Subgroups — C₂₃×#26192 #19173818

Order 5888.

734 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3097	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3046	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2951	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2912	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2097	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2089	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2079	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2038	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1930	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, complemented
12	C₂₃×#1882	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1876	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#1860	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1860	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#137	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
19	C₂₃×#536	C₂²	1	1	yes	characteristic
20	C₂₃×#520	C₂²	1	1	yes	characteristic
21	C₂₃×#520	C₂²	1	1	yes	characteristic
22	C₂₃×#520	C₂²	1	1	yes	characteristic
23	C₂₃×#485	—	2	1	no	—
24	C₂₃×#485	—	2	1	no	—
25	C₂₃×#484	—	2	1	no	—
26	C₂₃×#483	—	2	1	no	—
27	C₂₃×#464	—	2	1	no	—
28	C₂₃×#463	—	2	1	no	—
29	C₂₃×#462	—	2	1	no	—
30	C₂₃×#461	—	2	1	no	—
31	C₂₃×#440	—	2	1	no	—
32	C₂₃×#440	—	2	1	no	—
33	C₄×Q₁₆×C₂₃	—	2	1	no	—
34	C₄×D₈×C₂₃	—	2	1	no	—
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×#427	—	2	1	no	—
41	C₂₃×#427	C₂²	1	1	yes	characteristic
42	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
43	C₂×C₂₃×#106	—	2	1	no	—
44	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
45	C₂₃×#424	C₂²	1	1	yes	characteristic
46	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
47	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
48	C₂₃×#419	C₂²	1	1	yes	characteristic
49	C₂₃×#417	C₂²	1	1	yes	characteristic
50	C₂₃×#417	C₂²	1	1	yes	characteristic
51	C₂₃×#417	C₂²	1	1	yes	characteristic
52	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
53	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
54	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
55	C₂₃×#408	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×#341	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	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
70	C₂₃×#137	—	2	2	no	—
71	C₂₃×#137	—	2	2	no	—
72	C₂₃×#135	—	2	2	no	—
73	C₂₃×#135	—	2	2	no	—
74	C₂×Q₁₆×C₂₃	—	2	1	no	—
75	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
76	C₂×SD₁₆×C₂₃	—	2	1	no	—
77	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
78	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
79	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
80	C₂₃×#131	—	2	2	no	—
81	C₂₃×#131	—	2	2	no	—
82	C₂×M₁₆×C₂₃	—	2	1	no	—
83	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
