Subgroups — C₂₃×#8264 #19155890

Order 5888.

462 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1273	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1273	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₃×C₄⋊C₈	C₄	1	2	yes	—
9	C₂×C₂₃×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₄³×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#336	C₄	1	2	yes	—
18	C₂₃×#336	C₄	1	2	yes	—
19	C₂₃×#336	C₄	1	2	yes	—
20	C₂₃×#336	C₄	1	2	yes	—
21	C₂₃×#322	—	2	2	no	—
22	C₈²×C₂₃	—	2	2	no	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
36	C₂₃×C₄⋊C₈	—	2	4	no	—
37	C₂₃×C₄⋊C₈	C₈	1	8	yes	complemented
38	C₄×C₈×C₂₃	—	2	1	no	—
39	C₄×C₈×C₂₃	—	2	1	no	—
40	C₄×C₈×C₂₃	—	2	1	no	—
41	C₄×C₈×C₂₃	—	2	1	no	—
42	C₄×C₈×C₂₃	—	2	1	no	—
43	C₄×C₈×C₂₃	—	2	1	no	—
44	C₄×C₈×C₂₃	—	2	1	no	—
45	C₄×C₈×C₂₃	—	2	1	no	—
46	C₄×C₈×C₂₃	—	2	1	no	—
47	C₄×C₈×C₂₃	—	2	1	no	—
48	C₄×C₈×C₂₃	D₄	1	2	yes	—
49	C₄×C₈×C₂₃	Q₈	1	2	yes	—
50	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₂₃	—	2	2	no	—
58	C₂×C₈×C₂₃	—	2	2	no	—
59	C₂×C₈×C₂₃	—	2	2	no	—
60	C₂×C₈×C₂₃	—	2	2	no	—
61	C₂×C₈×C₂₃	—	2	2	no	—
62	C₂×C₈×C₂₃	—	2	2	no	—
63	C₂×C₈×C₂₃	—	2	2	no	—
64	C₂×C₈×C₂₃	—	2	2	no	—
65	C₂×C₈×C₂₃	—	2	2	no	—
66	C₂×C₈×C₂₃	—	2	2	no	—
67	C₂×C₈×C₂₃	—	2	2	no	—
68	C₂×C₈×C₂₃	—	2	2	no	—
69	C₂×C₈×C₂₃	—	2	2	no	—
70	C₂×C₈×C₂₃	—	2	2	no	—
71	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
72	C₂×C₈×C₂₃	C₂×C₈	1	4	yes	—
73	C₂×C₈×C₂₃	C₂×C₈	1	4	yes	—
74	C₂×C₈×C₂₃	M₁₆	1	4	yes	—
75	C₄²×C₂₃	—	2	2	no	—
76	C₄²×C₂₃	—	2	2	no	—
77	C₄²×C₂₃	—	2	2	no	—
78	C₄²×C₂₃	—	2	2	no	—
79	C₄²×C₂₃	—	2	2	no	—
80	C₄²×C₂₃	C₂×C₈	1	4	yes	—
81	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
83	C₄²×C₂₃	#41	1	1	yes	characteristic
84	C₄²×C₂₃	#41	1	1	yes	characteristic
85	#8264	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₂₃	C₂×C₄²	1	1	yes	characteristic
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	1	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	1	no	—
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	C₄×C₈	1	2	yes	—
98	C₂×C₄×C₂₃	C₄×C₈	1	2	yes	—
99	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₂₃	#131	1	2	yes	—
108	C₂×C₄×C₂₃	#131	1	2	yes	—
109	C₂×C₄×C₂₃	#131	1	2	yes	—
110	C₂×C₄×C₂₃	#131	1	2	yes	—
111	C₈×C₂₃	—	2	4	no	—
112	C₈×C₂₃	—	2	4	no	—
113	C₈×C₂₃	—	2	4	no	—
114	C₈×C₂₃	—	2	4	no	—
115	C₈×C₂₃	—	4	2	no	—
116	C₈×C₂₃	—	4	2	no	—
117	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
118	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
119	#1744	C₂×C₂₃	1	1	yes	characteristic
120	C₂×#322	C₂×C₂₃	1	1	yes	characteristic
121	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic
122	#1273	C₂×C₂₃	1	1	yes	characteristic
123	#1273	C₂×C₂₃	1	1	yes	characteristic
124	C₂²×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
126	C₂²×C₂₃	C₄×M₁₆	1	1	yes	characteristic
127	C₂²×C₂₃	#405	1	1	yes	characteristic
128	C₂²×C₂₃	#405	1	1	yes	characteristic
129	C₂²×C₂₃	#405	1	1	yes	characteristic
130	C₂²×C₂₃	#405	1	1	yes	characteristic
131	C₄×C₂₃	—	2	2	no	—
132	C₄×C₂₃	—	2	2	no	—
133	C₄×C₂₃	—	4	1	no	—
134	C₄×C₂₃	—	4	1	no	—
135	C₄×C₂₃	#424	1	2	yes	—
136	C₄×C₂₃	#433	1	2	yes	—
137	C₄×C₂₃	C₈×D₄	1	2	yes	—
138	C₄×C₂₃	#435	1	2	yes	—
139	C₄×C₂₃	C₈×Q₈	1	2	yes	—
140	C₄×C₂₃	#446	1	2	yes	—
141	C₂×C₄⋊C₈	C₄×C₂₃	1	2	yes	—
142	C₂×C₄⋊C₈	C₄×C₂₃	1	2	yes	—
143	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
149	C₄³	C₂²×C₂₃	1	1	yes	characteristic
150	#336	C₄×C₂₃	1	2	yes	—
151	#336	C₄×C₂₃	1	2	yes	—
152	#336	C₄×C₂₃	1	2	yes	—
153	#336	C₄×C₂₃	1	2	yes	—
154	#322	—	2	2	no	—
155	C₈²	—	2	2	no	—
156	C₂×C₂₃	C₈×M₁₆	1	1	yes	characteristic
157	C₂×C₂₃	#1450	1	1	yes	characteristic
158	C₂×C₂₃	#1453	1	1	yes	characteristic
159	C₂×C₂₃	#1454	1	1	yes	characteristic
160	C₂×C₂₃	#1768	1	1	yes	characteristic
