Subgroups — C₂×C₂₃×#2089 #19171298

Order 5888.

810 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#428	C₂	1	2	yes	maximal
2	C₂²×C₂₃×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#424	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2089	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₃×#343	C₂	1	2	yes	maximal
6	C₂×C₂₃×#341	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#428	—	2	4	no	—
13	C₂₃×#428	C₂²	1	8	yes	—
14	C₂×C₂₃×#107	—	2	2	no	—
15	C₂×C₂₃×#107	—	2	2	no	—
16	C₂×C₂₃×#107	—	2	2	no	—
17	C₂×C₂₃×#107	C₂²	1	4	yes	—
18	C₂×C₂₃×#106	—	2	2	no	—
19	C₂₃×#424	—	2	2	no	—
20	C₂₃×#424	C₂²	1	4	yes	—
21	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
22	C₂₃×#343	C₂²	1	8	yes	—
23	C₂₃×#341	C₂²	1	4	yes	—
24	C₂³×C₄×C₂₃	—	2	2	no	—
25	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₂₃	—	2	2	no	—
27	C₂×M₁₆×C₂₃	—	2	4	no	—
28	C₂×M₁₆×C₂₃	—	2	1	no	—
29	C₂×M₁₆×C₂₃	—	2	4	no	—
30	C₂×M₁₆×C₂₃	Q₈	1	8	yes	—
31	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
32	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
33	C₂²×C₈×C₂₃	—	2	2	no	—
34	C₂²×C₈×C₂₃	—	2	4	no	—
35	C₂²×C₈×C₂₃	—	2	1	no	—
36	C₂²×C₈×C₂₃	D₄	1	2	yes	—
37	C₂²×C₈×C₂₃	D₄	1	2	yes	—
38	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
39	C₂₃×#117	—	2	1	no	—
40	C₂₃×#117	—	2	1	no	—
41	C₂₃×#117	D₄	1	2	yes	—
42	C₂₃×#117	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
48	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
51	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
52	C₂×C₄²×C₂₃	—	2	1	no	—
53	C₂₃×#107	—	2	8	no	—
54	C₂₃×#107	—	2	8	no	—
55	C₂₃×#106	—	2	8	no	—
56	C₂₃×C₄⋊C₈	—	2	8	no	—
57	C₂⁴×C₂₃	#41	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	—	2	1	no	—
59	C₂²×C₄×C₂₃	—	2	1	no	—
60	C₂²×C₄×C₂₃	—	2	4	no	—
61	C₂²×C₄×C₂₃	—	4	2	no	—
62	C₂²×C₄×C₂₃	—	4	2	no	—
63	C₂²×C₄×C₂₃	—	8	4	no	—
64	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	#41	1	4	yes	—
71	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
72	M₁₆×C₂₃	—	2	8	no	—
73	M₁₆×C₂₃	—	2	4	no	—
74	M₁₆×C₂₃	—	2	4	no	—
75	M₁₆×C₂₃	—	4	2	no	—
76	M₁₆×C₂₃	C₂×Q₈	1	8	yes	—
77	C₂×C₈×C₂₃	—	2	8	no	—
78	C₂×C₈×C₂₃	—	2	4	no	—
79	C₂×C₈×C₂₃	—	2	4	no	—
80	C₂×C₈×C₂₃	—	2	4	no	—
81	C₂×C₈×C₂₃	—	2	4	no	—
82	C₂×C₈×C₂₃	—	2	2	no	—
83	C₂×C₈×C₂₃	—	4	2	no	—
84	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
85	C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₄⋊C₄×C₂₃	—	2	4	no	—
88	C₄⋊C₄×C₂₃	—	2	4	no	—
89	C₄⋊C₄×C₂₃	—	4	2	no	—
90	C₄⋊C₄×C₂₃	—	4	2	no	—
91	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
92	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
93	C₂²⋊C₄×C₂₃	—	2	2	no	—
94	C₂²⋊C₄×C₂₃	—	2	2	no	—
95	C₄²×C₂₃	—	2	2	no	—
96	C₄²×C₂₃	—	2	2	no	—
97	C₂×#2089	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₂₃	—	2	2	no	—
99	C₂³×C₂₃	—	2	2	no	—
100	C₂³×C₂₃	#121	1	1	yes	characteristic
101	C₂³×C₂₃	#121	1	1	yes	characteristic
102	C₂³×C₂₃	#125	1	4	yes	—
103	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	—	4	4	no	—
109	C₂×C₄×C₂₃	—	8	4	no	—
110	C₂×C₄×C₂₃	—	8	4	no	—
111	C₂×C₄×C₂₃	—	8	4	no	—
112	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
115	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
116	C₂×C₄×C₂₃	#123	1	2	yes	—
117	C₂×C₄×C₂₃	#123	1	2	yes	—
118	C₂×C₄×C₂₃	#125	1	4	yes	—
119	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
120	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
121	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
124	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
125	C₈×C₂₃	—	2	8	no	—
126	C₈×C₂₃	—	2	4	no	—
127	C₈×C₂₃	—	4	2	no	—
128	C₂×#428	C₂×C₂₃	1	2	yes	—
129	C₂²×#107	C₂×C₂₃	1	1	yes	characteristic
130	C₂×#424	C₂×C₂₃	1	1	yes	characteristic
131	#2089	C₂×C₂₃	1	8	yes	direct product factor
132	C₂×#343	C₂×C₂₃	1	2	yes	—
133	C₂×#341	C₂×C₂₃	1	1	yes	characteristic
134	C₂²×C₂₃	—	2	8	no	—
135	C₂²×C₂₃	—	2	1	no	—
136	C₂²×C₂₃	—	2	1	no	—
137	C₂²×C₂₃	—	2	1	no	—
138	C₂²×C₂₃	—	2	1	no	—
139	C₂²×C₂₃	#398	1	2	yes	—
140	C₂²×C₂₃	#398	1	2	yes	—
141	C₂²×C₂₃	#472	1	2	yes	—
142	C₂²×C₂₃	#473	1	2	yes	—
143	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
144	C₂²×C₂₃	C₂×#125	1	2	yes	—
145	C₄×C₂₃	—	4	2	no	—
146	C₄×C₂₃	—	4	2	no	—
147	C₄×C₂₃	—	8	4	no	—
148	C₄×C₂₃	#398	1	2	yes	—
149	C₄×C₂₃	#398	1	2	yes	—
150	C₄×C₂₃	C₂×C₂²⋊Q₈	1	2	yes	—
151	C₄×C₂₃	C₂×#123	1	1	yes	characteristic
