Subgroups — C₂₃×#13255 #19160881

Order 5888.

498 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#360	C₂	1	4	yes	maximal, complemented
4	C₄×C₂₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#135	—	2	1	no	—
6	C₂×C₂₃×#135	C₂²	1	2	yes	—
7	C₂₃×#532	—	2	1	no	—
8	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
9	C₂₃×#495	—	2	1	no	—
10	C₂₃×#495	C₄	1	2	yes	complemented
11	C₂₃×#494	C₄	1	1	yes	characteristic, complemented
12	C₂₃×#493	C₄	1	1	yes	characteristic, complemented
13	C₂₃×#492	—	2	1	no	—
14	C₂×C₂₃×#106	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
17	C₂₃×C₄⋊C₁₆	—	2	1	no	—
18	C₂₃×C₄⋊C₁₆	—	2	1	no	—
19	C₂₃×#360	—	2	8	no	—
20	C₂₃×#340	—	2	1	no	—
21	C₂×#41×C₂₃	—	2	2	no	—
22	C₂₃×#135	—	2	4	no	—
23	C₂₃×#135	—	2	4	no	—
24	C₂₃×#135	—	4	4	no	—
25	C₂×Q₁₆×C₂₃	—	2	1	no	—
26	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
27	C₂×SD₁₆×C₂₃	—	2	2	no	—
28	C₂×SD₁₆×C₂₃	—	2	2	no	—
29	C₂×D₈×C₂₃	—	2	1	no	—
30	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
33	C₂₃×C₄⋊Q₈	—	2	1	no	—
34	C₂₃×#127	—	2	1	no	—
35	C₂₃×#124	—	2	2	no	—
36	C₂₃×#121	—	4	1	no	—
37	C₂₃×#121	—	4	1	no	—
38	C₄×D₄×C₂₃	—	4	1	no	—
39	C₄×D₄×C₂₃	—	4	1	no	—
40	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
41	C₂×C₄²×C₂₃	—	4	1	no	—
42	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
43	C₂₃×M₃₂	—	2	4	no	—
44	C₂×C₁₆×C₂₃	—	2	2	no	—
45	C₂₃×#106	—	2	2	no	—
46	C₂₃×#106	D₄	1	4	yes	complemented
47	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
51	C₂₃×#95	—	4	1	no	—
52	#41×C₂₃	—	4	4	no	—
53	#41×C₂₃	—	4	4	no	—
54	C₂×Q₈×C₂₃	—	2	2	no	—
55	C₂×D₄×C₂₃	—	2	2	no	—
56	C₂×D₄×C₂₃	—	4	2	no	—
57	C₂×D₄×C₂₃	—	4	1	no	—
58	C₂×D₄×C₂₃	—	4	1	no	—
59	C₂²×C₄×C₂₃	—	4	2	no	—
60	C₂²×C₄×C₂₃	—	4	2	no	—
61	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
64	Q₁₆×C₂₃	—	2	4	no	—
65	Q₁₆×C₂₃	—	4	2	no	—
66	SD₁₆×C₂₃	—	4	4	no	—
67	SD₁₆×C₂₃	—	4	2	no	—
68	SD₁₆×C₂₃	—	4	2	no	—
69	D₈×C₂₃	—	2	4	no	—
70	D₈×C₂₃	—	4	2	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₂₃	#41	1	2	yes	—
78	C₄⋊C₄×C₂₃	—	2	4	no	—
79	C₄⋊C₄×C₂₃	—	4	1	no	—
80	C₄⋊C₄×C₂₃	—	4	1	no	—
81	C₄⋊C₄×C₂₃	—	4	2	no	—
82	C₂²⋊C₄×C₂₃	—	4	2	no	—
83	C₂²⋊C₄×C₂₃	—	4	2	no	—
84	C₄²×C₂₃	—	8	2	no	—
85	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
88	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
89	C₁₆×C₂₃	—	4	2	no	—
90	#13255	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₃	—	4	2	no	—
92	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
93	Q₈×C₂₃	—	4	2	no	—
94	Q₈×C₂₃	—	4	2	no	—
95	D₄×C₂₃	—	4	2	no	—
96	D₄×C₂₃	—	4	2	no	—
97	D₄×C₂₃	—	8	2	no	—
98	D₄×C₂₃	—	8	4	no	—
99	D₄×C₂₃	—	8	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	4	1	no	—
102	C₂×C₄×C₂₃	—	4	2	no	—
103	C₂×C₄×C₂₃	—	4	4	no	—
104	C₂×C₄×C₂₃	—	8	2	no	—
105	C₂×C₄×C₂₃	—	8	2	no	—
106	C₂×C₄×C₂₃	—	8	4	no	—
107	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
108	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
