Subgroups — C₂₃×#12933 #19160559

Order 5888.

380 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2916	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2117	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1815	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1385	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#568	—	2	1	no	—
9	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂₃×#430	—	2	1	no	—
13	C₂₃×#430	—	2	2	no	—
14	C₂×C₂₃×#108	—	2	2	no	—
15	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
19	C₂₃×#417	—	2	1	no	—
20	C₂₃×#365	—	2	2	no	—
21	C₂₃×#365	—	2	2	no	—
22	C₂₃×#350	C₄	1	4	yes	complemented
23	C₂₃×#343	—	2	1	no	—
24	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
25	C₂₃×#131	—	2	1	no	—
26	C₂₃×#131	—	2	1	no	—
27	C₂₃×#131	—	4	2	no	—
28	C₂₃×#131	—	4	2	no	—
29	C₂₃×#131	C₂×C₄	1	2	yes	—
30	C₂₃×#131	Q₈	1	2	yes	—
31	C₂×M₁₆×C₂₃	—	2	1	no	—
32	C₂×M₁₆×C₂₃	—	2	4	no	—
33	C₂×M₁₆×C₂₃	—	2	1	no	—
34	C₂×M₁₆×C₂₃	—	2	1	no	—
35	C₂×M₁₆×C₂₃	—	2	1	no	—
36	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₂₃	—	2	1	no	—
38	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
40	C₂₃×M₃₂	—	2	1	no	—
41	C₂₃×M₃₂	—	2	1	no	—
42	C₂₃×M₃₂	—	2	1	no	—
43	C₂₃×M₃₂	C₂×C₄	1	2	yes	—
44	C₂×C₁₆×C₂₃	—	2	1	no	—
45	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
46	C₂₃×#108	—	2	4	no	—
47	C₂₃×#108	—	2	1	no	—
48	C₂₃×#108	—	2	1	no	—
49	C₂₃×#108	—	2	1	no	—
50	C₂₃×#108	—	2	1	no	—
51	C₂₃×#108	D₄	1	2	yes	—
52	C₂₃×#108	D₄	1	2	yes	—
53	C₂₃×#106	—	2	1	no	—
54	C₂₃×#106	—	2	1	no	—
55	C₂₃×#103	—	4	1	no	—
56	C₂₃×#102	—	4	1	no	—
57	#41×C₂₃	—	2	1	no	—
58	#41×C₂₃	—	2	1	no	—
59	#41×C₂₃	C₄⋊C₄	1	2	yes	—
60	#41×C₂₃	SD₁₆	1	2	yes	—
61	C₂×Q₈×C₂₃	SD₁₆	1	1	yes	characteristic
62	C₂×D₄×C₂₃	—	2	1	no	—
63	C₂×D₄×C₂₃	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	—	2	1	no	—
65	C₂²×C₄×C₂₃	—	4	1	no	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	M₁₆×C₂₃	—	2	1	no	—
68	M₁₆×C₂₃	—	2	2	no	—
69	M₁₆×C₂₃	—	2	2	no	—
70	M₁₆×C₂₃	—	2	2	no	—
71	M₁₆×C₂₃	—	2	2	no	—
72	M₁₆×C₂₃	—	4	4	no	—
73	M₁₆×C₂₃	—	4	1	no	—
74	M₁₆×C₂₃	—	4	1	no	—
75	M₁₆×C₂₃	—	4	1	no	—
76	M₁₆×C₂₃	—	4	1	no	—
77	M₁₆×C₂₃	C₄⋊C₄	1	2	yes	—
78	C₂×C₈×C₂₃	—	2	1	no	—
79	C₂×C₈×C₂₃	—	2	2	no	—
80	C₂×C₈×C₂₃	—	2	2	no	—
81	C₂×C₈×C₂₃	—	4	4	no	—
82	C₂×C₈×C₂₃	—	4	1	no	—
83	C₂×C₈×C₂₃	—	4	1	no	—
84	C₂×C₈×C₂₃	—	4	1	no	—
85	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
90	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
91	C₄⋊C₄×C₂₃	—	4	1	no	—
92	C₄⋊C₄×C₂₃	—	4	1	no	—
93	C₁₆×C₂₃	—	2	2	no	—
94	C₁₆×C₂₃	—	2	2	no	—
95	#12933	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₂₃	—	2	1	no	—
97	C₂³×C₂₃	#137	1	1	yes	characteristic
98	Q₈×C₂₃	—	2	1	no	—
99	Q₈×C₂₃	#106	1	2	yes	—
100	D₄×C₂₃	—	2	2	no	—
101	D₄×C₂₃	—	2	2	no	—
102	D₄×C₂₃	—	2	1	no	—
103	D₄×C₂₃	#106	1	2	yes	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	8	1	no	—
108	C₂×C₄×C₂₃	—	16	1	no	—
109	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
111	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₈×C₂₃	—	2	2	no	—
116	C₈×C₂₃	—	4	2	no	—
117	C₈×C₂₃	—	4	2	no	—
118	C₈×C₂₃	—	8	1	no	—
119	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
120	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
121	C₈×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₈×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
123	#2916	C₂×C₂₃	1	1	yes	characteristic
124	C₂×#365	C₂×C₂₃	1	1	yes	characteristic, complemented
125	C₂×#365	C₂×C₂₃	1	1	yes	characteristic, complemented
126	#2117	C₂×C₂₃	1	1	yes	characteristic
127	#1815	C₂×C₂₃	1	1	yes	characteristic
128	#1388	C₂×C₂₃	1	1	yes	characteristic, complemented
129	#1385	C₂×C₂₃	1	1	yes	characteristic, complemented
130	C₂²×C₂₃	—	2	2	no	—
131	C₂²×C₂₃	—	4	1	no	—
132	C₂²×C₂₃	—	8	1	no	—
133	C₂²×C₂₃	#428	1	1	yes	characteristic
134	C₂²×C₂₃	#448	1	1	yes	characteristic
135	C₂²×C₂₃	#483	1	1	yes	characteristic
136	C₄×C₂₃	—	2	2	no	—
137	C₄×C₂₃	—	16	1	no	—
138	C₄×C₂₃	#385	1	1	yes	characteristic
139	C₄×C₂₃	C₂×#106	1	1	yes	characteristic
140	C₄×C₂₃	#450	1	1	yes	characteristic
141	C₄×C₂₃	#481	1	1	yes	characteristic
142	#568	—	2	1	no	—
143	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
