Subgroups — C₂₃×#21981 #19169607

Order 5888.

966 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#384	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1948	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1853	C₂	1	4	yes	maximal
6	C₂₃×#1844	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1834	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#117	—	2	1	no	—
9	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
10	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
11	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
12	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#431	C₄	1	8	yes	complemented
14	C₂×C₂₃×#106	C₂²	1	8	yes	—
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#384	—	2	1	no	—
19	C₂₃×#384	—	2	1	no	—
20	C₂₃×#384	C₂²	1	2	yes	—
21	C₂₃×#384	C₂²	1	2	yes	—
22	C₂₃×#384	C₂²	1	1	yes	characteristic
23	C₂₃×#384	C₂²	1	1	yes	characteristic
24	C₂₃×#382	—	2	1	no	—
25	C₂₃×#380	—	2	1	no	—
26	C₂₃×#380	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
28	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
29	C₂₃×#340	C₂²	1	8	yes	—
30	C₂³×C₄×C₂₃	—	2	2	no	—
31	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
35	C₂₃×#117	—	4	4	no	—
36	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
46	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
47	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
48	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
50	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
51	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
52	C₂×C₄²×C₂₃	—	2	2	no	—
53	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₂₃	Q₈	1	4	yes	—
57	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
59	C₂₃×#106	—	4	8	no	—
60	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
61	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
62	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	complemented
63	C₂₃×#95	—	2	2	no	—
64	C₂₃×#95	—	2	2	no	—
65	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	—	2	2	no	—
67	C₂²×C₄×C₂₃	—	2	2	no	—
68	C₂²×C₄×C₂₃	—	2	2	no	—
69	C₂²×C₄×C₂₃	—	2	2	no	—
70	C₂²×C₄×C₂₃	—	2	2	no	—
71	C₂²×C₄×C₂₃	—	2	2	no	—
72	C₂²×C₄×C₂₃	—	2	4	no	—
73	C₂²×C₄×C₂₃	—	4	4	no	—
74	C₂²×C₄×C₂₃	—	4	4	no	—
75	C₂²×C₄×C₂₃	—	4	4	no	—
76	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
77	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
78	C₂²×C₄×C₂₃	SD₁₆	1	4	yes	—
79	C₂²×C₄×C₂₃	SD₁₆	1	4	yes	—
80	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
87	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
88	C₂×C₈×C₂₃	—	2	4	no	—
89	C₂×C₈×C₂₃	—	2	4	no	—
90	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
91	C₄⋊C₄×C₂₃	—	2	4	no	—
92	C₄⋊C₄×C₂₃	—	2	8	no	—
93	C₄⋊C₄×C₂₃	—	2	4	no	—
94	C₄⋊C₄×C₂₃	—	4	4	no	—
95	C₄⋊C₄×C₂₃	—	4	4	no	—
96	C₄⋊C₄×C₂₃	—	4	4	no	—
97	C₄⋊C₄×C₂₃	—	4	4	no	—
98	C₄⋊C₄×C₂₃	—	4	2	no	—
99	C₄⋊C₄×C₂₃	—	4	2	no	—
100	C₄⋊C₄×C₂₃	—	4	2	no	—
101	C₄⋊C₄×C₂₃	—	4	4	no	—
102	C₄⋊C₄×C₂₃	—	4	2	no	—
103	C₂²⋊C₄×C₂₃	—	4	4	no	—
104	C₄²×C₂₃	—	4	4	no	—
105	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
106	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
107	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
108	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
109	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
110	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
111	#21981	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂³×C₂₃	—	2	1	no	—
113	C₂³×C₂₃	—	2	1	no	—
114	C₂³×C₂₃	—	2	1	no	—
115	C₂³×C₂₃	—	2	1	no	—
116	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂³×C₂₃	C₂×SD₁₆	1	2	yes	—
118	C₂³×C₂₃	C₂×SD₁₆	1	2	yes	—
119	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₂₃	—	2	1	no	—
121	C₂×C₄×C₂₃	—	2	1	no	—
122	C₂×C₄×C₂₃	—	2	1	no	—
123	C₂×C₄×C₂₃	—	2	1	no	—
124	C₂×C₄×C₂₃	—	2	4	no	—
125	C₂×C₄×C₂₃	—	2	4	no	—
126	C₂×C₄×C₂₃	—	2	8	no	—
127	C₂×C₄×C₂₃	—	4	4	no	—
128	C₂×C₄×C₂₃	—	4	2	no	—
129	C₂×C₄×C₂₃	—	4	2	no	—
130	C₂×C₄×C₂₃	—	4	2	no	—
131	C₂×C₄×C₂₃	—	4	2	no	—
132	C₂×C₄×C₂₃	—	4	4	no	—
133	C₂×C₄×C₂₃	—	4	4	no	—
134	C₂×C₄×C₂₃	—	4	2	no	—
135	C₂×C₄×C₂₃	—	4	2	no	—
