Subgroups — C₂₃×#22114 #19169740

Order 5888.

796 conjugacy classes of subgroups.

The subgroup lattice has 592 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
2	C₄×C₂₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2293	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1989	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1949	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1942	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1941	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1934	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1934	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#1926	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1925	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#1921	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1920	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1804	C₂	1	1	yes	maximal, characteristic
15	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#117	—	2	1	no	—
19	C₂₃×#478	C₄	1	8	yes	complemented
20	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#390	C₂²	1	1	yes	characteristic
33	C₂₃×#390	C₂²	1	1	yes	characteristic
34	C₂₃×#389	C₂²	1	1	yes	characteristic
35	C₂₃×#389	C₂²	1	1	yes	characteristic
36	C₂₃×#389	C₂²	1	1	yes	characteristic
37	C₂₃×#389	C₂²	1	1	yes	characteristic
38	C₂₃×#388	—	2	1	no	—
39	C₂₃×#388	—	2	1	no	—
40	C₂₃×#387	—	2	1	no	—
41	C₂₃×#387	—	2	1	no	—
42	C₂₃×#382	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
44	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
45	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
46	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
47	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
48	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
49	C₄³×C₂₃	C₂²	1	1	yes	characteristic
50	C₂₃×#340	C₂²	1	1	yes	characteristic
51	C₂₃×#340	C₂²	1	1	yes	characteristic
52	C₂₃×#340	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×#336	C₂²	1	1	yes	characteristic
59	C₂₃×#336	C₂²	1	1	yes	characteristic
60	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₂₃	—	2	1	no	—
62	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂₃×#125	—	2	2	no	—
67	C₂₃×#125	C₂×C₄	1	4	yes	—
68	C₄×D₄×C₂₃	—	2	2	no	—
69	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
70	C₂₃×#117	—	4	4	no	—
71	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
72	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
73	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
74	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
75	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
84	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
89	C₂×C₄²×C₂₃	—	2	1	no	—
90	C₂×C₄²×C₂₃	—	2	1	no	—
91	C₂×C₄²×C₂₃	—	2	1	no	—
92	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
93	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
96	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
99	C₂₃×#107	—	4	1	no	—
100	C₂₃×#107	C₂×C₄	1	4	yes	—
101	C₂₃×#106	—	4	1	no	—
102	C₂₃×#106	C₂×C₄	1	4	yes	—
103	C₂₃×C₄⋊C₈	—	2	2	no	—
104	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
105	C₂₃×#102	—	4	1	no	—
106	C₂₃×#102	—	4	1	no	—
107	C₂₃×#102	C₂×C₄	1	4	yes	—
108	C₂₃×#102	C₂×C₄	1	4	yes	—
109	C₄×C₈×C₂₃	—	4	1	no	—
110	C₄×C₈×C₂₃	—	4	1	no	—
111	C₂₃×#95	—	2	1	no	—
112	C₂₃×#95	—	2	1	no	—
113	C₂₃×#95	—	2	1	no	—
114	C₂₃×#95	—	2	1	no	—
115	C₂₃×#95	—	2	1	no	—
116	C₂₃×#95	—	2	1	no	—
117	C₂₃×#95	—	2	1	no	—
118	C₂₃×#95	—	2	1	no	—
119	C₂₃×#95	—	2	1	no	—
120	C₂⁴×C₂₃	—	2	1	no	—
121	C₂×D₄×C₂₃	—	2	1	no	—
122	C₂×D₄×C₂₃	—	2	1	no	—
123	C₂×D₄×C₂₃	—	2	1	no	—
124	C₂×D₄×C₂₃	—	2	1	no	—
125	C₂×D₄×C₂₃	—	2	1	no	—
126	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
