Subgroups — C₂₃×#25438 #19173064

Order 5888.

892 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#473	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3057	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3044	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#3043	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#2950	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2047	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2046	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#2005	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#2004	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1977	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1946	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1888	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1886	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1765	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#137	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#137	C₂²	1	1	yes	characteristic
18	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
20	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
21	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
23	C₂₃×#536	C₂²	1	1	yes	characteristic
24	C₂₃×#535	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
27	C₂₃×#519	C₂²	1	1	yes	characteristic
28	C₂₃×#519	C₂²	1	1	yes	characteristic
29	C₂₃×#497	—	2	4	no	—
30	C₂₃×#473	—	2	4	no	—
31	C₂₃×#468	—	2	2	no	—
32	C₂₃×#465	—	2	2	no	—
33	C₂₃×#464	—	2	2	no	—
34	C₂₃×#464	—	2	2	no	—
35	C₂₃×#462	—	2	2	no	—
36	C₂₃×#460	—	2	2	no	—
37	C₂₃×#441	—	2	2	no	—
38	C₄×Q₁₆×C₂₃	—	2	2	no	—
39	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
40	C₂₃×#427	C₂²	1	1	yes	characteristic
41	C₂₃×#424	C₂²	1	1	yes	characteristic
42	C₂₃×#424	C₂²	1	1	yes	characteristic
43	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
44	C₂₃×#419	C₂²	1	1	yes	characteristic
45	C₂₃×#419	C₂²	1	1	yes	characteristic
46	C₂₃×#419	C₂²	1	1	yes	characteristic
47	C₂₃×#419	C₂²	1	1	yes	characteristic
48	C₂₃×#418	C₂²	1	1	yes	characteristic
49	C₂₃×#418	C₂²	1	1	yes	characteristic
50	C₂₃×#416	C₂²	1	1	yes	characteristic
51	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
52	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
53	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
54	C₂×C₂₃×#100	C₂²	1	1	yes	characteristic
55	C₂₃×#405	C₂²	1	1	yes	characteristic
56	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
57	C₂₃×#341	C₂²	1	1	yes	characteristic
58	C₂₃×#339	C₂²	1	1	yes	characteristic
