Subgroups — C₂₃×#7899 #19155525

Order 5888.

638 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	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1279	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₂₃	—	2	1	no	—
6	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#431	—	2	2	no	—
8	C₂₃×#431	C₄	1	4	yes	complemented
9	C₂₃×#423	—	2	2	no	—
10	C₂₃×#423	C₄	1	4	yes	complemented
11	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
12	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#383	—	2	1	no	—
17	C₂₃×#382	C₄	1	2	yes	—
18	C₂₃×#381	—	2	1	no	—
19	C₂₃×#380	C₄	1	2	yes	—
20	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
22	C₂₃×#376	—	2	1	no	—
23	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₂₃	—	2	4	no	—
27	C₂×M₁₆×C₂₃	—	4	4	no	—
28	C₂²×C₈×C₂₃	—	2	2	no	—
29	C₂²×C₈×C₂₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
31	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
32	C₂×C₄²×C₂₃	—	2	2	no	—
33	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	D₄	1	2	yes	—
39	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
40	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
41	C₂₃×C₄⋊C₈	—	2	4	no	—
42	C₂₃×C₄⋊C₈	—	2	4	no	—
43	C₂₃×C₄⋊C₈	—	2	2	no	—
44	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
45	C₂₃×C₂²⋊C₈	—	2	4	no	—
46	C₄×C₈×C₂₃	—	2	2	no	—
47	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
48	C₂₃×#95	—	2	2	no	—
49	C₂₃×#95	—	2	2	no	—
50	C₂₃×#95	—	2	2	no	—
51	C₂₃×#95	—	2	2	no	—
52	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	—	2	4	no	—
54	C₂²×C₄×C₂₃	—	2	2	no	—
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	—	2	4	no	—
57	C₂²×C₄×C₂₃	—	4	2	no	—
58	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
59	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
70	M₁₆×C₂₃	—	4	8	no	—
71	C₂×C₈×C₂₃	—	2	4	no	—
72	C₂×C₈×C₂₃	—	2	4	no	—
73	C₂×C₈×C₂₃	—	4	4	no	—
74	C₂×C₈×C₂₃	—	4	4	no	—
75	C₄⋊C₄×C₂₃	—	2	8	no	—
76	C₂²⋊C₄×C₂₃	—	4	4	no	—
77	C₄²×C₂₃	—	4	2	no	—
78	C₄²×C₂₃	—	4	2	no	—
79	C₄²×C₂₃	C₄²	1	2	yes	—
80	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
85	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
89	#7899	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₃	—	2	2	no	—
91	C₂³×C₂₃	—	2	2	no	—
92	C₂³×C₂₃	—	2	2	no	—
93	C₂³×C₂₃	#95	1	2	yes	—
94	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	4	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	4	no	—
103	C₂×C₄×C₂₃	—	2	4	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	—	4	4	no	—
107	C₂×C₄×C₂₃	—	4	4	no	—
108	C₂×C₄×C₂₃	—	4	4	no	—
109	C₂×C₄×C₂₃	#95	1	2	yes	—
110	C₂×C₄×C₂₃	#95	1	2	yes	—
111	C₂×C₄×C₂₃	#95	1	2	yes	—
112	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	#131	1	4	yes	—
119	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₂₃	#137	1	2	yes	—
121	C₈×C₂₃	—	4	4	no	—
122	C₈×C₂₃	—	4	4	no	—
123	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
124	C₂×#423	C₂×C₂₃	1	1	yes	characteristic
125	#2283	C₂×C₂₃	1	1	yes	characteristic
126	#1279	C₂×C₂₃	1	4	yes	complemented
127	C₂²×C₂₃	—	2	2	no	—
128	C₂²×C₂₃	—	2	2	no	—
129	C₂²×C₂₃	—	2	2	no	—
130	C₂²×C₂₃	—	4	4	no	—
131	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
132	C₂²×C₂₃	#408	1	2	yes	—
133	C₂²×C₂₃	#417	1	2	yes	—
134	C₂²×C₂₃	#421	1	2	yes	—
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	—	2	1	no	—
137	C₄×C₂₃	—	2	2	no	—
138	C₄×C₂₃	—	2	1	no	—
139	C₄×C₂₃	—	2	2	no	—
140	C₄×C₂₃	—	8	4	no	—
141	C₄×C₂₃	#405	1	2	yes	—
142	C₄×C₂₃	#418	1	1	yes	characteristic
143	C₄×C₂₃	#419	1	1	yes	characteristic
144	C₄×C₂₃	#424	1	2	yes	—
145	C₄×C₂₃	#428	1	2	yes	—
146	C₂²×M₁₆	—	2	1	no	—
147	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
148	#431	—	2	2	no	—
149	#431	C₄×C₂₃	1	4	yes	complemented
150	#423	—	2	2	no	—
151	#423	C₄×C₂₃	1	4	yes	complemented
152	C₂×C₄⋊C₈	—	2	1	no	—
153	C₂×C₄⋊C₈	—	2	1	no	—
154	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
155	C₂×C₂²⋊C₈	—	2	1	no	—
156	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
157	#383	—	2	1	no	—
158	#382	C₄×C₂₃	1	2	yes	—
