Subgroups — C₂₃×#21968 #19169594

Order 5888.

754 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#424	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2314	C₂	1	2	yes	maximal
3	C₂₃×#1949	C₂	1	4	yes	maximal, complemented
4	C₂₃×#1947	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1809	C₂	1	2	yes	maximal
6	C₂₃×#1808	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#107	C₂²	1	2	yes	—
11	C₂×C₂₃×#107	C₂²	1	2	yes	—
12	C₂×C₂₃×#107	C₂²	1	2	yes	—
13	C₂×C₂₃×#106	C₂²	1	2	yes	—
14	C₂₃×#424	C₄	1	8	yes	complemented
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
16	C₂₃×#388	—	2	4	no	—
17	C₂₃×#387	—	2	4	no	—
18	C₂₃×#384	C₂²	1	2	yes	—
19	C₂₃×#382	C₂²	1	2	yes	—
20	C₂₃×#380	C₂²	1	4	yes	—
21	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
22	C₂₃×#340	C₂²	1	4	yes	—
23	C₂₃×#340	C₂²	1	4	yes	—
24	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	C₂×M₁₆×C₂₃	—	4	1	no	—
26	C₂×M₁₆×C₂₃	—	4	1	no	—
27	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₂₃	—	4	1	no	—
29	C₂²×C₈×C₂₃	—	4	1	no	—
30	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
32	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
33	C₂₃×#117	C₂×C₄	1	4	yes	—
34	C₂₃×#117	Q₈	1	4	yes	—
35	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
36	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
38	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
39	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
40	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
41	C₂×C₄²×C₂₃	—	2	4	no	—
42	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
43	C₂₃×#107	—	4	2	no	—
44	C₂₃×#107	—	4	2	no	—
45	C₂₃×#107	—	4	2	no	—
46	C₂₃×#106	—	4	2	no	—
47	C₂₃×C₄⋊C₈	C₂×C₄	1	16	yes	—
48	C₂₃×#95	—	2	4	no	—
49	C₂₃×#95	—	2	4	no	—
50	C₂₃×#95	—	2	4	no	—
51	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	—	2	1	no	—
53	C₂²×C₄×C₂₃	—	2	1	no	—
54	C₂²×C₄×C₂₃	—	2	1	no	—
55	C₂²×C₄×C₂₃	—	2	1	no	—
56	C₂²×C₄×C₂₃	—	2	4	no	—
57	C₂²×C₄×C₂₃	—	2	4	no	—
58	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
60	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
63	M₁₆×C₂₃	—	4	4	no	—
64	C₂×C₈×C₂₃	—	2	2	no	—
65	C₂×C₈×C₂₃	—	2	2	no	—
66	C₂×C₈×C₂₃	—	2	2	no	—
67	C₂×C₈×C₂₃	—	2	2	no	—
68	C₂×C₈×C₂₃	—	4	4	no	—
69	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
71	C₄⋊C₄×C₂₃	—	2	4	no	—
72	C₄⋊C₄×C₂₃	—	2	4	no	—
73	C₄⋊C₄×C₂₃	—	2	4	no	—
74	C₄⋊C₄×C₂₃	—	2	4	no	—
75	C₄⋊C₄×C₂₃	—	8	4	no	—
76	C₄⋊C₄×C₂₃	C₄⋊C₄	1	8	yes	—
77	C₂²⋊C₄×C₂₃	—	8	4	no	—
78	C₂²⋊C₄×C₂₃	C₄⋊C₄	1	8	yes	—
79	C₄²×C₂₃	—	8	4	no	—
80	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
81	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
82	#21968	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₂₃	—	2	1	no	—
84	C₂³×C₂₃	—	2	1	no	—
85	C₂³×C₂₃	—	2	2	no	—
86	C₂³×C₂₃	—	2	2	no	—
87	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
88	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	4	no	—
92	C₂×C₄×C₂₃	—	2	4	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	—	4	4	no	—
98	C₂×C₄×C₂₃	—	4	4	no	—
99	C₂×C₄×C₂₃	—	4	4	no	—
100	C₂×C₄×C₂₃	—	4	4	no	—
101	C₂×C₄×C₂₃	—	4	4	no	—
102	C₂×C₄×C₂₃	—	4	4	no	—
103	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
104	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
108	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
109	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
110	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
111	C₈×C₂₃	—	2	4	no	—
112	C₈×C₂₃	—	2	4	no	—
113	C₂×#424	C₂×C₂₃	1	1	yes	characteristic
114	#2314	C₂×C₂₃	1	2	yes	—
115	#1949	C₂×C₂₃	1	4	yes	complemented
116	#1947	C₂×C₂₃	1	4	yes	complemented
117	#1809	C₂×C₂₃	1	2	yes	—
118	#1808	C₂×C₂₃	1	2	yes	—
119	C₂²×C₂₃	—	2	2	no	—
120	C₂²×C₂₃	—	2	4	no	—
121	C₂²×C₂₃	—	4	1	no	—
122	C₂²×C₂₃	—	4	1	no	—
123	C₂²×C₂₃	—	4	2	no	—
124	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₂₃	#537	1	1	yes	characteristic
126	C₂²×C₂₃	#537	1	1	yes	characteristic
127	C₂²×C₂₃	#576	1	2	yes	—
128	C₂²×C₂₃	#578	1	2	yes	—
129	C₄×C₂₃	—	2	2	no	—
130	C₄×C₂₃	—	2	8	no	—
131	C₄×C₂₃	—	8	4	no	—
132	C₄×C₂₃	—	8	4	no	—
133	C₄×C₂₃	#518	1	1	yes	characteristic
134	C₄×C₂₃	#519	1	1	yes	characteristic
135	C₄×C₂₃	#544	1	1	yes	characteristic
136	C₄×C₂₃	#544	1	1	yes	characteristic
