Subgroups — C₂₃×#22659 #19170285

Order 5888.

1042 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#435	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2352	C₂	1	1	yes	maximal, characteristic
5	D₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1949	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1948	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1854	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1853	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1853	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1852	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1847	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1809	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1808	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1776	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₂₃	—	2	1	no	—
19	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#435	C₄	1	8	yes	complemented
25	C₂₃×#428	—	2	4	no	—
26	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#106	—	2	4	no	—
31	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
41	C₂₃×#389	—	2	1	no	—
42	C₂₃×#388	—	2	1	no	—
43	C₂₃×#387	—	2	1	no	—
44	C₂₃×#386	—	2	1	no	—
45	C₂₃×#385	—	2	1	no	—
46	C₂₃×#385	—	2	1	no	—
47	C₂₃×#384	C₂²	1	1	yes	characteristic
48	C₂₃×#382	C₂²	1	1	yes	characteristic
49	C₂₃×#380	C₂²	1	1	yes	characteristic
50	C₂₃×#380	C₂²	1	1	yes	characteristic
51	C₂₃×#380	C₂²	1	1	yes	characteristic
52	C₂₃×#380	C₂²	1	1	yes	characteristic
53	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
54	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
55	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#340	C₂²	1	1	yes	characteristic
58	C₂₃×#340	C₂²	1	1	yes	characteristic
59	C₂₃×#340	C₂²	1	1	yes	characteristic
60	C₂₃×#340	C₂²	1	1	yes	characteristic
61	C₂₃×#340	C₂²	1	1	yes	characteristic
62	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₂₃	—	2	1	no	—
64	C₂³×C₄×C₂₃	—	2	1	no	—
65	C₂³×C₄×C₂₃	—	2	1	no	—
66	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂×M₁₆×C₂₃	—	2	2	no	—
69	C₂×M₁₆×C₂₃	—	2	2	no	—
70	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
71	C₂²×C₈×C₂₃	—	2	2	no	—
72	C₂²×C₈×C₂₃	—	2	2	no	—
73	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	complemented
74	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₄×D₄×C₂₃	—	4	4	no	—
79	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
80	C₄×D₄×C₂₃	Q₈	1	4	yes	—
81	C₂₃×#117	—	2	2	no	—
82	C₂₃×#117	—	4	1	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
86	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
89	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
92	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
93	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
94	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
106	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
107	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
108	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
110	C₂×C₄²×C₂₃	—	2	1	no	—
111	C₂×C₄²×C₂₃	—	2	1	no	—
112	C₂×C₄²×C₂₃	—	2	1	no	—
113	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
114	C₂₃×#107	—	2	2	no	—
115	C₂₃×#107	—	2	2	no	—
116	C₂₃×#107	—	2	2	no	—
117	C₂₃×#107	—	4	1	no	—
118	C₂₃×#106	—	2	2	no	—
119	C₂₃×#106	—	2	2	no	—
120	C₂₃×#106	—	2	2	no	—
121	C₂₃×#106	—	2	2	no	—
