Subgroups — C₂₃×#19652 #19167278

Order 5888.

882 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3208	C₂	1	4	yes	maximal, complemented
2	C₂₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#493	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1715	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1706	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₂₃	—	2	2	no	—
8	C₂₃×#533	C₂²	1	2	yes	—
9	C₂×C₂₃×#127	C₂²	1	2	yes	—
10	C₂₃×#500	C₂²	1	2	yes	complemented
11	C₂₃×#499	—	2	2	no	—
12	C₂₃×#498	C₂²	1	2	yes	complemented
13	C₂₃×#493	—	2	2	no	—
14	C₂₃×#493	C₂²	1	2	yes	—
15	C₂₃×#493	C₂²	1	2	yes	—
16	C₂₃×#492	C₂²	1	8	yes	complemented
17	C₂₃×#486	C₂²	1	8	yes	complemented
18	C₂₃×#460	—	2	8	no	—
19	C₂₃×#450	—	2	8	no	—
20	C₂₃×#431	C₂²	1	2	yes	—
21	C₂₃×#427	—	2	2	no	—
22	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#322	C₂²	1	2	yes	complemented
24	C₈²×C₂₃	C₂²	1	2	yes	complemented
25	C₂²×D₄×C₂₃	—	2	2	no	—
26	C₂²×D₄×C₂₃	—	2	4	no	—
27	C₂×SD₁₆×C₂₃	—	2	16	no	—
28	C₂×D₈×C₂₃	—	2	4	no	—
29	C₂×D₈×C₂₃	—	2	4	no	—
30	C₂×D₈×C₂₃	—	4	8	no	—
31	C₂²×C₈×C₂₃	D₄	1	2	yes	—
32	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
33	C₂₃×#127	—	2	4	no	—
34	C₂₃×#127	—	2	2	no	—
35	C₂₃×#127	C₂³	1	4	yes	—
36	C₂₃×#125	—	2	2	no	—
37	C₂₃×C₂²⋊Q₈	—	2	4	no	—
38	C₄×Q₈×C₂₃	—	2	4	no	—
39	C₂₃×#117	—	2	2	no	—
40	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
41	C₂₃×#107	—	2	4	no	—
42	C₂₃×#106	—	2	4	no	—
43	C₂₃×C₄⋊C₈	D₄	1	4	yes	—
44	C₂₃×#102	—	2	16	no	—
45	C₂₃×C₂²⋊C₈	D₄	1	4	yes	—
46	C₄×C₈×C₂₃	D₄	1	2	yes	—
47	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₂₃	C₂³	1	4	yes	—
49	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₂⁴×C₂₃	—	4	2	no	—
51	C₂×Q₈×C₂₃	—	2	4	no	—
52	C₂×D₄×C₂₃	—	2	8	no	—
53	C₂×D₄×C₂₃	—	2	4	no	—
54	C₂×D₄×C₂₃	—	4	4	no	—
55	C₂×D₄×C₂₃	—	4	2	no	—
56	C₂×D₄×C₂₃	—	4	8	no	—
57	C₂×D₄×C₂₃	—	8	8	no	—
58	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
60	SD₁₆×C₂₃	—	8	16	no	—
61	D₈×C₂₃	—	4	8	no	—
62	D₈×C₂₃	—	4	8	no	—
63	C₂×C₈×C₂₃	SD₁₆	1	8	yes	complemented
64	C₂×C₈×C₂₃	C₂×D₄	1	8	yes	—
65	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
67	C₄⋊C₄×C₂₃	—	2	4	no	—
68	C₄⋊C₄×C₂₃	—	2	4	no	—
69	C₄⋊C₄×C₂₃	—	4	4	no	—
70	C₂²⋊C₄×C₂₃	—	4	2	no	—
71	C₄²×C₂₃	—	4	2	no	—
72	C₄²×C₂₃	C₂×D₄	1	2	yes	—
73	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
75	#19652	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₂₃	—	4	4	no	—
77	C₂³×C₂₃	—	8	8	no	—
78	C₂³×C₂₃	—	8	4	no	—
79	C₂³×C₂₃	#127	1	1	yes	characteristic
80	Q₈×C₂₃	—	4	8	no	—
81	D₄×C₂₃	—	4	8	no	—
82	D₄×C₂₃	—	4	16	no	—
83	D₄×C₂₃	—	8	8	no	—
