Subgroups — C₂₃×#8181 #19155807

Order 5888.

606 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×#2293	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×D₄⋊C₈	C₂	1	2	yes	maximal
5	C₂₃×#1279	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×#117	—	2	1	no	—
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₂₃×#97	—	2	2	no	—
10	C₂×C₄×C₈×C₂₃	—	2	1	no	—
11	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#390	C₄	1	2	yes	—
13	C₂₃×#389	C₄	1	2	yes	—
14	C₂₃×#388	—	2	2	no	—
15	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
17	C₄³×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#336	—	2	1	no	—
19	C₂₃×D₄⋊C₈	C₄	1	8	yes	complemented
20	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₂₃	—	2	1	no	—
22	C₂²×C₈×C₂₃	—	2	2	no	—
23	C₂²×C₈×C₂₃	—	2	2	no	—
24	C₄×D₄×C₂₃	—	2	2	no	—
25	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
26	C₂₃×#117	—	4	4	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
31	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
32	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
33	C₂×C₄²×C₂₃	—	2	1	no	—
34	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₂₃	D₄	1	2	yes	—
36	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
39	C₂₃×C₄⋊C₈	—	2	2	no	—
40	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
41	C₂₃×#97	—	2	8	no	—
42	C₄×C₈×C₂₃	—	2	2	no	—
43	C₄×C₈×C₂₃	—	4	2	no	—
44	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
45	C₂₃×#95	—	2	2	no	—
46	C₂₃×#95	—	2	1	no	—
47	C₂⁴×C₂₃	—	2	1	no	—
48	C₂×D₄×C₂₃	—	2	1	no	—
49	C₂×D₄×C₂₃	—	2	4	no	—
50	C₂×D₄×C₂₃	C₄²	1	2	yes	—
51	C₂²×C₄×C₂₃	—	2	1	no	—
52	C₂²×C₄×C₂₃	—	2	4	no	—
53	C₂²×C₄×C₂₃	—	2	2	no	—
54	C₂²×C₄×C₂₃	—	4	2	no	—
55	C₂²×C₄×C₂₃	—	4	1	no	—
56	C₂²×C₄×C₂₃	—	4	1	no	—
57	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₂₃	—	2	8	no	—
63	C₂×C₈×C₂₃	—	2	4	no	—
64	C₂×C₈×C₂₃	—	2	4	no	—
65	C₂×C₈×C₂₃	—	4	2	no	—
66	C₂×C₈×C₂₃	—	4	2	no	—
67	C₄⋊C₄×C₂₃	—	2	1	no	—
68	C₄⋊C₄×C₂₃	—	2	4	no	—
69	C₄⋊C₄×C₂₃	—	4	4	no	—
70	C₄⋊C₄×C₂₃	C₄²	1	2	yes	—
71	C₂²⋊C₄×C₂₃	—	2	4	no	—
72	C₂²⋊C₄×C₂₃	—	4	4	no	—
73	C₂²⋊C₄×C₂₃	—	4	4	no	—
74	C₄²×C₂₃	—	2	4	no	—
75	C₄²×C₂₃	—	2	4	no	—
76	C₄²×C₂₃	—	2	2	no	—
77	C₄²×C₂₃	—	2	1	no	—
78	C₄²×C₂₃	—	2	1	no	—
79	C₄²×C₂₃	—	4	2	no	—
80	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
81	C₄²×C₂₃	#41	1	2	yes	—
82	#8181	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₂₃	—	2	2	no	—
84	C₂³×C₂₃	—	4	2	no	—
85	C₂³×C₂₃	—	4	1	no	—
86	C₂³×C₂₃	—	4	1	no	—
87	C₂³×C₂₃	—	4	2	no	—
88	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	D₄×C₂₃	—	2	4	no	—
90	D₄×C₂₃	—	4	2	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	4	1	no	—
99	C₂×C₄×C₂₃	—	4	1	no	—
100	C₂×C₄×C₂₃	—	4	4	no	—
101	C₂×C₄×C₂₃	—	4	4	no	—
102	C₂×C₄×C₂₃	—	4	2	no	—
103	C₂×C₄×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	#104	1	4	yes	—
106	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₂₃	#131	1	4	yes	—
111	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
112	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
114	C₈×C₂₃	—	2	8	no	—
115	C₈×C₂₃	—	4	4	no	—
116	#2293	C₂×C₂₃	1	1	yes	characteristic
117	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
118	C₄×#97	C₂×C₂₃	1	1	yes	characteristic, complemented
119	C₂×D₄⋊C₈	C₂×C₂₃	1	2	yes	—
120	#1279	C₂×C₂₃	1	2	yes	complemented
121	C₂²×C₂₃	—	4	2	no	—
122	C₂²×C₂₃	—	4	2	no	—
123	C₂²×C₂₃	—	4	2	no	—
124	C₂²×C₂₃	—	8	2	no	—
125	C₂²×C₂₃	—	8	2	no	—
126	C₂²×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₂₃	#408	1	2	yes	—
128	C₂²×C₂₃	#417	1	1	yes	characteristic
129	C₂²×C₂₃	#418	1	1	yes	characteristic
130	C₂²×C₂₃	C₂×#104	1	2	yes	—
131	C₄×C₂₃	—	2	4	no	—
132	C₄×C₂₃	—	2	2	no	—
133	C₄×C₂₃	—	4	2	no	—
134	C₄×C₂₃	—	4	2	no	—
135	C₄×C₂₃	—	8	2	no	—
136	C₄×C₂₃	#405	1	2	yes	—
137	C₄×C₂₃	#433	1	2	yes	—
138	C₄×C₂₃	#440	1	2	yes	—
139	C₄×C₂₃	#442	1	2	yes	—
140	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
141	C₂×#117	—	2	1	no	—
142	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
143	C₂×#97	—	2	2	no	—
