Subgroups — C₂₃×#22615 #19170241

Order 5888.

838 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#106	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2345	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2344	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1944	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1920	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1919	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1824	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#1823	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1808	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#1776	C₂	1	2	yes	maximal, complemented
11	C₂₃×#1752	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#117	—	2	1	no	—
14	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#107	C₂²	1	2	yes	—
17	C₂×C₂₃×#106	C₄	1	8	yes	complemented
18	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#97	C₂²	1	2	yes	—
24	C₂₃×#389	C₂²	1	2	yes	—
25	C₂₃×#389	C₂²	1	2	yes	—
26	C₂₃×#387	—	2	2	no	—
27	C₂₃×#387	—	2	2	no	—
28	C₂₃×#381	—	2	1	no	—
29	C₂₃×#380	—	2	1	no	—
30	C₂₃×#380	C₂²	1	1	yes	characteristic
31	C₂₃×#380	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
34	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
35	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
36	C₂₃×#340	C₂²	1	2	yes	—
37	C₂₃×#340	C₂²	1	2	yes	—
38	C₂₃×#340	C₂²	1	2	yes	—
39	C₂₃×#340	C₂²	1	2	yes	—
40	C₂₃×#340	C₂²	1	2	yes	—
41	C₂₃×#336	C₂²	1	2	yes	—
42	C₂³×C₄×C₂₃	—	2	1	no	—
43	C₂³×C₄×C₂₃	—	2	1	no	—
44	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₂₃	—	2	4	no	—
46	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₂₃×#117	—	4	4	no	—
51	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
59	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
69	C₂×C₄²×C₂₃	—	2	2	no	—
70	C₂×C₄²×C₂₃	—	2	2	no	—
71	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
72	C₂₃×#107	—	2	4	no	—
73	C₂₃×#106	C₂×C₄	1	4	yes	—
74	C₂₃×#106	C₂×C₄	1	4	yes	—
75	C₂₃×#106	C₂×C₄	1	4	yes	—
76	C₂₃×#106	C₂×C₄	1	4	yes	—
77	C₂₃×C₂²⋊C₈	—	2	2	no	—
78	C₂₃×#97	—	2	4	no	—
79	C₂₃×#95	—	2	2	no	—
80	C₂₃×#95	—	2	2	no	—
81	C₂₃×#95	—	2	2	no	—
82	C₂₃×#95	—	2	1	no	—
83	C₂₃×#95	—	2	1	no	—
84	C₂₃×#95	—	2	2	no	—
85	C₂₃×#95	D₄	1	2	yes	—
86	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₂₃	—	2	2	no	—
88	C₂²×C₄×C₂₃	—	2	2	no	—
89	C₂²×C₄×C₂₃	—	2	4	no	—
90	C₂²×C₄×C₂₃	—	2	1	no	—
91	C₂²×C₄×C₂₃	—	2	1	no	—
92	C₂²×C₄×C₂₃	—	2	1	no	—
93	C₂²×C₄×C₂₃	—	2	1	no	—
94	C₂²×C₄×C₂₃	—	2	4	no	—
95	C₂²×C₄×C₂₃	—	2	2	no	—
96	C₂²×C₄×C₂₃	—	2	2	no	—
97	C₂²×C₄×C₂₃	—	4	2	no	—
98	C₂²×C₄×C₂₃	—	4	2	no	—
99	C₂²×C₄×C₂₃	—	8	2	no	—
100	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₂₃	#41	1	2	yes	—
105	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
106	C₂×C₈×C₂₃	—	2	2	no	—
107	C₂×C₈×C₂₃	—	2	8	no	—
108	C₂×C₈×C₂₃	—	2	2	no	—
109	C₂×C₈×C₂₃	—	2	2	no	—
110	C₂×C₈×C₂₃	—	2	2	no	—
111	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₂₃	#41	1	4	yes	—
114	C₄⋊C₄×C₂₃	—	2	4	no	—
115	C₄⋊C₄×C₂₃	—	2	4	no	—
116	C₄⋊C₄×C₂₃	—	2	1	no	—
117	C₄⋊C₄×C₂₃	—	2	1	no	—
118	C₄⋊C₄×C₂₃	—	2	1	no	—
119	C₄⋊C₄×C₂₃	—	2	1	no	—
120	C₄⋊C₄×C₂₃	—	2	2	no	—
121	C₄⋊C₄×C₂₃	—	2	2	no	—
122	C₄⋊C₄×C₂₃	—	2	2	no	—
