Subgroups — C₂₃×#21705 #19169331

Order 5888.

1046 conjugacy classes of subgroups.

The subgroup lattice has 268 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	C₂×C₂₃×#443	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2931	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1974	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1962	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#135	C₂²	1	4	yes	—
9	C₂×C₂₃×#131	C₂²	1	4	yes	—
10	C₂₃×#532	C₂²	1	2	yes	—
11	C₂₃×#495	C₄	1	4	yes	complemented
12	C₂₃×#494	C₄	1	1	yes	characteristic, complemented
13	C₂₃×#493	C₄	1	1	yes	characteristic, complemented
14	C₂₃×#492	C₄	1	2	yes	complemented
15	C₂₃×#443	—	2	16	no	—
16	C₂₃×#436	—	2	4	no	—
17	C₈×D₄×C₂₃	—	2	4	no	—
18	C₂×C₂₃×#108	C₂²	1	4	yes	—
19	C₂₃×#423	C₂²	1	2	yes	—
20	C₂×C₂₃×#104	C₂²	1	8	yes	—
21	C₂₃×#408	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₄³×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×#41×C₂₃	C₂³	1	4	yes	—
28	C₂₃×#135	—	2	8	no	—
29	C₂₃×#135	—	2	8	no	—
30	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	—
31	C₂×SD₁₆×C₂₃	C₂×C₄	1	8	yes	—
32	C₂×D₈×C₂₃	C₂×C₄	1	4	yes	—
33	C₂₃×#131	—	2	16	no	—
34	C₂×M₁₆×C₂₃	—	2	4	no	—
35	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
36	C₂²×C₈×C₂₃	—	2	4	no	—
37	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
41	C₂₃×#127	C₂×C₄	1	2	yes	—
42	C₂₃×#124	C₂×C₄	1	4	yes	—
43	C₂₃×#121	—	2	4	no	—
44	C₄×D₄×C₂₃	—	2	4	no	—
45	C₂×C₄²×C₂₃	—	2	2	no	—
46	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
47	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
48	C₂₃×#108	—	2	8	no	—
49	C₂₃×C₄⋊C₈	—	2	4	no	—
50	C₂₃×#104	—	2	16	no	—
51	C₂₃×C₂²⋊C₈	—	2	8	no	—
52	C₄×C₈×C₂₃	—	2	4	no	—
53	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₂₃	D₄	1	2	yes	—
57	C₄×C₈×C₂₃	D₄	1	2	yes	—
58	C₄×C₈×C₂₃	D₄	1	8	yes	complemented
59	#41×C₂₃	—	2	8	no	—
60	#41×C₂₃	—	2	8	no	—
61	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
62	C₂×D₄×C₂₃	—	2	4	no	—
63	C₂×D₄×C₂₃	—	2	4	no	—
64	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₂₃	—	2	2	no	—
66	C₂²×C₄×C₂₃	—	2	4	no	—
67	C₂²×C₄×C₂₃	#41	1	2	yes	—
68	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
69	Q₁₆×C₂₃	—	2	8	no	—
70	SD₁₆×C₂₃	—	2	16	no	—
71	D₈×C₂₃	—	2	8	no	—
72	M₁₆×C₂₃	—	2	8	no	—
73	M₁₆×C₂₃	—	4	8	no	—
74	C₂×C₈×C₂₃	—	2	2	no	—
75	C₂×C₈×C₂₃	—	2	4	no	—
76	C₂×C₈×C₂₃	—	4	8	no	—
77	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
81	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
83	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₂₃	#41	1	4	yes	—
85	C₄⋊C₄×C₂₃	—	2	4	no	—
86	C₂²⋊C₄×C₂₃	—	2	8	no	—
87	C₄²×C₂₃	—	2	8	no	—
88	C₄²×C₂₃	—	2	2	no	—
89	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₂₃	C₂×D₄	1	4	yes	—
92	C₄²×C₂₃	C₂×D₄	1	2	yes	—
93	#21705	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₂₃	—	2	4	no	—
95	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
96	Q₈×C₂₃	—	2	8	no	—
97	D₄×C₂₃	—	2	8	no	—
98	D₄×C₂₃	—	4	8	no	—
99	D₄×C₂₃	—	4	8	no	—
100	C₂×C₄×C₂₃	—	2	8	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	—	4	4	no	—
