Subgroups — C₂₃×#13260 #19160886

Order 5888.

482 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#495	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#361	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2141	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2140	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×SD₁₆×C₂₃	—	2	1	no	—
10	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#124	—	2	1	no	—
12	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂₃×#495	—	2	2	no	—
15	C₂₃×#495	C₄	1	4	yes	complemented
16	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
17	C₂₃×#427	C₂²	1	1	yes	characteristic
18	C₂₃×#405	—	2	1	no	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#364	—	2	2	no	—
21	C₂₃×#362	—	2	2	no	—
22	C₂₃×#361	—	2	2	no	—
23	C₂₃×#358	—	2	2	no	—
24	C₂₃×#357	—	2	2	no	—
25	C₂₃×#339	—	2	1	no	—
26	C₂²×Q₈×C₂₃	—	2	1	no	—
27	C₂²×D₄×C₂₃	—	2	1	no	—
28	C₂×Q₁₆×C₂₃	—	2	1	no	—
29	C₂×Q₁₆×C₂₃	—	2	1	no	—
30	C₂×Q₁₆×C₂₃	—	2	1	no	—
31	C₂×Q₁₆×C₂₃	—	2	1	no	—
32	C₂×Q₁₆×C₂₃	—	2	1	no	—
33	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
34	C₂×SD₁₆×C₂₃	—	2	4	no	—
35	C₂×SD₁₆×C₂₃	—	4	1	no	—
36	C₂×SD₁₆×C₂₃	—	4	1	no	—
37	C₂×SD₁₆×C₂₃	—	4	2	no	—
38	C₂×D₈×C₂₃	—	2	1	no	—
39	C₂×D₈×C₂₃	—	2	1	no	—
40	C₂×D₈×C₂₃	—	2	1	no	—
41	C₂×D₈×C₂₃	—	2	1	no	—
42	C₂×D₈×C₂₃	—	2	1	no	—
43	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
44	C₂×M₁₆×C₂₃	—	2	1	no	—
45	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂₃×#124	—	2	4	no	—
48	C₂₃×#124	—	4	2	no	—
49	C₂₃×#117	—	2	1	no	—
50	C₂×C₂²⋊C₄×C₂₃	—	4	2	no	—
51	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
52	C₂₃×M₃₂	—	2	1	no	—
53	C₂₃×M₃₂	—	2	1	no	—
54	C₂₃×M₃₂	—	2	1	no	—
55	C₂₃×M₃₂	—	2	1	no	—
56	C₂×C₁₆×C₂₃	—	2	1	no	—
57	C₂×C₁₆×C₂₃	—	2	1	no	—
58	C₂₃×#108	—	2	1	no	—
59	C₂₃×#108	D₄	1	2	yes	—
60	C₂₃×#107	D₄	1	2	yes	—
61	C₂₃×#106	—	2	1	no	—
62	C₂₃×#97	—	4	1	no	—
63	C₄×C₈×C₂₃	—	4	1	no	—
64	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₂₃	D₄	1	2	yes	—
66	C₂⁴×C₂₃	—	4	1	no	—
67	C₂×Q₈×C₂₃	—	2	2	no	—
68	C₂×Q₈×C₂₃	—	4	1	no	—
69	C₂×Q₈×C₂₃	—	4	1	no	—
70	C₂×Q₈×C₂₃	—	4	1	no	—
71	C₂×Q₈×C₂₃	—	4	1	no	—
72	C₂×Q₈×C₂₃	—	4	1	no	—
73	C₂×D₄×C₂₃	—	2	2	no	—
74	C₂×D₄×C₂₃	—	4	1	no	—
75	C₂×D₄×C₂₃	—	4	1	no	—
76	C₂×D₄×C₂₃	—	4	1	no	—
77	C₂×D₄×C₂₃	—	4	1	no	—
78	C₂×D₄×C₂₃	—	4	1	no	—
79	C₂²×C₄×C₂₃	—	2	1	no	—
80	C₂²×C₄×C₂₃	—	4	1	no	—
81	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
82	Q₁₆×C₂₃	—	2	2	no	—
83	Q₁₆×C₂₃	—	2	2	no	—
84	Q₁₆×C₂₃	—	4	1	no	—
85	Q₁₆×C₂₃	—	4	1	no	—
86	SD₁₆×C₂₃	—	4	4	no	—
87	SD₁₆×C₂₃	—	4	4	no	—
88	D₈×C₂₃	—	2	2	no	—
89	D₈×C₂₃	—	2	2	no	—
90	D₈×C₂₃	—	4	1	no	—
91	D₈×C₂₃	—	4	1	no	—
92	M₁₆×C₂₃	—	2	2	no	—
93	M₁₆×C₂₃	—	4	1	no	—
94	C₂×C₈×C₂₃	—	4	1	no	—
95	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
96	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
97	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
98	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
99	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
104	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
105	C₄⋊C₄×C₂₃	—	2	2	no	—
106	C₂²⋊C₄×C₂₃	—	4	8	no	—
107	C₂²⋊C₄×C₂₃	—	4	1	no	—
108	C₄²×C₂₃	—	2	1	no	—
109	C₄²×C₂₃	—	4	1	no	—
110	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
111	C₁₆×C₂₃	—	4	1	no	—
112	C₁₆×C₂₃	—	4	1	no	—
