Subgroups — C₂₃×#26403 #19174029

Order 5888.

706 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3084	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3082	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2947	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2913	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2097	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2089	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2064	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2061	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1951	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1922	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1908	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#1882	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1866	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#1860	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#1857	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#137	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
18	C₂₃×#544	C₂²	1	1	yes	characteristic
19	C₂₃×#520	C₂²	1	1	yes	characteristic
20	C₂₃×#520	C₂²	1	1	yes	characteristic
21	C₂₃×#519	C₂²	1	1	yes	characteristic
22	C₂₃×#479	—	2	1	no	—
23	C₂₃×#479	—	2	1	no	—
24	C₂₃×#479	—	2	1	no	—
25	C₂₃×#478	—	2	1	no	—
26	C₂₃×#477	—	2	1	no	—
27	C₂₃×#476	—	2	1	no	—
28	C₂₃×#475	—	2	1	no	—
29	C₂₃×#475	—	2	1	no	—
30	C₂₃×#441	—	2	1	no	—
31	C₂₃×#440	—	2	1	no	—
32	C₄×Q₁₆×C₂₃	—	2	1	no	—
33	C₄×SD₁₆×C₂₃	—	2	1	no	—
34	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
35	C₂₃×#428	—	2	1	no	—
36	C₂₃×#428	—	2	1	no	—
37	C₂₃×#428	C₂²	1	1	yes	characteristic
38	C₂₃×#428	C₂²	1	1	yes	characteristic
39	C₂₃×#428	C₂²	1	1	yes	characteristic
40	C₂₃×#428	C₂²	1	1	yes	characteristic
41	C₂₃×#428	C₂²	1	1	yes	characteristic
42	C₂₃×#427	—	2	1	no	—
43	C₂₃×#427	C₂²	1	1	yes	characteristic
44	C₂×C₂₃×#107	—	2	1	no	—
45	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
46	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
47	C₂₃×#424	C₂²	1	1	yes	characteristic
48	C₂₃×#424	C₂²	1	1	yes	characteristic
49	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
50	C₂₃×#419	C₂²	1	1	yes	characteristic
51	C₂₃×#417	C₂²	1	1	yes	characteristic
52	C₂₃×#416	C₂²	1	1	yes	characteristic
53	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
54	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
55	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
56	C₂₃×#405	C₂²	1	1	yes	characteristic
57	C₂₃×#391	C₂²	1	1	yes	characteristic
58	C₂₃×#382	C₂²	1	1	yes	characteristic
59	C₂₃×#343	C₂²	1	1	yes	characteristic
60	C₂₃×#343	C₂²	1	1	yes	characteristic
61	C₂₃×#343	C₂²	1	1	yes	characteristic
62	C₂₃×#341	C₂²	1	1	yes	characteristic
63	C₂₃×#339	C₂²	1	1	yes	characteristic
