Subgroups — C₂₃×#12650 #19160276

Order 5888.

438 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#444	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#360	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2139	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#347	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1776	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1387	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#131	—	2	1	no	—
10	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂₃×#444	—	2	2	no	—
13	C₂₃×#444	C₄	1	4	yes	complemented
14	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#104	—	2	1	no	—
18	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
20	C₂₃×#360	C₄	1	4	yes	complemented
21	C₂₃×#358	C₄	1	2	yes	complemented
22	C₂₃×#357	C₄	1	2	yes	complemented
23	C₂₃×#347	—	2	2	no	—
24	C₂₃×#340	—	2	1	no	—
25	C₂×#41×C₂₃	—	2	1	no	—
26	C₂₃×#135	—	2	1	no	—
27	C₂₃×#135	—	2	1	no	—
28	C₂₃×#135	C₂×C₄	1	2	yes	—
29	C₂₃×#135	C₂×C₄	1	2	yes	—
30	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂×SD₁₆×C₂₃	—	2	1	no	—
32	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂₃×#131	—	2	4	no	—
34	C₂₃×#131	—	4	2	no	—
35	C₂×M₁₆×C₂₃	—	2	1	no	—
36	C₂×M₁₆×C₂₃	—	4	1	no	—
37	C₂²×C₈×C₂₃	—	4	1	no	—
38	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
42	C₂×C₄²×C₂₃	—	4	1	no	—
43	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
44	C₂₃×M₃₂	—	2	1	no	—
45	C₂₃×M₃₂	—	2	1	no	—
46	C₂₃×M₃₂	—	2	1	no	—
47	C₂₃×M₃₂	C₂×C₄	1	2	yes	—
48	C₂×C₁₆×C₂₃	—	2	1	no	—
49	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
50	C₂₃×#108	—	2	1	no	—
51	C₂₃×#108	C₂×C₄	1	2	yes	—
52	C₂₃×#107	—	2	1	no	—
53	C₂₃×#107	C₂×C₄	1	2	yes	—
54	C₂₃×#106	—	2	1	no	—
55	C₂₃×#106	C₂×C₄	1	2	yes	—
56	C₂₃×#104	—	2	4	no	—
57	C₂₃×#97	C₂×C₄	1	2	yes	—
58	C₂₃×#97	D₄	1	2	yes	—
59	C₂₃×#95	—	4	1	no	—
60	#41×C₂₃	—	2	2	no	—
61	#41×C₂₃	—	2	2	no	—
62	#41×C₂₃	—	4	1	no	—
63	#41×C₂₃	—	4	1	no	—
64	C₂×Q₈×C₂₃	—	2	1	no	—
65	C₂×D₄×C₂₃	—	2	1	no	—
66	C₂×D₄×C₂₃	—	4	1	no	—
67	C₂²×C₄×C₂₃	—	2	1	no	—
68	C₂²×C₄×C₂₃	—	4	1	no	—
69	C₂²×C₄×C₂₃	—	4	1	no	—
70	C₂²×C₄×C₂₃	—	4	1	no	—
71	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
72	Q₁₆×C₂₃	—	2	1	no	—
73	Q₁₆×C₂₃	C₄²	1	2	yes	complemented
74	SD₁₆×C₂₃	—	2	2	no	—
75	SD₁₆×C₂₃	—	2	2	no	—
76	D₈×C₂₃	—	2	1	no	—
77	D₈×C₂₃	C₄²	1	2	yes	complemented
78	M₁₆×C₂₃	—	2	2	no	—
79	M₁₆×C₂₃	—	4	4	no	—
80	M₁₆×C₂₃	—	4	1	no	—
81	C₂×C₈×C₂₃	—	2	1	no	—
82	C₂×C₈×C₂₃	—	2	1	no	—
83	C₂×C₈×C₂₃	—	4	4	no	—
84	C₂×C₈×C₂₃	—	4	1	no	—
85	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
91	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
92	C₄⋊C₄×C₂₃	—	2	2	no	—
93	C₄⋊C₄×C₂₃	—	2	2	no	—
94	C₄⋊C₄×C₂₃	—	4	1	no	—
95	C₄⋊C₄×C₂₃	—	4	1	no	—
96	C₄²×C₂₃	—	2	1	no	—
97	C₄²×C₂₃	—	8	2	no	—
98	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
99	C₁₆×C₂₃	—	2	2	no	—
100	C₁₆×C₂₃	—	2	2	no	—
101	#12650	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂³×C₂₃	—	4	1	no	—
103	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
104	Q₈×C₂₃	—	2	2	no	—
105	Q₈×C₂₃	—	4	1	no	—
106	D₄×C₂₃	—	2	2	no	—
107	D₄×C₂₃	—	4	2	no	—
108	D₄×C₂₃	—	4	2	no	—
109	D₄×C₂₃	—	4	1	no	—
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	—	4	1	no	—
112	C₂×C₄×C₂₃	—	4	1	no	—
113	C₂×C₄×C₂₃	—	4	1	no	—
114	C₂×C₄×C₂₃	—	4	2	no	—
