Subgroups — C₂₃×#22678 #19170304

Order 5888.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#440	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#440	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
4	Q₈×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2349	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1943	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1942	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1865	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1864	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#1863	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1862	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#1847	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1805	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1802	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#1771	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#440	—	2	4	no	—
24	C₂₃×#440	C₄	1	8	yes	complemented
25	C₄×SD₁₆×C₂₃	—	2	4	no	—
26	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
39	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	C₂₃×#389	—	2	1	no	—
42	C₂₃×#389	—	2	1	no	—
43	C₂₃×#389	C₂²	1	1	yes	characteristic
44	C₂₃×#389	C₂²	1	1	yes	characteristic
45	C₂₃×#389	C₂²	1	1	yes	characteristic
46	C₂₃×#389	C₂²	1	1	yes	characteristic
47	C₂₃×#385	—	2	1	no	—
48	C₂₃×#385	—	2	1	no	—
49	C₂₃×#384	—	2	1	no	—
50	C₂₃×#384	C₂²	1	1	yes	characteristic
51	C₂₃×#382	C₂²	1	1	yes	characteristic
52	C₂₃×#380	—	2	1	no	—
53	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
54	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
55	C₂×C₂₃×#95	—	2	1	no	—
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#340	C₂²	1	1	yes	characteristic
58	C₂₃×#340	C₂²	1	1	yes	characteristic
59	C₂₃×#340	C₂²	1	1	yes	characteristic
60	C₂₃×#336	C₂²	1	1	yes	characteristic
61	C₂₃×#336	C₂²	1	1	yes	characteristic
62	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₂₃	—	2	1	no	—
65	C₂³×C₄×C₂₃	—	2	1	no	—
66	C₂³×C₄×C₂₃	—	2	1	no	—
67	C₂×SD₁₆×C₂₃	C₂×C₄	1	4	yes	—
68	C₂×SD₁₆×C₂₃	D₄	1	4	yes	—
69	C₂×SD₁₆×C₂₃	Q₈	1	4	yes	—
70	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₄×Q₈×C₂₃	—	2	2	no	—
75	C₄×Q₈×C₂₃	—	2	2	no	—
76	C₄×Q₈×C₂₃	—	2	2	no	—
77	C₄×Q₈×C₂₃	—	4	1	no	—
78	C₄×Q₈×C₂₃	—	4	1	no	—
79	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	complemented
80	C₄×D₄×C₂₃	—	2	2	no	—
81	C₄×D₄×C₂₃	—	2	2	no	—
82	C₄×D₄×C₂₃	—	2	2	no	—
83	C₄×D₄×C₂₃	—	4	1	no	—
84	C₄×D₄×C₂₃	—	4	1	no	—
85	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
86	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
89	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
94	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
106	C₂×C₄²×C₂₃	—	2	1	no	—
107	C₂×C₄²×C₂₃	—	2	1	no	—
108	C₂×C₄²×C₂₃	—	2	1	no	—
109	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
112	C₂₃×#107	—	2	2	no	—
113	C₂₃×#107	—	4	1	no	—
114	C₂₃×#107	C₂×C₄	1	4	yes	complemented
115	C₂₃×#106	—	2	2	no	—
116	C₂₃×#106	—	4	1	no	—
117	C₂₃×C₄⋊C₈	—	2	2	no	—
118	C₂₃×#103	—	2	2	no	—
119	C₂₃×#103	—	2	2	no	—
120	C₂₃×#103	C₂×C₄	1	4	yes	complemented
121	C₂₃×#102	—	2	2	no	—
122	C₂₃×#102	—	2	2	no	—
123	C₂₃×#102	C₂×C₄	1	4	yes	—
124	C₂₃×#97	—	2	2	no	—
125	C₂₃×#97	C₂×C₄	1	4	yes	complemented
126	C₄×C₈×C₂₃	—	2	2	no	—
127	C₂₃×#95	—	2	1	no	—
128	C₂₃×#95	—	2	1	no	—