84	D₄×C₂₃	—	8	4	no	—
85	C₂×C₄×C₂₃	—	2	4	no	—
86	C₂×C₄×C₂₃	—	4	4	no	—
87	C₂×C₄×C₂₃	#127	1	2	yes	—
88	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
90	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
92	C₈×C₂₃	—	2	8	no	—
93	C₈×C₂₃	C₂×SD₁₆	1	4	yes	—
94	C₈×C₂₃	C₂×SD₁₆	1	4	yes	—
95	#3208	C₂×C₂₃	1	4	yes	complemented
96	#3160	C₂×C₂₃	1	1	yes	characteristic, complemented
97	C₂×#493	C₂×C₂₃	1	1	yes	characteristic
98	#1715	C₂×C₂₃	1	4	yes	complemented
99	#1706	C₂×C₂₃	1	4	yes	complemented
100	#1450	C₂×C₂₃	1	1	yes	characteristic, complemented
101	C₂²×C₂₃	—	2	2	no	—
102	C₂²×C₂₃	—	8	4	no	—
103	C₂²×C₂₃	—	8	8	no	—
104	C₂²×C₂₃	—	16	8	no	—
105	C₂²×C₂₃	#492	1	2	yes	—
106	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
107	C₄×C₂₃	—	2	2	no	—
108	C₄×C₂₃	—	8	4	no	—
109	C₄×C₂₃	#492	1	2	yes	—
110	C₄×C₂₃	C₂²×SD₁₆	1	2	yes	—
111	C₄×C₂₃	#576	1	4	yes	—
112	C₂²×D₈	—	2	2	no	—
113	#533	C₂²×C₂₃	1	2	yes	—
114	C₂×#127	C₂²×C₂₃	1	2	yes	—
115	#500	C₂²×C₂₃	1	2	yes	complemented
116	#499	—	2	2	no	—
117	#498	C₂²×C₂₃	1	2	yes	complemented
118	#493	—	2	2	no	—
119	#493	C₂²×C₂₃	1	2	yes	—
120	#493	C₂²×C₂₃	1	2	yes	—
121	#492	C₂²×C₂₃	1	8	yes	complemented
122	#486	C₂²×C₂₃	1	8	yes	complemented
123	#460	—	2	8	no	—
124	#450	—	2	8	no	—
125	#431	C₂²×C₂₃	1	2	yes	—
126	#427	—	2	2	no	—
127	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
128	#322	C₂²×C₂₃	1	2	yes	complemented
129	C₈²	C₂²×C₂₃	1	2	yes	complemented
130	C₂×C₂₃	—	2	2	no	—
131	C₂×C₂₃	—	16	4	no	—
132	C₂×C₂₃	C₂×#492	1	1	yes	characteristic
133	C₂×C₂₃	#3155	1	2	yes	—
134	C₂²×D₄	—	2	2	no	—
135	C₂²×D₄	—	2	4	no	—
136	C₂×SD₁₆	—	2	16	no	—
137	C₂×D₈	—	2	4	no	—
138	C₂×D₈	—	2	4	no	—
139	C₂×D₈	—	4	8	no	—
140	C₂²×C₈	D₄×C₂₃	1	2	yes	—
141	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
142	#127	—	2	4	no	—
143	#127	—	2	2	no	—
144	#127	C₂³×C₂₃	1	4	yes	—
145	#125	—	2	2	no	—
146	C₂²⋊Q₈	—	2	4	no	—
147	C₄×Q₈	—	2	4	no	—
148	#117	—	2	2	no	—
149	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
150	#107	—	2	4	no	—
151	#106	—	2	4	no	—
152	C₄⋊C₈	D₄×C₂₃	1	4	yes	—
153	#102	—	2	16	no	—
154	C₂²⋊C₈	D₄×C₂₃	1	4	yes	—
155	C₄×C₈	D₄×C₂₃	1	2	yes	—
156	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
157	C₄×C₈	C₂³×C₂₃	1	4	yes	—
158	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
159	C₂₃	#19652	1	1	yes	characteristic, direct product factor, Sylow
160	C₂⁴	—	4	2	no	—
161	C₂×Q₈	—	2	4	no	—
162	C₂×D₄	—	2	8	no	—
163	C₂×D₄	—	2	4	no	—
164	C₂×D₄	—	4	4	no	—
165	C₂×D₄	—	4	2	no	—
166	C₂×D₄	—	4	8	no	—
167	C₂×D₄	—	8	8	no	—
168	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