127	C₂²×C₄×C₂₃	—	2	1	no	—
128	C₂²×C₄×C₂₃	—	2	1	no	—
129	C₂²×C₄×C₂₃	—	2	1	no	—
130	C₂²×C₄×C₂₃	—	2	4	no	—
131	C₂²×C₄×C₂₃	—	2	1	no	—
132	C₂²×C₄×C₂₃	—	2	1	no	—
133	C₂²×C₄×C₂₃	—	2	1	no	—
134	C₂²×C₄×C₂₃	—	2	1	no	—
135	C₂²×C₄×C₂₃	—	4	2	no	—
136	C₂²×C₄×C₂₃	—	4	2	no	—
137	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
144	C₂×C₈×C₂₃	—	2	1	no	—
145	C₂×C₈×C₂₃	—	2	1	no	—
146	C₂×C₈×C₂₃	—	2	1	no	—
147	C₂×C₈×C₂₃	—	2	1	no	—
148	C₂×C₈×C₂₃	—	2	1	no	—
149	C₂×C₈×C₂₃	—	2	1	no	—
150	C₂×C₈×C₂₃	—	2	1	no	—
151	C₂×C₈×C₂₃	—	2	1	no	—
152	C₂×C₈×C₂₃	—	2	1	no	—
153	C₂×C₈×C₂₃	—	2	1	no	—
154	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
155	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
156	C₄⋊C₄×C₂₃	—	2	4	no	—
157	C₄⋊C₄×C₂₃	—	2	4	no	—
158	C₄⋊C₄×C₂₃	—	2	1	no	—
159	C₄⋊C₄×C₂₃	—	2	1	no	—
160	C₄⋊C₄×C₂₃	—	2	1	no	—
161	C₄⋊C₄×C₂₃	—	2	1	no	—
162	C₄⋊C₄×C₂₃	—	2	1	no	—
163	C₄⋊C₄×C₂₃	—	2	1	no	—
164	C₄⋊C₄×C₂₃	—	2	1	no	—
165	C₄⋊C₄×C₂₃	—	2	1	no	—
166	C₄⋊C₄×C₂₃	—	2	1	no	—
167	C₄⋊C₄×C₂₃	—	2	1	no	—
168	C₄⋊C₄×C₂₃	—	2	1	no	—
169	C₄⋊C₄×C₂₃	—	4	2	no	—
170	C₄⋊C₄×C₂₃	—	4	2	no	—
171	C₄⋊C₄×C₂₃	—	4	2	no	—
172	C₄⋊C₄×C₂₃	—	4	2	no	—
173	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
174	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
175	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
176	C₂²⋊C₄×C₂₃	—	2	4	no	—
177	C₂²⋊C₄×C₂₃	—	4	2	no	—
178	C₂²⋊C₄×C₂₃	—	4	2	no	—
179	C₂²⋊C₄×C₂₃	—	4	2	no	—
180	C₂²⋊C₄×C₂₃	—	4	2	no	—
181	C₄²×C₂₃	—	2	2	no	—
182	C₄²×C₂₃	—	2	2	no	—
183	C₄²×C₂₃	—	2	2	no	—
184	C₄²×C₂₃	—	2	1	no	—
185	C₄²×C₂₃	—	2	1	no	—
186	C₄²×C₂₃	—	2	1	no	—
187	C₄²×C₂₃	—	2	1	no	—
188	C₄²×C₂₃	—	4	1	no	—
189	C₄²×C₂₃	—	4	2	no	—
190	C₄²×C₂₃	—	4	2	no	—
191	C₄²×C₂₃	—	4	2	no	—
192	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
193	C₄²×C₂₃	#41	1	2	yes	—
194	#22114	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂³×C₂₃	—	2	2	no	—
196	C₂³×C₂₃	—	4	1	no	—
197	C₂³×C₂₃	—	4	1	no	—
198	C₂³×C₂₃	—	4	1	no	—
199	C₂³×C₂₃	—	4	1	no	—
200	C₂³×C₂₃	—	4	1	no	—
201	C₂³×C₂₃	—	4	1	no	—
202	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
203	D₄×C₂₃	—	2	2	no	—
204	D₄×C₂₃	—	2	2	no	—
205	D₄×C₂₃	—	4	1	no	—
206	D₄×C₂₃	—	4	1	no	—
207	C₂×C₄×C₂₃	—	2	1	no	—
208	C₂×C₄×C₂₃	—	2	1	no	—
209	C₂×C₄×C₂₃	—	2	1	no	—
210	C₂×C₄×C₂₃	—	2	1	no	—
211	C₂×C₄×C₂₃	—	2	1	no	—
212	C₂×C₄×C₂₃	—	2	1	no	—
213	C₂×C₄×C₂₃	—	2	1	no	—
214	C₂×C₄×C₂₃	—	2	1	no	—
215	C₂×C₄×C₂₃	—	2	1	no	—
216	C₂×C₄×C₂₃	—	2	1	no	—
217	C₂×C₄×C₂₃	—	2	2	no	—
218	C₂×C₄×C₂₃	—	2	2	no	—
219	C₂×C₄×C₂₃	—	2	2	no	—
220	C₂×C₄×C₂₃	—	2	1	no	—
221	C₂×C₄×C₂₃	—	2	1	no	—
222	C₂×C₄×C₂₃	—	2	1	no	—
223	C₂×C₄×C₂₃	—	4	8	no	—
224	C₂×C₄×C₂₃	—	4	1	no	—
225	C₂×C₄×C₂₃	—	4	1	no	—
226	C₂×C₄×C₂₃	—	4	1	no	—
227	C₂×C₄×C₂₃	—	4	1	no	—
228	C₂×C₄×C₂₃	—	4	1	no	—
229	C₂×C₄×C₂₃	—	4	1	no	—
230	C₂×C₄×C₂₃	—	4	1	no	—
231	C₂×C₄×C₂₃	—	4	1	no	—
232	C₂×C₄×C₂₃	—	4	1	no	—
233	C₂×C₄×C₂₃	—	4	1	no	—
234	C₂×C₄×C₂₃	—	4	1	no	—
235	C₂×C₄×C₂₃	—	4	1	no	—
236	C₂×C₄×C₂₃	—	4	1	no	—
237	C₂×C₄×C₂₃	—	4	1	no	—
238	C₂×C₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	4	1	no	—
240	C₂×C₄×C₂₃	—	4	1	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
243	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
244	C₂×C₄×C₂₃	#135	1	2	yes	—
245	C₂×C₄×C₂₃	#135	1	2	yes	—
246	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
247	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
248	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
249	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
250	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
251	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
252	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