159	#381	—	2	1	no	—
160	#380	C₄×C₂₃	1	2	yes	—
161	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
162	C₄×C₂²⋊C₄	—	2	1	no	—
163	#376	—	2	1	no	—
164	C₂×C₂₃	—	4	2	no	—
165	C₂×C₂₃	#1495	1	4	yes	—
166	C₂×C₂₃	#1729	1	1	yes	characteristic
167	C₂×C₂₃	#1733	1	1	yes	characteristic
168	C₂×C₂₃	#1736	1	1	yes	characteristic
169	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
170	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
171	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
172	C₂×M₁₆	—	2	4	no	—
173	C₂×M₁₆	—	4	4	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
177	C₂×C₂²⋊C₄	—	2	2	no	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
180	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
181	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
182	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
184	C₂×C₄²	D₄×C₂₃	1	2	yes	—
185	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
186	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
187	C₄⋊C₈	—	2	4	no	—
188	C₄⋊C₈	—	2	4	no	—
189	C₄⋊C₈	—	2	2	no	—
190	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
191	C₂²⋊C₈	—	2	4	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
194	#95	—	2	2	no	—
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	#95	—	2	2	no	—
198	C₂₃	#7899	1	1	yes	characteristic, direct product factor, Sylow
199	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
206	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
207	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
211	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
212	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
213	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
217	M₁₆	—	4	8	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	4	4	no	—
221	C₂×C₈	—	4	4	no	—
222	C₄⋊C₄	—	2	8	no	—
223	C₂²⋊C₄	—	4	4	no	—
224	C₄²	—	4	2	no	—
225	C₄²	—	4	2	no	—
226	C₄²	C₄²×C₂₃	1	2	yes	—
227	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
228	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
229	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
230	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
231	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
232	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
233	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
234	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
235	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
236	C₂³	—	2	2	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	2	no	—
239	C₂³	C₂₃×#95	1	2	yes	—
240	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	C₂₃×#95	1	2	yes	—
256	C₂×C₄	C₂₃×#95	1	2	yes	—
257	C₂×C₄	C₂₃×#95	1	2	yes	—
258	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
260	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
264	C₂×C₄	C₂₃×#131	1	4	yes	—
265	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
266	C₂×C₄	C₂₃×#137	1	2	yes	—
267	C₈	—	4	4	no	—
268	C₈	—	4	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	4	4	no	—
273	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
274	C₂²	C₂₃×#408	1	2	yes	—
275	C₂²	C₂₃×#417	1	2	yes	—
276	C₂²	C₂₃×#421	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	8	4	no	—
283	C₄	C₂₃×#405	1	2	yes	—
284	C₄	C₂₃×#418	1	1	yes	characteristic
285	C₄	C₂₃×#419	1	1	yes	characteristic
286	C₄	C₂₃×#424	1	2	yes	—
287	C₄	C₂₃×#428	1	2	yes	—
288	C₂	—	4	2	no	—
289	C₂	C₂₃×#1495	1	4	yes	—
290	C₂	C₂₃×#1729	1	1	yes	characteristic
291	C₂	C₂₃×#1733	1	1	yes	characteristic
292	C₂	C₂₃×#1736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1279	C₂	semidirect
8. C₂₃×#431	C₄	semidirect
10. C₂₃×#423	C₄	semidirect
89. #7899	C₂₃	direct (N × H)
126. #1279	C₂×C₂₃	semidirect
149. #431	C₄×C₂₃	semidirect
151. #423	C₄×C₂₃	semidirect
198. C₂₃	#7899	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×#431	2
C₂×C₂₃×#423	2
C₂₃×#2283	2
C₂₃×#1279	2
#7899	23

Subgroups — C23×#7899 #19155525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7899 #19155525