Subgroups — C₂₃×#8183 #19155809

Order 5888.

574 conjugacy classes of subgroups.

The subgroup lattice has 284 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	C₂₃×#1744	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×C₄×C₈×C₂₃	—	2	1	no	—
10	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
11	C₂₃×#390	C₄	1	2	yes	—
12	C₂₃×#389	C₄	1	2	yes	—
13	C₂₃×#388	—	2	2	no	—
14	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
16	C₄³×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#336	—	2	2	no	—
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	2	no	—
27	C₂₃×#117	—	4	2	no	—
28	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
32	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
33	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
34	C₂×C₄²×C₂₃	—	2	1	no	—
35	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	D₄	1	2	yes	—
38	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×C₄⋊C₈	—	2	2	no	—
41	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
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	1	no	—
55	C₂²×C₄×C₂₃	—	4	1	no	—
56	C₂²×C₄×C₂₃	—	4	2	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	4	no	—
63	C₂×C₈×C₂₃	—	2	4	no	—
64	C₂×C₈×C₂₃	—	4	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	2	no	—
76	C₄²×C₂₃	—	4	2	no	—
77	C₄²×C₂₃	—	4	1	no	—
78	C₄²×C₂₃	—	4	1	no	—
79	C₄²×C₂₃	D₈	1	2	yes	—
80	C₄²×C₂₃	SD₁₆	1	2	yes	—
81	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
82	C₄²×C₂₃	#41	1	2	yes	—
83	#8183	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₂₃	—	2	2	no	—
85	C₂³×C₂₃	—	4	2	no	—
86	C₂³×C₂₃	—	4	1	no	—
87	C₂³×C₂₃	—	4	1	no	—
88	C₂³×C₂₃	—	4	2	no	—
89	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	D₄×C₂₃	—	2	4	no	—
91	D₄×C₂₃	—	4	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	—	2	2	no	—
99	C₂×C₄×C₂₃	—	4	1	no	—
100	C₂×C₄×C₂₃	—	4	1	no	—
101	C₂×C₄×C₂₃	—	4	4	no	—
102	C₂×C₄×C₂₃	—	4	4	no	—
103	C₂×C₄×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	#102	1	4	yes	—
107	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₂₃	#131	1	4	yes	—
112	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₂₃	#135	1	2	yes	—
115	C₈×C₂₃	—	4	4	no	—
116	C₈×C₂₃	—	4	4	no	—
117	#2293	C₂×C₂₃	1	1	yes	characteristic
118	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
119	#1744	C₂×C₂₃	1	1	yes	characteristic, complemented
120	C₂×D₄⋊C₈	C₂×C₂₃	1	2	yes	—
121	#1279	C₂×C₂₃	1	2	yes	complemented
122	C₂²×C₂₃	—	4	2	no	—
123	C₂²×C₂₃	—	4	2	no	—
124	C₂²×C₂₃	—	4	2	no	—
125	C₂²×C₂₃	—	8	2	no	—
126	C₂²×C₂₃	—	8	2	no	—
127	C₂²×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₂₃	#408	1	2	yes	—
129	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
130	C₂²×C₂₃	#416	1	1	yes	characteristic
131	C₂²×C₂₃	#421	1	2	yes	—
132	C₄×C₂₃	—	2	4	no	—
133	C₄×C₂₃	—	2	2	no	—
134	C₄×C₂₃	—	4	2	no	—
135	C₄×C₂₃	—	4	2	no	—
136	C₄×C₂₃	—	8	2	no	—
137	C₄×C₂₃	#405	1	2	yes	—
138	C₄×C₂₃	#433	1	2	yes	—
139	C₄×C₂₃	C₄×D₈	1	2	yes	—
140	C₄×C₂₃	C₄×SD₁₆	1	2	yes	—
141	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×#117	—	2	1	no	—
143	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
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	2	no	—
153	#336	—	2	1	no	—
154	D₄⋊C₈	C₄×C₂₃	1	8	yes	complemented
155	C₂×C₂₃	—	8	2	no	—
156	C₂×C₂₃	#1480	1	2	yes	—
157	C₂×C₂₃	#1482	1	2	yes	—
158	C₂×C₂₃	#1750	1	1	yes	characteristic
159	C₂×C₂₃	#1759	1	1	yes	characteristic
160	C₂×C₂₃	C₄×#102	1	1	yes	characteristic
