Subgroups — C₂₃×#26084 #19173710

Order 5888.

678 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3084	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3083	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2912	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2090	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2089	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2061	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2060	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1951	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1946	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1860	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#1859	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1847	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1816	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#1765	C₂	1	1	yes	maximal, characteristic
16	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
17	C₂₃×#544	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×#501	—	2	1	no	—
22	C₂₃×#501	—	2	1	no	—
23	C₂₃×#500	—	2	1	no	—
24	C₂₃×#499	—	2	1	no	—
25	C₂₃×#479	—	2	1	no	—
26	C₂₃×#479	—	2	1	no	—
27	C₂₃×#478	—	2	1	no	—
28	C₂₃×#478	—	2	1	no	—
29	C₂₃×#477	—	2	1	no	—
30	C₂₃×#476	—	2	1	no	—
31	C₂₃×#475	—	2	1	no	—
32	C₂₃×#474	—	2	1	no	—
33	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
34	C₂₃×#428	—	2	1	no	—
35	C₂₃×#428	—	2	1	no	—
36	C₂₃×#428	C₂²	1	1	yes	characteristic
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	C₂²	1	1	yes	characteristic
45	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
46	C₂×C₂₃×#106	—	2	1	no	—
47	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
48	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
49	C₂₃×#424	C₂²	1	1	yes	characteristic
50	C₂₃×#424	C₂²	1	1	yes	characteristic
51	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
52	C₂₃×#417	C₂²	1	1	yes	characteristic
53	C₂₃×#417	C₂²	1	1	yes	characteristic
54	C₂₃×#416	C₂²	1	1	yes	characteristic
55	C₂₃×#405	C₂²	1	1	yes	characteristic
56	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
57	C₂₃×#384	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×#343	C₂²	1	1	yes	characteristic
63	C₂₃×#341	C₂²	1	1	yes	characteristic
64	C₂₃×#339	C₂²	1	1	yes	characteristic
65	C₂₃×#339	C₂²	1	1	yes	characteristic
66	C₂₃×#339	C₂²	1	1	yes	characteristic
67	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
68	C₂₃×#131	—	2	2	no	—
69	C₂₃×#131	Q₈	1	4	yes	complemented
70	C₂×M₁₆×C₂₃	—	2	1	no	—
71	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
72	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
74	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₂₃	—	2	1	no	—
76	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₂₃×C₄⋊Q₈	—	2	1	no	—
79	C₂₃×C₄⋊Q₈	—	2	1	no	—
80	C₂₃×C₄⋊Q₈	—	2	1	no	—
81	C₂₃×C₄⋊Q₈	—	2	1	no	—
82	C₂₃×#125	—	2	1	no	—
83	C₂₃×#125	—	2	1	no	—
84	C₂₃×#125	—	2	1	no	—
85	C₂₃×#125	—	2	1	no	—
86	C₂₃×C₂²⋊Q₈	—	4	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×D₄×C₂₃	—	2	1	no	—
