Subgroups — C₂₃×#26355 #19173981

Order 5888.

848 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#428	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2440	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2362	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2096	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2088	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2066	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2058	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1947	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1918	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1894	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#1872	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1863	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1854	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#1851	C₂	1	1	yes	maximal, characteristic
16	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
19	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#477	—	2	4	no	—
24	C₂₃×#428	—	2	4	no	—
25	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#106	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×C₄⋊C₈	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×#103	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
38	C₂₃×#398	—	2	1	no	—
39	C₂₃×#395	—	2	1	no	—
40	C₂₃×#391	C₂²	1	1	yes	characteristic
41	C₂₃×#389	C₂²	1	1	yes	characteristic
42	C₂₃×#389	C₂²	1	1	yes	characteristic
43	C₂₃×#389	C₂²	1	1	yes	characteristic
44	C₂₃×#388	—	2	1	no	—
45	C₂₃×#388	—	2	1	no	—
46	C₂₃×#387	—	2	1	no	—
47	C₂₃×#385	—	2	1	no	—
48	C₂₃×#384	C₂²	1	1	yes	characteristic
49	C₂₃×#380	C₂²	1	1	yes	characteristic
50	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
51	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
52	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
53	C₂₃×#340	C₂²	1	1	yes	characteristic
54	C₂₃×#340	C₂²	1	1	yes	characteristic
55	C₂₃×#340	C₂²	1	1	yes	characteristic
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×#336	C₂²	1	1	yes	characteristic
60	C₂₃×#336	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₂₃	—	2	1	no	—
62	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₂₃	—	2	1	no	—
64	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂×M₁₆×C₂₃	—	2	2	no	—
66	C₂×M₁₆×C₂₃	—	2	2	no	—
67	C₂×M₁₆×C₂₃	—	2	2	no	—
68	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂₃×C₄⋊Q₈	—	2	2	no	—
73	C₂₃×C₄⋊Q₈	—	4	1	no	—
74	C₂₃×C₂²⋊Q₈	—	2	2	no	—
75	C₂₃×C₂²⋊Q₈	—	4	4	no	—
76	C₂₃×C₂²⋊Q₈	Q₈	1	4	yes	complemented
77	C₄×Q₈×C₂₃	—	2	2	no	—
78	C₄×Q₈×C₂₃	—	4	1	no	—
79	C₂₃×#117	—	2	2	no	—
80	C₂₃×#117	—	4	1	no	—
81	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
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	—	4	1	no	—
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₂₃	C₂³	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×C₂²⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
104	C₂×C₄²×C₂₃	—	2	1	no	—
105	C₂×C₄²×C₂₃	—	2	1	no	—
106	C₂×C₄²×C₂₃	—	2	1	no	—
107	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
