Subgroups — C₂₃×#17323 #19164949

Order 5888.

874 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2920	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1452	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1451	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1450	C₂	1	2	yes	maximal, complemented
6	C₈×M₁₆×C₂₃	C₂	1	4	yes	maximal, complemented
7	C₂³×C₈×C₂₃	C₄	1	2	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#432	C₄	1	4	yes	—
10	C₂₃×#432	C₄	1	4	yes	—
11	C₂₃×#431	C₂²	1	4	yes	—
12	C₂₃×#431	C₂²	1	4	yes	—
13	C₂₃×#423	C₄	1	4	yes	—
14	C₂₃×#407	C₄	1	4	yes	—
15	C₄×M₁₆×C₂₃	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₂₃	C₄	1	4	yes	—
17	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#322	C₂²	1	8	yes	complemented
23	C₂₃×#322	C₂²	1	4	yes	complemented
24	C₈²×C₂₃	C₂²	1	4	yes	complemented
25	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
27	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₂₃	—	2	8	no	—
29	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
39	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
41	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
42	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
43	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
44	C₂₃×#97	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
47	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
48	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₂₃	C₂³	1	8	yes	—
50	C₄×C₈×C₂₃	C₂³	1	2	yes	—
51	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂⁴×C₂₃	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	—	2	4	no	—
55	C₂²×C₄×C₂₃	—	2	8	no	—
56	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
58	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
59	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
60	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
61	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
66	M₁₆×C₂₃	—	2	16	no	—
67	C₂×C₈×C₂₃	—	2	4	no	—
68	C₂×C₈×C₂₃	—	2	8	no	—
69	C₂×C₈×C₂₃	—	2	8	no	—
70	C₂×C₈×C₂₃	M₁₆	1	8	yes	complemented
71	C₂×C₈×C₂₃	C₂²×C₄	1	16	yes	—
72	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
75	C₄²×C₂₃	C₄²	1	2	yes	—
76	C₄²×C₂₃	C₄²	1	1	yes	characteristic
77	C₄²×C₂₃	C₄²	1	2	yes	—
78	C₄²×C₂₃	C₄²	1	1	yes	characteristic
79	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
81	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
83	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
84	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
85	#17323	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₂₃	—	2	2	no	—
87	C₂³×C₂₃	—	2	4	no	—
88	C₂³×C₂₃	C₂×C₄²	1	1	yes	characteristic
89	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	4	no	—
92	C₂×C₄×C₂₃	—	2	4	no	—
93	C₂×C₄×C₂₃	—	2	4	no	—
94	C₂×C₄×C₂₃	—	2	4	no	—
95	C₂×C₄×C₂₃	—	2	4	no	—
96	C₂×C₄×C₂₃	—	2	4	no	—
97	C₂×C₄×C₂₃	—	2	4	no	—
98	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
99	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
100	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
101	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
104	C₂×C₄×C₂₃	#131	1	8	yes	—
105	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
107	C₈×C₂₃	—	2	4	no	—
108	C₈×C₂₃	—	2	16	no	—
109	C₈×C₂₃	C₂×M₁₆	1	4	yes	—
110	C₈×C₂₃	C₂×M₁₆	1	4	yes	—
111	#2920	C₂×C₂₃	1	2	yes	—
112	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
113	#1452	C₂×C₂₃	1	2	yes	complemented
114	#1451	C₂×C₂₃	1	4	yes	complemented
115	#1450	C₂×C₂₃	1	2	yes	complemented
116	C₈×M₁₆	C₂×C₂₃	1	4	yes	complemented
117	C₂²×C₂₃	—	2	8	no	—
118	C₂²×C₂₃	—	2	2	no	—
119	C₂²×C₂₃	—	2	2	no	—
120	C₂²×C₂₃	—	2	2	no	—
121	C₂²×C₂₃	C₄×M₁₆	1	2	yes	—
122	C₂²×C₂₃	#405	1	4	yes	—
123	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
124	C₄×C₂₃	—	2	2	no	—
125	C₄×C₂₃	—	2	2	no	—
126	C₄×C₂₃	—	2	2	no	—
127	C₄×C₂₃	C₄×M₁₆	1	2	yes	—
128	C₄×C₂₃	#405	1	4	yes	—
129	C₄×C₂₃	C₂²×M₁₆	1	2	yes	—
130	C₄×C₂₃	C₂×#131	1	4	yes	—
131	C₂³×C₈	C₄×C₂₃	1	2	yes	—
132	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
133	#432	C₄×C₂₃	1	4	yes	—
134	#432	C₄×C₂₃	1	4	yes	—
135	#431	C₂²×C₂₃	1	4	yes	—
136	#431	C₂²×C₂₃	1	4	yes	—
137	#423	C₄×C₂₃	1	4	yes	—
138	#407	C₄×C₂₃	1	4	yes	—
139	C₄×M₁₆	C₂²×C₂₃	1	4	yes	—
140	C₂×C₄×C₈	C₄×C₂₃	1	4	yes	—
141	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	—
142	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
143	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
145	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
146	#322	C₂²×C₂₃	1	8	yes	complemented
147	#322	C₂²×C₂₃	1	4	yes	complemented
148	C₈²	C₂²×C₂₃	1	4	yes	complemented
149	C₂×C₂₃	—	2	2	no	—
150	C₂×C₂₃	—	2	4	no	—
151	C₂×C₂₃	C₂×C₄×M₁₆	1	1	yes	characteristic
152	C₂×C₂₃	C₂×#405	1	2	yes	—
153	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
154	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
155	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	—
156	C₂²×C₈	—	2	8	no	—
157	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
158	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
159	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
160	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
161	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
162	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
163	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
164	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
165	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
166	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
167	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
168	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
169	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
170	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
171	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
172	#97	C₂×C₄×C₂₃	1	8	yes	—
173	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
174	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
175	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
176	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
177	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
178	C₄×C₈	C₂³×C₂₃	1	8	yes	—
179	C₄×C₈	C₂³×C₂₃	1	2	yes	—
180	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
181	C₂₃	#17323	1	1	yes	characteristic, direct product factor, Sylow
182	C₂⁴	C₄²×C₂₃	1	1	yes	characteristic
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
186	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
187	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
188	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
189	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
190	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
191	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
192	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
193	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
194	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
195	M₁₆	—	2	16	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	—	2	8	no	—
199	C₂×C₈	M₁₆×C₂₃	1	8	yes	complemented
200	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
201	C₂×C₈	C₂²×C₄×C₂₃	1	16	yes	—
202	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
203	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
204	C₄²	C₄²×C₂₃	1	2	yes	—
205	C₄²	C₄²×C₂₃	1	1	yes	characteristic
206	C₄²	C₄²×C₂₃	1	2	yes	—
207	C₄²	C₄²×C₂₃	1	1	yes	characteristic
208	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
209	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
210	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
211	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
212	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
213	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
214	C₂³	—	2	2	no	—
215	C₂³	—	2	4	no	—
216	C₂³	C₂×C₄²×C₂₃	1	1	yes	characteristic
217	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
227	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
228	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
229	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
230	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
231	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
232	C₂×C₄	C₂₃×#131	1	8	yes	—
233	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
234	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
235	C₈	—	2	4	no	—
236	C₈	—	2	16	no	—
237	C₈	C₂×M₁₆×C₂₃	1	4	yes	—
238	C₈	C₂×M₁₆×C₂₃	1	4	yes	—
239	C₂²	—	2	8	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	C₄×M₁₆×C₂₃	1	2	yes	—
244	C₂²	C₂₃×#405	1	4	yes	—
245	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	C₄×M₁₆×C₂₃	1	2	yes	—
250	C₄	C₂₃×#405	1	4	yes	—
251	C₄	C₂²×M₁₆×C₂₃	1	2	yes	—
252	C₄	C₂×C₂₃×#131	1	4	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	4	no	—
255	C₂	C₂×C₄×M₁₆×C₂₃	1	1	yes	characteristic
256	C₂	C₂×C₂₃×#405	1	2	yes	—