129	C₂₃×#95	—	2	1	no	—
130	C₂₃×#95	—	2	1	no	—
131	C₂₃×#95	—	2	1	no	—
132	C₂₃×#95	—	2	1	no	—
133	C₂₃×#95	D₄	1	1	yes	characteristic
134	C₂₃×#95	D₄	1	1	yes	characteristic
135	C₂⁴×C₂₃	—	2	1	no	—
136	C₂×Q₈×C₂₃	—	2	1	no	—
137	C₂×Q₈×C₂₃	—	2	1	no	—
138	C₂×Q₈×C₂₃	—	2	1	no	—
139	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
140	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
141	C₂×Q₈×C₂₃	C₂×Q₈	1	2	yes	—
142	C₂×D₄×C₂₃	—	2	1	no	—
143	C₂×D₄×C₂₃	—	2	1	no	—
144	C₂×D₄×C₂₃	—	2	1	no	—
145	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
146	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
147	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
148	C₂²×C₄×C₂₃	—	2	1	no	—
149	C₂²×C₄×C₂₃	—	2	1	no	—
150	C₂²×C₄×C₂₃	—	2	1	no	—
151	C₂²×C₄×C₂₃	—	2	4	no	—
152	C₂²×C₄×C₂₃	—	2	1	no	—
153	C₂²×C₄×C₂₃	—	2	1	no	—
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	—	2	1	no	—
156	C₂²×C₄×C₂₃	—	4	2	no	—
157	C₂²×C₄×C₂₃	—	4	2	no	—
158	C₂²×C₄×C₂₃	—	4	2	no	—
159	C₂²×C₄×C₂₃	—	4	2	no	—
160	C₂²×C₄×C₂₃	—	8	1	no	—
161	C₂²×C₄×C₂₃	—	8	1	no	—
162	C₂²×C₄×C₂₃	—	8	1	no	—
163	C₂²×C₄×C₂₃	—	8	1	no	—
164	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
171	SD₁₆×C₂₃	C₄⋊C₄	1	16	yes	complemented
172	C₂×C₈×C₂₃	—	2	1	no	—
173	C₂×C₈×C₂₃	—	2	1	no	—
174	C₂×C₈×C₂₃	—	2	1	no	—
175	C₂×C₈×C₂₃	—	2	1	no	—
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	C₂²×C₄	1	2	yes	—
181	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
182	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
183	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
184	C₄⋊C₄×C₂₃	—	2	4	no	—
185	C₄⋊C₄×C₂₃	—	2	2	no	—
186	C₄⋊C₄×C₂₃	—	2	1	no	—
187	C₄⋊C₄×C₂₃	—	2	1	no	—
188	C₄⋊C₄×C₂₃	—	2	1	no	—
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	—	4	1	no	—
197	C₄⋊C₄×C₂₃	—	4	2	no	—
198	C₄⋊C₄×C₂₃	—	4	2	no	—
199	C₄⋊C₄×C₂₃	—	4	2	no	—
200	C₄⋊C₄×C₂₃	—	8	1	no	—
201	C₄⋊C₄×C₂₃	—	8	1	no	—
202	C₄⋊C₄×C₂₃	—	8	1	no	—
203	C₄⋊C₄×C₂₃	—	8	1	no	—
204	C₄⋊C₄×C₂₃	—	8	1	no	—
205	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
206	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
207	C₂²⋊C₄×C₂₃	—	2	4	no	—
208	C₂²⋊C₄×C₂₃	—	4	2	no	—
209	C₂²⋊C₄×C₂₃	—	4	2	no	—
210	C₄²×C₂₃	—	2	4	no	—
211	C₄²×C₂₃	—	2	1	no	—
212	C₄²×C₂₃	—	2	1	no	—
213	C₄²×C₂₃	—	2	1	no	—
214	C₄²×C₂₃	—	2	1	no	—
215	C₄²×C₂₃	—	4	2	no	—
216	C₄²×C₂₃	—	4	2	no	—
217	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
218	C₄²×C₂₃	#41	1	2	yes	—
219	#22678	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂³×C₂₃	—	2	2	no	—
221	C₂³×C₂₃	—	2	2	no	—
222	C₂³×C₂₃	—	2	2	no	—
223	C₂³×C₂₃	—	4	1	no	—
224	C₂³×C₂₃	—	4	1	no	—
225	C₂³×C₂₃	—	4	1	no	—
226	C₂³×C₂₃	—	4	1	no	—
227	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
228	Q₈×C₂₃	—	2	2	no	—
229	Q₈×C₂₃	—	2	2	no	—
230	Q₈×C₂₃	—	2	2	no	—
231	Q₈×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
232	D₄×C₂₃	—	2	2	no	—
233	D₄×C₂₃	—	2	2	no	—
234	D₄×C₂₃	—	2	2	no	—
235	D₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
236	C₂×C₄×C₂₃	—	2	2	no	—
237	C₂×C₄×C₂₃	—	2	1	no	—
238	C₂×C₄×C₂₃	—	2	2	no	—
239	C₂×C₄×C₂₃	—	2	2	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	1	no	—
245	C₂×C₄×C₂₃	—	2	2	no	—
246	C₂×C₄×C₂₃	—	2	2	no	—
247	C₂×C₄×C₂₃	—	2	1	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	1	no	—
255	C₂×C₄×C₂₃	—	2	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	—	4	1	no	—