59	C₂₃×#339	C₂²	1	1	yes	characteristic
60	C₂₃×#339	C₂²	1	1	yes	characteristic
61	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂₃×#137	—	2	2	no	—
67	C₂₃×#137	—	2	2	no	—
68	C₂₃×#137	—	4	1	no	—
69	C₂₃×#137	—	4	1	no	—
70	C₂×Q₁₆×C₂₃	—	2	2	no	—
71	C₂×Q₁₆×C₂₃	—	2	1	no	—
72	C₂×Q₁₆×C₂₃	—	2	1	no	—
73	C₂×Q₁₆×C₂₃	—	2	1	no	—
74	C₂×Q₁₆×C₂₃	—	2	1	no	—
75	C₂×Q₁₆×C₂₃	—	2	2	no	—
76	C₂×Q₁₆×C₂₃	—	2	2	no	—
77	C₂×Q₁₆×C₂₃	—	4	1	no	—
78	C₂×Q₁₆×C₂₃	D₄	1	4	yes	complemented
79	C₂×SD₁₆×C₂₃	—	2	2	no	—
80	C₂×SD₁₆×C₂₃	—	2	1	no	—
81	C₂×SD₁₆×C₂₃	—	2	1	no	—
82	C₂×SD₁₆×C₂₃	—	2	2	no	—
83	C₂×SD₁₆×C₂₃	—	2	1	no	—
84	C₂×SD₁₆×C₂₃	—	2	1	no	—
85	C₂×SD₁₆×C₂₃	—	2	2	no	—
86	C₂×SD₁₆×C₂₃	—	2	2	no	—
87	C₂×SD₁₆×C₂₃	—	4	1	no	—
88	C₂₃×#131	—	2	2	no	—
89	C₂₃×#131	—	2	2	no	—
90	C₂×M₁₆×C₂₃	—	2	1	no	—
91	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
92	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
94	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
95	C₂²×C₈×C₂₃	—	2	1	no	—
96	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
97	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
98	C₂₃×C₄⋊Q₈	—	2	1	no	—
99	C₂₃×C₄⋊Q₈	—	2	2	no	—
100	C₂₃×C₄⋊Q₈	—	2	2	no	—
101	C₂₃×#127	—	2	1	no	—
102	C₂₃×#124	—	2	1	no	—
103	C₂₃×#124	—	2	2	no	—
104	C₂₃×#124	—	2	1	no	—
105	C₂₃×#124	—	2	2	no	—
106	C₂₃×#123	—	4	1	no	—
107	C₂₃×C₂²⋊Q₈	—	2	2	no	—
108	C₂₃×#121	—	2	2	no	—
109	C₂₃×#120	—	4	1	no	—
110	C₄×Q₈×C₂₃	—	2	2	no	—
111	C₄×Q₈×C₂₃	—	2	2	no	—
112	C₄×Q₈×C₂₃	—	2	2	no	—
113	C₄×Q₈×C₂₃	—	2	2	no	—
114	C₂₃×#117	—	4	1	no	—
115	C₂₃×#117	—	4	1	no	—
116	C₂₃×#117	C₂³	1	1	yes	characteristic
117	C₂₃×#117	C₂³	1	1	yes	characteristic
118	C₂₃×#117	C₂³	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
120	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
121	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
122	C₂₃×#108	—	2	2	no	—
123	C₂₃×#107	D₄	1	2	yes	—
124	C₂₃×#106	D₄	1	2	yes	—
125	C₂₃×C₄⋊C₈	—	2	2	no	—
126	C₂₃×C₄⋊C₈	—	2	2	no	—
127	C₂₃×#104	—	2	2	no	—
128	C₂₃×#103	—	2	2	no	—
129	C₂₃×#103	—	2	2	no	—
130	C₂₃×#103	—	2	2	no	—
131	C₂₃×#103	—	2	2	no	—
132	C₂₃×#103	D₄	1	2	yes	—
133	C₂₃×#102	—	2	2	no	—
134	C₂₃×#102	—	2	2	no	—
135	C₂₃×#102	D₄	1	2	yes	—
136	C₂₃×#101	—	2	2	no	—
137	C₂₃×#101	—	2	2	no	—
138	C₂₃×#101	—	4	1	no	—
139	C₂₃×#100	—	4	1	no	—
140	C₂₃×#97	—	2	2	no	—
141	C₂₃×#97	—	2	1	no	—
142	C₄×C₈×C₂₃	—	2	1	no	—
143	C₂⁴×C₂₃	—	2	1	no	—
144	#41×C₂₃	—	2	1	no	—
145	#41×C₂₃	—	2	1	no	—
146	#41×C₂₃	—	2	1	no	—
147	#41×C₂₃	—	2	1	no	—
148	C₂×Q₈×C₂₃	—	2	1	no	—
149	C₂×Q₈×C₂₃	—	2	1	no	—