253	C₈×C₂₃	—	2	2	no	—
254	C₈×C₂₃	—	2	2	no	—
255	C₈×C₂₃	—	4	1	no	—
256	C₈×C₂₃	—	4	1	no	—
257	C₂×#478	C₂×C₂₃	1	1	yes	characteristic
258	C₄×#125	C₂×C₂₃	1	1	yes	characteristic, complemented
259	#2293	C₂×C₂₃	1	1	yes	characteristic
260	#1989	C₂×C₂₃	1	1	yes	characteristic, complemented
261	#1949	C₂×C₂₃	1	1	yes	characteristic, complemented
262	#1942	C₂×C₂₃	1	1	yes	characteristic, complemented
263	#1941	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#1934	C₂×C₂₃	1	1	yes	characteristic
265	#1934	C₂×C₂₃	1	1	yes	characteristic
266	#1926	C₂×C₂₃	1	1	yes	characteristic
267	#1925	C₂×C₂₃	1	1	yes	characteristic
268	#1921	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1920	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#1804	C₂×C₂₃	1	1	yes	characteristic
271	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
272	C₂²×C₂₃	—	4	2	no	—
273	C₂²×C₂₃	—	4	2	no	—
274	C₂²×C₂₃	—	4	2	no	—
275	C₂²×C₂₃	—	8	1	no	—
276	C₂²×C₂₃	—	8	1	no	—
277	C₂²×C₂₃	—	8	1	no	—
278	C₂²×C₂₃	—	8	1	no	—
279	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
280	C₂²×C₂₃	#536	1	1	yes	characteristic
281	C₂²×C₂₃	#537	1	1	yes	characteristic
282	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
283	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
284	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
285	C₂²×C₂₃	#575	1	1	yes	characteristic
286	C₄×C₂₃	—	2	2	no	—
287	C₄×C₂₃	—	2	2	no	—
288	C₄×C₂₃	—	2	2	no	—
289	C₄×C₂₃	—	2	2	no	—
290	C₄×C₂₃	—	2	2	no	—
291	C₄×C₂₃	—	4	2	no	—
292	C₄×C₂₃	—	4	2	no	—
293	C₄×C₂₃	—	4	2	no	—
294	C₄×C₂₃	—	4	1	no	—
295	C₄×C₂₃	—	8	1	no	—
296	C₄×C₂₃	—	8	1	no	—
297	C₄×C₂₃	—	8	1	no	—
298	C₄×C₂₃	—	8	1	no	—
299	C₄×C₂₃	#519	1	1	yes	characteristic
300	C₄×C₂₃	#520	1	1	yes	characteristic
301	C₄×C₂₃	#541	1	1	yes	characteristic
302	C₄×C₂₃	#542	1	1	yes	characteristic
303	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
304	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
305	C₂×#117	—	2	1	no	—
306	#478	C₄×C₂₃	1	8	yes	complemented
307	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
308	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
309	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
310	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
311	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
312	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
313	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
314	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
315	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
316	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
317	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
318	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
319	#390	C₂²×C₂₃	1	1	yes	characteristic
320	#390	C₂²×C₂₃	1	1	yes	characteristic
321	#389	C₂²×C₂₃	1	1	yes	characteristic
322	#389	C₂²×C₂₃	1	1	yes	characteristic
323	#389	C₂²×C₂₃	1	1	yes	characteristic
324	#389	C₂²×C₂₃	1	1	yes	characteristic
325	#388	—	2	1	no	—
326	#388	—	2	1	no	—
327	#387	—	2	1	no	—
328	#387	—	2	1	no	—
329	#382	C₂²×C₂₃	1	1	yes	characteristic
330	C₄×C₄⋊C₄	—	2	1	no	—
331	C₄×C₄⋊C₄	—	2	1	no	—
332	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
333	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
334	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
335	C₄×C₂²⋊C₄	—	2	1	no	—
336	C₄³	C₂²×C₂₃	1	1	yes	characteristic
337	#340	C₂²×C₂₃	1	1	yes	characteristic
338	#340	C₂²×C₂₃	1	1	yes	characteristic
339	#340	C₂²×C₂₃	1	1	yes	characteristic
340	#340	C₂²×C₂₃	1	1	yes	characteristic
341	#340	C₂²×C₂₃	1	1	yes	characteristic
342	#340	C₂²×C₂₃	1	1	yes	characteristic
343	#340	C₂²×C₂₃	1	1	yes	characteristic
344	#340	C₂²×C₂₃	1	1	yes	characteristic
