Subgroups — C₂₃×#22642 #19170268

Order 5888.

680 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1940	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1924	C₂	1	2	yes	maximal
4	C₂₃×#1923	C₂	1	2	yes	maximal
5	C₂₃×#1840	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1797	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1771	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1768	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1749	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
15	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#97	C₂²	1	2	yes	—
18	C₂×C₂₃×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
20	C₂₃×#383	C₄	1	2	yes	—
21	C₂₃×#382	C₄	1	1	yes	characteristic
22	C₂₃×#382	C₄	1	1	yes	characteristic
23	C₂₃×#381	C₄	1	2	yes	—
24	C₂₃×#380	C₄	1	1	yes	characteristic
25	C₂₃×#380	C₄	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
29	C₂₃×#376	C₂²	1	2	yes	—
30	C₂₃×#336	C₂²	1	2	yes	—
31	C₂₃×#336	C₂²	1	2	yes	—
32	C₂₃×#336	C₂²	1	2	yes	—
33	C₂₃×#336	C₂²	1	2	yes	—
34	C₂₃×#336	C₂²	1	1	yes	characteristic
35	C₂₃×#336	C₂²	1	1	yes	characteristic
36	C₂₃×#336	C₂²	1	2	yes	—
37	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₂₃	—	2	4	no	—
51	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
57	C₂₃×C₄⋊C₈	—	2	4	no	—
58	C₂₃×C₄⋊C₈	—	2	4	no	—
59	C₂₃×C₄⋊C₈	—	2	4	no	—
60	C₂₃×C₂²⋊C₈	—	2	4	no	—
61	C₂₃×C₂²⋊C₈	—	4	1	no	—
62	C₂₃×C₂²⋊C₈	—	4	1	no	—
63	C₂₃×#97	—	2	4	no	—
64	C₂₃×#97	—	4	2	no	—
65	C₄×C₈×C₂₃	—	4	2	no	—
66	C₂₃×#95	C₂×C₄	1	2	yes	—
67	C₂₃×#95	C₂×C₄	1	2	yes	—
68	C₂₃×#95	C₂×C₄	1	2	yes	—
69	C₂₃×#95	C₂×C₄	1	2	yes	—
70	C₂₃×#95	C₂×C₄	1	2	yes	—
71	C₂₃×#95	C₂×C₄	1	2	yes	—
72	C₂₃×#95	C₂×C₄	1	2	yes	—
73	C₂₃×#95	C₂×C₄	1	2	yes	—
74	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	—	2	2	no	—
76	C₂²×C₄×C₂₃	—	2	2	no	—
77	C₂²×C₄×C₂₃	—	2	2	no	—
78	C₂²×C₄×C₂₃	—	2	2	no	—
79	C₂²×C₄×C₂₃	—	2	2	no	—
80	C₂²×C₄×C₂₃	—	2	1	no	—
81	C₂²×C₄×C₂₃	—	2	1	no	—
82	C₂²×C₄×C₂₃	—	4	1	no	—
83	C₂²×C₄×C₂₃	—	4	1	no	—
84	C₂²×C₄×C₂₃	—	4	1	no	—
85	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
95	C₂×C₈×C₂₃	—	2	2	no	—
96	C₂×C₈×C₂₃	—	2	2	no	—
97	C₂×C₈×C₂₃	—	2	2	no	—
98	C₂×C₈×C₂₃	—	2	2	no	—
99	C₂×C₈×C₂₃	—	2	2	no	—
100	C₂×C₈×C₂₃	—	2	2	no	—
101	C₂×C₈×C₂₃	—	2	2	no	—
102	C₂×C₈×C₂₃	—	2	2	no	—
103	C₂×C₈×C₂₃	—	2	2	no	—
104	C₂×C₈×C₂₃	—	2	2	no	—
105	C₂×C₈×C₂₃	—	2	2	no	—
106	C₂×C₈×C₂₃	#41	1	4	yes	—
107	C₄⋊C₄×C₂₃	—	4	1	no	—
108	C₄⋊C₄×C₂₃	—	4	1	no	—
109	C₄⋊C₄×C₂₃	—	4	1	no	—
110	C₄⋊C₄×C₂₃	—	4	1	no	—
111	C₂²⋊C₄×C₂₃	—	2	4	no	—
112	C₂²⋊C₄×C₂₃	—	4	2	no	—
113	C₄²×C₂₃	—	2	2	no	—
114	C₄²×C₂₃	—	2	2	no	—
115	C₄²×C₂₃	—	2	2	no	—
116	C₄²×C₂₃	—	2	2	no	—
117	C₄²×C₂₃	—	2	2	no	—
118	C₄²×C₂₃	—	2	2	no	—
119	C₄²×C₂₃	—	2	2	no	—
120	C₄²×C₂₃	#41	1	4	yes	—
121	#22642	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₂₃	—	2	1	no	—
123	C₂³×C₂₃	—	2	1	no	—
124	C₂³×C₂₃	—	2	1	no	—
