Subgroups — C₂₃×#13254 #19160880

Order 5888.

514 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#495	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#358	C₂	1	2	yes	maximal, complemented
4	C₂×C₂₃×#357	C₂	1	2	yes	maximal
5	C₄×C₂₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×SD₁₆×C₂₃	—	2	1	no	—
8	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#124	—	2	1	no	—
10	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂₃×#495	—	2	2	no	—
12	C₂₃×#495	C₄	1	4	yes	complemented
13	C₂×C₂₃×#107	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
16	C₂₃×C₄⋊C₁₆	—	2	2	no	—
17	C₂₃×#358	—	2	4	no	—
18	C₂₃×#357	—	2	4	no	—
19	C₂₃×#340	—	2	1	no	—
20	C₂²×Q₈×C₂₃	—	2	1	no	—
21	C₂²×D₄×C₂₃	—	2	1	no	—
22	C₂×Q₁₆×C₂₃	—	2	2	no	—
23	C₂×Q₁₆×C₂₃	—	2	1	no	—
24	C₂×Q₁₆×C₂₃	—	2	2	no	—
25	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
26	C₂×SD₁₆×C₂₃	—	2	4	no	—
27	C₂×SD₁₆×C₂₃	—	4	4	no	—
28	C₂×D₈×C₂₃	—	2	2	no	—
29	C₂×D₈×C₂₃	—	2	1	no	—
30	C₂×D₈×C₂₃	—	2	2	no	—
31	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂₃×#124	—	2	4	no	—
35	C₂₃×#124	—	4	2	no	—
36	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
38	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
39	C₂×C₄²×C₂₃	—	4	1	no	—
40	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₂₃	—	2	2	no	—
42	C₂×C₁₆×C₂₃	—	2	2	no	—
43	C₂×C₁₆×C₂₃	—	2	2	no	—
44	C₂₃×#107	—	2	2	no	—
45	C₂₃×#107	D₄	1	4	yes	—
46	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₂₃	D₄	1	2	yes	—
48	C₂₃×#95	—	4	1	no	—
49	C₂⁴×C₂₃	—	4	1	no	—
50	C₂×Q₈×C₂₃	—	2	2	no	—
51	C₂×Q₈×C₂₃	—	4	1	no	—
52	C₂×Q₈×C₂₃	—	4	2	no	—
53	C₂×Q₈×C₂₃	—	4	2	no	—
54	C₂×D₄×C₂₃	—	2	2	no	—
55	C₂×D₄×C₂₃	—	4	2	no	—
56	C₂×D₄×C₂₃	—	4	1	no	—
57	C₂×D₄×C₂₃	—	4	2	no	—
58	C₂²×C₄×C₂₃	—	4	2	no	—
59	C₂²×C₄×C₂₃	—	4	1	no	—
60	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₂₃	—	2	4	no	—
64	Q₁₆×C₂₃	—	4	1	no	—
65	Q₁₆×C₂₃	—	4	1	no	—
66	SD₁₆×C₂₃	—	4	8	no	—
67	D₈×C₂₃	—	2	4	no	—
68	D₈×C₂₃	—	4	1	no	—
69	D₈×C₂₃	—	4	1	no	—
70	C₂×C₈×C₂₃	—	2	2	no	—
71	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₂₃	#41	1	2	yes	—
76	C₄⋊C₄×C₂₃	—	2	4	no	—
77	C₄⋊C₄×C₂₃	—	4	2	no	—
78	C₂²⋊C₄×C₂₃	—	4	4	no	—
79	C₂²⋊C₄×C₂₃	—	4	4	no	—
80	C₄²×C₂₃	—	8	2	no	—
81	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₂₃	SD₁₆	1	2	yes	—
83	C₁₆×C₂₃	—	4	2	no	—
84	#13254	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₂₃	—	4	2	no	—
86	C₂³×C₂₃	—	8	2	no	—
87	C₂³×C₂₃	—	8	2	no	—
88	C₂³×C₂₃	—	8	1	no	—
89	C₂³×C₂₃	—	8	1	no	—
90	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
91	Q₈×C₂₃	—	4	2	no	—
92	Q₈×C₂₃	—	4	2	no	—
93	Q₈×C₂₃	—	8	2	no	—
94	D₄×C₂₃	—	4	2	no	—
95	D₄×C₂₃	—	4	2	no	—
96	D₄×C₂₃	—	8	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	4	2	no	—
101	C₂×C₄×C₂₃	—	4	4	no	—
102	C₂×C₄×C₂₃	—	8	2	no	—
