Subgroups — C₂₃×#12624 #19160250

Order 5888.

456 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2930	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2120	C₂	1	2	yes	maximal
3	C₂₃×#1757	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1392	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₂₃	C₄	1	1	yes	characteristic, complemented
8	C₂₃×#525	C₂²	1	1	yes	characteristic
9	C₂₃×#436	—	2	2	no	—
10	C₂₃×#435	—	2	2	no	—
11	C₂₃×#424	C₄	1	1	yes	characteristic, complemented
12	C₂₃×#413	C₂²	1	2	yes	—
13	C₂₃×#410	C₂²	1	2	yes	—
14	C₂₃×#408	C₂²	1	1	yes	characteristic
15	C₂₃×#407	C₄	1	1	yes	characteristic, complemented
16	C₂₃×#405	—	2	2	no	—
17	C₂₃×#405	C₄	1	1	yes	characteristic, complemented
18	C₂₃×#352	C₄	1	4	yes	complemented
19	C₂₃×AΣL(1,𝔽₁₆)	C₄	1	4	yes	complemented
20	C₂₃×#344	—	2	2	no	—
21	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
22	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂₃×#131	—	4	2	no	—
24	C₂×M₁₆×C₂₃	—	2	1	no	—
25	C₂×M₁₆×C₂₃	—	2	1	no	—
26	C₂×M₁₆×C₂₃	—	2	2	no	—
27	C₂×M₁₆×C₂₃	—	4	2	no	—
28	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
31	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	D₄	1	2	yes	—
34	C₂₃×#123	C₂×C₄	1	2	yes	—
35	C₂₃×C₂²⋊Q₈	—	2	1	no	—
36	C₂₃×#121	—	2	1	no	—
37	C₂₃×#120	C₂×C₄	1	2	yes	—
38	C₄×D₄×C₂₃	—	2	2	no	—
39	C₂₃×#117	—	2	2	no	—
40	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
41	C₂₃×C₄⋊C₈	—	2	2	no	—
42	C₂₃×#101	—	2	2	no	—
43	C₂₃×#100	C₂×C₄	1	4	yes	—
44	C₂₃×#99	—	2	2	no	—
45	C₂₃×#99	C₂×C₄	1	4	yes	—
46	C₂₃×C₂²⋊C₈	—	2	2	no	—
47	C₂₃×C₂²⋊C₈	—	2	2	no	—
48	C₂₃×#97	—	2	1	no	—
49	C₂₃×#97	—	4	2	no	—
50	C₄×C₈×C₂₃	—	2	1	no	—
51	C₄×C₈×C₂₃	—	4	2	no	—
52	C₂⁴×C₂₃	C₄²	1	1	yes	characteristic
53	#41×C₂₃	—	4	1	no	—
54	#41×C₂₃	—	4	1	no	—
55	C₂×Q₈×C₂₃	#41	1	1	yes	characteristic
56	C₂×D₄×C₂₃	—	2	2	no	—
57	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
58	C₂×D₄×C₂₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	—	2	1	no	—
60	C₂²×C₄×C₂₃	—	2	1	no	—
61	C₂²×C₄×C₂₃	—	4	1	no	—
62	C₂²×C₄×C₂₃	—	4	1	no	—
63	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
66	M₁₆×C₂₃	—	2	4	no	—
67	M₁₆×C₂₃	—	4	1	no	—
68	M₁₆×C₂₃	—	4	2	no	—
69	M₁₆×C₂₃	—	4	2	no	—
70	M₁₆×C₂₃	—	4	2	no	—
71	M₁₆×C₂₃	—	4	2	no	—
72	M₁₆×C₂₃	—	4	1	no	—
73	C₂×C₈×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	—	2	2	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	2	1	no	—
77	C₂×C₈×C₂₃	—	2	1	no	—
78	C₂×C₈×C₂₃	—	4	2	no	—
79	C₂×C₈×C₂₃	—	4	1	no	—
80	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₄⋊C₄×C₂₃	—	2	2	no	—
85	C₄⋊C₄×C₂₃	—	4	2	no	—
86	C₂²⋊C₄×C₂₃	—	2	2	no	—
87	C₂²⋊C₄×C₂₃	—	2	2	no	—
88	C₂²⋊C₄×C₂₃	—	2	4	no	—
89	C₂²⋊C₄×C₂₃	C₄²	1	2	yes	—
90	C₄²×C₂₃	—	2	1	no	—
91	C₄²×C₂₃	—	4	2	no	—
92	#12624	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₂₃	—	2	1	no	—
