Subgroups — C₂₃×#24706 #19172332

Order 5888.

868 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#478	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2851	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂₃×#121	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2077	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2075	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2034	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#2033	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1989	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1949	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1944	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1943	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1936	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1936	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1804	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₂₃	—	2	1	no	—
19	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#478	—	2	4	no	—
22	C₂₃×#432	—	2	4	no	—
23	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
37	C₂₃×#390	C₂²	1	1	yes	characteristic
38	C₂₃×#389	C₂²	1	1	yes	characteristic
39	C₂₃×#389	C₂²	1	1	yes	characteristic
40	C₂₃×#389	C₂²	1	1	yes	characteristic
41	C₂₃×#388	—	2	1	no	—
42	C₂₃×#387	—	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×#382	C₂²	1	1	yes	characteristic
48	C₂₃×#380	C₂²	1	1	yes	characteristic
49	C₂₃×#380	C₂²	1	1	yes	characteristic
50	C₂₃×#380	C₂²	1	1	yes	characteristic
51	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
52	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
53	C₂₃×#340	C₂²	1	1	yes	characteristic
54	C₂₃×#340	C₂²	1	1	yes	characteristic
55	C₂₃×#340	C₂²	1	1	yes	characteristic
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	C₂²×D₄×C₂₃	—	2	1	no	—
62	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₂₃	—	2	1	no	—
64	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
66	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂₃×#125	—	2	2	no	—
72	C₂₃×#125	—	2	2	no	—
73	C₂₃×#121	—	2	2	no	—
74	C₂₃×#121	—	2	2	no	—
75	C₄×D₄×C₂₃	—	2	2	no	—
76	C₄×D₄×C₂₃	—	2	2	no	—
77	C₄×D₄×C₂₃	—	4	4	no	—
78	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
97	C₂×C₄²×C₂₃	—	2	2	no	—
98	C₂×C₄²×C₂₃	—	2	2	no	—
99	C₂×C₄²×C₂₃	—	2	1	no	—
100	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
101	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
104	C₂₃×#107	—	2	2	no	—
105	C₂₃×#107	—	2	2	no	—
106	C₂₃×#106	—	2	2	no	—
107	C₂₃×#106	—	2	2	no	—
108	C₂₃×C₄⋊C₈	—	2	2	no	—
109	C₂₃×C₄⋊C₈	D₄	1	4	yes	complemented
110	C₂₃×#102	—	2	2	no	—
111	C₂₃×#102	—	2	2	no	—
112	C₂₃×#102	—	2	2	no	—
113	C₂₃×#102	—	2	2	no	—
114	C₂₃×C₂²⋊C₈	—	2	2	no	—
115	C₂₃×C₂²⋊C₈	—	2	2	no	—
116	C₂₃×#97	—	2	2	no	—
117	C₂₃×#97	—	2	2	no	—
118	C₂₃×#95	—	2	1	no	—
119	C₂₃×#95	—	2	1	no	—
120	C₂₃×#95	—	2	1	no	—
121	C₂₃×#95	—	2	1	no	—
122	C₂₃×#95	—	2	1	no	—
123	C₂₃×#95	—	2	1	no	—
124	C₂₃×#95	—	2	1	no	—
125	C₂₃×#95	—	2	1	no	—