125	C₂³×C₂₃	—	2	1	no	—
126	C₂³×C₂₃	—	2	1	no	—
127	C₂³×C₂₃	—	2	1	no	—
128	C₂³×C₂₃	#131	1	2	yes	—
129	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₂₃	—	2	1	no	—
131	C₂×C₄×C₂₃	—	2	1	no	—
132	C₂×C₄×C₂₃	—	2	1	no	—
133	C₂×C₄×C₂₃	—	2	1	no	—
134	C₂×C₄×C₂₃	—	2	1	no	—
135	C₂×C₄×C₂₃	—	2	1	no	—
136	C₂×C₄×C₂₃	—	2	1	no	—
137	C₂×C₄×C₂₃	—	2	2	no	—
138	C₂×C₄×C₂₃	—	2	2	no	—
139	C₂×C₄×C₂₃	—	2	1	no	—
140	C₂×C₄×C₂₃	—	2	2	no	—
141	C₂×C₄×C₂₃	—	2	2	no	—
142	C₂×C₄×C₂₃	—	2	1	no	—
143	C₂×C₄×C₂₃	—	2	2	no	—
144	C₂×C₄×C₂₃	—	2	2	no	—
145	C₂×C₄×C₂₃	—	2	1	no	—
146	C₂×C₄×C₂₃	—	2	2	no	—
147	C₂×C₄×C₂₃	—	2	2	no	—
148	C₂×C₄×C₂₃	—	2	2	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	1	no	—
153	C₂×C₄×C₂₃	—	2	1	no	—
154	C₂×C₄×C₂₃	—	2	1	no	—
155	C₂×C₄×C₂₃	—	4	4	no	—
156	C₂×C₄×C₂₃	—	4	4	no	—
157	C₂×C₄×C₂₃	—	4	2	no	—
158	C₂×C₄×C₂₃	—	4	2	no	—
159	C₂×C₄×C₂₃	#117	1	2	yes	—
160	C₂×C₄×C₂₃	#117	1	2	yes	—
161	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
162	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
163	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
165	C₈×C₂₃	—	2	4	no	—
166	C₈×C₂₃	—	4	2	no	—
167	C₈×C₂₃	—	4	2	no	—
168	C₈×C₂₃	—	4	2	no	—
169	#2283	C₂×C₂₃	1	1	yes	characteristic
170	#1940	C₂×C₂₃	1	2	yes	complemented
171	#1924	C₂×C₂₃	1	2	yes	—
172	#1923	C₂×C₂₃	1	2	yes	—
173	#1840	C₂×C₂₃	1	2	yes	complemented
174	#1797	C₂×C₂₃	1	1	yes	characteristic
175	#1771	C₂×C₂₃	1	2	yes	complemented
176	#1768	C₂×C₂₃	1	2	yes	complemented
177	#1749	C₂×C₂₃	1	1	yes	characteristic
178	C₂²×C₂₃	—	2	2	no	—
179	C₂²×C₂₃	—	2	2	no	—
180	C₂²×C₂₃	—	2	2	no	—
181	C₂²×C₂₃	—	4	1	no	—
182	C₂²×C₂₃	—	4	1	no	—
183	C₂²×C₂₃	—	4	1	no	—
184	C₂²×C₂₃	—	4	1	no	—
185	C₂²×C₂₃	C₂×#117	1	1	yes	characteristic
186	C₂²×C₂₃	C₄×#41	1	1	yes	characteristic
187	C₂²×C₂₃	#520	1	1	yes	characteristic
188	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
189	C₂²×C₂₃	#568	1	1	yes	characteristic
190	C₂²×C₂₃	#568	1	1	yes	characteristic
191	C₂²×C₂₃	#568	1	1	yes	characteristic
192	C₄×C₂₃	—	2	2	no	—
193	C₄×C₂₃	—	2	2	no	—
194	C₄×C₂₃	—	4	1	no	—
195	C₄×C₂₃	—	4	1	no	—
196	C₄×C₂₃	—	4	1	no	—
197	C₄×C₂₃	—	4	2	no	—
198	C₄×C₂₃	—	4	2	no	—
199	C₄×C₂₃	—	4	2	no	—
200	C₄×C₂₃	—	4	2	no	—
201	C₄×C₂₃	#553	1	2	yes	—
202	C₄×C₂₃	#556	1	2	yes	—
203	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
204	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
205	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
206	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
207	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
208	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
209	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
210	C₂×#97	C₂²×C₂₃	1	2	yes	—
211	C₂×#97	C₂²×C₂₃	1	2	yes	—
212	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
213	#383	C₄×C₂₃	1	2	yes	—
214	#382	C₄×C₂₃	1	1	yes	characteristic
215	#382	C₄×C₂₃	1	1	yes	characteristic