64	C₂₃×#339	C₂²	1	1	yes	characteristic
65	C₂₃×#339	C₂²	1	1	yes	characteristic
66	C₂₃×#336	C₂²	1	1	yes	characteristic
67	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
68	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₂₃×#137	—	2	2	no	—
70	C₂₃×#137	Q₈	1	4	yes	complemented
71	C₂×Q₁₆×C₂₃	—	2	1	no	—
72	C₂×SD₁₆×C₂₃	—	2	1	no	—
73	C₂₃×#131	—	2	2	no	—
74	C₂₃×#131	—	2	2	no	—
75	C₂×M₁₆×C₂₃	—	2	1	no	—
76	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
77	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
78	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
79	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₂₃	—	2	1	no	—
81	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
83	C₂₃×C₄⋊Q₈	—	2	1	no	—
84	C₂₃×C₄⋊Q₈	—	2	1	no	—
85	C₂₃×#125	—	2	1	no	—
86	C₂₃×#125	—	2	1	no	—
87	C₂₃×C₂²⋊Q₈	—	4	1	no	—
88	C₄×Q₈×C₂₃	—	2	1	no	—
89	C₄×Q₈×C₂₃	—	2	1	no	—
90	C₄×Q₈×C₂₃	—	2	1	no	—
91	C₄×Q₈×C₂₃	—	2	1	no	—
92	C₄×Q₈×C₂₃	—	2	1	no	—
93	C₄×Q₈×C₂₃	—	2	1	no	—
94	C₄×D₄×C₂₃	—	2	1	no	—
95	C₄×D₄×C₂₃	—	2	1	no	—
96	C₄×D₄×C₂₃	—	4	1	no	—
97	C₄×D₄×C₂₃	—	4	1	no	—
98	C₂₃×#117	—	2	1	no	—
99	C₂₃×#117	—	2	1	no	—
100	C₂₃×#117	—	4	1	no	—
101	C₂₃×#117	C₂³	1	1	yes	characteristic
102	C₂₃×#117	C₂³	1	1	yes	characteristic
103	C₂₃×#117	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
106	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
110	C₂₃×#108	—	2	2	no	—
111	C₂₃×#107	—	2	1	no	—
112	C₂₃×#107	—	2	1	no	—
113	C₂₃×#107	—	2	1	no	—
114	C₂₃×#107	—	2	1	no	—
115	C₂₃×#107	—	2	1	no	—
116	C₂₃×#107	—	2	1	no	—
117	C₂₃×#107	—	2	1	no	—
118	C₂₃×#107	D₄	1	1	yes	characteristic
119	C₂₃×#107	D₄	1	1	yes	characteristic
120	C₂₃×#106	—	2	1	no	—
121	C₂₃×#106	—	2	1	no	—
122	C₂₃×#106	—	2	1	no	—
123	C₂₃×#106	—	2	1	no	—
124	C₂₃×#106	—	2	1	no	—
125	C₂₃×#106	—	2	1	no	—
126	C₂₃×#106	—	2	1	no	—
127	C₂₃×#106	D₄	1	1	yes	characteristic
128	C₂₃×#106	D₄	1	1	yes	characteristic
129	C₂₃×C₄⋊C₈	—	2	1	no	—
130	C₂₃×C₄⋊C₈	—	2	1	no	—
131	C₂₃×C₄⋊C₈	—	2	1	no	—
132	C₂₃×C₄⋊C₈	—	2	1	no	—
133	C₂₃×#104	—	2	2	no	—
134	C₂₃×#103	—	2	1	no	—
135	C₂₃×#103	—	2	1	no	—
136	C₂₃×#103	—	2	1	no	—
137	C₂₃×#103	—	2	1	no	—
138	C₂₃×#103	—	2	1	no	—
139	C₂₃×#103	—	2	1	no	—
140	C₂₃×#103	—	2	1	no	—
141	C₂₃×#103	D₄	1	1	yes	characteristic
142	C₂₃×#103	D₄	1	1	yes	characteristic
143	C₂₃×#102	—	2	1	no	—
144	C₂₃×#102	D₄	1	1	yes	characteristic
145	C₂₃×#102	D₄	1	1	yes	characteristic
146	C₂₃×#101	—	2	2	no	—
147	C₂₃×#97	—	2	1	no	—
148	C₄×C₈×C₂₃	—	2	1	no	—
149	C₂₃×#95	—	4	1	no	—
150	C₂₃×#95	—	4	1	no	—
151	#41×C₂₃	—	2	1	no	—
152	#41×C₂₃	—	2	1	no	—
153	#41×C₂₃	C₂×Q₈	1	2	yes	—
154	#41×C₂₃	#41	1	2	yes	—
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	—	4	1	no	—
161	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
162	C₂×Q₈×C₂₃	C₂×Q₈	1	2	yes	—
163	C₂×D₄×C₂₃	—	2	1	no	—
