Subgroups — C₂₃×#2496 #15670115

Order 2944.

468 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#126	C₂	1	2	yes	maximal
2	C₂₃×#400	C₂	1	2	yes	maximal, complemented
3	C₂₃×#397	C₂	1	2	yes	maximal
4	C₂₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#387	C₂	1	2	yes	maximal, complemented
7	C₂₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#380	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#380	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#126	—	2	8	no	—
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
29	C₂₃×#95	C₂²	1	1	yes	characteristic
30	C₂₃×#95	C₂²	1	1	yes	characteristic
31	C₂₃×#95	C₂²	1	2	yes	—
32	C₂₃×#95	C₂²	1	2	yes	—
33	C₂₃×#95	C₂²	1	2	yes	—
34	C₂₃×#95	C₂²	1	2	yes	—
35	C₂₃×#95	C₂²	1	2	yes	—
36	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	—	2	2	no	—
38	C₂²×C₄×C₂₃	—	2	2	no	—
39	C₂²×C₄×C₂₃	—	2	2	no	—
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₂₃	—	2	4	no	—
51	C₄⋊C₄×C₂₃	—	2	4	no	—
52	C₄⋊C₄×C₂₃	—	2	4	no	—
53	C₄⋊C₄×C₂₃	—	2	4	no	—
54	C₄⋊C₄×C₂₃	—	2	2	no	—
55	C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₄⋊C₄×C₂₃	—	2	2	no	—
58	C₂²⋊C₄×C₂₃	—	2	4	no	—
59	C₂²⋊C₄×C₂₃	—	2	4	no	—
60	C₂²⋊C₄×C₂₃	—	2	2	no	—
61	C₂²⋊C₄×C₂₃	D₄	1	4	yes	—
62	C₄²×C₂₃	—	2	4	no	—
63	C₄²×C₂₃	—	2	2	no	—
64	C₄²×C₂₃	—	2	2	no	—
65	C₂³×C₂₃	—	2	1	no	—
66	C₂³×C₂₃	—	2	1	no	—
67	C₂³×C₂₃	—	2	1	no	—
68	C₂³×C₂₃	—	2	1	no	—
69	C₂³×C₂₃	—	2	1	no	—
70	C₂³×C₂₃	—	2	1	no	—
71	C₂³×C₂₃	C₂×D₄	1	2	yes	—
72	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
73	C₂×C₄×C₂₃	—	2	1	no	—
74	C₂×C₄×C₂₃	—	2	1	no	—
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	2	1	no	—
78	C₂×C₄×C₂₃	—	2	1	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	1	no	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	1	no	—
92	C₂×C₄×C₂₃	—	2	1	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	—	4	4	no	—
98	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₂₃	#41	1	2	yes	—
101	C₂×C₄×C₂₃	#41	1	2	yes	—
102	C₂×C₄×C₂₃	#41	1	2	yes	—
103	C₂×C₄×C₂₃	#41	1	2	yes	—
104	#2496	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₂₃	—	2	2	no	—
106	C₂²×C₂₃	—	2	2	no	—
107	C₂²×C₂₃	—	2	2	no	—
108	C₂²×C₂₃	—	4	1	no	—
109	C₂²×C₂₃	—	4	1	no	—
110	C₂²×C₂₃	—	4	1	no	—
111	C₂²×C₂₃	—	4	1	no	—
112	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
113	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
114	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
115	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
116	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
117	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
118	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
119	C₄×C₂₃	—	2	2	no	—
120	C₄×C₂₃	—	2	2	no	—
121	C₄×C₂₃	—	2	2	no	—
122	C₄×C₂₃	—	4	1	no	—
123	C₄×C₂₃	—	4	2	no	—
124	C₄×C₂₃	—	4	2	no	—
125	C₄×C₂₃	—	4	2	no	—
126	C₄×C₂₃	—	4	2	no	—
127	C₄×C₂₃	—	4	1	no	—
128	C₄×C₂₃	#126	1	4	yes	complemented
129	C₂×#126	C₂×C₂₃	1	2	yes	—
130	#400	C₂×C₂₃	1	2	yes	complemented
131	#397	C₂×C₂₃	1	2	yes	—
132	#389	C₂×C₂₃	1	1	yes	characteristic, complemented
133	#389	C₂×C₂₃	1	1	yes	characteristic, complemented
134	#387	C₂×C₂₃	1	2	yes	complemented
135	#384	C₂×C₂₃	1	1	yes	characteristic, complemented
136	#380	C₂×C₂₃	1	1	yes	characteristic
137	#380	C₂×C₂₃	1	1	yes	characteristic
138	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
139	C₄×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic
140	C₂×C₂₃	—	4	2	no	—
141	C₂×C₂₃	C₂×#126	1	1	yes	characteristic
142	C₂×C₂₃	#532	1	1	yes	characteristic
143	C₂×C₂₃	#541	1	1	yes	characteristic
144	C₂×C₂₃	#547	1	1	yes	characteristic
145	C₂×C₂₃	#547	1	1	yes	characteristic
