Subgroups — C₂×C₂₃×#1687 #19166757

Order 5888.

926 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#459	C₂	1	2	yes	maximal, complemented
3	C₂×C₂₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1687	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#331	C₂	1	2	yes	maximal, complemented
7	C₂²×D₈×C₂₃	—	2	2	no	—
8	C₂²×D₈×C₂₃	—	2	1	no	—
9	C₂²×D₈×C₂₃	—	2	1	no	—
10	C₂²×M₁₆×C₂₃	—	2	1	no	—
11	C₂×C₂₃×#127	C₂²	1	2	yes	—
12	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#493	—	2	2	no	—
14	C₂₃×#493	C₂²	1	4	yes	complemented
15	C₂₃×#459	—	2	4	no	—
16	C₂₃×#459	C₂²	1	8	yes	complemented
17	C₂₃×#436	—	2	2	no	—
18	C₂₃×#436	C₂²	1	4	yes	complemented
19	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₂₃×#102	—	2	2	no	—
22	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#334	C₂²	1	4	yes	complemented
25	C₂₃×#331	C₂²	1	8	yes	complemented
26	C₂²×D₄×C₂₃	—	2	2	no	—
27	C₂²×D₄×C₂₃	—	2	2	no	—
28	C₂²×D₄×C₂₃	—	2	2	no	—
29	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₂₃	—	2	1	no	—
31	C₂×D₈×C₂₃	—	2	8	no	—
32	C₂×D₈×C₂₃	—	2	4	no	—
33	C₂×D₈×C₂₃	—	2	4	no	—
34	C₂×D₈×C₂₃	—	4	4	no	—
35	C₂×D₈×C₂₃	—	4	8	no	—
36	C₂×D₈×C₂₃	—	4	4	no	—
37	C₂×M₁₆×C₂₃	—	2	4	no	—
38	C₂×M₁₆×C₂₃	—	4	4	no	—
39	C₂²×C₈×C₂₃	—	2	1	no	—
40	C₂²×C₈×C₂₃	—	2	2	no	—
41	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
43	C₂₃×#127	—	2	2	no	—
44	C₂₃×#127	—	2	2	no	—
45	C₂₃×#127	—	2	2	no	—
46	C₂₃×#127	C₂³	1	4	yes	—
47	C₄×D₄×C₂₃	—	2	1	no	—
48	C₄×D₄×C₂₃	—	2	1	no	—
49	C₄×D₄×C₂₃	D₄	1	2	yes	—
50	C₄×D₄×C₂₃	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
53	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
54	C₂₃×C₄⋊C₈	—	2	1	no	—
55	C₂₃×C₄⋊C₈	—	2	2	no	—
56	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
57	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
58	C₂₃×#102	—	2	8	no	—
59	C₂₃×C₂²⋊C₈	—	2	4	no	—
60	C₄×C₈×C₂₃	D₄	1	2	yes	—
61	C₄×C₈×C₂₃	C₂³	1	2	yes	—
62	C₂⁴×C₂₃	—	2	1	no	—
63	C₂⁴×C₂₃	—	4	2	no	—
64	C₂×D₄×C₂₃	—	2	1	no	—
65	C₂×D₄×C₂₃	—	2	4	no	—
66	C₂×D₄×C₂₃	—	2	4	no	—
67	C₂×D₄×C₂₃	—	2	4	no	—
68	C₂×D₄×C₂₃	—	2	4	no	—
69	C₂×D₄×C₂₃	—	4	2	no	—
70	C₂×D₄×C₂₃	—	4	2	no	—
71	C₂×D₄×C₂₃	—	4	2	no	—
72	C₂×D₄×C₂₃	—	4	4	no	—
73	C₂×D₄×C₂₃	—	4	4	no	—
74	C₂×D₄×C₂₃	—	4	4	no	—
75	C₂×D₄×C₂₃	—	4	4	no	—
76	C₂×D₄×C₂₃	—	8	4	no	—
77	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₂₃	—	2	1	no	—
79	C₂²×C₄×C₂₃	—	2	2	no	—
80	C₂²×C₄×C₂₃	—	2	2	no	—
81	C₂²×C₄×C₂₃	—	4	2	no	—
82	C₂²×C₄×C₂₃	—	4	2	no	—
83	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
86	D₈×C₂₃	—	4	8	no	—
87	D₈×C₂₃	—	4	16	no	—
88	D₈×C₂₃	—	4	8	no	—
89	M₁₆×C₂₃	—	4	8	no	—
90	C₂×C₈×C₂₃	—	2	2	no	—
91	C₂×C₈×C₂₃	—	2	4	no	—
92	C₂×C₈×C₂₃	—	2	2	no	—
93	C₂×C₈×C₂₃	—	4	2	no	—
94	C₂×C₈×C₂₃	—	4	4	no	—
95	C₂×C₈×C₂₃	D₈	1	4	yes	complemented
96	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
98	C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
100	C₂²⋊C₄×C₂₃	—	2	2	no	—
101	C₂²⋊C₄×C₂₃	—	2	2	no	—
102	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₂₃	#41	1	1	yes	characteristic
105	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
106	C₂×#1687	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₂₃	—	2	2	no	—
108	C₂³×C₂₃	—	4	2	no	—
109	C₂³×C₂₃	—	4	2	no	—
110	C₂³×C₂₃	—	4	2	no	—
111	C₂³×C₂₃	—	4	4	no	—
112	C₂³×C₂₃	—	8	4	no	—
113	C₂³×C₂₃	—	8	4	no	—
114	C₂³×C₂₃	—	8	4	no	—
115	C₂³×C₂₃	#121	1	1	yes	characteristic
116	D₄×C₂₃	—	2	4	no	—
117	D₄×C₂₃	—	4	4	no	—
118	D₄×C₂₃	—	4	4	no	—
119	D₄×C₂₃	—	4	4	no	—
