Subgroups — C₂₃×#17813 #19165439

Order 5888.

730 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#423	C₂	1	2	yes	maximal
3	C₂₃×#1570	C₂	1	8	yes	maximal, complemented
4	C₂₃×#1566	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₂₃	—	2	2	no	—
6	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#431	C₄	1	4	yes	—
8	C₂₃×#431	C₂²	1	4	yes	—
9	C₂₃×#423	—	2	2	no	—
10	C₂₃×#423	—	2	2	no	—
11	C₂₃×#423	C₂²	1	4	yes	—
12	C₂₃×#423	C₂²	1	4	yes	—
13	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
14	C₂×C₂₃×C₄⋊C₈	C₄	1	2	yes	—
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
16	C₂×C₂₃×C₂²⋊C₈	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#334	C₂²	1	16	yes	complemented
19	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₂₃	Q₈	1	1	yes	characteristic
22	C₂×M₁₆×C₂₃	—	2	4	no	—
23	C₂×M₁₆×C₂₃	—	2	4	no	—
24	C₂×M₁₆×C₂₃	—	4	8	no	—
25	C₂²×C₈×C₂₃	—	2	4	no	—
26	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₂₃	D₄	1	2	yes	—
30	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
33	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
35	C₂₃×C₄⋊C₈	—	2	4	no	—
36	C₂₃×C₄⋊C₈	—	2	4	no	—
37	C₂₃×C₄⋊C₈	—	2	4	no	—
38	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
39	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
40	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
41	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₂₃	C₂³	1	4	yes	—
43	C₂⁴×C₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	—	4	2	no	—
45	C₂²×C₄×C₂₃	—	4	2	no	—
46	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₂₃	SD₁₆	1	8	yes	—
53	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
59	M₁₆×C₂₃	—	4	8	no	—
60	M₁₆×C₂₃	—	4	8	no	—
61	C₂×C₈×C₂₃	—	2	8	no	—
62	C₂×C₈×C₂₃	—	4	4	no	—
63	C₂×C₈×C₂₃	—	4	8	no	—
64	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
65	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
66	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
67	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
68	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
69	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₂₃	C₂×D₄	1	2	yes	—
71	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
73	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
74	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
75	#17813	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₂₃	—	4	1	no	—
77	C₂³×C₂₃	—	4	1	no	—
78	C₂³×C₂₃	#106	1	4	yes	—
79	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
80	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	4	no	—
83	C₂×C₄×C₂₃	—	2	4	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	4	2	no	—
86	C₂×C₄×C₂₃	—	4	1	no	—
87	C₂×C₄×C₂₃	—	4	1	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	#106	1	4	yes	—
90	C₂×C₄×C₂₃	#106	1	8	yes	—
91	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
98	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
99	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
100	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
103	C₈×C₂₃	—	2	8	no	—
104	C₈×C₂₃	—	4	8	no	—
105	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
106	C₂×#423	C₂×C₂₃	1	2	yes	—
107	#1570	C₂×C₂₃	1	8	yes	complemented
108	#1566	C₂×C₂₃	1	4	yes	complemented
109	C₂²×C₂₃	—	2	1	no	—
110	C₂²×C₂₃	—	2	2	no	—
111	C₂²×C₂₃	—	2	2	no	—
112	C₂²×C₂₃	—	2	1	no	—
113	C₂²×C₂₃	—	4	4	no	—
114	C₂²×C₂₃	C₂×#106	1	4	yes	—
115	C₂²×C₂₃	C₂×#106	1	2	yes	—
116	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₂₃	—	2	1	no	—
118	C₄×C₂₃	—	2	2	no	—
119	C₄×C₂₃	—	2	2	no	—
120	C₄×C₂₃	—	2	1	no	—
121	C₄×C₂₃	C₂×#106	1	4	yes	—
122	C₄×C₂₃	C₂×#106	1	2	yes	—
123	C₄×C₂₃	#430	1	2	yes	—
124	C₄×C₂₃	#568	1	2	yes	—
125	C₄×C₂₃	C₂²×SD₁₆	1	2	yes	—
126	C₂²×M₁₆	—	2	2	no	—
127	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
128	#431	C₄×C₂₃	1	4	yes	—
129	#431	C₂²×C₂₃	1	4	yes	—
130	#423	—	2	2	no	—
131	#423	—	2	2	no	—
132	#423	C₂²×C₂₃	1	4	yes	—
133	#423	C₂²×C₂₃	1	4	yes	—
134	C₂×C₄⋊C₈	—	2	2	no	—
135	C₂×C₄⋊C₈	C₄×C₂₃	1	2	yes	—
136	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
137	C₂×C₂²⋊C₈	C₄×C₂₃	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
139	#334	C₂²×C₂₃	1	16	yes	complemented
140	C₂×C₂₃	—	2	2	no	—
141	C₂×C₂₃	—	2	4	no	—
142	C₂×C₂₃	#2905	1	1	yes	characteristic
143	C₂×C₂₃	C₂²×#106	1	1	yes	characteristic
144	C₂×C₂₃	C₂×#430	1	1	yes	characteristic
145	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