146	#430	—	2	1	no	—
147	#430	—	2	2	no	—
148	C₂×#108	—	2	2	no	—
149	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
153	#417	—	2	1	no	—
154	#365	—	2	2	no	—
155	#365	—	2	2	no	—
156	#350	C₄×C₂₃	1	4	yes	complemented
157	#343	—	2	1	no	—
158	C₂×C₂₃	—	2	1	no	—
159	C₂×C₂₃	—	2	1	no	—
160	C₂×C₂₃	—	2	1	no	—
161	C₂×C₂₃	—	4	2	no	—
162	C₂×C₂₃	#1853	1	1	yes	characteristic
163	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
164	#131	—	2	1	no	—
165	#131	—	2	1	no	—
166	#131	—	4	2	no	—
167	#131	—	4	2	no	—
168	#131	C₂×C₄×C₂₃	1	2	yes	—
169	#131	Q₈×C₂₃	1	2	yes	—
170	C₂×M₁₆	—	2	1	no	—
171	C₂×M₁₆	—	2	4	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
178	C₂×C₄⋊C₄	—	2	1	no	—
179	M₃₂	—	2	1	no	—
180	M₃₂	—	2	1	no	—
181	M₃₂	—	2	1	no	—
182	M₃₂	C₂×C₄×C₂₃	1	2	yes	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
185	#108	—	2	4	no	—
186	#108	—	2	1	no	—
187	#108	—	2	1	no	—
188	#108	—	2	1	no	—
189	#108	—	2	1	no	—
190	#108	D₄×C₂₃	1	2	yes	—
191	#108	D₄×C₂₃	1	2	yes	—
192	#106	—	2	1	no	—
193	#106	—	2	1	no	—
194	#103	—	4	1	no	—
195	#102	—	4	1	no	—
196	C₂₃	#12933	1	1	yes	characteristic, direct product factor, Sylow
197	#41	—	2	1	no	—
198	#41	—	2	1	no	—
199	#41	C₄⋊C₄×C₂₃	1	2	yes	—
200	#41	SD₁₆×C₂₃	1	2	yes	—
201	C₂×Q₈	SD₁₆×C₂₃	1	1	yes	characteristic
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	SD₁₆×C₂₃	1	1	yes	characteristic
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
207	M₁₆	—	2	1	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	2	no	—
212	M₁₆	—	4	4	no	—
213	M₁₆	—	4	1	no	—
214	M₁₆	—	4	1	no	—
215	M₁₆	—	4	1	no	—
216	M₁₆	—	4	1	no	—
217	M₁₆	C₄⋊C₄×C₂₃	1	2	yes	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
226	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
227	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
228	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
229	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
230	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₁₆	—	2	2	no	—
234	C₁₆	—	2	2	no	—
235	C₂³	—	2	1	no	—
236	C₂³	C₂₃×#137	1	1	yes	characteristic
237	Q₈	—	2	1	no	—
238	Q₈	C₂₃×#106	1	2	yes	—
239	D₄	—	2	2	no	—
240	D₄	—	2	2	no	—
241	D₄	—	2	1	no	—
242	D₄	C₂₃×#106	1	2	yes	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	8	1	no	—
247	C₂×C₄	—	16	1	no	—
248	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
249	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
250	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
251	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
253	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
254	C₈	—	2	2	no	—
255	C₈	—	4	2	no	—
256	C₈	—	4	2	no	—
257	C₈	—	8	1	no	—
258	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
259	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
260	C₈	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
261	C₈	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
262	C₂²	—	2	2	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	8	1	no	—
265	C₂²	C₂₃×#428	1	1	yes	characteristic
266	C₂²	C₂₃×#448	1	1	yes	characteristic
267	C₂²	C₂₃×#483	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	16	1	no	—
270	C₄	C₂₃×#385	1	1	yes	characteristic
271	C₄	C₂×C₂₃×#106	1	1	yes	characteristic
272	C₄	C₂₃×#450	1	1	yes	characteristic
273	C₄	C₂₃×#481	1	1	yes	characteristic
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	4	2	no	—
278	C₂	C₂₃×#1853	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#1853
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂×#41×C₂₃	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#365	C₂	semidirect
3. C₂×C₂₃×#365	C₂	semidirect
6. C₂₃×#1388	C₂	semidirect
7. C₂₃×#1385	C₂	semidirect
22. C₂₃×#350	C₄	semidirect
95. #12933	C₂₃	direct (N × H)
124. C₂×#365	C₂×C₂₃	semidirect
125. C₂×#365	C₂×C₂₃	semidirect
128. #1388	C₂×C₂₃	semidirect
129. #1385	C₂×C₂₃	semidirect
156. #350	C₄×C₂₃	semidirect
196. C₂₃	#12933	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×#2916	2
C₂×C₂₃×#365	2
C₂₃×#2117	2
C₂₃×#1815	2
C₂₃×#1388	2
C₂₃×#1385	2
#12933	23

Subgroups — C23×#12933 #19160559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12933 #19160559