164	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₂₃	—	2	1	no	—
166	C₂²×C₄×C₂₃	—	2	1	no	—
167	C₂²×C₄×C₂₃	—	2	1	no	—
168	C₂²×C₄×C₂₃	—	2	1	no	—
169	C₂²×C₄×C₂₃	—	2	1	no	—
170	C₂²×C₄×C₂₃	—	2	1	no	—
171	C₂²×C₄×C₂₃	—	4	1	no	—
172	C₂²×C₄×C₂₃	—	4	1	no	—
173	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
174	Q₁₆×C₂₃	—	2	4	no	—
175	SD₁₆×C₂₃	—	2	4	no	—
176	M₁₆×C₂₃	—	2	4	no	—
177	M₁₆×C₂₃	—	2	1	no	—
178	M₁₆×C₂₃	—	2	1	no	—
179	M₁₆×C₂₃	—	2	1	no	—
180	M₁₆×C₂₃	—	2	1	no	—
181	M₁₆×C₂₃	—	2	1	no	—
182	M₁₆×C₂₃	—	2	1	no	—
183	M₁₆×C₂₃	C₂×Q₈	1	2	yes	—
184	M₁₆×C₂₃	#41	1	2	yes	—
185	C₂×C₈×C₂₃	—	2	4	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	—	2	1	no	—
191	C₂×C₈×C₂₃	—	2	1	no	—
192	C₂×C₈×C₂₃	—	2	1	no	—
193	C₂×C₈×C₂₃	—	2	1	no	—
194	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
195	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
196	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
197	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
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	—	4	1	no	—
205	C₄⋊C₄×C₂₃	—	4	1	no	—
206	C₄⋊C₄×C₂₃	—	4	1	no	—
207	C₄⋊C₄×C₂₃	—	4	1	no	—
208	C₄⋊C₄×C₂₃	—	4	1	no	—
209	C₄⋊C₄×C₂₃	—	4	1	no	—
210	C₄⋊C₄×C₂₃	—	4	1	no	—
211	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
212	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
213	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
214	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
215	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
216	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
217	C₂²⋊C₄×C₂₃	—	2	1	no	—
218	C₂²⋊C₄×C₂₃	—	2	1	no	—
219	C₂²⋊C₄×C₂₃	—	2	1	no	—
220	C₂²⋊C₄×C₂₃	—	4	1	no	—
221	C₂²⋊C₄×C₂₃	—	4	1	no	—
222	C₄²×C₂₃	—	2	1	no	—
223	C₄²×C₂₃	—	2	1	no	—
224	C₄²×C₂₃	—	2	1	no	—
225	C₄²×C₂₃	—	2	1	no	—
226	C₄²×C₂₃	—	2	1	no	—
227	C₄²×C₂₃	—	4	1	no	—
228	C₄²×C₂₃	—	4	1	no	—
229	C₄²×C₂₃	—	4	1	no	—
230	C₄²×C₂₃	—	4	1	no	—
231	#26403	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂³×C₂₃	—	2	1	no	—
233	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
234	Q₈×C₂₃	—	2	1	no	—
235	Q₈×C₂₃	—	2	2	no	—
236	Q₈×C₂₃	—	2	2	no	—
237	Q₈×C₂₃	—	4	1	no	—
238	Q₈×C₂₃	—	4	1	no	—
239	Q₈×C₂₃	—	16	1	no	—
240	Q₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
241	D₄×C₂₃	—	2	2	no	—
242	D₄×C₂₃	—	2	2	no	—
243	D₄×C₂₃	—	2	1	no	—
244	D₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
245	C₂×C₄×C₂₃	—	2	1	no	—
246	C₂×C₄×C₂₃	—	2	2	no	—
247	C₂×C₄×C₂₃	—	2	2	no	—
248	C₂×C₄×C₂₃	—	2	1	no	—
249	C₂×C₄×C₂₃	—	2	1	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	2	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	—	8	1	no	—
264	C₂×C₄×C₂₃	—	8	1	no	—
265	C₂×C₄×C₂₃	—	8	1	no	—
266	C₂×C₄×C₂₃	—	16	1	no	—
267	C₂×C₄×C₂₃	—	16	1	no	—
268	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