91	C₄×D₄×C₂₃	—	2	1	no	—
92	C₄×D₄×C₂₃	—	4	1	no	—
93	C₄×D₄×C₂₃	—	4	1	no	—
94	C₂₃×#117	—	2	1	no	—
95	C₂₃×#117	—	2	1	no	—
96	C₂₃×#117	C₂³	1	1	yes	characteristic
97	C₂₃×#117	C₂³	1	1	yes	characteristic
98	C₂₃×#117	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
107	C₂₃×#108	—	2	2	no	—
108	C₂₃×#107	—	2	1	no	—
109	C₂₃×#107	—	2	1	no	—
110	C₂₃×#107	—	2	1	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	—	2	1	no	—
128	C₂₃×#106	—	2	1	no	—
129	C₂₃×#106	—	2	1	no	—
130	C₂₃×#106	D₄	1	1	yes	characteristic
131	C₂₃×#106	D₄	1	1	yes	characteristic
132	C₂₃×C₄⋊C₈	—	2	1	no	—
133	C₂₃×C₄⋊C₈	—	2	1	no	—
134	C₂₃×C₄⋊C₈	—	2	1	no	—
135	C₂₃×C₄⋊C₈	—	2	1	no	—
136	C₂₃×#104	Q₈	1	4	yes	complemented
137	C₂₃×#103	—	2	1	no	—
138	C₂₃×#103	—	2	1	no	—
139	C₂₃×#103	D₄	1	1	yes	characteristic
140	C₂₃×#103	D₄	1	1	yes	characteristic
141	C₂₃×#102	—	2	1	no	—
142	C₂₃×#102	—	2	1	no	—
143	C₂₃×#102	D₄	1	1	yes	characteristic
144	C₂₃×#102	D₄	1	1	yes	characteristic
145	C₂₃×#97	—	2	1	no	—
146	C₂₃×#97	—	2	2	no	—
147	C₄×C₈×C₂₃	—	2	1	no	—
148	C₂₃×#95	—	4	1	no	—
149	#41×C₂₃	—	2	1	no	—
150	#41×C₂₃	—	2	1	no	—
151	#41×C₂₃	C₂×Q₈	1	2	yes	—
152	#41×C₂₃	C₂×Q₈	1	2	yes	—
153	C₂×Q₈×C₂₃	—	4	1	no	—
154	C₂×Q₈×C₂₃	—	4	1	no	—
155	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₂×D₄×C₂₃	—	2	1	no	—
157	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	—	2	1	no	—
159	C₂²×C₄×C₂₃	—	2	1	no	—
160	C₂²×C₄×C₂₃	—	2	1	no	—
161	C₂²×C₄×C₂₃	—	2	1	no	—
162	C₂²×C₄×C₂₃	—	2	1	no	—
163	C₂²×C₄×C₂₃	—	2	1	no	—
164	C₂²×C₄×C₂₃	—	4	1	no	—
165	C₂²×C₄×C₂₃	—	4	1	no	—
166	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
167	M₁₆×C₂₃	—	2	1	no	—
168	M₁₆×C₂₃	—	2	1	no	—
169	M₁₆×C₂₃	—	2	4	no	—
170	M₁₆×C₂₃	—	2	1	no	—
171	M₁₆×C₂₃	—	2	1	no	—
172	M₁₆×C₂₃	—	2	1	no	—
173	M₁₆×C₂₃	—	2	1	no	—
174	M₁₆×C₂₃	C₂×Q₈	1	2	yes	—
175	M₁₆×C₂₃	C₂×Q₈	1	2	yes	—
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	4	no	—
181	C₂×C₈×C₂₃	—	2	1	no	—
182	C₂×C₈×C₂₃	—	2	1	no	—
183	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
185	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
186	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
188	C₂×C₈×C₂₃	#41	1	2	yes	—
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	—	2	1	no	—
196	C₄⋊C₄×C₂₃	—	2	1	no	—
197	C₄⋊C₄×C₂₃	—	4	1	no	—
198	C₄⋊C₄×C₂₃	—	4	1	no	—
199	C₄⋊C₄×C₂₃	—	4	1	no	—
200	C₄⋊C₄×C₂₃	—	4	1	no	—
201	C₄⋊C₄×C₂₃	—	4	1	no	—
202	C₄⋊C₄×C₂₃	—	4	1	no	—
203	C₄⋊C₄×C₂₃	—	4	1	no	—
204	C₄⋊C₄×C₂₃	—	4	1	no	—
205	C₄⋊C₄×C₂₃	—	8	2	no	—
206	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
207	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
208	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
209	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
210	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
211	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
212	C₂²⋊C₄×C₂₃	—	2	1	no	—
213	C₂²⋊C₄×C₂₃	—	2	1	no	—
214	C₂²⋊C₄×C₂₃	—	2	1	no	—
215	C₂²⋊C₄×C₂₃	—	4	1	no	—
216	C₂²⋊C₄×C₂₃	—	4	1	no	—
217	C₄²×C₂₃	—	2	1	no	—
