Subgroups — C₂₃×#19608 #19167234

Order 5888.

798 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3167	C₂	1	4	yes	maximal, complemented
2	C₂₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1716	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1715	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1714	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#135	—	2	2	no	—
9	C₂₃×#533	C₂²	1	2	yes	—
10	C₂₃×#532	C₂²	1	2	yes	—
11	C₂₃×#500	C₂²	1	2	yes	—
12	C₂₃×#499	—	2	2	no	—
13	C₂₃×#498	C₂²	1	4	yes	—
14	C₂₃×#498	C₂²	1	2	yes	—
15	C₂₃×#495	—	2	2	no	—
16	C₂₃×#494	C₂²	1	1	yes	characteristic
17	C₂₃×#493	C₂²	1	1	yes	characteristic
18	C₂₃×#492	C₂²	1	4	yes	—
19	C₂₃×#492	C₂²	1	2	yes	—
20	C₂₃×#487	C₂²	1	4	yes	—
21	C₂₃×#486	C₂²	1	4	yes	—
22	C₂₃×#465	—	2	8	no	—
23	C₄×SD₁₆×C₂₃	—	2	8	no	—
24	C₂₃×#427	—	2	2	no	—
25	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#322	C₂²	1	2	yes	—
28	C₂₃×#322	C₂²	1	2	yes	—
29	C₂×#41×C₂₃	—	2	2	no	—
30	C₂₃×#135	—	4	8	no	—
31	C₂×Q₁₆×C₂₃	—	2	2	no	—
32	C₂×SD₁₆×C₂₃	—	2	4	no	—
33	C₂×SD₁₆×C₂₃	—	2	8	no	—
34	C₂×D₈×C₂₃	—	2	2	no	—
35	C₂²×C₈×C₂₃	D₄	1	4	yes	—
36	C₂²×C₈×C₂₃	D₄	1	2	yes	—
37	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
38	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
39	C₂₃×#127	C₂³	1	2	yes	—
40	C₂₃×#125	—	2	2	no	—
41	C₂₃×#124	—	2	2	no	—
42	C₂₃×C₂²⋊Q₈	—	2	4	no	—
43	C₂₃×#121	—	2	4	no	—
44	C₄×Q₈×C₂₃	—	2	4	no	—
45	C₄×D₄×C₂₃	—	2	4	no	—
46	C₂₃×#117	—	2	2	no	—
47	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
48	C₂₃×#107	—	2	4	no	—
49	C₂₃×#106	—	2	4	no	—
50	C₂₃×#106	—	2	8	no	—
51	C₂₃×#103	—	2	8	no	—
52	C₂₃×#102	—	2	8	no	—
53	C₄×C₈×C₂₃	D₄	1	4	yes	—
54	C₄×C₈×C₂₃	D₄	1	2	yes	—
55	C₄×C₈×C₂₃	C₂³	1	4	yes	—
56	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
58	#41×C₂₃	—	4	8	no	—
59	C₂×Q₈×C₂₃	—	2	4	no	—
60	C₂×Q₈×C₂₃	—	2	2	no	—
61	C₂×D₄×C₂₃	—	2	4	no	—
62	C₂×D₄×C₂₃	—	2	2	no	—
63	C₂×D₄×C₂₃	—	4	2	no	—
64	C₂²×C₄×C₂₃	—	4	2	no	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₂₃	—	4	4	no	—
68	SD₁₆×C₂₃	—	4	16	no	—
69	SD₁₆×C₂₃	—	4	8	no	—
70	D₈×C₂₃	—	4	4	no	—
71	C₂×C₈×C₂₃	—	4	4	no	—
72	C₂×C₈×C₂₃	SD₁₆	1	8	yes	complemented
73	C₂×C₈×C₂₃	C₂×D₄	1	8	yes	—
74	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
76	C₄⋊C₄×C₂₃	—	2	4	no	—
77	C₄⋊C₄×C₂₃	—	2	4	no	—
78	C₄⋊C₄×C₂₃	—	2	4	no	—
79	C₄⋊C₄×C₂₃	—	4	4	no	—
80	C₂²⋊C₄×C₂₃	—	4	2	no	—
81	C₂²⋊C₄×C₂₃	—	4	4	no	—
82	C₄²×C₂₃	—	4	2	no	—
83	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₂₃	C₂×D₄	1	2	yes	—
85	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
86	#19608	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₂₃	—	4	2	no	—