113	#13260	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂³×C₂₃	—	4	2	no	—
115	C₂³×C₂₃	—	8	4	no	—
116	C₂³×C₂₃	—	8	1	no	—
117	C₂³×C₂₃	—	8	1	no	—
118	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
119	Q₈×C₂₃	—	4	2	no	—
120	Q₈×C₂₃	—	4	2	no	—
121	Q₈×C₂₃	—	8	1	no	—
122	Q₈×C₂₃	—	8	1	no	—
123	D₄×C₂₃	—	4	2	no	—
124	D₄×C₂₃	—	4	2	no	—
125	D₄×C₂₃	—	8	1	no	—
126	D₄×C₂₃	—	8	1	no	—
127	C₂×C₄×C₂₃	—	2	2	no	—
128	C₂×C₄×C₂₃	—	4	2	no	—
129	C₂×C₄×C₂₃	—	4	2	no	—
130	C₂×C₄×C₂₃	—	4	2	no	—
131	C₂×C₄×C₂₃	—	8	1	no	—
132	C₂×C₄×C₂₃	—	8	1	no	—
133	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₈×C₂₃	—	4	2	no	—
140	C₈×C₂₃	#102	1	2	yes	—
141	C₈×C₂₃	#102	1	2	yes	—
142	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
143	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
144	C₈×C₂₃	#124	1	1	yes	characteristic
145	C₈×C₂₃	#127	1	1	yes	characteristic
146	C₂×#495	C₂×C₂₃	1	1	yes	characteristic
147	C₂×#364	C₂×C₂₃	1	1	yes	characteristic, complemented
148	C₂×#362	C₂×C₂₃	1	1	yes	characteristic, complemented
149	C₂×#361	C₂×C₂₃	1	1	yes	characteristic
150	#2141	C₂×C₂₃	1	1	yes	characteristic, complemented
151	#2140	C₂×C₂₃	1	1	yes	characteristic
152	#1772	C₂×C₂₃	1	1	yes	characteristic, complemented
153	C₂²×C₂₃	—	2	2	no	—
154	C₂²×C₂₃	—	8	4	no	—
155	C₂²×C₂₃	—	8	2	no	—
156	C₂²×C₂₃	—	16	2	no	—
157	C₂²×C₂₃	—	16	2	no	—
158	C₂²×C₂₃	#418	1	1	yes	characteristic
159	C₂²×C₂₃	#459	1	1	yes	characteristic
160	C₂²×C₂₃	#460	1	1	yes	characteristic
161	C₄×C₂₃	—	4	2	no	—
162	C₄×C₂₃	—	8	2	no	—
163	C₄×C₂₃	—	8	2	no	—
164	C₄×C₂₃	#390	1	1	yes	characteristic
165	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
166	C₄×C₂₃	#459	1	1	yes	characteristic
167	C₄×C₂₃	#460	1	1	yes	characteristic
168	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
169	C₂²×SD₁₆	—	2	1	no	—
170	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
171	C₂×#124	—	2	1	no	—
172	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
173	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
174	#495	—	2	2	no	—
175	#495	C₄×C₂₃	1	4	yes	complemented
176	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
177	#427	C₂²×C₂₃	1	1	yes	characteristic
178	#405	—	2	1	no	—
179	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
180	#364	—	2	2	no	—
181	#362	—	2	2	no	—
182	#361	—	2	2	no	—
183	#358	—	2	2	no	—
184	#357	—	2	2	no	—
185	#339	—	2	1	no	—
186	C₂×C₂₃	—	2	1	no	—
187	C₂×C₂₃	—	2	1	no	—
188	C₂×C₂₃	—	16	2	no	—
189	C₂×C₂₃	#1982	1	1	yes	characteristic
190	C₂×C₂₃	#2212	1	2	yes	—
191	C₂²×Q₈	—	2	1	no	—
192	C₂²×D₄	—	2	1	no	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	—	2	1	no	—
195	C₂×Q₁₆	—	2	1	no	—
196	C₂×Q₁₆	—	2	1	no	—
197	C₂×Q₁₆	—	2	1	no	—
198	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
199	C₂×SD₁₆	—	2	4	no	—
200	C₂×SD₁₆	—	4	1	no	—
201	C₂×SD₁₆	—	4	1	no	—
202	C₂×SD₁₆	—	4	2	no	—
203	C₂×D₈	—	2	1	no	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	—	2	1	no	—
207	C₂×D₈	—	2	1	no	—
208	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
209	C₂×M₁₆	—	2	1	no	—
210	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
212	#124	—	2	4	no	—
213	#124	—	4	2	no	—
214	#117	—	2	1	no	—
215	C₂×C₂²⋊C₄	—	4	2	no	—
216	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
217	M₃₂	—	2	1	no	—
218	M₃₂	—	2	1	no	—
219	M₃₂	—	2	1	no	—
