	no	—
108	Q₈×C₂₃	—	2	6	no	—
109	D₄×C₂₃	—	2	12	no	—
110	D₄×C₂₃	—	2	12	no	—
111	D₄×C₂₃	—	2	6	no	—
112	D₄×C₂₃	—	8	6	no	—
113	C₂×C₄×C₂₃	—	2	12	no	—
114	C₂×C₄×C₂₃	—	2	6	no	—
115	C₂×C₄×C₂₃	—	2	6	no	—
116	C₂×C₄×C₂₃	—	2	6	no	—
117	C₂×C₄×C₂₃	—	2	3	no	—
118	C₂×C₄×C₂₃	—	4	1	no	—
119	C₂×C₄×C₂₃	—	4	3	no	—
120	C₂×C₄×C₂₃	C₄×D₄	1	6	yes	—
121	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
122	C₂×C₄×C₂₃	C₂×#41	1	3	yes	—
123	C₈×C₂₃	—	2	6	no	—
124	C₈×C₂₃	—	4	1	no	—
125	C₈×C₂₃	—	4	3	no	—
126	C₈×C₂₃	#120	1	1	yes	characteristic
127	C₈×C₂₃	#120	1	3	yes	—
128	#3573	C₂×C₂₃	1	1	yes	characteristic, complemented
129	#3025	C₂×C₂₃	1	1	yes	characteristic
130	#2930	C₂×C₂₃	1	3	yes	complemented
131	#1891	C₂×C₂₃	1	3	yes	complemented
132	#1868	C₂×C₂₃	1	3	yes	complemented
133	#1791	C₂×C₂₃	1	1	yes	characteristic, complemented
134	#1757	C₂×C₂₃	1	3	yes	complemented
135	C₂²×C₂₃	—	2	6	no	—
136	C₂²×C₂₃	—	4	1	no	—
137	C₂²×C₂₃	—	4	6	no	—
138	C₂²×C₂₃	—	4	3	no	—
139	C₂²×C₂₃	—	4	3	no	—
140	C₂²×C₂₃	—	4	1	no	—
141	C₂²×C₂₃	—	8	4	no	—
142	C₂²×C₂₃	C₂×C₄×D₄	1	3	yes	—
143	C₄×C₂₃	—	2	6	no	—
144	C₄×C₂₃	—	4	6	no	—
145	C₄×C₂₃	—	4	1	no	—
146	C₄×C₂₃	C₂×#120	1	1	yes	characteristic
147	C₄×C₂₃	#525	1	3	yes	—
148	#585	C₂²×C₂₃	1	1	yes	characteristic
149	#568	—	2	6	no	—
150	#568	C₂²×C₂₃	1	3	yes	—
151	C₂×#131	—	2	6	no	—
152	C₂×#131	C₂²×C₂₃	1	3	yes	—
153	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
154	#525	C₂²×C₂₃	1	3	yes	—
155	#458	C₄×C₂₃	1	4	yes	—
156	B(4,𝔽₂)	C₄×C₂₃	1	4	yes	—
157	#436	—	2	6	no	—
158	#435	—	2	6	no	—
159	#424	C₂²×C₂₃	1	3	yes	complemented
160	#413	C₂²×C₂₃	1	3	yes	—
161	#410	C₂²×C₂₃	1	3	yes	—
162	#408	C₂²×C₂₃	1	3	yes	—
163	#407	C₂²×C₂₃	1	3	yes	complemented
164	#405	C₂²×C₂₃	1	3	yes	complemented
165	#344	C₂²×C₂₃	1	6	yes	complemented
166	C₂×C₂₃	—	2	3	no	—
167	C₂×C₂₃	—	4	6	no	—
168	C₂×C₂₃	—	8	1	no	—
169	C₂×C₂₃	C₄×#120	1	1	yes	characteristic
170	2₋¹⁺⁴	—	2	3	no	—
171	H(5,𝔽₂)	—	2	3	no	—
172	H(5,𝔽₂)	C₂×C₄×C₂₃	1	4	yes	—
173	C₂×#41	—	2	6	no	—
174	C₂×#41	C₂³×C₂₃	1	3	yes	—
175	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
176	#131	—	2	4	no	—
177	#131	—	2	12	no	—
178	#131	—	2	6	no	—
179	#131	—	2	12	no	—
180	#131	—	2	6	no	—
181	C₂×M₁₆	—	2	12	no	—
182	C₂×M₁₆	—	2	6	no	—
183	C₂×M₁₆	—	2	3	no	—
184	C₂×M₁₆	—	2	3	no	—
185	C₂×M₁₆	D₄×C₂₃	1	6	yes	complemented
186	C₂×M₁₆	C₂³×C₂₃	1	3	yes	—
187	C₂×M₁₆	C₂³×C₂₃	1	3	yes	—
188	C₂²×C₈	—	2	4	no	—
189	C₂²×C₈	D₄×C₂₃	1	6	yes	complemented
190	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
192	#123	C₂×C₄×C₂₃	1	6	yes	—
193	C₂²⋊Q₈	—	2	3	no	—
194	#121	—	2	3	no	—
195	#120	C₂×C₄×C₂₃	1	6	yes	—
196	C₄×D₄	—	2	6	no	—
197	#117	C₂³×C₂₃	1	3	yes	—
198	C₄⋊C₈	—	2	6	no	—
199	#101	—	2	3	no	—
200	#100	—	2	3	no	—
201	#99	C₂×C₄×C₂₃	1	12	yes	—
202	C₂²⋊C₈	—	2	6	no	—
203	C₂²⋊C₈	—	2	6	no	—
204	#97	—	2	3	no	—
205	C₄×C₈	—	2	3	no	—
206	C₂₃	#21484	1	1	yes	characteristic, direct product factor, Sylow
207	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
208	#41	—	2	4	no	—
209	#41	—	2	12	no	—
210	#41	—	2	12	no	—
211	#41	—	2	6	no	—
212	#41	—	2	6	no	—
213	C₂×Q₈	—	2	6	no	—
214	C₂×Q₈	#41×C₂₃	1	3	yes	—
215	C₂×D₄	—	2	6	no	—
