Subgroups — C₂₃×#19653 #19167279

Order 5888.

730 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	C₂₃×#3229	C₂	1	4	yes	maximal
2	C₂₃×#3160	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1716	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1707	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1450	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₂₃	—	2	2	no	—
8	C₂₃×#533	C₂²	1	2	yes	—
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂₃×#500	C₂²	1	2	yes	—
11	C₂₃×#499	—	2	2	no	—
12	C₂₃×#498	C₂²	1	8	yes	—
13	C₂₃×#498	C₂²	1	2	yes	—
14	C₂₃×#494	—	2	2	no	—
15	C₂₃×#494	C₂²	1	2	yes	—
16	C₂₃×#494	C₂²	1	2	yes	—
17	C₂₃×#487	C₂²	1	8	yes	—
18	C₂₃×#483	—	2	8	no	—
19	C₂₃×#475	—	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	—
24	C₈²×C₂₃	C₂²	1	2	yes	—
25	C₂²×Q₈×C₂₃	—	2	2	no	—
26	C₂×Q₁₆×C₂₃	—	2	4	no	—
27	C₂×Q₁₆×C₂₃	—	2	4	no	—
28	C₂×Q₁₆×C₂₃	—	4	8	no	—
29	C₂²×C₈×C₂₃	D₄	1	2	yes	—
30	C₂₃×C₄⋊Q₈	—	2	4	no	—
31	C₂₃×C₄⋊Q₈	—	2	2	no	—
32	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
33	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
34	C₂₃×#125	—	2	2	no	—
35	C₂₃×C₂²⋊Q₈	—	2	4	no	—
36	C₄×Q₈×C₂₃	—	2	4	no	—
37	C₂₃×#117	—	2	2	no	—
38	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
39	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×#107	—	2	4	no	—
41	C₂₃×#106	—	2	4	no	—
42	C₂₃×#106	—	2	16	no	—
43	C₂₃×C₄⋊C₈	D₄	1	4	yes	—
44	C₂₃×#103	—	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	C₂×Q₈×C₂₃	—	2	4	no	—
51	C₂×Q₈×C₂₃	—	2	4	no	—
52	C₂×Q₈×C₂₃	—	4	2	no	—
53	C₂×Q₈×C₂₃	—	4	8	no	—
54	C₂²×C₄×C₂₃	—	4	2	no	—
55	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
57	Q₁₆×C₂₃	—	4	8	no	—
58	Q₁₆×C₂₃	—	4	8	no	—
59	C₂×C₈×C₂₃	SD₁₆	1	8	yes	—
60	C₂×C₈×C₂₃	C₂×D₄	1	8	yes	—
61	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
63	C₄⋊C₄×C₂₃	—	2	8	no	—
64	C₄⋊C₄×C₂₃	—	2	4	no	—
65	C₄⋊C₄×C₂₃	—	2	4	no	—
66	C₄⋊C₄×C₂₃	—	4	4	no	—
67	C₄⋊C₄×C₂₃	—	4	4	no	—
68	C₂²⋊C₄×C₂₃	—	4	2	no	—
69	C₄²×C₂₃	—	4	2	no	—
70	C₄²×C₂₃	C₂×D₄	1	2	yes	—
71	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
73	#19653	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₂₃	#127	1	1	yes	characteristic
75	Q₈×C₂₃	—	4	8	no	—
76	Q₈×C₂₃	—	4	8	no	—
77	Q₈×C₂₃	—	8	4	no	—
78	C₂×C₄×C₂₃	—	2	4	no	—
79	C₂×C₄×C₂₃	—	4	4	no	—
80	C₂×C₄×C₂₃	—	4	4	no	—
81	C₂×C₄×C₂₃	—	8	4	no	—
82	C₂×C₄×C₂₃	#127	1	2	yes	—
83	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
84	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
85	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
87	C₈×C₂₃	—	2	8	no	—
88	C₈×C₂₃	C₂×SD₁₆	1	4	yes	—
89	C₈×C₂₃	C₂×SD₁₆	1	4	yes	—
90	#3229	C₂×C₂₃	1	4	yes	—
91	#3160	C₂×C₂₃	1	1	yes	characteristic
92	C₂×#494	C₂×C₂₃	1	1	yes	characteristic
93	#1716	C₂×C₂₃	1	4	yes	complemented
94	#1707	C₂×C₂₃	1	4	yes	complemented
95	#1450	C₂×C₂₃	1	1	yes	characteristic
96	C₂²×C₂₃	—	2	2	no	—
97	C₂²×C₂₃	#492	1	2	yes	—
98	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
99	C₄×C₂₃	—	2	2	no	—
100	C₄×C₂₃	—	8	4	no	—
101	C₄×C₂₃	—	8	4	no	—
102	C₄×C₂₃	#492	1	2	yes	—
103	C₄×C₂₃	C₂²×SD₁₆	1	2	yes	—
104	C₄×C₂₃	#578	1	4	yes	—
105	C₂²×Q₁₆	—	2	2	no	—
106	#533	C₂²×C₂₃	1	2	yes	—
107	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
108	#500	C₂²×C₂₃	1	2	yes	—
109	#499	—	2	2	no	—
