Subgroups — C₂₃×#22969 #19170595

Order 5888.

886 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#476	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2369	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1947	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1944	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1920	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1862	C₂	1	2	yes	maximal
9	C₂₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1762	C₂	1	2	yes	maximal
11	C₄×C₂₃×#102	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
16	C₂₃×#476	C₄	1	8	yes	complemented
17	C₂×C₂₃×#107	C₂²	1	2	yes	—
18	C₂×C₂₃×#106	C₂²	1	2	yes	—
19	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#102	C₂²	1	2	yes	—
22	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#97	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#391	—	2	2	no	—
26	C₂₃×#389	—	2	1	no	—
27	C₂₃×#389	C₂²	1	2	yes	—
28	C₂₃×#389	C₂²	1	2	yes	—
29	C₂₃×#385	—	2	2	no	—
30	C₂₃×#384	C₂²	1	1	yes	characteristic
31	C₂₃×#380	—	2	1	no	—
32	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
33	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
36	C₂₃×#340	C₂²	1	1	yes	characteristic
37	C₂₃×#340	C₂²	1	2	yes	—
38	C₂₃×#340	C₂²	1	2	yes	—
39	C₂₃×#340	C₂²	1	1	yes	characteristic
40	C₂₃×#340	C₂²	1	2	yes	—
41	C₂₃×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₂₃	—	2	1	no	—
43	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₂₃	—	2	1	no	—
45	C₂³×C₄×C₂₃	—	2	2	no	—
46	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₂₃×C₄⋊Q₈	—	2	2	no	—
50	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
51	C₄×D₄×C₂₃	—	2	4	no	—
52	C₄×D₄×C₂₃	—	2	2	no	—
53	C₄×D₄×C₂₃	—	4	2	no	—
54	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
55	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
61	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
70	C₂×C₄²×C₂₃	—	2	1	no	—
71	C₂×C₄²×C₂₃	—	2	2	no	—
72	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
73	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
74	C₂₃×#107	—	4	2	no	—
75	C₂₃×#106	C₂×C₄	1	8	yes	complemented
76	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
77	C₂₃×#102	—	2	2	no	—
78	C₂₃×#102	C₂×C₄	1	8	yes	—
79	C₂₃×#102	Q₈	1	4	yes	complemented
80	C₂₃×#97	—	4	2	no	—
81	C₄×C₈×C₂₃	—	2	2	no	—
82	C₂₃×#95	—	2	2	no	—
83	C₂₃×#95	—	2	2	no	—
84	C₂₃×#95	—	2	1	no	—
85	C₂₃×#95	—	2	2	no	—
86	C₂₃×#95	D₄	1	2	yes	—
87	C₂⁴×C₂₃	—	2	1	no	—
88	C₂×Q₈×C₂₃	—	2	4	no	—
89	C₂×Q₈×C₂₃	—	8	1	no	—
90	C₂×Q₈×C₂₃	—	8	1	no	—
91	C₂×D₄×C₂₃	—	2	1	no	—
92	C₂×D₄×C₂₃	—	2	1	no	—
93	C₂×D₄×C₂₃	—	2	1	no	—
94	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
95	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
96	C₂×D₄×C₂₃	#41	1	2	yes	—
97	C₂²×C₄×C₂₃	—	2	1	no	—
98	C₂²×C₄×C₂₃	—	2	2	no	—
99	C₂²×C₄×C₂₃	—	2	1	no	—
100	C₂²×C₄×C₂₃	—	2	4	no	—
101	C₂²×C₄×C₂₃	—	2	1	no	—
102	C₂²×C₄×C₂₃	—	2	2	no	—
103	C₂²×C₄×C₂₃	—	4	4	no	—
104	C₂²×C₄×C₂₃	—	4	4	no	—
105	C₂²×C₄×C₂₃	—	4	2	no	—