84	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
85	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
86	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
87	C₂²×C₈×C₂₃	—	2	1	no	—
88	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
90	C₂₃×C₄⋊Q₈	—	2	1	no	—
91	C₂₃×#124	—	2	1	no	—
92	C₂₃×#123	—	4	1	no	—
93	C₂₃×C₂²⋊Q₈	—	2	1	no	—
94	C₂₃×C₂²⋊Q₈	—	2	1	no	—
95	C₄×Q₈×C₂₃	—	2	1	no	—
96	C₄×Q₈×C₂₃	—	2	1	no	—
97	C₄×Q₈×C₂₃	—	2	1	no	—
98	C₄×D₄×C₂₃	—	2	1	no	—
99	C₄×D₄×C₂₃	—	2	1	no	—
100	C₄×D₄×C₂₃	—	2	1	no	—
101	C₄×D₄×C₂₃	—	4	1	no	—
102	C₄×D₄×C₂₃	—	4	1	no	—
103	C₄×D₄×C₂₃	—	4	1	no	—
104	C₂₃×#117	—	2	1	no	—
105	C₂₃×#117	—	2	1	no	—
106	C₂₃×#117	—	2	1	no	—
107	C₂₃×#117	C₂³	1	1	yes	characteristic
108	C₂₃×#117	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
110	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
111	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
112	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
116	C₂₃×#107	—	2	1	no	—
117	C₂₃×#107	—	2	1	no	—
118	C₂₃×#107	—	2	1	no	—
119	C₂₃×#107	—	2	1	no	—
120	C₂₃×#107	—	2	1	no	—
121	C₂₃×#107	D₄	1	1	yes	characteristic
122	C₂₃×#107	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
129	C₂₃×#106	D₄	1	1	yes	characteristic
130	C₂₃×C₄⋊C₈	—	2	1	no	—
131	C₂₃×C₄⋊C₈	—	2	1	no	—
132	C₂₃×#104	—	2	2	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	—	2	1	no	—
142	C₂₃×#103	—	2	1	no	—
143	C₂₃×#102	—	2	1	no	—
144	C₂₃×#102	—	2	1	no	—
145	C₂₃×#102	—	2	1	no	—
146	C₂₃×#101	—	2	2	no	—
147	C₂₃×C₂²⋊C₈	—	2	1	no	—
148	C₂₃×C₂²⋊C₈	—	2	1	no	—
149	C₂₃×#97	—	2	1	no	—
150	C₄×C₈×C₂₃	—	2	1	no	—
151	C₂₃×#95	—	4	1	no	—
152	C₂₃×#95	—	4	1	no	—
153	#41×C₂₃	—	2	1	no	—
154	#41×C₂₃	—	2	1	no	—
155	#41×C₂₃	—	2	1	no	—
156	#41×C₂₃	—	2	1	no	—
157	#41×C₂₃	—	2	1	no	—
158	#41×C₂₃	—	2	1	no	—
159	#41×C₂₃	#41	1	2	yes	—
160	#41×C₂₃	#41	1	2	yes	—
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	—	2	1	no	—
165	C₂×Q₈×C₂₃	—	2	1	no	—
166	C₂×Q₈×C₂₃	—	2	1	no	—
167	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
168	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₂×D₄×C₂₃	—	2	1	no	—
170	C₂×D₄×C₂₃	—	2	1	no	—
171	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
172	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₂₃	—	2	1	no	—
174	C₂²×C₄×C₂₃	—	2	1	no	—
175	C₂²×C₄×C₂₃	—	2	1	no	—
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	—	4	1	no	—
181	C₂²×C₄×C₂₃	—	4	1	no	—
182	C₂²×C₄×C₂₃	—	4	1	no	—
183	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
184	Q₁₆×C₂₃	—	2	2	no	—
185	Q₁₆×C₂₃	—	4	2	no	—
186	SD₁₆×C₂₃	—	2	2	no	—
187	SD₁₆×C₂₃	—	2	2	no	—
188	SD₁₆×C₂₃	—	4	2	no	—
189	D₈×C₂₃	—	2	2	no	—
190	M₁₆×C₂₃	—	2	4	no	—
191	M₁₆×C₂₃	—	2	1	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	#41	1	2	yes	—
199	C₂×C₈×C₂₃	—	2	4	no	—
200	C₂×C₈×C₂₃	—	2	1	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	—	2	1	no	—
