Subgroups — C₂₃×#22676 #19170302

Order 5888.

1086 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#442	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#442	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2349	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1944	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1941	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1864	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1864	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1862	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1862	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1847	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1804	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1802	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1771	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
24	C₂₃×#442	—	2	4	no	—
25	C₂₃×#442	C₄	1	8	yes	complemented
26	C₄×D₈×C₂₃	—	2	4	no	—
27	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
39	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
40	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
42	C₂₃×#389	C₂²	1	1	yes	characteristic
43	C₂₃×#389	C₂²	1	1	yes	characteristic
44	C₂₃×#389	C₂²	1	1	yes	characteristic
45	C₂₃×#389	C₂²	1	1	yes	characteristic
46	C₂₃×#385	—	2	1	no	—
47	C₂₃×#385	—	2	1	no	—
48	C₂₃×#385	—	2	1	no	—
49	C₂₃×#385	—	2	1	no	—
50	C₂₃×#384	C₂²	1	1	yes	characteristic
51	C₂₃×#382	C₂²	1	1	yes	characteristic
52	C₂₃×#380	—	2	1	no	—
53	C₂₃×#380	—	2	1	no	—
54	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
55	C₂×C₂₃×#95	—	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×#336	C₂²	1	1	yes	characteristic
61	C₂₃×#336	C₂²	1	1	yes	characteristic
62	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₂₃	—	2	1	no	—
65	C₂³×C₄×C₂₃	—	2	1	no	—
66	C₂³×C₄×C₂₃	—	2	1	no	—
67	C₂³×C₄×C₂₃	—	2	1	no	—
68	C₂³×C₄×C₂₃	—	2	1	no	—
69	C₂³×C₄×C₂₃	—	2	1	no	—
70	C₂×D₈×C₂₃	C₂×C₄	1	4	yes	—
71	C₂×D₈×C₂₃	D₄	1	4	yes	—
72	C₂×D₈×C₂₃	Q₈	1	4	yes	—
73	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
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	C₄×D₄×C₂₃	—	2	2	no	—
78	C₄×D₄×C₂₃	—	2	2	no	—
79	C₄×D₄×C₂₃	—	2	2	no	—
80	C₄×D₄×C₂₃	—	2	2	no	—
81	C₄×D₄×C₂₃	—	2	2	no	—
82	C₄×D₄×C₂₃	—	2	2	no	—
83	C₄×D₄×C₂₃	—	4	1	no	—
84	C₄×D₄×C₂₃	—	4	1	no	—
85	C₄×D₄×C₂₃	—	4	1	no	—
86	C₄×D₄×C₂₃	—	4	1	no	—
87	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	complemented
88	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	complemented
89	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
94	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
96	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₂₃	D₄	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×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
109	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
110	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
113	C₂₃×#107	—	2	2	no	—
114	C₂₃×#107	—	4	1	no	—
115	C₂₃×#106	—	2	2	no	—
116	C₂₃×#106	—	4	1	no	—
117	C₂₃×#106	C₂×C₄	1	4	yes	complemented
118	C₂₃×C₄⋊C₈	—	2	2	no	—
119	C₂₃×#102	—	2	2	no	—
120	C₂₃×#102	—	2	2	no	—
121	C₂₃×#102	—	2	2	no	—
122	C₂₃×#102	—	2	2	no	—