109	C₂₃×#107	—	2	2	no	—
110	C₂₃×#107	—	2	2	no	—
111	C₂₃×#107	D₄	1	4	yes	—
112	C₂₃×#106	—	2	2	no	—
113	C₂₃×#106	—	2	2	no	—
114	C₂₃×#106	—	4	1	no	—
115	C₂₃×C₄⋊C₈	—	2	2	no	—
116	C₂₃×C₄⋊C₈	—	2	2	no	—
117	C₂₃×#103	—	2	2	no	—
118	C₂₃×#103	—	2	2	no	—
119	C₂₃×#103	—	2	2	no	—
120	C₂₃×#103	—	4	1	no	—
121	C₂₃×C₂²⋊C₈	—	2	2	no	—
122	C₂₃×#95	—	2	1	no	—
123	C₂₃×#95	—	2	1	no	—
124	C₂₃×#95	—	2	1	no	—
125	C₂₃×#95	—	2	1	no	—
126	C₂₃×#95	—	2	1	no	—
127	C₂₃×#95	—	2	1	no	—
128	C₂₃×#95	—	2	1	no	—
129	C₂₃×#95	D₄	1	1	yes	characteristic
130	C₂₃×#95	D₄	1	1	yes	characteristic
131	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Q₈×C₂₃	—	2	1	no	—
133	C₂×Q₈×C₂₃	—	2	1	no	—
134	C₂×Q₈×C₂₃	—	2	1	no	—
135	C₂×Q₈×C₂₃	—	2	1	no	—
136	C₂×Q₈×C₂₃	—	2	1	no	—
137	C₂×Q₈×C₂₃	—	4	2	no	—
138	C₂×Q₈×C₂₃	—	4	2	no	—
139	C₂×Q₈×C₂₃	—	8	1	no	—
140	C₂×Q₈×C₂₃	C₂×Q₈	1	2	yes	—
141	C₂²×C₄×C₂₃	—	2	1	no	—
142	C₂²×C₄×C₂₃	—	2	1	no	—
143	C₂²×C₄×C₂₃	—	2	1	no	—
144	C₂²×C₄×C₂₃	—	2	1	no	—
145	C₂²×C₄×C₂₃	—	2	1	no	—
146	C₂²×C₄×C₂₃	—	2	1	no	—
147	C₂²×C₄×C₂₃	—	2	1	no	—
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	1	no	—
152	C₂²×C₄×C₂₃	—	2	1	no	—
153	C₂²×C₄×C₂₃	—	4	2	no	—
154	C₂²×C₄×C₂₃	—	4	2	no	—
155	C₂²×C₄×C₂₃	—	8	1	no	—
156	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
162	C₂²×C₄×C₂₃	#41	1	2	yes	—
163	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
164	M₁₆×C₂₃	—	2	8	no	—
165	C₂×C₈×C₂₃	—	2	1	no	—
166	C₂×C₈×C₂₃	—	2	1	no	—
167	C₂×C₈×C₂₃	—	2	1	no	—
168	C₂×C₈×C₂₃	—	2	1	no	—
169	C₂×C₈×C₂₃	—	2	1	no	—
170	C₂×C₈×C₂₃	—	2	1	no	—
171	C₂×C₈×C₂₃	—	2	1	no	—
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	C₂×D₄	1	2	yes	—
176	C₂×C₈×C₂₃	#41	1	2	yes	—
177	C₄⋊C₄×C₂₃	—	2	4	no	—
178	C₄⋊C₄×C₂₃	—	2	1	no	—
179	C₄⋊C₄×C₂₃	—	2	1	no	—
180	C₄⋊C₄×C₂₃	—	2	1	no	—
181	C₄⋊C₄×C₂₃	—	2	1	no	—
182	C₄⋊C₄×C₂₃	—	2	1	no	—
183	C₄⋊C₄×C₂₃	—	2	1	no	—
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	2	1	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	—	4	2	no	—
190	C₄⋊C₄×C₂₃	—	4	2	no	—
191	C₄⋊C₄×C₂₃	—	4	2	no	—
192	C₄⋊C₄×C₂₃	—	4	2	no	—
193	C₄⋊C₄×C₂₃	—	4	2	no	—
194	C₄⋊C₄×C₂₃	—	4	2	no	—
195	C₄⋊C₄×C₂₃	—	4	2	no	—
196	C₄⋊C₄×C₂₃	—	8	1	no	—
197	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
198	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
199	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
200	C₂²⋊C₄×C₂₃	—	2	4	no	—
201	C₂²⋊C₄×C₂₃	—	2	2	no	—
202	C₂²⋊C₄×C₂₃	—	2	2	no	—
203	C₂²⋊C₄×C₂₃	—	4	2	no	—
204	C₂²⋊C₄×C₂₃	—	4	2	no	—
205	C₄²×C₂₃	—	2	1	no	—
206	C₄²×C₂₃	—	2	1	no	—
207	C₄²×C₂₃	—	2	1	no	—
208	C₄²×C₂₃	—	2	1	no	—
209	C₄²×C₂₃	—	4	2	no	—
210	C₄²×C₂₃	—	4	2	no	—
211	C₄²×C₂₃	—	4	2	no	—
212	#26355	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂³×C₂₃	—	2	1	no	—
214	C₂³×C₂₃	—	2	1	no	—
215	C₂³×C₂₃	—	2	1	no	—
216	C₂³×C₂₃	—	2	1	no	—
217	C₂³×C₂₃	—	2	1	no	—
218	C₂³×C₂₃	—	2	1	no	—
219	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
220	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
221	Q₈×C₂₃	—	2	2	no	—
222	Q₈×C₂₃	—	2	2	no	—
223	Q₈×C₂₃	—	4	1	no	—
224	Q₈×C₂₃	—	4	1	no	—
225	Q₈×C₂₃	—	8	4	no	—