123	C₄⋊C₄×C₂₃	—	2	2	no	—
124	C₄⋊C₄×C₂₃	—	4	4	no	—
125	C₄⋊C₄×C₂₃	—	8	2	no	—
126	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
128	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
129	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
130	C₂²⋊C₄×C₂₃	—	2	2	no	—
131	C₂²⋊C₄×C₂₃	—	2	2	no	—
132	C₂²⋊C₄×C₂₃	—	4	2	no	—
133	C₂²⋊C₄×C₂₃	—	4	2	no	—
134	C₄²×C₂₃	—	2	2	no	—
135	C₄²×C₂₃	—	2	2	no	—
136	C₄²×C₂₃	—	4	4	no	—
137	C₄²×C₂₃	—	8	2	no	—
138	#22615	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂³×C₂₃	—	2	2	no	—
140	C₂³×C₂₃	—	2	2	no	—
141	C₂³×C₂₃	C₄×D₄	1	2	yes	—
142	C₂³×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
143	C₂³×C₂₃	#137	1	2	yes	—
144	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₂₃	—	2	2	no	—
146	C₂×C₄×C₂₃	—	2	2	no	—
147	C₂×C₄×C₂₃	—	2	2	no	—
148	C₂×C₄×C₂₃	—	2	4	no	—
149	C₂×C₄×C₂₃	—	2	2	no	—
150	C₂×C₄×C₂₃	—	2	2	no	—
151	C₂×C₄×C₂₃	—	2	2	no	—
152	C₂×C₄×C₂₃	—	2	2	no	—
153	C₂×C₄×C₂₃	—	2	2	no	—
154	C₂×C₄×C₂₃	—	2	2	no	—
155	C₂×C₄×C₂₃	—	2	2	no	—
156	C₂×C₄×C₂₃	—	2	2	no	—
157	C₂×C₄×C₂₃	—	2	2	no	—
158	C₂×C₄×C₂₃	—	2	2	no	—
159	C₂×C₄×C₂₃	—	4	1	no	—
160	C₂×C₄×C₂₃	—	4	1	no	—
161	C₂×C₄×C₂₃	—	4	1	no	—
162	C₂×C₄×C₂₃	—	4	1	no	—
163	C₂×C₄×C₂₃	—	4	2	no	—
164	C₂×C₄×C₂₃	—	4	2	no	—
165	C₂×C₄×C₂₃	—	4	8	no	—
166	C₂×C₄×C₂₃	—	4	2	no	—
167	C₂×C₄×C₂₃	—	4	2	no	—
168	C₂×C₄×C₂₃	—	4	2	no	—
169	C₂×C₄×C₂₃	—	4	2	no	—
170	C₂×C₄×C₂₃	—	4	2	no	—
171	C₂×C₄×C₂₃	—	4	2	no	—
172	C₂×C₄×C₂₃	—	8	4	no	—
173	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
174	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
175	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
176	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
177	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
178	C₈×C₂₃	—	2	2	no	—
179	C₈×C₂₃	—	4	2	no	—
180	C₈×C₂₃	#117	1	4	yes	complemented
181	C₂²×#106	C₂×C₂₃	1	1	yes	characteristic
182	#2345	C₂×C₂₃	1	1	yes	characteristic
183	#2344	C₂×C₂₃	1	1	yes	characteristic
184	#1944	C₂×C₂₃	1	2	yes	complemented
185	#1920	C₂×C₂₃	1	2	yes	complemented
186	#1919	C₂×C₂₃	1	2	yes	complemented
187	#1824	C₂×C₂₃	1	1	yes	characteristic
188	#1823	C₂×C₂₃	1	1	yes	characteristic
189	#1808	C₂×C₂₃	1	1	yes	characteristic
190	#1776	C₂×C₂₃	1	2	yes	complemented
191	#1752	C₂×C₂₃	1	1	yes	characteristic
192	C₂²×C₂₃	—	2	4	no	—
193	C₂²×C₂₃	—	2	4	no	—
194	C₂²×C₂₃	—	2	2	no	—
195	C₂²×C₂₃	—	2	2	no	—
196	C₂²×C₂₃	#440	1	4	yes	—
197	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
198	C₂²×C₂₃	#547	1	2	yes	—
199	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
201	C₂²×C₂₃	#577	1	2	yes	—
202	C₄×C₂₃	—	2	2	no	—
203	C₄×C₂₃	—	4	2	no	—
204	C₄×C₂₃	—	4	2	no	—
205	C₄×C₂₃	—	4	2	no	—
206	C₄×C₂₃	—	4	2	no	—
207	C₄×C₂₃	—	4	2	no	—
208	C₄×C₂₃	—	4	2	no	—
209	C₄×C₂₃	—	8	2	no	—
210	C₄×C₂₃	—	8	2	no	—
211	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
212	C₄×C₂₃	#519	1	1	yes	characteristic
213	C₄×C₂₃	#552	1	2	yes	—
214	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
215	C₂×#117	—	2	1	no	—
216	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
217	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
218	C₂×#107	C₂²×C₂₃	1	2	yes	—
219	C₂×#106	C₄×C₂₃	1	8	yes	complemented
220	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