216	#381	C₄×C₂₃	1	2	yes	—
217	#380	C₄×C₂₃	1	1	yes	characteristic
218	#380	C₄×C₂₃	1	1	yes	characteristic
219	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
220	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
221	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
222	#376	C₂²×C₂₃	1	2	yes	—
223	#336	C₂²×C₂₃	1	2	yes	—
224	#336	C₂²×C₂₃	1	1	yes	characteristic
225	#336	C₂²×C₂₃	1	2	yes	—
226	#336	C₂²×C₂₃	1	2	yes	—
227	#336	C₂²×C₂₃	1	2	yes	—
228	#336	C₂²×C₂₃	1	2	yes	—
229	#336	C₂²×C₂₃	1	1	yes	characteristic
230	C₂×C₂₃	—	4	2	no	—
231	C₂×C₂₃	#2347	1	1	yes	characteristic
232	C₂×C₂₃	#2921	1	1	yes	characteristic
233	C₂×C₂₃	#2924	1	1	yes	characteristic
234	C₂×C₂₃	#2967	1	1	yes	characteristic
235	C₂×C₂₃	#2968	1	1	yes	characteristic
236	C₂×C₂₃	#2992	1	1	yes	characteristic
237	C₂×C₂₃	#2995	1	1	yes	characteristic
238	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
239	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
240	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
241	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
242	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
243	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
244	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
245	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
246	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
247	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
248	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
249	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
250	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
251	C₂×C₄²	—	2	4	no	—
252	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
253	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
254	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
255	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
256	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
257	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
258	C₄⋊C₈	—	2	4	no	—
259	C₄⋊C₈	—	2	4	no	—
260	C₄⋊C₈	—	2	4	no	—
261	C₂²⋊C₈	—	2	4	no	—
262	C₂²⋊C₈	—	4	1	no	—
263	C₂²⋊C₈	—	4	1	no	—
264	#97	—	2	4	no	—
265	#97	—	4	2	no	—
266	C₄×C₈	—	4	2	no	—
267	#95	C₂×C₄×C₂₃	1	2	yes	—
268	#95	C₂×C₄×C₂₃	1	2	yes	—
269	#95	C₂×C₄×C₂₃	1	2	yes	—
270	#95	C₂×C₄×C₂₃	1	2	yes	—
271	#95	C₂×C₄×C₂₃	1	2	yes	—
272	#95	C₂×C₄×C₂₃	1	2	yes	—
273	#95	C₂×C₄×C₂₃	1	2	yes	—
274	#95	C₂×C₄×C₂₃	1	2	yes	—
275	C₂₃	#22642	1	1	yes	characteristic, direct product factor, Sylow
276	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	1	no	—
283	C₂²×C₄	—	2	1	no	—
284	C₂²×C₄	—	4	1	no	—
285	C₂²×C₄	—	4	1	no	—
286	C₂²×C₄	—	4	1	no	—
287	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
288	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
289	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
290	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