105	C₂×C₄×C₂₃	—	4	4	no	—
106	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
110	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
112	C₈×C₂₃	—	2	2	no	—
113	C₈×C₂₃	—	4	4	no	—
114	C₈×C₂₃	C₄×D₄	1	4	yes	—
115	C₈×C₂₃	C₄×D₄	1	4	yes	—
116	C₈×C₂₃	#127	1	2	yes	—
117	C₈×C₂₃	#127	1	2	yes	—
118	#3147	C₂×C₂₃	1	1	yes	characteristic
119	C₂×#443	C₂×C₂₃	1	4	yes	complemented
120	#2931	C₂×C₂₃	1	2	yes	complemented
121	#1974	C₂×C₂₃	1	4	yes	complemented
122	#1962	C₂×C₂₃	1	2	yes	complemented
123	#1848	C₂×C₂₃	1	1	yes	characteristic, complemented
124	C₄²×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
125	C₂²×C₂₃	—	2	2	no	—
126	C₂²×C₂₃	—	4	4	no	—
127	C₂²×C₂₃	—	4	8	no	—
128	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂²×C₂₃	#532	1	2	yes	—
130	C₄×C₂₃	—	2	8	no	—
131	C₄×C₂₃	—	2	2	no	—
132	C₄×C₂₃	—	4	4	no	—
133	C₄×C₂₃	#443	1	4	yes	complemented
134	C₄×C₂₃	C₂×C₄×D₄	1	2	yes	—
135	C₄×C₂₃	C₂×#127	1	1	yes	characteristic
136	C₄×C₂₃	#532	1	1	yes	characteristic
137	C₂×#135	C₂²×C₂₃	1	4	yes	—
138	C₂×#131	C₂²×C₂₃	1	4	yes	—
139	#532	C₂²×C₂₃	1	2	yes	—
140	#495	C₄×C₂₃	1	4	yes	complemented
141	#494	C₄×C₂₃	1	1	yes	characteristic, complemented
142	#493	C₄×C₂₃	1	1	yes	characteristic, complemented
143	#492	C₄×C₂₃	1	2	yes	complemented
144	#443	—	2	16	no	—
145	#436	—	2	4	no	—
146	C₈×D₄	—	2	4	no	—
147	C₂×#108	C₂²×C₂₃	1	4	yes	—
148	#423	C₂²×C₂₃	1	2	yes	—
149	C₂×#104	C₂²×C₂₃	1	8	yes	—
150	#408	C₂²×C₂₃	1	4	yes	—
151	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
152	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
155	C₄³	C₂²×C₂₃	1	1	yes	characteristic
156	C₂×C₂₃	—	2	2	no	—
157	C₂×C₂₃	—	8	4	no	—
158	C₂×C₂₃	C₄×#127	1	1	yes	characteristic
159	C₂×C₂₃	C₂×#443	1	2	yes	—
160	C₂×#41	C₂³×C₂₃	1	4	yes	—
161	#135	—	2	8	no	—
162	#135	—	2	8	no	—
163	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	—
164	C₂×SD₁₆	C₂×C₄×C₂₃	1	8	yes	—
165	C₂×D₈	C₂×C₄×C₂₃	1	4	yes	—
166	#131	—	2	16	no	—
167	C₂×M₁₆	—	2	4	no	—
168	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
169	C₂²×C₈	—	2	4	no	—
170	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
172	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
173	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
174	#127	C₂×C₄×C₂₃	1	2	yes	—
175	#124	C₂×C₄×C₂₃	1	4	yes	—
176	#121	—	2	4	no	—
177	C₄×D₄	—	2	4	no	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
180	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
181	#108	—	2	8	no	—
182	C₄⋊C₈	—	2	4	no	—
183	#104	—	2	16	no	—
184	C₂²⋊C₈	—	2	8	no	—
185	C₄×C₈	—	2	4	no	—
186	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
187	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
188	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
189	C₄×C₈	D₄×C₂₃	1	2	yes	—
190	C₄×C₈	D₄×C₂₃	1	2	yes	—
191	C₄×C₈	D₄×C₂₃	1	8	yes	complemented
192	C₂₃	#21705	1	1	yes	characteristic, direct product factor, Sylow
193	#41	—	2	8	no	—
194	#41	—	2	8	no	—
195	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	#41×C₂₃	1	2	yes	—
202	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