144	C₂×C₄×C₈	—	2	1	no	—
145	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
146	#390	C₄×C₂₃	1	2	yes	—
147	#389	C₄×C₂₃	1	2	yes	—
148	#388	—	2	2	no	—
149	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
150	C₄×C₂²⋊C₄	—	2	1	no	—
151	C₄³	C₂²×C₂₃	1	1	yes	characteristic
152	#336	—	2	1	no	—
153	D₄⋊C₈	C₄×C₂₃	1	8	yes	complemented
154	C₂×C₂₃	—	8	2	no	—
155	C₂×C₂₃	#1483	1	2	yes	—
156	C₂×C₂₃	#1484	1	2	yes	—
157	C₂×C₂₃	#1750	1	1	yes	characteristic
158	C₂×C₂₃	C₄×#104	1	1	yes	characteristic
159	C₂×C₂₃	#1762	1	1	yes	characteristic
160	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
161	C₂³×C₄	—	2	1	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₄×D₄	—	2	2	no	—
165	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
166	#117	—	4	4	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
169	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	—	2	1	no	—
171	C₂×C₂²⋊C₄	—	2	2	no	—
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₄²	—	2	1	no	—
174	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
175	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
176	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
177	C₂×C₄²	D₄×C₂₃	1	2	yes	—
178	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
181	#97	—	2	8	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	—	4	2	no	—
184	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
185	#95	—	2	2	no	—
186	#95	—	2	1	no	—
187	C₂₃	#8181	1	1	yes	characteristic, direct product factor, Sylow
188	C₂⁴	—	2	1	no	—
189	C₂×D₄	—	2	1	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	C₄²×C₂₃	1	2	yes	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
199	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
200	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
203	C₂×C₈	—	2	8	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	C₄²×C₂₃	1	2	yes	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₂²⋊C₄	—	4	4	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	2	1	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	4	2	no	—
221	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
222	C₄²	#41×C₂₃	1	2	yes	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	1	no	—
226	C₂³	—	4	1	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
229	D₄	—	2	4	no	—
230	D₄	—	4	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	C₂₃×#104	1	4	yes	—
246	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
247	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
248	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
249	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
250	C₂×C₄	C₂₃×#131	1	4	yes	—
251	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
252	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
254	C₈	—	2	8	no	—
255	C₈	—	4	4	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
262	C₂²	C₂₃×#408	1	2	yes	—
263	C₂²	C₂₃×#417	1	1	yes	characteristic
264	C₂²	C₂₃×#418	1	1	yes	characteristic
265	C₂²	C₂×C₂₃×#104	1	2	yes	—
266	C₄	—	2	4	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	8	2	no	—
271	C₄	C₂₃×#405	1	2	yes	—
272	C₄	C₂₃×#433	1	2	yes	—
273	C₄	C₂₃×#440	1	2	yes	—
274	C₄	C₂₃×#442	1	2	yes	—
275	C₂	—	8	2	no	—
276	C₂	C₂₃×#1483	1	2	yes	—
277	C₂	C₂₃×#1484	1	2	yes	—
278	C₂	C₂₃×#1750	1	1	yes	characteristic
279	C₂	C₄×C₂₃×#104	1	1	yes	characteristic
280	C₂	C₂₃×#1762	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₂₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₂₃×C₄⋊C₈	C₂	semidirect
3. C₄×C₂₃×#97	C₂	semidirect
5. C₂₃×#1279	C₂	semidirect
19. C₂₃×D₄⋊C₈	C₄	semidirect
82. #8181	C₂₃	direct (N × H)
117. C₄×C₄⋊C₈	C₂×C₂₃	semidirect
118. C₄×#97	C₂×C₂₃	semidirect
120. #1279	C₂×C₂₃	semidirect
153. D₄⋊C₈	C₄×C₂₃	semidirect
187. C₂₃	#8181	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×#2293	2
C₄×C₂₃×C₄⋊C₈	2
C₄×C₂₃×#97	2
C₂×C₂₃×D₄⋊C₈	2
C₂₃×#1279	2
#8181	23

Subgroups — C23×#8181 #19155807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8181 #19155807