120	D₄×C₂₃	—	4	4	no	—
121	D₄×C₂₃	—	4	2	no	—
122	D₄×C₂₃	—	8	4	no	—
123	D₄×C₂₃	—	8	4	no	—
124	D₄×C₂₃	—	8	4	no	—
125	D₄×C₂₃	—	8	4	no	—
126	C₂×C₄×C₂₃	—	2	2	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	—	4	4	no	—
132	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
137	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
138	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
141	C₈×C₂₃	—	2	2	no	—
142	C₈×C₂₃	—	4	2	no	—
143	C₈×C₂₃	—	4	4	no	—
144	C₈×C₂₃	C₂×D₈	1	4	yes	complemented
145	C₂×#493	C₂×C₂₃	1	1	yes	characteristic, complemented
146	C₂×#459	C₂×C₂₃	1	2	yes	complemented
147	C₂×#436	C₂×C₂₃	1	1	yes	characteristic, complemented
148	#1687	C₂×C₂₃	1	8	yes	direct product factor
149	C₂×#334	C₂×C₂₃	1	1	yes	characteristic, complemented
150	C₂×#331	C₂×C₂₃	1	2	yes	complemented
151	C₂²×C₂₃	—	4	2	no	—
152	C₂²×C₂₃	—	4	2	no	—
153	C₂²×C₂₃	—	4	2	no	—
154	C₂²×C₂₃	—	8	4	no	—
155	C₂²×C₂₃	—	8	4	no	—
156	C₂²×C₂₃	—	8	4	no	—
157	C₂²×C₂₃	—	8	4	no	—
158	C₂²×C₂₃	—	16	8	no	—
159	C₂²×C₂₃	#459	1	2	yes	—
160	C₂²×C₂₃	#469	1	2	yes	—
161	C₂²×C₂₃	#472	1	2	yes	—
162	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
163	C₄×C₂₃	—	2	1	no	—
164	C₄×C₂₃	—	2	1	no	—
165	C₄×C₂₃	—	4	2	no	—
166	C₄×C₂₃	#459	1	2	yes	—
167	C₄×C₂₃	#469	1	2	yes	—
168	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
169	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
171	C₂²×D₈	—	2	2	no	—
172	C₂²×D₈	—	2	1	no	—
173	C₂²×D₈	—	2	1	no	—
174	C₂²×M₁₆	—	2	1	no	—
175	C₂×#127	C₂²×C₂₃	1	2	yes	—
176	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
177	#493	—	2	2	no	—
178	#493	C₂²×C₂₃	1	4	yes	complemented
179	#459	—	2	4	no	—
180	#459	C₂²×C₂₃	1	8	yes	complemented
181	#436	—	2	2	no	—
182	#436	C₂²×C₂₃	1	4	yes	complemented
183	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
184	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
185	C₂×#102	—	2	2	no	—
186	C₂×C₂²⋊C₈	—	2	1	no	—
187	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
188	#334	C₂²×C₂₃	1	4	yes	complemented
189	#331	C₂²×C₂₃	1	8	yes	complemented
190	C₂×C₂₃	—	8	2	no	—
191	C₂×C₂₃	—	16	4	no	—
192	C₂×C₂₃	#1687	1	4	yes	direct product factor
193	C₂×C₂₃	C₂×#459	1	1	yes	characteristic
194	C₂×C₂₃	C₂×#469	1	1	yes	characteristic
195	C₂×C₂₃	C₂×#472	1	1	yes	characteristic
196	C₂²×D₄	—	2	2	no	—
197	C₂²×D₄	—	2	2	no	—
198	C₂²×D₄	—	2	2	no	—
199	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
200	C₂³×C₄	—	2	1	no	—
201	C₂×D₈	—	2	8	no	—
202	C₂×D₈	—	2	4	no	—
203	C₂×D₈	—	2	4	no	—
204	C₂×D₈	—	4	4	no	—
205	C₂×D₈	—	4	8	no	—
206	C₂×D₈	—	4	4	no	—
207	C₂×M₁₆	—	2	4	no	—
208	C₂×M₁₆	—	4	4	no	—
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	2	2	no	—
211	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
212	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
213	#127	—	2	2	no	—
214	#127	—	2	2	no	—
215	#127	—	2	2	no	—
216	#127	C₂³×C₂₃	1	4	yes	—
217	C₄×D₄	—	2	1	no	—
218	C₄×D₄	—	2	1	no	—
219	C₄×D₄	D₄×C₂₃	1	2	yes	—
220	C₄×D₄	C₂³×C₂₃	1	2	yes	—
221	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
222	C₂×C₂²⋊C₄	—	2	1	no	—
223	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
227	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
228	#102	—	2	8	no	—
229	C₂²⋊C₈	—	2	4	no	—
230	C₄×C₈	D₄×C₂₃	1	2	yes	—
231	C₄×C₈	C₂³×C₂₃	1	2	yes	—
232	C₂₃	C₂×#1687	1	1	yes	characteristic, direct product factor, Sylow
233	C₂⁴	—	2	1	no	—
234	C₂⁴	—	4	2	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	2	4	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	4	no	—