150	C₂×Q₈×C₂₃	—	2	1	no	—
151	C₂×Q₈×C₂₃	—	2	1	no	—
152	C₂×Q₈×C₂₃	—	2	1	no	—
153	C₂×Q₈×C₂₃	—	2	1	no	—
154	C₂×Q₈×C₂₃	—	2	1	no	—
155	C₂×Q₈×C₂₃	—	2	1	no	—
156	C₂×Q₈×C₂₃	—	2	1	no	—
157	C₂×Q₈×C₂₃	—	2	1	no	—
158	C₂×Q₈×C₂₃	—	2	1	no	—
159	C₂×Q₈×C₂₃	—	2	1	no	—
160	C₂×Q₈×C₂₃	—	2	1	no	—
161	C₂×Q₈×C₂₃	—	2	1	no	—
162	C₂×Q₈×C₂₃	—	2	1	no	—
163	C₂×Q₈×C₂₃	—	2	1	no	—
164	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
165	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
166	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
167	C₂×D₄×C₂₃	—	2	1	no	—
168	C₂×D₄×C₂₃	—	2	1	no	—
169	C₂×D₄×C₂₃	—	2	1	no	—
170	C₂×D₄×C₂₃	—	2	1	no	—
171	C₂×D₄×C₂₃	—	2	1	no	—
172	C₂×D₄×C₂₃	—	2	1	no	—
173	C₂×D₄×C₂₃	—	2	1	no	—
174	C₂×D₄×C₂₃	—	4	2	no	—
175	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₂₃	—	2	1	no	—
177	C₂²×C₄×C₂₃	—	2	1	no	—
178	C₂²×C₄×C₂₃	—	2	1	no	—
179	C₂²×C₄×C₂₃	—	2	1	no	—
180	C₂²×C₄×C₂₃	—	2	1	no	—
181	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
182	Q₁₆×C₂₃	—	2	4	no	—
183	Q₁₆×C₂₃	—	2	4	no	—
184	Q₁₆×C₂₃	—	4	4	no	—
185	Q₁₆×C₂₃	—	4	2	no	—
186	Q₁₆×C₂₃	—	4	2	no	—
187	SD₁₆×C₂₃	—	4	4	no	—
188	SD₁₆×C₂₃	—	4	2	no	—
189	SD₁₆×C₂₃	—	4	2	no	—
190	SD₁₆×C₂₃	—	4	2	no	—
191	SD₁₆×C₂₃	—	4	2	no	—
192	M₁₆×C₂₃	—	2	1	no	—
193	M₁₆×C₂₃	—	2	1	no	—
194	M₁₆×C₂₃	—	2	1	no	—
195	M₁₆×C₂₃	—	2	1	no	—
196	M₁₆×C₂₃	—	2	1	no	—
197	M₁₆×C₂₃	—	2	1	no	—
198	M₁₆×C₂₃	—	2	1	no	—
199	M₁₆×C₂₃	—	2	1	no	—
200	M₁₆×C₂₃	—	4	2	no	—
201	C₂×C₈×C₂₃	—	2	1	no	—
202	C₂×C₈×C₂₃	—	2	1	no	—
203	C₂×C₈×C₂₃	—	2	1	no	—
204	C₂×C₈×C₂₃	—	2	1	no	—
205	C₂×C₈×C₂₃	—	2	1	no	—
206	C₂×C₈×C₂₃	—	2	1	no	—
207	C₂×C₈×C₂₃	—	4	2	no	—
208	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
209	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
210	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
211	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
212	C₂×C₈×C₂₃	#41	1	2	yes	—
213	C₄⋊C₄×C₂₃	—	4	4	no	—
214	C₄⋊C₄×C₂₃	—	4	2	no	—
215	C₄⋊C₄×C₂₃	—	4	2	no	—
216	C₄⋊C₄×C₂₃	—	4	2	no	—
217	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
218	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
219	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
220	C₂²⋊C₄×C₂₃	—	2	1	no	—
221	C₂²⋊C₄×C₂₃	—	2	1	no	—
222	C₂²⋊C₄×C₂₃	—	2	1	no	—
223	C₂²⋊C₄×C₂₃	—	4	4	no	—
224	C₂²⋊C₄×C₂₃	—	4	2	no	—
225	C₂²⋊C₄×C₂₃	—	4	2	no	—
226	C₄²×C₂₃	—	2	1	no	—
227	C₄²×C₂₃	—	2	1	no	—
228	C₄²×C₂₃	—	2	1	no	—
229	C₄²×C₂₃	—	2	1	no	—
230	C₄²×C₂₃	—	4	2	no	—
231	C₄²×C₂₃	—	4	2	no	—
232	C₄²×C₂₃	#41	1	2	yes	—
233	#25438	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₂³×C₂₃	—	2	1	no	—
