Subgroups — C₂×C₂₃×#2213 #19181218

Order 5888.

790 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#509	C₂	1	2	yes	maximal, complemented
2	C₂×C₂₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2213	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#361	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
8	C₂²×D₈×C₂₃	—	2	1	no	—
9	C₂²×D₈×C₂₃	C₂²	1	2	yes	—
10	C₂×C₂₃×#131	—	2	1	no	—
11	C₂×C₂₃×#127	—	2	1	no	—
12	C₂₃×#509	—	2	4	no	—
13	C₂₃×#509	C₂²	1	8	yes	complemented
14	C₂×C₂₃×SD₃₂	—	2	2	no	—
15	C₂×C₂₃×D₁₆	—	2	2	no	—
16	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
17	C₂₃×#493	—	2	2	no	—
18	C₂₃×#493	C₂²	1	4	yes	complemented
19	C₂₃×#443	—	2	2	no	—
20	C₂₃×#443	C₂²	1	4	yes	complemented
21	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#104	—	2	1	no	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#364	C₂²	1	4	yes	complemented
25	C₂₃×#361	C₂²	1	8	yes	complemented
26	C₂×#41×C₂₃	—	2	1	no	—
27	C₂²×D₄×C₂₃	—	2	2	no	—
28	C₂²×D₄×C₂₃	—	4	2	no	—
29	C₂₃×#135	—	2	1	no	—
30	C₂₃×#135	—	2	1	no	—
31	C₂₃×#135	D₄	1	2	yes	complemented
32	C₂₃×#135	C₂³	1	2	yes	—
33	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic, complemented
34	C₂×SD₁₆×C₂₃	—	2	1	no	—
35	C₂×D₈×C₂₃	—	2	2	no	—
36	C₂×D₈×C₂₃	—	2	2	no	—
37	C₂×D₈×C₂₃	—	2	4	no	—
38	C₂×D₈×C₂₃	—	2	2	no	—
39	C₂×D₈×C₂₃	—	2	2	no	—
40	C₂×D₈×C₂₃	—	2	2	no	—
41	C₂×D₈×C₂₃	—	4	1	no	—
42	C₂×D₈×C₂₃	—	4	1	no	—
43	C₂×D₈×C₂₃	—	4	2	no	—
44	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic, complemented
45	C₂×D₈×C₂₃	C₂³	1	4	yes	—
46	C₂₃×#131	—	2	4	no	—
47	C₂₃×#131	—	4	2	no	—
48	C₂×M₁₆×C₂₃	—	2	1	no	—
49	C₂×M₁₆×C₂₃	—	4	1	no	—
50	C₂²×C₈×C₂₃	—	4	1	no	—
51	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂₃×#127	—	2	4	no	—
54	C₂₃×#127	—	4	2	no	—
55	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
56	C₂₃×SD₃₂	—	2	8	no	—
57	C₂₃×D₁₆	—	2	8	no	—
58	C₂₃×M₃₂	—	2	2	no	—
59	C₂₃×M₃₂	C₂³	1	4	yes	—
60	C₂×C₁₆×C₂₃	—	2	2	no	—
61	C₂₃×#108	—	2	1	no	—
62	C₂₃×#108	C₂³	1	2	yes	—
63	C₂₃×#104	—	2	4	no	—
64	C₄×C₈×C₂₃	D₄	1	2	yes	—
65	C₄×C₈×C₂₃	C₂³	1	2	yes	—
66	C₂⁴×C₂₃	—	4	2	no	—
67	#41×C₂₃	—	2	2	no	—
68	#41×C₂₃	—	2	2	no	—
69	#41×C₂₃	—	4	1	no	—
70	#41×C₂₃	—	4	1	no	—
71	C₂×Q₈×C₂₃	—	2	1	no	—
72	C₂×D₄×C₂₃	—	2	1	no	—
73	C₂×D₄×C₂₃	—	2	4	no	—
74	C₂×D₄×C₂₃	—	4	8	no	—
75	C₂×D₄×C₂₃	—	4	1	no	—
76	C₂×D₄×C₂₃	—	4	2	no	—
77	C₂×D₄×C₂₃	—	4	2	no	—
78	C₂×D₄×C₂₃	—	4	2	no	—
79	C₂×D₄×C₂₃	—	4	2	no	—
80	C₂×D₄×C₂₃	—	4	2	no	—
81	C₂×D₄×C₂₃	—	8	8	no	—
82	C₂²×C₄×C₂₃	—	2	1	no	—
83	C₂²×C₄×C₂₃	—	4	1	no	—
84	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
85	Q₁₆×C₂₃	—	2	1	no	—
86	Q₁₆×C₂₃	C₂×D₄	1	2	yes	complemented
87	SD₁₆×C₂₃	—	2	2	no	—
88	SD₁₆×C₂₃	—	2	2	no	—
89	D₈×C₂₃	—	2	1	no	—
90	D₈×C₂₃	—	2	4	no	—
91	D₈×C₂₃	—	2	4	no	—
92	D₈×C₂₃	—	4	2	no	—
93	D₈×C₂₃	—	4	2	no	—
94	D₈×C₂₃	—	4	4	no	—
95	D₈×C₂₃	—	4	4	no	—
96	D₈×C₂₃	C₂×D₄	1	2	yes	complemented
97	M₁₆×C₂₃	—	2	2	no	—
