Subgroups — C₂₃×#10264 #19157890

Order 5888.

626 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	C₂×C₂₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#461	C₂	1	2	yes	maximal
3	C₂₃×#1954	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#332	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×SD₁₆×C₂₃	—	2	2	no	—
7	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#500	—	2	4	no	—
10	C₂₃×#461	—	2	4	no	—
11	C₂₃×#461	—	2	4	no	—
12	C₂×C₂₃×#107	—	2	2	no	—
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₂₃×#103	—	2	2	no	—
16	C₂×C₂₃×C₂²⋊C₈	—	2	2	no	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#332	—	2	4	no	—
19	C₂²×Q₈×C₂₃	—	2	2	no	—
20	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₂₃	—	2	1	no	—
22	C₂×SD₁₆×C₂₃	—	2	8	no	—
23	C₂×SD₁₆×C₂₃	—	4	4	no	—
24	C₂×SD₁₆×C₂₃	—	4	4	no	—
25	C₂²×C₈×C₂₃	—	2	2	no	—
26	C₂²×C₈×C₂₃	—	2	1	no	—
27	C₂²×C₈×C₂₃	D₄	1	2	yes	—
28	C₂₃×C₄⋊Q₈	—	2	2	no	—
29	C₂₃×C₄⋊Q₈	—	2	2	no	—
30	C₂₃×C₄⋊Q₈	—	2	2	no	—
31	C₂₃×C₄⋊Q₈	D₄	1	4	yes	—
32	C₄×D₄×C₂₃	—	2	2	no	—
33	C₄×D₄×C₂₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
38	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
39	C₂₃×#107	—	2	8	no	—
40	C₂₃×C₄⋊C₈	—	2	2	no	—
41	C₂₃×C₄⋊C₈	—	2	2	no	—
42	C₂₃×C₄⋊C₈	D₄	1	4	yes	—
43	C₂₃×#103	—	2	8	no	—
44	C₂₃×C₂²⋊C₈	—	4	4	no	—
45	C₄×C₈×C₂₃	—	2	2	no	—
46	C₂⁴×C₂₃	—	2	1	no	—
47	C₂×Q₈×C₂₃	—	2	4	no	—
48	C₂×Q₈×C₂₃	—	4	2	no	—
49	C₂×Q₈×C₂₃	—	4	2	no	—
50	C₂×Q₈×C₂₃	—	4	2	no	—
51	C₂×Q₈×C₂₃	—	4	2	no	—
52	C₂×Q₈×C₂₃	—	4	2	no	—
53	C₂×D₄×C₂₃	—	2	2	no	—
54	C₂×D₄×C₂₃	—	2	1	no	—
55	C₂×D₄×C₂₃	—	2	2	no	—
56	C₂×D₄×C₂₃	D₈	1	2	yes	—
57	C₂²×C₄×C₂₃	—	2	1	no	—
58	C₂²×C₄×C₂₃	—	4	2	no	—
59	C₂²×C₄×C₂₃	—	4	1	no	—
60	C₂²×C₄×C₂₃	—	4	1	no	—
61	C₂²×C₄×C₂₃	—	4	2	no	—
62	C₂²×C₄×C₂₃	—	4	2	no	—
63	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
66	SD₁₆×C₂₃	—	4	16	no	—
67	C₂×C₈×C₂₃	—	2	2	no	—
68	C₂×C₈×C₂₃	—	2	2	no	—
69	C₂×C₈×C₂₃	—	2	4	no	—
70	C₂×C₈×C₂₃	—	4	2	no	—
71	C₂×C₈×C₂₃	—	4	2	no	—
72	C₂×C₈×C₂₃	—	4	1	no	—
73	C₂×C₈×C₂₃	—	4	2	no	—
74	C₂×C₈×C₂₃	SD₁₆	1	4	yes	complemented
75	C₄⋊C₄×C₂₃	—	2	2	no	—
76	C₄⋊C₄×C₂₃	—	2	4	no	—
77	C₄⋊C₄×C₂₃	—	2	4	no	—
78	C₄⋊C₄×C₂₃	—	4	2	no	—
79	C₄⋊C₄×C₂₃	—	4	2	no	—
80	C₂²⋊C₄×C₂₃	—	4	2	no	—
81	C₄²×C₂₃	C₂×D₄	1	2	yes	—
82	C₄²×C₂₃	C₂×D₄	1	2	yes	—
83	#10264	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₂₃	—	2	2	no	—
85	C₂³×C₂₃	—	4	1	no	—
86	C₂³×C₂₃	—	4	2	no	—
87	C₂³×C₂₃	—	4	1	no	—
88	C₂³×C₂₃	—	4	2	no	—
89	C₂³×C₂₃	#121	1	1	yes	characteristic
90	Q₈×C₂₃	—	4	4	no	—
91	Q₈×C₂₃	—	4	4	no	—
92	Q₈×C₂₃	—	8	2	no	—
93	Q₈×C₂₃	—	8	2	no	—
94	D₄×C₂₃	—	2	4	no	—
95	D₄×C₂₃	—	4	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	4	1	no	—
98	C₂×C₄×C₂₃	—	4	4	no	—