137	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
138	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
139	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
140	C₂×#107	C₂²×C₂₃	1	2	yes	—
141	C₂×#107	C₂²×C₂₃	1	2	yes	—
142	C₂×#107	C₂²×C₂₃	1	2	yes	—
143	C₂×#106	C₂²×C₂₃	1	2	yes	—
144	#424	C₄×C₂₃	1	8	yes	complemented
145	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
146	#388	—	2	4	no	—
147	#387	—	2	4	no	—
148	#384	C₂²×C₂₃	1	2	yes	—
149	#382	C₂²×C₂₃	1	2	yes	—
150	#380	C₂²×C₂₃	1	4	yes	—
151	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
152	#340	C₂²×C₂₃	1	4	yes	—
153	#340	C₂²×C₂₃	1	4	yes	—
154	C₂×C₂₃	—	4	2	no	—
155	C₂×C₂₃	#2316	1	1	yes	characteristic
156	C₂×C₂₃	#2913	1	2	yes	—
157	C₂×C₂₃	#3259	1	1	yes	characteristic
158	C₂×C₂₃	#3261	1	1	yes	characteristic
159	C₂×C₂₃	#3262	1	2	yes	—
160	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
161	C₂×M₁₆	—	4	1	no	—
162	C₂×M₁₆	—	4	1	no	—
163	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
164	C₂²×C₈	—	4	1	no	—
165	C₂²×C₈	—	4	1	no	—
166	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
167	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
168	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
169	#117	C₂×C₄×C₂₃	1	4	yes	—
170	#117	Q₈×C₂₃	1	4	yes	—
171	C₂×C₄⋊C₄	—	2	4	no	—
172	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
173	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
174	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
175	C₂×C₂²⋊C₄	—	2	4	no	—
176	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
177	C₂×C₄²	—	2	4	no	—
178	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
179	#107	—	4	2	no	—
180	#107	—	4	2	no	—
181	#107	—	4	2	no	—
182	#106	—	4	2	no	—
183	C₄⋊C₈	C₂×C₄×C₂₃	1	16	yes	—
184	#95	—	2	4	no	—
185	#95	—	2	4	no	—
186	#95	—	2	4	no	—
187	C₂₃	#21968	1	1	yes	characteristic, direct product factor, Sylow
188	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
196	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
198	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
199	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
200	M₁₆	—	4	4	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	4	no	—
206	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
207	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	8	4	no	—
213	C₄⋊C₄	C₄⋊C₄×C₂₃	1	8	yes	—
214	C₂²⋊C₄	—	8	4	no	—
215	C₂²⋊C₄	C₄⋊C₄×C₂₃	1	8	yes	—
216	C₄²	—	8	4	no	—
217	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
218	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
224	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
240	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
241	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
242	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
243	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
244	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
245	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
246	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
247	C₈	—	2	4	no	—
248	C₈	—	2	4	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
255	C₂²	C₂₃×#537	1	1	yes	characteristic
256	C₂²	C₂₃×#537	1	1	yes	characteristic
257	C₂²	C₂₃×#576	1	2	yes	—
258	C₂²	C₂₃×#578	1	2	yes	—
259	C₄	—	2	8	no	—
260	C₄	—	2	2	no	—
261	C₄	—	8	4	no	—
262	C₄	—	8	4	no	—
263	C₄	C₂₃×#518	1	1	yes	characteristic
264	C₄	C₂₃×#519	1	1	yes	characteristic
265	C₄	C₂₃×#544	1	1	yes	characteristic
266	C₄	C₂₃×#544	1	1	yes	characteristic
267	C₂	—	4	2	no	—
268	C₂	C₂₃×#2316	1	1	yes	characteristic
269	C₂	C₂₃×#2913	1	2	yes	—
270	C₂	C₂₃×#3259	1	1	yes	characteristic
271	C₂	C₂₃×#3261	1	1	yes	characteristic
272	C₂	C₂₃×#3262	1	2	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 ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1949	C₂	semidirect
4. C₂₃×#1947	C₂	semidirect
14. C₂₃×#424	C₄	semidirect
82. #21968	C₂₃	direct (N × H)
115. #1949	C₂×C₂₃	semidirect
116. #1947	C₂×C₂₃	semidirect
144. #424	C₄×C₂₃	semidirect
187. C₂₃	#21968	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×#424	2
C₂₃×#2314	2
C₂₃×#1949	2
C₂₃×#1947	2
C₂₃×#1809	2
C₂₃×#1808	2
#21968	23

Subgroups — C23×#21968 #19169594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21968 #19169594