226	C₂×C₄×C₂₃	—	2	1	no	—
227	C₂×C₄×C₂₃	—	2	1	no	—
228	C₂×C₄×C₂₃	—	2	1	no	—
229	C₂×C₄×C₂₃	—	2	1	no	—
230	C₂×C₄×C₂₃	—	2	1	no	—
231	C₂×C₄×C₂₃	—	2	1	no	—
232	C₂×C₄×C₂₃	—	2	2	no	—
233	C₂×C₄×C₂₃	—	2	2	no	—
234	C₂×C₄×C₂₃	—	2	2	no	—
235	C₂×C₄×C₂₃	—	2	2	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	2	no	—
240	C₂×C₄×C₂₃	—	2	2	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	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	—	4	1	no	—
252	C₂×C₄×C₂₃	—	4	1	no	—
253	C₂×C₄×C₂₃	—	4	1	no	—
254	C₂×C₄×C₂₃	—	4	1	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	—	4	1	no	—
264	C₂×C₄×C₂₃	—	4	1	no	—
265	C₂×C₄×C₂₃	—	4	1	no	—
266	C₂×C₄×C₂₃	—	8	4	no	—
267	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
268	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
269	C₂×C₄×C₂₃	#137	1	2	yes	—
270	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
272	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
273	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
274	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
275	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
276	C₈×C₂₃	—	2	2	no	—
277	C₈×C₂₃	—	2	2	no	—
278	C₈×C₂₃	—	4	1	no	—
279	C₈×C₂₃	—	4	1	no	—
280	C₂×#477	C₂×C₂₃	1	1	yes	characteristic, complemented
281	C₂×#428	C₂×C₂₃	1	1	yes	characteristic
282	#2440	C₂×C₂₃	1	1	yes	characteristic
283	#2362	C₂×C₂₃	1	1	yes	characteristic
284	#2096	C₂×C₂₃	1	1	yes	characteristic, complemented
285	#2088	C₂×C₂₃	1	1	yes	characteristic, complemented
286	#2066	C₂×C₂₃	1	1	yes	characteristic, complemented
287	#2058	C₂×C₂₃	1	1	yes	characteristic, complemented
288	#1947	C₂×C₂₃	1	1	yes	characteristic, complemented
289	#1918	C₂×C₂₃	1	1	yes	characteristic, complemented
290	#1894	C₂×C₂₃	1	1	yes	characteristic
291	#1872	C₂×C₂₃	1	1	yes	characteristic
292	#1863	C₂×C₂₃	1	1	yes	characteristic, complemented
293	#1854	C₂×C₂₃	1	1	yes	characteristic
294	#1851	C₂×C₂₃	1	1	yes	characteristic
295	C₂²×C₂₃	—	2	2	no	—
296	C₂²×C₂₃	—	2	2	no	—
297	C₂²×C₂₃	—	2	2	no	—
298	C₂²×C₂₃	—	4	1	no	—
299	C₂²×C₂₃	—	4	1	no	—
300	C₂²×C₂₃	—	4	1	no	—
301	C₂²×C₂₃	—	4	1	no	—
302	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
303	C₂²×C₂₃	D₄²	1	1	yes	characteristic
304	C₂²×C₂₃	#547	1	1	yes	characteristic
305	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
307	C₂²×C₂₃	#577	1	1	yes	characteristic
308	C₂²×C₂₃	#578	1	1	yes	characteristic
309	C₄×C₂₃	—	2	2	no	—
310	C₄×C₂₃	—	2	2	no	—
311	C₄×C₂₃	—	2	2	no	—
312	C₄×C₂₃	—	4	2	no	—
313	C₄×C₂₃	—	4	1	no	—
314	C₄×C₂₃	—	4	2	no	—
315	C₄×C₂₃	—	4	1	no	—
316	C₄×C₂₃	—	4	2	no	—
317	C₄×C₂₃	—	4	2	no	—
318	C₄×C₂₃	—	8	1	no	—
319	C₄×C₂₃	—	8	1	no	—
320	C₄×C₂₃	—	8	1	no	—
321	C₄×C₂₃	—	8	1	no	—
322	C₄×C₂₃	#525	1	1	yes	characteristic
323	C₄×C₂₃	#539	1	1	yes	characteristic
324	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
325	C₄×C₂₃	#554	1	1	yes	characteristic
326	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
327	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
328	C₂×C₂²⋊Q₈	—	2	1	no	—
329	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
