Subgroups — C₂₃×#17989 #19165615

Order 5888.

624 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2981	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1586	C₂	1	4	yes	maximal, complemented
4	C₂₃×#1581	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1496	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#532	C₂²	1	2	yes	—
9	C₂₃×#500	C₄	1	1	yes	characteristic
10	C₂₃×#493	C₄	1	1	yes	characteristic
11	C₂₃×#486	C₄	1	2	yes	—
12	C₂₃×#466	C₄	1	2	yes	—
13	C₂₃×#466	C₄	1	2	yes	—
14	C₄×D₈×C₂₃	C₂²	1	2	yes	—
15	C₄×D₈×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#436	C₂²	1	4	yes	—
17	C₂₃×#435	—	2	4	no	—
18	C₈×D₄×C₂₃	C₂²	1	4	yes	—
19	C₂₃×#432	C₂²	1	2	yes	—
20	C₂₃×#431	C₂²	1	1	yes	characteristic
21	C₂₃×#431	C₂²	1	1	yes	characteristic
22	C₂₃×#408	—	2	4	no	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#335	C₂²	1	2	yes	—
25	C₂₃×#334	C₂²	1	2	yes	—
26	C₂₃×D₄⋊C₈	C₂²	1	8	yes	—
27	C₂₃×#322	C₂²	1	2	yes	—
28	C₂₃×#322	C₂²	1	2	yes	—
29	C₂×#41×C₂₃	—	2	2	no	—
30	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
31	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₂₃	—	2	4	no	—
33	C₂²×C₈×C₂₃	—	2	4	no	—
34	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
37	C₂₃×#127	C₂×C₄	1	2	yes	—
38	C₂₃×#124	—	2	2	no	—
39	C₂₃×#121	—	2	2	no	—
40	C₂₃×#121	C₂×C₄	1	4	yes	—
41	C₄×D₄×C₂₃	—	2	2	no	—
42	C₄×D₄×C₂₃	C₂³	1	4	yes	—
43	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
44	C₂₃×#107	C₂×C₄	1	2	yes	—
45	C₂₃×#107	C₂×C₄	1	2	yes	—
46	C₂₃×C₄⋊C₈	—	2	1	no	—
47	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
48	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
49	C₂₃×#102	C₂×C₄	1	4	yes	—
50	C₂₃×#102	C₂×C₄	1	4	yes	—
51	C₂₃×C₂²⋊C₈	—	2	2	no	—
52	C₂₃×C₂²⋊C₈	—	2	1	no	—
53	C₂₃×C₂²⋊C₈	—	2	4	no	—
54	C₂₃×C₂²⋊C₈	—	2	4	no	—
55	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
56	C₂₃×#97	—	2	2	no	—
57	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₂₃	C₂³	1	2	yes	—
60	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₂₃	C₂³	1	2	yes	—
63	#41×C₂₃	—	8	4	no	—
64	C₂×Q₈×C₂₃	—	2	2	no	—
65	C₂×D₄×C₂₃	—	2	4	no	—
66	C₂×D₄×C₂₃	—	2	2	no	—
67	C₂×D₄×C₂₃	—	2	2	no	—
68	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₂₃	—	2	4	no	—
70	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
73	D₈×C₂₃	—	4	2	no	—
74	D₈×C₂₃	—	4	2	no	—
75	M₁₆×C₂₃	—	8	4	no	—
76	C₂×C₈×C₂₃	—	2	4	no	—
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	—	4	1	no	—
79	C₂×C₈×C₂₃	—	4	1	no	—
80	C₂×C₈×C₂₃	—	8	4	no	—
81	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
86	C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
88	C₂²⋊C₄×C₂₃	—	2	4	no	—
89	C₂²⋊C₄×C₂₃	—	2	4	no	—
90	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₂₃	#41	1	1	yes	characteristic
93	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
94	#17989	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₂₃	—	2	4	no	—
96	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
97	Q₈×C₂₃	—	8	2	no	—
98	D₄×C₂₃	—	2	4	no	—
99	D₄×C₂₃	—	2	4	no	—
100	D₄×C₂₃	—	8	2	no	—
101	D₄×C₂₃	—	8	2	no	—
102	D₄×C₂₃	—	8	4	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	2	4	no	—
107	C₂×C₄×C₂₃	—	4	1	no	—
108	C₂×C₄×C₂₃	—	4	4	no	—
109	C₂×C₄×C₂₃	—	4	4	no	—
110	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
116	C₈×C₂₃	—	2	1	no	—
117	C₈×C₂₃	—	2	1	no	—
118	C₈×C₂₃	—	2	1	no	—
119	C₈×C₂₃	—	2	1	no	—
120	C₈×C₂₃	—	2	4	no	—
121	C₈×C₂₃	—	4	2	no	—
122	C₈×C₂₃	—	4	4	no	—
123	#3169	C₂×C₂₃	1	1	yes	characteristic
124	#2981	C₂×C₂₃	1	2	yes	complemented
125	#1586	C₂×C₂₃	1	4	yes	complemented
126	#1581	C₂×C₂₃	1	4	yes	complemented
127	#1568	C₂×C₂₃	1	1	yes	characteristic, complemented
128	#1496	C₂×C₂₃	1	2	yes	complemented
129	#1452	C₂×C₂₃	1	1	yes	characteristic, complemented
130	C₂²×C₂₃	—	2	1	no	—
131	C₂²×C₂₃	—	2	1	no	—
132	C₂²×C₂₃	—	2	1	no	—
133	C₂²×C₂₃	—	4	4	no	—
134	C₂²×C₂₃	—	4	4	no	—
135	C₂²×C₂₃	—	4	4	no	—
136	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
137	C₄×C₂₃	—	2	1	no	—
138	C₄×C₂₃	—	2	1	no	—
139	C₄×C₂₃	—	2	1	no	—
140	C₄×C₂₃	—	2	1	no	—
141	C₄×C₂₃	—	4	4	no	—
142	C₄×C₂₃	#568	1	1	yes	characteristic
143	C₄×C₂₃	#568	1	1	yes	characteristic
144	C₄×C₂₃	#576	1	2	yes	—
145	#532	C₂²×C₂₃	1	2	yes	—
