Subgroups — C₂×C₂₃×#2483 #19182372

Order 5888.

4382 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
2	C₂₃×#2483	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₂₃×#397	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×#385	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×#381	C₂	1	4	yes	maximal, complemented
6	C₂×C₂₃×#379	C₂	1	1	yes	maximal, characteristic
7	C₂⁶×C₂₃	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	complemented
11	C₂²×C₂²⋊C₄×C₂₃	—	2	16	no	—
12	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	16	yes	—
13	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
14	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
15	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
16	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂₃×#397	C₂²	1	32	yes	complemented
18	C₂₃×#385	C₂²	1	32	yes	complemented
19	C₂₃×#381	C₂²	1	32	yes	complemented
20	C₂₃×#379	C₂²	1	8	yes	—
21	C₂×C₂₃×#95	C₂²	1	4	yes	complemented
22	C₂×C₂₃×#95	C₂²	1	8	yes	complemented
23	C₂⁵×C₂₃	—	2	16	no	—
24	C₂⁵×C₂₃	—	4	4	no	—
25	C₂⁵×C₂₃	D₄	1	8	yes	—
26	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
27	C₂⁵×C₂₃	C₂³	1	4	yes	—
28	C₂⁵×C₂₃	C₂³	1	2	yes	—
29	C₂³×C₄×C₂₃	—	2	8	no	—
30	C₂³×C₄×C₂₃	—	2	8	no	—
31	C₂³×C₄×C₂₃	—	2	16	no	—
32	C₂³×C₄×C₂₃	—	4	8	no	—
33	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
34	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
35	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
36	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
37	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
38	C₂×C₄⋊C₄×C₂₃	—	4	8	no	—
39	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	complemented
40	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
41	C₂×C₂²⋊C₄×C₂₃	—	2	64	no	—
42	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
43	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
44	C₂×C₂²⋊C₄×C₂₃	—	4	4	no	—
45	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
46	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
47	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
48	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
49	C₂₃×#95	C₂³	1	16	yes	complemented
50	C₂₃×#95	C₂³	1	32	yes	complemented
51	C₂⁴×C₂₃	—	2	8	no	—
52	C₂⁴×C₂₃	—	2	8	no	—
53	C₂⁴×C₂₃	—	2	64	no	—
54	C₂⁴×C₂₃	—	2	32	no	—
55	C₂⁴×C₂₃	—	2	8	no	—
56	C₂⁴×C₂₃	—	2	4	no	—
57	C₂⁴×C₂₃	—	4	16	no	—
58	C₂⁴×C₂₃	—	4	8	no	—
59	C₂⁴×C₂₃	—	4	64	no	—
60	C₂⁴×C₂₃	C₂×D₄	1	4	yes	—
61	C₂⁴×C₂₃	C₂×D₄	1	16	yes	—
62	C₂⁴×C₂₃	C₂×D₄	1	8	yes	—
63	C₂⁴×C₂₃	#41	1	16	yes	—
64	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
65	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
66	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	—	2	16	no	—
68	C₂²×C₄×C₂₃	—	2	16	no	—
69	C₂²×C₄×C₂₃	—	2	8	no	—
70	C₂²×C₄×C₂₃	—	2	8	no	—