235	C₂³×C₂₃	—	4	1	no	—
236	C₂³×C₂₃	—	4	1	no	—
237	C₂³×C₂₃	—	4	1	no	—
238	C₂³×C₂₃	—	8	2	no	—
239	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
240	Q₈×C₂₃	—	2	2	no	—
241	Q₈×C₂₃	—	2	2	no	—
242	Q₈×C₂₃	—	2	1	no	—
243	Q₈×C₂₃	—	2	2	no	—
244	Q₈×C₂₃	—	2	2	no	—
245	Q₈×C₂₃	—	2	1	no	—
246	Q₈×C₂₃	—	4	1	no	—
247	Q₈×C₂₃	—	4	1	no	—
248	Q₈×C₂₃	—	4	1	no	—
249	Q₈×C₂₃	—	4	1	no	—
250	Q₈×C₂₃	—	4	1	no	—
251	Q₈×C₂₃	—	4	1	no	—
252	Q₈×C₂₃	—	4	1	no	—
253	Q₈×C₂₃	—	4	1	no	—
254	D₄×C₂₃	—	2	1	no	—
255	D₄×C₂₃	—	2	1	no	—
256	D₄×C₂₃	—	4	1	no	—
257	D₄×C₂₃	—	4	1	no	—
258	D₄×C₂₃	—	4	1	no	—
259	D₄×C₂₃	—	4	1	no	—
260	D₄×C₂₃	—	4	1	no	—
261	D₄×C₂₃	—	4	1	no	—
262	D₄×C₂₃	—	16	2	no	—
263	C₂×C₄×C₂₃	—	2	2	no	—
264	C₂×C₄×C₂₃	—	2	2	no	—
265	C₂×C₄×C₂₃	—	2	2	no	—
266	C₂×C₄×C₂₃	—	2	1	no	—
267	C₂×C₄×C₂₃	—	2	2	no	—
268	C₂×C₄×C₂₃	—	2	2	no	—
269	C₂×C₄×C₂₃	—	2	2	no	—
270	C₂×C₄×C₂₃	—	4	2	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	C₂²×D₄	1	1	yes	characteristic
281	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
284	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
285	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
286	C₈×C₂₃	—	2	2	no	—
287	C₈×C₂₃	—	4	1	no	—
288	C₈×C₂₃	—	4	1	no	—
289	C₈×C₂₃	—	4	1	no	—
290	C₈×C₂₃	—	4	1	no	—
291	C₈×C₂₃	#121	1	2	yes	—
292	C₈×C₂₃	#121	1	2	yes	—
293	C₂×#497	C₂×C₂₃	1	1	yes	characteristic, complemented
294	C₂×#473	C₂×C₂₃	1	1	yes	characteristic, complemented
295	#3057	C₂×C₂₃	1	1	yes	characteristic
296	#3044	C₂×C₂₃	1	1	yes	characteristic, complemented
297	#3043	C₂×C₂₃	1	1	yes	characteristic
298	#2950	C₂×C₂₃	1	1	yes	characteristic, complemented
299	#2047	C₂×C₂₃	1	1	yes	characteristic, complemented
300	#2046	C₂×C₂₃	1	1	yes	characteristic, complemented
301	#2005	C₂×C₂₃	1	1	yes	characteristic, complemented
302	#2004	C₂×C₂₃	1	1	yes	characteristic
303	#1977	C₂×C₂₃	1	1	yes	characteristic, complemented
304	#1946	C₂×C₂₃	1	1	yes	characteristic, complemented
305	#1888	C₂×C₂₃	1	1	yes	characteristic, complemented
306	#1886	C₂×C₂₃	1	1	yes	characteristic, complemented
307	#1765	C₂×C₂₃	1	1	yes	characteristic, complemented
308	C₂²×C₂₃	—	2	2	no	—
309	C₂²×C₂₃	—	4	1	no	—
310	C₂²×C₂₃	—	4	2	no	—
311	C₂²×C₂₃	—	8	2	no	—
312	C₂²×C₂₃	—	8	1	no	—
313	C₂²×C₂₃	—	8	2	no	—
314	C₂²×C₂₃	—	16	1	no	—
315	C₂²×C₂₃	#540	1	1	yes	characteristic
316	C₂²×C₂₃	D₄²	1	1	yes	characteristic
317	C₂²×C₂₃	#546	1	1	yes	characteristic
318	C₄×C₂₃	—	4	2	no	—
319	C₄×C₂₃	—	4	1	no	—
320	C₄×C₂₃	—	4	2	no	—
321	C₄×C₂₃	—	4	2	no	—
322	C₄×C₂₃	—	4	2	no	—
