Subgroups — C₂₃×#1879 #15669498

Order 2944.

438 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#102	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#102	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₃	—	2	1	no	—
11	C₂×D₈×C₂₃	—	2	2	no	—
12	C₂×D₈×C₂₃	—	2	2	no	—
13	C₂×D₈×C₂₃	C₄	1	4	yes	complemented
14	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₄×D₄×C₂₃	—	2	2	no	—
17	C₄×D₄×C₂₃	—	2	2	no	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
21	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
22	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
23	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#102	—	2	2	no	—
25	C₂₃×#102	—	2	2	no	—
26	C₂⁴×C₂₃	—	2	1	no	—
27	C₂⁴×C₂₃	—	2	1	no	—
28	C₂×D₄×C₂₃	—	2	1	no	—
29	C₂×D₄×C₂₃	—	2	1	no	—
30	C₂×D₄×C₂₃	—	2	1	no	—
31	C₂×D₄×C₂₃	—	2	1	no	—
32	C₂×D₄×C₂₃	—	2	1	no	—
33	C₂×D₄×C₂₃	—	2	1	no	—
34	C₂×D₄×C₂₃	—	2	1	no	—
35	C₂×D₄×C₂₃	—	2	1	no	—
36	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	complemented
38	C₂×D₄×C₂₃	D₄	1	2	yes	—
39	C₂×D₄×C₂₃	D₄	1	2	yes	—
40	C₂²×C₄×C₂₃	—	2	4	no	—
41	C₂²×C₄×C₂₃	—	2	1	no	—
42	C₂²×C₄×C₂₃	—	2	1	no	—
43	C₂²×C₄×C₂₃	—	4	2	no	—
44	C₂²×C₄×C₂₃	—	4	2	no	—
45	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	D₈×C₂₃	—	2	8	no	—
49	C₂×C₈×C₂₃	—	2	1	no	—
50	C₂×C₈×C₂₃	—	2	1	no	—
51	C₂×C₈×C₂₃	—	2	1	no	—
52	C₂×C₈×C₂₃	—	2	1	no	—
53	C₂×C₈×C₂₃	—	2	1	no	—
54	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	complemented
55	C₄⋊C₄×C₂₃	—	2	1	no	—
56	C₄⋊C₄×C₂₃	—	2	1	no	—
57	C₄⋊C₄×C₂₃	—	2	1	no	—
58	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
59	C₂²⋊C₄×C₂₃	—	2	4	no	—
60	C₂²⋊C₄×C₂₃	—	4	2	no	—
61	C₂²⋊C₄×C₂₃	—	4	2	no	—
62	C₄²×C₂₃	—	2	1	no	—
63	C₄²×C₂₃	—	2	1	no	—
64	C₂³×C₂₃	—	2	2	no	—
65	C₂³×C₂₃	—	2	2	no	—
66	C₂³×C₂₃	—	2	2	no	—
67	C₂³×C₂₃	—	2	2	no	—
68	C₂³×C₂₃	—	4	1	no	—
69	C₂³×C₂₃	—	4	1	no	—
70	C₂³×C₂₃	—	4	1	no	—
71	C₂³×C₂₃	—	4	1	no	—
72	C₂³×C₂₃	—	4	1	no	—
73	C₂³×C₂₃	—	4	1	no	—
74	C₂³×C₂₃	—	4	1	no	—
75	C₂³×C₂₃	—	4	1	no	—
76	C₂³×C₂₃	—	4	1	no	—
77	C₂³×C₂₃	—	4	1	no	—
78	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
79	D₄×C₂₃	—	2	2	no	—
80	D₄×C₂₃	—	2	2	no	—
81	D₄×C₂₃	—	2	2	no	—
82	D₄×C₂₃	—	2	2	no	—
83	D₄×C₂₃	—	2	2	no	—
84	D₄×C₂₃	—	4	1	no	—
85	D₄×C₂₃	—	4	1	no	—
86	D₄×C₂₃	C₂²⋊C₄	1	4	yes	complemented
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	1	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	—	4	8	no	—
94	C₂×C₄×C₂₃	—	4	1	no	—
95	C₂×C₄×C₂₃	—	4	1	no	—
96	C₂×C₄×C₂₃	—	4	1	no	—
97	C₂×C₄×C₂₃	—	4	1	no	—
98	C₂×C₄×C₂₃	—	4	1	no	—
99	C₂×C₄×C₂₃	D₈	1	2	yes	complemented
100	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
105	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
106	C₈×C₂₃	—	2	2	no	—
107	C₈×C₂₃	—	4	1	no	—
108	#1879	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂²×C₂₃	—	2	4	no	—
110	C₂²×C₂₃	—	4	2	no	—
111	C₂²×C₂₃	—	4	2	no	—