291	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
292	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
293	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
294	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
295	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
296	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	#41×C₂₃	1	4	yes	—
309	C₄⋊C₄	—	4	1	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₂²⋊C₄	—	2	4	no	—
314	C₂²⋊C₄	—	4	2	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	2	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	2	2	no	—
320	C₄²	—	2	2	no	—
321	C₄²	—	2	2	no	—
322	C₄²	#41×C₂₃	1	4	yes	—
323	C₂³	—	2	1	no	—
324	C₂³	—	2	1	no	—
325	C₂³	—	2	1	no	—
326	C₂³	—	2	1	no	—
327	C₂³	—	2	1	no	—
328	C₂³	—	2	1	no	—
329	C₂³	C₂₃×#131	1	2	yes	—
330	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	4	4	no	—
357	C₂×C₄	—	4	4	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	C₂₃×#117	1	2	yes	—
361	C₂×C₄	C₂₃×#117	1	2	yes	—
362	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
363	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
364	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
365	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
366	C₈	—	2	4	no	—
367	C₈	—	4	2	no	—
368	C₈	—	4	2	no	—
369	C₈	—	4	2	no	—
370	C₂²	—	2	2	no	—
371	C₂²	—	2	2	no	—
372	C₂²	—	2	2	no	—
373	C₂²	—	4	1	no	—
374	C₂²	—	4	1	no	—
375	C₂²	—	4	1	no	—
376	C₂²	—	4	1	no	—
377	C₂²	C₂×C₂₃×#117	1	1	yes	characteristic
378	C₂²	C₄×#41×C₂₃	1	1	yes	characteristic
379	C₂²	C₂₃×#520	1	1	yes	characteristic
380	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
381	C₂²	C₂₃×#568	1	1	yes	characteristic
382	C₂²	C₂₃×#568	1	1	yes	characteristic
383	C₂²	C₂₃×#568	1	1	yes	characteristic
384	C₄	—	2	2	no	—
385	C₄	—	2	2	no	—
386	C₄	—	4	1	no	—
387	C₄	—	4	1	no	—
388	C₄	—	4	2	no	—
389	C₄	—	4	2	no	—
390	C₄	—	4	2	no	—
391	C₄	—	4	2	no	—
392	C₄	—	4	1	no	—
393	C₄	C₂₃×#553	1	2	yes	—
394	C₄	C₂₃×#556	1	2	yes	—
395	C₂	—	4	2	no	—
396	C₂	C₂₃×#2347	1	1	yes	characteristic
397	C₂	C₂₃×#2921	1	1	yes	characteristic
398	C₂	C₂₃×#2924	1	1	yes	characteristic
399	C₂	C₂₃×#2967	1	1	yes	characteristic
400	C₂	C₂₃×#2968	1	1	yes	characteristic
401	C₂	C₂₃×#2992	1	1	yes	characteristic
402	C₂	C₂₃×#2995	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1940	C₂	semidirect
5. C₂₃×#1840	C₂	semidirect
7. C₂₃×#1771	C₂	semidirect
8. C₂₃×#1768	C₂	semidirect
121. #22642	C₂₃	direct (N × H)
170. #1940	C₂×C₂₃	semidirect
173. #1840	C₂×C₂₃	semidirect
175. #1771	C₂×C₂₃	semidirect
176. #1768	C₂×C₂₃	semidirect
275. C₂₃	#22642	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×#2283	2
C₂₃×#1940	2
C₂₃×#1924	2
C₂₃×#1923	2
C₂₃×#1840	2
C₂₃×#1797	2
C₂₃×#1771	2
C₂₃×#1768	2
C₂₃×#1749	2
#22642	23

Subgroups — C23×#22642 #19170268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22642 #19170268