Subgroups — C₂₃×#12667 #19160293

Order 5888.

296 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#364	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2110	C₂	1	2	yes	maximal
3	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1772	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1378	C₂	1	2	yes	maximal
6	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
7	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#108	C₄	1	1	yes	characteristic
10	C₂₃×#427	C₄	1	1	yes	characteristic
11	C₂₃×#405	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#364	C₄	1	2	yes	complemented
14	C₂₃×#364	C₄	1	2	yes	complemented
15	C₂₃×#347	—	2	4	no	—
16	C₂₃×#346	C₄	1	2	yes	complemented
17	C₄×C₁₆×C₂₃	C₄	1	2	yes	complemented
18	C₂₃×#339	C₄	1	2	yes	—
19	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
22	C₂₃×#117	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂₃×M₃₂	—	2	2	no	—
25	C₂₃×M₃₂	—	2	1	no	—
26	C₂₃×M₃₂	—	2	1	no	—
27	C₂₃×M₃₂	C₂×C₄	1	4	yes	—
28	C₂₃×M₃₂	D₄	1	2	yes	—
29	C₂₃×M₃₂	Q₈	1	2	yes	—
30	C₂×C₁₆×C₂₃	—	2	2	no	—
31	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
33	C₂₃×#108	—	2	1	no	—
34	C₂₃×#108	—	2	1	no	—
35	C₂₃×#107	—	2	1	no	—
36	C₂₃×#106	—	2	1	no	—
37	C₂₃×#97	—	2	2	no	—
38	C₄×C₈×C₂₃	—	2	2	no	—
39	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	—	2	1	no	—
44	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	M₁₆×C₂₃	—	2	2	no	—
46	M₁₆×C₂₃	—	2	2	no	—
47	C₂×C₈×C₂₃	—	2	2	no	—
48	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₂₃	M₁₆	1	1	yes	characteristic
51	C₂×C₈×C₂₃	M₁₆	1	1	yes	characteristic
52	C₂×C₈×C₂₃	M₁₆	1	1	yes	characteristic
53	C₂×C₈×C₂₃	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
59	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₂²⋊C₄×C₂₃	—	2	2	no	—
62	C₄²×C₂₃	—	2	1	no	—
63	C₄²×C₂₃	—	2	2	no	—
64	C₄²×C₂₃	C₄²	1	1	yes	characteristic
65	C₄²×C₂₃	C₄²	1	1	yes	characteristic
66	C₁₆×C₂₃	—	2	2	no	—
67	C₁₆×C₂₃	—	2	4	no	—
68	C₁₆×C₂₃	C₄⋊C₄	1	4	yes	—
69	#12667	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₂₃	#131	1	1	yes	characteristic
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	1	no	—
73	C₂×C₄×C₂₃	—	4	2	no	—
74	C₂×C₄×C₂₃	—	4	2	no	—
75	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
81	C₈×C₂₃	—	4	2	no	—
82	C₈×C₂₃	#97	1	1	yes	characteristic
83	C₈×C₂₃	#97	1	1	yes	characteristic
84	C₈×C₂₃	#97	1	1	yes	characteristic
85	C₈×C₂₃	#97	1	1	yes	characteristic
86	C₈×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₈×C₂₃	#117	1	1	yes	characteristic
88	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
89	C₈×C₂₃	C₄×Q₈	1	1	yes	characteristic
90	C₂×#364	C₂×C₂₃	1	1	yes	characteristic
91	#2110	C₂×C₂₃	1	2	yes	—
92	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic
93	#1772	C₂×C₂₃	1	1	yes	characteristic
94	#1378	C₂×C₂₃	1	2	yes	—
95	C₂²×C₂₃	—	4	1	no	—
96	C₂²×C₂₃	#405	1	1	yes	characteristic
97	C₂²×C₂₃	#436	1	1	yes	characteristic
98	C₂²×C₂₃	#446	1	1	yes	characteristic
99	C₄×C₂₃	—	2	2	no	—
100	C₄×C₂₃	—	4	1	no	—
101	C₄×C₂₃	—	8	2	no	—
102	C₄×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
103	C₄×C₂₃	C₂×#97	1	1	yes	characteristic
104	C₄×C₂₃	#436	1	1	yes	characteristic
105	C₄×C₂₃	#446	1	1	yes	characteristic
106	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
107	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
108	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
109	C₂×#108	C₄×C₂₃	1	1	yes	characteristic
110	#427	C₄×C₂₃	1	1	yes	characteristic
111	#405	C₂²×C₂₃	1	2	yes	—
112	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
113	#364	C₄×C₂₃	1	2	yes	complemented
114	#364	C₄×C₂₃	1	2	yes	complemented
115	#347	—	2	4	no	—
116	#346	C₄×C₂₃	1	2	yes	complemented
117	C₄×C₁₆	C₄×C₂₃	1	2	yes	complemented
118	#339	C₄×C₂₃	1	2	yes	—