123	C₂₃×#102	C₂×C₄	1	4	yes	complemented
124	C₂₃×#102	C₂×C₄	1	4	yes	complemented
125	C₂₃×#97	—	2	2	no	—
126	C₂₃×#97	C₂×C₄	1	4	yes	complemented
127	C₄×C₈×C₂₃	—	2	2	no	—
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	D₄	1	1	yes	characteristic
135	C₂₃×#95	D₄	1	1	yes	characteristic
136	C₂⁴×C₂₃	—	2	1	no	—
137	C₂⁴×C₂₃	—	2	1	no	—
138	C₂×D₄×C₂₃	—	2	1	no	—
139	C₂×D₄×C₂₃	—	2	1	no	—
140	C₂×D₄×C₂₃	—	2	1	no	—
141	C₂×D₄×C₂₃	—	2	1	no	—
142	C₂×D₄×C₂₃	—	2	1	no	—
143	C₂×D₄×C₂₃	—	2	1	no	—
144	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
145	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
146	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
147	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
148	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
149	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
150	C₂²×C₄×C₂₃	—	2	1	no	—
151	C₂²×C₄×C₂₃	—	2	1	no	—
152	C₂²×C₄×C₂₃	—	2	4	no	—
153	C₂²×C₄×C₂₃	—	2	4	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	—	2	1	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	—	4	2	no	—
164	C₂²×C₄×C₂₃	—	4	2	no	—
165	C₂²×C₄×C₂₃	—	4	2	no	—
166	C₂²×C₄×C₂₃	—	4	2	no	—
167	C₂²×C₄×C₂₃	—	4	2	no	—
168	C₂²×C₄×C₂₃	—	8	1	no	—
169	C₂²×C₄×C₂₃	—	8	1	no	—
170	C₂²×C₄×C₂₃	—	8	1	no	—
171	C₂²×C₄×C₂₃	—	8	1	no	—
172	C₂²×C₄×C₂₃	—	8	1	no	—
173	C₂²×C₄×C₂₃	—	8	1	no	—
174	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
181	D₈×C₂₃	C₄⋊C₄	1	16	yes	complemented
182	C₂×C₈×C₂₃	—	2	1	no	—
183	C₂×C₈×C₂₃	—	2	1	no	—
184	C₂×C₈×C₂₃	—	2	1	no	—
185	C₂×C₈×C₂₃	—	2	1	no	—
186	C₂×C₈×C₂₃	—	2	1	no	—
187	C₂×C₈×C₂₃	—	2	1	no	—
188	C₂×C₈×C₂₃	—	2	1	no	—
189	C₂×C₈×C₂₃	—	2	1	no	—
190	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
191	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
192	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
193	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
194	C₄⋊C₄×C₂₃	—	2	2	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	—	2	1	no	—
204	C₄⋊C₄×C₂₃	—	2	1	no	—
205	C₄⋊C₄×C₂₃	—	4	1	no	—
206	C₄⋊C₄×C₂₃	—	4	2	no	—
207	C₄⋊C₄×C₂₃	—	4	2	no	—
208	C₄⋊C₄×C₂₃	—	4	2	no	—
209	C₄⋊C₄×C₂₃	—	8	1	no	—
210	C₄⋊C₄×C₂₃	—	8	1	no	—
211	C₄⋊C₄×C₂₃	—	8	1	no	—
212	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
213	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
214	C₂²⋊C₄×C₂₃	—	2	4	no	—
215	C₂²⋊C₄×C₂₃	—	2	4	no	—
216	C₂²⋊C₄×C₂₃	—	4	2	no	—
217	C₂²⋊C₄×C₂₃	—	4	2	no	—
218	C₂²⋊C₄×C₂₃	—	4	2	no	—
219	C₂²⋊C₄×C₂₃	—	4	2	no	—
220	C₄²×C₂₃	—	2	1	no	—
221	C₄²×C₂₃	—	2	1	no	—
222	C₄²×C₂₃	—	2	1	no	—
223	C₄²×C₂₃	—	2	1	no	—
224	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
225	C₄²×C₂₃	#41	1	2	yes	—
226	#22676	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
227	C₂³×C₂₃	—	2	2	no	—
228	C₂³×C₂₃	—	2	2	no	—
229	C₂³×C₂₃	—	2	2	no	—
230	C₂³×C₂₃	—	2	2	no	—
231	C₂³×C₂₃	—	2	2	no	—
232	C₂³×C₂₃	—	2	2	no	—
233	C₂³×C₂₃	—	4	1	no	—
234	C₂³×C₂₃	—	4	1	no	—
235	C₂³×C₂₃	—	4	1	no	—
236	C₂³×C₂₃	—	4	1	no	—
237	C₂³×C₂₃	—	4	1	no	—
238	C₂³×C₂₃	—	4	1	no	—
239	C₂³×C₂₃	—	4	1	no	—
240	C₂³×C₂₃	—	4	1	no	—
241	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
242	D₄×C₂₃	—	2	2	no	—
243	D₄×C₂₃	—	2	2	no	—
244	D₄×C₂₃	—	2	2	no	—
245	D₄×C₂₃	—	2	2	no	—