218	C₄²×C₂₃	—	2	1	no	—
219	C₄²×C₂₃	—	2	1	no	—
220	C₄²×C₂₃	—	2	1	no	—
221	C₄²×C₂₃	—	4	1	no	—
222	C₄²×C₂₃	—	4	1	no	—
223	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
224	#26084	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂³×C₂₃	—	2	1	no	—
226	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
227	Q₈×C₂₃	—	2	1	no	—
228	Q₈×C₂₃	—	16	1	no	—
229	Q₈×C₂₃	—	16	1	no	—
230	Q₈×C₂₃	C₄⋊Q₈	1	2	yes	—
231	D₄×C₂₃	—	2	2	no	—
232	D₄×C₂₃	—	2	2	no	—
233	D₄×C₂₃	—	2	1	no	—
234	D₄×C₂₃	C₄⋊Q₈	1	2	yes	—
235	C₂×C₄×C₂₃	—	2	1	no	—
236	C₂×C₄×C₂₃	—	2	2	no	—
237	C₂×C₄×C₂₃	—	2	2	no	—
238	C₂×C₄×C₂₃	—	2	1	no	—
239	C₂×C₄×C₂₃	—	2	1	no	—
240	C₂×C₄×C₂₃	—	2	1	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	2	no	—
245	C₂×C₄×C₂₃	—	4	1	no	—
246	C₂×C₄×C₂₃	—	4	1	no	—
247	C₂×C₄×C₂₃	—	4	1	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	—	8	1	no	—
252	C₂×C₄×C₂₃	—	8	1	no	—
253	C₂×C₄×C₂₃	—	8	1	no	—
254	C₂×C₄×C₂₃	—	8	1	no	—
255	C₂×C₄×C₂₃	—	16	1	no	—
256	C₂×C₄×C₂₃	—	16	1	no	—
257	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
260	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
261	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
262	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
263	C₈×C₂₃	—	2	2	no	—
264	C₈×C₂₃	—	2	1	no	—
265	C₈×C₂₃	—	2	1	no	—
266	C₈×C₂₃	—	2	2	no	—
267	C₈×C₂₃	—	4	1	no	—
268	C₈×C₂₃	—	4	1	no	—
269	C₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
270	C₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
271	#3164	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#3084	C₂×C₂₃	1	1	yes	characteristic
273	#3083	C₂×C₂₃	1	1	yes	characteristic
274	#2912	C₂×C₂₃	1	1	yes	characteristic
275	#2090	C₂×C₂₃	1	1	yes	characteristic, complemented
276	#2089	C₂×C₂₃	1	1	yes	characteristic, complemented
277	#2061	C₂×C₂₃	1	1	yes	characteristic, complemented
278	#2060	C₂×C₂₃	1	1	yes	characteristic, complemented
279	#1951	C₂×C₂₃	1	1	yes	characteristic, complemented
280	#1946	C₂×C₂₃	1	1	yes	characteristic, complemented
281	#1860	C₂×C₂₃	1	1	yes	characteristic
282	#1859	C₂×C₂₃	1	1	yes	characteristic
283	#1847	C₂×C₂₃	1	1	yes	characteristic, complemented
284	#1816	C₂×C₂₃	1	1	yes	characteristic
285	#1765	C₂×C₂₃	1	1	yes	characteristic
286	C₂²×C₂₃	—	2	1	no	—
287	C₂²×C₂₃	—	2	1	no	—
288	C₂²×C₂₃	—	2	2	no	—
289	C₂²×C₂₃	—	4	1	no	—
290	C₂²×C₂₃	—	4	1	no	—
291	C₂²×C₂₃	#544	1	1	yes	characteristic
292	C₂²×C₂₃	D₄²	1	1	yes	characteristic
293	C₂²×C₂₃	#554	1	1	yes	characteristic
294	C₄×C₂₃	—	2	2	no	—
295	C₄×C₂₃	—	4	2	no	—
296	C₄×C₂₃	—	4	1	no	—
297	C₄×C₂₃	—	4	1	no	—
298	C₄×C₂₃	—	4	1	no	—
299	C₄×C₂₃	—	4	1	no	—
300	C₄×C₂₃	—	4	1	no	—
301	C₄×C₂₃	—	4	1	no	—
302	C₄×C₂₃	—	8	1	no	—
303	C₄×C₂₃	—	8	1	no	—
304	C₄×C₂₃	—	8	1	no	—
305	C₄×C₂₃	—	8	1	no	—
306	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
307	C₄×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
308	C₄×C₂₃	#547	1	1	yes	characteristic
309	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
310	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