106	C₂²×C₄×C₂₃	—	4	2	no	—
107	C₂²×C₄×C₂₃	—	8	2	no	—
108	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
113	C₂×C₈×C₂₃	—	2	1	no	—
114	C₂×C₈×C₂₃	—	2	2	no	—
115	C₂×C₈×C₂₃	—	2	1	no	—
116	C₂×C₈×C₂₃	—	2	2	no	—
117	C₂×C₈×C₂₃	—	2	1	no	—
118	C₂×C₈×C₂₃	—	2	1	no	—
119	C₂×C₈×C₂₃	—	2	1	no	—
120	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
121	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
122	C₄⋊C₄×C₂₃	—	2	4	no	—
123	C₄⋊C₄×C₂₃	—	2	2	no	—
124	C₄⋊C₄×C₂₃	—	2	2	no	—
125	C₄⋊C₄×C₂₃	—	2	1	no	—
126	C₄⋊C₄×C₂₃	—	2	2	no	—
127	C₄⋊C₄×C₂₃	—	2	1	no	—
128	C₄⋊C₄×C₂₃	—	2	1	no	—
129	C₄⋊C₄×C₂₃	—	2	2	no	—
130	C₄⋊C₄×C₂₃	—	2	1	no	—
131	C₄⋊C₄×C₂₃	—	2	2	no	—
132	C₄⋊C₄×C₂₃	—	4	4	no	—
133	C₄⋊C₄×C₂₃	—	4	2	no	—
134	C₄⋊C₄×C₂₃	—	8	1	no	—
135	C₄⋊C₄×C₂₃	—	8	2	no	—
136	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
137	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
138	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
139	C₂²⋊C₄×C₂₃	—	2	4	no	—
140	C₂²⋊C₄×C₂₃	—	4	4	no	—
141	C₄²×C₂₃	—	2	2	no	—
142	C₄²×C₂₃	—	2	2	no	—
143	C₄²×C₂₃	—	4	2	no	—
144	C₄²×C₂₃	—	8	2	no	—
145	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
146	C₄²×C₂₃	#41	1	2	yes	—
147	#22969	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂³×C₂₃	—	2	2	no	—
149	C₂³×C₂₃	—	2	2	no	—
150	C₂³×C₂₃	—	2	2	no	—
151	C₂³×C₂₃	—	4	1	no	—
152	C₂³×C₂₃	—	4	1	no	—
153	C₂³×C₂₃	—	4	1	no	—
154	C₂³×C₂₃	—	4	1	no	—
155	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
156	Q₈×C₂₃	—	8	4	no	—
157	D₄×C₂₃	—	2	2	no	—
158	D₄×C₂₃	—	2	2	no	—
159	D₄×C₂₃	—	2	2	no	—
160	D₄×C₂₃	C₄×Q₈	1	4	yes	—
161	C₂×C₄×C₂₃	—	2	1	no	—
162	C₂×C₄×C₂₃	—	2	1	no	—
163	C₂×C₄×C₂₃	—	2	2	no	—
164	C₂×C₄×C₂₃	—	2	4	no	—
165	C₂×C₄×C₂₃	—	2	2	no	—
166	C₂×C₄×C₂₃	—	2	2	no	—
167	C₂×C₄×C₂₃	—	2	2	no	—
168	C₂×C₄×C₂₃	—	2	2	no	—
169	C₂×C₄×C₂₃	—	2	2	no	—
170	C₂×C₄×C₂₃	—	2	2	no	—
171	C₂×C₄×C₂₃	—	2	2	no	—
172	C₂×C₄×C₂₃	—	4	1	no	—
173	C₂×C₄×C₂₃	—	4	1	no	—
174	C₂×C₄×C₂₃	—	4	1	no	—
175	C₂×C₄×C₂₃	—	4	2	no	—
176	C₂×C₄×C₂₃	—	4	1	no	—
177	C₂×C₄×C₂₃	—	4	2	no	—
178	C₂×C₄×C₂₃	—	4	8	no	—
179	C₂×C₄×C₂₃	—	4	2	no	—
180	C₂×C₄×C₂₃	—	4	1	no	—
181	C₂×C₄×C₂₃	—	4	1	no	—
182	C₂×C₄×C₂₃	—	4	2	no	—
183	C₂×C₄×C₂₃	—	4	1	no	—
184	C₂×C₄×C₂₃	—	4	2	no	—
185	C₂×C₄×C₂₃	—	8	8	no	—
186	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
187	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
188	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
189	C₂×C₄×C₂₃	#137	1	2	yes	—
190	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
191	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
192	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
195	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
196	C₈×C₂₃	—	2	4	no	—
197	C₈×C₂₃	—	2	2	no	—
198	C₈×C₂₃	—	4	1	no	—
199	C₂×#476	C₂×C₂₃	1	1	yes	characteristic
200	#2369	C₂×C₂₃	1	1	yes	characteristic, complemented
201	D₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
202	#1990	C₂×C₂₃	1	1	yes	characteristic, complemented
203	#1947	C₂×C₂₃	1	1	yes	characteristic, complemented