109	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
110	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	#121	1	2	yes	—
113	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
115	C₈×C₂₃	—	2	1	no	—
116	C₈×C₂₃	—	2	1	no	—
117	C₈×C₂₃	C₄×D₄	1	2	yes	—
118	C₈×C₂₃	#124	1	1	yes	characteristic
119	C₈×C₂₃	#127	1	1	yes	characteristic
120	#3147	C₂×C₂₃	1	1	yes	characteristic
121	#2150	C₂×C₂₃	1	1	yes	characteristic, complemented
122	C₂×#360	C₂×C₂₃	1	4	yes	complemented
123	C₄×#106	C₂×C₂₃	1	1	yes	characteristic, complemented
124	C₂²×C₂₃	—	2	1	no	—
125	C₂²×C₂₃	—	2	1	no	—
126	C₂²×C₂₃	—	8	2	no	—
127	C₂²×C₂₃	—	8	2	no	—
128	C₂²×C₂₃	—	8	2	no	—
129	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
130	C₂²×C₂₃	#459	1	2	yes	—
131	C₄×C₂₃	—	2	1	no	—
132	C₄×C₂₃	—	2	1	no	—
133	C₄×C₂₃	—	2	1	no	—
134	C₄×C₂₃	—	8	4	no	—
135	C₄×C₂₃	—	8	2	no	—
136	C₄×C₂₃	#360	1	2	yes	complemented
137	C₄×C₂₃	#390	1	1	yes	characteristic
138	C₄×C₂₃	#418	1	1	yes	characteristic
139	C₄×C₂₃	#460	1	2	yes	—
140	C₂×#135	—	2	1	no	—
141	C₂×#135	C₂²×C₂₃	1	2	yes	—
142	#532	—	2	1	no	—
143	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
144	#495	—	2	1	no	—
145	#495	C₄×C₂₃	1	2	yes	complemented
146	#494	C₄×C₂₃	1	1	yes	characteristic, complemented
147	#493	C₄×C₂₃	1	1	yes	characteristic, complemented
148	#492	—	2	1	no	—
149	C₂×#106	C₂²×C₂₃	1	2	yes	—
150	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
151	C₄×C₄⋊C₄	—	2	1	no	—
152	C₄⋊C₁₆	—	2	1	no	—
153	C₄⋊C₁₆	—	2	1	no	—
154	#360	—	2	8	no	—
155	#340	—	2	1	no	—
156	C₂×C₂₃	—	2	2	no	—
157	C₂×C₂₃	—	16	2	no	—
158	C₂×C₂₃	#1982	1	1	yes	characteristic
159	C₂×C₂₃	C₂×#360	1	1	yes	characteristic
160	C₂×C₂₃	#2145	1	1	yes	characteristic
161	C₂×#41	—	2	2	no	—
162	#135	—	2	4	no	—
163	#135	—	2	4	no	—
164	#135	—	4	4	no	—
165	C₂×Q₁₆	—	2	1	no	—
166	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
167	C₂×SD₁₆	—	2	2	no	—
168	C₂×SD₁₆	—	2	2	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
171	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
173	C₄⋊Q₈	—	2	1	no	—
174	#127	—	2	1	no	—
175	#124	—	2	2	no	—
176	#121	—	4	1	no	—
177	#121	—	4	1	no	—
178	C₄×D₄	—	4	1	no	—
179	C₄×D₄	—	4	1	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄²	—	4	1	no	—
182	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
183	M₃₂	—	2	4	no	—
184	C₂×C₁₆	—	2	2	no	—
185	#106	—	2	2	no	—
186	#106	D₄×C₂₃	1	4	yes	complemented
187	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
188	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
189	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
190	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
191	#95	—	4	1	no	—
192	C₂₃	#13255	1	1	yes	characteristic, direct product factor, Sylow
193	#41	—	4	4	no	—
194	#41	—	4	4	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
205	Q₁₆	—	2	4	no	—
206	Q₁₆	—	4	2	no	—
207	SD₁₆	—	4	4	no	—
208	SD₁₆	—	4	2	no	—
