Subgroups — C₂₃×#21555 #19169181

Order 5888.

938 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#472	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂₃×#442	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2875	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1978	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1951	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1931	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1907	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1815	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1778	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂₃×#100	C₂	1	2	yes	maximal, complemented
12	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	2	yes	—
13	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
16	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#472	C₄	1	8	yes	complemented
19	C₂₃×#442	—	2	1	no	—
20	C₂₃×#442	—	2	1	no	—
21	C₂₃×#442	—	2	1	no	—
22	C₂₃×#442	—	2	1	no	—
23	C₂₃×#442	—	2	2	no	—
24	C₂₃×#440	—	2	2	no	—
25	C₄×SD₁₆×C₂₃	—	2	2	no	—
26	C₄×D₈×C₂₃	—	2	2	no	—
27	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
29	C₂₃×#428	C₂²	1	2	yes	—
30	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
31	C₂₃×#418	C₂²	1	2	yes	—
32	C₂₃×#417	C₂²	1	2	yes	—
33	C₂×C₂₃×#102	C₂²	1	2	yes	—
34	C₂×C₂₃×#100	C₂²	1	2	yes	—
35	C₂×C₂₃×#100	C₂²	1	2	yes	—
36	C₂₃×#405	—	2	1	no	—
37	C₂₃×#405	—	2	1	no	—
38	C₂₃×#405	C₂²	1	1	yes	characteristic
39	C₂₃×#405	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₂₃	—	2	1	no	—
41	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
42	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
43	C₂×C₂₃×#97	—	2	1	no	—
44	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
45	C₂₃×#343	C₂²	1	2	yes	—
46	C₂₃×#343	C₂²	1	2	yes	—
47	C₂₃×#341	C₂²	1	2	yes	—
48	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
50	C₂³×C₄×C₂₃	—	2	2	no	—
51	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
52	C₂₃×AGL(1,ℤ/8)	C₂×C₄	1	8	yes	complemented
53	C₂×SD₁₆×C₂₃	—	2	2	no	—
54	C₂×D₈×C₂₃	—	2	2	no	—
55	C₂×D₈×C₂₃	—	2	1	no	—
56	C₂×D₈×C₂₃	—	2	1	no	—
57	C₂×D₈×C₂₃	—	2	2	no	—
58	C₂×D₈×C₂₃	C₂×C₄	1	4	yes	complemented
59	C₂₃×#131	—	2	2	no	—
60	C₂₃×#131	—	2	2	no	—
61	C₂₃×#131	—	2	2	no	—
62	C₂₃×#131	C₂×C₄	1	4	yes	complemented
63	C₂×M₁₆×C₂₃	—	2	1	no	—
64	C₂×M₁₆×C₂₃	—	2	1	no	—
65	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
66	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
68	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₂₃	—	2	1	no	—
70	C₂²×C₈×C₂₃	—	2	1	no	—
71	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₄×Q₈×C₂₃	D₄	1	2	yes	—
74	C₄×D₄×C₂₃	—	2	2	no	—
75	C₄×D₄×C₂₃	—	2	2	no	—
76	C₄×D₄×C₂₃	—	2	2	no	—
77	C₄×D₄×C₂₃	—	2	2	no	—
78	C₄×D₄×C₂₃	—	2	2	no	—
79	C₄×D₄×C₂₃	D₄	1	2	yes	—
80	C₂₃×#117	—	2	1	no	—
81	C₂₃×#117	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
83	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
84	C₂×C₄²×C₂₃	—	2	1	no	—
85	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