122	C₂₃×#106	—	4	1	no	—
123	C₂₃×#106	D₄	1	4	yes	complemented
124	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
125	C₂₃×C₂²⋊C₈	C₂×C₄	1	4	yes	complemented
126	C₂₃×C₂²⋊C₈	C₂×C₄	1	4	yes	complemented
127	C₂₃×#97	C₂×C₄	1	4	yes	complemented
128	C₂₃×#95	—	2	1	no	—
129	C₂₃×#95	—	2	1	no	—
130	C₂₃×#95	—	2	1	no	—
131	C₂₃×#95	—	2	1	no	—
132	C₂₃×#95	—	2	1	no	—
133	C₂₃×#95	—	2	1	no	—
134	C₂₃×#95	—	2	1	no	—
135	C₂₃×#95	—	2	1	no	—
136	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
137	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂×D₄×C₂₃	—	2	2	no	—
139	C₂×D₄×C₂₃	—	2	2	no	—
140	C₂×D₄×C₂₃	C₄⋊C₄	1	4	yes	—
141	C₂²×C₄×C₂₃	—	2	1	no	—
142	C₂²×C₄×C₂₃	—	2	1	no	—
143	C₂²×C₄×C₂₃	—	2	1	no	—
144	C₂²×C₄×C₂₃	—	2	1	no	—
145	C₂²×C₄×C₂₃	—	2	1	no	—
146	C₂²×C₄×C₂₃	—	2	1	no	—
147	C₂²×C₄×C₂₃	—	2	1	no	—
148	C₂²×C₄×C₂₃	—	2	1	no	—
149	C₂²×C₄×C₂₃	—	2	1	no	—
150	C₂²×C₄×C₂₃	—	2	1	no	—
151	C₂²×C₄×C₂₃	—	2	1	no	—
152	C₂²×C₄×C₂₃	—	2	1	no	—
153	C₂²×C₄×C₂₃	—	2	1	no	—
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	—	2	1	no	—
156	C₂²×C₄×C₂₃	—	2	1	no	—
157	C₂²×C₄×C₂₃	—	4	2	no	—
158	C₂²×C₄×C₂₃	—	4	2	no	—
159	C₂²×C₄×C₂₃	—	4	2	no	—
160	C₂²×C₄×C₂₃	—	4	2	no	—
161	C₂²×C₄×C₂₃	—	4	2	no	—
162	C₂²×C₄×C₂₃	—	4	2	no	—
163	C₂²×C₄×C₂₃	—	8	1	no	—
164	C₂²×C₄×C₂₃	—	8	1	no	—
165	C₂²×C₄×C₂₃	—	8	1	no	—
166	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
167	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
168	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
173	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
174	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
175	M₁₆×C₂₃	—	2	8	no	—
176	C₂×C₈×C₂₃	—	2	1	no	—
177	C₂×C₈×C₂₃	—	2	1	no	—
178	C₂×C₈×C₂₃	—	2	1	no	—
179	C₂×C₈×C₂₃	—	2	1	no	—
180	C₂×C₈×C₂₃	—	2	1	no	—
181	C₂×C₈×C₂₃	—	2	1	no	—
182	C₂×C₈×C₂₃	—	2	8	no	—
183	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
184	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
185	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
186	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
187	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
188	C₂×C₈×C₂₃	#41	1	2	yes	—
189	C₄⋊C₄×C₂₃	—	2	1	no	—
190	C₄⋊C₄×C₂₃	—	2	1	no	—
191	C₄⋊C₄×C₂₃	—	2	1	no	—
192	C₄⋊C₄×C₂₃	—	2	1	no	—
193	C₄⋊C₄×C₂₃	—	2	1	no	—
194	C₄⋊C₄×C₂₃	—	2	1	no	—
195	C₄⋊C₄×C₂₃	—	2	1	no	—
196	C₄⋊C₄×C₂₃	—	2	1	no	—
197	C₄⋊C₄×C₂₃	—	2	1	no	—
198	C₄⋊C₄×C₂₃	—	2	1	no	—
199	C₄⋊C₄×C₂₃	—	2	1	no	—
200	C₄⋊C₄×C₂₃	—	2	1	no	—
201	C₄⋊C₄×C₂₃	—	2	1	no	—
202	C₄⋊C₄×C₂₃	—	2	1	no	—
203	C₄⋊C₄×C₂₃	—	4	2	no	—
204	C₄⋊C₄×C₂₃	—	4	2	no	—
205	C₄⋊C₄×C₂₃	—	4	2	no	—
206	C₄⋊C₄×C₂₃	—	4	2	no	—
207	C₄⋊C₄×C₂₃	—	8	1	no	—
208	C₄⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
209	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
210	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
211	C₂²⋊C₄×C₂₃	—	4	2	no	—
212	C₂²⋊C₄×C₂₃	—	4	2	no	—
213	C₂²⋊C₄×C₂₃	—	4	2	no	—
214	C₂²⋊C₄×C₂₃	—	4	2	no	—
215	C₂²⋊C₄×C₂₃	—	8	1	no	—
216	C₂²⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
217	C₂²⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
218	C₄²×C₂₃	—	4	2	no	—
219	C₄²×C₂₃	—	4	2	no	—
220	C₄²×C₂₃	—	8	1	no	—
221	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
222	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
223	#22659	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂³×C₂₃	—	2	1	no	—