208	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
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	1	yes	characteristic
212	C₄⋊C₄×C₂₃	—	2	1	no	—
213	C₄⋊C₄×C₂₃	—	2	1	no	—
214	C₄⋊C₄×C₂₃	—	2	1	no	—
215	C₄⋊C₄×C₂₃	—	2	1	no	—
216	C₄⋊C₄×C₂₃	—	2	1	no	—
217	C₄⋊C₄×C₂₃	—	2	1	no	—
218	C₄⋊C₄×C₂₃	—	4	1	no	—
219	C₄⋊C₄×C₂₃	—	4	1	no	—
220	C₄⋊C₄×C₂₃	—	4	1	no	—
221	C₄⋊C₄×C₂₃	—	4	1	no	—
222	C₄⋊C₄×C₂₃	—	4	1	no	—
223	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
224	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
225	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
226	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
227	C₂²⋊C₄×C₂₃	—	2	1	no	—
228	C₂²⋊C₄×C₂₃	—	2	1	no	—
229	C₂²⋊C₄×C₂₃	—	4	1	no	—
230	C₂²⋊C₄×C₂₃	—	4	1	no	—
231	C₂²⋊C₄×C₂₃	—	4	1	no	—
232	C₂²⋊C₄×C₂₃	—	4	1	no	—
233	C₂²⋊C₄×C₂₃	—	4	1	no	—
234	C₄²×C₂₃	—	2	1	no	—
235	C₄²×C₂₃	—	2	1	no	—
236	C₄²×C₂₃	—	2	1	no	—
237	C₄²×C₂₃	—	2	1	no	—
238	C₄²×C₂₃	—	4	1	no	—
239	C₄²×C₂₃	—	4	1	no	—
240	C₄²×C₂₃	—	4	1	no	—
241	#26192	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
242	C₂³×C₂₃	—	2	1	no	—
243	C₂³×C₂₃	—	2	1	no	—
244	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
245	Q₈×C₂₃	—	2	1	no	—
246	Q₈×C₂₃	—	2	1	no	—
247	Q₈×C₂₃	—	2	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	#123	1	2	yes	—
253	D₄×C₂₃	—	2	2	no	—
254	D₄×C₂₃	—	2	2	no	—
255	D₄×C₂₃	—	2	1	no	—
256	D₄×C₂₃	—	2	1	no	—
257	D₄×C₂₃	—	2	1	no	—
258	D₄×C₂₃	—	4	1	no	—
259	D₄×C₂₃	—	4	1	no	—
260	D₄×C₂₃	#123	1	2	yes	—
261	C₂×C₄×C₂₃	—	2	1	no	—
262	C₂×C₄×C₂₃	—	2	2	no	—
263	C₂×C₄×C₂₃	—	2	2	no	—
264	C₂×C₄×C₂₃	—	2	1	no	—
265	C₂×C₄×C₂₃	—	2	1	no	—
266	C₂×C₄×C₂₃	—	2	1	no	—
267	C₂×C₄×C₂₃	—	2	1	no	—
268	C₂×C₄×C₂₃	—	2	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	—	4	1	no	—
278	C₂×C₄×C₂₃	—	4	1	no	—
279	C₂×C₄×C₂₃	—	4	1	no	—
280	C₂×C₄×C₂₃	—	8	1	no	—
281	C₂×C₄×C₂₃	—	8	1	no	—
282	C₂×C₄×C₂₃	—	8	1	no	—
283	C₂×C₄×C₂₃	—	16	1	no	—
284	C₂×C₄×C₂₃	—	16	1	no	—
285	C₂×C₄×C₂₃	—	16	1	no	—
286	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
288	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
289	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
290	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
291	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
292	C₈×C₂₃	—	2	1	no	—
293	C₈×C₂₃	—	2	1	no	—
294	C₈×C₂₃	—	2	1	no	—
295	C₈×C₂₃	—	2	2	no	—
296	C₈×C₂₃	—	2	1	no	—
297	C₈×C₂₃	—	4	1	no	—
298	C₈×C₂₃	—	4	1	no	—
299	C₈×C₂₃	—	4	1	no	—
300	#3097	C₂×C₂₃	1	1	yes	characteristic, complemented
301	#3046	C₂×C₂₃	1	1	yes	characteristic
302	#2951	C₂×C₂₃	1	1	yes	characteristic
303	#2912	C₂×C₂₃	1	1	yes	characteristic, complemented
304	#2097	C₂×C₂₃	1	1	yes	characteristic, complemented
305	#2089	C₂×C₂₃	1	1	yes	characteristic, complemented
306	#2079	C₂×C₂₃	1	1	yes	characteristic, complemented