126	C₂₃×#95	—	2	1	no	—
127	C₂⁴×C₂₃	—	2	1	no	—
128	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
129	C₂×D₄×C₂₃	—	2	4	no	—
130	C₂×D₄×C₂₃	—	2	1	no	—
131	C₂×D₄×C₂₃	—	2	1	no	—
132	C₂×D₄×C₂₃	—	2	1	no	—
133	C₂×D₄×C₂₃	—	2	1	no	—
134	C₂×D₄×C₂₃	—	2	1	no	—
135	C₂×D₄×C₂₃	—	4	2	no	—
136	C₂×D₄×C₂₃	—	4	2	no	—
137	C₂×D₄×C₂₃	#41	1	2	yes	—
138	C₂²×C₄×C₂₃	—	2	2	no	—
139	C₂²×C₄×C₂₃	—	2	1	no	—
140	C₂²×C₄×C₂₃	—	2	2	no	—
141	C₂²×C₄×C₂₃	—	2	2	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	—	4	1	no	—
155	C₂²×C₄×C₂₃	—	4	1	no	—
156	C₂²×C₄×C₂₃	—	4	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	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
166	C₂×C₈×C₂₃	—	2	1	no	—
167	C₂×C₈×C₂₃	—	2	1	no	—
168	C₂×C₈×C₂₃	—	2	1	no	—
169	C₂×C₈×C₂₃	—	2	1	no	—
170	C₂×C₈×C₂₃	—	2	1	no	—
171	C₂×C₈×C₂₃	—	2	1	no	—
172	C₂×C₈×C₂₃	—	2	1	no	—
173	C₂×C₈×C₂₃	—	2	1	no	—
174	C₂×C₈×C₂₃	—	2	1	no	—
175	C₂×C₈×C₂₃	—	2	1	no	—
176	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
177	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
178	C₄⋊C₄×C₂₃	—	2	1	no	—
179	C₄⋊C₄×C₂₃	—	2	1	no	—
180	C₄⋊C₄×C₂₃	—	2	1	no	—
181	C₄⋊C₄×C₂₃	—	2	1	no	—
182	C₄⋊C₄×C₂₃	—	2	1	no	—
183	C₄⋊C₄×C₂₃	—	2	1	no	—
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	2	1	no	—
186	C₄⋊C₄×C₂₃	—	2	1	no	—
187	C₄⋊C₄×C₂₃	—	2	1	no	—
188	C₄⋊C₄×C₂₃	—	2	1	no	—
189	C₄⋊C₄×C₂₃	—	2	1	no	—
190	C₄⋊C₄×C₂₃	—	2	1	no	—
191	C₄⋊C₄×C₂₃	—	4	2	no	—
192	C₄⋊C₄×C₂₃	—	4	2	no	—
193	C₄⋊C₄×C₂₃	—	4	2	no	—
194	C₄⋊C₄×C₂₃	—	4	2	no	—
195	C₄⋊C₄×C₂₃	—	4	2	no	—
196	C₄⋊C₄×C₂₃	#41	1	2	yes	—
197	C₂²⋊C₄×C₂₃	—	2	4	no	—
198	C₂²⋊C₄×C₂₃	—	4	2	no	—
199	C₂²⋊C₄×C₂₃	—	4	2	no	—
200	C₂²⋊C₄×C₂₃	—	4	2	no	—
201	C₂²⋊C₄×C₂₃	—	4	2	no	—
202	C₂²⋊C₄×C₂₃	—	4	2	no	—
203	C₂²⋊C₄×C₂₃	—	4	2	no	—
204	C₄²×C₂₃	—	2	2	no	—
205	C₄²×C₂₃	—	2	2	no	—
206	C₄²×C₂₃	—	2	1	no	—
207	C₄²×C₂₃	—	2	1	no	—
208	C₄²×C₂₃	—	4	2	no	—
209	C₄²×C₂₃	C₂×D₄	1	2	yes	—
210	C₄²×C₂₃	#41	1	2	yes	—
211	#24706	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂³×C₂₃	—	2	1	no	—
213	C₂³×C₂₃	—	2	1	no	—
214	C₂³×C₂₃	—	2	1	no	—
215	C₂³×C₂₃	—	2	1	no	—
216	C₂³×C₂₃	—	2	1	no	—
217	C₂³×C₂₃	—	2	1	no	—
218	C₂³×C₂₃	—	2	2	no	—
219	C₂³×C₂₃	—	4	1	no	—
220	C₂³×C₂₃	—	4	1	no	—
221	C₂³×C₂₃	—	4	1	no	—
222	C₂³×C₂₃	—	4	1	no	—
223	C₂³×C₂₃	—	4	1	no	—
224	C₂³×C₂₃	—	4	1	no	—
225	C₂³×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
226	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
227	D₄×C₂₃	—	2	2	no	—
228	D₄×C₂₃	—	2	2	no	—
229	D₄×C₂₃	—	4	8	no	—
230	D₄×C₂₃	—	4	1	no	—
231	D₄×C₂₃	—	4	1	no	—
232	C₂×C₄×C₂₃	—	2	1	no	—
233	C₂×C₄×C₂₃	—	2	1	no	—
234	C₂×C₄×C₂₃	—	2	1	no	—
235	C₂×C₄×C₂₃	—	2	1	no	—
236	C₂×C₄×C₂₃	—	2	1	no	—
237	C₂×C₄×C₂₃	—	2	1	no	—
238	C₂×C₄×C₂₃	—	2	1	no	—
239	C₂×C₄×C₂₃	—	2	1	no	—
240	C₂×C₄×C₂₃	—	2	2	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	—	4	4	no	—
247	C₂×C₄×C₂₃	—	4	4	no	—
248	C₂×C₄×C₂₃	—	4	1	no	—
249	C₂×C₄×C₂₃	—	4	1	no	—
250	C₂×C₄×C₂₃	—	4	1	no	—
251	C₂×C₄×C₂₃	—	4	1	no	—
252	C₂×C₄×C₂₃	—	4	1	no	—
253	C₂×C₄×C₂₃	—	4	1	no	—
254	C₂×C₄×C₂₃	—	4	1	no	—
