Subgroups — C₂₃×#20831 #19168457

Order 5888.

758 conjugacy classes of subgroups.

The subgroup lattice has 214 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	2	yes	maximal
2	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1848	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1837	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1811	C₂	1	4	yes	maximal
6	C₂₃×#1778	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#432	—	2	4	no	—
10	C₂₃×#431	—	2	4	no	—
11	C₂×C₂₃×#108	C₂²	1	8	yes	—
12	C₂₃×#423	C₂²	1	4	yes	—
13	C₂₃×#407	C₂²	1	4	yes	—
14	C₄×M₁₆×C₂₃	C₄	1	4	yes	—
15	C₄×M₁₆×C₂₃	C₄	1	4	yes	—
16	C₄×M₁₆×C₂₃	C₂²	1	4	yes	—
17	C₂×C₂₃×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#341	C₂²	1	8	yes	—
21	C₂³×C₄×C₂₃	Q₈	1	2	yes	—
22	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₂₃	—	2	8	no	—
24	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	—
25	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
26	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
27	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
28	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₂₃	D₄	1	4	yes	—
31	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
32	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
34	C₂₃×#108	—	2	16	no	—
35	C₂₃×C₄⋊C₈	—	2	8	no	—
36	C₂₃×C₂²⋊C₈	—	2	8	no	—
37	C₂₃×#97	—	2	4	no	—
38	C₂₃×#97	C₂×C₄	1	8	yes	—
39	C₄×C₈×C₂₃	—	2	4	no	—
40	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
44	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	—	2	4	no	—
46	C₂²×C₄×C₂₃	—	4	4	no	—
47	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
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	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
54	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
55	M₁₆×C₂₃	—	2	16	no	—
56	M₁₆×C₂₃	—	2	16	no	—
57	C₂×C₈×C₂₃	—	2	4	no	—
58	C₂×C₈×C₂₃	—	2	8	no	—
59	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
60	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₂₃	#41	1	8	yes	—
62	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
63	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
64	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
67	C₄²×C₂₃	C₂×D₄	1	2	yes	—
68	C₄²×C₂₃	C₂×D₄	1	2	yes	—
69	#20831	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₂₃	—	2	2	no	—
71	C₂³×C₂₃	—	4	2	no	—
72	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
73	C₂³×C₂₃	C₂²×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	4	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	8	no	—
78	C₂×C₄×C₂₃	—	2	4	no	—
79	C₂×C₄×C₂₃	—	4	4	no	—
80	C₂×C₄×C₂₃	—	4	2	no	—
81	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
83	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
84	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
88	C₈×C₂₃	—	2	4	no	—
89	C₈×C₂₃	—	4	8	no	—
90	C₈×C₂₃	#117	1	8	yes	—
91	#2920	C₂×C₂₃	1	2	yes	—
92	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic
93	#1848	C₂×C₂₃	1	2	yes	complemented
94	#1837	C₂×C₂₃	1	2	yes	complemented
95	#1811	C₂×C₂₃	1	4	yes	—
96	#1778	C₂×C₂₃	1	4	yes	complemented
97	C₂²×C₂₃	—	2	2	no	—
98	C₂²×C₂₃	—	2	4	no	—
99	C₂²×C₂₃	—	2	2	no	—
100	C₂²×C₂₃	—	4	4	no	—
101	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₂₃	#533	1	2	yes	—
103	C₄×C₂₃	—	2	4	no	—
104	C₄×C₂₃	—	2	2	no	—
105	C₄×C₂₃	—	2	2	no	—
106	C₄×C₂₃	#430	1	4	yes	—
107	C₄×C₂₃	C₂×#117	1	2	yes	—
108	C₄×C₂₃	#532	1	2	yes	—
109	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
110	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
111	#432	—	2	4	no	—
112	#431	—	2	4	no	—
113	C₂×#108	C₂²×C₂₃	1	8	yes	—
114	#423	C₂²×C₂₃	1	4	yes	—
115	#407	C₂²×C₂₃	1	4	yes	—
116	C₄×M₁₆	C₄×C₂₃	1	4	yes	—
117	C₄×M₁₆	C₄×C₂₃	1	4	yes	—
118	C₄×M₁₆	C₂²×C₂₃	1	4	yes	—
119	C₂×#97	C₂²×C₂₃	1	2	yes	—
120	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
122	#341	C₂²×C₂₃	1	8	yes	—
123	C₂×C₂₃	—	2	2	no	—
124	C₂×C₂₃	—	4	2	no	—
125	C₂×C₂₃	#2285	1	1	yes	characteristic
126	C₂×C₂₃	C₂×#430	1	2	yes	—
127	C₂³×C₄	Q₈×C₂₃	1	2	yes	—
128	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
129	C₂×M₁₆	—	2	8	no	—
130	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	—
131	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
132	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
133	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
134	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
135	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
136	C₂×C₄²	D₄×C₂₃	1	4	yes	—
137	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
138	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
139	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
140	#108	—	2	16	no	—
141	C₄⋊C₈	—	2	8	no	—
142	C₂²⋊C₈	—	2	8	no	—
143	#97	—	2	4	no	—
144	#97	C₂×C₄×C₂₃	1	8	yes	—
145	C₄×C₈	—	2	4	no	—
146	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
147	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
148	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
149	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
150	C₂₃	#20831	1	1	yes	characteristic, direct product factor, Sylow
151	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	—	4	4	no	—
154	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
155	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
156	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
157	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
158	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
159	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
160	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
161	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
162	M₁₆	—	2	16	no	—
163	M₁₆	—	2	16	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	8	no	—
166	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
167	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
168	C₂×C₈	#41×C₂₃	1	8	yes	—
169	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
170	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
171	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
172	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
173	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
174	C₄²	C₂×D₄×C₂₃	1	2	yes	—
175	C₄²	C₂×D₄×C₂₃	1	2	yes	—
176	C₂³	—	2	2	no	—
177	C₂³	—	4	2	no	—
178	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
179	C₂³	C₂²×Q₈×C₂₃	1	1	yes	characteristic
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
188	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
189	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
190	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
191	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
192	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
193	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
194	C₈	—	2	4	no	—
195	C₈	—	4	8	no	—
196	C₈	C₂₃×#117	1	8	yes	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	4	no	—
201	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
202	C₂²	C₂₃×#533	1	2	yes	—
203	C₄	—	2	4	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	2	no	—
206	C₄	C₂₃×#430	1	4	yes	—
207	C₄	C₂×C₂₃×#117	1	2	yes	—
208	C₄	C₂₃×#532	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	4	2	no	—
211	C₂	C₂₃×#2285	1	1	yes	characteristic
212	C₂	C₂×C₂₃×#430	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₂₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
3. C₂₃×#1848	C₂	semidirect
4. C₂₃×#1837	C₂	semidirect
6. C₂₃×#1778	C₂	semidirect
69. #20831	C₂₃	direct (N × H)
93. #1848	C₂×C₂₃	semidirect
94. #1837	C₂×C₂₃	semidirect
96. #1778	C₂×C₂₃	semidirect
150. C₂₃	#20831	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
C₂×C₄×M₁₆×C₂₃	2
C₂₃×#1848	2
C₂₃×#1837	2
C₂₃×#1811	2
C₂₃×#1778	2
#20831	23

Subgroups — C23×#20831 #19168457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20831 #19168457