146	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
147	C₂³×C₄	Q₈×C₂₃	1	1	yes	characteristic
148	C₂×M₁₆	—	2	4	no	—
149	C₂×M₁₆	—	2	4	no	—
150	C₂×M₁₆	—	4	8	no	—
151	C₂²×C₈	—	2	4	no	—
152	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
153	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
154	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
155	C₂×C₄²	D₄×C₂₃	1	2	yes	—
156	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
157	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
158	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
159	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
160	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
161	C₄⋊C₈	—	2	4	no	—
162	C₄⋊C₈	—	2	4	no	—
163	C₄⋊C₈	—	2	4	no	—
164	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
165	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
166	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
167	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
168	C₄×C₈	C₂³×C₂₃	1	4	yes	—
169	C₂₃	#17813	1	1	yes	characteristic, direct product factor, Sylow
170	C₂⁴	C₄⋊C₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
174	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
175	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
176	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
177	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
179	C₂²×C₄	SD₁₆×C₂₃	1	8	yes	—
180	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
181	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
183	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
184	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
185	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
186	M₁₆	—	4	8	no	—
187	M₁₆	—	4	8	no	—
188	C₂×C₈	—	2	8	no	—
189	C₂×C₈	—	4	4	no	—
190	C₂×C₈	—	4	8	no	—
191	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
192	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
193	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
194	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
195	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
196	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
197	C₄²	C₂×D₄×C₂₃	1	2	yes	—
198	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
199	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
200	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
201	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
202	C₂³	—	4	1	no	—
203	C₂³	—	4	1	no	—
204	C₂³	C₂₃×#106	1	4	yes	—
205	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
206	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	C₂₃×#106	1	4	yes	—
216	C₂×C₄	C₂₃×#106	1	8	yes	—
217	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
218	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
219	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
220	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
222	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
223	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
224	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
225	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
226	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
227	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
228	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
229	C₈	—	2	8	no	—
230	C₈	—	4	8	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	4	no	—
236	C₂²	C₂×C₂₃×#106	1	4	yes	—
237	C₂²	C₂×C₂₃×#106	1	2	yes	—
238	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	2	no	—
243	C₄	C₂×C₂₃×#106	1	4	yes	—
244	C₄	C₂×C₂₃×#106	1	2	yes	—
245	C₄	C₂₃×#430	1	2	yes	—
246	C₄	C₂₃×#568	1	2	yes	—
247	C₄	C₂²×SD₁₆×C₂₃	1	2	yes	—
248	C₂	—	2	4	no	—
249	C₂	—	2	2	no	—
250	C₂	C₂₃×#2905	1	1	yes	characteristic
251	C₂	C₂²×C₂₃×#106	1	1	yes	characteristic
252	C₂	C₂×C₂₃×#430	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	C₂⁴×C₂₃	C₄⋊C₄
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×#1570	C₂	semidirect
4. C₂₃×#1566	C₂	semidirect
18. C₂₃×#334	C₂²	semidirect
75. #17813	C₂₃	direct (N × H)
107. #1570	C₂×C₂₃	semidirect
108. #1566	C₂×C₂₃	semidirect
139. #334	C₂²×C₂₃	semidirect
169. C₂₃	#17813	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×#431	2
C₂×C₂₃×#423	2
C₂₃×#1570	2
C₂₃×#1566	2
#17813	23

Subgroups — C23×#17813 #19165439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17813 #19165439