307	#2038	C₂×C₂₃	1	1	yes	characteristic, complemented
308	#1930	C₂×C₂₃	1	1	yes	characteristic, complemented
309	#1922	C₂×C₂₃	1	1	yes	characteristic, complemented
310	#1908	C₂×C₂₃	1	1	yes	characteristic, complemented
311	#1882	C₂×C₂₃	1	1	yes	characteristic, complemented
312	#1876	C₂×C₂₃	1	1	yes	characteristic
313	#1860	C₂×C₂₃	1	1	yes	characteristic, complemented
314	#1860	C₂×C₂₃	1	1	yes	characteristic, complemented
315	C₂²×C₂₃	—	2	1	no	—
316	C₂²×C₂₃	—	2	1	no	—
317	C₂²×C₂₃	—	2	2	no	—
318	C₂²×C₂₃	—	4	1	no	—
319	C₂²×C₂₃	—	4	1	no	—
320	C₂²×C₂₃	—	4	1	no	—
321	C₂²×C₂₃	—	4	1	no	—
322	C₂²×C₂₃	—	4	1	no	—
323	C₂²×C₂₃	#539	1	1	yes	characteristic
324	C₂²×C₂₃	D₄²	1	1	yes	characteristic
325	C₂²×C₂₃	#553	1	1	yes	characteristic
326	C₄×C₂₃	—	2	2	no	—
327	C₄×C₂₃	—	4	1	no	—
328	C₄×C₂₃	—	4	1	no	—
329	C₄×C₂₃	—	4	1	no	—
330	C₄×C₂₃	—	4	1	no	—
331	C₄×C₂₃	—	4	1	no	—
332	C₄×C₂₃	—	8	1	no	—
333	C₄×C₂₃	—	8	1	no	—
334	C₄×C₂₃	—	8	1	no	—
335	C₄×C₂₃	—	8	1	no	—
336	C₄×C₂₃	—	8	1	no	—
337	C₄×C₂₃	C₂×#123	1	1	yes	characteristic
338	C₄×C₂₃	#525	1	1	yes	characteristic
339	C₄×C₂₃	#547	1	1	yes	characteristic
340	C₄×C₂₃	#548	1	1	yes	characteristic
341	C₂×#137	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
343	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
344	#536	C₂²×C₂₃	1	1	yes	characteristic
345	#520	C₂²×C₂₃	1	1	yes	characteristic
346	#520	C₂²×C₂₃	1	1	yes	characteristic
347	#520	C₂²×C₂₃	1	1	yes	characteristic
348	#485	—	2	1	no	—
349	#485	—	2	1	no	—
350	#484	—	2	1	no	—
351	#483	—	2	1	no	—
352	#464	—	2	1	no	—
353	#463	—	2	1	no	—
354	#462	—	2	1	no	—
355	#461	—	2	1	no	—
356	#440	—	2	1	no	—
357	#440	—	2	1	no	—
358	C₄×Q₁₆	—	2	1	no	—
359	C₄×D₈	—	2	1	no	—
360	#428	—	2	1	no	—
361	#428	—	2	1	no	—
362	#428	C₂²×C₂₃	1	1	yes	characteristic
363	#428	C₂²×C₂₃	1	1	yes	characteristic
364	#428	C₂²×C₂₃	1	1	yes	characteristic
365	#427	—	2	1	no	—
366	#427	C₂²×C₂₃	1	1	yes	characteristic
367	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
368	C₂×#106	—	2	1	no	—
369	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
370	#424	C₂²×C₂₃	1	1	yes	characteristic
371	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
372	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
373	#419	C₂²×C₂₃	1	1	yes	characteristic
374	#417	C₂²×C₂₃	1	1	yes	characteristic
375	#417	C₂²×C₂₃	1	1	yes	characteristic
376	#417	C₂²×C₂₃	1	1	yes	characteristic
377	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
378	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
379	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
380	#408	C₂²×C₂₃	1	1	yes	characteristic
381	#405	C₂²×C₂₃	1	1	yes	characteristic
382	#391	C₂²×C₂₃	1	1	yes	characteristic
383	#382	C₂²×C₂₃	1	1	yes	characteristic
384	#343	C₂²×C₂₃	1	1	yes	characteristic
385	#343	C₂²×C₂₃	1	1	yes	characteristic
386	#343	C₂²×C₂₃	1	1	yes	characteristic