71	C₂²×C₄×C₂₃	—	2	16	no	—
72	C₂²×C₄×C₂₃	—	2	32	no	—
73	C₂²×C₄×C₂₃	—	2	16	no	—
74	C₂²×C₄×C₂₃	—	2	8	no	—
75	C₂²×C₄×C₂₃	—	2	16	no	—
76	C₂²×C₄×C₂₃	—	4	16	no	—
77	C₂²×C₄×C₂₃	—	4	32	no	—
78	C₂²×C₄×C₂₃	—	4	32	no	—
79	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
80	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
81	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
82	C₄⋊C₄×C₂₃	—	4	16	no	—
83	C₂²⋊C₄×C₂₃	—	2	32	no	—
84	C₂²⋊C₄×C₂₃	—	2	32	no	—
85	C₂²⋊C₄×C₂₃	—	2	32	no	—
86	C₂²⋊C₄×C₂₃	—	4	8	no	—
87	C₂×#2483	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₂₃	—	2	16	no	—
89	C₂³×C₂₃	—	2	16	no	—
90	C₂³×C₂₃	—	2	64	no	—
91	C₂³×C₂₃	—	2	8	no	—
92	C₂³×C₂₃	—	2	8	no	—
93	C₂³×C₂₃	—	2	64	no	—
94	C₂³×C₂₃	—	2	16	no	—
95	C₂³×C₂₃	—	2	8	no	—
96	C₂³×C₂₃	—	2	8	no	—
97	C₂³×C₂₃	—	2	4	no	—
98	C₂³×C₂₃	—	4	16	no	—
99	C₂³×C₂₃	—	4	64	no	—
100	C₂³×C₂₃	—	4	128	no	—
101	C₂³×C₂₃	—	4	16	no	—
102	C₂³×C₂₃	#123	1	32	yes	—
103	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
104	C₂³×C₂₃	C₂²×D₄	1	8	yes	—
105	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
106	C₂³×C₂₃	C₂×#41	1	16	yes	—
107	C₂³×C₂₃	C₂×#41	1	8	yes	—
108	C₂³×C₂₃	H(5,𝔽₂)	1	4	yes	—
109	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
110	C₂×C₄×C₂₃	—	2	8	no	—
111	C₂×C₄×C₂₃	—	2	8	no	—
112	C₂×C₄×C₂₃	—	2	16	no	—
113	C₂×C₄×C₂₃	—	2	16	no	—
114	C₂×C₄×C₂₃	—	2	8	no	—
115	C₂×C₄×C₂₃	—	2	16	no	—
116	C₂×C₄×C₂₃	—	4	32	no	—
117	C₂×C₄×C₂₃	—	4	32	no	—
118	C₂×C₄×C₂₃	—	4	16	no	—
119	C₂×C₄×C₂₃	—	4	16	no	—
120	C₂×C₄×C₂₃	—	4	16	no	—
121	C₂³×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
122	#2483	C₂×C₂₃	1	16	yes	direct product factor
123	C₂×#397	C₂×C₂₃	1	4	yes	complemented
124	C₂×#385	C₂×C₂₃	1	4	yes	complemented
125	C₂×#381	C₂×C₂₃	1	4	yes	complemented
126	C₂×#379	C₂×C₂₃	1	1	yes	characteristic
127	C₂²×C₂₃	—	2	8	no	—
128	C₂²×C₂₃	—	2	8	no	—
129	C₂²×C₂₃	—	2	32	no	—
130	C₂²×C₂₃	—	2	64	no	—
131	C₂²×C₂₃	—	2	8	no	—
132	C₂²×C₂₃	—	2	4	no	—
133	C₂²×C₂₃	—	4	64	no	—
134	C₂²×C₂₃	—	4	8	no	—
135	C₂²×C₂₃	—	4	16	no	—
136	C₂²×C₂₃	C₂×#123	1	16	yes	—
137	C₂²×C₂₃	C₂×#123	1	8	yes	—
138	C₂²×C₂₃	#534	1	4	yes	—
139	C₂²×C₂₃	#551	1	16	yes	—
140	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
141	C₂²×C₂₃	C₂²×#41	1	4	yes	—
142	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
143	C₄×C₂₃	—	4	8	no	—
144	C₄×C₂₃	—	4	8	no	—
145	C₄×C₂₃	—	4	8	no	—
146	C₂⁶	C₂²×C₂₃	1	1	yes	characteristic
147	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
148	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
149	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	complemented
150	C₂²×C₂²⋊C₄	—	2	16	no	—
151	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
152	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	16	yes	—
153	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