88	C₂³×C₂₃	#127	1	1	yes	characteristic
89	Q₈×C₂₃	—	4	4	no	—
90	Q₈×C₂₃	—	4	8	no	—
91	D₄×C₂₃	—	4	4	no	—
92	D₄×C₂₃	—	4	8	no	—
93	D₄×C₂₃	—	8	4	no	—
94	C₂×C₄×C₂₃	—	2	4	no	—
95	C₂×C₄×C₂₃	—	4	4	no	—
96	C₂×C₄×C₂₃	—	4	2	no	—
97	C₂×C₄×C₂₃	—	8	4	no	—
98	C₂×C₄×C₂₃	#127	1	2	yes	—
99	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
100	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
101	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
103	C₈×C₂₃	—	2	8	no	—
104	C₈×C₂₃	C₂×SD₁₆	1	4	yes	—
105	C₈×C₂₃	C₂×SD₁₆	1	4	yes	—
106	#3167	C₂×C₂₃	1	4	yes	complemented
107	#3160	C₂×C₂₃	1	1	yes	characteristic, complemented
108	#3147	C₂×C₂₃	1	1	yes	characteristic
109	#1716	C₂×C₂₃	1	2	yes	complemented
110	#1715	C₂×C₂₃	1	2	yes	complemented
111	#1714	C₂×C₂₃	1	4	yes	complemented
112	C₂×#322	C₂×C₂₃	1	1	yes	characteristic, complemented
113	C₂²×C₂₃	—	2	2	no	—
114	C₂²×C₂₃	—	8	2	no	—
115	C₂²×C₂₃	—	8	4	no	—
116	C₂²×C₂₃	#492	1	2	yes	—
117	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
118	C₄×C₂₃	—	2	2	no	—
119	C₄×C₂₃	—	8	2	no	—
120	C₄×C₂₃	—	8	4	no	—
121	C₄×C₂₃	#492	1	2	yes	—
122	C₄×C₂₃	C₂²×SD₁₆	1	2	yes	—
123	C₄×C₂₃	#575	1	4	yes	—
124	C₂×#135	—	2	2	no	—
125	#533	C₂²×C₂₃	1	2	yes	—
126	#532	C₂²×C₂₃	1	2	yes	—
127	#500	C₂²×C₂₃	1	2	yes	—
128	#499	—	2	2	no	—
129	#498	C₂²×C₂₃	1	2	yes	—
130	#498	C₂²×C₂₃	1	4	yes	—
131	#495	—	2	2	no	—
132	#494	C₂²×C₂₃	1	1	yes	characteristic
133	#493	C₂²×C₂₃	1	1	yes	characteristic
134	#492	C₂²×C₂₃	1	2	yes	—
135	#492	C₂²×C₂₃	1	4	yes	—
136	#487	C₂²×C₂₃	1	4	yes	—
137	#486	C₂²×C₂₃	1	4	yes	—
138	#465	—	2	8	no	—
139	C₄×SD₁₆	—	2	8	no	—
140	#427	—	2	2	no	—
141	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
143	#322	C₂²×C₂₃	1	2	yes	—
144	#322	C₂²×C₂₃	1	2	yes	—
145	C₂×C₂₃	—	2	2	no	—
146	C₂×C₂₃	—	16	2	no	—
147	C₂×C₂₃	C₂×#492	1	1	yes	characteristic
148	C₂×C₂₃	#3150	1	2	yes	—
149	C₂×#41	—	2	2	no	—
150	#135	—	4	8	no	—
151	C₂×Q₁₆	—	2	2	no	—
152	C₂×SD₁₆	—	2	4	no	—
153	C₂×SD₁₆	—	2	8	no	—
154	C₂×D₈	—	2	2	no	—
155	C₂²×C₈	D₄×C₂₃	1	4	yes	—
156	C₂²×C₈	D₄×C₂₃	1	2	yes	—
157	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
158	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
159	#127	C₂³×C₂₃	1	2	yes	—
160	#125	—	2	2	no	—
161	#124	—	2	2	no	—
162	C₂²⋊Q₈	—	2	4	no	—
163	#121	—	2	4	no	—
164	C₄×Q₈	—	2	4	no	—
165	C₄×D₄	—	2	4	no	—
166	#117	—	2	2	no	—
167	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
168	#107	—	2	4	no	—
169	#106	—	2	4	no	—
170	#106	—	2	8	no	—
171	#103	—	2	8	no	—
172	#102	—	2	8	no	—
173	C₄×C₈	D₄×C₂₃	1	4	yes	—
174	C₄×C₈	D₄×C₂₃	1	2	yes	—
