Subgroups — C₂₃×#8050 #19155676

Order 5888.

604 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2920	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2283	C₂	1	2	yes	maximal
3	C₂₃×#1297	C₂	1	4	yes	maximal
4	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
5	C₂₃×#432	—	2	2	no	—
6	C₂₃×#431	—	2	2	no	—
7	C₂₃×#423	C₄	1	2	yes	complemented
8	C₂₃×#407	C₂²	1	2	yes	—
9	C₄×M₁₆×C₂₃	C₄	1	2	yes	complemented
10	C₂₃×#383	C₄	1	4	yes	complemented
11	C₂₃×#382	C₄	1	4	yes	complemented
12	C₂₃×#381	—	2	2	no	—
13	C₂₃×#380	—	2	2	no	—
14	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
15	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
16	C₂₃×#376	—	2	2	no	—
17	C₂³×C₄×C₂₃	D₄	1	2	yes	—
18	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
20	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
21	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
22	C₂×C₄²×C₂₃	—	2	4	no	—
23	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₂₃	D₄	1	2	yes	—
28	C₂×C₄²×C₂₃	D₄	1	2	yes	—
29	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
30	C₂₃×C₄⋊C₈	—	2	4	no	—
31	C₂₃×C₂²⋊C₈	—	2	4	no	—
32	C₂₃×#97	—	2	2	no	—
33	C₄×C₈×C₂₃	—	2	2	no	—
34	C₂₃×#95	—	2	4	no	—
35	C₂₃×#95	—	2	4	no	—
36	C₂₃×#95	—	2	4	no	—
37	C₂₃×#95	C₂×C₄	1	8	yes	—
38	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	—	2	8	no	—
40	C₂²×C₄×C₂₃	—	4	2	no	—
41	C₂²×C₄×C₂₃	—	4	4	no	—
42	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
52	M₁₆×C₂₃	—	4	4	no	—
53	C₂×C₈×C₂₃	—	2	4	no	—
54	C₄⋊C₄×C₂₃	—	4	8	no	—
55	C₂²⋊C₄×C₂₃	—	4	8	no	—
56	C₄²×C₂₃	—	4	8	no	—
57	C₄²×C₂₃	C₄²	1	2	yes	—
58	C₄²×C₂₃	C₄²	1	1	yes	characteristic
59	C₄²×C₂₃	C₄²	1	1	yes	characteristic
60	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
61	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
64	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
65	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
66	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
67	#8050	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₂₃	—	2	4	no	—
69	C₂³×C₂₃	—	4	1	no	—
70	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
71	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₃	—	2	4	no	—
73	C₂×C₄×C₂₃	—	2	4	no	—
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	4	2	no	—
77	C₂×C₄×C₂₃	—	4	2	no	—
78	C₂×C₄×C₂₃	—	4	2	no	—
79	C₂×C₄×C₂₃	—	4	2	no	—
80	C₂×C₄×C₂₃	—	4	2	no	—
81	C₂×C₄×C₂₃	—	4	1	no	—
82	C₂×C₄×C₂₃	—	4	8	no	—
83	C₂×C₄×C₂₃	—	4	8	no	—
84	C₂×C₄×C₂₃	—	4	8	no	—
85	C₂×C₄×C₂₃	#95	1	2	yes	—
86	C₂×C₄×C₂₃	#95	1	2	yes	—
87	C₂×C₄×C₂₃	#95	1	2	yes	—
88	C₂×C₄×C₂₃	#95	1	2	yes	—
89	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₈×C₂₃	—	4	4	no	—
93	#2920	C₂×C₂₃	1	1	yes	characteristic
94	#2283	C₂×C₂₃	1	2	yes	—
95	#1297	C₂×C₂₃	1	4	yes	—
96	C₂²×C₂₃	—	2	2	no	—
97	C₂²×C₂₃	—	2	2	no	—
98	C₂²×C₂₃	—	2	2	no	—
99	C₂²×C₂₃	—	2	1	no	—
100	C₂²×C₂₃	—	4	4	no	—
101	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
102	C₂²×C₂₃	#421	1	4	yes	—
103	C₄×C₂₃	—	2	2	no	—
104	C₄×C₂₃	—	2	2	no	—
105	C₄×C₂₃	—	2	2	no	—
106	C₄×C₂₃	—	2	2	no	—
107	C₄×C₂₃	—	2	1	no	—
108	C₄×C₂₃	—	2	2	no	—
109	C₄×C₂₃	—	8	8	no	—
110	C₄×C₂₃	#410	1	2	yes	—
111	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
112	#432	—	2	2	no	—
113	#431	—	2	2	no	—
114	#423	C₄×C₂₃	1	2	yes	complemented
115	#407	C₂²×C₂₃	1	2	yes	—
116	C₄×M₁₆	C₄×C₂₃	1	2	yes	complemented
117	#383	C₄×C₂₃	1	4	yes	complemented
118	#382	C₄×C₂₃	1	4	yes	complemented
119	#381	—	2	2	no	—
120	#380	—	2	2	no	—
121	C₄×C₄⋊C₄	—	2	2	no	—
122	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
123	#376	—	2	2	no	—
124	C₂×C₂₃	—	2	4	no	—