119	C₂×C₂₃	—	2	1	no	—
120	C₂×C₂₃	—	2	1	no	—
121	C₂×C₂₃	—	2	1	no	—
122	C₂×C₂₃	#1770	1	1	yes	characteristic
123	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
124	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
125	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
126	#117	C₂×C₄×C₂₃	1	2	yes	—
127	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
128	M₃₂	—	2	2	no	—
129	M₃₂	—	2	1	no	—
130	M₃₂	—	2	1	no	—
131	M₃₂	C₂×C₄×C₂₃	1	4	yes	—
132	M₃₂	D₄×C₂₃	1	2	yes	—
133	M₃₂	Q₈×C₂₃	1	2	yes	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
136	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
137	#108	—	2	1	no	—
138	#108	—	2	1	no	—
139	#107	—	2	1	no	—
140	#106	—	2	1	no	—
141	#97	—	2	2	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
144	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
145	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
146	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
147	C₂₃	#12667	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
154	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
155	C₂×C₈	M₁₆×C₂₃	1	1	yes	characteristic
156	C₂×C₈	M₁₆×C₂₃	1	1	yes	characteristic
157	C₂×C₈	M₁₆×C₂₃	1	1	yes	characteristic
158	C₂×C₈	M₁₆×C₂₃	1	1	yes	characteristic
159	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
160	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
161	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
162	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
163	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
164	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
165	C₄⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	2	no	—
169	C₄²	C₄²×C₂₃	1	1	yes	characteristic
170	C₄²	C₄²×C₂₃	1	1	yes	characteristic
171	C₁₆	—	2	2	no	—
172	C₁₆	—	2	4	no	—
173	C₁₆	C₄⋊C₄×C₂₃	1	4	yes	—
174	C₂³	C₂₃×#131	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
180	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
181	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
182	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
183	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
184	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
185	C₈	—	4	2	no	—
186	C₈	C₂₃×#97	1	1	yes	characteristic
187	C₈	C₂₃×#97	1	1	yes	characteristic
188	C₈	C₂₃×#97	1	1	yes	characteristic
189	C₈	C₂₃×#97	1	1	yes	characteristic
190	C₈	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
191	C₈	C₂₃×#117	1	1	yes	characteristic
192	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
193	C₈	C₄×Q₈×C₂₃	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	C₂₃×#405	1	1	yes	characteristic
196	C₂²	C₂₃×#436	1	1	yes	characteristic
197	C₂²	C₂₃×#446	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	8	2	no	—
201	C₄	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
202	C₄	C₂×C₂₃×#97	1	1	yes	characteristic
203	C₄	C₂₃×#436	1	1	yes	characteristic
204	C₄	C₂₃×#446	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	C₂₃×#1770	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#117
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#1770
prime radical	C₂³×C₂₃	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂×C₂₃×M₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂₃×#364	C₄	semidirect
14. C₂₃×#364	C₄	semidirect
16. C₂₃×#346	C₄	semidirect
17. C₄×C₁₆×C₂₃	C₄	semidirect
69. #12667	C₂₃	direct (N × H)
113. #364	C₄×C₂₃	semidirect
114. #364	C₄×C₂₃	semidirect
116. #346	C₄×C₂₃	semidirect
117. C₄×C₁₆	C₄×C₂₃	semidirect
147. C₂₃	#12667	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×#364	2
C₂₃×#2110	2
C₄×C₂₃×M₃₂	2
C₂₃×#1772	2
C₂₃×#1378	2
#12667	23

Subgroups — C23×#12667 #19160293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12667 #19160293