103	C₂×C₄×C₂₃	—	8	2	no	—
104	C₂×C₄×C₂₃	—	8	2	no	—
105	C₂×C₄×C₂₃	#102	1	2	yes	—
106	C₂×C₄×C₂₃	#102	1	2	yes	—
107	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₂₃	#121	1	2	yes	—
109	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
111	C₈×C₂₃	—	2	2	no	—
112	C₈×C₂₃	C₄×D₄	1	2	yes	—
113	C₈×C₂₃	#124	1	1	yes	characteristic
114	C₈×C₂₃	#127	1	1	yes	characteristic
115	C₂×#495	C₂×C₂₃	1	1	yes	characteristic
116	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
117	C₂×#358	C₂×C₂₃	1	2	yes	complemented
118	C₂×#357	C₂×C₂₃	1	2	yes	—
119	C₄×#107	C₂×C₂₃	1	1	yes	characteristic, complemented
120	C₂²×C₂₃	—	8	2	no	—
121	C₂²×C₂₃	—	8	2	no	—
122	C₂²×C₂₃	—	8	2	no	—
123	C₂²×C₂₃	—	16	2	no	—
124	C₂²×C₂₃	—	16	2	no	—
125	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
126	C₂²×C₂₃	#459	1	2	yes	—
127	C₂²×C₂₃	#508	1	2	yes	—
128	C₂²×C₂₃	#509	1	1	yes	characteristic
129	C₂²×C₂₃	#510	1	1	yes	characteristic
130	C₄×C₂₃	—	2	2	no	—
131	C₄×C₂₃	—	2	2	no	—
132	C₄×C₂₃	—	8	4	no	—
133	C₄×C₂₃	—	8	2	no	—
134	C₄×C₂₃	#390	1	1	yes	characteristic
135	C₄×C₂₃	#418	1	1	yes	characteristic
136	C₄×C₂₃	#460	1	2	yes	—
137	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
138	C₂²×SD₁₆	—	2	1	no	—
139	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
140	C₂×#124	—	2	1	no	—
141	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
142	#495	—	2	2	no	—
143	#495	C₄×C₂₃	1	4	yes	complemented
144	C₂×#107	C₂²×C₂₃	1	2	yes	—
145	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
146	C₄×C₄⋊C₄	—	2	1	no	—
147	C₄⋊C₁₆	—	2	2	no	—
148	#358	—	2	4	no	—
149	#357	—	2	4	no	—
150	#340	—	2	1	no	—
151	C₂×C₂₃	—	16	2	no	—
152	C₂×C₂₃	#1982	1	1	yes	characteristic
153	C₂×C₂₃	#2138	1	1	yes	characteristic
154	C₂×C₂₃	#2139	1	1	yes	characteristic
155	C₂×C₂₃	#2210	1	2	yes	—
156	C₂×C₂₃	#2211	1	2	yes	—
157	C₂²×Q₈	—	2	1	no	—
158	C₂²×D₄	—	2	1	no	—
159	C₂×Q₁₆	—	2	2	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×Q₁₆	—	2	2	no	—
162	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
163	C₂×SD₁₆	—	2	4	no	—
164	C₂×SD₁₆	—	4	4	no	—
165	C₂×D₈	—	2	2	no	—
166	C₂×D₈	—	2	1	no	—
167	C₂×D₈	—	2	2	no	—
168	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
169	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
171	#124	—	2	4	no	—
172	#124	—	4	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₂²⋊C₄	—	4	1	no	—
175	C₂×C₂²⋊C₄	—	4	1	no	—
176	C₂×C₄²	—	4	1	no	—
177	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
178	C₂×C₁₆	—	2	2	no	—
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	2	2	no	—
181	#107	—	2	2	no	—
182	#107	D₄×C₂₃	1	4	yes	—
183	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
184	C₄×C₈	D₄×C₂₃	1	2	yes	—
185	#95	—	4	1	no	—
