Subgroups — C₂₃×#17826 #19165452

Order 5888.

702 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2906	C₂	1	2	yes	maximal
3	C₂₃×#1570	C₂	1	6	yes	maximal, complemented
4	C₂₃×#1566	C₂	1	3	yes	maximal, complemented
5	C₂×C₂₃×#334	C₂	1	3	yes	maximal, complemented
6	C₂×C₂₃×#131	—	2	2	no	—
7	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₂₃	C₄	1	1	yes	characteristic
9	C₈×Q₈×C₂₃	C₄	1	1	yes	characteristic
10	C₈×D₄×C₂₃	C₄	1	3	yes	—
11	C₈×D₄×C₂₃	C₄	1	3	yes	—
12	C₂₃×#431	C₂²	1	3	yes	—
13	C₂₃×#424	—	2	6	no	—
14	C₂₃×#423	C₂²	1	6	yes	—
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	6	yes	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	3	yes	—
17	C₂₃×#334	C₂²	1	1	yes	characteristic
18	C₂₃×#334	C₂²	1	3	yes	—
19	C₂₃×#334	C₂²	1	6	yes	—
20	C₂₃×#334	C₂²	1	6	yes	—
21	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
22	C₂₃×#131	—	4	8	no	—
23	C₂×M₁₆×C₂₃	—	2	6	no	—
24	C₂²×C₈×C₂₃	—	2	6	no	—
25	C₂²×C₈×C₂₃	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₂₃	C₂×C₄	1	3	yes	—
27	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₄×Q₈×C₂₃	Q₈	1	1	yes	characteristic
29	C₄×D₄×C₂₃	C₂×C₄	1	3	yes	—
30	C₄×D₄×C₂₃	Q₈	1	3	yes	—
31	C₂₃×#117	D₄	1	3	yes	—
32	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
33	C₂₃×C₄⋊C₈	—	2	6	no	—
34	C₂₃×C₄⋊C₈	—	2	6	no	—
35	C₂₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
36	C₂₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
37	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
38	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
39	C₂₃×C₂²⋊C₈	C₂×C₄	1	6	yes	—
40	C₄×C₈×C₂₃	C₂×C₄	1	3	yes	—
41	C₄×C₈×C₂₃	C₂×C₄	1	3	yes	—
42	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₂₃	C₂³	1	3	yes	—
44	#41×C₂₃	SD₁₆	1	8	yes	—
45	C₂×Q₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×D₄×C₂₃	C₄⋊C₄	1	3	yes	—
47	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
48	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
49	C₂²×C₄×C₂₃	C₂×Q₈	1	3	yes	—
50	M₁₆×C₂₃	—	4	12	no	—
51	C₂×C₈×C₂₃	—	2	6	no	—
52	C₂×C₈×C₂₃	—	2	2	no	—
53	C₂×C₈×C₂₃	—	2	6	no	—
54	C₂×C₈×C₂₃	—	4	3	no	—
55	C₂×C₈×C₂₃	—	4	12	no	—
56	C₂×C₈×C₂₃	C₂²×C₄	1	3	yes	—
57	C₂×C₈×C₂₃	C₂²×C₄	1	3	yes	—
58	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₄⋊C₄×C₂₃	C₄⋊C₄	1	3	yes	—
61	C₄⋊C₄×C₂₃	C₄⋊C₄	1	3	yes	—
62	C₂²⋊C₄×C₂₃	C₄⋊C₄	1	6	yes	—
63	C₄²×C₂₃	C₂²×C₄	1	3	yes	—
64	C₄²×C₂₃	C₂×D₄	1	3	yes	—
65	C₄²×C₂₃	C₂×Q₈	1	3	yes	—
66	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
67	#17826	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
69	Q₈×C₂₃	#106	1	4	yes	—
70	D₄×C₂₃	#106	1	12	yes	—
71	C₂×C₄×C₂₃	—	2	3	no	—
72	C₂×C₄×C₂₃	—	2	3	no	—
73	C₂×C₄×C₂₃	—	4	3	no	—
74	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
77	C₂×C₄×C₂₃	C₂×SD₁₆	1	12	yes	—
78	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
81	C₈×C₂₃	—	2	3	no	—
82	C₈×C₂₃	—	2	1	no	—
83	C₈×C₂₃	—	2	3	no	—
84	C₈×C₂₃	—	4	2	no	—
85	C₈×C₂₃	—	4	6	no	—
86	C₈×C₂₃	#131	1	2	yes	complemented
87	C₈×#41	C₂×C₂₃	1	1	yes	characteristic
88	#2906	C₂×C₂₃	1	2	yes	—
89	#1570	C₂×C₂₃	1	6	yes	complemented
90	#1566	C₂×C₂₃	1	3	yes	complemented
91	C₂×#334	C₂×C₂₃	1	3	yes	complemented
92	C₂²×C₂₃	—	2	3	no	—
93	C₂²×C₂₃	—	2	3	no	—
94	C₂²×C₂₃	C₂×#106	1	6	yes	—
95	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₂₃	—	2	3	no	—
97	C₄×C₂₃	—	2	1	no	—
98	C₄×C₂₃	—	2	3	no	—
99	C₄×C₂₃	C₂×#106	1	6	yes	—
100	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
101	C₄×C₂₃	#568	1	1	yes	characteristic
102	C₄×C₂₃	C₂²×SD₁₆	1	2	yes	—
103	C₂×#131	—	2	2	no	—
104	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
105	C₈×Q₈	C₄×C₂₃	1	1	yes	characteristic
106	C₈×Q₈	C₄×C₂₃	1	1	yes	characteristic
107	C₈×D₄	C₄×C₂₃	1	3	yes	—
108	C₈×D₄	C₄×C₂₃	1	3	yes	—
109	#431	C₂²×C₂₃	1	3	yes	—
110	#424	—	2	6	no	—
111	#423	C₂²×C₂₃	1	6	yes	—
112	C₂×C₄⋊C₈	C₂²×C₂₃	1	6	yes	—
113	C₂×C₄×C₈	C₂²×C₂₃	1	3	yes	—