221	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
222	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
223	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
224	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
225	C₂×#97	C₂²×C₂₃	1	2	yes	—
226	#389	C₂²×C₂₃	1	2	yes	—
227	#389	C₂²×C₂₃	1	2	yes	—
228	#387	—	2	2	no	—
229	#387	—	2	2	no	—
230	#381	—	2	1	no	—
231	#380	—	2	1	no	—
232	#380	C₂²×C₂₃	1	1	yes	characteristic
233	#380	C₂²×C₂₃	1	1	yes	characteristic
234	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
235	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
236	C₄×C₂²⋊C₄	—	2	1	no	—
237	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
238	#340	C₂²×C₂₃	1	2	yes	—
239	#340	C₂²×C₂₃	1	2	yes	—
240	#340	C₂²×C₂₃	1	2	yes	—
241	#340	C₂²×C₂₃	1	2	yes	—
242	#340	C₂²×C₂₃	1	2	yes	—
243	#336	C₂²×C₂₃	1	2	yes	—
244	C₂×C₂₃	—	2	4	no	—
245	C₂×C₂₃	#2344	1	1	yes	characteristic
246	C₂×C₂₃	C₂×#440	1	1	yes	characteristic
247	C₂×C₂₃	#2949	1	1	yes	characteristic
248	C₂×C₂₃	#3344	1	2	yes	—
249	C₂×C₂₃	#3345	1	2	yes	—
250	C₂³×C₄	—	2	1	no	—
251	C₂³×C₄	—	2	1	no	—
252	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
253	C₂²×C₈	—	2	4	no	—
254	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
255	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
256	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
257	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
258	#117	—	4	4	no	—
259	C₂×C₄⋊C₄	—	2	1	no	—
260	C₂×C₄⋊C₄	—	2	1	no	—
261	C₂×C₄⋊C₄	—	2	2	no	—
262	C₂×C₄⋊C₄	—	2	2	no	—
263	C₂×C₄⋊C₄	—	2	2	no	—
264	C₂×C₄⋊C₄	—	2	2	no	—
265	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
266	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
267	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
268	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
269	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
270	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
271	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
272	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
273	C₂×C₂²⋊C₄	—	2	1	no	—
274	C₂×C₂²⋊C₄	—	2	1	no	—
275	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
276	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
277	C₂×C₄²	—	2	2	no	—
278	C₂×C₄²	—	2	2	no	—
279	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
280	#107	—	2	4	no	—
281	#106	C₂×C₄×C₂₃	1	4	yes	—
282	#106	C₂×C₄×C₂₃	1	4	yes	—
283	#106	C₂×C₄×C₂₃	1	4	yes	—
284	#106	C₂×C₄×C₂₃	1	4	yes	—
285	C₂²⋊C₈	—	2	2	no	—
286	#97	—	2	4	no	—
287	#95	—	2	2	no	—
288	#95	—	2	2	no	—
289	#95	—	2	2	no	—
290	#95	—	2	1	no	—
291	#95	—	2	1	no	—
292	#95	—	2	2	no	—
293	#95	D₄×C₂₃	1	2	yes	—
294	C₂₃	#22615	1	1	yes	characteristic, direct product factor, Sylow
295	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	4	no	—
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	2	1	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	4	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	4	2	no	—
307	C₂²×C₄	—	4	2	no	—
308	C₂²×C₄	—	8	2	no	—
309	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
310	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
311	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