220	M₃₂	—	2	1	no	—
221	C₂×C₁₆	—	2	1	no	—
222	C₂×C₁₆	—	2	1	no	—
223	#108	—	2	1	no	—
224	#108	D₄×C₂₃	1	2	yes	—
225	#107	D₄×C₂₃	1	2	yes	—
226	#106	—	2	1	no	—
227	#97	—	4	1	no	—
228	C₄×C₈	—	4	1	no	—
229	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
230	C₄×C₈	D₄×C₂₃	1	2	yes	—
231	C₂₃	#13260	1	1	yes	characteristic, direct product factor, Sylow
232	C₂⁴	—	4	1	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
248	Q₁₆	—	2	2	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	—	4	1	no	—
251	Q₁₆	—	4	1	no	—
252	SD₁₆	—	4	4	no	—
253	SD₁₆	—	4	4	no	—
254	D₈	—	2	2	no	—
255	D₈	—	2	2	no	—
256	D₈	—	4	1	no	—
257	D₈	—	4	1	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
262	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
263	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
264	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
265	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
266	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
269	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
270	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	4	8	no	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
277	C₁₆	—	4	1	no	—
278	C₁₆	—	4	1	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	8	4	no	—
281	C₂³	—	8	1	no	—
282	C₂³	—	8	1	no	—
283	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	8	1	no	—
287	Q₈	—	8	1	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	8	1	no	—
291	D₄	—	8	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
299	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
300	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
302	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
303	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
304	C₈	—	4	2	no	—
305	C₈	C₂₃×#102	1	2	yes	—
306	C₈	C₂₃×#102	1	2	yes	—
307	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
308	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
309	C₈	C₂₃×#124	1	1	yes	characteristic
310	C₈	C₂₃×#127	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	8	4	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	16	2	no	—
315	C₂²	—	16	2	no	—
316	C₂²	C₂₃×#418	1	1	yes	characteristic
317	C₂²	C₂₃×#459	1	1	yes	characteristic
318	C₂²	C₂₃×#460	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	C₂₃×#390	1	1	yes	characteristic
323	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
324	C₄	C₂₃×#459	1	1	yes	characteristic
325	C₄	C₂₃×#460	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	16	2	no	—
329	C₂	C₂₃×#1982	1	1	yes	characteristic
330	C₂	C₂₃×#2212	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	#459
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#364	C₂	semidirect
3. C₂×C₂₃×#362	C₂	semidirect
5. C₂₃×#2141	C₂	semidirect
7. C₂₃×#1772	C₂	semidirect
15. C₂₃×#495	C₄	semidirect
113. #13260	C₂₃	direct (N × H)
147. C₂×#364	C₂×C₂₃	semidirect
148. C₂×#362	C₂×C₂₃	semidirect
150. #2141	C₂×C₂₃	semidirect
152. #1772	C₂×C₂₃	semidirect
175. #495	C₄×C₂₃	semidirect
231. C₂₃	#13260	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×#495	2
C₂×C₂₃×#364	2
C₂×C₂₃×#362	2
C₂×C₂₃×#361	2
C₂₃×#2141	2
C₂₃×#2140	2
C₂₃×#1772	2
#13260	23

Subgroups — C23×#13260 #19160886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13260 #19160886