114	#334	C₂²×C₂₃	1	1	yes	characteristic
115	#334	C₂²×C₂₃	1	3	yes	—
116	#334	C₂²×C₂₃	1	6	yes	—
117	#334	C₂²×C₂₃	1	6	yes	—
118	C₂×C₂₃	—	2	6	no	—
119	C₂×C₂₃	#2906	1	1	yes	characteristic
120	C₂×C₂₃	C₂²×#106	1	1	yes	characteristic
121	C₂×C₂₃	#2916	1	1	yes	characteristic
122	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
123	#131	—	4	8	no	—
124	C₂×M₁₆	—	2	6	no	—
125	C₂²×C₈	—	2	6	no	—
126	C₂²×C₈	C₂×C₄×C₂₃	1	3	yes	—
127	C₂²×C₈	C₂×C₄×C₂₃	1	3	yes	—
128	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
129	C₄×Q₈	Q₈×C₂₃	1	1	yes	characteristic
130	C₄×D₄	C₂×C₄×C₂₃	1	3	yes	—
131	C₄×D₄	Q₈×C₂₃	1	3	yes	—
132	#117	D₄×C₂₃	1	3	yes	—
133	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
134	C₄⋊C₈	—	2	6	no	—
135	C₄⋊C₈	—	2	6	no	—
136	C₄⋊C₈	C₂×C₄×C₂₃	1	3	yes	—
137	C₄⋊C₈	C₂×C₄×C₂₃	1	3	yes	—
138	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
139	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
140	C₂²⋊C₈	C₂×C₄×C₂₃	1	6	yes	—
141	C₄×C₈	C₂×C₄×C₂₃	1	3	yes	—
142	C₄×C₈	C₂×C₄×C₂₃	1	3	yes	—
143	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
144	C₄×C₈	C₂³×C₂₃	1	3	yes	—
145	C₂₃	#17826	1	1	yes	characteristic, direct product factor, Sylow
146	#41	SD₁₆×C₂₃	1	8	yes	—
147	C₂×Q₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
148	C₂×D₄	C₄⋊C₄×C₂₃	1	3	yes	—
149	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
150	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
151	C₂²×C₄	C₂×Q₈×C₂₃	1	3	yes	—
152	M₁₆	—	4	12	no	—
153	C₂×C₈	—	2	6	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	6	no	—
156	C₂×C₈	—	4	3	no	—
157	C₂×C₈	—	4	12	no	—
158	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
159	C₂×C₈	C₂²×C₄×C₂₃	1	3	yes	—
160	C₂×C₈	C₂²×C₄×C₂₃	1	3	yes	—
161	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
162	C₄⋊C₄	C₄⋊C₄×C₂₃	1	3	yes	—
163	C₄⋊C₄	C₄⋊C₄×C₂₃	1	3	yes	—
164	C₂²⋊C₄	C₄⋊C₄×C₂₃	1	6	yes	—
165	C₄²	C₂²×C₄×C₂₃	1	3	yes	—
166	C₄²	C₂×D₄×C₂₃	1	3	yes	—
167	C₄²	C₂×Q₈×C₂₃	1	3	yes	—
168	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
169	C₂³	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
170	Q₈	C₂₃×#106	1	4	yes	—
171	D₄	C₂₃×#106	1	12	yes	—
172	C₂×C₄	—	2	3	no	—
173	C₂×C₄	—	2	3	no	—
174	C₂×C₄	—	4	3	no	—
175	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
176	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
177	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
178	C₂×C₄	C₂×SD₁₆×C₂₃	1	12	yes	—
179	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
180	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
181	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
182	C₈	—	2	3	no	—
183	C₈	—	2	1	no	—
184	C₈	—	2	3	no	—
185	C₈	—	4	2	no	—
186	C₈	—	4	6	no	—
187	C₈	C₂₃×#131	1	2	yes	complemented
188	C₂²	—	2	3	no	—
189	C₂²	—	2	3	no	—
190	C₂²	C₂×C₂₃×#106	1	6	yes	—
191	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	3	no	—
194	C₄	—	2	3	no	—
195	C₄	C₂×C₂₃×#106	1	6	yes	—
196	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
197	C₄	C₂₃×#568	1	1	yes	characteristic
198	C₄	C₂²×SD₁₆×C₂₃	1	2	yes	—
199	C₂	—	2	6	no	—
200	C₂	C₂₃×#2906	1	1	yes	characteristic
201	C₂	C₂²×C₂₃×#106	1	1	yes	characteristic
202	C₂	C₂₃×#2916	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	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₂₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1570	C₂	semidirect
4. C₂₃×#1566	C₂	semidirect
5. C₂×C₂₃×#334	C₂	semidirect
67. #17826	C₂₃	direct (N × H)
86. C₈×C₂₃	#131	semidirect
89. #1570	C₂×C₂₃	semidirect
90. #1566	C₂×C₂₃	semidirect
91. C₂×#334	C₂×C₂₃	semidirect
145. C₂₃	#17826	direct (N × H)
187. C₈	C₂₃×#131	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₈×#41×C₂₃	2
C₂₃×#2906	2
C₂₃×#1570	2
C₂₃×#1566	2
C₂×C₂₃×#334	2
#17826	23

Subgroups — C23×#17826 #19165452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17826 #19165452