264	C₂×C₄×C₂₃	—	4	1	no	—
265	C₂×C₄×C₂₃	—	4	1	no	—
266	C₂×C₄×C₂₃	—	4	1	no	—
267	C₂×C₄×C₂₃	—	4	1	no	—
268	C₂×C₄×C₂₃	—	4	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	—	8	4	no	—
273	C₂×C₄×C₂₃	—	8	4	no	—
274	C₂×C₄×C₂₃	—	8	4	no	—
275	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
276	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
277	C₂×C₄×C₂₃	#137	1	2	yes	—
278	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
279	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
280	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
281	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
282	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
283	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
284	C₈×C₂₃	—	2	2	no	—
285	C₈×C₂₃	—	4	1	no	—
286	C₈×C₂₃	—	4	1	no	—
287	C₈×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
288	C₂×#440	C₂×C₂₃	1	1	yes	characteristic
289	C₂×#440	C₂×C₂₃	1	1	yes	characteristic
290	C₂×C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic
291	Q₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
292	#2349	C₂×C₂₃	1	1	yes	characteristic
293	#1943	C₂×C₂₃	1	1	yes	characteristic, complemented
294	#1942	C₂×C₂₃	1	1	yes	characteristic, complemented
295	#1865	C₂×C₂₃	1	1	yes	characteristic, complemented
296	#1864	C₂×C₂₃	1	1	yes	characteristic
297	#1863	C₂×C₂₃	1	1	yes	characteristic, complemented
298	#1862	C₂×C₂₃	1	1	yes	characteristic
299	#1847	C₂×C₂₃	1	1	yes	characteristic, complemented
300	#1805	C₂×C₂₃	1	1	yes	characteristic, complemented
301	#1802	C₂×C₂₃	1	1	yes	characteristic
302	#1771	C₂×C₂₃	1	1	yes	characteristic, complemented
303	C₂²×C₂₃	—	2	4	no	—
304	C₂²×C₂₃	—	4	2	no	—
305	C₂²×C₂₃	—	4	2	no	—
306	C₂²×C₂₃	—	4	2	no	—
307	C₂²×C₂₃	—	4	2	no	—
308	C₂²×C₂₃	—	4	2	no	—
309	C₂²×C₂₃	—	4	2	no	—
310	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
311	C₂²×C₂₃	D₄²	1	1	yes	characteristic
312	C₂²×C₂₃	D₄×Q₈	1	1	yes	characteristic
313	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
314	C₂²×C₂₃	#575	1	1	yes	characteristic
315	C₂²×C₂₃	#576	1	1	yes	characteristic
316	C₂²×C₂₃	#577	1	1	yes	characteristic
317	C₄×C₂₃	—	2	4	no	—
318	C₄×C₂₃	—	2	2	no	—
319	C₄×C₂₃	—	2	2	no	—
320	C₄×C₂₃	—	2	2	no	—
321	C₄×C₂₃	—	4	2	no	—
322	C₄×C₂₃	—	4	2	no	—
323	C₄×C₂₃	—	4	1	no	—
324	C₄×C₂₃	—	4	1	no	—
325	C₄×C₂₃	—	4	1	no	—
326	C₄×C₂₃	—	8	1	no	—
327	C₄×C₂₃	—	8	1	no	—
328	C₄×C₂₃	—	8	1	no	—
329	C₄×C₂₃	—	8	1	no	—
330	C₄×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
331	C₄×C₂₃	#520	1	1	yes	characteristic
332	C₄×C₂₃	#547	1	1	yes	characteristic
333	C₄×C₂₃	#554	1	1	yes	characteristic
334	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
335	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
336	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
337	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
338	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
339	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
340	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
341	#440	—	2	4	no	—
342	#440	C₄×C₂₃	1	8	yes	complemented
343	C₄×SD₁₆	—	2	4	no	—
344	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
347	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
348	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