161	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
162	C₂³×C₄	—	2	1	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₄×D₄	—	2	2	no	—
166	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
167	#117	—	4	2	no	—
168	#117	—	4	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
171	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	—	2	1	no	—
173	C₂×C₂²⋊C₄	—	2	2	no	—
174	C₂×C₂²⋊C₄	—	2	2	no	—
175	C₂×C₄²	—	2	1	no	—
176	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
177	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
178	C₂×C₄²	D₄×C₂₃	1	2	yes	—
179	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	—	4	2	no	—
185	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
186	#95	—	2	2	no	—
187	#95	—	2	1	no	—
188	C₂₃	#8183	1	1	yes	characteristic, direct product factor, Sylow
189	C₂⁴	—	2	1	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	C₄²×C₂₃	1	2	yes	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
200	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
201	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₄⋊C₄	C₄²×C₂₃	1	2	yes	—
213	C₂²⋊C₄	—	2	4	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₂²⋊C₄	—	4	4	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	4	2	no	—
219	C₄²	—	4	1	no	—
220	C₄²	—	4	1	no	—
221	C₄²	D₈×C₂₃	1	2	yes	—
222	C₄²	SD₁₆×C₂₃	1	2	yes	—
223	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
224	C₄²	#41×C₂₃	1	2	yes	—
225	C₂³	—	2	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	1	no	—
228	C₂³	—	4	1	no	—
229	C₂³	—	4	2	no	—
230	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
231	D₄	—	2	4	no	—
232	D₄	—	4	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	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	C₂₃×#102	1	4	yes	—
248	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
249	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
250	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
251	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
252	C₂×C₄	C₂₃×#131	1	4	yes	—
253	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
254	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₂₃×#135	1	2	yes	—
256	C₈	—	4	4	no	—
257	C₈	—	4	4	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	—	8	2	no	—
263	C₂²	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
264	C₂²	C₂₃×#408	1	2	yes	—
265	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
266	C₂²	C₂₃×#416	1	1	yes	characteristic
267	C₂²	C₂₃×#421	1	2	yes	—
268	C₄	—	2	4	no	—
269	C₄	—	2	2	no	—
270	C₄	—	4	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	8	2	no	—
273	C₄	C₂₃×#405	1	2	yes	—
274	C₄	C₂₃×#433	1	2	yes	—
275	C₄	C₄×D₈×C₂₃	1	2	yes	—
276	C₄	C₄×SD₁₆×C₂₃	1	2	yes	—
277	C₂	—	8	2	no	—
278	C₂	C₂₃×#1480	1	2	yes	—
279	C₂	C₂₃×#1482	1	2	yes	—
280	C₂	C₂₃×#1750	1	1	yes	characteristic
281	C₂	C₂₃×#1759	1	1	yes	characteristic
282	C₂	C₄×C₂₃×#102	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. C₂₃×#1744	C₂	semidirect
5. C₂₃×#1279	C₂	semidirect
19. C₂₃×D₄⋊C₈	C₄	semidirect
83. #8183	C₂₃	direct (N × H)
118. C₄×C₄⋊C₈	C₂×C₂₃	semidirect
119. #1744	C₂×C₂₃	semidirect
121. #1279	C₂×C₂₃	semidirect
154. D₄⋊C₈	C₄×C₂₃	semidirect
188. C₂₃	#8183	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
C₂₃×#1744	2
C₂×C₂₃×D₄⋊C₈	2
C₂₃×#1279	2
#8183	23

Subgroups — C23×#8183 #19155809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8183 #19155809