330	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
331	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
332	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
333	#477	—	2	4	no	—
334	#428	—	2	4	no	—
335	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
336	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
337	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
338	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
339	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
340	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
341	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
343	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
344	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
347	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
348	#398	—	2	1	no	—
349	#395	—	2	1	no	—
350	#391	C₂²×C₂₃	1	1	yes	characteristic
351	#389	C₂²×C₂₃	1	1	yes	characteristic
352	#389	C₂²×C₂₃	1	1	yes	characteristic
353	#389	C₂²×C₂₃	1	1	yes	characteristic
354	#388	—	2	1	no	—
355	#388	—	2	1	no	—
356	#387	—	2	1	no	—
357	#385	—	2	1	no	—
358	#384	C₂²×C₂₃	1	1	yes	characteristic
359	#380	C₂²×C₂₃	1	1	yes	characteristic
360	C₄×C₄⋊C₄	—	2	1	no	—
361	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
362	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
363	#340	C₂²×C₂₃	1	1	yes	characteristic
364	#340	C₂²×C₂₃	1	1	yes	characteristic
365	#340	C₂²×C₂₃	1	1	yes	characteristic
366	#340	C₂²×C₂₃	1	1	yes	characteristic
367	#340	C₂²×C₂₃	1	1	yes	characteristic
368	#340	C₂²×C₂₃	1	1	yes	characteristic
369	#336	C₂²×C₂₃	1	1	yes	characteristic
370	#336	C₂²×C₂₃	1	1	yes	characteristic
371	C₂×C₂₃	—	4	2	no	—
372	C₂×C₂₃	#2449	1	1	yes	characteristic
373	C₂×C₂₃	#3079	1	1	yes	characteristic
374	C₂×C₂₃	#3082	1	1	yes	characteristic
375	C₂×C₂₃	#3270	1	1	yes	characteristic
376	C₂×C₂₃	#3272	1	1	yes	characteristic
377	C₂×C₂₃	#3343	1	1	yes	characteristic
378	C₂×C₂₃	#3356	1	1	yes	characteristic
379	C₂²×Q₈	—	2	1	no	—
380	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
381	C₂³×C₄	—	2	1	no	—
382	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
383	C₂×M₁₆	—	2	2	no	—
384	C₂×M₁₆	—	2	2	no	—
385	C₂×M₁₆	—	2	2	no	—
386	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
387	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
388	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
389	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
390	C₄⋊Q₈	—	2	2	no	—
391	C₄⋊Q₈	—	4	1	no	—
392	C₂²⋊Q₈	—	2	2	no	—
393	C₂²⋊Q₈	—	4	4	no	—
394	C₂²⋊Q₈	Q₈×C₂₃	1	4	yes	complemented
395	C₄×Q₈	—	2	2	no	—
396	C₄×Q₈	—	4	1	no	—
397	#117	—	2	2	no	—
398	#117	—	4	1	no	—
399	C₂×C₄⋊C₄	—	2	1	no	—
400	C₂×C₄⋊C₄	—	2	2	no	—
401	C₂×C₄⋊C₄	—	2	1	no	—
402	C₂×C₄⋊C₄	—	2	1	no	—
403	C₂×C₄⋊C₄	—	2	1	no	—
404	C₂×C₄⋊C₄	—	2	1	no	—
405	C₂×C₄⋊C₄	—	2	1	no	—
406	C₂×C₄⋊C₄	—	2	1	no	—
407	C₂×C₄⋊C₄	—	2	1	no	—
408	C₂×C₄⋊C₄	—	2	1	no	—
409	C₂×C₄⋊C₄	—	4	1	no	—
410	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
411	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
412	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
