Subgroups — C₂₃×#19633 #19167259

Order 5888.

710 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3209	C₂	1	2	yes	maximal
2	C₂₃×#3170	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3165	C₂	1	2	yes	maximal
4	C₂×C₂₃×#500	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1723	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1713	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1707	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#533	C₂²	1	2	yes	—
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂₃×#501	—	2	4	no	—
12	C₂₃×#500	—	2	2	no	—
13	C₂₃×#500	C₂²	1	4	yes	—
14	C₂₃×#500	C₂²	1	1	yes	characteristic
15	C₂₃×#500	C₂²	1	2	yes	—
16	C₂₃×#500	C₂²	1	2	yes	—
17	C₂₃×#498	C₂²	1	4	yes	—
18	C₂₃×#494	C₂²	1	1	yes	characteristic
19	C₂₃×#494	C₂²	1	4	yes	—
20	C₂₃×#487	C₂²	1	2	yes	—
21	C₂₃×#487	C₂²	1	4	yes	—
22	C₂₃×#467	—	2	2	no	—
23	C₂₃×#462	—	2	4	no	—
24	C₂₃×#451	—	2	4	no	—
25	C₄×Q₁₆×C₂₃	—	2	2	no	—
26	C₂₃×#431	C₂²	1	1	yes	characteristic
27	C₂₃×#428	—	2	4	no	—
28	C₂×C₂₃×#107	—	2	2	no	—
29	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#322	C₂²	1	2	yes	—
32	C₈²×C₂₃	C₂²	1	2	yes	—
33	C₂²×Q₈×C₂₃	—	2	2	no	—
34	C₂×Q₁₆×C₂₃	—	2	2	no	—
35	C₂×Q₁₆×C₂₃	—	2	8	no	—
36	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
37	C₂²×C₈×C₂₃	D₄	1	2	yes	—
38	C₂₃×C₄⋊Q₈	—	2	2	no	—
39	C₂₃×C₄⋊Q₈	—	2	2	no	—
40	C₂₃×C₄⋊Q₈	—	2	2	no	—
41	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
42	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
43	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
44	C₂₃×#125	—	2	2	no	—
45	C₂₃×C₂²⋊Q₈	—	2	2	no	—
46	C₂₃×C₂²⋊Q₈	—	2	2	no	—
47	C₄×Q₈×C₂₃	—	2	2	no	—
48	C₄×Q₈×C₂₃	—	2	2	no	—
49	C₂₃×#117	—	2	2	no	—
50	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
53	C₂₃×#107	—	2	4	no	—
54	C₂₃×#107	—	2	4	no	—
55	C₂₃×#107	—	2	8	no	—
56	C₂₃×#107	—	2	2	no	—
57	C₂₃×#106	—	2	8	no	—
58	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
59	C₂₃×#103	—	2	4	no	—
60	C₂₃×#103	—	2	8	no	—
61	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
62	C₂₃×#97	D₄	1	2	yes	—
63	C₄×C₈×C₂₃	D₄	1	2	yes	—
64	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₂₃	C₂³	1	2	yes	—
66	C₄×C₈×C₂₃	C₂³	1	2	yes	—
67	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₂×Q₈×C₂₃	—	2	2	no	—
69	C₂×Q₈×C₂₃	—	2	4	no	—
70	C₂×Q₈×C₂₃	—	2	2	no	—
71	C₂×Q₈×C₂₃	—	4	2	no	—
72	C₂×Q₈×C₂₃	—	8	4	no	—
73	C₂²×C₄×C₂₃	—	4	2	no	—
74	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₂₃	—	4	4	no	—
78	Q₁₆×C₂₃	—	8	8	no	—
79	M₁₆×C₂₃	Q₁₆	1	8	yes	complemented
80	C₂×C₈×C₂₃	—	2	4	no	—
81	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
82	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
84	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
85	C₄⋊C₄×C₂₃	—	2	2	no	—
86	C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₄⋊C₄×C₂₃	—	2	2	no	—
88	C₄⋊C₄×C₂₃	—	2	4	no	—
89	C₄⋊C₄×C₂₃	—	2	4	no	—
