Subgroups — C₂₃×#24823 #19172449

Order 5888.

928 conjugacy classes of subgroups.

The subgroup lattice has 678 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×#442	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2566	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2085	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2084	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2067	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2065	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1992	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1940	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1879	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#1879	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1804	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1802	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1802	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#125	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	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
23	C₂₃×#478	—	2	4	no	—
24	C₂₃×#442	—	2	4	no	—
25	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×C₄⋊C₈	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×#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×#97	C₂²	1	1	yes	characteristic
38	C₂₃×#396	—	2	1	no	—
39	C₂₃×#396	—	2	1	no	—
40	C₂₃×#390	C₂²	1	1	yes	characteristic
41	C₂₃×#390	C₂²	1	1	yes	characteristic
42	C₂₃×#389	C₂²	1	1	yes	characteristic
43	C₂₃×#389	C₂²	1	1	yes	characteristic
44	C₂₃×#385	—	2	1	no	—
45	C₂₃×#385	—	2	1	no	—
46	C₂₃×#385	—	2	1	no	—
47	C₂₃×#385	—	2	1	no	—
48	C₂₃×#384	C₂²	1	1	yes	characteristic
49	C₂₃×#384	C₂²	1	1	yes	characteristic
50	C₂₃×#382	C₂²	1	1	yes	characteristic
51	C₂₃×#380	—	2	1	no	—
52	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
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×#336	C₂²	1	1	yes	characteristic
60	C₂₃×#336	C₂²	1	1	yes	characteristic
61	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₂₃	C₂³	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	—	2	1	no	—
67	C₂×D₈×C₂₃	—	2	2	no	—
68	C₂×D₈×C₂₃	—	2	2	no	—
69	C₂×D₈×C₂₃	Q₈	1	4	yes	—
70	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₂₃×#125	—	2	2	no	—
75	C₂₃×#125	—	2	2	no	—
76	C₄×D₄×C₂₃	—	2	2	no	—
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	—	4	1	no	—
82	C₄×D₄×C₂₃	—	4	1	no	—
83	C₄×D₄×C₂₃	D₄	1	4	yes	—
84	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	1	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	—	2	1	no	—
92	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
93	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
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×C₂²⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
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	—	2	1	no	—
109	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
112	C₂₃×#107	—	2	2	no	—
113	C₂₃×#106	—	2	2	no	—
114	C₂₃×#106	—	2	2	no	—
115	C₂₃×C₄⋊C₈	—	2	2	no	—
116	C₂₃×C₄⋊C₈	—	4	1	no	—
117	C₂₃×C₄⋊C₈	—	4	1	no	—
118	C₂₃×#102	—	2	2	no	—
119	C₂₃×#102	—	2	2	no	—
120	C₂₃×#102	—	2	2	no	—
121	C₂₃×#102	—	2	2	no	—
122	C₂₃×#102	—	4	1	no	—
123	C₂₃×#102	—	4	1	no	—