255	C₂×C₄×C₂₃	—	4	1	no	—
256	C₂×C₄×C₂₃	—	4	1	no	—
257	C₂×C₄×C₂₃	—	4	1	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	—	8	4	no	—
273	C₂×C₄×C₂₃	#121	1	2	yes	—
274	C₂×C₄×C₂₃	#121	1	2	yes	—
275	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
277	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
278	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
279	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
280	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
281	C₈×C₂₃	—	2	2	no	—
282	C₈×C₂₃	—	2	2	no	—
283	C₈×C₂₃	—	4	1	no	—
284	C₈×C₂₃	—	4	1	no	—
285	C₂×#478	C₂×C₂₃	1	1	yes	characteristic, complemented
286	C₂×#432	C₂×C₂₃	1	1	yes	characteristic, complemented
287	#2851	C₂×C₂₃	1	1	yes	characteristic, complemented
288	C₄×#121	C₂×C₂₃	1	1	yes	characteristic
289	#2077	C₂×C₂₃	1	1	yes	characteristic, complemented
290	#2075	C₂×C₂₃	1	1	yes	characteristic, complemented
291	#2034	C₂×C₂₃	1	1	yes	characteristic
292	#2033	C₂×C₂₃	1	1	yes	characteristic
293	#1989	C₂×C₂₃	1	1	yes	characteristic, complemented
294	#1949	C₂×C₂₃	1	1	yes	characteristic, complemented
295	#1944	C₂×C₂₃	1	1	yes	characteristic, complemented
296	#1943	C₂×C₂₃	1	1	yes	characteristic, complemented
297	#1936	C₂×C₂₃	1	1	yes	characteristic, complemented
298	#1936	C₂×C₂₃	1	1	yes	characteristic, complemented
299	#1804	C₂×C₂₃	1	1	yes	characteristic, complemented
300	C₂²×C₂₃	—	2	2	no	—
301	C₂²×C₂₃	—	2	2	no	—
302	C₂²×C₂₃	—	2	2	no	—
303	C₂²×C₂₃	—	4	1	no	—
304	C₂²×C₂₃	—	4	1	no	—
305	C₂²×C₂₃	—	4	1	no	—
306	C₂²×C₂₃	—	4	1	no	—
307	C₂²×C₂₃	—	4	2	no	—
308	C₂²×C₂₃	—	4	2	no	—
309	C₂²×C₂₃	—	4	2	no	—
310	C₂²×C₂₃	—	8	1	no	—
311	C₂²×C₂₃	—	8	1	no	—
312	C₂²×C₂₃	—	8	1	no	—
313	C₂²×C₂₃	—	8	1	no	—
314	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
315	C₂²×C₂₃	#525	1	1	yes	characteristic
316	C₂²×C₂₃	#536	1	1	yes	characteristic
317	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
318	C₂²×C₂₃	#575	1	1	yes	characteristic
319	C₂²×C₂₃	#575	1	1	yes	characteristic
320	C₂²×C₂₃	#575	1	1	yes	characteristic
321	C₄×C₂₃	—	2	2	no	—
322	C₄×C₂₃	—	2	2	no	—
323	C₄×C₂₃	—	4	2	no	—
324	C₄×C₂₃	—	4	1	no	—
325	C₄×C₂₃	—	4	1	no	—
326	C₄×C₂₃	—	4	1	no	—
327	C₄×C₂₃	—	8	1	no	—
328	C₄×C₂₃	—	8	1	no	—
329	C₄×C₂₃	—	8	1	no	—
330	C₄×C₂₃	—	8	1	no	—
331	C₄×C₂₃	—	8	1	no	—
332	C₄×C₂₃	—	8	1	no	—
333	C₄×C₂₃	#547	1	1	yes	characteristic
334	C₄×C₂₃	#548	1	1	yes	characteristic
335	C₄×C₂₃	#552	1	1	yes	characteristic
336	C₄×C₂₃	#556	1	1	yes	characteristic
337	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
338	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
339	C₂×C₄×D₄	—	2	1	no	—
340	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
341	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
342	#478	—	2	4	no	—
343	#432	—	2	4	no	—
344	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
347	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
348	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
349	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