311	#544	C₂²×C₂₃	1	1	yes	characteristic
312	#544	C₂²×C₂₃	1	1	yes	characteristic
313	#520	C₂²×C₂₃	1	1	yes	characteristic
314	#520	C₂²×C₂₃	1	1	yes	characteristic
315	#501	—	2	1	no	—
316	#501	—	2	1	no	—
317	#500	—	2	1	no	—
318	#499	—	2	1	no	—
319	#479	—	2	1	no	—
320	#479	—	2	1	no	—
321	#478	—	2	1	no	—
322	#478	—	2	1	no	—
323	#477	—	2	1	no	—
324	#476	—	2	1	no	—
325	#475	—	2	1	no	—
326	#474	—	2	1	no	—
327	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
328	#428	—	2	1	no	—
329	#428	—	2	1	no	—
330	#428	C₂²×C₂₃	1	1	yes	characteristic
331	#428	C₂²×C₂₃	1	1	yes	characteristic
332	#428	C₂²×C₂₃	1	1	yes	characteristic
333	#428	C₂²×C₂₃	1	1	yes	characteristic
334	#428	C₂²×C₂₃	1	1	yes	characteristic
335	#428	C₂²×C₂₃	1	1	yes	characteristic
336	#427	—	2	1	no	—
337	#427	C₂²×C₂₃	1	1	yes	characteristic
338	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
339	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
340	C₂×#106	—	2	1	no	—
341	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
343	#424	C₂²×C₂₃	1	1	yes	characteristic
344	#424	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
346	#417	C₂²×C₂₃	1	1	yes	characteristic
347	#417	C₂²×C₂₃	1	1	yes	characteristic
348	#416	C₂²×C₂₃	1	1	yes	characteristic
349	#405	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
351	#384	C₂²×C₂₃	1	1	yes	characteristic
352	#382	C₂²×C₂₃	1	1	yes	characteristic
353	#343	C₂²×C₂₃	1	1	yes	characteristic
354	#343	C₂²×C₂₃	1	1	yes	characteristic
355	#343	C₂²×C₂₃	1	1	yes	characteristic
356	#343	C₂²×C₂₃	1	1	yes	characteristic
357	#341	C₂²×C₂₃	1	1	yes	characteristic
358	#339	C₂²×C₂₃	1	1	yes	characteristic
359	#339	C₂²×C₂₃	1	1	yes	characteristic
360	#339	C₂²×C₂₃	1	1	yes	characteristic
361	C₂×C₂₃	—	2	1	no	—
362	C₂×C₂₃	—	2	1	no	—
363	C₂×C₂₃	—	4	2	no	—
364	C₂×C₂₃	#2440	1	1	yes	characteristic
365	C₂×C₂₃	#3288	1	1	yes	characteristic
366	C₂×C₂₃	#3289	1	1	yes	characteristic
367	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
368	#131	—	2	2	no	—
369	#131	Q₈×C₂₃	1	4	yes	complemented
370	C₂×M₁₆	—	2	1	no	—
371	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
372	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
373	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
374	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
375	C₂²×C₈	—	2	1	no	—
376	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
377	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
378	C₄⋊Q₈	—	2	1	no	—
379	C₄⋊Q₈	—	2	1	no	—
380	C₄⋊Q₈	—	2	1	no	—
381	C₄⋊Q₈	—	2	1	no	—
382	#125	—	2	1	no	—
383	#125	—	2	1	no	—
384	#125	—	2	1	no	—
385	#125	—	2	1	no	—
386	C₂²⋊Q₈	—	4	1	no	—
387	C₂²⋊Q₈	—	4	1	no	—
388	C₄×Q₈	—	2	1	no	—
389	C₄×Q₈	—	2	1	no	—
390	C₄×D₄	—	2	1	no	—
391	C₄×D₄	—	2	1	no	—
392	C₄×D₄	—	4	1	no	—
393	C₄×D₄	—	4	1	no	—
394	#117	—	2	1	no	—
395	#117	—	2	1	no	—
396	#117	C₂³×C₂₃	1	1	yes	characteristic
397	#117	C₂³×C₂₃	1	1	yes	characteristic
398	#117	C₂³×C₂₃	1	1	yes	characteristic
399	C₂×C₄⋊C₄	—	2	1	no	—
400	C₂×C₄⋊C₄	—	2	1	no	—
401	C₂×C₄⋊C₄	—	4	1	no	—