98	M₁₆×C₂₃	—	4	1	no	—
99	M₁₆×C₂₃	—	4	4	no	—
100	C₂×C₈×C₂₃	—	4	1	no	—
101	C₂×C₈×C₂₃	—	4	4	no	—
102	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
103	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
104	C₂×C₈×C₂₃	D₈	1	2	yes	—
105	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
110	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
111	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
112	C₄²×C₂₃	—	2	1	no	—
113	C₄²×C₂₃	C₂×D₄	1	2	yes	—
114	C₁₆×C₂₃	—	2	4	no	—
115	C₂×#2213	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₂₃	—	4	1	no	—
117	C₂³×C₂₃	—	4	4	no	—
118	C₂³×C₂₃	—	8	8	no	—
119	C₂³×C₂₃	—	8	2	no	—
120	C₂³×C₂₃	—	8	2	no	—
121	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
122	Q₈×C₂₃	—	2	2	no	—
123	Q₈×C₂₃	—	4	1	no	—
124	D₄×C₂₃	—	2	2	no	—
125	D₄×C₂₃	—	4	2	no	—
126	D₄×C₂₃	—	4	2	no	—
127	D₄×C₂₃	—	4	4	no	—
128	D₄×C₂₃	—	4	4	no	—
129	D₄×C₂₃	—	4	1	no	—
130	D₄×C₂₃	—	8	16	no	—
131	D₄×C₂₃	—	8	2	no	—
132	D₄×C₂₃	—	8	2	no	—
133	C₂×C₄×C₂₃	—	2	4	no	—
134	C₂×C₄×C₂₃	—	2	2	no	—
135	C₂×C₄×C₂₃	—	4	1	no	—
136	C₂×C₄×C₂₃	—	4	2	no	—
137	C₂×C₄×C₂₃	—	4	2	no	—
138	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
144	C₈×C₂₃	—	4	2	no	—
145	C₈×C₂₃	#121	1	1	yes	characteristic
146	C₈×C₂₃	#121	1	1	yes	characteristic
147	C₈×C₂₃	C₂×D₈	1	2	yes	—
148	C₈×C₂₃	C₂×D₈	1	2	yes	—
149	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
150	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
151	C₂×#509	C₂×C₂₃	1	2	yes	complemented
152	C₂×#493	C₂×C₂₃	1	1	yes	characteristic, complemented
153	C₂×#443	C₂×C₂₃	1	1	yes	characteristic, complemented
154	#2213	C₂×C₂₃	1	8	yes	direct product factor
155	C₂×#364	C₂×C₂₃	1	1	yes	characteristic, complemented
156	C₂×#361	C₂×C₂₃	1	2	yes	complemented
157	C₂²×C₂₃	—	2	2	no	—
158	C₂²×C₂₃	—	4	2	no	—
159	C₂²×C₂₃	—	8	8	no	—
160	C₂²×C₂₃	—	8	2	no	—
161	C₂²×C₂₃	—	8	4	no	—
162	C₂²×C₂₃	—	16	4	no	—
163	C₂²×C₂₃	—	16	4	no	—
164	C₂²×C₂₃	#459	1	1	yes	characteristic
165	C₂²×C₂₃	#459	1	1	yes	characteristic
166	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₄×C₂₃	—	2	4	no	—
168	C₄×C₂₃	—	4	2	no	—
169	C₄×C₂₃	#459	1	1	yes	characteristic
170	C₄×C₂₃	#459	1	1	yes	characteristic
171	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
172	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
173	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
174	C₂²×D₈	—	2	1	no	—
175	C₂²×D₈	C₂²×C₂₃	1	2	yes	—
176	C₂×#131	—	2	1	no	—
177	C₂×#127	—	2	1	no	—
178	#509	—	2	4	no	—
179	#509	C₂²×C₂₃	1	8	yes	complemented
180	C₂×SD₃₂	—	2	2	no	—
181	C₂×D₁₆	—	2	2	no	—
182	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
183	#493	—	2	2	no	—
184	#493	C₂²×C₂₃	1	4	yes	complemented
185	#443	—	2	2	no	—
186	#443	C₂²×C₂₃	1	4	yes	complemented
187	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
188	C₂×#104	—	2	1	no	—
189	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