323	C₄×C₂₃	—	4	2	no	—
324	C₄×C₂₃	—	4	2	no	—
325	C₄×C₂₃	—	8	1	no	—
326	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
327	C₄×C₂₃	#532	1	1	yes	characteristic
328	C₄×C₂₃	D₄²	1	1	yes	characteristic
329	C₄×C₂₃	#550	1	1	yes	characteristic
330	C₂×#137	C₂²×C₂₃	1	1	yes	characteristic
331	C₂×#137	C₂²×C₂₃	1	1	yes	characteristic
332	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
333	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
334	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
335	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
336	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
337	#536	C₂²×C₂₃	1	1	yes	characteristic
338	#535	C₂²×C₂₃	1	1	yes	characteristic
339	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
340	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
341	#519	C₂²×C₂₃	1	1	yes	characteristic
342	#519	C₂²×C₂₃	1	1	yes	characteristic
343	#497	—	2	4	no	—
344	#473	—	2	4	no	—
345	#468	—	2	2	no	—
346	#465	—	2	2	no	—
347	#464	—	2	2	no	—
348	#464	—	2	2	no	—
349	#462	—	2	2	no	—
350	#460	—	2	2	no	—
351	#441	—	2	2	no	—
352	C₄×Q₁₆	—	2	2	no	—
353	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
354	#427	C₂²×C₂₃	1	1	yes	characteristic
355	#424	C₂²×C₂₃	1	1	yes	characteristic
356	#424	C₂²×C₂₃	1	1	yes	characteristic
357	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
358	#419	C₂²×C₂₃	1	1	yes	characteristic
359	#419	C₂²×C₂₃	1	1	yes	characteristic
360	#419	C₂²×C₂₃	1	1	yes	characteristic
361	#419	C₂²×C₂₃	1	1	yes	characteristic
362	#418	C₂²×C₂₃	1	1	yes	characteristic
363	#418	C₂²×C₂₃	1	1	yes	characteristic
364	#416	C₂²×C₂₃	1	1	yes	characteristic
365	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
366	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
367	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
368	C₂×#100	C₂²×C₂₃	1	1	yes	characteristic
369	#405	C₂²×C₂₃	1	1	yes	characteristic
370	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
371	#341	C₂²×C₂₃	1	1	yes	characteristic
372	#339	C₂²×C₂₃	1	1	yes	characteristic
373	#339	C₂²×C₂₃	1	1	yes	characteristic
374	#339	C₂²×C₂₃	1	1	yes	characteristic
375	C₂×C₂₃	—	2	1	no	—
376	C₂×C₂₃	—	2	1	no	—
377	C₂×C₂₃	—	8	1	no	—
378	C₂×C₂₃	—	16	1	no	—
379	C₂×C₂₃	#2428	1	1	yes	characteristic
380	C₂×C₂₃	#3289	1	2	yes	—
381	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
382	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
383	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
384	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