402	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
403	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
404	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
405	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
406	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
407	#108	—	2	2	no	—
408	#107	—	2	1	no	—
409	#107	—	2	1	no	—
410	#107	—	2	1	no	—
411	#107	—	2	1	no	—
412	#107	—	2	1	no	—
413	#107	—	2	1	no	—
414	#107	—	2	1	no	—
415	#107	—	2	1	no	—
416	#107	—	2	1	no	—
417	#107	—	2	1	no	—
418	#107	D₄×C₂₃	1	1	yes	characteristic
419	#107	D₄×C₂₃	1	1	yes	characteristic
420	#106	—	2	1	no	—
421	#106	—	2	1	no	—
422	#106	—	2	1	no	—
423	#106	—	2	1	no	—
424	#106	—	2	1	no	—
425	#106	—	2	1	no	—
426	#106	—	2	1	no	—
427	#106	—	2	1	no	—
428	#106	—	2	1	no	—
429	#106	—	2	1	no	—
430	#106	D₄×C₂₃	1	1	yes	characteristic
431	#106	D₄×C₂₃	1	1	yes	characteristic
432	C₄⋊C₈	—	2	1	no	—
433	C₄⋊C₈	—	2	1	no	—
434	C₄⋊C₈	—	2	1	no	—
435	C₄⋊C₈	—	2	1	no	—
436	#104	Q₈×C₂₃	1	4	yes	complemented
437	#103	—	2	1	no	—
438	#103	—	2	1	no	—
439	#103	D₄×C₂₃	1	1	yes	characteristic
440	#103	D₄×C₂₃	1	1	yes	characteristic
441	#102	—	2	1	no	—
442	#102	—	2	1	no	—
443	#102	D₄×C₂₃	1	1	yes	characteristic
444	#102	D₄×C₂₃	1	1	yes	characteristic
445	#97	—	2	1	no	—
446	#97	—	2	2	no	—
447	C₄×C₈	—	2	1	no	—
448	#95	—	4	1	no	—
449	C₂₃	#26084	1	1	yes	characteristic, direct product factor, Sylow
450	#41	—	2	1	no	—
451	#41	—	2	1	no	—
452	#41	C₂×Q₈×C₂₃	1	2	yes	—
453	#41	C₂×Q₈×C₂₃	1	2	yes	—
454	C₂×Q₈	—	4	1	no	—
455	C₂×Q₈	—	4	1	no	—
456	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	4	1	no	—
466	C₂²×C₄	—	4	1	no	—
467	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
468	M₁₆	—	2	1	no	—
469	M₁₆	—	2	1	no	—
470	M₁₆	—	2	4	no	—
471	M₁₆	—	2	1	no	—
472	M₁₆	—	2	1	no	—
473	M₁₆	—	2	1	no	—
474	M₁₆	—	2	1	no	—
475	M₁₆	C₂×Q₈×C₂₃	1	2	yes	—
476	M₁₆	C₂×Q₈×C₂₃	1	2	yes	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	4	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
485	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
486	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
487	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
488	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
489	C₂×C₈	#41×C₂₃	1	2	yes	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	4	1	no	—
499	C₄⋊C₄	—	4	1	no	—
500	C₄⋊C₄	—	4	1	no	—
501	C₄⋊C₄	—	4	1	no	—
502	C₄⋊C₄	—	4	1	no	—
503	C₄⋊C₄	—	4	1	no	—
504	C₄⋊C₄	—	4	1	no	—
505	C₄⋊C₄	—	4	1	no	—
506	C₄⋊C₄	—	8	2	no	—
507	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
508	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
509	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
510	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
511	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
512	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
513	C₂²⋊C₄	—	2	1	no	—
514	C₂²⋊C₄	—	2	1	no	—
515	C₂²⋊C₄	—	2	1	no	—
516	C₂²⋊C₄	—	4	1	no	—
517	C₂²⋊C₄	—	4	1	no	—
518	C₄²	—	2	1	no	—
519	C₄²	—	2	1	no	—