124	C₂₃×#97	—	2	2	no	—
125	C₂₃×#95	—	2	1	no	—
126	C₂₃×#95	—	2	1	no	—
127	C₂₃×#95	—	2	1	no	—
128	C₂₃×#95	—	2	1	no	—
129	C₂₃×#95	D₄	1	1	yes	characteristic
130	C₂₃×#95	D₄	1	1	yes	characteristic
131	C₂⁴×C₂₃	—	2	1	no	—
132	C₂⁴×C₂₃	—	2	1	no	—
133	C₂×D₄×C₂₃	—	2	1	no	—
134	C₂×D₄×C₂₃	—	2	1	no	—
135	C₂×D₄×C₂₃	—	2	1	no	—
136	C₂×D₄×C₂₃	—	2	1	no	—
137	C₂×D₄×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	C₂×D₄	1	2	yes	—
142	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
143	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
144	C₂×D₄×C₂₃	#41	1	2	yes	—
145	C₂²×C₄×C₂₃	—	2	1	no	—
146	C₂²×C₄×C₂₃	—	2	4	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	—	4	2	no	—
153	C₂²×C₄×C₂₃	—	4	2	no	—
154	C₂²×C₄×C₂₃	—	4	2	no	—
155	C₂²×C₄×C₂₃	—	4	2	no	—
156	C₂²×C₄×C₂₃	—	4	2	no	—
157	C₂²×C₄×C₂₃	—	4	2	no	—
158	C₂²×C₄×C₂₃	—	4	2	no	—
159	C₂²×C₄×C₂₃	—	8	1	no	—
160	C₂²×C₄×C₂₃	—	8	1	no	—
161	C₂²×C₄×C₂₃	—	8	1	no	—
162	C₂²×C₄×C₂₃	—	8	1	no	—
163	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
170	D₈×C₂₃	—	2	8	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	—	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	C₂×Q₈	1	2	yes	—
183	C₄⋊C₄×C₂₃	—	2	2	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	—	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	—	4	1	no	—
196	C₄⋊C₄×C₂₃	—	4	2	no	—
197	C₄⋊C₄×C₂₃	—	4	2	no	—
198	C₄⋊C₄×C₂₃	—	4	2	no	—
199	C₄⋊C₄×C₂₃	—	4	2	no	—
200	C₄⋊C₄×C₂₃	—	4	2	no	—
201	C₄⋊C₄×C₂₃	—	8	1	no	—
202	C₄⋊C₄×C₂₃	—	8	1	no	—
203	C₄⋊C₄×C₂₃	—	8	1	no	—
204	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
205	C₂²⋊C₄×C₂₃	—	2	4	no	—
206	C₂²⋊C₄×C₂₃	—	4	2	no	—
207	C₂²⋊C₄×C₂₃	—	4	2	no	—
208	C₂²⋊C₄×C₂₃	—	4	2	no	—
209	C₂²⋊C₄×C₂₃	—	4	2	no	—
210	C₂²⋊C₄×C₂₃	—	4	2	no	—
211	C₂²⋊C₄×C₂₃	—	8	1	no	—
212	C₂²⋊C₄×C₂₃	—	8	1	no	—
213	C₄²×C₂₃	—	2	1	no	—
214	C₄²×C₂₃	—	2	1	no	—
215	C₄²×C₂₃	—	2	1	no	—
216	C₄²×C₂₃	—	2	1	no	—
217	C₄²×C₂₃	C₂×D₄	1	2	yes	—
218	C₄²×C₂₃	#41	1	2	yes	—
219	#24823	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂³×C₂₃	—	2	2	no	—
221	C₂³×C₂₃	—	2	2	no	—
222	C₂³×C₂₃	—	2	2	no	—
223	C₂³×C₂₃	—	2	2	no	—
224	C₂³×C₂₃	—	4	1	no	—
225	C₂³×C₂₃	—	4	1	no	—
226	C₂³×C₂₃	—	4	1	no	—
227	C₂³×C₂₃	—	4	1	no	—
228	C₂³×C₂₃	—	4	1	no	—
229	C₂³×C₂₃	—	4	1	no	—
230	C₂³×C₂₃	—	4	1	no	—
231	C₂³×C₂₃	—	4	1	no	—
232	C₂³×C₂₃	—	4	1	no	—
233	C₂³×C₂₃	—	4	1	no	—
234	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
235	D₄×C₂₃	—	2	2	no	—
236	D₄×C₂₃	—	2	2	no	—
237	D₄×C₂₃	—	2	2	no	—
238	D₄×C₂₃	—	2	2	no	—
239	D₄×C₂₃	—	2	2	no	—
240	D₄×C₂₃	—	4	1	no	—
241	D₄×C₂₃	—	4	1	no	—
242	D₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
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	1	no	—
249	C₂×C₄×C₂₃	—	2	2	no	—
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	1	no	—
257	C₂×C₄×C₂₃	—	2	1	no	—
258	C₂×C₄×C₂₃	—	2	1	no	—
259	C₂×C₄×C₂₃	—	4	8	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	—	8	4	no	—
278	C₂×C₄×C₂₃	—	8	4	no	—
279	C₂×C₄×C₂₃	—	8	4	no	—
280	C₂×C₄×C₂₃	#121	1	2	yes	—
281	C₂×C₄×C₂₃	#121	1	2	yes	—
282	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
283	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
285	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