125	C₂×C₂₃	—	4	1	no	—
126	C₂×C₂₃	#1733	1	2	yes	—
127	C₂×C₂₃	#1740	1	1	yes	characteristic
128	C₂³×C₄	D₄×C₂₃	1	2	yes	—
129	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
130	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
131	C₂×C₄⋊C₄	—	2	4	no	—
132	C₂×C₂²⋊C₄	—	2	4	no	—
133	C₂×C₄²	—	2	4	no	—
134	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
135	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
136	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
137	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
138	C₂×C₄²	D₄×C₂₃	1	2	yes	—
139	C₂×C₄²	D₄×C₂₃	1	2	yes	—
140	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
141	C₄⋊C₈	—	2	4	no	—
142	C₂²⋊C₈	—	2	4	no	—
143	#97	—	2	2	no	—
144	C₄×C₈	—	2	2	no	—
145	#95	—	2	4	no	—
146	#95	—	2	4	no	—
147	#95	—	2	4	no	—
148	#95	C₂×C₄×C₂₃	1	8	yes	—
149	C₂₃	#8050	1	1	yes	characteristic, direct product factor, Sylow
150	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	—	4	4	no	—
154	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
155	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
156	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
157	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
158	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
159	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
160	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
161	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
162	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
163	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
164	M₁₆	—	4	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₄⋊C₄	—	4	8	no	—
167	C₂²⋊C₄	—	4	8	no	—
168	C₄²	—	4	8	no	—
169	C₄²	C₄²×C₂₃	1	2	yes	—
170	C₄²	C₄²×C₂₃	1	1	yes	characteristic
171	C₄²	C₄²×C₂₃	1	1	yes	characteristic
172	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
173	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
174	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
175	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
176	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
177	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
178	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
179	C₂³	—	2	4	no	—
180	C₂³	—	4	1	no	—
181	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
182	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	8	no	—
194	C₂×C₄	—	4	8	no	—
195	C₂×C₄	—	4	8	no	—
196	C₂×C₄	C₂₃×#95	1	2	yes	—
197	C₂×C₄	C₂₃×#95	1	2	yes	—
198	C₂×C₄	C₂₃×#95	1	2	yes	—
199	C₂×C₄	C₂₃×#95	1	2	yes	—
200	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
201	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
203	C₈	—	4	4	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	4	4	no	—
209	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
210	C₂²	C₂₃×#421	1	4	yes	—
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	—	8	8	no	—
218	C₄	C₂₃×#410	1	2	yes	—
219	C₂	—	2	4	no	—
220	C₂	—	4	1	no	—
221	C₂	C₂₃×#1733	1	2	yes	—
222	C₂	C₂₃×#1740	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#95
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₂₃×#423	C₄	semidirect
9. C₄×M₁₆×C₂₃	C₄	semidirect
10. C₂₃×#383	C₄	semidirect
11. C₂₃×#382	C₄	semidirect
67. #8050	C₂₃	direct (N × H)
114. #423	C₄×C₂₃	semidirect
116. C₄×M₁₆	C₄×C₂₃	semidirect
117. #383	C₄×C₂₃	semidirect
118. #382	C₄×C₂₃	semidirect
149. C₂₃	#8050	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×#2920	2
C₂₃×#2283	2
C₂₃×#1297	2
#8050	23

Subgroups — C23×#8050 #19155676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8050 #19155676