204	#1944	C₂×C₂₃	1	2	yes	complemented
205	#1920	C₂×C₂₃	1	2	yes	complemented
206	#1862	C₂×C₂₃	1	2	yes	—
207	#1770	C₂×C₂₃	1	1	yes	characteristic, complemented
208	#1762	C₂×C₂₃	1	2	yes	—
209	C₄×#102	C₂×C₂₃	1	1	yes	characteristic
210	C₂²×C₂₃	—	2	4	no	—
211	C₂²×C₂₃	—	4	2	no	—
212	C₂²×C₂₃	—	4	2	no	—
213	C₂²×C₂₃	—	4	2	no	—
214	C₂²×C₂₃	—	4	2	no	—
215	C₂²×C₂₃	—	4	2	no	—
216	C₂²×C₂₃	—	4	2	no	—
217	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
218	C₂²×C₂₃	#546	1	1	yes	characteristic
219	C₂²×C₂₃	D₄×Q₈	1	1	yes	characteristic
220	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
221	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
222	C₂²×C₂₃	#576	1	1	yes	characteristic
223	C₂²×C₂₃	#577	1	1	yes	characteristic
224	C₄×C₂₃	—	2	2	no	—
225	C₄×C₂₃	—	4	2	no	—
226	C₄×C₂₃	—	4	2	no	—
227	C₄×C₂₃	—	4	2	no	—
228	C₄×C₂₃	—	4	4	no	—
229	C₄×C₂₃	—	4	2	no	—
230	C₄×C₂₃	—	8	1	no	—
231	C₄×C₂₃	—	8	2	no	—
232	C₄×C₂₃	#440	1	4	yes	—
233	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
234	C₄×C₂₃	#520	1	1	yes	characteristic
235	C₄×C₂₃	#552	1	1	yes	characteristic
236	C₄×C₂₃	#554	1	1	yes	characteristic
237	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
238	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
239	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
240	C₂²×C₄⋊C₄	—	2	1	no	—
241	#476	C₄×C₂₃	1	8	yes	complemented
242	C₂×#107	C₂²×C₂₃	1	2	yes	—
243	C₂×#106	C₂²×C₂₃	1	2	yes	—
244	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
245	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
246	C₂×#102	C₂²×C₂₃	1	2	yes	—
247	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
248	C₂×#97	C₂²×C₂₃	1	2	yes	—
249	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
250	#391	—	2	2	no	—
251	#389	—	2	1	no	—
252	#389	C₂²×C₂₃	1	2	yes	—
253	#389	C₂²×C₂₃	1	2	yes	—
254	#385	—	2	2	no	—
255	#384	C₂²×C₂₃	1	1	yes	characteristic
256	#380	—	2	1	no	—
257	C₄×C₄⋊C₄	—	2	1	no	—
258	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
259	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
260	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
261	#340	C₂²×C₂₃	1	2	yes	—
262	#340	C₂²×C₂₃	1	1	yes	characteristic
263	#340	C₂²×C₂₃	1	2	yes	—
264	#340	C₂²×C₂₃	1	1	yes	characteristic
265	#340	C₂²×C₂₃	1	2	yes	—
266	#336	C₂²×C₂₃	1	2	yes	—
267	C₂×C₂₃	—	4	4	no	—
268	C₂×C₂₃	#2360	1	1	yes	characteristic
269	C₂×C₂₃	C₂×#440	1	1	yes	characteristic
270	C₂×C₂₃	#2946	1	1	yes	characteristic
271	C₂×C₂₃	#3325	1	1	yes	characteristic
272	C₂×C₂₃	#3331	1	1	yes	characteristic
273	C₂×C₂₃	#3384	1	1	yes	characteristic
274	C₂×C₂₃	#3385	1	1	yes	characteristic
275	C₂²×Q₈	—	2	1	no	—
276	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
277	C₂³×C₄	—	2	1	no	—
278	C₂³×C₄	—	2	2	no	—
279	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
280	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
281	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
282	C₄⋊Q₈	—	2	2	no	—
283	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