225	C₂³×C₂₃	—	2	1	no	—
226	C₂³×C₂₃	—	2	1	no	—
227	C₂³×C₂₃	—	2	1	no	—
228	C₂³×C₂₃	—	2	1	no	—
229	C₂³×C₂₃	—	2	1	no	—
230	C₂³×C₂₃	—	2	1	no	—
231	C₂³×C₂₃	—	2	1	no	—
232	C₂³×C₂₃	—	2	1	no	—
233	C₂³×C₂₃	—	2	1	no	—
234	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
235	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
236	C₂³×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
237	C₂³×C₂₃	#137	1	2	yes	—
238	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
239	D₄×C₂₃	—	2	8	no	—
240	C₂×C₄×C₂₃	—	2	1	no	—
241	C₂×C₄×C₂₃	—	2	1	no	—
242	C₂×C₄×C₂₃	—	2	1	no	—
243	C₂×C₄×C₂₃	—	2	1	no	—
244	C₂×C₄×C₂₃	—	2	1	no	—
245	C₂×C₄×C₂₃	—	2	1	no	—
246	C₂×C₄×C₂₃	—	2	1	no	—
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	2	2	no	—
249	C₂×C₄×C₂₃	—	2	2	no	—
250	C₂×C₄×C₂₃	—	2	2	no	—
251	C₂×C₄×C₂₃	—	2	2	no	—
252	C₂×C₄×C₂₃	—	2	2	no	—
253	C₂×C₄×C₂₃	—	2	2	no	—
254	C₂×C₄×C₂₃	—	2	2	no	—
255	C₂×C₄×C₂₃	—	2	2	no	—
256	C₂×C₄×C₂₃	—	2	2	no	—
257	C₂×C₄×C₂₃	—	2	2	no	—
258	C₂×C₄×C₂₃	—	4	1	no	—
259	C₂×C₄×C₂₃	—	4	1	no	—
260	C₂×C₄×C₂₃	—	4	1	no	—
261	C₂×C₄×C₂₃	—	4	1	no	—
262	C₂×C₄×C₂₃	—	4	1	no	—
263	C₂×C₄×C₂₃	—	4	1	no	—
264	C₂×C₄×C₂₃	—	4	1	no	—
265	C₂×C₄×C₂₃	—	4	1	no	—
266	C₂×C₄×C₂₃	—	4	1	no	—
267	C₂×C₄×C₂₃	—	4	1	no	—
268	C₂×C₄×C₂₃	—	4	1	no	—
269	C₂×C₄×C₂₃	—	4	1	no	—
270	C₂×C₄×C₂₃	—	4	1	no	—
271	C₂×C₄×C₂₃	—	4	1	no	—
272	C₂×C₄×C₂₃	—	4	1	no	—
273	C₂×C₄×C₂₃	—	4	1	no	—
274	C₂×C₄×C₂₃	—	4	1	no	—
275	C₂×C₄×C₂₃	—	4	1	no	—
276	C₂×C₄×C₂₃	—	4	1	no	—
277	C₂×C₄×C₂₃	—	4	1	no	—
278	C₂×C₄×C₂₃	—	4	1	no	—
279	C₂×C₄×C₂₃	—	4	1	no	—
280	C₂×C₄×C₂₃	—	8	4	no	—
281	C₂×C₄×C₂₃	—	8	4	no	—
282	C₂×C₄×C₂₃	—	8	4	no	—
283	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
284	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
285	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
286	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
287	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
288	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
289	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
290	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
291	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
292	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
293	C₈×C₂₃	—	2	2	no	—
294	C₈×C₂₃	—	2	2	no	—
295	C₈×C₂₃	—	2	2	no	—
296	C₈×C₂₃	C₄×D₄	1	4	yes	complemented
297	C₂×#435	C₂×C₂₃	1	1	yes	characteristic
298	C₂×#428	C₂×C₂₃	1	1	yes	characteristic, complemented
299	C₂²×#106	C₂×C₂₃	1	1	yes	characteristic, complemented
300	#2352	C₂×C₂₃	1	1	yes	characteristic
301	D₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
302	#1949	C₂×C₂₃	1	1	yes	characteristic, complemented
303	#1948	C₂×C₂₃	1	1	yes	characteristic, complemented
304	#1854	C₂×C₂₃	1	1	yes	characteristic, complemented
305	#1853	C₂×C₂₃	1	1	yes	characteristic, complemented
306	#1853	C₂×C₂₃	1	1	yes	characteristic, complemented
307	#1852	C₂×C₂₃	1	1	yes	characteristic, complemented
308	#1847	C₂×C₂₃	1	1	yes	characteristic, complemented
309	#1809	C₂×C₂₃	1	1	yes	characteristic, complemented
310	#1808	C₂×C₂₃	1	1	yes	characteristic, complemented
311	#1776	C₂×C₂₃	1	1	yes	characteristic, complemented
312	C₂²×C₂₃	—	2	2	no	—
313	C₂²×C₂₃	—	2	2	no	—
314	C₂²×C₂₃	—	2	2	no	—
315	C₂²×C₂₃	—	2	2	no	—
316	C₂²×C₂₃	—	2	2	no	—
317	C₂²×C₂₃	—	2	2	no	—
318	C₂²×C₂₃	—	2	2	no	—
319	C₂²×C₂₃	—	2	2	no	—