349	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
351	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
352	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
353	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
354	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
356	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
357	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
358	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
359	#389	—	2	1	no	—
360	#389	—	2	1	no	—
361	#389	C₂²×C₂₃	1	1	yes	characteristic
362	#389	C₂²×C₂₃	1	1	yes	characteristic
363	#389	C₂²×C₂₃	1	1	yes	characteristic
364	#389	C₂²×C₂₃	1	1	yes	characteristic
365	#385	—	2	1	no	—
366	#385	—	2	1	no	—
367	#384	—	2	1	no	—
368	#384	C₂²×C₂₃	1	1	yes	characteristic
369	#382	C₂²×C₂₃	1	1	yes	characteristic
370	#380	—	2	1	no	—
371	C₄×C₄⋊C₄	—	2	1	no	—
372	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
373	C₂×#95	—	2	1	no	—
374	#340	C₂²×C₂₃	1	1	yes	characteristic
375	#340	C₂²×C₂₃	1	1	yes	characteristic
376	#340	C₂²×C₂₃	1	1	yes	characteristic
377	#340	C₂²×C₂₃	1	1	yes	characteristic
378	#336	C₂²×C₂₃	1	1	yes	characteristic
379	#336	C₂²×C₂₃	1	1	yes	characteristic
380	C₂×C₂₃	—	4	4	no	—
381	C₂×C₂₃	D₄×C₄⋊C₄	1	1	yes	characteristic
382	C₂×C₂₃	C₄×#137	1	1	yes	characteristic
383	C₂×C₂₃	#2946	1	1	yes	characteristic
384	C₂×C₂₃	#3266	1	1	yes	characteristic
385	C₂×C₂₃	#3268	1	1	yes	characteristic
386	C₂×C₂₃	#3385	1	1	yes	characteristic
387	C₂×C₂₃	#3388	1	1	yes	characteristic
388	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
389	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
390	C₂³×C₄	—	2	1	no	—
391	C₂³×C₄	—	2	1	no	—
392	C₂³×C₄	—	2	1	no	—
393	C₂×SD₁₆	C₂×C₄×C₂₃	1	4	yes	—
394	C₂×SD₁₆	D₄×C₂₃	1	4	yes	—
395	C₂×SD₁₆	Q₈×C₂₃	1	4	yes	—
396	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
397	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
398	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
399	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
400	C₄×Q₈	—	2	2	no	—
401	C₄×Q₈	—	2	2	no	—
402	C₄×Q₈	—	2	2	no	—
403	C₄×Q₈	—	4	1	no	—
404	C₄×Q₈	—	4	1	no	—
405	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
406	C₄×D₄	—	2	2	no	—
407	C₄×D₄	—	2	2	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	4	1	no	—
410	C₄×D₄	—	4	1	no	—
411	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	—	2	1	no	—
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	—	2	1	no	—
416	C₂×C₄⋊C₄	—	2	1	no	—
417	C₂×C₄⋊C₄	—	2	1	no	—
418	C₂×C₄⋊C₄	—	2	1	no	—
419	C₂×C₄⋊C₄	—	2	1	no	—
420	C₂×C₄⋊C₄	—	2	1	no	—
421	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
422	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
423	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
424	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
425	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
426	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
427	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
428	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
429	C₂×C₂²⋊C₄	—	2	1	no	—
430	C₂×C₂²⋊C₄	—	2	1	no	—
431	C₂×C₂²⋊C₄	—	2	1	no	—
432	C₂×C₄²	—	2	1	no	—
433	C₂×C₄²	—	2	1	no	—
434	C₂×C₄²	—	2	1	no	—
435	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
436	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