246	D₄×C₂₃	—	2	2	no	—
247	D₄×C₂₃	—	2	2	no	—
248	D₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
249	D₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	complemented
250	C₂×C₄×C₂₃	—	2	1	no	—
251	C₂×C₄×C₂₃	—	2	1	no	—
252	C₂×C₄×C₂₃	—	2	1	no	—
253	C₂×C₄×C₂₃	—	2	1	no	—
254	C₂×C₄×C₂₃	—	2	1	no	—
255	C₂×C₄×C₂₃	—	2	1	no	—
256	C₂×C₄×C₂₃	—	2	2	no	—
257	C₂×C₄×C₂₃	—	2	2	no	—
258	C₂×C₄×C₂₃	—	2	1	no	—
259	C₂×C₄×C₂₃	—	2	1	no	—
260	C₂×C₄×C₂₃	—	2	1	no	—
261	C₂×C₄×C₂₃	—	2	1	no	—
262	C₂×C₄×C₂₃	—	2	1	no	—
263	C₂×C₄×C₂₃	—	2	1	no	—
264	C₂×C₄×C₂₃	—	2	1	no	—
265	C₂×C₄×C₂₃	—	2	1	no	—
266	C₂×C₄×C₂₃	—	2	1	no	—
267	C₂×C₄×C₂₃	—	4	8	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	—	4	1	no	—
281	C₂×C₄×C₂₃	—	4	1	no	—
282	C₂×C₄×C₂₃	—	4	1	no	—
283	C₂×C₄×C₂₃	—	4	1	no	—
284	C₂×C₄×C₂₃	—	8	4	no	—
285	C₂×C₄×C₂₃	—	8	4	no	—
286	C₂×C₄×C₂₃	—	8	4	no	—
287	C₂×C₄×C₂₃	—	8	4	no	—
288	C₂×C₄×C₂₃	—	8	4	no	—
289	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
290	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
291	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
292	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
293	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
294	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
295	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
296	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
297	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
298	C₈×C₂₃	—	2	2	no	—
299	C₈×C₂₃	—	4	1	no	—
300	C₈×C₂₃	—	4	1	no	—
301	C₈×C₂₃	C₂×C₄⋊C₄	1	4	yes	complemented
302	C₂×#442	C₂×C₂₃	1	1	yes	characteristic
303	C₂×#442	C₂×C₂₃	1	1	yes	characteristic
304	C₂×C₄×D₈	C₂×C₂₃	1	1	yes	characteristic
305	#2349	C₂×C₂₃	1	1	yes	characteristic, complemented
306	D₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
307	#1944	C₂×C₂₃	1	1	yes	characteristic, complemented
308	#1941	C₂×C₂₃	1	1	yes	characteristic, complemented
309	#1864	C₂×C₂₃	1	1	yes	characteristic, complemented
310	#1864	C₂×C₂₃	1	1	yes	characteristic, complemented
311	#1862	C₂×C₂₃	1	1	yes	characteristic, complemented
312	#1862	C₂×C₂₃	1	1	yes	characteristic, complemented
313	#1847	C₂×C₂₃	1	1	yes	characteristic, complemented
314	#1804	C₂×C₂₃	1	1	yes	characteristic, complemented
315	#1802	C₂×C₂₃	1	1	yes	characteristic, complemented
316	#1771	C₂×C₂₃	1	1	yes	characteristic, complemented
317	C₂²×C₂₃	—	2	4	no	—
318	C₂²×C₂₃	—	2	4	no	—
319	C₂²×C₂₃	—	4	2	no	—
320	C₂²×C₂₃	—	4	2	no	—
321	C₂²×C₂₃	—	4	2	no	—
322	C₂²×C₂₃	—	4	2	no	—
323	C₂²×C₂₃	—	4	2	no	—
324	C₂²×C₂₃	—	4	2	no	—
325	C₂²×C₂₃	—	4	2	no	—
326	C₂²×C₂₃	—	4	2	no	—
327	C₂²×C₂₃	—	4	2	no	—
328	C₂²×C₂₃	—	4	2	no	—
329	C₂²×C₂₃	—	4	2	no	—
330	C₂²×C₂₃	—	4	2	no	—
331	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
332	C₂²×C₂₃	D₄²	1	1	yes	characteristic
333	C₂²×C₂₃	D₄×Q₈	1	1	yes	characteristic
334	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
335	C₂²×C₂₃	#575	1	1	yes	characteristic
336	C₂²×C₂₃	#576	1	1	yes	characteristic
337	C₂²×C₂₃	#577	1	1	yes	characteristic
338	C₄×C₂₃	—	2	2	no	—
339	C₄×C₂₃	—	2	2	no	—
340	C₄×C₂₃	—	2	2	no	—
341	C₄×C₂₃	—	4	2	no	—
342	C₄×C₂₃	—	4	2	no	—
343	C₄×C₂₃	—	4	1	no	—
344	C₄×C₂₃	—	4	1	no	—
345	C₄×C₂₃	—	4	1	no	—