345	#336	C₂²×C₂₃	1	1	yes	characteristic
346	#336	C₂²×C₂₃	1	1	yes	characteristic
347	C₂×C₂₃	—	8	2	no	—
348	C₂×C₂₃	#2323	1	1	yes	characteristic
349	C₂×C₂₃	C₄×#135	1	1	yes	characteristic
350	C₂×C₂₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
351	C₂×C₂₃	#3228	1	1	yes	characteristic
352	C₂×C₂₃	#3232	1	1	yes	characteristic
353	C₂×C₂₃	#3247	1	1	yes	characteristic
354	C₂×C₂₃	#3250	1	1	yes	characteristic
355	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
356	C₂³×C₄	—	2	1	no	—
357	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
358	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
359	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
360	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
361	#125	—	2	2	no	—
362	#125	C₂×C₄×C₂₃	1	4	yes	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
365	#117	—	4	4	no	—
366	C₂×C₄⋊C₄	—	2	1	no	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	—	2	1	no	—
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	—	2	1	no	—
372	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
373	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
374	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
375	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
376	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
377	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
378	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
379	C₂×C₂²⋊C₄	—	2	1	no	—
380	C₂×C₂²⋊C₄	—	2	1	no	—
381	C₂×C₂²⋊C₄	—	2	1	no	—
382	C₂×C₂²⋊C₄	—	2	1	no	—
383	C₂×C₂²⋊C₄	—	2	1	no	—
384	C₂×C₄²	—	2	1	no	—
385	C₂×C₄²	—	2	1	no	—
386	C₂×C₄²	—	2	1	no	—
387	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
388	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
389	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
390	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
391	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
392	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
393	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
394	#107	—	4	1	no	—
395	#107	C₂×C₄×C₂₃	1	4	yes	—
396	#106	—	4	1	no	—
397	#106	C₂×C₄×C₂₃	1	4	yes	—
398	C₄⋊C₈	—	2	2	no	—
399	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
400	#102	—	4	1	no	—
401	#102	—	4	1	no	—
402	#102	C₂×C₄×C₂₃	1	4	yes	—
403	#102	C₂×C₄×C₂₃	1	4	yes	—
404	C₄×C₈	—	4	1	no	—
405	C₄×C₈	—	4	1	no	—
406	#95	—	2	1	no	—
407	#95	—	2	1	no	—
408	#95	—	2	1	no	—
409	#95	—	2	1	no	—
410	#95	—	2	1	no	—
411	#95	—	2	1	no	—
412	#95	—	2	1	no	—
413	#95	—	2	1	no	—
414	#95	—	2	1	no	—
415	C₂₃	#22114	1	1	yes	characteristic, direct product factor, Sylow
416	C₂⁴	—	2	1	no	—
417	C₂×D₄	—	2	1	no	—
418	C₂×D₄	—	2	1	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	4	2	no	—
432	C₂²×C₄	—	4	2	no	—
433	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
434	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
436	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
437	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
438	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
439	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
440	C₂×C₈	—	2	1	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	1	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	1	no	—
446	C₂×C₈	—	2	1	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
451	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
452	C₄⋊C₄	—	2	4	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	4	2	no	—
466	C₄⋊C₄	—	4	2	no	—
467	C₄⋊C₄	—	4	2	no	—
468	C₄⋊C₄	—	4	2	no	—
469	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
470	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