209	SD₁₆	—	4	2	no	—
210	D₈	—	2	4	no	—
211	D₈	—	4	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
214	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
215	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
216	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
217	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
218	C₂×C₈	#41×C₂₃	1	2	yes	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₄⋊C₄	—	4	1	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₂²⋊C₄	—	4	2	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₄²	—	8	2	no	—
226	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
227	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
228	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
229	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
230	C₁₆	—	4	2	no	—
231	C₂³	—	4	2	no	—
232	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
233	Q₈	—	4	2	no	—
234	Q₈	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	8	2	no	—
238	D₄	—	8	4	no	—
239	D₄	—	8	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
248	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
249	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
250	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂₃×#121	1	2	yes	—
253	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
254	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	C₄×D₄×C₂₃	1	2	yes	—
258	C₈	C₂₃×#124	1	1	yes	characteristic
259	C₈	C₂₃×#127	1	1	yes	characteristic
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	8	2	no	—
263	C₂²	—	8	2	no	—
264	C₂²	—	8	2	no	—
265	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
266	C₂²	C₂₃×#459	1	2	yes	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	8	4	no	—
271	C₄	—	8	2	no	—
272	C₄	C₂₃×#360	1	2	yes	complemented
273	C₄	C₂₃×#390	1	1	yes	characteristic
274	C₄	C₂₃×#418	1	1	yes	characteristic
275	C₄	C₂₃×#460	1	2	yes	—
276	C₂	—	2	2	no	—
277	C₂	—	16	2	no	—
278	C₂	C₂₃×#1982	1	1	yes	characteristic
279	C₂	C₂×C₂₃×#360	1	1	yes	characteristic
280	C₂	C₂₃×#2145	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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. C₂₃×#2150	C₂	semidirect
3. C₂×C₂₃×#360	C₂	semidirect
4. C₄×C₂₃×#106	C₂	semidirect
10. C₂₃×#495	C₄	semidirect
11. C₂₃×#494	C₄	semidirect
12. C₂₃×#493	C₄	semidirect
46. C₂₃×#106	D₄	semidirect
90. #13255	C₂₃	direct (N × H)
121. #2150	C₂×C₂₃	semidirect
122. C₂×#360	C₂×C₂₃	semidirect
123. C₄×#106	C₂×C₂₃	semidirect
136. C₄×C₂₃	#360	semidirect
145. #495	C₄×C₂₃	semidirect
146. #494	C₄×C₂₃	semidirect
147. #493	C₄×C₂₃	semidirect
186. #106	D₄×C₂₃	semidirect
192. C₂₃	#13255	direct (N × H)
272. C₄	C₂₃×#360	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3147	2
C₂₃×#2150	2
C₂×C₂₃×#360	2
C₄×C₂₃×#106	2
#13255	23

Subgroups — C23×#13255 #19160881

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13255 #19160881