152	C₄×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
154	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
155	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
156	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
157	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
158	#428	—	2	4	no	—
159	#428	C₂²×C₂₃	1	8	yes	—
160	C₂×#107	—	2	2	no	—
161	C₂×#107	—	2	2	no	—
162	C₂×#107	—	2	2	no	—
163	C₂×#107	C₂²×C₂₃	1	4	yes	—
164	C₂×#106	—	2	2	no	—
165	#424	—	2	2	no	—
166	#424	C₂²×C₂₃	1	4	yes	—
167	C₂×C₄⋊C₈	—	2	2	no	—
168	#343	C₂²×C₂₃	1	8	yes	—
169	#341	C₂²×C₂₃	1	4	yes	—
170	C₂×C₂₃	—	2	2	no	—
171	C₂×C₂₃	—	2	2	no	—
172	C₂×C₂₃	#2089	1	4	yes	direct product factor
173	C₂×C₂₃	C₂×#398	1	1	yes	characteristic
174	C₂×C₂₃	C₂×#472	1	1	yes	characteristic
175	C₂×C₂₃	C₂×#473	1	1	yes	characteristic
176	C₂³×C₄	—	2	2	no	—
177	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	4	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	Q₈×C₂₃	1	8	yes	—
183	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
184	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	4	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	D₄×C₂₃	1	2	yes	—
189	C₂²×C₈	D₄×C₂₃	1	2	yes	—
190	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
191	#117	—	2	1	no	—
192	#117	—	2	1	no	—
193	#117	D₄×C₂₃	1	2	yes	—
194	#117	C₂³×C₂₃	1	2	yes	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
201	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
202	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
203	C₂×C₂²⋊C₄	—	2	1	no	—
204	C₂×C₄²	—	2	1	no	—
205	#107	—	2	8	no	—
206	#107	—	2	8	no	—
207	#106	—	2	8	no	—
208	C₄⋊C₈	—	2	8	no	—
209	C₂₃	C₂×#2089	1	1	yes	characteristic, direct product factor, Sylow
210	C₂⁴	#41×C₂₃	1	1	yes	characteristic
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	—	8	4	no	—
217	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
222	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
223	C₂²×C₄	#41×C₂₃	1	4	yes	—
224	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
225	M₁₆	—	2	8	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	C₂×Q₈×C₂₃	1	8	yes	—
230	C₂×C₈	—	2	8	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
245	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	C₂₃×#121	1	1	yes	characteristic
253	C₂³	C₂₃×#121	1	1	yes	characteristic
254	C₂³	C₂₃×#125	1	4	yes	—
255	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	8	4	no	—
262	C₂×C₄	—	8	4	no	—
263	C₂×C₄	—	8	4	no	—
264	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
265	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
266	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
267	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
268	C₂×C₄	C₂₃×#123	1	2	yes	—
269	C₂×C₄	C₂₃×#123	1	2	yes	—
270	C₂×C₄	C₂₃×#125	1	4	yes	—
271	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
272	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
273	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
275	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
276	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
277	C₈	—	2	8	no	—
278	C₈	—	2	4	no	—
279	C₈	—	4	2	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	8	no	—
285	C₂²	C₂₃×#398	1	2	yes	—
286	C₂²	C₂₃×#398	1	2	yes	—
287	C₂²	C₂₃×#472	1	2	yes	—
288	C₂²	C₂₃×#473	1	2	yes	—
289	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
290	C₂²	C₂×C₂₃×#125	1	2	yes	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	8	4	no	—
294	C₄	C₂₃×#398	1	2	yes	—
295	C₄	C₂₃×#398	1	2	yes	—
296	C₄	C₂×C₂₃×C₂²⋊Q₈	1	2	yes	—
297	C₄	C₂×C₂₃×#123	1	1	yes	characteristic
298	C₄	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	C₂₃×#2089	1	4	yes	direct product factor
302	C₂	C₂×C₂₃×#398	1	1	yes	characteristic
303	C₂	C₂×C₂₃×#472	1	1	yes	characteristic
304	C₂	C₂×C₂₃×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#2089	C₂	direct (N × H)
97. C₂×#2089	C₂₃	direct (N × H)
131. #2089	C₂×C₂₃	direct (N × H)
172. C₂×C₂₃	#2089	direct (N × H)
209. C₂₃	C₂×#2089	direct (N × H)
301. C₂	C₂₃×#2089	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×#428	2
C₂²×C₂₃×#107	2
C₂×C₂₃×#424	2
C₂₃×#2089	2
C₂×C₂₃×#343	2
C₂×C₂₃×#341	2
C₂×#2089	23

Subgroups — C2×C23×#2089 #19171298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2089 #19171298