286	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
287	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
288	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
289	C₈×C₂₃	—	2	2	no	—
290	C₈×C₂₃	—	4	1	no	—
291	C₈×C₂₃	—	4	1	no	—
292	C₈×C₂₃	—	4	1	no	—
293	C₂×#478	C₂×C₂₃	1	1	yes	characteristic, complemented
294	C₂×#442	C₂×C₂₃	1	1	yes	characteristic
295	#2566	C₂×C₂₃	1	1	yes	characteristic, complemented
296	D₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
297	#2085	C₂×C₂₃	1	1	yes	characteristic, complemented
298	#2084	C₂×C₂₃	1	1	yes	characteristic, complemented
299	#2067	C₂×C₂₃	1	1	yes	characteristic, complemented
300	#2065	C₂×C₂₃	1	1	yes	characteristic, complemented
301	#1992	C₂×C₂₃	1	1	yes	characteristic, complemented
302	#1940	C₂×C₂₃	1	1	yes	characteristic, complemented
303	#1879	C₂×C₂₃	1	1	yes	characteristic
304	#1879	C₂×C₂₃	1	1	yes	characteristic
305	#1804	C₂×C₂₃	1	1	yes	characteristic, complemented
306	#1802	C₂×C₂₃	1	1	yes	characteristic, complemented
307	#1802	C₂×C₂₃	1	1	yes	characteristic, complemented
308	C₂²×C₂₃	—	2	4	no	—
309	C₂²×C₂₃	—	4	2	no	—
310	C₂²×C₂₃	—	4	2	no	—
311	C₂²×C₂₃	—	4	2	no	—
312	C₂²×C₂₃	—	4	2	no	—
313	C₂²×C₂₃	—	4	2	no	—
314	C₂²×C₂₃	—	4	2	no	—
315	C₂²×C₂₃	—	4	2	no	—
316	C₂²×C₂₃	—	4	2	no	—
317	C₂²×C₂₃	—	4	2	no	—
318	C₂²×C₂₃	—	8	1	no	—
319	C₂²×C₂₃	—	8	1	no	—
320	C₂²×C₂₃	—	8	1	no	—
321	C₂²×C₂₃	—	8	1	no	—
322	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
323	C₂²×C₂₃	#546	1	1	yes	characteristic
324	C₂²×C₂₃	D₄×Q₈	1	1	yes	characteristic
325	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
326	C₂²×C₂₃	#575	1	1	yes	characteristic
327	C₂²×C₂₃	#576	1	1	yes	characteristic
328	C₂²×C₂₃	#577	1	1	yes	characteristic
329	C₄×C₂₃	—	2	2	no	—
330	C₄×C₂₃	—	2	2	no	—
331	C₄×C₂₃	—	2	2	no	—
332	C₄×C₂₃	—	4	2	no	—
333	C₄×C₂₃	—	4	1	no	—
334	C₄×C₂₃	—	4	1	no	—
335	C₄×C₂₃	—	4	1	no	—
336	C₄×C₂₃	—	8	1	no	—
337	C₄×C₂₃	—	8	1	no	—
338	C₄×C₂₃	—	8	1	no	—
339	C₄×C₂₃	—	8	1	no	—
340	C₄×C₂₃	—	8	1	no	—
341	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
342	C₄×C₂₃	#537	1	1	yes	characteristic
343	C₄×C₂₃	#552	1	1	yes	characteristic
344	C₄×C₂₃	#554	1	1	yes	characteristic
345	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
347	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
348	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
349	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
351	C₂²×C₄⋊C₄	—	2	1	no	—
352	#478	—	2	4	no	—
353	#442	—	2	4	no	—
354	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
356	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
357	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
358	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
359	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
360	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
361	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
362	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
363	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
364	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
365	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