190	#364	C₂²×C₂₃	1	4	yes	complemented
191	#361	C₂²×C₂₃	1	8	yes	complemented
192	C₂×C₂₃	—	2	1	no	—
193	C₂×C₂₃	—	2	1	no	—
194	C₂×C₂₃	—	8	2	no	—
195	C₂×C₂₃	—	16	4	no	—
196	C₂×C₂₃	#2213	1	2	yes	direct product factor
197	C₂×C₂₃	C₂×#459	1	1	yes	characteristic
198	C₂×#41	—	2	1	no	—
199	C₂²×D₄	—	2	2	no	—
200	C₂²×D₄	—	4	2	no	—
201	#135	—	2	1	no	—
202	#135	—	2	1	no	—
203	#135	D₄×C₂₃	1	2	yes	complemented
204	#135	C₂³×C₂₃	1	2	yes	—
205	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic, complemented
206	C₂×SD₁₆	—	2	1	no	—
207	C₂×D₈	—	2	2	no	—
208	C₂×D₈	—	2	2	no	—
209	C₂×D₈	—	2	4	no	—
210	C₂×D₈	—	2	2	no	—
211	C₂×D₈	—	2	2	no	—
212	C₂×D₈	—	2	2	no	—
213	C₂×D₈	—	4	1	no	—
214	C₂×D₈	—	4	1	no	—
215	C₂×D₈	—	4	2	no	—
216	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic, complemented
217	C₂×D₈	C₂³×C₂₃	1	4	yes	—
218	#131	—	2	4	no	—
219	#131	—	4	2	no	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	4	1	no	—
222	C₂²×C₈	—	4	1	no	—
223	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
224	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
225	#127	—	2	4	no	—
226	#127	—	4	2	no	—
227	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
228	SD₃₂	—	2	8	no	—
229	D₁₆	—	2	8	no	—
230	M₃₂	—	2	2	no	—
231	M₃₂	C₂³×C₂₃	1	4	yes	—
232	C₂×C₁₆	—	2	2	no	—
233	#108	—	2	1	no	—
234	#108	C₂³×C₂₃	1	2	yes	—
235	#104	—	2	4	no	—
236	C₄×C₈	D₄×C₂₃	1	2	yes	—
237	C₄×C₈	C₂³×C₂₃	1	2	yes	—
238	C₂₃	C₂×#2213	1	1	yes	characteristic, direct product factor, Sylow
239	C₂⁴	—	4	2	no	—
240	#41	—	2	2	no	—
241	#41	—	2	2	no	—
242	#41	—	4	1	no	—
243	#41	—	4	1	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	2	4	no	—
247	C₂×D₄	—	4	8	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	—	8	8	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
258	Q₁₆	—	2	1	no	—
259	Q₁₆	C₂×D₄×C₂₃	1	2	yes	complemented
260	SD₁₆	—	2	2	no	—
261	SD₁₆	—	2	2	no	—
262	D₈	—	2	1	no	—
263	D₈	—	2	4	no	—
264	D₈	—	2	4	no	—
265	D₈	—	4	2	no	—
266	D₈	—	4	2	no	—
267	D₈	—	4	4	no	—
268	D₈	—	4	4	no	—
269	D₈	C₂×D₄×C₂₃	1	2	yes	complemented
270	M₁₆	—	2	2	no	—
271	M₁₆	—	4	1	no	—
272	M₁₆	—	4	4	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	4	no	—
275	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
276	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
277	C₂×C₈	D₈×C₂₃	1	2	yes	—
278	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
279	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
280	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
281	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
282	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
283	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
284	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
285	C₄²	—	2	1	no	—
286	C₄²	C₂×D₄×C₂₃	1	2	yes	—
287	C₁₆	—	2	4	no	—
288	C₂³	—	4	1	no	—
289	C₂³	—	4	4	no	—
290	C₂³	—	8	8	no	—
291	C₂³	—	8	2	no	—
292	C₂³	—	8	2	no	—