387	#341	C₂²×C₂₃	1	1	yes	characteristic
388	#341	C₂²×C₂₃	1	1	yes	characteristic
389	#339	C₂²×C₂₃	1	1	yes	characteristic
390	#339	C₂²×C₂₃	1	1	yes	characteristic
391	#336	C₂²×C₂₃	1	1	yes	characteristic
392	C₂×C₂₃	—	2	1	no	—
393	C₂×C₂₃	—	2	1	no	—
394	C₂×C₂₃	—	4	2	no	—
395	C₂×C₂₃	—	8	1	no	—
396	C₂×C₂₃	#2443	1	1	yes	characteristic
397	C₂×C₂₃	#3289	1	1	yes	characteristic
398	C₂×C₂₃	#3290	1	1	yes	characteristic
399	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
400	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
401	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
402	#137	—	2	2	no	—
403	#137	—	2	2	no	—
404	#135	—	2	2	no	—
405	#135	—	2	2	no	—
406	C₂×Q₁₆	—	2	1	no	—
407	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
408	C₂×SD₁₆	—	2	1	no	—
409	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
410	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
411	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
412	#131	—	2	2	no	—
413	#131	—	2	2	no	—
414	C₂×M₁₆	—	2	1	no	—
415	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
416	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
417	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
418	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
419	C₂²×C₈	—	2	1	no	—
420	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
421	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
422	C₄⋊Q₈	—	2	1	no	—
423	#124	—	2	1	no	—
424	#123	—	4	1	no	—
425	C₂²⋊Q₈	—	2	1	no	—
426	C₂²⋊Q₈	—	2	1	no	—
427	C₄×Q₈	—	2	1	no	—
428	C₄×Q₈	—	2	1	no	—
429	C₄×Q₈	—	2	1	no	—
430	C₄×D₄	—	2	1	no	—
431	C₄×D₄	—	2	1	no	—
432	C₄×D₄	—	2	1	no	—
433	C₄×D₄	—	4	1	no	—
434	C₄×D₄	—	4	1	no	—
435	C₄×D₄	—	4	1	no	—
436	#117	—	2	1	no	—
437	#117	—	2	1	no	—
438	#117	—	2	1	no	—
439	#117	C₂³×C₂₃	1	1	yes	characteristic
440	#117	C₂³×C₂₃	1	1	yes	characteristic
441	C₂×C₄⋊C₄	—	2	1	no	—
442	C₂×C₄⋊C₄	—	2	1	no	—
443	C₂×C₄⋊C₄	—	2	1	no	—
444	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
445	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
446	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
447	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
448	#107	—	2	1	no	—
449	#107	—	2	1	no	—
450	#107	—	2	1	no	—
451	#107	—	2	1	no	—
452	#107	—	2	1	no	—
453	#107	D₄×C₂₃	1	1	yes	characteristic
454	#107	D₄×C₂₃	1	1	yes	characteristic
455	#106	—	2	1	no	—
456	#106	—	2	1	no	—
457	#106	—	2	1	no	—
458	#106	—	2	1	no	—
459	#106	—	2	1	no	—
460	#106	D₄×C₂₃	1	1	yes	characteristic
461	#106	D₄×C₂₃	1	1	yes	characteristic
462	C₄⋊C₈	—	2	1	no	—
463	C₄⋊C₈	—	2	1	no	—
464	#104	—	2	2	no	—
465	#104	—	2	2	no	—
466	#103	—	2	1	no	—
467	#103	—	2	1	no	—
468	#103	—	2	1	no	—
469	#103	—	2	1	no	—
470	#103	—	2	1	no	—
471	#103	—	2	1	no	—
472	#103	—	2	1	no	—
473	#103	—	2	1	no	—
474	#103	—	2	1	no	—
475	#102	—	2	1	no	—
476	#102	—	2	1	no	—
477	#102	—	2	1	no	—