90	C₄⋊C₄×C₂₃	—	2	2	no	—
91	C₄⋊C₄×C₂₃	—	4	2	no	—
92	C₄⋊C₄×C₂₃	—	4	2	no	—
93	C₄⋊C₄×C₂₃	—	4	2	no	—
94	C₄⋊C₄×C₂₃	—	4	2	no	—
95	C₂²⋊C₄×C₂₃	—	4	2	no	—
96	C₄²×C₂₃	—	4	2	no	—
97	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
101	#19633	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂³×C₂₃	#127	1	1	yes	characteristic
103	Q₈×C₂₃	—	4	4	no	—
104	Q₈×C₂₃	—	4	4	no	—
105	Q₈×C₂₃	—	4	8	no	—
106	Q₈×C₂₃	—	8	4	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	4	2	no	—
110	C₂×C₄×C₂₃	—	4	2	no	—
111	C₂×C₄×C₂₃	—	4	4	no	—
112	C₂×C₄×C₂₃	—	8	4	no	—
113	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
114	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
115	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂×Q₁₆	1	4	yes	—
117	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₂₃	—	2	2	no	—
121	C₈×C₂₃	—	2	4	no	—
122	C₈×C₂₃	C₂×Q₁₆	1	8	yes	—
123	C₈×C₂₃	#137	1	4	yes	—
124	#3209	C₂×C₂₃	1	2	yes	—
125	#3170	C₂×C₂₃	1	1	yes	characteristic
126	#3165	C₂×C₂₃	1	2	yes	—
127	C₂×#500	C₂×C₂₃	1	1	yes	characteristic
128	#1723	C₂×C₂₃	1	4	yes	complemented
129	#1713	C₂×C₂₃	1	2	yes	complemented
130	#1707	C₂×C₂₃	1	2	yes	complemented
131	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic
132	C₂²×C₂₃	—	2	1	no	—
133	C₂²×C₂₃	—	2	1	no	—
134	C₂²×C₂₃	#494	1	2	yes	—
135	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
136	C₄×C₂₃	—	2	1	no	—
137	C₄×C₂₃	—	2	1	no	—
138	C₄×C₂₃	—	8	2	no	—
139	C₄×C₂₃	—	8	4	no	—
140	C₄×C₂₃	—	8	2	no	—
141	C₄×C₂₃	#494	1	2	yes	—
142	C₄×C₂₃	C₂²×Q₁₆	1	2	yes	—
143	C₄×C₂₃	C₂×#137	1	2	yes	—
144	C₄×C₂₃	#576	1	2	yes	—
145	#533	C₂²×C₂₃	1	2	yes	—
146	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
147	#501	—	2	4	no	—
148	#500	—	2	2	no	—
149	#500	C₂²×C₂₃	1	2	yes	—
150	#500	C₂²×C₂₃	1	2	yes	—
151	#500	C₂²×C₂₃	1	4	yes	—
152	#500	C₂²×C₂₃	1	1	yes	characteristic
153	#498	C₂²×C₂₃	1	4	yes	—
154	#494	C₂²×C₂₃	1	1	yes	characteristic
155	#494	C₂²×C₂₃	1	4	yes	—
156	#487	C₂²×C₂₃	1	2	yes	—
157	#487	C₂²×C₂₃	1	4	yes	—
158	#467	—	2	2	no	—
159	#462	—	2	4	no	—
160	#451	—	2	4	no	—
161	C₄×Q₁₆	—	2	2	no	—
162	#431	C₂²×C₂₃	1	1	yes	characteristic
163	#428	—	2	4	no	—
164	C₂×#107	—	2	2	no	—
165	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
167	#322	C₂²×C₂₃	1	2	yes	—
168	C₈²	C₂²×C₂₃	1	2	yes	—
169	C₂×C₂₃	—	2	2	no	—
170	C₂×C₂₃	C₂×#494	1	1	yes	characteristic
171	C₂×C₂₃	#3152	1	1	yes	characteristic
172	C₂×C₂₃	#3155	1	1	yes	characteristic
173	C₂²×Q₈	—	2	2	no	—
174	C₂×Q₁₆	—	2	2	no	—
175	C₂×Q₁₆	—	2	8	no	—
176	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
177	C₂²×C₈	D₄×C₂₃	1	2	yes	—
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
182	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
183	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
184	#125	—	2	2	no	—
185	C₂²⋊Q₈	—	2	2	no	—
186	C₂²⋊Q₈	—	2	2	no	—
187	C₄×Q₈	—	2	2	no	—
188	C₄×Q₈	—	2	2	no	—
189	#117	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