99	C₂×C₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	—	8	2	no	—
101	C₂×C₄×C₂₃	—	8	2	no	—
102	C₂×C₄×C₂₃	—	8	2	no	—
103	C₂×C₄×C₂₃	—	8	1	no	—
104	C₂×C₄×C₂₃	—	8	1	no	—
105	C₂×C₄×C₂₃	#120	1	2	yes	—
106	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
107	C₂×C₄×C₂₃	#121	1	2	yes	—
108	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
111	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
112	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
113	C₂×C₄×C₂₃	#137	1	2	yes	—
114	C₈×C₂₃	—	2	4	no	—
115	C₈×C₂₃	—	4	4	no	—
116	C₈×C₂₃	—	8	1	no	—
117	C₂×#500	C₂×C₂₃	1	1	yes	characteristic, complemented
118	C₂×#461	C₂×C₂₃	1	2	yes	—
119	#1954	C₂×C₂₃	1	1	yes	characteristic
120	C₂×#332	C₂×C₂₃	1	2	yes	complemented
121	#1277	C₂×C₂₃	1	1	yes	characteristic, complemented
122	C₂²×C₂₃	—	4	4	no	—
123	C₂²×C₂₃	—	4	2	no	—
124	C₂²×C₂₃	—	8	2	no	—
125	C₂²×C₂₃	—	8	1	no	—
126	C₂²×C₂₃	—	8	1	no	—
127	C₂²×C₂₃	#392	1	1	yes	characteristic
128	C₂²×C₂₃	#461	1	2	yes	—
129	C₂²×C₂₃	#466	1	1	yes	characteristic
130	C₂²×C₂₃	#469	1	1	yes	characteristic
131	C₂²×C₂₃	#473	1	2	yes	—
132	C₄×C₂₃	—	2	2	no	—
133	C₄×C₂₃	—	8	4	no	—
134	C₄×C₂₃	—	8	1	no	—
135	C₄×C₂₃	#447	1	2	yes	—
136	C₄×C₂₃	#448	1	2	yes	—
137	C₄×C₂₃	#449	1	2	yes	—
138	C₄×C₂₃	#492	1	1	yes	characteristic
139	C₄×C₂₃	#497	1	1	yes	characteristic
140	C₂²×SD₁₆	—	2	2	no	—
141	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
142	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
143	#500	—	2	4	no	—
144	#461	—	2	4	no	—
145	#461	—	2	4	no	—
146	C₂×#107	—	2	2	no	—
147	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
149	C₂×#103	—	2	2	no	—
150	C₂×C₂²⋊C₈	—	2	2	no	—
151	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
152	#332	—	2	4	no	—
153	C₂×C₂₃	—	8	2	no	—
154	C₂×C₂₃	#1673	1	2	yes	—
155	C₂×C₂₃	#1693	1	2	yes	—
156	C₂×C₂₃	#1999	1	1	yes	characteristic
157	C₂×C₂₃	#2013	1	1	yes	characteristic
158	C₂×C₂₃	#2020	1	1	yes	characteristic
159	C₂²×Q₈	—	2	2	no	—
160	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
161	C₂³×C₄	—	2	1	no	—
162	C₂×SD₁₆	—	2	8	no	—
163	C₂×SD₁₆	—	4	4	no	—
164	C₂×SD₁₆	—	4	4	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	1	no	—
167	C₂²×C₈	D₄×C₂₃	1	2	yes	—
168	C₄⋊Q₈	—	2	2	no	—
169	C₄⋊Q₈	—	2	2	no	—
170	C₄⋊Q₈	—	2	2	no	—
171	C₄⋊Q₈	D₄×C₂₃	1	4	yes	—
172	C₄×D₄	—	2	2	no	—
173	C₄×D₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
177	C₂×C₂²⋊C₄	—	2	1	no	—
178	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
179	#107	—	2	8	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	D₄×C₂₃	1	4	yes	—
183	#103	—	2	8	no	—
184	C₂²⋊C₈	—	4	4	no	—
185	C₄×C₈	—	2	2	no	—