	subgroup	G/H
center	C₄×C₈×C₂₃	C₂³
commutator	C₂²	C₂²×C₄²×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₄²
prime radical	C₂⁴×C₂₃	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1452	C₂	semidirect
4. C₂₃×#1451	C₂	semidirect
5. C₂₃×#1450	C₂	semidirect
6. C₈×M₁₆×C₂₃	C₂	semidirect
22. C₂₃×#322	C₂²	semidirect
23. C₂₃×#322	C₂²	semidirect
24. C₈²×C₂₃	C₂²	semidirect
70. C₂×C₈×C₂₃	M₁₆	semidirect
85. #17323	C₂₃	direct (N × H)
113. #1452	C₂×C₂₃	semidirect
114. #1451	C₂×C₂₃	semidirect
115. #1450	C₂×C₂₃	semidirect
116. C₈×M₁₆	C₂×C₂₃	semidirect
146. #322	C₂²×C₂₃	semidirect
147. #322	C₂²×C₂₃	semidirect
148. C₈²	C₂²×C₂₃	semidirect
181. C₂₃	#17323	direct (N × H)
199. C₂×C₈	M₁₆×C₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#2920	2
C₂²×C₄×C₈×C₂₃	2
C₂₃×#1452	2
C₂₃×#1451	2
C₂₃×#1450	2
C₈×M₁₆×C₂₃	2
#17323	23

Subgroups — C23×#17323 #19164949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17323 #19164949