136	C₂×C₄×C₂₃	—	8	8	no	—
137	C₂×C₄×C₂₃	#106	1	16	yes	complemented
138	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
143	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
144	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
145	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
146	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
147	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₂₃	#137	1	2	yes	—
149	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
151	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
152	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
153	C₈×C₂₃	—	2	8	no	—
154	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
155	#2285	C₂×C₂₃	1	1	yes	characteristic
156	C₂×#384	C₂×C₂₃	1	1	yes	characteristic
157	#1948	C₂×C₂₃	1	4	yes	complemented
158	#1853	C₂×C₂₃	1	4	yes	—
159	#1844	C₂×C₂₃	1	2	yes	complemented
160	#1834	C₂×C₂₃	1	2	yes	complemented
161	C₂²×C₂₃	—	2	2	no	—
162	C₂²×C₂₃	—	2	2	no	—
163	C₂²×C₂₃	—	2	2	no	—
164	C₂²×C₂₃	—	2	2	no	—
165	C₂²×C₂₃	—	2	2	no	—
166	C₂²×C₂₃	—	2	2	no	—
167	C₂²×C₂₃	C₂×#106	1	4	yes	—
168	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₂²×C₂₃	#535	1	1	yes	characteristic
170	C₂²×C₂₃	#536	1	1	yes	characteristic
171	C₂²×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
172	C₂²×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
173	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
175	C₄×C₂₃	—	2	2	no	—
176	C₄×C₂₃	—	2	2	no	—
177	C₄×C₂₃	—	8	2	no	—
178	C₄×C₂₃	—	8	4	no	—
179	C₄×C₂₃	—	8	2	no	—
180	C₄×C₂₃	C₂×#106	1	4	yes	—
181	C₄×C₂₃	C₂×#106	1	4	yes	—
182	C₄×C₂₃	#428	1	4	yes	—
183	C₄×C₂₃	#520	1	2	yes	—
184	C₄×C₂₃	#544	1	2	yes	—
185	C₂×#117	—	2	1	no	—
186	C₂²×C₄⋊C₄	—	2	1	no	—
187	C₂²×C₄⋊C₄	—	2	1	no	—
188	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
189	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
190	#431	C₄×C₂₃	1	8	yes	complemented
191	C₂×#106	C₂²×C₂₃	1	8	yes	—
192	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
193	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
194	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
195	#384	—	2	1	no	—
196	#384	—	2	1	no	—
197	#384	C₂²×C₂₃	1	1	yes	characteristic
198	#384	C₂²×C₂₃	1	2	yes	—
199	#384	C₂²×C₂₃	1	1	yes	characteristic
200	#384	C₂²×C₂₃	1	2	yes	—
201	#382	—	2	1	no	—
202	#380	—	2	1	no	—
203	#380	C₂²×C₂₃	1	2	yes	—
204	C₄×C₄⋊C₄	—	2	1	no	—
205	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
206	#340	C₂²×C₂₃	1	8	yes	—
207	C₂×C₂₃	—	2	4	no	—
208	C₂×C₂₃	#2317	1	1	yes	characteristic
209	C₂×C₂₃	C₂²×#106	1	1	yes	characteristic
210	C₂×C₂₃	C₂×#428	1	1	yes	characteristic
211	C₂×C₂₃	#3206	1	1	yes	characteristic
212	C₂×C₂₃	#3208	1	1	yes	characteristic
213	C₂×C₂₃	#3227	1	1	yes	characteristic
214	C₂×C₂₃	#3229	1	1	yes	characteristic
215	C₂³×C₄	—	2	2	no	—
216	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
217	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
218	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
219	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
220	#117	—	4	4	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	—	4	4	no	—
231	C₂×C₄⋊C₄	—	4	4	no	—
232	C₂×C₄⋊C₄	—	4	4	no	—
233	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
234	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
235	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
236	C₂×C₂²⋊C₄	—	2	2	no	—
237	C₂×C₄²	—	2	2	no	—
238	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
239	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
240	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
241	C₂×C₄²	Q₈×C₂₃	1	4	yes	—
242	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
243	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
244	#106	—	4	8	no	—
245	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	complemented
246	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
247	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	complemented
248	#95	—	2	2	no	—
249	#95	—	2	2	no	—
250	C₂₃	#21981	1	1	yes	characteristic, direct product factor, Sylow
251	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	4	4	no	—