110	#498	C₂²×C₂₃	1	2	yes	—
111	#498	C₂²×C₂₃	1	8	yes	—
112	#494	—	2	2	no	—
113	#494	C₂²×C₂₃	1	2	yes	—
114	#494	C₂²×C₂₃	1	2	yes	—
115	#487	C₂²×C₂₃	1	8	yes	—
116	#483	—	2	8	no	—
117	#475	—	2	8	no	—
118	#431	C₂²×C₂₃	1	2	yes	—
119	#427	—	2	2	no	—
120	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
121	#322	C₂²×C₂₃	1	2	yes	—
122	C₈²	C₂²×C₂₃	1	2	yes	—
123	C₂×C₂₃	—	2	2	no	—
124	C₂×C₂₃	C₂×#492	1	1	yes	characteristic
125	C₂×C₂₃	#3156	1	2	yes	—
126	C₂²×Q₈	—	2	2	no	—
127	C₂×Q₁₆	—	2	4	no	—
128	C₂×Q₁₆	—	2	4	no	—
129	C₂×Q₁₆	—	4	8	no	—
130	C₂²×C₈	D₄×C₂₃	1	2	yes	—
131	C₄⋊Q₈	—	2	4	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
134	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
135	#125	—	2	2	no	—
136	C₂²⋊Q₈	—	2	4	no	—
137	C₄×Q₈	—	2	4	no	—
138	#117	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	4	no	—
140	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
141	#107	—	2	4	no	—
142	#106	—	2	4	no	—
143	#106	—	2	16	no	—
144	C₄⋊C₈	D₄×C₂₃	1	4	yes	—
145	#103	—	2	16	no	—
146	C₂²⋊C₈	D₄×C₂₃	1	4	yes	—
147	C₄×C₈	D₄×C₂₃	1	2	yes	—
148	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
149	C₄×C₈	C₂³×C₂₃	1	4	yes	—
150	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
151	C₂₃	#19653	1	1	yes	characteristic, direct product factor, Sylow
152	C₂×Q₈	—	2	4	no	—
153	C₂×Q₈	—	2	4	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	—	4	8	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
158	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
159	Q₁₆	—	4	8	no	—
160	Q₁₆	—	4	8	no	—
161	C₂×C₈	SD₁₆×C₂₃	1	8	yes	—
162	C₂×C₈	C₂×D₄×C₂₃	1	8	yes	—
163	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
164	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	4	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₂²⋊C₄	—	4	2	no	—
171	C₄²	—	4	2	no	—
172	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
173	C₄²	C₂×D₄×C₂₃	1	2	yes	—
174	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
175	C₂³	C₂₃×#127	1	1	yes	characteristic
176	Q₈	—	4	8	no	—
177	Q₈	—	4	8	no	—
178	Q₈	—	8	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	8	4	no	—
183	C₂×C₄	C₂₃×#127	1	2	yes	—
184	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
185	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
186	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
187	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
188	C₈	—	2	8	no	—
189	C₈	C₂×SD₁₆×C₂₃	1	4	yes	—
190	C₈	C₂×SD₁₆×C₂₃	1	4	yes	—
191	C₂²	—	2	2	no	—
192	C₂²	C₂₃×#492	1	2	yes	—
193	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	8	4	no	—
196	C₄	—	8	4	no	—
197	C₄	C₂₃×#492	1	2	yes	—
198	C₄	C₂²×SD₁₆×C₂₃	1	2	yes	—
199	C₄	C₂₃×#578	1	4	yes	—
200	C₂	—	2	2	no	—
201	C₂	C₂×C₂₃×#492	1	1	yes	characteristic
202	C₂	C₂₃×#3156	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂³×C₂₃	#127
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
4. C₂₃×#1716	C₂	semidirect
5. C₂₃×#1707	C₂	semidirect
73. #19653	C₂₃	direct (N × H)
93. #1716	C₂×C₂₃	semidirect
94. #1707	C₂×C₂₃	semidirect
151. C₂₃	#19653	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×#3229	2
C₂₃×#3160	2
C₂×C₂₃×#494	2
C₂₃×#1716	2
C₂₃×#1707	2
C₂₃×#1450	2
#19653	23

Subgroups — C23×#19653 #19167279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19653 #19167279