520	C₄²	—	2	1	no	—
521	C₄²	—	2	1	no	—
522	C₄²	—	4	1	no	—
523	C₄²	—	4	1	no	—
524	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
525	C₂³	—	2	1	no	—
526	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
527	Q₈	—	2	1	no	—
528	Q₈	—	16	1	no	—
529	Q₈	—	16	1	no	—
530	Q₈	C₂₃×C₄⋊Q₈	1	2	yes	—
531	D₄	—	2	2	no	—
532	D₄	—	2	2	no	—
533	D₄	—	2	1	no	—
534	D₄	C₂₃×C₄⋊Q₈	1	2	yes	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	8	1	no	—
552	C₂×C₄	—	8	1	no	—
553	C₂×C₄	—	8	1	no	—
554	C₂×C₄	—	8	1	no	—
555	C₂×C₄	—	16	1	no	—
556	C₂×C₄	—	16	1	no	—
557	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
558	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
559	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
560	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
561	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
562	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
563	C₈	—	2	1	no	—
564	C₈	—	2	1	no	—
565	C₈	—	2	2	no	—
566	C₈	—	2	2	no	—
567	C₈	—	4	1	no	—
568	C₈	—	4	1	no	—
569	C₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
570	C₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
571	C₂²	—	2	1	no	—
572	C₂²	—	2	1	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	4	1	no	—
575	C₂²	—	4	1	no	—
576	C₂²	C₂₃×#544	1	1	yes	characteristic
577	C₂²	D₄²×C₂₃	1	1	yes	characteristic
578	C₂²	C₂₃×#554	1	1	yes	characteristic
579	C₄	—	2	2	no	—
580	C₄	—	4	2	no	—
581	C₄	—	4	1	no	—
582	C₄	—	4	1	no	—
583	C₄	—	4	1	no	—
584	C₄	—	4	1	no	—
585	C₄	—	4	1	no	—
586	C₄	—	4	1	no	—
587	C₄	—	8	1	no	—
588	C₄	—	8	1	no	—
589	C₄	—	8	1	no	—
590	C₄	—	8	1	no	—
591	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
592	C₄	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
593	C₄	C₂₃×#547	1	1	yes	characteristic
594	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
595	C₂	—	2	1	no	—
596	C₂	—	2	1	no	—
597	C₂	—	4	2	no	—
598	C₂	C₂₃×#2440	1	1	yes	characteristic
599	C₂	C₂₃×#3288	1	1	yes	characteristic
600	C₂	C₂₃×#3289	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
1. C₂₃×#3164	C₂	semidirect
5. C₂₃×#2090	C₂	semidirect
6. C₂₃×#2089	C₂	semidirect
7. C₂₃×#2061	C₂	semidirect
8. C₂₃×#2060	C₂	semidirect
9. C₂₃×#1951	C₂	semidirect
10. C₂₃×#1946	C₂	semidirect
13. C₂₃×#1847	C₂	semidirect
69. C₂₃×#131	Q₈	semidirect
136. C₂₃×#104	Q₈	semidirect
224. #26084	C₂₃	direct (N × H)
271. #3164	C₂×C₂₃	semidirect
275. #2090	C₂×C₂₃	semidirect
276. #2089	C₂×C₂₃	semidirect
277. #2061	C₂×C₂₃	semidirect
278. #2060	C₂×C₂₃	semidirect
279. #1951	C₂×C₂₃	semidirect
280. #1946	C₂×C₂₃	semidirect
283. #1847	C₂×C₂₃	semidirect
369. #131	Q₈×C₂₃	semidirect
436. #104	Q₈×C₂₃	semidirect
449. C₂₃	#26084	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×#3164	2
C₂₃×#3084	2
C₂₃×#3083	2
C₂₃×#2912	2
C₂₃×#2090	2
C₂₃×#2089	2
C₂₃×#2061	2
C₂₃×#2060	2
C₂₃×#1951	2
C₂₃×#1946	2
C₂₃×#1860	2
C₂₃×#1859	2
C₂₃×#1847	2
C₂₃×#1816	2
C₂₃×#1765	2
#26084	23

Subgroups — C23×#26084 #19173710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26084 #19173710