284	C₄×D₄	—	2	4	no	—
285	C₄×D₄	—	2	2	no	—
286	C₄×D₄	—	4	2	no	—
287	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	—	2	2	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	—	2	2	no	—
293	C₂×C₄⋊C₄	—	4	4	no	—
294	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
295	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
296	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
297	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
298	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
299	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
300	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
301	C₂×C₂²⋊C₄	—	2	1	no	—
302	C₂×C₂²⋊C₄	—	2	2	no	—
303	C₂×C₄²	—	2	1	no	—
304	C₂×C₄²	—	2	2	no	—
305	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
306	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
307	#107	—	4	2	no	—
308	#106	C₂×C₄×C₂₃	1	8	yes	complemented
309	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
310	#102	—	2	2	no	—
311	#102	C₂×C₄×C₂₃	1	8	yes	—
312	#102	Q₈×C₂₃	1	4	yes	complemented
313	#97	—	4	2	no	—
314	C₄×C₈	—	2	2	no	—
315	#95	—	2	2	no	—
316	#95	—	2	2	no	—
317	#95	—	2	1	no	—
318	#95	—	2	2	no	—
319	#95	D₄×C₂₃	1	2	yes	—
320	C₂₃	#22969	1	1	yes	characteristic, direct product factor, Sylow
321	C₂⁴	—	2	1	no	—
322	C₂×Q₈	—	2	4	no	—
323	C₂×Q₈	—	8	1	no	—
324	C₂×Q₈	—	8	1	no	—
325	C₂×D₄	—	2	1	no	—
326	C₂×D₄	—	2	1	no	—
327	C₂×D₄	—	2	1	no	—
328	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
329	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
330	C₂×D₄	#41×C₂₃	1	2	yes	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	4	4	no	—
338	C₂²×C₄	—	4	4	no	—
339	C₂²×C₄	—	4	2	no	—
340	C₂²×C₄	—	4	2	no	—
341	C₂²×C₄	—	8	2	no	—
342	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
343	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
344	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
345	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
346	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
355	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
356	C₄⋊C₄	—	2	4	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	4	4	no	—
367	C₄⋊C₄	—	4	2	no	—
368	C₄⋊C₄	—	8	1	no	—
369	C₄⋊C₄	—	8	2	no	—
370	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
371	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
372	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
373	C₂²⋊C₄	—	2	4	no	—
374	C₂²⋊C₄	—	4	4	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	4	2	no	—
378	C₄²	—	8	2	no	—
379	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
380	C₄²	#41×C₂₃	1	2	yes	—
381	C₂³	—	2	2	no	—
382	C₂³	—	2	2	no	—
383	C₂³	—	2	2	no	—
384	C₂³	—	4	1	no	—
385	C₂³	—	4	1	no	—
386	C₂³	—	4	1	no	—
387	C₂³	—	4	1	no	—
388	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
389	Q₈	—	8	4	no	—
390	D₄	—	2	2	no	—
391	D₄	—	2	2	no	—
392	D₄	—	2	2	no	—
393	D₄	C₄×Q₈×C₂₃	1	4	yes	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	8	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	8	8	no	—