366	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
367	#396	—	2	1	no	—
368	#396	—	2	1	no	—
369	#390	C₂²×C₂₃	1	1	yes	characteristic
370	#390	C₂²×C₂₃	1	1	yes	characteristic
371	#389	C₂²×C₂₃	1	1	yes	characteristic
372	#389	C₂²×C₂₃	1	1	yes	characteristic
373	#385	—	2	1	no	—
374	#385	—	2	1	no	—
375	#385	—	2	1	no	—
376	#385	—	2	1	no	—
377	#384	C₂²×C₂₃	1	1	yes	characteristic
378	#384	C₂²×C₂₃	1	1	yes	characteristic
379	#382	C₂²×C₂₃	1	1	yes	characteristic
380	#380	—	2	1	no	—
381	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
382	#340	C₂²×C₂₃	1	1	yes	characteristic
383	#340	C₂²×C₂₃	1	1	yes	characteristic
384	#340	C₂²×C₂₃	1	1	yes	characteristic
385	#340	C₂²×C₂₃	1	1	yes	characteristic
386	#340	C₂²×C₂₃	1	1	yes	characteristic
387	#340	C₂²×C₂₃	1	1	yes	characteristic
388	#336	C₂²×C₂₃	1	1	yes	characteristic
389	#336	C₂²×C₂₃	1	1	yes	characteristic
390	C₂×C₂₃	—	4	4	no	—
391	C₂×C₂₃	—	8	2	no	—
392	C₂×C₂₃	#2417	1	1	yes	characteristic
393	C₂×C₂₃	#3036	1	1	yes	characteristic
394	C₂×C₂₃	#3045	1	1	yes	characteristic
395	C₂×C₂₃	#3307	1	1	yes	characteristic
396	C₂×C₂₃	#3318	1	1	yes	characteristic
397	C₂×C₂₃	#3384	1	1	yes	characteristic
398	C₂×C₂₃	#3389	1	1	yes	characteristic
399	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
400	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
401	C₂³×C₄	—	2	1	no	—
402	C₂³×C₄	—	2	1	no	—
403	C₂³×C₄	—	2	1	no	—
404	C₂³×C₄	—	2	1	no	—
405	C₂×D₈	—	2	2	no	—
406	C₂×D₈	—	2	2	no	—
407	C₂×D₈	Q₈×C₂₃	1	4	yes	—
408	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
409	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
410	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
411	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
412	#125	—	2	2	no	—
413	#125	—	2	2	no	—
414	C₄×D₄	—	2	2	no	—
415	C₄×D₄	—	2	2	no	—
416	C₄×D₄	—	2	2	no	—
417	C₄×D₄	—	2	2	no	—
418	C₄×D₄	—	2	2	no	—
419	C₄×D₄	—	4	1	no	—
420	C₄×D₄	—	4	1	no	—
421	C₄×D₄	D₄×C₂₃	1	4	yes	—
422	C₂×C₄⋊C₄	—	2	1	no	—
423	C₂×C₄⋊C₄	—	2	1	no	—
424	C₂×C₄⋊C₄	—	2	1	no	—
425	C₂×C₄⋊C₄	—	2	1	no	—
426	C₂×C₄⋊C₄	—	2	1	no	—
427	C₂×C₄⋊C₄	—	2	1	no	—
428	C₂×C₄⋊C₄	—	2	1	no	—
429	C₂×C₄⋊C₄	—	2	1	no	—
430	C₂×C₄⋊C₄	—	4	4	no	—
431	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
432	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
433	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
434	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
435	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
436	C₂×C₄⋊C₄	C₂³×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×C₂²⋊C₄	—	2	1	no	—
440	C₂×C₂²⋊C₄	—	2	1	no	—
441	C₂×C₂²⋊C₄	—	2	1	no	—
442	C₂×C₂²⋊C₄	—	2	1	no	—
443	C₂×C₂²⋊C₄	—	2	1	no	—
444	C₂×C₂²⋊C₄	—	2	1	no	—
445	C₂×C₂²⋊C₄	—	2	1	no	—
446	C₂×C₂²⋊C₄	—	2	1	no	—
447	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
448	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
449	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
450	#107	—	2	2	no	—
451	#106	—	2	2	no	—
452	#106	—	2	2	no	—
453	C₄⋊C₈	—	2	2	no	—
454	C₄⋊C₈	—	4	1	no	—
455	C₄⋊C₈	—	4	1	no	—
456	#102	—	2	2	no	—
457	#102	—	2	2	no	—
458	#102	—	2	2	no	—
459	#102	—	2	2	no	—
460	#102	—	4	1	no	—
461	#102	—	4	1	no	—
462	#97	—	2	2	no	—
463	#95	—	2	1	no	—