269	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
271	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
272	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
273	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
274	C₈×C₂₃	—	2	2	no	—
275	C₈×C₂₃	—	2	1	no	—
276	C₈×C₂₃	—	2	2	no	—
277	C₈×C₂₃	—	2	1	no	—
278	C₈×C₂₃	—	2	1	no	—
279	C₈×C₂₃	—	2	1	no	—
280	C₈×C₂₃	—	4	1	no	—
281	C₈×C₂₃	—	4	1	no	—
282	#3084	C₂×C₂₃	1	1	yes	characteristic
283	#3082	C₂×C₂₃	1	1	yes	characteristic, complemented
284	#2947	C₂×C₂₃	1	1	yes	characteristic
285	#2913	C₂×C₂₃	1	1	yes	characteristic
286	#2097	C₂×C₂₃	1	1	yes	characteristic, complemented
287	#2089	C₂×C₂₃	1	1	yes	characteristic, complemented
288	#2064	C₂×C₂₃	1	1	yes	characteristic, complemented
289	#2061	C₂×C₂₃	1	1	yes	characteristic, complemented
290	#1951	C₂×C₂₃	1	1	yes	characteristic, complemented
291	#1922	C₂×C₂₃	1	1	yes	characteristic, complemented
292	#1908	C₂×C₂₃	1	1	yes	characteristic
293	#1882	C₂×C₂₃	1	1	yes	characteristic
294	#1866	C₂×C₂₃	1	1	yes	characteristic
295	#1860	C₂×C₂₃	1	1	yes	characteristic
296	#1857	C₂×C₂₃	1	1	yes	characteristic, complemented
297	C₂²×C₂₃	—	2	1	no	—
298	C₂²×C₂₃	—	2	1	no	—
299	C₂²×C₂₃	—	2	2	no	—
300	C₂²×C₂₃	—	4	1	no	—
301	C₂²×C₂₃	—	4	1	no	—
302	C₂²×C₂₃	#539	1	1	yes	characteristic
303	C₂²×C₂₃	D₄²	1	1	yes	characteristic
304	C₂²×C₂₃	#554	1	1	yes	characteristic
305	C₄×C₂₃	—	2	2	no	—
306	C₄×C₂₃	—	4	2	no	—
307	C₄×C₂₃	—	4	1	no	—
308	C₄×C₂₃	—	4	1	no	—
309	C₄×C₂₃	—	4	1	no	—
310	C₄×C₂₃	—	4	1	no	—
311	C₄×C₂₃	—	4	1	no	—
312	C₄×C₂₃	—	4	1	no	—
313	C₄×C₂₃	—	8	1	no	—
314	C₄×C₂₃	—	8	1	no	—
315	C₄×C₂₃	—	8	1	no	—
316	C₄×C₂₃	—	8	1	no	—
317	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
318	C₄×C₂₃	#525	1	1	yes	characteristic
319	C₄×C₂₃	#547	1	1	yes	characteristic
320	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
321	C₂×#137	C₂²×C₂₃	1	1	yes	characteristic
322	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
323	#544	C₂²×C₂₃	1	1	yes	characteristic
324	#520	C₂²×C₂₃	1	1	yes	characteristic
325	#520	C₂²×C₂₃	1	1	yes	characteristic
326	#519	C₂²×C₂₃	1	1	yes	characteristic
327	#479	—	2	1	no	—
328	#479	—	2	1	no	—
329	#479	—	2	1	no	—
330	#478	—	2	1	no	—
331	#477	—	2	1	no	—
332	#476	—	2	1	no	—
333	#475	—	2	1	no	—
334	#475	—	2	1	no	—
335	#441	—	2	1	no	—
336	#440	—	2	1	no	—
337	C₄×Q₁₆	—	2	1	no	—
338	C₄×SD₁₆	—	2	1	no	—
339	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
340	#428	—	2	1	no	—
341	#428	—	2	1	no	—
342	#428	C₂²×C₂₃	1	1	yes	characteristic
343	#428	C₂²×C₂₃	1	1	yes	characteristic
344	#428	C₂²×C₂₃	1	1	yes	characteristic
345	#428	C₂²×C₂₃	1	1	yes	characteristic
346	#428	C₂²×C₂₃	1	1	yes	characteristic
347	#427	—	2	1	no	—
348	#427	C₂²×C₂₃	1	1	yes	characteristic
349	C₂×#107	—	2	1	no	—
350	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
