Subgroups — C₂₃×#20551 #19168177

Order 5888.

866 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#432	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×#1771	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1768	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1753	C₂	1	4	yes	maximal
7	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#432	C₄	1	8	yes	complemented
10	C₂₃×#423	C₄	1	8	yes	complemented
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
15	C₂×C₂₃×#97	C₂²	1	4	yes	—
16	C₂×C₂₃×#97	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
18	C₂₃×#383	C₄	1	2	yes	—
19	C₂₃×#382	C₄	1	2	yes	—
20	C₂₃×#381	C₄	1	2	yes	—
21	C₂₃×#380	C₄	1	2	yes	—
22	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂₃×#376	C₂²	1	2	yes	—
25	C₂₃×#336	C₂²	1	8	yes	—
26	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₂₃	—	4	2	no	—
30	C₂×M₁₆×C₂₃	—	4	2	no	—
31	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
33	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
34	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
43	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
44	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
45	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
46	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
47	C₂₃×#97	—	4	4	no	—
48	C₂₃×#97	C₂×C₄	1	8	yes	—
49	C₄×C₈×C₂₃	—	4	4	no	—
50	C₂₃×#95	C₂×C₄	1	4	yes	—
51	C₂₃×#95	C₂×C₄	1	4	yes	—
52	C₂₃×#95	C₂×C₄	1	4	yes	—
53	C₂₃×#95	C₂×C₄	1	4	yes	—
54	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	—	4	2	no	—
58	C₂²×C₄×C₂₃	—	4	2	no	—
59	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
60	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
61	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
62	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₂₃	C₂⁴	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	—	2	4	no	—
74	C₂×C₈×C₂₃	—	2	4	no	—
75	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
76	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
77	C₄⋊C₄×C₂₃	—	4	4	no	—
78	C₂²⋊C₄×C₂₃	—	4	4	no	—
79	C₄²×C₂₃	—	2	4	no	—
80	C₄²×C₂₃	—	2	4	no	—
81	C₄²×C₂₃	C₄²	1	4	yes	—
82	C₄²×C₂₃	C₄²	1	4	yes	—
83	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
84	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
85	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
86	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
87	#20551	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₂₃	—	2	2	no	—
89	C₂³×C₂₃	—	2	2	no	—
90	C₂³×C₂₃	—	2	2	no	—
91	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
92	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
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	2	no	—
100	C₂×C₄×C₂₃	—	2	4	no	—
101	C₂×C₄×C₂₃	—	2	8	no	—
102	C₂×C₄×C₂₃	—	2	8	no	—
103	C₂×C₄×C₂₃	—	2	8	no	—
104	C₂×C₄×C₂₃	—	4	4	no	—
105	C₂×C₄×C₂₃	—	4	4	no	—
106	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
114	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
115	C₈×C₂₃	—	2	8	no	—
116	C₈×C₂₃	—	2	8	no	—
117	C₂×#432	C₂×C₂₃	1	1	yes	characteristic
118	C₂×#423	C₂×C₂₃	1	1	yes	characteristic
119	#2283	C₂×C₂₃	1	1	yes	characteristic
120	#1771	C₂×C₂₃	1	4	yes	complemented
121	#1768	C₂×C₂₃	1	4	yes	complemented
122	#1753	C₂×C₂₃	1	4	yes	—
123	C₂²×C₂₃	—	2	2	no	—
124	C₂²×C₂₃	—	2	2	no	—
125	C₂²×C₂₃	—	2	2	no	—
126	C₂²×C₂₃	—	4	4	no	—
127	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
128	C₂²×C₂₃	#520	1	2	yes	—
129	C₂²×C₂₃	#568	1	2	yes	—
130	C₂²×C₂₃	#568	1	2	yes	—
131	C₄×C₂₃	—	2	2	no	—
132	C₄×C₂₃	—	2	2	no	—
133	C₄×C₂₃	—	2	2	no	—
134	C₄×C₂₃	—	2	2	no	—
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	—	4	8	no	—
137	C₄×C₂₃	#518	1	1	yes	characteristic
138	C₄×C₂₃	#519	1	1	yes	characteristic
139	C₄×C₂₃	#520	1	2	yes	—
140	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
141	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
142	#432	C₄×C₂₃	1	8	yes	complemented
143	#423	C₄×C₂₃	1	8	yes	complemented
144	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
145	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
146	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
147	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
148	C₂×#97	C₂²×C₂₃	1	4	yes	—
149	C₂×#97	C₂²×C₂₃	1	2	yes	—