346	C₄×C₂₃	—	8	1	no	—
347	C₄×C₂₃	—	8	1	no	—
348	C₄×C₂₃	—	8	1	no	—
349	C₄×C₂₃	—	8	1	no	—
350	C₄×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
351	C₄×C₂₃	#520	1	1	yes	characteristic
352	C₄×C₂₃	#547	1	1	yes	characteristic
353	C₄×C₂₃	#554	1	1	yes	characteristic
354	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
356	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
357	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
358	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
359	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
360	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
361	C₂²×C₄⋊C₄	—	2	1	no	—
362	#442	—	2	4	no	—
363	#442	C₄×C₂₃	1	8	yes	complemented
364	C₄×D₈	—	2	4	no	—
365	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
366	C₂×#107	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×#102	C₂²×C₂₃	1	1	yes	characteristic
372	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
373	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
374	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
375	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
376	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
377	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
378	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
379	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
380	#389	C₂²×C₂₃	1	1	yes	characteristic
381	#389	C₂²×C₂₃	1	1	yes	characteristic
382	#389	C₂²×C₂₃	1	1	yes	characteristic
383	#389	C₂²×C₂₃	1	1	yes	characteristic
384	#385	—	2	1	no	—
385	#385	—	2	1	no	—
386	#385	—	2	1	no	—
387	#385	—	2	1	no	—
388	#384	C₂²×C₂₃	1	1	yes	characteristic
389	#382	C₂²×C₂₃	1	1	yes	characteristic
390	#380	—	2	1	no	—
391	#380	—	2	1	no	—
392	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
393	C₂×#95	—	2	1	no	—
394	#340	C₂²×C₂₃	1	1	yes	characteristic
395	#340	C₂²×C₂₃	1	1	yes	characteristic
396	#340	C₂²×C₂₃	1	1	yes	characteristic
397	#340	C₂²×C₂₃	1	1	yes	characteristic
398	#336	C₂²×C₂₃	1	1	yes	characteristic
399	#336	C₂²×C₂₃	1	1	yes	characteristic
400	C₂×C₂₃	—	4	4	no	—
401	C₂×C₂₃	—	4	4	no	—
402	C₂×C₂₃	D₄×C₄⋊C₄	1	1	yes	characteristic
403	C₂×C₂₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
404	C₂×C₂₃	#2946	1	1	yes	characteristic
405	C₂×C₂₃	#3264	1	1	yes	characteristic
406	C₂×C₂₃	#3267	1	1	yes	characteristic
407	C₂×C₂₃	#3384	1	1	yes	characteristic
408	C₂×C₂₃	#3389	1	1	yes	characteristic
409	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
410	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
411	C₂³×C₄	—	2	1	no	—
412	C₂³×C₄	—	2	1	no	—
413	C₂³×C₄	—	2	1	no	—
414	C₂³×C₄	—	2	1	no	—
415	C₂³×C₄	—	2	1	no	—
416	C₂³×C₄	—	2	1	no	—
417	C₂×D₈	C₂×C₄×C₂₃	1	4	yes	—
418	C₂×D₈	D₄×C₂₃	1	4	yes	—
419	C₂×D₈	Q₈×C₂₃	1	4	yes	—
420	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
421	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
422	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
423	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
424	C₄×D₄	—	2	2	no	—
425	C₄×D₄	—	2	2	no	—
426	C₄×D₄	—	2	2	no	—
427	C₄×D₄	—	2	2	no	—
428	C₄×D₄	—	2	2	no	—
429	C₄×D₄	—	2	2	no	—
430	C₄×D₄	—	4	1	no	—
431	C₄×D₄	—	4	1	no	—