464	#95	—	2	1	no	—
465	#95	—	2	1	no	—
466	#95	—	2	1	no	—
467	#95	D₄×C₂₃	1	1	yes	characteristic
468	#95	D₄×C₂₃	1	1	yes	characteristic
469	C₂₃	#24823	1	1	yes	characteristic, direct product factor, Sylow
470	C₂⁴	—	2	1	no	—
471	C₂⁴	—	2	1	no	—
472	C₂×D₄	—	2	1	no	—
473	C₂×D₄	—	2	1	no	—
474	C₂×D₄	—	2	1	no	—
475	C₂×D₄	—	2	1	no	—
476	C₂×D₄	—	2	1	no	—
477	C₂×D₄	—	2	1	no	—
478	C₂×D₄	—	2	1	no	—
479	C₂×D₄	—	2	1	no	—
480	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
481	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
482	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
483	C₂×D₄	#41×C₂₃	1	2	yes	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	4	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	—	4	2	no	—
492	C₂²×C₄	—	4	2	no	—
493	C₂²×C₄	—	4	2	no	—
494	C₂²×C₄	—	4	2	no	—
495	C₂²×C₄	—	4	2	no	—
496	C₂²×C₄	—	4	2	no	—
497	C₂²×C₄	—	4	2	no	—
498	C₂²×C₄	—	8	1	no	—
499	C₂²×C₄	—	8	1	no	—
500	C₂²×C₄	—	8	1	no	—
501	C₂²×C₄	—	8	1	no	—
502	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
503	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
504	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
505	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
506	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
507	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
508	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
509	D₈	—	2	8	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	—	2	1	no	—
512	C₂×C₈	—	2	1	no	—
513	C₂×C₈	—	2	1	no	—
514	C₂×C₈	—	2	1	no	—
515	C₂×C₈	—	2	1	no	—
516	C₂×C₈	—	2	1	no	—
517	C₂×C₈	—	2	1	no	—
518	C₂×C₈	—	2	1	no	—
519	C₂×C₈	—	2	1	no	—
520	C₂×C₈	—	2	1	no	—
521	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
522	C₄⋊C₄	—	2	2	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	1	no	—
529	C₄⋊C₄	—	2	1	no	—
530	C₄⋊C₄	—	2	1	no	—
531	C₄⋊C₄	—	2	1	no	—
532	C₄⋊C₄	—	2	1	no	—
533	C₄⋊C₄	—	2	1	no	—
534	C₄⋊C₄	—	4	1	no	—
535	C₄⋊C₄	—	4	2	no	—
536	C₄⋊C₄	—	4	2	no	—
537	C₄⋊C₄	—	4	2	no	—
538	C₄⋊C₄	—	4	2	no	—
539	C₄⋊C₄	—	4	2	no	—
540	C₄⋊C₄	—	8	1	no	—
541	C₄⋊C₄	—	8	1	no	—
542	C₄⋊C₄	—	8	1	no	—
543	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
544	C₂²⋊C₄	—	2	4	no	—
545	C₂²⋊C₄	—	4	2	no	—
546	C₂²⋊C₄	—	4	2	no	—
547	C₂²⋊C₄	—	4	2	no	—
548	C₂²⋊C₄	—	4	2	no	—
549	C₂²⋊C₄	—	4	2	no	—
550	C₂²⋊C₄	—	8	1	no	—
551	C₂²⋊C₄	—	8	1	no	—
552	C₄²	—	2	1	no	—
553	C₄²	—	2	1	no	—
554	C₄²	—	2	1	no	—
555	C₄²	—	2	1	no	—
556	C₄²	C₂×D₄×C₂₃	1	2	yes	—
557	C₄²	#41×C₂₃	1	2	yes	—
558	C₂³	—	2	2	no	—
559	C₂³	—	2	2	no	—
560	C₂³	—	2	2	no	—
561	C₂³	—	2	2	no	—
562	C₂³	—	4	1	no	—
563	C₂³	—	4	1	no	—
564	C₂³	—	4	1	no	—
565	C₂³	—	4	1	no	—
566	C₂³	—	4	1	no	—
567	C₂³	—	4	1	no	—
568	C₂³	—	4	1	no	—
569	C₂³	—	4	1	no	—
570	C₂³	—	4	1	no	—
571	C₂³	—	4	1	no	—
572	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
573	D₄	—	2	2	no	—
574	D₄	—	2	2	no	—
575	D₄	—	2	2	no	—
576	D₄	—	2	2	no	—
577	D₄	—	2	2	no	—
578	D₄	—	4	1	no	—
579	D₄	—	4	1	no	—
580	D₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
581	C₂×C₄	—	2	1	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	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	4	8	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	—	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	—	8	4	no	—