94	C₂³×C₂₃	—	4	1	no	—
95	C₂³×C₂₃	—	8	1	no	—
96	C₂³×C₂₃	C₂×C₄²	1	1	yes	characteristic
97	C₂³×C₂₃	C₄×D₄	1	2	yes	—
98	Q₈×C₂₃	—	4	1	no	—
99	D₄×C₂₃	—	4	2	no	—
100	D₄×C₂₃	—	4	1	no	—
101	D₄×C₂₃	—	8	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	2	1	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	4	1	no	—
110	C₂×C₄×C₂₃	—	4	1	no	—
111	C₂×C₄×C₂₃	—	4	1	no	—
112	C₂×C₄×C₂₃	—	4	4	no	—
113	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
116	C₈×C₂₃	—	2	2	no	—
117	C₈×C₂₃	—	2	1	no	—
118	C₈×C₂₃	—	4	4	no	—
119	C₈×C₂₃	—	4	1	no	—
120	C₈×C₂₃	—	4	1	no	—
121	C₈×C₂₃	#99	1	2	yes	—
122	#2930	C₂×C₂₃	1	1	yes	characteristic
123	#2120	C₂×C₂₃	1	2	yes	—
124	#1757	C₂×C₂₃	1	2	yes	complemented
125	#1392	C₂×C₂₃	1	1	yes	characteristic, complemented
126	#1391	C₂×C₂₃	1	1	yes	characteristic, complemented
127	C₂²×C₂₃	—	2	2	no	—
128	C₂²×C₂₃	—	2	1	no	—
129	C₂²×C₂₃	—	4	2	no	—
130	C₂²×C₂₃	—	4	1	no	—
131	C₂²×C₂₃	—	4	1	no	—
132	C₂²×C₂₃	—	4	1	no	—
133	C₂²×C₂₃	—	4	1	no	—
134	C₂²×C₂₃	—	8	2	no	—
135	C₂²×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₂₃	—	2	2	no	—
137	C₄×C₂₃	—	2	1	no	—
138	C₄×C₂₃	—	4	2	no	—
139	C₄×C₂₃	—	4	1	no	—
140	C₄×C₂₃	—	8	4	no	—
141	C₄×C₂₃	C₂×#99	1	1	yes	characteristic
142	C₄×C₂₃	#410	1	1	yes	characteristic
143	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
144	C₂²×M₁₆	C₄×C₂₃	1	1	yes	characteristic, complemented
145	#525	C₂²×C₂₃	1	1	yes	characteristic
146	#436	—	2	2	no	—
147	#435	—	2	2	no	—
148	#424	C₄×C₂₃	1	1	yes	characteristic, complemented
149	#413	C₂²×C₂₃	1	2	yes	—
150	#410	C₂²×C₂₃	1	2	yes	—
151	#408	C₂²×C₂₃	1	1	yes	characteristic
152	#407	C₄×C₂₃	1	1	yes	characteristic, complemented
153	#405	—	2	2	no	—
154	#405	C₄×C₂₃	1	1	yes	characteristic, complemented
155	#352	C₄×C₂₃	1	4	yes	complemented
156	AΣL(1,𝔽₁₆)	C₄×C₂₃	1	4	yes	complemented
157	#344	—	2	2	no	—
158	C₂×C₂₃	—	2	1	no	—
159	C₂×C₂₃	—	4	2	no	—
160	C₂×C₂₃	—	4	1	no	—
161	C₂×C₂₃	—	8	1	no	—
162	C₂×C₂₃	C₄×#99	1	1	yes	characteristic
163	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
164	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
165	#131	—	4	2	no	—
166	C₂×M₁₆	—	2	1	no	—
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	4	2	no	—
170	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
171	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
172	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
173	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
174	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
175	C₂²×C₈	D₄×C₂₃	1	2	yes	—
176	#123	C₂×C₄×C₂₃	1	2	yes	—
177	C₂²⋊Q₈	—	2	1	no	—
178	#121	—	2	1	no	—
179	#120	C₂×C₄×C₂₃	1	2	yes	—
180	C₄×D₄	—	2	2	no	—
181	#117	—	2	2	no	—
182	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
183	C₄⋊C₈	—	2	2	no	—
184	#101	—	2	2	no	—
