Subgroups — C₂₃×#22620 #19170246

Order 5888.

766 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#427	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2347	C₂	1	2	yes	maximal
3	C₂₃×#1943	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1920	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1825	C₂	1	2	yes	maximal
6	C₂₃×#1807	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1776	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1752	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#117	C₂²	1	2	yes	—
11	C₂₃×#427	C₄	1	8	yes	complemented
12	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#106	C₂²	1	2	yes	—
15	C₂×C₂₃×#106	C₂²	1	2	yes	—
16	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#97	C₂²	1	2	yes	—
18	C₂₃×#389	C₂²	1	4	yes	—
19	C₂₃×#388	—	2	4	no	—
20	C₂₃×#383	—	2	2	no	—
21	C₂₃×#380	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
24	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#340	C₂²	1	2	yes	—
26	C₂₃×#340	C₂²	1	4	yes	—
27	C₂₃×#340	C₂²	1	4	yes	—
28	C₂₃×#336	C₂²	1	2	yes	—
29	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₂₃	—	2	4	no	—
31	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂₃×#117	—	2	4	no	—
36	C₂₃×#117	C₂×C₄	1	8	yes	—
37	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
38	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
41	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
44	C₂×C₄²×C₂₃	—	2	2	no	—
45	C₂×C₄²×C₂₃	—	2	4	no	—
46	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
47	C₂₃×#107	C₂×C₄	1	4	yes	—
48	C₂₃×#107	C₂×C₄	1	4	yes	—
49	C₂₃×#106	—	2	4	no	—
50	C₂₃×#106	C₂×C₄	1	8	yes	—
51	C₂₃×C₂²⋊C₈	—	2	2	no	—
52	C₂₃×#97	—	2	4	no	—
53	C₂₃×#95	—	2	2	no	—
54	C₂₃×#95	—	2	4	no	—
55	C₂₃×#95	—	2	4	no	—
56	C₂₃×#95	D₄	1	2	yes	—
57	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	—	2	2	no	—
59	C₂²×C₄×C₂₃	—	2	2	no	—
60	C₂²×C₄×C₂₃	—	2	2	no	—
61	C₂²×C₄×C₂₃	—	2	2	no	—
62	C₂²×C₄×C₂₃	—	2	4	no	—
63	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	#41	1	2	yes	—
67	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₂₃	—	2	2	no	—
69	C₂×C₈×C₂₃	—	2	8	no	—
70	C₂×C₈×C₂₃	—	2	2	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₂₃	#41	1	4	yes	—
76	C₄⋊C₄×C₂₃	—	2	2	no	—
77	C₄⋊C₄×C₂₃	—	2	2	no	—
78	C₄⋊C₄×C₂₃	—	2	4	no	—
79	C₄⋊C₄×C₂₃	—	2	4	no	—
80	C₄⋊C₄×C₂₃	—	4	8	no	—
81	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
82	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
83	C₂²⋊C₄×C₂₃	—	2	8	no	—
84	C₂²⋊C₄×C₂₃	—	4	2	no	—
85	C₂²⋊C₄×C₂₃	—	8	2	no	—
86	C₄²×C₂₃	—	2	8	no	—
87	C₄²×C₂₃	—	2	2	no	—
88	C₄²×C₂₃	—	2	2	no	—
89	C₄²×C₂₃	—	8	4	no	—
90	#22620	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₃	—	2	2	no	—
92	C₂³×C₂₃	—	2	2	no	—
93	C₂³×C₂₃	—	2	2	no	—
94	C₂³×C₂₃	C₄×D₄	1	2	yes	—
95	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
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	—	2	4	no	—
100	C₂×C₄×C₂₃	—	2	4	no	—
101	C₂×C₄×C₂₃	—	2	4	no	—
102	C₂×C₄×C₂₃	—	2	4	no	—
103	C₂×C₄×C₂₃	—	2	4	no	—
104	C₂×C₄×C₂₃	—	2	4	no	—
105	C₂×C₄×C₂₃	—	4	4	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	—	4	4	no	—
109	C₂×C₄×C₂₃	—	4	4	no	—
110	C₂×C₄×C₂₃	—	4	4	no	—
111	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
114	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
116	C₈×C₂₃	—	2	2	no	—
117	C₈×C₂₃	—	4	2	no	—
118	C₈×C₂₃	#117	1	4	yes	—
119	C₂×#427	C₂×C₂₃	1	1	yes	characteristic
120	#2347	C₂×C₂₃	1	2	yes	—
121	#1943	C₂×C₂₃	1	2	yes	complemented
122	#1920	C₂×C₂₃	1	4	yes	complemented
123	#1825	C₂×C₂₃	1	2	yes	—
124	#1807	C₂×C₂₃	1	1	yes	characteristic
125	#1776	C₂×C₂₃	1	2	yes	complemented
126	#1752	C₂×C₂₃	1	1	yes	characteristic
127	C₂²×C₂₃	—	2	2	no	—
128	C₂²×C₂₃	—	2	2	no	—
129	C₂²×C₂₃	—	2	2	no	—
130	C₂²×C₂₃	—	4	4	no	—
131	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂²×C₂₃	#547	1	2	yes	—
133	C₂²×C₂₃	#575	1	2	yes	—
134	C₂²×C₂₃	#577	1	2	yes	—
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	—	4	4	no	—
137	C₄×C₂₃	—	4	4	no	—
138	C₄×C₂₃	—	4	4	no	—
139	C₄×C₂₃	—	8	4	no	—
140	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
141	C₄×C₂₃	#519	1	1	yes	characteristic