432	C₄×D₄	—	4	1	no	—
433	C₄×D₄	—	4	1	no	—
434	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	complemented
435	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	complemented
436	C₂×C₄⋊C₄	—	2	1	no	—
437	C₂×C₄⋊C₄	—	2	1	no	—
438	C₂×C₄⋊C₄	—	2	1	no	—
439	C₂×C₄⋊C₄	—	2	1	no	—
440	C₂×C₄⋊C₄	—	2	1	no	—
441	C₂×C₄⋊C₄	—	2	1	no	—
442	C₂×C₄⋊C₄	—	4	4	no	—
443	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
444	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
445	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
446	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
447	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
448	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
449	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
450	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
451	C₂×C₂²⋊C₄	—	2	1	no	—
452	C₂×C₂²⋊C₄	—	2	1	no	—
453	C₂×C₂²⋊C₄	—	2	1	no	—
454	C₂×C₂²⋊C₄	—	2	1	no	—
455	C₂×C₂²⋊C₄	—	2	1	no	—
456	C₂×C₂²⋊C₄	—	2	1	no	—
457	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
458	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
459	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
460	#107	—	2	2	no	—
461	#107	—	4	1	no	—
462	#106	—	2	2	no	—
463	#106	—	4	1	no	—
464	#106	C₂×C₄×C₂₃	1	4	yes	complemented
465	C₄⋊C₈	—	2	2	no	—
466	#102	—	2	2	no	—
467	#102	—	2	2	no	—
468	#102	—	2	2	no	—
469	#102	—	2	2	no	—
470	#102	C₂×C₄×C₂₃	1	4	yes	complemented
471	#102	C₂×C₄×C₂₃	1	4	yes	complemented
472	#97	—	2	2	no	—
473	#97	C₂×C₄×C₂₃	1	4	yes	complemented
474	C₄×C₈	—	2	2	no	—
475	#95	—	2	1	no	—
476	#95	—	2	1	no	—
477	#95	—	2	1	no	—
478	#95	—	2	1	no	—
479	#95	—	2	1	no	—
480	#95	—	2	1	no	—
481	#95	D₄×C₂₃	1	1	yes	characteristic
482	#95	D₄×C₂₃	1	1	yes	characteristic
483	C₂₃	#22676	1	1	yes	characteristic, direct product factor, Sylow
484	C₂⁴	—	2	1	no	—
485	C₂⁴	—	2	1	no	—
486	C₂×D₄	—	2	1	no	—
487	C₂×D₄	—	2	1	no	—
488	C₂×D₄	—	2	1	no	—
489	C₂×D₄	—	2	1	no	—
490	C₂×D₄	—	2	1	no	—
491	C₂×D₄	—	2	1	no	—
492	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
493	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
494	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
495	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
496	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
497	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
498	C₂²×C₄	—	2	1	no	—
499	C₂²×C₄	—	2	1	no	—
500	C₂²×C₄	—	2	4	no	—
501	C₂²×C₄	—	2	4	no	—
502	C₂²×C₄	—	2	1	no	—
503	C₂²×C₄	—	2	1	no	—
504	C₂²×C₄	—	2	1	no	—
505	C₂²×C₄	—	2	1	no	—
506	C₂²×C₄	—	4	2	no	—
507	C₂²×C₄	—	4	2	no	—
508	C₂²×C₄	—	4	2	no	—
509	C₂²×C₄	—	4	2	no	—
510	C₂²×C₄	—	4	2	no	—
511	C₂²×C₄	—	4	2	no	—
512	C₂²×C₄	—	4	2	no	—
513	C₂²×C₄	—	4	2	no	—
514	C₂²×C₄	—	4	2	no	—
515	C₂²×C₄	—	4	2	no	—
516	C₂²×C₄	—	8	1	no	—
517	C₂²×C₄	—	8	1	no	—
518	C₂²×C₄	—	8	1	no	—
519	C₂²×C₄	—	8	1	no	—
520	C₂²×C₄	—	8	1	no	—
521	C₂²×C₄	—	8	1	no	—
522	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
523	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
524	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
525	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
526	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