351	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
352	#424	C₂²×C₂₃	1	1	yes	characteristic
353	#424	C₂²×C₂₃	1	1	yes	characteristic
354	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
355	#419	C₂²×C₂₃	1	1	yes	characteristic
356	#417	C₂²×C₂₃	1	1	yes	characteristic
357	#416	C₂²×C₂₃	1	1	yes	characteristic
358	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
359	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
360	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
361	#405	C₂²×C₂₃	1	1	yes	characteristic
362	#391	C₂²×C₂₃	1	1	yes	characteristic
363	#382	C₂²×C₂₃	1	1	yes	characteristic
364	#343	C₂²×C₂₃	1	1	yes	characteristic
365	#343	C₂²×C₂₃	1	1	yes	characteristic
366	#343	C₂²×C₂₃	1	1	yes	characteristic
367	#341	C₂²×C₂₃	1	1	yes	characteristic
368	#339	C₂²×C₂₃	1	1	yes	characteristic
369	#339	C₂²×C₂₃	1	1	yes	characteristic
370	#339	C₂²×C₂₃	1	1	yes	characteristic
371	#336	C₂²×C₂₃	1	1	yes	characteristic
372	C₂×C₂₃	—	2	1	no	—
373	C₂×C₂₃	—	2	1	no	—
374	C₂×C₂₃	—	4	2	no	—
375	C₂×C₂₃	#2449	1	1	yes	characteristic
376	C₂×C₂₃	#3289	1	1	yes	characteristic
377	C₂×C₂₃	#3290	1	1	yes	characteristic
378	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
379	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
380	#137	—	2	2	no	—
381	#137	Q₈×C₂₃	1	4	yes	complemented
382	C₂×Q₁₆	—	2	1	no	—
383	C₂×SD₁₆	—	2	1	no	—
384	#131	—	2	2	no	—
385	#131	—	2	2	no	—
386	C₂×M₁₆	—	2	1	no	—
387	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
388	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
389	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
390	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
391	C₂²×C₈	—	2	1	no	—
392	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
393	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
394	C₄⋊Q₈	—	2	1	no	—
395	C₄⋊Q₈	—	2	1	no	—
396	#125	—	2	1	no	—
397	#125	—	2	1	no	—
398	C₂²⋊Q₈	—	4	1	no	—
399	C₄×Q₈	—	2	1	no	—
400	C₄×Q₈	—	2	1	no	—
401	C₄×Q₈	—	2	1	no	—
402	C₄×Q₈	—	2	1	no	—
403	C₄×Q₈	—	2	1	no	—
404	C₄×Q₈	—	2	1	no	—
405	C₄×D₄	—	2	1	no	—
406	C₄×D₄	—	2	1	no	—
407	C₄×D₄	—	4	1	no	—
408	C₄×D₄	—	4	1	no	—
409	#117	—	2	1	no	—
410	#117	—	2	1	no	—
411	#117	—	4	1	no	—
412	#117	C₂³×C₂₃	1	1	yes	characteristic
413	#117	C₂³×C₂₃	1	1	yes	characteristic
414	#117	C₂³×C₂₃	1	1	yes	characteristic
415	C₂×C₄⋊C₄	—	2	1	no	—
416	C₂×C₄⋊C₄	—	2	1	no	—
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×C₄²	C₂³×C₂₃	1	1	yes	characteristic
421	#108	—	2	2	no	—
422	#107	—	2	1	no	—
423	#107	—	2	1	no	—
424	#107	—	2	1	no	—
425	#107	—	2	1	no	—
426	#107	—	2	1	no	—
427	#107	—	2	1	no	—
428	#107	—	2	1	no	—
429	#107	D₄×C₂₃	1	1	yes	characteristic
430	#107	D₄×C₂₃	1	1	yes	characteristic
431	#106	—	2	1	no	—
432	#106	—	2	1	no	—
433	#106	—	2	1	no	—