112	C₂²×C₂₃	—	4	2	no	—
113	C₂²×C₂₃	—	4	2	no	—
114	C₂²×C₂₃	—	4	2	no	—
115	C₂²×C₂₃	—	4	2	no	—
116	C₂²×C₂₃	—	4	2	no	—
117	C₂²×C₂₃	—	4	2	no	—
118	C₂²×C₂₃	—	4	2	no	—
119	C₂²×C₂₃	—	8	1	no	—
120	C₂²×C₂₃	—	8	1	no	—
121	C₂²×C₂₃	—	8	1	no	—
122	C₂²×C₂₃	—	8	1	no	—
123	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₂₃	#120	1	1	yes	characteristic
125	C₂²×C₂₃	#121	1	1	yes	characteristic
126	C₂²×C₂₃	C₂×D₈	1	1	yes	characteristic
127	C₂²×C₂₃	#135	1	1	yes	characteristic
128	C₂²×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂²×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₄×C₂₃	—	2	2	no	—
131	C₄×C₂₃	—	4	2	no	—
132	C₄×C₂₃	—	4	1	no	—
133	C₄×C₂₃	—	8	1	no	—
134	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
136	C₄×C₂₃	#121	1	1	yes	characteristic
137	C₄×C₂₃	#123	1	1	yes	characteristic
138	C₂²×D₈	C₂×C₂₃	1	1	yes	characteristic
139	C₂×C₄×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
140	C₂×#102	C₂×C₂₃	1	1	yes	characteristic, complemented
141	C₂×#102	C₂×C₂₃	1	1	yes	characteristic, complemented
142	#390	C₂×C₂₃	1	1	yes	characteristic, complemented
143	#340	C₂×C₂₃	1	1	yes	characteristic, complemented
144	#336	C₂×C₂₃	1	1	yes	characteristic, complemented
145	C₂×C₂₃	—	4	4	no	—
146	C₂×C₂₃	—	8	2	no	—
147	C₂×C₂₃	#386	1	1	yes	characteristic
148	C₂×C₂₃	C₄×D₈	1	1	yes	characteristic
149	C₂×C₂₃	#442	1	1	yes	characteristic
150	C₂×C₂₃	#447	1	1	yes	characteristic
151	C₂×C₂₃	#449	1	1	yes	characteristic
152	C₂×C₂₃	#459	1	1	yes	characteristic
153	C₂×C₂₃	#463	1	1	yes	characteristic
154	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
155	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
156	C₂³×C₄	—	2	1	no	—
157	C₂×D₈	—	2	2	no	—
158	C₂×D₈	—	2	2	no	—
159	C₂×D₈	C₄×C₂₃	1	4	yes	complemented
160	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
161	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
162	C₄×D₄	—	2	2	no	—
163	C₄×D₄	—	2	2	no	—
164	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
166	C₂×C₂²⋊C₄	—	2	1	no	—
167	C₂×C₂²⋊C₄	—	2	1	no	—
168	C₂×C₂²⋊C₄	—	2	1	no	—
169	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
170	#102	—	2	2	no	—
171	#102	—	2	2	no	—
172	C₂₃	#1879	1	1	yes	characteristic, direct product factor, Sylow
173	C₂⁴	—	2	1	no	—
174	C₂⁴	—	2	1	no	—
175	C₂×D₄	—	2	1	no	—
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	—	2	1	no	—
182	C₂×D₄	—	2	1	no	—
183	C₂×D₄	C₂×C₄×C₂₃	1	2	yes	—
184	C₂×D₄	C₂×C₄×C₂₃	1	2	yes	complemented
185	C₂×D₄	D₄×C₂₃	1	2	yes	—
186	C₂×D₄	D₄×C₂₃	1	2	yes	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
193	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
194	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
195	D₈	—	2	8	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	C₂×C₄×C₂₃	1	2	yes	complemented
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
206	C₂²⋊C₄	—	2	4	no	—
207	C₂²⋊C₄	—	4	2	no	—