413	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
414	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
415	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
416	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
417	C₂×C₂²⋊C₄	—	2	1	no	—
418	C₂×C₂²⋊C₄	—	2	1	no	—
419	C₂×C₂²⋊C₄	—	2	1	no	—
420	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
421	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
422	C₂×C₄²	—	2	1	no	—
423	C₂×C₄²	—	2	1	no	—
424	C₂×C₄²	—	2	1	no	—
425	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
426	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
427	#107	—	2	2	no	—
428	#107	—	2	2	no	—
429	#107	D₄×C₂₃	1	4	yes	—
430	#106	—	2	2	no	—
431	#106	—	2	2	no	—
432	#106	—	4	1	no	—
433	C₄⋊C₈	—	2	2	no	—
434	C₄⋊C₈	—	2	2	no	—
435	#103	—	2	2	no	—
436	#103	—	2	2	no	—
437	#103	—	2	2	no	—
438	#103	—	4	1	no	—
439	C₂²⋊C₈	—	2	2	no	—
440	#95	—	2	1	no	—
441	#95	—	2	1	no	—
442	#95	—	2	1	no	—
443	#95	—	2	1	no	—
444	#95	—	2	1	no	—
445	#95	—	2	1	no	—
446	#95	—	2	1	no	—
447	#95	D₄×C₂₃	1	1	yes	characteristic
448	#95	D₄×C₂₃	1	1	yes	characteristic
449	C₂₃	#26355	1	1	yes	characteristic, direct product factor, Sylow
450	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
451	C₂×Q₈	—	2	1	no	—
452	C₂×Q₈	—	2	1	no	—
453	C₂×Q₈	—	2	1	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	1	no	—
456	C₂×Q₈	—	4	2	no	—
457	C₂×Q₈	—	4	2	no	—
458	C₂×Q₈	—	8	1	no	—
459	C₂×Q₈	C₂×Q₈×C₂₃	1	2	yes	—
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	—	2	1	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	4	2	no	—
473	C₂²×C₄	—	4	2	no	—
474	C₂²×C₄	—	8	1	no	—
475	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
479	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
480	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
481	C₂²×C₄	#41×C₂₃	1	2	yes	—
482	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
483	M₁₆	—	2	8	no	—
484	C₂×C₈	—	2	1	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	1	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	2	1	no	—
494	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
495	C₂×C₈	#41×C₂₃	1	2	yes	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	2	1	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	4	2	no	—
509	C₄⋊C₄	—	4	2	no	—
510	C₄⋊C₄	—	4	2	no	—
511	C₄⋊C₄	—	4	2	no	—
512	C₄⋊C₄	—	4	2	no	—
513	C₄⋊C₄	—	4	2	no	—
514	C₄⋊C₄	—	4	2	no	—
515	C₄⋊C₄	—	8	1	no	—
516	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
517	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
518	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
519	C₂²⋊C₄	—	2	4	no	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	4	2	no	—
523	C₂²⋊C₄	—	4	2	no	—
524	C₄²	—	2	1	no	—
525	C₄²	—	2	1	no	—
526	C₄²	—	2	1	no	—
527	C₄²	—	2	1	no	—
528	C₄²	—	4	2	no	—
529	C₄²	—	4	2	no	—
530	C₄²	—	4	2	no	—
531	C₂³	—	2	1	no	—
532	C₂³	—	2	1	no	—
533	C₂³	—	2	1	no	—
534	C₂³	—	2	1	no	—
535	C₂³	—	2	1	no	—
536	C₂³	—	2	1	no	—
537	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
538	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