434	#106	—	2	1	no	—
435	#106	—	2	1	no	—
436	#106	—	2	1	no	—
437	#106	—	2	1	no	—
438	#106	D₄×C₂₃	1	1	yes	characteristic
439	#106	D₄×C₂₃	1	1	yes	characteristic
440	C₄⋊C₈	—	2	1	no	—
441	C₄⋊C₈	—	2	1	no	—
442	C₄⋊C₈	—	2	1	no	—
443	C₄⋊C₈	—	2	1	no	—
444	#104	—	2	2	no	—
445	#103	—	2	1	no	—
446	#103	—	2	1	no	—
447	#103	—	2	1	no	—
448	#103	—	2	1	no	—
449	#103	—	2	1	no	—
450	#103	—	2	1	no	—
451	#103	—	2	1	no	—
452	#103	D₄×C₂₃	1	1	yes	characteristic
453	#103	D₄×C₂₃	1	1	yes	characteristic
454	#102	—	2	1	no	—
455	#102	D₄×C₂₃	1	1	yes	characteristic
456	#102	D₄×C₂₃	1	1	yes	characteristic
457	#101	—	2	2	no	—
458	#97	—	2	1	no	—
459	C₄×C₈	—	2	1	no	—
460	#95	—	4	1	no	—
461	#95	—	4	1	no	—
462	C₂₃	#26403	1	1	yes	characteristic, direct product factor, Sylow
463	#41	—	2	1	no	—
464	#41	—	2	1	no	—
465	#41	C₂×Q₈×C₂₃	1	2	yes	—
466	#41	#41×C₂₃	1	2	yes	—
467	C₂×Q₈	—	2	1	no	—
468	C₂×Q₈	—	2	1	no	—
469	C₂×Q₈	—	2	1	no	—
470	C₂×Q₈	—	2	1	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	4	1	no	—
473	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
474	C₂×Q₈	C₂×Q₈×C₂₃	1	2	yes	—
475	C₂×D₄	—	2	1	no	—
476	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
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	—	4	1	no	—
484	C₂²×C₄	—	4	1	no	—
485	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
486	Q₁₆	—	2	4	no	—
487	SD₁₆	—	2	4	no	—
488	M₁₆	—	2	4	no	—
489	M₁₆	—	2	1	no	—
490	M₁₆	—	2	1	no	—
491	M₁₆	—	2	1	no	—
492	M₁₆	—	2	1	no	—
493	M₁₆	—	2	1	no	—
494	M₁₆	—	2	1	no	—
495	M₁₆	C₂×Q₈×C₂₃	1	2	yes	—
496	M₁₆	#41×C₂₃	1	2	yes	—
497	C₂×C₈	—	2	4	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	C₂×D₄×C₂₃	1	1	yes	characteristic
507	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
508	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
509	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
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	—	4	1	no	—
517	C₄⋊C₄	—	4	1	no	—
518	C₄⋊C₄	—	4	1	no	—
519	C₄⋊C₄	—	4	1	no	—
520	C₄⋊C₄	—	4	1	no	—
521	C₄⋊C₄	—	4	1	no	—
522	C₄⋊C₄	—	4	1	no	—
523	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
524	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
525	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
526	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
527	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
528	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
529	C₂²⋊C₄	—	2	1	no	—
530	C₂²⋊C₄	—	2	1	no	—
531	C₂²⋊C₄	—	2	1	no	—
532	C₂²⋊C₄	—	4	1	no	—
533	C₂²⋊C₄	—	4	1	no	—
534	C₄²	—	2	1	no	—
535	C₄²	—	2	1	no	—
536	C₄²	—	2	1	no	—
537	C₄²	—	2	1	no	—
538	C₄²	—	2	1	no	—
539	C₄²	—	4	1	no	—
540	C₄²	—	4	1	no	—
541	C₄²	—	4	1	no	—
542	C₄²	—	4	1	no	—
543	C₂³	—	2	1	no	—
544	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
545	Q₈	—	2	1	no	—
546	Q₈	—	2	2	no	—