175	C₄×C₈	C₂³×C₂₃	1	4	yes	—
176	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
177	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
178	C₂₃	#19608	1	1	yes	characteristic, direct product factor, Sylow
179	#41	—	4	8	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	4	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
187	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
188	Q₁₆	—	4	4	no	—
189	SD₁₆	—	4	16	no	—
190	SD₁₆	—	4	8	no	—
191	D₈	—	4	4	no	—
192	C₂×C₈	—	4	4	no	—
193	C₂×C₈	SD₁₆×C₂₃	1	8	yes	complemented
194	C₂×C₈	C₂×D₄×C₂₃	1	8	yes	—
195	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
196	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₂²⋊C₄	—	4	2	no	—
202	C₂²⋊C₄	—	4	4	no	—
203	C₄²	—	4	2	no	—
204	C₄²	C₂×D₄×C₂₃	1	2	yes	—
205	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
206	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
207	C₂³	—	4	2	no	—
208	C₂³	C₂₃×#127	1	1	yes	characteristic
209	Q₈	—	4	4	no	—
210	Q₈	—	4	8	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	8	no	—
213	D₄	—	8	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	8	4	no	—
218	C₂×C₄	C₂₃×#127	1	2	yes	—
219	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
220	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
221	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
223	C₈	—	2	8	no	—
224	C₈	C₂×SD₁₆×C₂₃	1	4	yes	—
225	C₈	C₂×SD₁₆×C₂₃	1	4	yes	—
226	C₂²	—	2	2	no	—
227	C₂²	—	8	2	no	—
228	C₂²	—	8	4	no	—
229	C₂²	C₂₃×#492	1	2	yes	—
230	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	8	2	no	—
233	C₄	—	8	4	no	—
234	C₄	C₂₃×#492	1	2	yes	—
235	C₄	C₂²×SD₁₆×C₂₃	1	2	yes	—
236	C₄	C₂₃×#575	1	4	yes	—
237	C₂	—	2	2	no	—
238	C₂	—	16	2	no	—
239	C₂	C₂×C₂₃×#492	1	1	yes	characteristic
240	C₂	C₂₃×#3150	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×#127
prime radical	C₂₃×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3167	C₂	semidirect
2. C₂₃×#3160	C₂	semidirect
4. C₂₃×#1716	C₂	semidirect
5. C₂₃×#1715	C₂	semidirect
6. C₂₃×#1714	C₂	semidirect
7. C₂×C₂₃×#322	C₂	semidirect
72. C₂×C₈×C₂₃	SD₁₆	semidirect
86. #19608	C₂₃	direct (N × H)
106. #3167	C₂×C₂₃	semidirect
107. #3160	C₂×C₂₃	semidirect
109. #1716	C₂×C₂₃	semidirect
110. #1715	C₂×C₂₃	semidirect
111. #1714	C₂×C₂₃	semidirect
112. C₂×#322	C₂×C₂₃	semidirect
178. C₂₃	#19608	direct (N × H)
193. 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×#3167	2
C₂₃×#3160	2
C₂₃×#3147	2
C₂₃×#1716	2
C₂₃×#1715	2
C₂₃×#1714	2
C₂×C₂₃×#322	2
#19608	23

Subgroups — C23×#19608 #19167234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19608 #19167234