208	C₂²⋊C₄	—	4	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	4	1	no	—
216	C₂³	—	4	1	no	—
217	C₂³	—	4	1	no	—
218	C₂³	—	4	1	no	—
219	C₂³	—	4	1	no	—
220	C₂³	—	4	1	no	—
221	C₂³	—	4	1	no	—
222	C₂³	—	4	1	no	—
223	C₂³	—	4	1	no	—
224	C₂³	—	4	1	no	—
225	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
226	D₄	—	2	2	no	—
227	D₄	—	2	2	no	—
228	D₄	—	2	2	no	—
229	D₄	—	2	2	no	—
230	D₄	—	2	2	no	—
231	D₄	—	4	1	no	—
232	D₄	—	4	1	no	—
233	D₄	C₂²⋊C₄×C₂₃	1	4	yes	complemented
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	D₈×C₂₃	1	2	yes	complemented
247	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
248	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
249	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
251	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
252	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	4	1	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	8	1	no	—
266	C₂²	—	8	1	no	—
267	C₂²	—	8	1	no	—
268	C₂²	—	8	1	no	—
269	C₂²	C₄×D₄×C₂₃	1	1	yes	characteristic
270	C₂²	C₂₃×#120	1	1	yes	characteristic
271	C₂²	C₂₃×#121	1	1	yes	characteristic
272	C₂²	C₂×D₈×C₂₃	1	1	yes	characteristic
273	C₂²	C₂₃×#135	1	1	yes	characteristic
274	C₂²	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
275	C₂²	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	4	2	no	—
278	C₄	—	4	1	no	—
279	C₄	—	8	1	no	—
280	C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
281	C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
282	C₄	C₂₃×#121	1	1	yes	characteristic
283	C₄	C₂₃×#123	1	1	yes	characteristic
284	C₂	—	4	4	no	—
285	C₂	—	8	2	no	—
286	C₂	C₂₃×#386	1	1	yes	characteristic
287	C₂	C₄×D₈×C₂₃	1	1	yes	characteristic
288	C₂	C₂₃×#442	1	1	yes	characteristic
289	C₂	C₂₃×#447	1	1	yes	characteristic
290	C₂	C₂₃×#449	1	1	yes	characteristic
291	C₂	C₂₃×#459	1	1	yes	characteristic
292	C₂	C₂₃×#463	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
2. C₂×C₄×D₄×C₂₃	C₂	semidirect
3. C₂×C₂₃×#102	C₂	semidirect
4. C₂×C₂₃×#102	C₂	semidirect
5. C₂₃×#390	C₂	semidirect
6. C₂₃×#340	C₂	semidirect
7. C₂₃×#336	C₂	semidirect
13. C₂×D₈×C₂₃	C₄	semidirect
37. C₂×D₄×C₂₃	C₂×C₄	semidirect
54. C₂×C₈×C₂₃	C₂×C₄	semidirect
86. D₄×C₂₃	C₂²⋊C₄	semidirect
99. C₂×C₄×C₂₃	D₈	semidirect
108. #1879	C₂₃	direct (N × H)
139. C₂×C₄×D₄	C₂×C₂₃	semidirect
140. C₂×#102	C₂×C₂₃	semidirect
141. C₂×#102	C₂×C₂₃	semidirect
142. #390	C₂×C₂₃	semidirect
143. #340	C₂×C₂₃	semidirect
144. #336	C₂×C₂₃	semidirect
159. C₂×D₈	C₄×C₂₃	semidirect
172. C₂₃	#1879	direct (N × H)
184. C₂×D₄	C₂×C₄×C₂₃	semidirect
201. C₂×C₈	C₂×C₄×C₂₃	semidirect
233. D₄	C₂²⋊C₄×C₂₃	semidirect
246. C₂×C₄	D₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₈×C₂₃	2
C₂×C₄×D₄×C₂₃	2
C₂×C₂₃×#102	2
C₂₃×#390	2
C₂₃×#340	2
C₂₃×#336	2
#1879	23

Subgroups — C23×#1879 #15669498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1879 #15669498