539	Q₈	—	2	2	no	—
540	Q₈	—	2	2	no	—
541	Q₈	—	4	1	no	—
542	Q₈	—	4	1	no	—
543	Q₈	—	8	4	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	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	—	4	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	8	4	no	—
585	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
586	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
587	C₂×C₄	C₂₃×#137	1	2	yes	—
588	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
589	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
590	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
591	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
592	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
593	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
594	C₈	—	2	2	no	—
595	C₈	—	2	2	no	—
596	C₈	—	4	1	no	—
597	C₈	—	4	1	no	—
598	C₂²	—	2	2	no	—
599	C₂²	—	2	2	no	—
600	C₂²	—	2	2	no	—
601	C₂²	—	4	1	no	—
602	C₂²	—	4	1	no	—
603	C₂²	—	4	1	no	—
604	C₂²	—	4	1	no	—
605	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
606	C₂²	D₄²×C₂₃	1	1	yes	characteristic
607	C₂²	C₂₃×#547	1	1	yes	characteristic
608	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
609	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
610	C₂²	C₂₃×#577	1	1	yes	characteristic
611	C₂²	C₂₃×#578	1	1	yes	characteristic
612	C₄	—	2	2	no	—
613	C₄	—	2	2	no	—
614	C₄	—	2	2	no	—
615	C₄	—	4	2	no	—
616	C₄	—	4	1	no	—
617	C₄	—	4	2	no	—
618	C₄	—	4	2	no	—
619	C₄	—	4	2	no	—
620	C₄	—	4	1	no	—
621	C₄	—	8	1	no	—
622	C₄	—	8	1	no	—
623	C₄	—	8	1	no	—
624	C₄	—	8	1	no	—
625	C₄	C₂₃×#525	1	1	yes	characteristic
626	C₄	C₂₃×#539	1	1	yes	characteristic
627	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
628	C₄	C₂₃×#554	1	1	yes	characteristic
629	C₂	—	4	2	no	—
630	C₂	C₂₃×#2449	1	1	yes	characteristic
631	C₂	C₂₃×#3079	1	1	yes	characteristic
632	C₂	C₂₃×#3082	1	1	yes	characteristic
633	C₂	C₂₃×#3270	1	1	yes	characteristic
634	C₂	C₂₃×#3272	1	1	yes	characteristic
635	C₂	C₂₃×#3343	1	1	yes	characteristic
636	C₂	C₂₃×#3356	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₂²⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#477	C₂	semidirect
5. C₂₃×#2096	C₂	semidirect
6. C₂₃×#2088	C₂	semidirect
7. C₂₃×#2066	C₂	semidirect
8. C₂₃×#2058	C₂	semidirect
9. C₂₃×#1947	C₂	semidirect
10. C₂₃×#1918	C₂	semidirect
13. C₂₃×#1863	C₂	semidirect
76. C₂₃×C₂²⋊Q₈	Q₈	semidirect
212. #26355	C₂₃	direct (N × H)
280. C₂×#477	C₂×C₂₃	semidirect
284. #2096	C₂×C₂₃	semidirect
285. #2088	C₂×C₂₃	semidirect
286. #2066	C₂×C₂₃	semidirect
287. #2058	C₂×C₂₃	semidirect
288. #1947	C₂×C₂₃	semidirect
289. #1918	C₂×C₂₃	semidirect
292. #1863	C₂×C₂₃	semidirect
394. C₂²⋊Q₈	Q₈×C₂₃	semidirect
449. C₂₃	#26355	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×#477	2
C₂×C₂₃×#428	2
C₂₃×#2440	2
C₂₃×#2362	2
C₂₃×#2096	2
C₂₃×#2088	2
C₂₃×#2066	2
C₂₃×#2058	2
C₂₃×#1947	2
C₂₃×#1918	2
C₂₃×#1894	2
C₂₃×#1872	2
C₂₃×#1863	2
C₂₃×#1854	2
C₂₃×#1851	2
#26355	23

Subgroups — C23×#26355 #19173981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26355 #19173981