437	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
438	#107	—	2	2	no	—
439	#107	—	4	1	no	—
440	#107	C₂×C₄×C₂₃	1	4	yes	complemented
441	#106	—	2	2	no	—
442	#106	—	4	1	no	—
443	C₄⋊C₈	—	2	2	no	—
444	#103	—	2	2	no	—
445	#103	—	2	2	no	—
446	#103	C₂×C₄×C₂₃	1	4	yes	complemented
447	#102	—	2	2	no	—
448	#102	—	2	2	no	—
449	#102	C₂×C₄×C₂₃	1	4	yes	—
450	#97	—	2	2	no	—
451	#97	C₂×C₄×C₂₃	1	4	yes	complemented
452	C₄×C₈	—	2	2	no	—
453	#95	—	2	1	no	—
454	#95	—	2	1	no	—
455	#95	—	2	1	no	—
456	#95	—	2	1	no	—
457	#95	—	2	1	no	—
458	#95	—	2	1	no	—
459	#95	D₄×C₂₃	1	1	yes	characteristic
460	#95	D₄×C₂₃	1	1	yes	characteristic
461	C₂₃	#22678	1	1	yes	characteristic, direct product factor, Sylow
462	C₂⁴	—	2	1	no	—
463	C₂×Q₈	—	2	1	no	—
464	C₂×Q₈	—	2	1	no	—
465	C₂×Q₈	—	2	1	no	—
466	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
467	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
468	C₂×Q₈	C₂×Q₈×C₂₃	1	2	yes	—
469	C₂×D₄	—	2	1	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	2	1	no	—
472	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
473	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
474	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	4	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	2	no	—
484	C₂²×C₄	—	4	2	no	—
485	C₂²×C₄	—	4	2	no	—
486	C₂²×C₄	—	4	2	no	—
487	C₂²×C₄	—	8	1	no	—
488	C₂²×C₄	—	8	1	no	—
489	C₂²×C₄	—	8	1	no	—
490	C₂²×C₄	—	8	1	no	—
491	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
492	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
493	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
494	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
495	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
496	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
497	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
498	SD₁₆	C₄⋊C₄×C₂₃	1	16	yes	complemented
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	—	2	1	no	—
507	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
508	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
509	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
510	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
511	C₄⋊C₄	—	2	4	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	1	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	1	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	4	1	no	—
524	C₄⋊C₄	—	4	2	no	—
525	C₄⋊C₄	—	4	2	no	—
526	C₄⋊C₄	—	4	2	no	—
527	C₄⋊C₄	—	8	1	no	—
528	C₄⋊C₄	—	8	1	no	—
529	C₄⋊C₄	—	8	1	no	—
530	C₄⋊C₄	—	8	1	no	—
531	C₄⋊C₄	—	8	1	no	—
532	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
533	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
534	C₂²⋊C₄	—	2	4	no	—
535	C₂²⋊C₄	—	4	2	no	—
536	C₂²⋊C₄	—	4	2	no	—
537	C₄²	—	2	4	no	—
538	C₄²	—	2	1	no	—
539	C₄²	—	2	1	no	—
540	C₄²	—	2	1	no	—
541	C₄²	—	2	1	no	—
542	C₄²	—	4	2	no	—
543	C₄²	—	4	2	no	—
544	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
545	C₄²	#41×C₂₃	1	2	yes	—
546	C₂³	—	2	2	no	—
547	C₂³	—	2	2	no	—
548	C₂³	—	2	2	no	—
549	C₂³	—	4	1	no	—
550	C₂³	—	4	1	no	—
551	C₂³	—	4	1	no	—
552	C₂³	—	4	1	no	—
553	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
554	Q₈	—	2	2	no	—
555	Q₈	—	2	2	no	—
556	Q₈	—	2	2	no	—
557	Q₈	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
558	D₄	—	2	2	no	—
559	D₄	—	2	2	no	—
560	D₄	—	2	2	no	—