154	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
155	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
156	#397	C₂²×C₂₃	1	32	yes	complemented
157	#385	C₂²×C₂₃	1	32	yes	complemented
158	#381	C₂²×C₂₃	1	32	yes	complemented
159	#379	C₂²×C₂₃	1	8	yes	—
160	C₂×#95	C₂²×C₂₃	1	4	yes	complemented
161	C₂×#95	C₂²×C₂₃	1	8	yes	complemented
162	C₂×C₂₃	—	2	16	no	—
163	C₂×C₂₃	—	4	4	no	—
164	C₂×C₂₃	#2483	1	8	yes	direct product factor
165	C₂×C₂₃	C₂²×#123	1	2	yes	—
166	C₂×C₂₃	C₂×#534	1	1	yes	characteristic
167	C₂×C₂₃	C₂×#551	1	4	yes	—
168	C₂⁵	—	2	16	no	—
169	C₂⁵	—	4	4	no	—
170	C₂⁵	D₄×C₂₃	1	8	yes	—
171	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
172	C₂⁵	C₂³×C₂₃	1	4	yes	—
173	C₂⁵	C₂³×C₂₃	1	2	yes	—
174	C₂³×C₄	—	2	8	no	—
175	C₂³×C₄	—	2	8	no	—
176	C₂³×C₄	—	2	16	no	—
177	C₂³×C₄	—	4	8	no	—
178	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
179	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
180	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
181	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
182	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
183	C₂×C₄⋊C₄	—	4	8	no	—
184	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	complemented
185	C₂×C₂²⋊C₄	—	2	16	no	—
186	C₂×C₂²⋊C₄	—	2	64	no	—
187	C₂×C₂²⋊C₄	—	2	16	no	—
188	C₂×C₂²⋊C₄	—	2	16	no	—
189	C₂×C₂²⋊C₄	—	4	4	no	—
190	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
191	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
192	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
193	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
194	#95	C₂³×C₂₃	1	16	yes	complemented
195	#95	C₂³×C₂₃	1	32	yes	complemented
196	C₂₃	C₂×#2483	1	1	yes	characteristic, direct product factor, Sylow
197	C₂⁴	—	2	8	no	—
198	C₂⁴	—	2	8	no	—
199	C₂⁴	—	2	64	no	—
200	C₂⁴	—	2	32	no	—
201	C₂⁴	—	2	8	no	—
202	C₂⁴	—	2	4	no	—
203	C₂⁴	—	4	16	no	—
204	C₂⁴	—	4	8	no	—
205	C₂⁴	—	4	64	no	—
206	C₂⁴	C₂×D₄×C₂₃	1	4	yes	—
207	C₂⁴	C₂×D₄×C₂₃	1	16	yes	—
208	C₂⁴	C₂×D₄×C₂₃	1	8	yes	—
209	C₂⁴	#41×C₂₃	1	16	yes	—
210	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
211	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
212	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
213	C₂²×C₄	—	2	16	no	—
214	C₂²×C₄	—	2	16	no	—
215	C₂²×C₄	—	2	8	no	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	2	16	no	—
218	C₂²×C₄	—	2	32	no	—
219	C₂²×C₄	—	2	16	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	16	no	—
222	C₂²×C₄	—	4	16	no	—
223	C₂²×C₄	—	4	32	no	—
224	C₂²×C₄	—	4	32	no	—
225	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
226	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
227	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
228	C₄⋊C₄	—	4	16	no	—
229	C₂²⋊C₄	—	2	32	no	—
230	C₂²⋊C₄	—	2	32	no	—
231	C₂²⋊C₄	—	2	32	no	—
232	C₂²⋊C₄	—	4	8	no	—
233	C₂³	—	2	64	no	—
234	C₂³	—	2	8	no	—
