Subgroups — C₂₃×#8640 #19156266

Order 5888.

404 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2931	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1560	C₂	1	2	yes	maximal
5	C₂₃×#1309	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#131	C₄	1	2	yes	—
7	C₂₃×#532	C₂²	1	1	yes	characteristic
8	C₂₃×#436	—	2	1	no	—
9	C₂₃×#436	—	2	1	no	—
10	C₈×D₄×C₂₃	—	2	1	no	—
11	C₈×D₄×C₂₃	—	2	1	no	—
12	C₂₃×#423	C₂²	1	2	yes	—
13	C₂₃×#423	C₂²	1	2	yes	—
14	C₂₃×#423	C₂²	1	1	yes	characteristic
15	C₂₃×#408	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#335	—	2	1	no	—
18	C₂₃×#335	—	2	1	no	—
19	C₂₃×#334	—	2	1	no	—
20	C₂₃×#334	—	2	1	no	—
21	C₂₃×#333	—	2	2	no	—
22	C₂₃×#331	—	2	2	no	—
23	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
24	C₂₃×#131	—	4	4	no	—
25	C₂×M₁₆×C₂₃	—	2	2	no	—
26	C₂×M₁₆×C₂₃	—	2	2	no	—
27	C₂×M₁₆×C₂₃	—	2	2	no	—
28	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₂₃	—	2	2	no	—
30	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
33	C₂₃×#127	D₄	1	1	yes	characteristic
34	C₂₃×#124	Q₈	1	2	yes	—
35	C₂₃×#121	—	2	1	no	—
36	C₂₃×#121	—	2	1	no	—
37	C₄×D₄×C₂₃	—	2	1	no	—
38	C₄×D₄×C₂₃	—	2	1	no	—
39	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×C₄⋊C₈	—	2	1	no	—
41	C₂₃×C₄⋊C₈	—	2	1	no	—
42	C₂₃×C₄⋊C₈	—	2	2	no	—
43	C₂₃×C₄⋊C₈	—	2	2	no	—
44	C₂₃×C₄⋊C₈	—	2	2	no	—
45	C₂₃×C₄⋊C₈	—	2	2	no	—
46	C₂₃×C₂²⋊C₈	—	2	2	no	—
47	C₂₃×C₂²⋊C₈	—	2	2	no	—
48	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
52	#41×C₂₃	—	4	2	no	—
53	#41×C₂₃	—	4	2	no	—
54	C₂×Q₈×C₂₃	C₄⋊C₄	1	2	yes	—
55	C₂×D₄×C₂₃	—	2	2	no	—
56	C₂×D₄×C₂₃	—	2	1	no	—
57	C₂×D₄×C₂₃	—	2	1	no	—
58	C₂×D₄×C₂₃	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₂₃	—	2	2	no	—
60	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
63	M₁₆×C₂₃	—	4	4	no	—
64	M₁₆×C₂₃	—	4	2	no	—
65	M₁₆×C₂₃	—	8	2	no	—
66	M₁₆×C₂₃	—	8	2	no	—
67	C₂×C₈×C₂₃	—	2	2	no	—
68	C₂×C₈×C₂₃	—	2	2	no	—
69	C₂×C₈×C₂₃	—	4	1	no	—
70	C₂×C₈×C₂₃	—	4	4	no	—
71	C₂×C₈×C₂₃	—	4	2	no	—
72	C₂×C₈×C₂₃	—	4	2	no	—
73	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₄⋊C₄×C₂₃	—	2	1	no	—
78	C₄⋊C₄×C₂₃	—	2	1	no	—
79	C₂²⋊C₄×C₂₃	—	2	2	no	—
80	C₂²⋊C₄×C₂₃	—	2	2	no	—
81	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
85	#8640	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₂₃	—	2	2	no	—
87	C₂³×C₂₃	#117	1	1	yes	characteristic
88	Q₈×C₂₃	—	4	2	no	—
89	D₄×C₂₃	—	4	2	no	—
90	D₄×C₂₃	—	4	4	no	—
91	D₄×C₂₃	—	4	2	no	—
92	D₄×C₂₃	—	4	1	no	—
93	D₄×C₂₃	—	4	1	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	4	2	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
105	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
108	C₈×C₂₃	—	2	1	no	—
109	C₈×C₂₃	—	2	1	no	—
110	C₈×C₂₃	—	2	1	no	—
111	C₈×C₂₃	—	4	2	no	—
112	C₈×C₂₃	—	8	2	no	—
113	C₈×C₂₃	—	8	2	no	—
114	C₈×C₂₃	#100	1	2	yes	—
115	#2931	C₂×C₂₃	1	1	yes	characteristic
116	#1567	C₂×C₂₃	1	1	yes	characteristic, complemented
117	#1566	C₂×C₂₃	1	1	yes	characteristic, complemented
118	#1560	C₂×C₂₃	1	2	yes	—
119	#1309	C₂×C₂₃	1	2	yes	complemented
120	C₂²×C₂₃	—	2	1	no	—
121	C₂²×C₂₃	—	2	1	no	—
122	C₂²×C₂₃	—	2	1	no	—
123	C₂²×C₂₃	—	4	2	no	—
124	C₂²×C₂₃	—	4	2	no	—
125	C₂²×C₂₃	—	4	2	no	—
126	C₂²×C₂₃	#380	1	1	yes	characteristic
127	C₄×C₂₃	—	2	1	no	—
128	C₄×C₂₃	—	2	1	no	—
129	C₄×C₂₃	—	2	1	no	—
130	C₄×C₂₃	—	4	2	no	—
131	C₄×C₂₃	C₂×#100	1	1	yes	characteristic
132	C₄×C₂₃	#413	1	1	yes	characteristic
133	C₄×C₂₃	#471	1	2	yes	—
134	C₄×C₂₃	#472	1	1	yes	characteristic
135	C₄×C₂₃	#473	1	1	yes	characteristic