216	C₂×D₄	—	2	12	no	—
217	C₂×D₄	—	2	6	no	—
218	C₂×D₄	C₂²×C₄×C₂₃	1	6	yes	—
219	C₂×D₄	#41×C₂₃	1	3	yes	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	3	no	—
222	C₂²×C₄	—	2	3	no	—
223	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
225	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
226	M₁₆	—	2	12	no	—
227	M₁₆	—	2	6	no	—
228	M₁₆	—	2	12	no	—
229	M₁₆	—	2	6	no	—
230	M₁₆	—	4	1	no	—
231	M₁₆	—	4	3	no	—
232	C₂×C₈	—	2	12	no	—
233	C₂×C₈	—	2	6	no	—
234	C₂×C₈	—	2	6	no	—
235	C₂×C₈	—	2	3	no	—
236	C₂×C₈	—	2	3	no	—
237	C₂×C₈	C₂×D₄×C₂₃	1	6	yes	—
238	C₂×C₈	C₂×D₄×C₂₃	1	3	yes	—
239	C₂×C₈	C₂×D₄×C₂₃	1	3	yes	—
240	C₄⋊C₄	—	2	6	no	—
241	C₄⋊C₄	—	2	3	no	—
242	C₂²⋊C₄	—	2	6	no	—
243	C₂²⋊C₄	—	2	6	no	—
244	C₂²⋊C₄	C₂²×C₄×C₂₃	1	6	yes	—
245	C₄²	—	2	3	no	—
246	C₂³	—	2	4	no	—
247	C₂³	—	2	3	no	—
248	C₂³	—	8	1	no	—
249	C₂³	C₄×D₄×C₂₃	1	6	yes	—
250	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
251	Q₈	—	2	4	no	—
252	Q₈	—	2	6	no	—
253	D₄	—	2	12	no	—
254	D₄	—	2	12	no	—
255	D₄	—	2	6	no	—
256	D₄	—	8	6	no	—
257	C₂×C₄	—	2	12	no	—
258	C₂×C₄	—	2	6	no	—
259	C₂×C₄	—	2	6	no	—
260	C₂×C₄	—	2	6	no	—
261	C₂×C₄	—	2	3	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	3	no	—
264	C₂×C₄	C₄×D₄×C₂₃	1	6	yes	—
265	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
266	C₂×C₄	C₂×#41×C₂₃	1	3	yes	—
267	C₈	—	2	6	no	—
268	C₈	—	4	1	no	—
269	C₈	—	4	3	no	—
270	C₈	C₂₃×#120	1	1	yes	characteristic
271	C₈	C₂₃×#120	1	3	yes	—
272	C₂²	—	2	6	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	6	no	—
275	C₂²	—	4	3	no	—
276	C₂²	—	4	3	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	8	4	no	—
279	C₂²	C₂×C₄×D₄×C₂₃	1	3	yes	—
280	C₄	—	2	6	no	—
281	C₄	—	4	6	no	—
282	C₄	—	4	1	no	—
283	C₄	C₂×C₂₃×#120	1	1	yes	characteristic
284	C₄	C₂₃×#525	1	3	yes	—
285	C₂	—	2	3	no	—
286	C₂	—	4	6	no	—
287	C₂	—	8	1	no	—
288	C₂	C₄×C₂₃×#120	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	C₂×#120
commutator	C₂³	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂×C₂₃	C₄×#120
prime radical	C₂₃×#3025	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3573	C₂	semidirect
3. C₂₃×#2930	C₂	semidirect
4. C₂₃×#1891	C₂	semidirect
5. C₂₃×#1868	C₂	semidirect
6. C₂₃×#1791	C₂	semidirect
7. C₂₃×#1757	C₂	semidirect
13. C₂²×M₁₆×C₂₃	C₂²	semidirect
19. C₂₃×#424	C₂²	semidirect
23. C₂₃×#407	C₂²	semidirect
24. C₂₃×#405	C₂²	semidirect
25. C₂₃×#344	C₂²	semidirect
41. C₂×M₁₆×C₂₃	D₄	semidirect
45. C₂²×C₈×C₂₃	D₄	semidirect
101. #21484	C₂₃	direct (N × H)
128. #3573	C₂×C₂₃	semidirect
130. #2930	C₂×C₂₃	semidirect
131. #1891	C₂×C₂₃	semidirect
132. #1868	C₂×C₂₃	semidirect
133. #1791	C₂×C₂₃	semidirect
134. #1757	C₂×C₂₃	semidirect
153. C₂²×M₁₆	C₂²×C₂₃	semidirect
159. #424	C₂²×C₂₃	semidirect
163. #407	C₂²×C₂₃	semidirect
164. #405	C₂²×C₂₃	semidirect
165. #344	C₂²×C₂₃	semidirect
185. C₂×M₁₆	D₄×C₂₃	semidirect
189. C₂²×C₈	D₄×C₂₃	semidirect
206. C₂₃	#21484	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×#3573	2
C₂₃×#3025	2
C₂₃×#2930	2
C₂₃×#1891	2
C₂₃×#1868	2
C₂₃×#1791	2
C₂₃×#1757	2
#21484	23

Subgroups — C23×#21484 #19169110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21484 #19169110