150	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
151	#383	C₄×C₂₃	1	2	yes	—
152	#382	C₄×C₂₃	1	2	yes	—
153	#381	C₄×C₂₃	1	2	yes	—
154	#380	C₄×C₂₃	1	2	yes	—
155	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
156	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
157	#376	C₂²×C₂₃	1	2	yes	—
158	#336	C₂²×C₂₃	1	8	yes	—
159	C₂×C₂₃	—	4	2	no	—
160	C₂×C₂₃	#2274	1	1	yes	characteristic
161	C₂×C₂₃	#2873	1	1	yes	characteristic
162	C₂×C₂₃	#2873	1	1	yes	characteristic
163	C₂×C₂₃	#2995	1	4	yes	—
164	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
165	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
166	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
167	C₂×M₁₆	—	4	2	no	—
168	C₂×M₁₆	—	4	2	no	—
169	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	—
170	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
171	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
172	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
173	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
174	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
176	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
177	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
178	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
179	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
181	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
182	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
183	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
184	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
185	#97	—	4	4	no	—
186	#97	C₂×C₄×C₂₃	1	8	yes	—
187	C₄×C₈	—	4	4	no	—
188	#95	C₂×C₄×C₂₃	1	4	yes	—
189	#95	C₂×C₄×C₂₃	1	4	yes	—
190	#95	C₂×C₄×C₂₃	1	4	yes	—
191	#95	C₂×C₄×C₂₃	1	4	yes	—
192	C₂₃	#20551	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
199	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
200	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
201	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
204	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
207	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
208	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
209	M₁₆	—	4	8	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
215	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
216	C₄⋊C₄	—	4	4	no	—
217	C₂²⋊C₄	—	4	4	no	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	4	no	—
220	C₄²	C₄²×C₂₃	1	4	yes	—
221	C₄²	C₄²×C₂₃	1	4	yes	—
222	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
223	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
224	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
225	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
230	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
245	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
246	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
247	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
248	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
249	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
250	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
252	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
253	C₈	—	2	8	no	—
254	C₈	—	2	8	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	4	no	—
259	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
260	C₂²	C₂₃×#520	1	2	yes	—
261	C₂²	C₂₃×#568	1	2	yes	—
262	C₂²	C₂₃×#568	1	2	yes	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	8	no	—
269	C₄	C₂₃×#518	1	1	yes	characteristic
270	C₄	C₂₃×#519	1	1	yes	characteristic
271	C₄	C₂₃×#520	1	2	yes	—
272	C₂	—	4	2	no	—
273	C₂	C₂₃×#2274	1	1	yes	characteristic
274	C₂	C₂₃×#2873	1	1	yes	characteristic
275	C₂	C₂₃×#2873	1	1	yes	characteristic
276	C₂	C₂₃×#2995	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1771	C₂	semidirect
5. C₂₃×#1768	C₂	semidirect
9. C₂₃×#432	C₄	semidirect
10. C₂₃×#423	C₄	semidirect
87. #20551	C₂₃	direct (N × H)
120. #1771	C₂×C₂₃	semidirect
121. #1768	C₂×C₂₃	semidirect
142. #432	C₄×C₂₃	semidirect
143. #423	C₄×C₂₃	semidirect
192. C₂₃	#20551	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×#432	2
C₂×C₂₃×#423	2
C₂₃×#2283	2
C₂₃×#1771	2
C₂₃×#1768	2
C₂₃×#1753	2
#20551	23

Subgroups — C23×#20551 #19168177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20551 #19168177