385	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
386	#137	—	2	2	no	—
387	#137	—	2	2	no	—
388	#137	—	4	1	no	—
389	#137	—	4	1	no	—
390	C₂×Q₁₆	—	2	2	no	—
391	C₂×Q₁₆	—	2	1	no	—
392	C₂×Q₁₆	—	2	1	no	—
393	C₂×Q₁₆	—	2	1	no	—
394	C₂×Q₁₆	—	2	1	no	—
395	C₂×Q₁₆	—	2	2	no	—
396	C₂×Q₁₆	—	2	2	no	—
397	C₂×Q₁₆	—	4	1	no	—
398	C₂×Q₁₆	D₄×C₂₃	1	4	yes	complemented
399	C₂×SD₁₆	—	2	2	no	—
400	C₂×SD₁₆	—	2	1	no	—
401	C₂×SD₁₆	—	2	1	no	—
402	C₂×SD₁₆	—	2	2	no	—
403	C₂×SD₁₆	—	2	1	no	—
404	C₂×SD₁₆	—	2	1	no	—
405	C₂×SD₁₆	—	2	2	no	—
406	C₂×SD₁₆	—	2	2	no	—
407	C₂×SD₁₆	—	4	1	no	—
408	#131	—	2	2	no	—
409	#131	—	2	2	no	—
410	C₂×M₁₆	—	2	1	no	—
411	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
412	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
413	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
414	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
415	C₂²×C₈	—	2	1	no	—
416	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
417	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
418	C₄⋊Q₈	—	2	2	no	—
419	C₄⋊Q₈	—	2	1	no	—
420	C₄⋊Q₈	—	2	2	no	—
421	#127	—	2	1	no	—
422	#124	—	2	2	no	—
423	#124	—	2	1	no	—
424	#124	—	2	1	no	—
425	#124	—	2	2	no	—
426	#123	—	4	1	no	—
427	C₂²⋊Q₈	—	2	2	no	—
428	#121	—	2	2	no	—
429	#120	—	4	1	no	—
430	C₄×Q₈	—	2	2	no	—
431	C₄×Q₈	—	2	2	no	—
432	C₄×Q₈	—	2	2	no	—
433	C₄×Q₈	—	2	2	no	—
434	#117	—	4	1	no	—
435	#117	—	4	1	no	—
436	#117	C₂³×C₂₃	1	1	yes	characteristic
437	#117	C₂³×C₂₃	1	1	yes	characteristic
438	#117	C₂³×C₂₃	1	1	yes	characteristic
439	C₂×C₄⋊C₄	—	4	1	no	—
440	C₂×C₂²⋊C₄	—	4	1	no	—
441	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
442	#108	—	2	2	no	—
443	#107	D₄×C₂₃	1	2	yes	—
444	#106	D₄×C₂₃	1	2	yes	—
445	C₄⋊C₈	—	2	2	no	—
446	C₄⋊C₈	—	2	2	no	—
447	#104	—	2	2	no	—
448	#103	—	2	2	no	—
449	#103	—	2	2	no	—
450	#103	—	2	2	no	—
451	#103	—	2	2	no	—
452	#103	D₄×C₂₃	1	2	yes	—
453	#102	—	2	2	no	—
454	#102	—	2	2	no	—
455	#102	D₄×C₂₃	1	2	yes	—
456	#101	—	2	2	no	—
457	#101	—	2	2	no	—
458	#101	—	4	1	no	—
459	#100	—	4	1	no	—
460	#97	—	2	2	no	—
461	#97	—	2	1	no	—
462	C₄×C₈	—	2	1	no	—
463	C₂₃	#25438	1	1	yes	characteristic, direct product factor, Sylow
464	C₂⁴	—	2	1	no	—
465	#41	—	2	1	no	—
466	#41	—	2	1	no	—
467	#41	—	2	1	no	—
468	#41	—	2	1	no	—
469	C₂×Q₈	—	2	1	no	—
470	C₂×Q₈	—	2	1	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	1	no	—
473	C₂×Q₈	—	2	1	no	—
474	C₂×Q₈	—	2	1	no	—
475	C₂×Q₈	—	2	1	no	—
476	C₂×Q₈	—	2	1	no	—
477	C₂×Q₈	—	2	1	no	—
478	C₂×Q₈	—	2	1	no	—
479	C₂×Q₈	—	2	1	no	—
480	C₂×Q₈	—	2	1	no	—