260	C₂²×C₄	—	4	4	no	—
261	C₂²×C₄	—	4	4	no	—
262	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
263	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
264	C₂²×C₄	SD₁₆×C₂₃	1	4	yes	—
265	C₂²×C₄	SD₁₆×C₂₃	1	4	yes	—
266	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
267	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
272	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
273	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	2	8	no	—
279	C₄⋊C₄	—	2	4	no	—
280	C₄⋊C₄	—	4	4	no	—
281	C₄⋊C₄	—	4	4	no	—
282	C₄⋊C₄	—	4	4	no	—
283	C₄⋊C₄	—	4	4	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	4	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	4	no	—
290	C₄²	—	4	4	no	—
291	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
292	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
293	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
294	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
295	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
296	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
297	C₂³	—	2	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	2	1	no	—
301	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
302	C₂³	C₂×SD₁₆×C₂₃	1	2	yes	—
303	C₂³	C₂×SD₁₆×C₂₃	1	2	yes	—
304	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	8	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	8	8	no	—
322	C₂×C₄	C₂₃×#106	1	16	yes	complemented
323	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
324	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
325	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
326	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
327	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
328	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
329	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
330	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
331	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
332	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
333	C₂×C₄	C₂₃×#137	1	2	yes	—
334	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
335	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
336	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
337	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
338	C₈	—	2	8	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	C₂×C₂₃×#106	1	4	yes	—
346	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
347	C₂²	C₂₃×#535	1	1	yes	characteristic
348	C₂²	C₂₃×#536	1	1	yes	characteristic
349	C₂²	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
350	C₂²	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
351	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
352	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	8	2	no	—
356	C₄	—	8	4	no	—
357	C₄	—	8	2	no	—
358	C₄	C₂×C₂₃×#106	1	4	yes	—
359	C₄	C₂×C₂₃×#106	1	4	yes	—
360	C₄	C₂₃×#428	1	4	yes	—
361	C₄	C₂₃×#520	1	2	yes	—
362	C₄	C₂₃×#544	1	2	yes	—
363	C₂	—	2	4	no	—
364	C₂	C₂₃×#2317	1	1	yes	characteristic
365	C₂	C₂²×C₂₃×#106	1	1	yes	characteristic
366	C₂	C₂×C₂₃×#428	1	1	yes	characteristic
367	C₂	C₂₃×#3206	1	1	yes	characteristic
368	C₂	C₂₃×#3208	1	1	yes	characteristic
369	C₂	C₂₃×#3227	1	1	yes	characteristic
370	C₂	C₂₃×#3229	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	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1948	C₂	semidirect
6. C₂₃×#1844	C₂	semidirect
7. C₂₃×#1834	C₂	semidirect
13. C₂₃×#431	C₄	semidirect
60. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
62. C₄×C₈×C₂₃	C₂×C₄	semidirect
111. #21981	C₂₃	direct (N × H)
137. C₂×C₄×C₂₃	#106	semidirect
157. #1948	C₂×C₂₃	semidirect
159. #1844	C₂×C₂₃	semidirect
160. #1834	C₂×C₂₃	semidirect
190. #431	C₄×C₂₃	semidirect
245. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
247. C₄×C₈	C₂×C₄×C₂₃	semidirect
250. C₂₃	#21981	direct (N × H)
322. C₂×C₄	C₂₃×#106	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#431	2
C₂₃×#2285	2
C₂×C₂₃×#384	2
C₂₃×#1948	2
C₂₃×#1853	2
C₂₃×#1844	2
C₂₃×#1834	2
#21981	23

Subgroups — C23×#21981 #19169607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21981 #19169607