185	#100	C₂×C₄×C₂₃	1	4	yes	—
186	#99	—	2	2	no	—
187	#99	C₂×C₄×C₂₃	1	4	yes	—
188	C₂²⋊C₈	—	2	2	no	—
189	C₂²⋊C₈	—	2	2	no	—
190	#97	—	2	1	no	—
191	#97	—	4	2	no	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	—	4	2	no	—
194	C₂₃	#12624	1	1	yes	characteristic, direct product factor, Sylow
195	C₂⁴	C₄²×C₂₃	1	1	yes	characteristic
196	#41	—	4	1	no	—
197	#41	—	4	1	no	—
198	C₂×Q₈	#41×C₂₃	1	1	yes	characteristic
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
201	C₂×D₄	#41×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	4	1	no	—
211	M₁₆	—	4	2	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	1	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
225	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	C₄²×C₂₃	1	2	yes	—
233	C₄²	—	2	1	no	—
234	C₄²	—	4	2	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	4	1	no	—
237	C₂³	—	8	1	no	—
238	C₂³	C₂×C₄²×C₂₃	1	1	yes	characteristic
239	C₂³	C₄×D₄×C₂₃	1	2	yes	—
240	Q₈	—	4	1	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	1	no	—
243	D₄	—	8	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
256	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
257	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
258	C₈	—	2	1	no	—
259	C₈	—	2	2	no	—
260	C₈	—	4	4	no	—
261	C₈	—	4	1	no	—
262	C₈	—	4	1	no	—
263	C₈	C₂₃×#99	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	8	2	no	—
272	C₂²	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	1	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	1	no	—
277	C₄	—	8	4	no	—
278	C₄	C₂×C₂₃×#99	1	1	yes	characteristic
279	C₄	C₂₃×#410	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	4	2	no	—
282	C₂	—	4	1	no	—
283	C₂	—	8	1	no	—
284	C₂	C₄×C₂₃×#99	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	C₂×#99
commutator	C₂³	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×#41	C₂³×C₂₃
socle	C₂×C₂₃	C₄×#99
prime radical	C₂₃×#525	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂×C₈×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1757	C₂	semidirect
4. C₂₃×#1392	C₂	semidirect
5. C₂₃×#1391	C₂	semidirect
7. C₂²×M₁₆×C₂₃	C₄	semidirect
11. C₂₃×#424	C₄	semidirect
15. C₂₃×#407	C₄	semidirect
17. C₂₃×#405	C₄	semidirect
18. C₂₃×#352	C₄	semidirect
19. C₂₃×AΣL(1,𝔽₁₆)	C₄	semidirect
92. #12624	C₂₃	direct (N × H)
124. #1757	C₂×C₂₃	semidirect
125. #1392	C₂×C₂₃	semidirect
126. #1391	C₂×C₂₃	semidirect
144. C₂²×M₁₆	C₄×C₂₃	semidirect
148. #424	C₄×C₂₃	semidirect
152. #407	C₄×C₂₃	semidirect
154. #405	C₄×C₂₃	semidirect
155. #352	C₄×C₂₃	semidirect
156. AΣL(1,𝔽₁₆)	C₄×C₂₃	semidirect
194. C₂₃	#12624	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×#2930	2
C₂₃×#2120	2
C₂₃×#1757	2
C₂₃×#1392	2
C₂₃×#1391	2
#12624	23

Subgroups — C23×#12624 #19160250

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12624 #19160250