561	D₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	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	—	8	4	no	—
599	C₂×C₄	—	8	4	no	—
600	C₂×C₄	—	8	4	no	—
601	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
602	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
603	C₂×C₄	C₂₃×#137	1	2	yes	—
604	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
605	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
606	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
607	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
608	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
609	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
610	C₈	—	2	2	no	—
611	C₈	—	4	1	no	—
612	C₈	—	4	1	no	—
613	C₈	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
614	C₂²	—	2	4	no	—
615	C₂²	—	4	2	no	—
616	C₂²	—	4	2	no	—
617	C₂²	—	4	2	no	—
618	C₂²	—	4	2	no	—
619	C₂²	—	4	2	no	—
620	C₂²	—	4	2	no	—
621	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
622	C₂²	D₄²×C₂₃	1	1	yes	characteristic
623	C₂²	D₄×Q₈×C₂₃	1	1	yes	characteristic
624	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
625	C₂²	C₂₃×#575	1	1	yes	characteristic
626	C₂²	C₂₃×#576	1	1	yes	characteristic
627	C₂²	C₂₃×#577	1	1	yes	characteristic
628	C₄	—	2	2	no	—
629	C₄	—	2	2	no	—
630	C₄	—	2	2	no	—
631	C₄	—	2	4	no	—
632	C₄	—	4	2	no	—
633	C₄	—	4	2	no	—
634	C₄	—	4	1	no	—
635	C₄	—	4	1	no	—
636	C₄	—	4	1	no	—
637	C₄	—	8	1	no	—
638	C₄	—	8	1	no	—
639	C₄	—	8	1	no	—
640	C₄	—	8	1	no	—
641	C₄	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
642	C₄	C₂₃×#520	1	1	yes	characteristic
643	C₄	C₂₃×#547	1	1	yes	characteristic
644	C₄	C₂₃×#554	1	1	yes	characteristic
645	C₂	—	4	4	no	—
646	C₂	D₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
647	C₂	C₄×C₂₃×#137	1	1	yes	characteristic
648	C₂	C₂₃×#2946	1	1	yes	characteristic
649	C₂	C₂₃×#3266	1	1	yes	characteristic
650	C₂	C₂₃×#3268	1	1	yes	characteristic
651	C₂	C₂₃×#3385	1	1	yes	characteristic
652	C₂	C₂₃×#3388	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 ◁ C₄×C₄⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. Q₈×C₄⋊C₄×C₂₃	C₂	semidirect
6. C₂₃×#1943	C₂	semidirect
7. C₂₃×#1942	C₂	semidirect
8. C₂₃×#1865	C₂	semidirect
10. C₂₃×#1863	C₂	semidirect
12. C₂₃×#1847	C₂	semidirect
13. C₂₃×#1805	C₂	semidirect
15. C₂₃×#1771	C₂	semidirect
24. C₂₃×#440	C₄	semidirect
79. C₄×Q₈×C₂₃	C₂×C₄	semidirect
114. C₂₃×#107	C₂×C₄	semidirect
120. C₂₃×#103	C₂×C₄	semidirect
125. C₂₃×#97	C₂×C₄	semidirect
171. SD₁₆×C₂₃	C₄⋊C₄	semidirect
219. #22678	C₂₃	direct (N × H)
291. Q₈×C₄⋊C₄	C₂×C₂₃	semidirect
293. #1943	C₂×C₂₃	semidirect
294. #1942	C₂×C₂₃	semidirect
295. #1865	C₂×C₂₃	semidirect
297. #1863	C₂×C₂₃	semidirect
299. #1847	C₂×C₂₃	semidirect
300. #1805	C₂×C₂₃	semidirect
302. #1771	C₂×C₂₃	semidirect
342. #440	C₄×C₂₃	semidirect
405. C₄×Q₈	C₂×C₄×C₂₃	semidirect
440. #107	C₂×C₄×C₂₃	semidirect
446. #103	C₂×C₄×C₂₃	semidirect
451. #97	C₂×C₄×C₂₃	semidirect
461. C₂₃	#22678	direct (N × H)
498. SD₁₆	C₄⋊C₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#440	2
C₂×C₄×SD₁₆×C₂₃	2
Q₈×C₄⋊C₄×C₂₃	2
C₂₃×#2349	2
C₂₃×#1943	2
C₂₃×#1942	2
C₂₃×#1865	2
C₂₃×#1864	2
C₂₃×#1863	2
C₂₃×#1862	2
C₂₃×#1847	2
C₂₃×#1805	2
C₂₃×#1802	2
C₂₃×#1771	2
#22678	23

Subgroups — C23×#22678 #19170304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22678 #19170304