478	#101	—	2	2	no	—
479	C₂²⋊C₈	—	2	1	no	—
480	C₂²⋊C₈	—	2	1	no	—
481	#97	—	2	1	no	—
482	C₄×C₈	—	2	1	no	—
483	#95	—	4	1	no	—
484	#95	—	4	1	no	—
485	C₂₃	#26192	1	1	yes	characteristic, direct product factor, Sylow
486	#41	—	2	1	no	—
487	#41	—	2	1	no	—
488	#41	—	2	1	no	—
489	#41	—	2	1	no	—
490	#41	—	2	1	no	—
491	#41	—	2	1	no	—
492	#41	#41×C₂₃	1	2	yes	—
493	#41	#41×C₂₃	1	2	yes	—
494	C₂×Q₈	—	2	1	no	—
495	C₂×Q₈	—	2	1	no	—
496	C₂×Q₈	—	2	1	no	—
497	C₂×Q₈	—	2	1	no	—
498	C₂×Q₈	—	2	1	no	—
499	C₂×Q₈	—	2	1	no	—
500	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
501	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	2	1	no	—
504	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
505	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	2	1	no	—
508	C₂²×C₄	—	2	1	no	—
509	C₂²×C₄	—	2	1	no	—
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	—	2	1	no	—
512	C₂²×C₄	—	2	1	no	—
513	C₂²×C₄	—	4	1	no	—
514	C₂²×C₄	—	4	1	no	—
515	C₂²×C₄	—	4	1	no	—
516	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
517	Q₁₆	—	2	2	no	—
518	Q₁₆	—	4	2	no	—
519	SD₁₆	—	2	2	no	—
520	SD₁₆	—	2	2	no	—
521	SD₁₆	—	4	2	no	—
522	D₈	—	2	2	no	—
523	M₁₆	—	2	4	no	—
524	M₁₆	—	2	1	no	—
525	M₁₆	—	2	1	no	—
526	M₁₆	—	2	1	no	—
527	M₁₆	—	2	1	no	—
528	M₁₆	—	2	1	no	—
529	M₁₆	—	2	1	no	—
530	M₁₆	—	2	1	no	—
531	M₁₆	#41×C₂₃	1	2	yes	—
532	C₂×C₈	—	2	4	no	—
533	C₂×C₈	—	2	1	no	—
534	C₂×C₈	—	2	1	no	—
535	C₂×C₈	—	2	1	no	—
536	C₂×C₈	—	2	1	no	—
537	C₂×C₈	—	2	1	no	—
538	C₂×C₈	—	2	1	no	—
539	C₂×C₈	—	2	1	no	—
540	C₂×C₈	—	2	1	no	—
541	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
542	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
543	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
544	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
545	C₄⋊C₄	—	2	1	no	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	2	1	no	—
548	C₄⋊C₄	—	2	1	no	—
549	C₄⋊C₄	—	2	1	no	—
550	C₄⋊C₄	—	2	1	no	—
551	C₄⋊C₄	—	4	1	no	—
552	C₄⋊C₄	—	4	1	no	—
553	C₄⋊C₄	—	4	1	no	—
554	C₄⋊C₄	—	4	1	no	—
555	C₄⋊C₄	—	4	1	no	—
556	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
557	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
558	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
559	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
560	C₂²⋊C₄	—	2	1	no	—
561	C₂²⋊C₄	—	2	1	no	—
562	C₂²⋊C₄	—	4	1	no	—
563	C₂²⋊C₄	—	4	1	no	—
564	C₂²⋊C₄	—	4	1	no	—
565	C₂²⋊C₄	—	4	1	no	—
566	C₂²⋊C₄	—	4	1	no	—
567	C₄²	—	2	1	no	—
568	C₄²	—	2	1	no	—
569	C₄²	—	2	1	no	—
570	C₄²	—	2	1	no	—
571	C₄²	—	4	1	no	—
572	C₄²	—	4	1	no	—
573	C₄²	—	4	1	no	—
574	C₂³	—	2	1	no	—
575	C₂³	—	2	1	no	—
576	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
577	Q₈	—	2	1	no	—
578	Q₈	—	2	1	no	—
579	Q₈	—	2	1	no	—
580	Q₈	—	4	1	no	—
581	Q₈	—	4	1	no	—
582	Q₈	—	4	1	no	—
583	Q₈	—	4	1	no	—
584	Q₈	C₂₃×#123	1	2	yes	—
585	D₄	—	2	2	no	—
586	D₄	—	2	2	no	—
587	D₄	—	2	1	no	—