115	C₂×C₄×C₂₃	—	4	2	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	—	4	2	no	—
118	C₂×C₄×C₂₃	—	8	1	no	—
119	C₂×C₄×C₂₃	—	8	1	no	—
120	C₂×C₄×C₂₃	—	8	1	no	—
121	C₂×C₄×C₂₃	—	8	1	no	—
122	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
128	C₈×C₂₃	—	2	2	no	—
129	C₈×C₂₃	—	4	2	no	—
130	C₈×C₂₃	C₂×C₄²	1	1	yes	characteristic
131	C₈×C₂₃	#117	1	1	yes	characteristic
132	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
133	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
134	C₂×#444	C₂×C₂₃	1	1	yes	characteristic
135	C₂×#360	C₂×C₂₃	1	1	yes	characteristic
136	#2139	C₂×C₂₃	1	1	yes	characteristic
137	C₂×#347	C₂×C₂₃	1	1	yes	characteristic, complemented
138	#1776	C₂×C₂₃	1	1	yes	characteristic, complemented
139	#1387	C₂×C₂₃	1	1	yes	characteristic, complemented
140	#1383	C₂×C₂₃	1	1	yes	characteristic, complemented
141	C₂²×C₂₃	—	2	1	no	—
142	C₂²×C₂₃	—	2	1	no	—
143	C₂²×C₂₃	—	4	2	no	—
144	C₂²×C₂₃	—	8	1	no	—
145	C₂²×C₂₃	—	8	1	no	—
146	C₂²×C₂₃	#417	1	1	yes	characteristic
147	C₂²×C₂₃	#442	1	1	yes	characteristic
148	C₂²×C₂₃	#442	1	1	yes	characteristic
149	C₄×C₂₃	—	4	2	no	—
150	C₄×C₂₃	—	4	2	no	—
151	C₄×C₂₃	—	8	2	no	—
152	C₄×C₂₃	—	8	2	no	—
153	C₄×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
154	C₄×C₂₃	#418	1	1	yes	characteristic
155	C₄×C₂₃	#440	1	1	yes	characteristic
156	C₄×C₂₃	#440	1	1	yes	characteristic
157	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
158	C₂×#131	—	2	1	no	—
159	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
160	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
161	#444	—	2	2	no	—
162	#444	C₄×C₂₃	1	4	yes	complemented
163	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
164	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
165	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
166	C₂×#104	—	2	1	no	—
167	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
168	C₄×C₄⋊C₄	—	2	1	no	—
169	#360	C₄×C₂₃	1	4	yes	complemented
170	#358	C₄×C₂₃	1	2	yes	complemented
171	#357	C₄×C₂₃	1	2	yes	complemented
172	#347	—	2	2	no	—
173	#340	—	2	1	no	—
174	C₂×C₂₃	—	2	1	no	—
175	C₂×C₂₃	—	2	1	no	—
176	C₂×C₂₃	—	8	2	no	—
177	C₂×C₂₃	#1762	1	1	yes	characteristic
178	C₂×C₂₃	#2180	1	1	yes	characteristic
179	C₂×C₂₃	AGL(1,ℤ/16)	1	1	yes	characteristic
180	C₂×#41	—	2	1	no	—
181	#135	—	2	1	no	—
182	#135	—	2	1	no	—
183	#135	C₂×C₄×C₂₃	1	2	yes	—
184	#135	C₂×C₄×C₂₃	1	2	yes	—
185	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
186	C₂×SD₁₆	—	2	1	no	—
187	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
188	#131	—	2	4	no	—
189	#131	—	4	2	no	—
190	C₂×M₁₆	—	2	1	no	—
191	C₂×M₁₆	—	4	1	no	—
192	C₂²×C₈	—	4	1	no	—
193	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄²	—	4	1	no	—
198	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
199	M₃₂	—	2	1	no	—
200	M₃₂	—	2	1	no	—
201	M₃₂	—	2	1	no	—
202	M₃₂	C₂×C₄×C₂₃	1	2	yes	—
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
205	#108	—	2	1	no	—
206	#108	C₂×C₄×C₂₃	1	2	yes	—
207	#107	—	2	1	no	—
208	#107	C₂×C₄×C₂₃	1	2	yes	—
209	#106	—	2	1	no	—
210	#106	C₂×C₄×C₂₃	1	2	yes	—
211	#104	—	2	4	no	—
212	#97	C₂×C₄×C₂₃	1	2	yes	—
213	#97	D₄×C₂₃	1	2	yes	—
214	#95	—	4	1	no	—
215	C₂₃	#12650	1	1	yes	characteristic, direct product factor, Sylow
216	#41	—	2	2	no	—
217	#41	—	2	2	no	—
218	#41	—	4	1	no	—
219	#41	—	4	1	no	—
220	C₂×Q₈	—	2	1	no	—
221	C₂×D₄	—	2	1	no	—