320	C₂²×C₂₃	—	2	2	no	—
321	C₂²×C₂₃	—	4	1	no	—
322	C₂²×C₂₃	—	4	1	no	—
323	C₂²×C₂₃	—	4	1	no	—
324	C₂²×C₂₃	—	4	1	no	—
325	C₂²×C₂₃	#428	1	4	yes	complemented
326	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
327	C₂²×C₂₃	D₄²	1	1	yes	characteristic
328	C₂²×C₂₃	#547	1	1	yes	characteristic
329	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
330	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
331	C₂²×C₂₃	#577	1	1	yes	characteristic
332	C₂²×C₂₃	#577	1	1	yes	characteristic
333	C₄×C₂₃	—	2	2	no	—
334	C₄×C₂₃	—	2	2	no	—
335	C₄×C₂₃	—	2	2	no	—
336	C₄×C₂₃	—	2	2	no	—
337	C₄×C₂₃	—	4	2	no	—
338	C₄×C₂₃	—	4	2	no	—
339	C₄×C₂₃	—	8	1	no	—
340	C₄×C₂₃	—	8	1	no	—
341	C₄×C₂₃	—	8	1	no	—
342	C₄×C₂₃	—	8	1	no	—
343	C₄×C₂₃	—	8	1	no	—
344	C₄×C₂₃	—	8	1	no	—
345	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
346	C₄×C₂₃	#520	1	1	yes	characteristic
347	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
348	C₄×C₂₃	#554	1	1	yes	characteristic
349	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
350	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
351	C₂×C₄×D₄	—	2	1	no	—
352	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
353	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
354	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
355	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
356	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
357	#435	C₄×C₂₃	1	8	yes	complemented
358	#428	—	2	4	no	—
359	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
360	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
361	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
362	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
363	C₂×#106	—	2	4	no	—
364	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
365	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
366	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
367	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
368	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
369	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
370	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
371	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
372	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
373	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
374	#389	—	2	1	no	—
375	#388	—	2	1	no	—
376	#387	—	2	1	no	—
377	#386	—	2	1	no	—
378	#385	—	2	1	no	—
379	#385	—	2	1	no	—
380	#384	C₂²×C₂₃	1	1	yes	characteristic
381	#382	C₂²×C₂₃	1	1	yes	characteristic
382	#380	C₂²×C₂₃	1	1	yes	characteristic
383	#380	C₂²×C₂₃	1	1	yes	characteristic
384	#380	C₂²×C₂₃	1	1	yes	characteristic
385	#380	C₂²×C₂₃	1	1	yes	characteristic
386	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
387	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
388	C₄×C₂²⋊C₄	—	2	1	no	—
389	#340	C₂²×C₂₃	1	1	yes	characteristic
390	#340	C₂²×C₂₃	1	1	yes	characteristic
391	#340	C₂²×C₂₃	1	1	yes	characteristic
392	#340	C₂²×C₂₃	1	1	yes	characteristic
393	#340	C₂²×C₂₃	1	1	yes	characteristic
394	#340	C₂²×C₂₃	1	1	yes	characteristic
395	C₂×C₂₃	—	2	4	no	—
396	C₂×C₂₃	—	4	2	no	—
397	C₂×C₂₃	D₄×C₄⋊C₄	1	1	yes	characteristic
398	C₂×C₂₃	C₂×#428	1	1	yes	characteristic
399	C₂×C₂₃	#2912	1	1	yes	characteristic
400	C₂×C₂₃	#3265	1	1	yes	characteristic
401	C₂×C₂₃	#3266	1	1	yes	characteristic