169	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
170	SD₁₆	—	8	16	no	—
171	D₈	—	4	8	no	—
172	D₈	—	4	8	no	—
173	C₂×C₈	SD₁₆×C₂₃	1	8	yes	complemented
174	C₂×C₈	C₂×D₄×C₂₃	1	8	yes	—
175	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
176	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	4	4	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₄²	—	4	2	no	—
182	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
183	C₄²	C₂×D₄×C₂₃	1	2	yes	—
184	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
185	C₂³	—	4	4	no	—
186	C₂³	—	8	8	no	—
187	C₂³	—	8	4	no	—
188	C₂³	C₂₃×#127	1	1	yes	characteristic
189	Q₈	—	4	8	no	—
190	D₄	—	4	8	no	—
191	D₄	—	4	16	no	—
192	D₄	—	8	8	no	—
193	D₄	—	8	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₂₃×#127	1	2	yes	—
197	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
198	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
199	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
200	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
201	C₈	—	2	8	no	—
202	C₈	C₂×SD₁₆×C₂₃	1	4	yes	—
203	C₈	C₂×SD₁₆×C₂₃	1	4	yes	—
204	C₂²	—	2	2	no	—
205	C₂²	—	8	4	no	—
206	C₂²	—	8	8	no	—
207	C₂²	—	16	8	no	—
208	C₂²	C₂₃×#492	1	2	yes	—
209	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	8	4	no	—
212	C₄	C₂₃×#492	1	2	yes	—
213	C₄	C₂²×SD₁₆×C₂₃	1	2	yes	—
214	C₄	C₂₃×#576	1	4	yes	—
215	C₂	—	2	2	no	—
216	C₂	—	16	4	no	—
217	C₂	C₂×C₂₃×#492	1	1	yes	characteristic
218	C₂	C₂₃×#3155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#493	C₂
LCS	G ▷ C₄² ▷ 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
1. C₂₃×#3208	C₂	semidirect
2. C₂₃×#3160	C₂	semidirect
4. C₂₃×#1715	C₂	semidirect
5. C₂₃×#1706	C₂	semidirect
6. C₂₃×#1450	C₂	semidirect
10. C₂₃×#500	C₂²	semidirect
12. C₂₃×#498	C₂²	semidirect
16. C₂₃×#492	C₂²	semidirect
17. C₂₃×#486	C₂²	semidirect
23. C₂₃×#322	C₂²	semidirect
24. C₈²×C₂₃	C₂²	semidirect
63. C₂×C₈×C₂₃	SD₁₆	semidirect
75. #19652	C₂₃	direct (N × H)
95. #3208	C₂×C₂₃	semidirect
96. #3160	C₂×C₂₃	semidirect
98. #1715	C₂×C₂₃	semidirect
99. #1706	C₂×C₂₃	semidirect
100. #1450	C₂×C₂₃	semidirect
115. #500	C₂²×C₂₃	semidirect
117. #498	C₂²×C₂₃	semidirect
121. #492	C₂²×C₂₃	semidirect
122. #486	C₂²×C₂₃	semidirect
128. #322	C₂²×C₂₃	semidirect
129. C₈²	C₂²×C₂₃	semidirect
159. C₂₃	#19652	direct (N × H)
173. 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
C₂₃×#3208	2
C₂₃×#3160	2
C₂×C₂₃×#493	2
C₂₃×#1715	2
C₂₃×#1706	2
C₂₃×#1450	2
#19652	23

Subgroups — C23×#19652 #19167278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19652 #19167278