186	C₂₃	#13254	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁴	—	4	1	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	4	1	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	—	4	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
199	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
201	Q₁₆	—	2	4	no	—
202	Q₁₆	—	4	1	no	—
203	Q₁₆	—	4	1	no	—
204	SD₁₆	—	4	8	no	—
205	D₈	—	2	4	no	—
206	D₈	—	4	1	no	—
207	D₈	—	4	1	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
210	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
211	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
212	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
213	C₂×C₈	#41×C₂₃	1	2	yes	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₂²⋊C₄	—	4	4	no	—
217	C₂²⋊C₄	—	4	4	no	—
218	C₄²	—	8	2	no	—
219	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
220	C₄²	SD₁₆×C₂₃	1	2	yes	—
221	C₁₆	—	4	2	no	—
222	C₂³	—	4	2	no	—
223	C₂³	—	8	2	no	—
224	C₂³	—	8	2	no	—
225	C₂³	—	8	1	no	—
226	C₂³	—	8	1	no	—
227	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
228	Q₈	—	4	2	no	—
229	Q₈	—	4	2	no	—
230	Q₈	—	8	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	4	2	no	—
233	D₄	—	8	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	C₂₃×#102	1	2	yes	—
243	C₂×C₄	C₂₃×#102	1	2	yes	—
244	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
245	C₂×C₄	C₂₃×#121	1	2	yes	—
246	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
247	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
248	C₈	—	2	2	no	—
249	C₈	C₄×D₄×C₂₃	1	2	yes	—
250	C₈	C₂₃×#124	1	1	yes	characteristic
251	C₈	C₂₃×#127	1	1	yes	characteristic
252	C₂²	—	8	2	no	—
253	C₂²	—	8	2	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	16	2	no	—
256	C₂²	—	16	2	no	—
257	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
258	C₂²	C₂₃×#459	1	2	yes	—
259	C₂²	C₂₃×#508	1	2	yes	—
260	C₂²	C₂₃×#509	1	1	yes	characteristic
261	C₂²	C₂₃×#510	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	8	4	no	—
265	C₄	—	8	2	no	—
266	C₄	C₂₃×#390	1	1	yes	characteristic
267	C₄	C₂₃×#418	1	1	yes	characteristic
268	C₄	C₂₃×#460	1	2	yes	—
269	C₂	—	16	2	no	—
270	C₂	C₂₃×#1982	1	1	yes	characteristic
271	C₂	C₂₃×#2138	1	1	yes	characteristic
272	C₂	C₂₃×#2139	1	1	yes	characteristic
273	C₂	C₂₃×#2210	1	2	yes	—
274	C₂	C₂₃×#2211	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₂₃×#358	C₂	semidirect
5. C₄×C₂₃×#107	C₂	semidirect
12. C₂₃×#495	C₄	semidirect
84. #13254	C₂₃	direct (N × H)
116. C₂×C₄⋊C₁₆	C₂×C₂₃	semidirect
117. C₂×#358	C₂×C₂₃	semidirect
119. C₄×#107	C₂×C₂₃	semidirect
143. #495	C₄×C₂₃	semidirect
186. C₂₃	#13254	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×#495	2
C₂×C₂₃×C₄⋊C₁₆	2
C₂×C₂₃×#358	2
C₂×C₂₃×#357	2
C₄×C₂₃×#107	2
#13254	23

Subgroups — C23×#13254 #19160880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13254 #19160880