527	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
528	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
529	D₈	C₄⋊C₄×C₂₃	1	16	yes	complemented
530	C₂×C₈	—	2	1	no	—
531	C₂×C₈	—	2	1	no	—
532	C₂×C₈	—	2	1	no	—
533	C₂×C₈	—	2	1	no	—
534	C₂×C₈	—	2	1	no	—
535	C₂×C₈	—	2	1	no	—
536	C₂×C₈	—	2	1	no	—
537	C₂×C₈	—	2	1	no	—
538	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
539	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
540	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
541	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
542	C₄⋊C₄	—	2	2	no	—
543	C₄⋊C₄	—	2	1	no	—
544	C₄⋊C₄	—	2	1	no	—
545	C₄⋊C₄	—	2	1	no	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	2	1	no	—
548	C₄⋊C₄	—	2	1	no	—
549	C₄⋊C₄	—	2	1	no	—
550	C₄⋊C₄	—	2	1	no	—
551	C₄⋊C₄	—	2	1	no	—
552	C₄⋊C₄	—	2	1	no	—
553	C₄⋊C₄	—	4	1	no	—
554	C₄⋊C₄	—	4	2	no	—
555	C₄⋊C₄	—	4	2	no	—
556	C₄⋊C₄	—	4	2	no	—
557	C₄⋊C₄	—	8	1	no	—
558	C₄⋊C₄	—	8	1	no	—
559	C₄⋊C₄	—	8	1	no	—
560	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
561	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
562	C₂²⋊C₄	—	2	4	no	—
563	C₂²⋊C₄	—	2	4	no	—
564	C₂²⋊C₄	—	4	2	no	—
565	C₂²⋊C₄	—	4	2	no	—
566	C₂²⋊C₄	—	4	2	no	—
567	C₂²⋊C₄	—	4	2	no	—
568	C₄²	—	2	1	no	—
569	C₄²	—	2	1	no	—
570	C₄²	—	2	1	no	—
571	C₄²	—	2	1	no	—
572	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
573	C₄²	#41×C₂₃	1	2	yes	—
574	C₂³	—	2	2	no	—
575	C₂³	—	2	2	no	—
576	C₂³	—	2	2	no	—
577	C₂³	—	2	2	no	—
578	C₂³	—	2	2	no	—
579	C₂³	—	2	2	no	—
580	C₂³	—	4	1	no	—
581	C₂³	—	4	1	no	—
582	C₂³	—	4	1	no	—
583	C₂³	—	4	1	no	—
584	C₂³	—	4	1	no	—
585	C₂³	—	4	1	no	—
586	C₂³	—	4	1	no	—
587	C₂³	—	4	1	no	—
588	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
589	D₄	—	2	2	no	—
590	D₄	—	2	2	no	—
591	D₄	—	2	2	no	—
592	D₄	—	2	2	no	—
593	D₄	—	2	2	no	—
594	D₄	—	2	2	no	—
595	D₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
596	D₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	complemented
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	1	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	2	no	—
611	C₂×C₄	—	2	1	no	—
612	C₂×C₄	—	2	1	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	4	8	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	—	4	1	no	—
623	C₂×C₄	—	4	1	no	—
624	C₂×C₄	—	4	1	no	—
625	C₂×C₄	—	4	1	no	—
626	C₂×C₄	—	4	1	no	—
627	C₂×C₄	—	4	1	no	—
628	C₂×C₄	—	4	1	no	—
629	C₂×C₄	—	4	1	no	—
630	C₂×C₄	—	4	1	no	—
631	C₂×C₄	—	8	4	no	—
632	C₂×C₄	—	8	4	no	—
633	C₂×C₄	—	8	4	no	—
634	C₂×C₄	—	8	4	no	—
635	C₂×C₄	—	8	4	no	—
636	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
637	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
638	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
639	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
640	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
641	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
642	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
643	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
644	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
645	C₈	—	2	2	no	—
646	C₈	—	4	1	no	—
647	C₈	—	4	1	no	—
648	C₈	C₂×C₄⋊C₄×C₂₃	1	4	yes	complemented
649	C₂²	—	2	4	no	—