136	C₂×#131	C₄×C₂₃	1	2	yes	—
137	#532	C₂²×C₂₃	1	1	yes	characteristic
138	#436	—	2	1	no	—
139	#436	—	2	1	no	—
140	C₈×D₄	—	2	1	no	—
141	C₈×D₄	—	2	1	no	—
142	#423	C₂²×C₂₃	1	2	yes	—
143	#423	C₂²×C₂₃	1	2	yes	—
144	#423	C₂²×C₂₃	1	1	yes	characteristic
145	#408	C₄×C₂₃	1	2	yes	—
146	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
147	#335	—	2	1	no	—
148	#335	—	2	1	no	—
149	#334	—	2	1	no	—
150	#334	—	2	1	no	—
151	#333	—	2	2	no	—
152	#331	—	2	2	no	—
153	C₂×C₂₃	—	4	1	no	—
154	C₂×C₂₃	—	8	2	no	—
155	C₂×C₂₃	#1800	1	1	yes	characteristic
156	C₂×C₂₃	#1815	1	1	yes	characteristic
157	C₂×C₂₃	#1816	1	1	yes	characteristic
158	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
159	#131	—	4	4	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
166	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
167	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
168	#127	D₄×C₂₃	1	1	yes	characteristic
169	#124	Q₈×C₂₃	1	2	yes	—
170	#121	—	2	1	no	—
171	#121	—	2	1	no	—
172	C₄×D₄	—	2	1	no	—
173	C₄×D₄	—	2	1	no	—
174	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₂²⋊C₈	—	2	2	no	—
182	C₂²⋊C₈	—	2	2	no	—
183	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
184	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
185	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
186	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
187	C₂₃	#8640	1	1	yes	characteristic, direct product factor, Sylow
188	#41	—	4	2	no	—
189	#41	—	4	2	no	—
190	C₂×Q₈	C₄⋊C₄×C₂₃	1	2	yes	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	C₄⋊C₄×C₂₃	1	2	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
198	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
199	M₁₆	—	4	4	no	—
200	M₁₆	—	4	2	no	—
201	M₁₆	—	8	2	no	—
202	M₁₆	—	8	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
218	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
219	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
220	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	C₂³	C₂₃×#117	1	1	yes	characteristic
223	Q₈	—	4	2	no	—
224	D₄	—	4	2	no	—
225	D₄	—	4	4	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	1	no	—
228	D₄	—	4	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
239	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
240	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
241	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
242	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₈	—	8	2	no	—
248	C₈	—	8	2	no	—
249	C₈	C₂₃×#100	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	C₂₃×#380	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	4	2	no	—
261	C₄	C₂×C₂₃×#100	1	1	yes	characteristic
262	C₄	C₂₃×#413	1	1	yes	characteristic
263	C₄	C₂₃×#471	1	2	yes	—
264	C₄	C₂₃×#472	1	1	yes	characteristic
265	C₄	C₂₃×#473	1	1	yes	characteristic
266	C₂	—	4	1	no	—
267	C₂	—	8	2	no	—
268	C₂	C₂₃×#1800	1	1	yes	characteristic
269	C₂	C₂₃×#1815	1	1	yes	characteristic
270	C₂	C₂₃×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#380
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1567	C₂	semidirect
3. C₂₃×#1566	C₂	semidirect
5. C₂₃×#1309	C₂	semidirect
85. #8640	C₂₃	direct (N × H)
116. #1567	C₂×C₂₃	semidirect
117. #1566	C₂×C₂₃	semidirect
119. #1309	C₂×C₂₃	semidirect
187. C₂₃	#8640	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2931	2
C₂₃×#1567	2
C₂₃×#1566	2
C₂₃×#1560	2
C₂₃×#1309	2
#8640	23

Subgroups — C23×#8640 #19156266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8640 #19156266