351	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
352	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
353	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
354	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
356	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
357	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
358	#390	C₂²×C₂₃	1	1	yes	characteristic
359	#389	C₂²×C₂₃	1	1	yes	characteristic
360	#389	C₂²×C₂₃	1	1	yes	characteristic
361	#389	C₂²×C₂₃	1	1	yes	characteristic
362	#388	—	2	1	no	—
363	#387	—	2	1	no	—
364	#387	—	2	1	no	—
365	#386	—	2	1	no	—
366	#385	—	2	1	no	—
367	#385	—	2	1	no	—
368	#382	C₂²×C₂₃	1	1	yes	characteristic
369	#380	C₂²×C₂₃	1	1	yes	characteristic
370	#380	C₂²×C₂₃	1	1	yes	characteristic
371	#380	C₂²×C₂₃	1	1	yes	characteristic
372	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
373	C₄×C₂²⋊C₄	—	2	1	no	—
374	#340	C₂²×C₂₃	1	1	yes	characteristic
375	#340	C₂²×C₂₃	1	1	yes	characteristic
376	#340	C₂²×C₂₃	1	1	yes	characteristic
377	#340	C₂²×C₂₃	1	1	yes	characteristic
378	#340	C₂²×C₂₃	1	1	yes	characteristic
379	#340	C₂²×C₂₃	1	1	yes	characteristic
380	#340	C₂²×C₂₃	1	1	yes	characteristic
381	#340	C₂²×C₂₃	1	1	yes	characteristic
382	C₂×C₂₃	—	4	2	no	—
383	C₂×C₂₃	—	8	2	no	—
384	C₂×C₂₃	#2415	1	1	yes	characteristic
385	C₂×C₂₃	#3054	1	1	yes	characteristic
386	C₂×C₂₃	#3055	1	1	yes	characteristic
387	C₂×C₂₃	#3111	1	1	yes	characteristic
388	C₂×C₂₃	#3114	1	1	yes	characteristic
389	C₂×C₂₃	#3234	1	1	yes	characteristic
390	C₂×C₂₃	#3240	1	1	yes	characteristic
391	C₂²×D₄	—	2	1	no	—
392	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
393	C₂³×C₄	—	2	1	no	—
394	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
395	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
396	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
397	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
398	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
399	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
400	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
401	#125	—	2	2	no	—
402	#125	—	2	2	no	—
403	#121	—	2	2	no	—
404	#121	—	2	2	no	—
405	C₄×D₄	—	2	2	no	—
406	C₄×D₄	—	2	2	no	—
407	C₄×D₄	—	4	4	no	—
408	C₂×C₄⋊C₄	—	2	1	no	—
409	C₂×C₄⋊C₄	—	2	1	no	—
410	C₂×C₄⋊C₄	—	2	1	no	—
411	C₂×C₄⋊C₄	—	2	1	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
414	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
415	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
416	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
417	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
418	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
419	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
420	C₂×C₂²⋊C₄	—	2	1	no	—
421	C₂×C₂²⋊C₄	—	2	1	no	—
422	C₂×C₂²⋊C₄	—	2	1	no	—
423	C₂×C₂²⋊C₄	—	2	1	no	—
424	C₂×C₂²⋊C₄	—	2	1	no	—
425	C₂×C₂²⋊C₄	—	2	1	no	—
426	C₂×C₂²⋊C₄	—	2	1	no	—
427	C₂×C₄²	—	2	2	no	—
428	C₂×C₄²	—	2	2	no	—
429	C₂×C₄²	—	2	1	no	—