235	C₂³	—	2	8	no	—
236	C₂³	—	2	16	no	—
237	C₂³	—	2	16	no	—
238	C₂³	—	2	8	no	—
239	C₂³	—	2	4	no	—
240	C₂³	—	2	64	no	—
241	C₂³	—	2	16	no	—
242	C₂³	—	2	8	no	—
243	C₂³	—	4	16	no	—
244	C₂³	—	4	64	no	—
245	C₂³	—	4	128	no	—
246	C₂³	—	4	16	no	—
247	C₂³	C₂₃×#123	1	32	yes	—
248	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
249	C₂³	C₂²×D₄×C₂₃	1	8	yes	—
250	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
251	C₂³	C₂×#41×C₂₃	1	16	yes	—
252	C₂³	C₂×#41×C₂₃	1	8	yes	—
253	C₂³	C₂₃×H(5,𝔽₂)	1	4	yes	—
254	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	16	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	16	no	—
261	C₂×C₄	—	4	32	no	—
262	C₂×C₄	—	4	16	no	—
263	C₂×C₄	—	4	32	no	—
264	C₂×C₄	—	4	16	no	—
265	C₂×C₄	—	4	16	no	—
266	C₂²	—	2	8	no	—
267	C₂²	—	2	8	no	—
268	C₂²	—	2	32	no	—
269	C₂²	—	2	64	no	—
270	C₂²	—	2	8	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	4	64	no	—
273	C₂²	—	4	8	no	—
274	C₂²	—	4	16	no	—
275	C₂²	C₂×C₂₃×#123	1	16	yes	—
276	C₂²	C₂×C₂₃×#123	1	8	yes	—
277	C₂²	C₂₃×#534	1	4	yes	—
278	C₂²	C₂₃×#551	1	16	yes	—
279	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
280	C₂²	C₂²×#41×C₂₃	1	4	yes	—
281	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
282	C₄	—	4	8	no	—
283	C₄	—	4	8	no	—
284	C₄	—	4	8	no	—
285	C₂	—	2	16	no	—
286	C₂	—	4	4	no	—
287	C₂	C₂₃×#2483	1	8	yes	direct product factor
288	C₂	C₂²×C₂₃×#123	1	2	yes	—
289	C₂	C₂×C₂₃×#534	1	1	yes	characteristic
290	C₂	C₂×C₂₃×#551	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂⁶×C₂₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2483	C₂	direct (N × H)
3. C₂×C₂₃×#397	C₂	semidirect
4. C₂×C₂₃×#385	C₂	semidirect
5. C₂×C₂₃×#381	C₂	semidirect
10. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
17. C₂₃×#397	C₂²	semidirect
18. C₂₃×#385	C₂²	semidirect
19. C₂₃×#381	C₂²	semidirect
21. C₂×C₂₃×#95	C₂²	semidirect
22. C₂×C₂₃×#95	C₂²	semidirect
39. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
49. C₂₃×#95	C₂³	semidirect
50. C₂₃×#95	C₂³	semidirect
87. C₂×#2483	C₂₃	direct (N × H)
122. #2483	C₂×C₂₃	direct (N × H)
123. C₂×#397	C₂×C₂₃	semidirect
124. C₂×#385	C₂×C₂₃	semidirect
125. C₂×#381	C₂×C₂₃	semidirect
149. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
156. #397	C₂²×C₂₃	semidirect
157. #385	C₂²×C₂₃	semidirect
158. #381	C₂²×C₂₃	semidirect
160. C₂×#95	C₂²×C₂₃	semidirect
161. C₂×#95	C₂²×C₂₃	semidirect
164. C₂×C₂₃	#2483	direct (N × H)
184. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
194. #95	C₂³×C₂₃	semidirect
195. #95	C₂³×C₂₃	semidirect
196. C₂₃	C₂×#2483	direct (N × H)
287. C₂	C₂₃×#2483	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₂²⋊C₄×C₂₃	2
C₂₃×#2483	2
C₂×C₂₃×#397	2
C₂×C₂₃×#385	2
C₂×C₂₃×#381	2
C₂×C₂₃×#379	2
C₂×#2483	23

Subgroups — C2×C23×#2483 #19182372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2483 #19182372