402	C₂×C₂₃	#3344	1	1	yes	characteristic
403	C₂×C₂₃	#3345	1	1	yes	characteristic
404	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
405	C₂³×C₄	—	2	1	no	—
406	C₂³×C₄	—	2	1	no	—
407	C₂³×C₄	—	2	1	no	—
408	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
409	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
410	C₂×M₁₆	—	2	2	no	—
411	C₂×M₁₆	—	2	2	no	—
412	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
413	C₂²×C₈	—	2	2	no	—
414	C₂²×C₈	—	2	2	no	—
415	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	complemented
416	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
417	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
418	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
419	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
420	C₄×D₄	—	4	4	no	—
421	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
422	C₄×D₄	Q₈×C₂₃	1	4	yes	—
423	#117	—	2	2	no	—
424	#117	—	4	1	no	—
425	C₂×C₄⋊C₄	—	2	2	no	—
426	C₂×C₄⋊C₄	—	2	2	no	—
427	C₂×C₄⋊C₄	—	2	2	no	—
428	C₂×C₄⋊C₄	—	2	1	no	—
429	C₂×C₄⋊C₄	—	2	1	no	—
430	C₂×C₄⋊C₄	—	2	1	no	—
431	C₂×C₄⋊C₄	—	2	1	no	—
432	C₂×C₄⋊C₄	—	2	1	no	—
433	C₂×C₄⋊C₄	—	4	1	no	—
434	C₂×C₄⋊C₄	—	4	1	no	—
435	C₂×C₄⋊C₄	—	4	1	no	—
436	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
437	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
438	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
439	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
440	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
441	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
442	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
443	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
444	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
445	C₂×C₂²⋊C₄	—	2	1	no	—
446	C₂×C₂²⋊C₄	—	2	1	no	—
447	C₂×C₂²⋊C₄	—	2	1	no	—
448	C₂×C₂²⋊C₄	—	2	1	no	—
449	C₂×C₂²⋊C₄	—	2	1	no	—
450	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
451	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
452	C₂×C₄²	—	2	1	no	—
453	C₂×C₄²	—	2	1	no	—
454	C₂×C₄²	—	2	1	no	—
455	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
456	#107	—	2	2	no	—
457	#107	—	2	2	no	—
458	#107	—	2	2	no	—
459	#107	—	4	1	no	—
460	#106	—	2	2	no	—
461	#106	—	2	2	no	—
462	#106	—	2	2	no	—
463	#106	—	2	2	no	—
464	#106	—	4	1	no	—
465	#106	D₄×C₂₃	1	4	yes	complemented
466	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
467	C₂²⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
468	C₂²⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
469	#97	C₂×C₄×C₂₃	1	4	yes	complemented
470	#95	—	2	1	no	—
471	#95	—	2	1	no	—
472	#95	—	2	1	no	—
473	#95	—	2	1	no	—
474	#95	—	2	1	no	—
475	#95	—	2	1	no	—
476	#95	—	2	1	no	—
477	#95	—	2	1	no	—
478	C₂₃	#22659	1	1	yes	characteristic, direct product factor, Sylow
479	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
480	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
481	C₂×D₄	—	2	2	no	—
482	C₂×D₄	—	2	2	no	—
483	C₂×D₄	C₄⋊C₄×C₂₃	1	4	yes	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	1	no	—
497	C₂²×C₄	—	2	1	no	—
498	C₂²×C₄	—	2	1	no	—
499	C₂²×C₄	—	2	1	no	—
500	C₂²×C₄	—	4	2	no	—
501	C₂²×C₄	—	4	2	no	—
502	C₂²×C₄	—	4	2	no	—
503	C₂²×C₄	—	4	2	no	—
504	C₂²×C₄	—	4	2	no	—
505	C₂²×C₄	—	4	2	no	—
506	C₂²×C₄	—	8	1	no	—
507	C₂²×C₄	—	8	1	no	—
508	C₂²×C₄	—	8	1	no	—