161	C₂×C₂₃	C₈×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₂₃	#1771	1	1	yes	characteristic
163	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
164	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
165	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
166	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
167	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
168	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
169	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
170	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
171	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
172	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
173	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
174	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
175	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	C₈×C₂₃	1	8	yes	complemented
178	C₄×C₈	—	2	1	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	D₄×C₂₃	1	2	yes	—
189	C₄×C₈	Q₈×C₂₃	1	2	yes	—
190	C₂₃	#8264	1	1	yes	characteristic, direct product factor, Sylow
191	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
192	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
193	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
194	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
196	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
213	C₂×C₈	C₂×C₈×C₂₃	1	4	yes	—
214	C₂×C₈	C₂×C₈×C₂₃	1	4	yes	—
215	C₂×C₈	M₁₆×C₂₃	1	4	yes	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	C₂×C₈×C₂₃	1	4	yes	—
222	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
223	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
224	C₄²	#41×C₂₃	1	1	yes	characteristic
225	C₄²	#41×C₂₃	1	1	yes	characteristic
226	C₂³	C₂×C₄²×C₂₃	1	1	yes	characteristic
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	C₄×C₈×C₂₃	1	2	yes	—
238	C₂×C₄	C₄×C₈×C₂₃	1	2	yes	—
239	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
240	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
241	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
242	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
243	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
244	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
246	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
247	C₂×C₄	C₂₃×#131	1	2	yes	—
248	C₂×C₄	C₂₃×#131	1	2	yes	—
249	C₂×C₄	C₂₃×#131	1	2	yes	—
250	C₂×C₄	C₂₃×#131	1	2	yes	—
251	C₈	—	2	4	no	—
252	C₈	—	2	4	no	—
253	C₈	—	2	4	no	—
254	C₈	—	2	4	no	—
255	C₈	—	4	2	no	—
256	C₈	—	4	2	no	—
257	C₂²	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
258	C₂²	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
259	C₂²	C₄×M₁₆×C₂₃	1	1	yes	characteristic
260	C₂²	C₂₃×#405	1	1	yes	characteristic
261	C₂²	C₂₃×#405	1	1	yes	characteristic
262	C₂²	C₂₃×#405	1	1	yes	characteristic
263	C₂²	C₂₃×#405	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	C₂₃×#424	1	2	yes	—
269	C₄	C₂₃×#433	1	2	yes	—
270	C₄	C₈×D₄×C₂₃	1	2	yes	—
271	C₄	C₂₃×#435	1	2	yes	—
272	C₄	C₈×Q₈×C₂₃	1	2	yes	—
273	C₄	C₂₃×#446	1	2	yes	—
274	C₂	C₈×M₁₆×C₂₃	1	1	yes	characteristic
275	C₂	C₂₃×#1450	1	1	yes	characteristic
276	C₂	C₂₃×#1453	1	1	yes	characteristic
277	C₂	C₂₃×#1454	1	1	yes	characteristic
278	C₂	C₂₃×#1768	1	1	yes	characteristic
279	C₂	C₈×C₄⋊C₄×C₂₃	1	1	yes	characteristic
280	C₂	C₂₃×#1771	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₂₃	C₂³
commutator	C₂²	C₂×C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
37. C₂₃×C₄⋊C₈	C₈	semidirect
85. #8264	C₂₃	direct (N × H)
177. C₄⋊C₈	C₈×C₂₃	semidirect
190. C₂₃	#8264	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×C₄⋊C₈	2
C₂₃×#1744	2
C₂×C₂₃×#322	2
C₂×C₈²×C₂₃	2
C₂₃×#1273	2
#8264	23

Subgroups — C23×#8264 #19155890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8264 #19155890