146	C₂×C₂₃	#552	1	1	yes	characteristic
147	C₂×C₂₃	#556	1	1	yes	characteristic
148	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
149	#126	—	2	8	no	—
150	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
151	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
152	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
153	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
157	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
158	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
159	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
160	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
161	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
162	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
163	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
164	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
165	#95	C₂²×C₂₃	1	1	yes	characteristic
166	#95	C₂²×C₂₃	1	1	yes	characteristic
167	#95	C₂²×C₂₃	1	2	yes	—
168	#95	C₂²×C₂₃	1	2	yes	—
169	#95	C₂²×C₂₃	1	2	yes	—
170	#95	C₂²×C₂₃	1	2	yes	—
171	#95	C₂²×C₂₃	1	2	yes	—
172	C₂₃	#2496	1	1	yes	characteristic, direct product factor, Sylow
173	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
180	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
181	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
182	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
183	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
184	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
185	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
186	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	2	4	no	—
196	C₂²⋊C₄	—	2	4	no	—
197	C₂²⋊C₄	—	2	2	no	—
198	C₂²⋊C₄	D₄×C₂₃	1	4	yes	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	1	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	C₂×D₄×C₂₃	1	2	yes	—
209	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
236	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
237	C₂×C₄	#41×C₂₃	1	2	yes	—
238	C₂×C₄	#41×C₂₃	1	2	yes	—
239	C₂×C₄	#41×C₂₃	1	2	yes	—
240	C₂×C₄	#41×C₂₃	1	2	yes	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
249	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
250	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
251	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
252	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
253	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
254	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	2	no	—
264	C₄	C₂₃×#126	1	4	yes	complemented
265	C₂	—	4	2	no	—
266	C₂	C₂×C₂₃×#126	1	1	yes	characteristic
267	C₂	C₂₃×#532	1	1	yes	characteristic
268	C₂	C₂₃×#541	1	1	yes	characteristic
269	C₂	C₂₃×#547	1	1	yes	characteristic
270	C₂	C₂₃×#547	1	1	yes	characteristic
271	C₂	C₂₃×#552	1	1	yes	characteristic
272	C₂	C₂₃×#556	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂₃×#400	C₂	semidirect
4. C₂₃×#389	C₂	semidirect
5. C₂₃×#389	C₂	semidirect
6. C₂₃×#387	C₂	semidirect
7. C₂₃×#384	C₂	semidirect
10. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
104. #2496	C₂₃	direct (N × H)
128. C₄×C₂₃	#126	semidirect
130. #400	C₂×C₂₃	semidirect
132. #389	C₂×C₂₃	semidirect
133. #389	C₂×C₂₃	semidirect
134. #387	C₂×C₂₃	semidirect
135. #384	C₂×C₂₃	semidirect
138. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
172. C₂₃	#2496	direct (N × H)
264. C₄	C₂₃×#126	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#126	2
C₂₃×#400	2
C₂₃×#397	2
C₂₃×#389	2
C₂₃×#387	2
C₂₃×#384	2
C₂₃×#380	2
C₄×C₄⋊C₄×C₂₃	2
C₄×C₂²⋊C₄×C₂₃	2
#2496	23

Subgroups — C23×#2496 #15670115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2496 #15670115