509	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
510	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
511	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
512	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
513	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
514	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
515	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
516	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
517	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
518	M₁₆	—	2	8	no	—
519	C₂×C₈	—	2	1	no	—
520	C₂×C₈	—	2	1	no	—
521	C₂×C₈	—	2	1	no	—
522	C₂×C₈	—	2	1	no	—
523	C₂×C₈	—	2	1	no	—
524	C₂×C₈	—	2	1	no	—
525	C₂×C₈	—	2	8	no	—
526	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
527	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
528	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
529	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
530	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
531	C₂×C₈	#41×C₂₃	1	2	yes	—
532	C₄⋊C₄	—	2	1	no	—
533	C₄⋊C₄	—	2	1	no	—
534	C₄⋊C₄	—	2	1	no	—
535	C₄⋊C₄	—	2	1	no	—
536	C₄⋊C₄	—	2	1	no	—
537	C₄⋊C₄	—	2	1	no	—
538	C₄⋊C₄	—	2	1	no	—
539	C₄⋊C₄	—	2	1	no	—
540	C₄⋊C₄	—	2	1	no	—
541	C₄⋊C₄	—	2	1	no	—
542	C₄⋊C₄	—	2	1	no	—
543	C₄⋊C₄	—	2	1	no	—
544	C₄⋊C₄	—	2	1	no	—
545	C₄⋊C₄	—	2	1	no	—
546	C₄⋊C₄	—	4	2	no	—
547	C₄⋊C₄	—	4	2	no	—
548	C₄⋊C₄	—	4	2	no	—
549	C₄⋊C₄	—	4	2	no	—
550	C₄⋊C₄	—	8	1	no	—
551	C₄⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
552	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
553	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
554	C₂²⋊C₄	—	4	2	no	—
555	C₂²⋊C₄	—	4	2	no	—
556	C₂²⋊C₄	—	4	2	no	—
557	C₂²⋊C₄	—	4	2	no	—
558	C₂²⋊C₄	—	8	1	no	—
559	C₂²⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
560	C₂²⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
561	C₄²	—	4	2	no	—
562	C₄²	—	4	2	no	—
563	C₄²	—	8	1	no	—
564	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
565	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
566	C₂³	—	2	1	no	—
567	C₂³	—	2	1	no	—
568	C₂³	—	2	1	no	—
569	C₂³	—	2	1	no	—
570	C₂³	—	2	1	no	—
571	C₂³	—	2	1	no	—
572	C₂³	—	2	1	no	—
573	C₂³	—	2	1	no	—
574	C₂³	—	2	1	no	—
575	C₂³	—	2	1	no	—
576	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
577	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
578	C₂³	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
579	C₂³	C₂₃×#137	1	2	yes	—
580	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
581	D₄	—	2	8	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	4	1	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	1	no	—
615	C₂×C₄	—	4	1	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	4	1	no	—
622	C₂×C₄	—	8	4	no	—
623	C₂×C₄	—	8	4	no	—
624	C₂×C₄	—	8	4	no	—
625	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
626	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
627	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
628	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
629	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
630	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
631	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
632	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
633	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