203	Q₁₆	—	2	8	no	—
204	SD₁₆	—	2	16	no	—
205	D₈	—	2	8	no	—
206	M₁₆	—	2	8	no	—
207	M₁₆	—	4	8	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	4	8	no	—
211	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
212	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
213	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
214	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
215	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
217	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
218	C₂×C₈	#41×C₂₃	1	4	yes	—
219	C₄⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	2	8	no	—
221	C₄²	—	2	8	no	—
222	C₄²	—	2	2	no	—
223	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
224	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
225	C₄²	C₂×D₄×C₂₃	1	4	yes	—
226	C₄²	C₂×D₄×C₂₃	1	2	yes	—
227	C₂³	—	2	4	no	—
228	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
229	Q₈	—	2	8	no	—
230	D₄	—	2	8	no	—
231	D₄	—	4	8	no	—
232	D₄	—	4	8	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
240	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
241	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
242	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
243	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
244	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
245	C₈	—	2	2	no	—
246	C₈	—	4	4	no	—
247	C₈	C₄×D₄×C₂₃	1	4	yes	—
248	C₈	C₄×D₄×C₂₃	1	4	yes	—
249	C₈	C₂₃×#127	1	2	yes	—
250	C₈	C₂₃×#127	1	2	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	4	8	no	—
253	C₂²	—	4	4	no	—
254	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
255	C₂²	C₂₃×#532	1	2	yes	—
256	C₄	—	2	8	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	4	no	—
259	C₄	C₂₃×#443	1	4	yes	complemented
260	C₄	C₂×C₄×D₄×C₂₃	1	2	yes	—
261	C₄	C₂×C₂₃×#127	1	1	yes	characteristic
262	C₄	C₂₃×#532	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	8	4	no	—
265	C₂	C₄×C₂₃×#127	1	1	yes	characteristic
266	C₂	C₂×C₂₃×#443	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂₃×#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#443	C₂	semidirect
3. C₂₃×#2931	C₂	semidirect
4. C₂₃×#1974	C₂	semidirect
5. C₂₃×#1962	C₂	semidirect
6. C₂₃×#1848	C₂	semidirect
7. C₄²×C₈×C₂₃	C₂	semidirect
11. C₂₃×#495	C₄	semidirect
12. C₂₃×#494	C₄	semidirect
13. C₂₃×#493	C₄	semidirect
14. C₂₃×#492	C₄	semidirect
58. C₄×C₈×C₂₃	D₄	semidirect
93. #21705	C₂₃	direct (N × H)
119. C₂×#443	C₂×C₂₃	semidirect
120. #2931	C₂×C₂₃	semidirect
121. #1974	C₂×C₂₃	semidirect
122. #1962	C₂×C₂₃	semidirect
123. #1848	C₂×C₂₃	semidirect
124. C₄²×C₈	C₂×C₂₃	semidirect
133. C₄×C₂₃	#443	semidirect
140. #495	C₄×C₂₃	semidirect
141. #494	C₄×C₂₃	semidirect
142. #493	C₄×C₂₃	semidirect
143. #492	C₄×C₂₃	semidirect
191. C₄×C₈	D₄×C₂₃	semidirect
192. C₂₃	#21705	direct (N × H)
259. C₄	C₂₃×#443	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
C₂×C₂₃×#443	2
C₂₃×#2931	2
C₂₃×#1974	2
C₂₃×#1962	2
C₂₃×#1848	2
C₄²×C₈×C₂₃	2
#21705	23

Subgroups — C23×#21705 #19169331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21705 #19169331