547	Q₈	—	2	2	no	—
548	Q₈	—	4	1	no	—
549	Q₈	—	4	1	no	—
550	Q₈	—	16	1	no	—
551	Q₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
552	D₄	—	2	2	no	—
553	D₄	—	2	2	no	—
554	D₄	—	2	1	no	—
555	D₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	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	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	8	1	no	—
575	C₂×C₄	—	8	1	no	—
576	C₂×C₄	—	8	1	no	—
577	C₂×C₄	—	16	1	no	—
578	C₂×C₄	—	16	1	no	—
579	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
580	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
581	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
582	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
583	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
584	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
585	C₈	—	2	1	no	—
586	C₈	—	2	1	no	—
587	C₈	—	2	2	no	—
588	C₈	—	2	1	no	—
589	C₈	—	2	2	no	—
590	C₈	—	2	1	no	—
591	C₈	—	4	1	no	—
592	C₈	—	4	1	no	—
593	C₂²	—	2	1	no	—
594	C₂²	—	2	1	no	—
595	C₂²	—	2	2	no	—
596	C₂²	—	4	1	no	—
597	C₂²	—	4	1	no	—
598	C₂²	C₂₃×#539	1	1	yes	characteristic
599	C₂²	D₄²×C₂₃	1	1	yes	characteristic
600	C₂²	C₂₃×#554	1	1	yes	characteristic
601	C₄	—	2	2	no	—
602	C₄	—	4	2	no	—
603	C₄	—	4	1	no	—
604	C₄	—	4	1	no	—
605	C₄	—	4	1	no	—
606	C₄	—	4	1	no	—
607	C₄	—	4	1	no	—
608	C₄	—	4	1	no	—
609	C₄	—	8	1	no	—
610	C₄	—	8	1	no	—
611	C₄	—	8	1	no	—
612	C₄	—	8	1	no	—
613	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
614	C₄	C₂₃×#525	1	1	yes	characteristic
615	C₄	C₂₃×#547	1	1	yes	characteristic
616	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
617	C₂	—	2	1	no	—
618	C₂	—	2	1	no	—
619	C₂	—	4	2	no	—
620	C₂	C₂₃×#2449	1	1	yes	characteristic
621	C₂	C₂₃×#3289	1	1	yes	characteristic
622	C₂	C₂₃×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×#41×C₂₃	C₂³
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
2. C₂₃×#3082	C₂	semidirect
5. C₂₃×#2097	C₂	semidirect
6. C₂₃×#2089	C₂	semidirect
7. C₂₃×#2064	C₂	semidirect
8. C₂₃×#2061	C₂	semidirect
9. C₂₃×#1951	C₂	semidirect
10. C₂₃×#1922	C₂	semidirect
15. C₂₃×#1857	C₂	semidirect
70. C₂₃×#137	Q₈	semidirect
231. #26403	C₂₃	direct (N × H)
283. #3082	C₂×C₂₃	semidirect
286. #2097	C₂×C₂₃	semidirect
287. #2089	C₂×C₂₃	semidirect
288. #2064	C₂×C₂₃	semidirect
289. #2061	C₂×C₂₃	semidirect
290. #1951	C₂×C₂₃	semidirect
291. #1922	C₂×C₂₃	semidirect
296. #1857	C₂×C₂₃	semidirect
381. #137	Q₈×C₂₃	semidirect
462. C₂₃	#26403	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3084	2
C₂₃×#3082	2
C₂₃×#2947	2
C₂₃×#2913	2
C₂₃×#2097	2
C₂₃×#2089	2
C₂₃×#2064	2
C₂₃×#2061	2
C₂₃×#1951	2
C₂₃×#1922	2
C₂₃×#1908	2
C₂₃×#1882	2
C₂₃×#1866	2
C₂₃×#1860	2
C₂₃×#1857	2
#26403	23

Subgroups — C23×#26403 #19174029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26403 #19174029