430	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
431	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
433	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
434	#107	—	2	2	no	—
435	#107	—	2	2	no	—
436	#106	—	2	2	no	—
437	#106	—	2	2	no	—
438	C₄⋊C₈	—	2	2	no	—
439	C₄⋊C₈	D₄×C₂₃	1	4	yes	complemented
440	#102	—	2	2	no	—
441	#102	—	2	2	no	—
442	#102	—	2	2	no	—
443	#102	—	2	2	no	—
444	C₂²⋊C₈	—	2	2	no	—
445	C₂²⋊C₈	—	2	2	no	—
446	#97	—	2	2	no	—
447	#97	—	2	2	no	—
448	#95	—	2	1	no	—
449	#95	—	2	1	no	—
450	#95	—	2	1	no	—
451	#95	—	2	1	no	—
452	#95	—	2	1	no	—
453	#95	—	2	1	no	—
454	#95	—	2	1	no	—
455	#95	—	2	1	no	—
456	#95	—	2	1	no	—
457	C₂₃	#24706	1	1	yes	characteristic, direct product factor, Sylow
458	C₂⁴	—	2	1	no	—
459	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
460	C₂×D₄	—	2	4	no	—
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	2	1	no	—
463	C₂×D₄	—	2	1	no	—
464	C₂×D₄	—	2	1	no	—
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	—	4	2	no	—
467	C₂×D₄	—	4	2	no	—
468	C₂×D₄	#41×C₂₃	1	2	yes	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	4	1	no	—
486	C₂²×C₄	—	4	1	no	—
487	C₂²×C₄	—	4	1	no	—
488	C₂²×C₄	—	4	2	no	—
489	C₂²×C₄	—	4	2	no	—
490	C₂²×C₄	—	4	2	no	—
491	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
492	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
493	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
494	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
495	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
497	C₂×C₈	—	2	1	no	—
498	C₂×C₈	—	2	1	no	—
499	C₂×C₈	—	2	1	no	—
500	C₂×C₈	—	2	1	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	1	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
508	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
509	C₄⋊C₄	—	2	1	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	—	2	1	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	1	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	1	no	—
522	C₄⋊C₄	—	4	2	no	—
523	C₄⋊C₄	—	4	2	no	—
524	C₄⋊C₄	—	4	2	no	—
525	C₄⋊C₄	—	4	2	no	—
526	C₄⋊C₄	—	4	2	no	—
527	C₄⋊C₄	#41×C₂₃	1	2	yes	—
528	C₂²⋊C₄	—	2	4	no	—
529	C₂²⋊C₄	—	4	2	no	—
530	C₂²⋊C₄	—	4	2	no	—
531	C₂²⋊C₄	—	4	2	no	—
532	C₂²⋊C₄	—	4	2	no	—
533	C₂²⋊C₄	—	4	2	no	—
534	C₂²⋊C₄	—	4	2	no	—
535	C₄²	—	2	2	no	—
536	C₄²	—	2	2	no	—
537	C₄²	—	2	1	no	—
538	C₄²	—	2	1	no	—
539	C₄²	—	4	2	no	—
540	C₄²	C₂×D₄×C₂₃	1	2	yes	—
541	C₄²	#41×C₂₃	1	2	yes	—
542	C₂³	—	2	1	no	—
543	C₂³	—	2	1	no	—
544	C₂³	—	2	1	no	—
545	C₂³	—	2	1	no	—
546	C₂³	—	2	1	no	—
547	C₂³	—	2	1	no	—
548	C₂³	—	2	2	no	—
549	C₂³	—	4	1	no	—
550	C₂³	—	4	1	no	—
551	C₂³	—	4	1	no	—
552	C₂³	—	4	1	no	—
553	C₂³	—	4	1	no	—
554	C₂³	—	4	1	no	—
555	C₂³	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
556	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
557	D₄	—	2	2	no	—
558	D₄	—	2	2	no	—
559	D₄	—	4	8	no	—
560	D₄	—	4	1	no	—
561	D₄	—	4	1	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	4	4	no	—
577	C₂×C₄	—	4	4	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	8	4	no	—