481	C₂×Q₈	—	2	1	no	—
482	C₂×Q₈	—	2	1	no	—
483	C₂×Q₈	—	2	1	no	—
484	C₂×Q₈	—	2	1	no	—
485	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
486	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
487	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
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	—	2	1	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	—	2	1	no	—
495	C₂×D₄	—	4	2	no	—
496	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
497	C₂²×C₄	—	2	1	no	—
498	C₂²×C₄	—	2	1	no	—
499	C₂²×C₄	—	2	1	no	—
500	C₂²×C₄	—	2	1	no	—
501	C₂²×C₄	—	2	1	no	—
502	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
503	Q₁₆	—	2	4	no	—
504	Q₁₆	—	2	4	no	—
505	Q₁₆	—	4	4	no	—
506	Q₁₆	—	4	2	no	—
507	Q₁₆	—	4	2	no	—
508	SD₁₆	—	4	4	no	—
509	SD₁₆	—	4	2	no	—
510	SD₁₆	—	4	2	no	—
511	SD₁₆	—	4	2	no	—
512	SD₁₆	—	4	2	no	—
513	M₁₆	—	2	1	no	—
514	M₁₆	—	2	1	no	—
515	M₁₆	—	2	1	no	—
516	M₁₆	—	2	1	no	—
517	M₁₆	—	2	1	no	—
518	M₁₆	—	2	1	no	—
519	M₁₆	—	2	1	no	—
520	M₁₆	—	2	1	no	—
521	M₁₆	—	4	2	no	—
522	C₂×C₈	—	2	1	no	—
523	C₂×C₈	—	2	1	no	—
524	C₂×C₈	—	2	1	no	—
525	C₂×C₈	—	2	1	no	—
526	C₂×C₈	—	2	1	no	—
527	C₂×C₈	—	2	1	no	—
528	C₂×C₈	—	4	2	no	—
529	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
530	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
531	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
532	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
533	C₂×C₈	#41×C₂₃	1	2	yes	—
534	C₄⋊C₄	—	4	4	no	—
535	C₄⋊C₄	—	4	2	no	—
536	C₄⋊C₄	—	4	2	no	—
537	C₄⋊C₄	—	4	2	no	—
538	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
539	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
540	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
541	C₂²⋊C₄	—	2	1	no	—
542	C₂²⋊C₄	—	2	1	no	—
543	C₂²⋊C₄	—	2	1	no	—
544	C₂²⋊C₄	—	4	4	no	—
545	C₂²⋊C₄	—	4	2	no	—
546	C₂²⋊C₄	—	4	2	no	—
547	C₄²	—	2	1	no	—
548	C₄²	—	2	1	no	—
549	C₄²	—	2	1	no	—
550	C₄²	—	2	1	no	—
551	C₄²	—	4	2	no	—
552	C₄²	—	4	2	no	—
553	C₄²	#41×C₂₃	1	2	yes	—
554	C₂³	—	2	1	no	—
555	C₂³	—	4	1	no	—
556	C₂³	—	4	1	no	—
557	C₂³	—	4	1	no	—
558	C₂³	—	8	2	no	—
559	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
560	Q₈	—	2	2	no	—
561	Q₈	—	2	2	no	—
562	Q₈	—	2	1	no	—
563	Q₈	—	2	2	no	—
564	Q₈	—	2	2	no	—
565	Q₈	—	2	1	no	—
566	Q₈	—	4	1	no	—
567	Q₈	—	4	1	no	—
568	Q₈	—	4	1	no	—
569	Q₈	—	4	1	no	—
570	Q₈	—	4	1	no	—
571	Q₈	—	4	1	no	—
572	Q₈	—	4	1	no	—
573	Q₈	—	4	1	no	—
574	D₄	—	2	1	no	—
575	D₄	—	2	1	no	—
576	D₄	—	4	1	no	—
577	D₄	—	4	1	no	—
578	D₄	—	4	1	no	—
579	D₄	—	4	1	no	—
580	D₄	—	4	1	no	—
581	D₄	—	4	1	no	—