86	C₂₃×#108	—	2	2	no	—
87	C₂₃×#108	C₂×C₄	1	4	yes	complemented
88	C₂₃×#107	—	2	2	no	—
89	C₂₃×#106	—	2	1	no	—
90	C₂₃×#106	—	2	1	no	—
91	C₂₃×#106	—	2	2	no	—
92	C₂₃×#103	—	2	2	no	—
93	C₂₃×#102	—	2	2	no	—
94	C₂₃×#102	—	2	2	no	—
95	C₂₃×#102	—	2	2	no	—
96	C₂₃×#102	—	2	2	no	—
97	C₂₃×#102	—	2	2	no	—
98	C₂₃×#100	—	2	4	no	—
99	C₂₃×#100	C₂×C₄	1	8	yes	complemented
100	C₂₃×#97	—	2	1	no	—
101	C₂₃×#97	—	2	1	no	—
102	C₂₃×#97	—	2	1	no	—
103	C₂₃×#97	—	2	2	no	—
104	C₂₃×#97	D₄	1	1	yes	characteristic
105	C₂₃×#97	D₄	1	1	yes	characteristic
106	C₂₃×#97	D₄	1	1	yes	characteristic
107	C₂₃×#97	D₄	1	1	yes	characteristic
108	C₄×C₈×C₂₃	—	2	1	no	—
109	C₄×C₈×C₂₃	—	2	1	no	—
110	C₄×C₈×C₂₃	—	2	1	no	—
111	C₄×C₈×C₂₃	—	2	2	no	—
112	C₂⁴×C₂₃	—	2	2	no	—
113	#41×C₂₃	—	2	1	no	—
114	#41×C₂₃	—	2	1	no	—
115	#41×C₂₃	C₂²×C₄	1	2	yes	—
116	#41×C₂₃	#41	1	2	yes	—
117	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₂×D₄×C₂₃	—	2	1	no	—
119	C₂×D₄×C₂₃	—	2	2	no	—
120	C₂×D₄×C₂₃	—	2	2	no	—
121	C₂×D₄×C₂₃	—	2	2	no	—
122	C₂×D₄×C₂₃	—	2	2	no	—
123	C₂×D₄×C₂₃	—	2	2	no	—
124	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
125	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₂₃	—	2	1	no	—
127	C₂²×C₄×C₂₃	—	2	2	no	—
128	C₂²×C₄×C₂₃	—	2	2	no	—
129	C₂²×C₄×C₂₃	—	4	2	no	—
130	C₂²×C₄×C₂₃	—	4	2	no	—
131	C₂²×C₄×C₂₃	—	8	4	no	—
132	C₂²×C₄×C₂₃	#41	1	2	yes	—
133	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
134	SD₁₆×C₂₃	—	2	8	no	—
135	D₈×C₂₃	—	2	8	no	—
136	D₈×C₂₃	—	2	4	no	—
137	D₈×C₂₃	—	2	4	no	—
138	M₁₆×C₂₃	—	2	4	no	—
139	M₁₆×C₂₃	—	2	1	no	—
140	M₁₆×C₂₃	—	2	1	no	—
141	M₁₆×C₂₃	—	2	2	no	—
142	M₁₆×C₂₃	—	2	4	no	—
143	M₁₆×C₂₃	—	2	2	no	—
144	M₁₆×C₂₃	—	2	2	no	—
145	M₁₆×C₂₃	C₂²×C₄	1	2	yes	—
146	M₁₆×C₂₃	C₂²×C₄	1	4	yes	—
147	M₁₆×C₂₃	#41	1	2	yes	—
148	C₂×C₈×C₂₃	—	2	4	no	—
149	C₂×C₈×C₂₃	—	2	2	no	—
150	C₂×C₈×C₂₃	—	2	1	no	—
151	C₂×C₈×C₂₃	—	2	1	no	—
152	C₂×C₈×C₂₃	—	2	1	no	—
153	C₂×C₈×C₂₃	—	2	1	no	—
154	C₂×C₈×C₂₃	—	2	4	no	—
155	C₂×C₈×C₂₃	—	2	2	no	—
156	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
157	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
159	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
160	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
161	C₂×C₈×C₂₃	#41	1	2	yes	—
162	C₄⋊C₄×C₂₃	—	2	2	no	—
163	C₄⋊C₄×C₂₃	—	2	2	no	—
164	C₄⋊C₄×C₂₃	—	2	2	no	—
165	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
166	C₂²⋊C₄×C₂₃	—	2	1	no	—
167	C₂²⋊C₄×C₂₃	—	2	2	no	—
168	C₂²⋊C₄×C₂₃	—	4	2	no	—
169	C₂²⋊C₄×C₂₃	—	4	2	no	—
170	C₄²×C₂₃	—	2	2	no	—
171	C₄²×C₂₃	—	2	1	no	—
172	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₂₃	C₂×D₄	1	2	yes	—
175	#21555	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂³×C₂₃	—	2	1	no	—
177	C₂³×C₂₃	—	2	4	no	—
178	C₂³×C₂₃	—	4	2	no	—
179	C₂³×C₂₃	—	4	2	no	—
180	C₂³×C₂₃	—	8	2	no	—
181	C₂³×C₂₃	—	8	2	no	—
182	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
183	Q₈×C₂₃	—	2	1	no	—
184	Q₈×C₂₃	C₄×D₄	1	2	yes	—
185	D₄×C₂₃	—	2	2	no	—
186	D₄×C₂₃	—	2	2	no	—
187	D₄×C₂₃	—	2	4	no	—
188	D₄×C₂₃	—	2	4	no	—
189	D₄×C₂₃	—	2	1	no	—
190	D₄×C₂₃	—	4	2	no	—