603	C₂×C₄	C₂₃×#121	1	2	yes	—
604	C₂×C₄	C₂₃×#121	1	2	yes	—
605	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
606	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
607	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
608	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
609	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
610	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
611	C₈	—	2	2	no	—
612	C₈	—	2	2	no	—
613	C₈	—	4	1	no	—
614	C₈	—	4	1	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	2	2	no	—
617	C₂²	—	2	2	no	—
618	C₂²	—	4	1	no	—
619	C₂²	—	4	1	no	—
620	C₂²	—	4	1	no	—
621	C₂²	—	4	1	no	—
622	C₂²	—	4	2	no	—
623	C₂²	—	4	2	no	—
624	C₂²	—	4	2	no	—
625	C₂²	—	8	1	no	—
626	C₂²	—	8	1	no	—
627	C₂²	—	8	1	no	—
628	C₂²	—	8	1	no	—
629	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
630	C₂²	C₂₃×#525	1	1	yes	characteristic
631	C₂²	C₂₃×#536	1	1	yes	characteristic
632	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
633	C₂²	C₂₃×#575	1	1	yes	characteristic
634	C₂²	C₂₃×#575	1	1	yes	characteristic
635	C₂²	C₂₃×#575	1	1	yes	characteristic
636	C₄	—	2	2	no	—
637	C₄	—	2	2	no	—
638	C₄	—	4	1	no	—
639	C₄	—	4	1	no	—
640	C₄	—	4	1	no	—
641	C₄	—	4	2	no	—
642	C₄	—	8	1	no	—
643	C₄	—	8	1	no	—
644	C₄	—	8	1	no	—
645	C₄	—	8	1	no	—
646	C₄	—	8	1	no	—
647	C₄	—	8	1	no	—
648	C₄	C₂₃×#547	1	1	yes	characteristic
649	C₄	C₂₃×#548	1	1	yes	characteristic
650	C₄	C₂₃×#552	1	1	yes	characteristic
651	C₄	C₂₃×#556	1	1	yes	characteristic
652	C₂	—	4	2	no	—
653	C₂	—	8	2	no	—
654	C₂	C₂₃×#2415	1	1	yes	characteristic
655	C₂	C₂₃×#3054	1	1	yes	characteristic
656	C₂	C₂₃×#3055	1	1	yes	characteristic
657	C₂	C₂₃×#3111	1	1	yes	characteristic
658	C₂	C₂₃×#3114	1	1	yes	characteristic
659	C₂	C₂₃×#3234	1	1	yes	characteristic
660	C₂	C₂₃×#3240	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#478	C₂	semidirect
2. C₂×C₂₃×#432	C₂	semidirect
3. C₂₃×#2851	C₂	semidirect
5. C₂₃×#2077	C₂	semidirect
6. C₂₃×#2075	C₂	semidirect
9. C₂₃×#1989	C₂	semidirect
10. C₂₃×#1949	C₂	semidirect
11. C₂₃×#1944	C₂	semidirect
12. C₂₃×#1943	C₂	semidirect
13. C₂₃×#1936	C₂	semidirect
14. C₂₃×#1936	C₂	semidirect
15. C₂₃×#1804	C₂	semidirect
109. C₂₃×C₄⋊C₈	D₄	semidirect
211. #24706	C₂₃	direct (N × H)
285. C₂×#478	C₂×C₂₃	semidirect
286. C₂×#432	C₂×C₂₃	semidirect
287. #2851	C₂×C₂₃	semidirect
289. #2077	C₂×C₂₃	semidirect
290. #2075	C₂×C₂₃	semidirect
293. #1989	C₂×C₂₃	semidirect
294. #1949	C₂×C₂₃	semidirect
295. #1944	C₂×C₂₃	semidirect
296. #1943	C₂×C₂₃	semidirect
297. #1936	C₂×C₂₃	semidirect
298. #1936	C₂×C₂₃	semidirect
299. #1804	C₂×C₂₃	semidirect
439. C₄⋊C₈	D₄×C₂₃	semidirect
457. C₂₃	#24706	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#478	2
C₂×C₂₃×#432	2
C₂₃×#2851	2
C₄×C₂₃×#121	2
C₂₃×#2077	2
C₂₃×#2075	2
C₂₃×#2034	2
C₂₃×#2033	2
C₂₃×#1989	2
C₂₃×#1949	2
C₂₃×#1944	2
C₂₃×#1943	2
C₂₃×#1936	2
C₂₃×#1804	2
#24706	23

Subgroups — C23×#24706 #19172332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#24706 #19172332