650	C₂²	—	2	4	no	—
651	C₂²	—	4	2	no	—
652	C₂²	—	4	2	no	—
653	C₂²	—	4	2	no	—
654	C₂²	—	4	2	no	—
655	C₂²	—	4	2	no	—
656	C₂²	—	4	2	no	—
657	C₂²	—	4	2	no	—
658	C₂²	—	4	2	no	—
659	C₂²	—	4	2	no	—
660	C₂²	—	4	2	no	—
661	C₂²	—	4	2	no	—
662	C₂²	—	4	2	no	—
663	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
664	C₂²	D₄²×C₂₃	1	1	yes	characteristic
665	C₂²	D₄×Q₈×C₂₃	1	1	yes	characteristic
666	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
667	C₂²	C₂₃×#575	1	1	yes	characteristic
668	C₂²	C₂₃×#576	1	1	yes	characteristic
669	C₂²	C₂₃×#577	1	1	yes	characteristic
670	C₄	—	2	2	no	—
671	C₄	—	2	2	no	—
672	C₄	—	2	2	no	—
673	C₄	—	4	2	no	—
674	C₄	—	4	2	no	—
675	C₄	—	4	1	no	—
676	C₄	—	4	1	no	—
677	C₄	—	4	1	no	—
678	C₄	—	8	1	no	—
679	C₄	—	8	1	no	—
680	C₄	—	8	1	no	—
681	C₄	—	8	1	no	—
682	C₄	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
683	C₄	C₂₃×#520	1	1	yes	characteristic
684	C₄	C₂₃×#547	1	1	yes	characteristic
685	C₄	C₂₃×#554	1	1	yes	characteristic
686	C₂	—	4	4	no	—
687	C₂	—	4	4	no	—
688	C₂	D₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
689	C₂	C₄×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
690	C₂	C₂₃×#2946	1	1	yes	characteristic
691	C₂	C₂₃×#3264	1	1	yes	characteristic
692	C₂	C₂₃×#3267	1	1	yes	characteristic
693	C₂	C₂₃×#3384	1	1	yes	characteristic
694	C₂	C₂₃×#3389	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×D₈×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₄⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#2349	C₂	semidirect
5. D₄×C₄⋊C₄×C₂₃	C₂	semidirect
6. C₂₃×#1944	C₂	semidirect
7. C₂₃×#1941	C₂	semidirect
8. C₂₃×#1864	C₂	semidirect
9. C₂₃×#1864	C₂	semidirect
10. C₂₃×#1862	C₂	semidirect
11. C₂₃×#1862	C₂	semidirect
12. C₂₃×#1847	C₂	semidirect
13. C₂₃×#1804	C₂	semidirect
14. C₂₃×#1802	C₂	semidirect
15. C₂₃×#1771	C₂	semidirect
25. C₂₃×#442	C₄	semidirect
87. C₄×D₄×C₂₃	C₂×C₄	semidirect
88. C₄×D₄×C₂₃	C₂×C₄	semidirect
117. C₂₃×#106	C₂×C₄	semidirect
123. C₂₃×#102	C₂×C₄	semidirect
124. C₂₃×#102	C₂×C₄	semidirect
126. C₂₃×#97	C₂×C₄	semidirect
181. D₈×C₂₃	C₄⋊C₄	semidirect
226. #22676	C₂₃	direct (N × H)
249. D₄×C₂₃	C₂×C₄⋊C₄	semidirect
301. C₈×C₂₃	C₂×C₄⋊C₄	semidirect
305. #2349	C₂×C₂₃	semidirect
306. D₄×C₄⋊C₄	C₂×C₂₃	semidirect
307. #1944	C₂×C₂₃	semidirect
308. #1941	C₂×C₂₃	semidirect
309. #1864	C₂×C₂₃	semidirect
310. #1864	C₂×C₂₃	semidirect
311. #1862	C₂×C₂₃	semidirect
312. #1862	C₂×C₂₃	semidirect
313. #1847	C₂×C₂₃	semidirect
314. #1804	C₂×C₂₃	semidirect
315. #1802	C₂×C₂₃	semidirect
316. #1771	C₂×C₂₃	semidirect
363. #442	C₄×C₂₃	semidirect
434. C₄×D₄	C₂×C₄×C₂₃	semidirect
435. C₄×D₄	C₂×C₄×C₂₃	semidirect
464. #106	C₂×C₄×C₂₃	semidirect
470. #102	C₂×C₄×C₂₃	semidirect
471. #102	C₂×C₄×C₂₃	semidirect
473. #97	C₂×C₄×C₂₃	semidirect
483. C₂₃	#22676	direct (N × H)
529. D₈	C₄⋊C₄×C₂₃	semidirect
596. D₄	C₂×C₄⋊C₄×C₂₃	semidirect
648. C₈	C₂×C₄⋊C₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#442	2
C₂×C₄×D₈×C₂₃	2
C₂₃×#2349	2
D₄×C₄⋊C₄×C₂₃	2
C₂₃×#1944	2
C₂₃×#1941	2
C₂₃×#1864	2
C₂₃×#1862	2
C₂₃×#1847	2
C₂₃×#1804	2
C₂₃×#1802	2
C₂₃×#1771	2
#22676	23

Subgroups — C23×#22676 #19170302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22676 #19170302