191	D₄×C₂₃	—	4	2	no	—
192	D₄×C₂₃	C₄×D₄	1	2	yes	—
193	C₂×C₄×C₂₃	—	2	1	no	—
194	C₂×C₄×C₂₃	—	2	2	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	2	no	—
197	C₂×C₄×C₂₃	—	4	2	no	—
198	C₂×C₄×C₂₃	—	4	2	no	—
199	C₂×C₄×C₂₃	—	8	2	no	—
200	C₂×C₄×C₂₃	—	8	4	no	—
201	C₂×C₄×C₂₃	—	8	4	no	—
202	C₂×C₄×C₂₃	—	8	2	no	—
203	C₂×C₄×C₂₃	#121	1	2	yes	—
204	C₂×C₄×C₂₃	#121	1	2	yes	—
205	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
206	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
209	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
211	C₈×C₂₃	—	2	2	no	—
212	C₈×C₂₃	—	2	4	no	—
213	C₈×C₂₃	—	2	1	no	—
214	C₈×C₂₃	—	2	2	no	—
215	C₈×C₂₃	—	2	1	no	—
216	C₈×C₂₃	—	4	2	no	—
217	C₈×C₂₃	C₄×D₄	1	2	yes	—
218	C₈×C₂₃	C₄×D₄	1	2	yes	—
219	C₂×#472	C₂×C₂₃	1	1	yes	characteristic
220	C₄×AGL(1,ℤ/8)	C₂×C₂₃	1	2	yes	complemented
221	C₂×#442	C₂×C₂₃	1	1	yes	characteristic, complemented
222	#2875	C₂×C₂₃	1	1	yes	characteristic, complemented
223	#1978	C₂×C₂₃	1	1	yes	characteristic, complemented
224	#1951	C₂×C₂₃	1	1	yes	characteristic, complemented
225	#1931	C₂×C₂₃	1	2	yes	complemented
226	#1907	C₂×C₂₃	1	2	yes	complemented
227	#1815	C₂×C₂₃	1	1	yes	characteristic, complemented
228	#1778	C₂×C₂₃	1	1	yes	characteristic, complemented
229	C₄×#100	C₂×C₂₃	1	2	yes	complemented
230	C₂²×C₂₃	—	2	2	no	—
231	C₂²×C₂₃	—	2	2	no	—
232	C₂²×C₂₃	—	4	2	no	—
233	C₂²×C₂₃	—	4	4	no	—
234	C₂²×C₂₃	—	8	4	no	—
235	C₂²×C₂₃	—	8	4	no	—
236	C₂²×C₂₃	—	8	2	no	—
237	C₂²×C₂₃	—	8	2	no	—
238	C₂²×C₂₃	C₄×#41	1	1	yes	characteristic
239	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
240	C₂²×C₂₃	#529	1	1	yes	characteristic
241	C₄×C₂₃	—	2	2	no	—
242	C₄×C₂₃	—	2	2	no	—
243	C₄×C₂₃	—	2	2	no	—
244	C₄×C₂₃	—	4	2	no	—
245	C₄×C₂₃	—	8	2	no	—
246	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
247	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
248	C₄×C₂₃	#525	1	1	yes	characteristic
249	C₄×C₂₃	#532	1	1	yes	characteristic
250	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	2	yes	—
251	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
252	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
253	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
254	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
255	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
256	#472	C₄×C₂₃	1	8	yes	complemented
257	#442	—	2	2	no	—
258	#442	—	2	1	no	—
259	#442	—	2	1	no	—
260	#442	—	2	1	no	—
261	#442	—	2	1	no	—
262	#440	—	2	2	no	—
263	C₄×SD₁₆	—	2	2	no	—
264	C₄×D₈	—	2	2	no	—
265	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
266	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
267	#428	C₂²×C₂₃	1	2	yes	—
268	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
269	#418	C₂²×C₂₃	1	2	yes	—
270	#417	C₂²×C₂₃	1	2	yes	—
271	C₂×#102	C₂²×C₂₃	1	2	yes	—
272	C₂×#100	C₂²×C₂₃	1	2	yes	—
273	C₂×#100	C₂²×C₂₃	1	2	yes	—
274	#405	—	2	1	no	—
275	#405	—	2	1	no	—
276	#405	C₂²×C₂₃	1	1	yes	characteristic
277	#405	C₂²×C₂₃	1	1	yes	characteristic
278	C₄×M₁₆	—	2	1	no	—
279	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