244	C₂×D₄	—	4	4	no	—
245	C₂×D₄	—	4	4	no	—
246	C₂×D₄	—	4	4	no	—
247	C₂×D₄	—	8	4	no	—
248	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
256	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
257	D₈	—	4	8	no	—
258	D₈	—	4	16	no	—
259	D₈	—	4	8	no	—
260	M₁₆	—	4	8	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	4	2	no	—
265	C₂×C₈	—	4	4	no	—
266	C₂×C₈	D₈×C₂₃	1	4	yes	complemented
267	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
268	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
275	C₄²	#41×C₂₃	1	1	yes	characteristic
276	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
277	C₂³	—	2	2	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	4	no	—
282	C₂³	—	8	4	no	—
283	C₂³	—	8	4	no	—
284	C₂³	—	8	4	no	—
285	C₂³	C₂₃×#121	1	1	yes	characteristic
286	D₄	—	2	4	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	2	no	—
292	D₄	—	8	4	no	—
293	D₄	—	8	4	no	—
294	D₄	—	8	4	no	—
295	D₄	—	8	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
303	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
304	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
305	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
306	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
307	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
308	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
310	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	4	2	no	—
313	C₈	—	4	4	no	—
314	C₈	C₂×D₈×C₂₃	1	4	yes	complemented
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	8	4	no	—
319	C₂²	—	8	4	no	—
320	C₂²	—	8	4	no	—
321	C₂²	—	8	4	no	—
322	C₂²	—	16	8	no	—
323	C₂²	C₂₃×#459	1	2	yes	—
324	C₂²	C₂₃×#469	1	2	yes	—
325	C₂²	C₂₃×#472	1	2	yes	—
326	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	4	2	no	—
330	C₄	C₂₃×#459	1	2	yes	—
331	C₄	C₂₃×#469	1	2	yes	—
332	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
333	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
334	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
335	C₂	—	8	2	no	—
336	C₂	—	16	4	no	—
337	C₂	C₂₃×#1687	1	4	yes	direct product factor
338	C₂	C₂×C₂₃×#459	1	1	yes	characteristic
339	C₂	C₂×C₂₃×#469	1	1	yes	characteristic
340	C₂	C₂×C₂₃×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#493	C₂	semidirect
2. C₂×C₂₃×#459	C₂	semidirect
3. C₂×C₂₃×#436	C₂	semidirect
4. C₂₃×#1687	C₂	direct (N × H)
5. C₂×C₂₃×#334	C₂	semidirect
6. C₂×C₂₃×#331	C₂	semidirect
14. C₂₃×#493	C₂²	semidirect
16. C₂₃×#459	C₂²	semidirect
18. C₂₃×#436	C₂²	semidirect
24. C₂₃×#334	C₂²	semidirect
25. C₂₃×#331	C₂²	semidirect
95. C₂×C₈×C₂₃	D₈	semidirect
106. C₂×#1687	C₂₃	direct (N × H)
144. C₈×C₂₃	C₂×D₈	semidirect
145. C₂×#493	C₂×C₂₃	semidirect
146. C₂×#459	C₂×C₂₃	semidirect
147. C₂×#436	C₂×C₂₃	semidirect
148. #1687	C₂×C₂₃	direct (N × H)
149. C₂×#334	C₂×C₂₃	semidirect
150. C₂×#331	C₂×C₂₃	semidirect
178. #493	C₂²×C₂₃	semidirect
180. #459	C₂²×C₂₃	semidirect
182. #436	C₂²×C₂₃	semidirect
188. #334	C₂²×C₂₃	semidirect
189. #331	C₂²×C₂₃	semidirect
192. C₂×C₂₃	#1687	direct (N × H)
232. C₂₃	C₂×#1687	direct (N × H)
266. C₂×C₈	D₈×C₂₃	semidirect
314. C₈	C₂×D₈×C₂₃	semidirect
337. C₂	C₂₃×#1687	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×#493	2
C₂×C₂₃×#459	2
C₂×C₂₃×#436	2
C₂₃×#1687	2
C₂×C₂₃×#334	2
C₂×C₂₃×#331	2
C₂×#1687	23

Subgroups — C2×C23×#1687 #19166757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1687 #19166757