186	C₂₃	#10264	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁴	—	2	1	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	—	4	2	no	—
192	C₂×Q₈	—	4	2	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	D₈×C₂₃	1	2	yes	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
207	SD₁₆	—	4	16	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	SD₁₆×C₂₃	1	4	yes	complemented
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	—	4	2	no	—
222	C₄²	C₂×D₄×C₂₃	1	2	yes	—
223	C₄²	C₂×D₄×C₂₃	1	2	yes	—
224	C₂³	—	2	2	no	—
225	C₂³	—	4	1	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	1	no	—
228	C₂³	—	4	2	no	—
229	C₂³	C₂₃×#121	1	1	yes	characteristic
230	Q₈	—	4	4	no	—
231	Q₈	—	4	4	no	—
232	Q₈	—	8	2	no	—
233	Q₈	—	8	2	no	—
234	D₄	—	2	4	no	—
235	D₄	—	4	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	—	8	2	no	—
243	C₂×C₄	—	8	1	no	—
244	C₂×C₄	—	8	1	no	—
245	C₂×C₄	C₂₃×#120	1	2	yes	—
246	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
247	C₂×C₄	C₂₃×#121	1	2	yes	—
248	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
249	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
250	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
251	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
252	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
253	C₂×C₄	C₂₃×#137	1	2	yes	—
254	C₈	—	2	4	no	—
255	C₈	—	4	4	no	—
256	C₈	—	8	1	no	—
257	C₂²	—	4	4	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	1	no	—
261	C₂²	—	8	1	no	—
262	C₂²	C₂₃×#392	1	1	yes	characteristic
263	C₂²	C₂₃×#461	1	2	yes	—
264	C₂²	C₂₃×#466	1	1	yes	characteristic
265	C₂²	C₂₃×#469	1	1	yes	characteristic
266	C₂²	C₂₃×#473	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	8	4	no	—
269	C₄	—	8	1	no	—
270	C₄	C₂₃×#447	1	2	yes	—
271	C₄	C₂₃×#448	1	2	yes	—
272	C₄	C₂₃×#449	1	2	yes	—
273	C₄	C₂₃×#492	1	1	yes	characteristic
274	C₄	C₂₃×#497	1	1	yes	characteristic
275	C₂	—	8	2	no	—
276	C₂	C₂₃×#1673	1	2	yes	—
277	C₂	C₂₃×#1693	1	2	yes	—
278	C₂	C₂₃×#1999	1	1	yes	characteristic
279	C₂	C₂₃×#2013	1	1	yes	characteristic
280	C₂	C₂₃×#2020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂²×D₄×C₂₃	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ 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
1. C₂×C₂₃×#500	C₂	semidirect
4. C₂×C₂₃×#332	C₂	semidirect
5. C₂₃×#1277	C₂	semidirect
74. C₂×C₈×C₂₃	SD₁₆	semidirect
83. #10264	C₂₃	direct (N × H)
117. C₂×#500	C₂×C₂₃	semidirect
120. C₂×#332	C₂×C₂₃	semidirect
121. #1277	C₂×C₂₃	semidirect
186. C₂₃	#10264	direct (N × H)
215. C₂×C₈	SD₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#500	2
C₂×C₂₃×#461	2
C₂₃×#1954	2
C₂×C₂₃×#332	2
C₂₃×#1277	2
#10264	23

Subgroups — C23×#10264 #19157890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10264 #19157890