588	D₄	—	2	1	no	—
589	D₄	—	2	1	no	—
590	D₄	—	4	1	no	—
591	D₄	—	4	1	no	—
592	D₄	C₂₃×#123	1	2	yes	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	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	—	8	1	no	—
613	C₂×C₄	—	8	1	no	—
614	C₂×C₄	—	8	1	no	—
615	C₂×C₄	—	16	1	no	—
616	C₂×C₄	—	16	1	no	—
617	C₂×C₄	—	16	1	no	—
618	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
619	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
620	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
621	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
622	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
623	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
624	C₈	—	2	1	no	—
625	C₈	—	2	1	no	—
626	C₈	—	2	1	no	—
627	C₈	—	2	2	no	—
628	C₈	—	2	1	no	—
629	C₈	—	4	1	no	—
630	C₈	—	4	1	no	—
631	C₈	—	4	1	no	—
632	C₂²	—	2	1	no	—
633	C₂²	—	2	1	no	—
634	C₂²	—	2	2	no	—
635	C₂²	—	4	1	no	—
636	C₂²	—	4	1	no	—
637	C₂²	—	4	1	no	—
638	C₂²	—	4	1	no	—
639	C₂²	—	4	1	no	—
640	C₂²	C₂₃×#539	1	1	yes	characteristic
641	C₂²	D₄²×C₂₃	1	1	yes	characteristic
642	C₂²	C₂₃×#553	1	1	yes	characteristic
643	C₄	—	2	2	no	—
644	C₄	—	4	1	no	—
645	C₄	—	4	1	no	—
646	C₄	—	4	1	no	—
647	C₄	—	4	1	no	—
648	C₄	—	4	1	no	—
649	C₄	—	8	1	no	—
650	C₄	—	8	1	no	—
651	C₄	—	8	1	no	—
652	C₄	—	8	1	no	—
653	C₄	—	8	1	no	—
654	C₄	C₂×C₂₃×#123	1	1	yes	characteristic
655	C₄	C₂₃×#525	1	1	yes	characteristic
656	C₄	C₂₃×#547	1	1	yes	characteristic
657	C₄	C₂₃×#548	1	1	yes	characteristic
658	C₂	—	2	1	no	—
659	C₂	—	2	1	no	—
660	C₂	—	4	2	no	—
661	C₂	—	8	1	no	—
662	C₂	C₂₃×#2443	1	1	yes	characteristic
663	C₂	C₂₃×#3289	1	1	yes	characteristic
664	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×C₂₃×#135	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. C₂₃×#3097	C₂	semidirect
4. C₂₃×#2912	C₂	semidirect
5. C₂₃×#2097	C₂	semidirect
6. C₂₃×#2089	C₂	semidirect
7. C₂₃×#2079	C₂	semidirect
8. C₂₃×#2038	C₂	semidirect
9. C₂₃×#1930	C₂	semidirect
10. C₂₃×#1922	C₂	semidirect
11. C₂₃×#1908	C₂	semidirect
12. C₂₃×#1882	C₂	semidirect
14. C₂₃×#1860	C₂	semidirect
15. C₂₃×#1860	C₂	semidirect
241. #26192	C₂₃	direct (N × H)
300. #3097	C₂×C₂₃	semidirect
303. #2912	C₂×C₂₃	semidirect
304. #2097	C₂×C₂₃	semidirect
305. #2089	C₂×C₂₃	semidirect
306. #2079	C₂×C₂₃	semidirect
307. #2038	C₂×C₂₃	semidirect
308. #1930	C₂×C₂₃	semidirect
309. #1922	C₂×C₂₃	semidirect
310. #1908	C₂×C₂₃	semidirect
311. #1882	C₂×C₂₃	semidirect
313. #1860	C₂×C₂₃	semidirect
314. #1860	C₂×C₂₃	semidirect
485. C₂₃	#26192	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×#3097	2
C₂₃×#3046	2
C₂₃×#2951	2
C₂₃×#2912	2
C₂₃×#2097	2
C₂₃×#2089	2
C₂₃×#2079	2
C₂₃×#2038	2
C₂₃×#1930	2
C₂₃×#1922	2
C₂₃×#1908	2
C₂₃×#1882	2
C₂₃×#1876	2
C₂₃×#1860	2
#26192	23

Subgroups — C23×#26192 #19173818

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26192 #19173818