293	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
294	Q₈	—	2	2	no	—
295	Q₈	—	4	1	no	—
296	D₄	—	2	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	4	no	—
300	D₄	—	4	1	no	—
301	D₄	—	4	4	no	—
302	D₄	—	8	16	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
311	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
312	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
313	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
314	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
315	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
316	C₈	—	4	2	no	—
317	C₈	C₂₃×#121	1	1	yes	characteristic
318	C₈	C₂₃×#121	1	1	yes	characteristic
319	C₈	C₂×D₈×C₂₃	1	2	yes	—
320	C₈	C₂×D₈×C₂₃	1	2	yes	—
321	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
322	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	8	8	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	4	no	—
328	C₂²	—	16	4	no	—
329	C₂²	—	16	4	no	—
330	C₂²	C₂₃×#459	1	1	yes	characteristic
331	C₂²	C₂₃×#459	1	1	yes	characteristic
332	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	4	2	no	—
335	C₄	C₂₃×#459	1	1	yes	characteristic
336	C₄	C₂₃×#459	1	1	yes	characteristic
337	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
338	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	8	2	no	—
342	C₂	—	16	4	no	—
343	C₂	C₂₃×#2213	1	2	yes	direct product factor
344	C₂	C₂×C₂₃×#459	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#459
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
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
1. C₂×C₂₃×#509	C₂	semidirect
2. C₂×C₂₃×#493	C₂	semidirect
3. C₂×C₂₃×#443	C₂	semidirect
4. C₂₃×#2213	C₂	direct (N × H)
5. C₂×C₂₃×#364	C₂	semidirect
6. C₂×C₂₃×#361	C₂	semidirect
13. C₂₃×#509	C₂²	semidirect
18. C₂₃×#493	C₂²	semidirect
20. C₂₃×#443	C₂²	semidirect
24. C₂₃×#364	C₂²	semidirect
25. C₂₃×#361	C₂²	semidirect
31. C₂₃×#135	D₄	semidirect
33. C₂×Q₁₆×C₂₃	D₄	semidirect
44. C₂×D₈×C₂₃	D₄	semidirect
86. Q₁₆×C₂₃	C₂×D₄	semidirect
96. D₈×C₂₃	C₂×D₄	semidirect
115. C₂×#2213	C₂₃	direct (N × H)
151. C₂×#509	C₂×C₂₃	semidirect
152. C₂×#493	C₂×C₂₃	semidirect
153. C₂×#443	C₂×C₂₃	semidirect
154. #2213	C₂×C₂₃	direct (N × H)
155. C₂×#364	C₂×C₂₃	semidirect
156. C₂×#361	C₂×C₂₃	semidirect
179. #509	C₂²×C₂₃	semidirect
184. #493	C₂²×C₂₃	semidirect
186. #443	C₂²×C₂₃	semidirect
190. #364	C₂²×C₂₃	semidirect
191. #361	C₂²×C₂₃	semidirect
196. C₂×C₂₃	#2213	direct (N × H)
203. #135	D₄×C₂₃	semidirect
205. C₂×Q₁₆	D₄×C₂₃	semidirect
216. C₂×D₈	D₄×C₂₃	semidirect
238. C₂₃	C₂×#2213	direct (N × H)
259. Q₁₆	C₂×D₄×C₂₃	semidirect
269. D₈	C₂×D₄×C₂₃	semidirect
343. C₂	C₂₃×#2213	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×#509	2
C₂×C₂₃×#493	2
C₂×C₂₃×#443	2
C₂₃×#2213	2
C₂×C₂₃×#364	2
C₂×C₂₃×#361	2
C₂×#2213	23

Subgroups — C2×C23×#2213 #19181218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2213 #19181218