471	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
472	C₂²⋊C₄	—	2	4	no	—
473	C₂²⋊C₄	—	4	2	no	—
474	C₂²⋊C₄	—	4	2	no	—
475	C₂²⋊C₄	—	4	2	no	—
476	C₂²⋊C₄	—	4	2	no	—
477	C₄²	—	2	2	no	—
478	C₄²	—	2	2	no	—
479	C₄²	—	2	2	no	—
480	C₄²	—	2	1	no	—
481	C₄²	—	2	1	no	—
482	C₄²	—	2	1	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	4	1	no	—
485	C₄²	—	4	2	no	—
486	C₄²	—	4	2	no	—
487	C₄²	—	4	2	no	—
488	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
489	C₄²	#41×C₂₃	1	2	yes	—
490	C₂³	—	2	2	no	—
491	C₂³	—	4	1	no	—
492	C₂³	—	4	1	no	—
493	C₂³	—	4	1	no	—
494	C₂³	—	4	1	no	—
495	C₂³	—	4	1	no	—
496	C₂³	—	4	1	no	—
497	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
498	D₄	—	2	2	no	—
499	D₄	—	2	2	no	—
500	D₄	—	4	1	no	—
501	D₄	—	4	1	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	4	8	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
538	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
539	C₂×C₄	C₂₃×#135	1	2	yes	—
540	C₂×C₄	C₂₃×#135	1	2	yes	—
541	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
542	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
543	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
544	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
545	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
546	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
547	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
548	C₈	—	2	2	no	—
549	C₈	—	2	2	no	—
550	C₈	—	4	1	no	—
551	C₈	—	4	1	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	4	2	no	—
555	C₂²	—	8	1	no	—
556	C₂²	—	8	1	no	—
557	C₂²	—	8	1	no	—
558	C₂²	—	8	1	no	—
559	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
560	C₂²	C₂₃×#536	1	1	yes	characteristic
561	C₂²	C₂₃×#537	1	1	yes	characteristic
562	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
563	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
564	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
565	C₂²	C₂₃×#575	1	1	yes	characteristic
566	C₄	—	2	2	no	—
567	C₄	—	2	2	no	—
568	C₄	—	2	2	no	—
569	C₄	—	2	2	no	—
570	C₄	—	2	2	no	—
571	C₄	—	4	2	no	—
572	C₄	—	4	2	no	—
573	C₄	—	4	2	no	—
574	C₄	—	4	1	no	—
575	C₄	—	8	1	no	—
576	C₄	—	8	1	no	—
577	C₄	—	8	1	no	—
578	C₄	—	8	1	no	—
579	C₄	C₂₃×#519	1	1	yes	characteristic
580	C₄	C₂₃×#520	1	1	yes	characteristic
581	C₄	C₂₃×#541	1	1	yes	characteristic
582	C₄	C₂₃×#542	1	1	yes	characteristic
583	C₂	—	8	2	no	—
584	C₂	C₂₃×#2323	1	1	yes	characteristic
585	C₂	C₄×C₂₃×#135	1	1	yes	characteristic
586	C₂	C₄×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
587	C₂	C₂₃×#3228	1	1	yes	characteristic
588	C₂	C₂₃×#3232	1	1	yes	characteristic
589	C₂	C₂₃×#3247	1	1	yes	characteristic
590	C₂	C₂₃×#3250	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	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₂₃×#125	C₂	semidirect
4. C₂₃×#1989	C₂	semidirect
5. C₂₃×#1949	C₂	semidirect
6. C₂₃×#1942	C₂	semidirect
7. C₂₃×#1941	C₂	semidirect
12. C₂₃×#1921	C₂	semidirect
13. C₂₃×#1920	C₂	semidirect
15. C₄×C₂₃×C₄⋊C₈	C₂	semidirect
19. C₂₃×#478	C₄	semidirect
194. #22114	C₂₃	direct (N × H)
258. C₄×#125	C₂×C₂₃	semidirect
260. #1989	C₂×C₂₃	semidirect
261. #1949	C₂×C₂₃	semidirect
262. #1942	C₂×C₂₃	semidirect
263. #1941	C₂×C₂₃	semidirect
268. #1921	C₂×C₂₃	semidirect
269. #1920	C₂×C₂₃	semidirect
271. C₄×C₄⋊C₈	C₂×C₂₃	semidirect
306. #478	C₄×C₂₃	semidirect
415. C₂₃	#22114	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
C₄×C₂₃×#125	2
C₂₃×#2293	2
C₂₃×#1989	2
C₂₃×#1949	2
C₂₃×#1942	2
C₂₃×#1941	2
C₂₃×#1934	2
C₂₃×#1926	2
C₂₃×#1925	2
C₂₃×#1921	2
C₂₃×#1920	2
C₂₃×#1804	2
C₄×C₂₃×C₄⋊C₈	2
#22114	23

Subgroups — C23×#22114 #19169740

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22114 #19169740