582	D₄	—	16	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	4	2	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	C₂²×D₄×C₂₃	1	1	yes	characteristic
601	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
602	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
603	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
604	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
605	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
606	C₈	—	2	2	no	—
607	C₈	—	4	1	no	—
608	C₈	—	4	1	no	—
609	C₈	—	4	1	no	—
610	C₈	—	4	1	no	—
611	C₈	C₂₃×#121	1	2	yes	—
612	C₈	C₂₃×#121	1	2	yes	—
613	C₂²	—	2	2	no	—
614	C₂²	—	4	1	no	—
615	C₂²	—	4	2	no	—
616	C₂²	—	8	2	no	—
617	C₂²	—	8	1	no	—
618	C₂²	—	8	2	no	—
619	C₂²	—	16	1	no	—
620	C₂²	C₂₃×#540	1	1	yes	characteristic
621	C₂²	D₄²×C₂₃	1	1	yes	characteristic
622	C₂²	C₂₃×#546	1	1	yes	characteristic
623	C₄	—	4	2	no	—
624	C₄	—	4	1	no	—
625	C₄	—	4	2	no	—
626	C₄	—	4	2	no	—
627	C₄	—	4	2	no	—
628	C₄	—	4	2	no	—
629	C₄	—	4	2	no	—
630	C₄	—	8	1	no	—
631	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
632	C₄	C₂₃×#532	1	1	yes	characteristic
633	C₄	D₄²×C₂₃	1	1	yes	characteristic
634	C₄	C₂₃×#550	1	1	yes	characteristic
635	C₂	—	2	1	no	—
636	C₂	—	2	1	no	—
637	C₂	—	8	1	no	—
638	C₂	—	16	1	no	—
639	C₂	C₂₃×#2428	1	1	yes	characteristic
640	C₂	C₂₃×#3289	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#497	C₂	semidirect
2. C₂×C₂₃×#473	C₂	semidirect
4. C₂₃×#3044	C₂	semidirect
6. C₂₃×#2950	C₂	semidirect
7. C₂₃×#2047	C₂	semidirect
8. C₂₃×#2046	C₂	semidirect
9. C₂₃×#2005	C₂	semidirect
11. C₂₃×#1977	C₂	semidirect
12. C₂₃×#1946	C₂	semidirect
13. C₂₃×#1888	C₂	semidirect
14. C₂₃×#1886	C₂	semidirect
15. C₂₃×#1765	C₂	semidirect
78. C₂×Q₁₆×C₂₃	D₄	semidirect
233. #25438	C₂₃	direct (N × H)
293. C₂×#497	C₂×C₂₃	semidirect
294. C₂×#473	C₂×C₂₃	semidirect
296. #3044	C₂×C₂₃	semidirect
298. #2950	C₂×C₂₃	semidirect
299. #2047	C₂×C₂₃	semidirect
300. #2046	C₂×C₂₃	semidirect
301. #2005	C₂×C₂₃	semidirect
303. #1977	C₂×C₂₃	semidirect
304. #1946	C₂×C₂₃	semidirect
305. #1888	C₂×C₂₃	semidirect
306. #1886	C₂×C₂₃	semidirect
307. #1765	C₂×C₂₃	semidirect
398. C₂×Q₁₆	D₄×C₂₃	semidirect
463. C₂₃	#25438	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×#497	2
C₂×C₂₃×#473	2
C₂₃×#3057	2
C₂₃×#3044	2
C₂₃×#3043	2
C₂₃×#2950	2
C₂₃×#2047	2
C₂₃×#2046	2
C₂₃×#2005	2
C₂₃×#2004	2
C₂₃×#1977	2
C₂₃×#1946	2
C₂₃×#1888	2
C₂₃×#1886	2
C₂₃×#1765	2
#25438	23

Subgroups — C23×#25438 #19173064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#25438 #19173064