222	C₂×D₄	—	4	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
228	Q₁₆	—	2	1	no	—
229	Q₁₆	C₄²×C₂₃	1	2	yes	complemented
230	SD₁₆	—	2	2	no	—
231	SD₁₆	—	2	2	no	—
232	D₈	—	2	1	no	—
233	D₈	C₄²×C₂₃	1	2	yes	complemented
234	M₁₆	—	2	2	no	—
235	M₁₆	—	4	4	no	—
236	M₁₆	—	4	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	4	4	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
242	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
243	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
244	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
245	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
246	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
247	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	8	2	no	—
254	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
255	C₁₆	—	2	2	no	—
256	C₁₆	—	2	2	no	—
257	C₂³	—	4	1	no	—
258	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
259	Q₈	—	2	2	no	—
260	Q₈	—	4	1	no	—
261	D₄	—	2	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	4	2	no	—
264	D₄	—	4	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	—	8	1	no	—
277	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
278	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
279	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
280	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
281	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
282	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	4	2	no	—
285	C₈	C₂×C₄²×C₂₃	1	1	yes	characteristic
286	C₈	C₂₃×#117	1	1	yes	characteristic
287	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
288	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	8	1	no	—
293	C₂²	—	8	1	no	—
294	C₂²	C₂₃×#417	1	1	yes	characteristic
295	C₂²	C₂₃×#442	1	1	yes	characteristic
296	C₂²	C₂₃×#442	1	1	yes	characteristic
297	C₄	—	4	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	2	no	—
301	C₄	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
302	C₄	C₂₃×#418	1	1	yes	characteristic
303	C₄	C₂₃×#440	1	1	yes	characteristic
304	C₄	C₂₃×#440	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	8	2	no	—
308	C₂	C₂₃×#1762	1	1	yes	characteristic
309	C₂	C₂₃×#2180	1	1	yes	characteristic
310	C₂	C₂₃×AGL(1,ℤ/16)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#442
commutator	C₈	C₂×C₄²×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂×C₂₃×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₂₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₂₃×#347	C₂	semidirect
5. C₂₃×#1776	C₂	semidirect
6. C₂₃×#1387	C₂	semidirect
7. C₂₃×#1383	C₂	semidirect
13. C₂₃×#444	C₄	semidirect
20. C₂₃×#360	C₄	semidirect
21. C₂₃×#358	C₄	semidirect
22. C₂₃×#357	C₄	semidirect
73. Q₁₆×C₂₃	C₄²	semidirect
77. D₈×C₂₃	C₄²	semidirect
101. #12650	C₂₃	direct (N × H)
137. C₂×#347	C₂×C₂₃	semidirect
138. #1776	C₂×C₂₃	semidirect
139. #1387	C₂×C₂₃	semidirect
140. #1383	C₂×C₂₃	semidirect
162. #444	C₄×C₂₃	semidirect
169. #360	C₄×C₂₃	semidirect
170. #358	C₄×C₂₃	semidirect
171. #357	C₄×C₂₃	semidirect
215. C₂₃	#12650	direct (N × H)
229. Q₁₆	C₄²×C₂₃	semidirect
233. D₈	C₄²×C₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#444	2
C₂×C₂₃×#360	2
C₂₃×#2139	2
C₂×C₂₃×#347	2
C₂₃×#1776	2
C₂₃×#1387	2
C₂₃×#1383	2
#12650	23

Subgroups — C23×#12650 #19160276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12650 #19160276