634	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
635	C₈	—	2	2	no	—
636	C₈	—	2	2	no	—
637	C₈	—	2	2	no	—
638	C₈	C₄×D₄×C₂₃	1	4	yes	complemented
639	C₂²	—	2	2	no	—
640	C₂²	—	2	2	no	—
641	C₂²	—	2	2	no	—
642	C₂²	—	2	2	no	—
643	C₂²	—	2	2	no	—
644	C₂²	—	2	2	no	—
645	C₂²	—	2	2	no	—
646	C₂²	—	2	2	no	—
647	C₂²	—	2	2	no	—
648	C₂²	—	4	1	no	—
649	C₂²	—	4	1	no	—
650	C₂²	—	4	1	no	—
651	C₂²	—	4	1	no	—
652	C₂²	C₂₃×#428	1	4	yes	complemented
653	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
654	C₂²	D₄²×C₂₃	1	1	yes	characteristic
655	C₂²	C₂₃×#547	1	1	yes	characteristic
656	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
657	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
658	C₂²	C₂₃×#577	1	1	yes	characteristic
659	C₂²	C₂₃×#577	1	1	yes	characteristic
660	C₄	—	2	2	no	—
661	C₄	—	2	2	no	—
662	C₄	—	2	2	no	—
663	C₄	—	2	2	no	—
664	C₄	—	4	2	no	—
665	C₄	—	4	2	no	—
666	C₄	—	8	1	no	—
667	C₄	—	8	1	no	—
668	C₄	—	8	1	no	—
669	C₄	—	8	1	no	—
670	C₄	—	8	1	no	—
671	C₄	—	8	1	no	—
672	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
673	C₄	C₂₃×#520	1	1	yes	characteristic
674	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
675	C₄	C₂₃×#554	1	1	yes	characteristic
676	C₂	—	2	4	no	—
677	C₂	—	4	2	no	—
678	C₂	D₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
679	C₂	C₂×C₂₃×#428	1	1	yes	characteristic
680	C₂	C₂₃×#2912	1	1	yes	characteristic
681	C₂	C₂₃×#3265	1	1	yes	characteristic
682	C₂	C₂₃×#3266	1	1	yes	characteristic
683	C₂	C₂₃×#3344	1	1	yes	characteristic
684	C₂	C₂₃×#3345	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#428	C₂	semidirect
3. C₂²×C₂₃×#106	C₂	semidirect
6. C₂₃×#1949	C₂	semidirect
7. C₂₃×#1948	C₂	semidirect
8. C₂₃×#1854	C₂	semidirect
9. C₂₃×#1853	C₂	semidirect
10. C₂₃×#1853	C₂	semidirect
11. C₂₃×#1852	C₂	semidirect
12. C₂₃×#1847	C₂	semidirect
13. C₂₃×#1809	C₂	semidirect
14. C₂₃×#1808	C₂	semidirect
15. C₂₃×#1776	C₂	semidirect
24. C₂₃×#435	C₄	semidirect
70. C₂×M₁₆×C₂₃	C₂×C₄	semidirect
73. C₂²×C₈×C₂₃	C₂×C₄	semidirect
123. C₂₃×#106	D₄	semidirect
124. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
125. C₂₃×C₂²⋊C₈	C₂×C₄	semidirect
126. C₂₃×C₂²⋊C₈	C₂×C₄	semidirect
127. C₂₃×#97	C₂×C₄	semidirect
223. #22659	C₂₃	direct (N × H)
296. C₈×C₂₃	C₄×D₄	semidirect
298. C₂×#428	C₂×C₂₃	semidirect
299. C₂²×#106	C₂×C₂₃	semidirect
302. #1949	C₂×C₂₃	semidirect
303. #1948	C₂×C₂₃	semidirect
304. #1854	C₂×C₂₃	semidirect
305. #1853	C₂×C₂₃	semidirect
306. #1853	C₂×C₂₃	semidirect
307. #1852	C₂×C₂₃	semidirect
308. #1847	C₂×C₂₃	semidirect
309. #1809	C₂×C₂₃	semidirect
310. #1808	C₂×C₂₃	semidirect
311. #1776	C₂×C₂₃	semidirect
325. C₂²×C₂₃	#428	semidirect
357. #435	C₄×C₂₃	semidirect
412. C₂×M₁₆	C₂×C₄×C₂₃	semidirect
415. C₂²×C₈	C₂×C₄×C₂₃	semidirect
465. #106	D₄×C₂₃	semidirect
466. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
467. C₂²⋊C₈	C₂×C₄×C₂₃	semidirect
468. C₂²⋊C₈	C₂×C₄×C₂₃	semidirect
469. #97	C₂×C₄×C₂₃	semidirect
478. C₂₃	#22659	direct (N × H)
638. C₈	C₄×D₄×C₂₃	semidirect
652. C₂²	C₂₃×#428	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#435	2
C₂×C₂₃×#428	2
C₂²×C₂₃×#106	2
C₂₃×#2352	2
D₄×C₄⋊C₄×C₂₃	2
C₂₃×#1949	2
C₂₃×#1948	2
C₂₃×#1854	2
C₂₃×#1853	2
C₂₃×#1852	2
C₂₃×#1847	2
C₂₃×#1809	2
C₂₃×#1808	2
C₂₃×#1776	2
#22659	23

Subgroups — C23×#22659 #19170285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22659 #19170285