146	#500	C₄×C₂₃	1	1	yes	characteristic
147	#493	C₄×C₂₃	1	1	yes	characteristic
148	#486	C₄×C₂₃	1	2	yes	—
149	#466	C₄×C₂₃	1	2	yes	—
150	#466	C₄×C₂₃	1	2	yes	—
151	C₄×D₈	C₂²×C₂₃	1	2	yes	—
152	C₄×D₈	C₂²×C₂₃	1	2	yes	—
153	#436	C₂²×C₂₃	1	4	yes	—
154	#435	—	2	4	no	—
155	C₈×D₄	C₂²×C₂₃	1	4	yes	—
156	#432	C₂²×C₂₃	1	2	yes	—
157	#431	C₂²×C₂₃	1	1	yes	characteristic
158	#431	C₂²×C₂₃	1	1	yes	characteristic
159	#408	—	2	4	no	—
160	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
161	#335	C₂²×C₂₃	1	2	yes	—
162	#334	C₂²×C₂₃	1	2	yes	—
163	D₄⋊C₈	C₂²×C₂₃	1	8	yes	—
164	#322	C₂²×C₂₃	1	2	yes	—
165	#322	C₂²×C₂₃	1	2	yes	—
166	C₂×C₂₃	—	4	1	no	—
167	C₂×C₂₃	—	8	4	no	—
168	C₂×C₂₃	#2932	1	1	yes	characteristic
169	C₂×C₂₃	#2948	1	1	yes	characteristic
170	C₂×C₂₃	#2957	1	1	yes	characteristic
171	C₂×#41	—	2	2	no	—
172	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
173	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
174	C₂×M₁₆	—	2	4	no	—
175	C₂²×C₈	—	2	4	no	—
176	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
177	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
178	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
179	#127	C₂×C₄×C₂₃	1	2	yes	—
180	#124	—	2	2	no	—
181	#121	—	2	2	no	—
182	#121	C₂×C₄×C₂₃	1	4	yes	—
183	C₄×D₄	—	2	2	no	—
184	C₄×D₄	C₂³×C₂₃	1	4	yes	—
185	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
186	#107	C₂×C₄×C₂₃	1	2	yes	—
187	#107	C₂×C₄×C₂₃	1	2	yes	—
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
190	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
191	#102	C₂×C₄×C₂₃	1	4	yes	—
192	#102	C₂×C₄×C₂₃	1	4	yes	—
193	C₂²⋊C₈	—	2	2	no	—
194	C₂²⋊C₈	—	2	1	no	—
195	C₂²⋊C₈	—	2	4	no	—
196	C₂²⋊C₈	—	2	4	no	—
197	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
198	#97	—	2	2	no	—
199	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
200	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
201	C₄×C₈	C₂³×C₂₃	1	2	yes	—
202	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
203	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
204	C₄×C₈	C₂³×C₂₃	1	2	yes	—
205	C₂₃	#17989	1	1	yes	characteristic, direct product factor, Sylow
206	#41	—	8	4	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
216	D₈	—	4	2	no	—
217	D₈	—	4	2	no	—
218	M₁₆	—	8	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	8	4	no	—
224	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
226	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
234	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
235	C₄²	#41×C₂₃	1	1	yes	characteristic
236	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
237	C₂³	—	2	4	no	—
238	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
239	Q₈	—	8	2	no	—
240	D₄	—	2	4	no	—
241	D₄	—	2	4	no	—
242	D₄	—	8	2	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	4	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
253	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
256	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
257	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	4	no	—
263	C₈	—	4	2	no	—
264	C₈	—	4	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	4	no	—
271	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	4	no	—
277	C₄	C₂₃×#568	1	1	yes	characteristic
278	C₄	C₂₃×#568	1	1	yes	characteristic
279	C₄	C₂₃×#576	1	2	yes	—
280	C₂	—	4	1	no	—
281	C₂	—	8	4	no	—
282	C₂	C₂₃×#2932	1	1	yes	characteristic
283	C₂	C₂₃×#2948	1	1	yes	characteristic
284	C₂	C₂₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂₃×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2981	C₂	semidirect
3. C₂₃×#1586	C₂	semidirect
4. C₂₃×#1581	C₂	semidirect
5. C₂₃×#1568	C₂	semidirect
6. C₂₃×#1496	C₂	semidirect
7. C₂₃×#1452	C₂	semidirect
94. #17989	C₂₃	direct (N × H)
124. #2981	C₂×C₂₃	semidirect
125. #1586	C₂×C₂₃	semidirect
126. #1581	C₂×C₂₃	semidirect
127. #1568	C₂×C₂₃	semidirect
128. #1496	C₂×C₂₃	semidirect
129. #1452	C₂×C₂₃	semidirect
205. C₂₃	#17989	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×#3169	2
C₂₃×#2981	2
C₂₃×#1586	2
C₂₃×#1581	2
C₂₃×#1568	2
C₂₃×#1496	2
C₂₃×#1452	2
#17989	23

Subgroups — C23×#17989 #19165615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17989 #19165615