312	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
313	C₂²×C₄	#41×C₂₃	1	2	yes	—
314	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	8	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
321	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
322	C₂×C₈	#41×C₂₃	1	4	yes	—
323	C₄⋊C₄	—	2	4	no	—
324	C₄⋊C₄	—	2	4	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	2	1	no	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	4	4	no	—
334	C₄⋊C₄	—	8	2	no	—
335	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
336	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
337	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
338	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
339	C₂²⋊C₄	—	2	2	no	—
340	C₂²⋊C₄	—	2	2	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₂²⋊C₄	—	4	2	no	—
343	C₄²	—	2	2	no	—
344	C₄²	—	2	2	no	—
345	C₄²	—	4	4	no	—
346	C₄²	—	8	2	no	—
347	C₂³	—	2	2	no	—
348	C₂³	—	2	2	no	—
349	C₂³	C₄×D₄×C₂₃	1	2	yes	—
350	C₂³	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
351	C₂³	C₂₃×#137	1	2	yes	—
352	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	4	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	8	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	8	4	no	—
381	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
382	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
383	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
384	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
385	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
386	C₈	—	2	2	no	—
387	C₈	—	4	2	no	—
388	C₈	C₂₃×#117	1	4	yes	complemented
389	C₂²	—	2	2	no	—
390	C₂²	—	2	4	no	—
391	C₂²	—	2	4	no	—
392	C₂²	—	2	2	no	—
393	C₂²	C₂₃×#440	1	4	yes	—
394	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
395	C₂²	C₂₃×#547	1	2	yes	—
396	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
397	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
398	C₂²	C₂₃×#577	1	2	yes	—
399	C₄	—	2	2	no	—
400	C₄	—	4	2	no	—
401	C₄	—	4	2	no	—
402	C₄	—	4	2	no	—
403	C₄	—	4	2	no	—
404	C₄	—	4	2	no	—
405	C₄	—	4	2	no	—
406	C₄	—	8	2	no	—
407	C₄	—	8	2	no	—
408	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
409	C₄	C₂₃×#519	1	1	yes	characteristic
410	C₄	C₂₃×#552	1	2	yes	—
411	C₂	—	2	4	no	—
412	C₂	C₂₃×#2344	1	1	yes	characteristic
413	C₂	C₂×C₂₃×#440	1	1	yes	characteristic
414	C₂	C₂₃×#2949	1	1	yes	characteristic
415	C₂	C₂₃×#3344	1	2	yes	—
416	C₂	C₂₃×#3345	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂×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
4. C₂₃×#1944	C₂	semidirect
5. C₂₃×#1920	C₂	semidirect
6. C₂₃×#1919	C₂	semidirect
10. C₂₃×#1776	C₂	semidirect
17. C₂×C₂₃×#106	C₄	semidirect
138. #22615	C₂₃	direct (N × H)
180. C₈×C₂₃	#117	semidirect
184. #1944	C₂×C₂₃	semidirect
185. #1920	C₂×C₂₃	semidirect
186. #1919	C₂×C₂₃	semidirect
190. #1776	C₂×C₂₃	semidirect
219. C₂×#106	C₄×C₂₃	semidirect
294. C₂₃	#22615	direct (N × H)
388. C₈	C₂₃×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#106	2
C₂₃×#2345	2
C₂₃×#2344	2
C₂₃×#1944	2
C₂₃×#1920	2
C₂₃×#1919	2
C₂₃×#1824	2
C₂₃×#1823	2
C₂₃×#1808	2
C₂₃×#1776	2
C₂₃×#1752	2
#22615	23

Subgroups — C23×#22615 #19170241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22615 #19170241