142	C₄×C₂₃	#552	1	2	yes	—
143	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×#117	C₂²×C₂₃	1	2	yes	—
145	#427	C₄×C₂₃	1	8	yes	complemented
146	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
147	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×#106	C₂²×C₂₃	1	2	yes	—
149	C₂×#106	C₂²×C₂₃	1	2	yes	—
150	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×#97	C₂²×C₂₃	1	2	yes	—
152	#389	C₂²×C₂₃	1	4	yes	—
153	#388	—	2	4	no	—
154	#383	—	2	2	no	—
155	#380	C₂²×C₂₃	1	2	yes	—
156	C₄×C₄⋊C₄	—	2	2	no	—
157	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
158	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
159	#340	C₂²×C₂₃	1	2	yes	—
160	#340	C₂²×C₂₃	1	4	yes	—
161	#340	C₂²×C₂₃	1	4	yes	—
162	#336	C₂²×C₂₃	1	2	yes	—
163	C₂×C₂₃	—	4	2	no	—
164	C₂×C₂₃	#2344	1	1	yes	characteristic
165	C₂×C₂₃	#2943	1	1	yes	characteristic
166	C₂×C₂₃	#2952	1	1	yes	characteristic
167	C₂×C₂₃	#3350	1	4	yes	—
168	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
169	C₂²×C₈	—	2	4	no	—
170	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
171	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
172	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
174	#117	—	2	4	no	—
175	#117	C₂×C₄×C₂₃	1	8	yes	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
178	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
179	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
180	C₂×C₂²⋊C₄	—	2	2	no	—
181	C₂×C₂²⋊C₄	—	2	2	no	—
182	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
183	C₂×C₄²	—	2	2	no	—
184	C₂×C₄²	—	2	4	no	—
185	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
186	#107	C₂×C₄×C₂₃	1	4	yes	—
187	#107	C₂×C₄×C₂₃	1	4	yes	—
188	#106	—	2	4	no	—
189	#106	C₂×C₄×C₂₃	1	8	yes	—
190	C₂²⋊C₈	—	2	2	no	—
191	#97	—	2	4	no	—
192	#95	—	2	2	no	—
193	#95	—	2	4	no	—
194	#95	—	2	4	no	—
195	#95	D₄×C₂₃	1	2	yes	—
196	C₂₃	#22620	1	1	yes	characteristic, direct product factor, Sylow
197	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
205	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	#41×C₂₃	1	2	yes	—
207	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	8	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
214	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
215	C₂×C₈	#41×C₂₃	1	4	yes	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	8	no	—
221	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
222	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
223	C₂²⋊C₄	—	2	8	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₂²⋊C₄	—	8	2	no	—
226	C₄²	—	2	8	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	8	4	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	C₄×D₄×C₂₃	1	2	yes	—
234	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
251	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
253	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
254	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	4	2	no	—
257	C₈	C₂₃×#117	1	4	yes	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
263	C₂²	C₂₃×#547	1	2	yes	—
264	C₂²	C₂₃×#575	1	2	yes	—
265	C₂²	C₂₃×#577	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	4	4	no	—
268	C₄	—	4	4	no	—
269	C₄	—	4	4	no	—
270	C₄	—	8	4	no	—
271	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
272	C₄	C₂₃×#519	1	1	yes	characteristic
273	C₄	C₂₃×#552	1	2	yes	—
274	C₂	—	4	2	no	—
275	C₂	C₂₃×#2344	1	1	yes	characteristic
276	C₂	C₂₃×#2943	1	1	yes	characteristic
277	C₂	C₂₃×#2952	1	1	yes	characteristic
278	C₂	C₂₃×#3350	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₂²⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1943	C₂	semidirect
4. C₂₃×#1920	C₂	semidirect
7. C₂₃×#1776	C₂	semidirect
11. C₂₃×#427	C₄	semidirect
90. #22620	C₂₃	direct (N × H)
121. #1943	C₂×C₂₃	semidirect
122. #1920	C₂×C₂₃	semidirect
125. #1776	C₂×C₂₃	semidirect
145. #427	C₄×C₂₃	semidirect
196. C₂₃	#22620	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×#427	2
C₂₃×#2347	2
C₂₃×#1943	2
C₂₃×#1920	2
C₂₃×#1825	2
C₂₃×#1807	2
C₂₃×#1776	2
C₂₃×#1752	2
#22620	23

Subgroups — C23×#22620 #19170246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22620 #19170246