193	#107	—	2	4	no	—
194	#107	—	2	4	no	—
195	#107	—	2	8	no	—
196	#107	—	2	2	no	—
197	#106	—	2	8	no	—
198	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
199	#103	—	2	4	no	—
200	#103	—	2	8	no	—
201	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
202	#97	D₄×C₂₃	1	2	yes	—
203	C₄×C₈	D₄×C₂₃	1	2	yes	—
204	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
205	C₄×C₈	C₂³×C₂₃	1	2	yes	—
206	C₄×C₈	C₂³×C₂₃	1	2	yes	—
207	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
208	C₂₃	#19633	1	1	yes	characteristic, direct product factor, Sylow
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	4	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	4	2	no	—
213	C₂×Q₈	—	8	4	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
218	Q₁₆	—	4	4	no	—
219	Q₁₆	—	8	8	no	—
220	M₁₆	Q₁₆×C₂₃	1	8	yes	complemented
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
223	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
224	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
225	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
239	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
240	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
241	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
242	C₂³	C₂₃×#127	1	1	yes	characteristic
243	Q₈	—	4	4	no	—
244	Q₈	—	4	4	no	—
245	Q₈	—	4	8	no	—
246	Q₈	—	8	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	8	4	no	—
253	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
254	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
255	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
256	C₂×C₄	C₂×Q₁₆×C₂₃	1	4	yes	—
257	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
258	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	4	no	—
262	C₈	C₂×Q₁₆×C₂₃	1	8	yes	—
263	C₈	C₂₃×#137	1	4	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	C₂₃×#494	1	2	yes	—
267	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	8	2	no	—
271	C₄	—	8	4	no	—
272	C₄	—	8	2	no	—
273	C₄	C₂₃×#494	1	2	yes	—
274	C₄	C₂²×Q₁₆×C₂₃	1	2	yes	—
275	C₄	C₂×C₂₃×#137	1	2	yes	—
276	C₄	C₂₃×#576	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	C₂×C₂₃×#494	1	1	yes	characteristic
279	C₂	C₂₃×#3152	1	1	yes	characteristic
280	C₂	C₂₃×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂³×C₂₃	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₂₃×#1723	C₂	semidirect
6. C₂₃×#1713	C₂	semidirect
7. C₂₃×#1707	C₂	semidirect
79. M₁₆×C₂₃	Q₁₆	semidirect
101. #19633	C₂₃	direct (N × H)
128. #1723	C₂×C₂₃	semidirect
129. #1713	C₂×C₂₃	semidirect
130. #1707	C₂×C₂₃	semidirect
208. C₂₃	#19633	direct (N × H)
220. M₁₆	Q₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3209	2
C₂₃×#3170	2
C₂₃×#3165	2
C₂×C₂₃×#500	2
C₂₃×#1723	2
C₂₃×#1713	2
C₂₃×#1707	2
C₈×M₁₆×C₂₃	2
#19633	23

Subgroups — C23×#19633 #19167259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19633 #19167259