616	C₂×C₄	—	8	4	no	—
617	C₂×C₄	—	8	4	no	—
618	C₂×C₄	C₂₃×#121	1	2	yes	—
619	C₂×C₄	C₂₃×#121	1	2	yes	—
620	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
621	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
622	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
623	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
624	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
625	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
626	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
627	C₈	—	2	2	no	—
628	C₈	—	4	1	no	—
629	C₈	—	4	1	no	—
630	C₈	—	4	1	no	—
631	C₂²	—	2	4	no	—
632	C₂²	—	4	2	no	—
633	C₂²	—	4	2	no	—
634	C₂²	—	4	2	no	—
635	C₂²	—	4	2	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	4	2	no	—
638	C₂²	—	4	2	no	—
639	C₂²	—	4	2	no	—
640	C₂²	—	4	2	no	—
641	C₂²	—	8	1	no	—
642	C₂²	—	8	1	no	—
643	C₂²	—	8	1	no	—
644	C₂²	—	8	1	no	—
645	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
646	C₂²	C₂₃×#546	1	1	yes	characteristic
647	C₂²	D₄×Q₈×C₂₃	1	1	yes	characteristic
648	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
649	C₂²	C₂₃×#575	1	1	yes	characteristic
650	C₂²	C₂₃×#576	1	1	yes	characteristic
651	C₂²	C₂₃×#577	1	1	yes	characteristic
652	C₄	—	2	2	no	—
653	C₄	—	2	2	no	—
654	C₄	—	2	2	no	—
655	C₄	—	4	2	no	—
656	C₄	—	4	1	no	—
657	C₄	—	4	1	no	—
658	C₄	—	4	1	no	—
659	C₄	—	8	1	no	—
660	C₄	—	8	1	no	—
661	C₄	—	8	1	no	—
662	C₄	—	8	1	no	—
663	C₄	—	8	1	no	—
664	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
665	C₄	C₂₃×#537	1	1	yes	characteristic
666	C₄	C₂₃×#552	1	1	yes	characteristic
667	C₄	C₂₃×#554	1	1	yes	characteristic
668	C₂	—	4	4	no	—
669	C₂	—	8	2	no	—
670	C₂	C₂₃×#2417	1	1	yes	characteristic
671	C₂	C₂₃×#3036	1	1	yes	characteristic
672	C₂	C₂₃×#3045	1	1	yes	characteristic
673	C₂	C₂₃×#3307	1	1	yes	characteristic
674	C₂	C₂₃×#3318	1	1	yes	characteristic
675	C₂	C₂₃×#3384	1	1	yes	characteristic
676	C₂	C₂₃×#3389	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₂²×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
1. C₂×C₂₃×#478	C₂	semidirect
3. C₂₃×#2566	C₂	semidirect
4. D₄×C₄⋊C₄×C₂₃	C₂	semidirect
5. C₂₃×#2085	C₂	semidirect
6. C₂₃×#2084	C₂	semidirect
7. C₂₃×#2067	C₂	semidirect
8. C₂₃×#2065	C₂	semidirect
9. C₂₃×#1992	C₂	semidirect
10. C₂₃×#1940	C₂	semidirect
13. C₂₃×#1804	C₂	semidirect
14. C₂₃×#1802	C₂	semidirect
15. C₂₃×#1802	C₂	semidirect
219. #24823	C₂₃	direct (N × H)
293. C₂×#478	C₂×C₂₃	semidirect
295. #2566	C₂×C₂₃	semidirect
296. D₄×C₄⋊C₄	C₂×C₂₃	semidirect
297. #2085	C₂×C₂₃	semidirect
298. #2084	C₂×C₂₃	semidirect
299. #2067	C₂×C₂₃	semidirect
300. #2065	C₂×C₂₃	semidirect
301. #1992	C₂×C₂₃	semidirect
302. #1940	C₂×C₂₃	semidirect
305. #1804	C₂×C₂₃	semidirect
306. #1802	C₂×C₂₃	semidirect
307. #1802	C₂×C₂₃	semidirect
469. C₂₃	#24823	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×#442	2
C₂₃×#2566	2
D₄×C₄⋊C₄×C₂₃	2
C₂₃×#2085	2
C₂₃×#2084	2
C₂₃×#2067	2
C₂₃×#2065	2
C₂₃×#1992	2
C₂₃×#1940	2
C₂₃×#1879	2
C₂₃×#1804	2
C₂₃×#1802	2
#24823	23

Subgroups — C23×#24823 #19172449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#24823 #19172449