419	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
420	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
421	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
422	C₂×C₄	C₂₃×#137	1	2	yes	—
423	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
424	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
425	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
426	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
427	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
428	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
429	C₈	—	2	2	no	—
430	C₈	—	2	4	no	—
431	C₈	—	4	1	no	—
432	C₂²	—	2	4	no	—
433	C₂²	—	4	2	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	4	2	no	—
436	C₂²	—	4	2	no	—
437	C₂²	—	4	2	no	—
438	C₂²	—	4	2	no	—
439	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
440	C₂²	C₂₃×#546	1	1	yes	characteristic
441	C₂²	D₄×Q₈×C₂₃	1	1	yes	characteristic
442	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
443	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
444	C₂²	C₂₃×#576	1	1	yes	characteristic
445	C₂²	C₂₃×#577	1	1	yes	characteristic
446	C₄	—	2	2	no	—
447	C₄	—	4	2	no	—
448	C₄	—	4	4	no	—
449	C₄	—	4	2	no	—
450	C₄	—	4	2	no	—
451	C₄	—	4	2	no	—
452	C₄	—	8	2	no	—
453	C₄	—	8	1	no	—
454	C₄	C₂₃×#440	1	4	yes	—
455	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
456	C₄	C₂₃×#520	1	1	yes	characteristic
457	C₄	C₂₃×#552	1	1	yes	characteristic
458	C₄	C₂₃×#554	1	1	yes	characteristic
459	C₂	—	4	4	no	—
460	C₂	C₂₃×#2360	1	1	yes	characteristic
461	C₂	C₂×C₂₃×#440	1	1	yes	characteristic
462	C₂	C₂₃×#2946	1	1	yes	characteristic
463	C₂	C₂₃×#3325	1	1	yes	characteristic
464	C₂	C₂₃×#3331	1	1	yes	characteristic
465	C₂	C₂₃×#3384	1	1	yes	characteristic
466	C₂	C₂₃×#3385	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
2. C₂₃×#2369	C₂	semidirect
4. C₂₃×#1990	C₂	semidirect
5. C₂₃×#1947	C₂	semidirect
6. C₂₃×#1944	C₂	semidirect
7. C₂₃×#1920	C₂	semidirect
9. C₂₃×#1770	C₂	semidirect
16. C₂₃×#476	C₄	semidirect
50. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
75. C₂₃×#106	C₂×C₄	semidirect
76. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
79. C₂₃×#102	Q₈	semidirect
147. #22969	C₂₃	direct (N × H)
200. #2369	C₂×C₂₃	semidirect
202. #1990	C₂×C₂₃	semidirect
203. #1947	C₂×C₂₃	semidirect
204. #1944	C₂×C₂₃	semidirect
205. #1920	C₂×C₂₃	semidirect
207. #1770	C₂×C₂₃	semidirect
241. #476	C₄×C₂₃	semidirect
283. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
308. #106	C₂×C₄×C₂₃	semidirect
309. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
312. #102	Q₈×C₂₃	semidirect
320. C₂₃	#22969	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×#476	2
C₂₃×#2369	2
D₄×C₄⋊C₄×C₂₃	2
C₂₃×#1990	2
C₂₃×#1947	2
C₂₃×#1944	2
C₂₃×#1920	2
C₂₃×#1862	2
C₂₃×#1770	2
C₂₃×#1762	2
C₄×C₂₃×#102	2
#22969	23

Subgroups — C23×#22969 #19170595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22969 #19170595