280	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
281	C₂×#97	—	2	1	no	—
282	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
283	#343	C₂²×C₂₃	1	2	yes	—
284	#343	C₂²×C₂₃	1	2	yes	—
285	#341	C₂²×C₂₃	1	2	yes	—
286	C₂×C₂₃	—	2	1	no	—
287	C₂×C₂₃	—	2	1	no	—
288	C₂×C₂₃	—	4	2	no	—
289	C₂×C₂₃	—	8	4	no	—
290	C₂×C₂₃	C₄×#121	1	1	yes	characteristic
291	C₂×C₂₃	#3068	1	2	yes	—
292	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
293	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
294	C₂³×C₄	—	2	2	no	—
295	AGL(1,ℤ/8)	—	2	4	no	—
296	AGL(1,ℤ/8)	C₂×C₄×C₂₃	1	8	yes	complemented
297	C₂×SD₁₆	—	2	2	no	—
298	C₂×D₈	—	2	2	no	—
299	C₂×D₈	—	2	1	no	—
300	C₂×D₈	—	2	1	no	—
301	C₂×D₈	—	2	2	no	—
302	C₂×D₈	C₂×C₄×C₂₃	1	4	yes	complemented
303	#131	—	2	2	no	—
304	#131	—	2	2	no	—
305	#131	—	2	2	no	—
306	#131	C₂×C₄×C₂₃	1	4	yes	complemented
307	C₂×M₁₆	—	2	1	no	—
308	C₂×M₁₆	—	2	1	no	—
309	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
310	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
311	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
312	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
313	C₂²×C₈	—	2	1	no	—
314	C₂²×C₈	—	2	1	no	—
315	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
316	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
317	C₄×Q₈	D₄×C₂₃	1	2	yes	—
318	C₄×D₄	—	2	2	no	—
319	C₄×D₄	—	2	2	no	—
320	C₄×D₄	—	2	2	no	—
321	C₄×D₄	—	2	2	no	—
322	C₄×D₄	—	2	2	no	—
323	C₄×D₄	D₄×C₂₃	1	2	yes	—
324	#117	—	2	1	no	—
325	#117	C₂³×C₂₃	1	1	yes	characteristic
326	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
327	C₂×C₂²⋊C₄	—	2	2	no	—
328	C₂×C₄²	—	2	1	no	—
329	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
330	#108	—	2	2	no	—
331	#108	C₂×C₄×C₂₃	1	4	yes	complemented
332	#107	—	2	2	no	—
333	#106	—	2	1	no	—
334	#106	—	2	1	no	—
335	#106	—	2	2	no	—
336	#103	—	2	2	no	—
337	#102	—	2	2	no	—
338	#102	—	2	2	no	—
339	#102	—	2	2	no	—
340	#102	—	2	2	no	—
341	#102	—	2	2	no	—
342	#100	—	2	4	no	—
343	#100	C₂×C₄×C₂₃	1	8	yes	complemented
344	#97	—	2	1	no	—
345	#97	—	2	1	no	—
346	#97	—	2	1	no	—
347	#97	—	2	2	no	—
348	#97	D₄×C₂₃	1	1	yes	characteristic
349	#97	D₄×C₂₃	1	1	yes	characteristic
350	#97	D₄×C₂₃	1	1	yes	characteristic
351	#97	D₄×C₂₃	1	1	yes	characteristic
352	C₄×C₈	—	2	1	no	—
353	C₄×C₈	—	2	1	no	—
354	C₄×C₈	—	2	1	no	—
355	C₄×C₈	—	2	2	no	—
356	C₂₃	#21555	1	1	yes	characteristic, direct product factor, Sylow
357	C₂⁴	—	2	2	no	—
358	#41	—	2	1	no	—
359	#41	—	2	1	no	—
360	#41	C₂²×C₄×C₂₃	1	2	yes	—
361	#41	#41×C₂₃	1	2	yes	—
362	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
363	C₂×D₄	—	2	1	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
370	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	4	2	no	—
375	C₂²×C₄	—	4	2	no	—
376	C₂²×C₄	—	8	4	no	—
377	C₂²×C₄	#41×C₂₃	1	2	yes	—
378	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
379	SD₁₆	—	2	8	no	—
380	D₈	—	2	8	no	—
381	D₈	—	2	4	no	—
382	D₈	—	2	4	no	—
383	M₁₆	—	2	4	no	—
384	M₁₆	—	2	1	no	—
385	M₁₆	—	2	1	no	—
386	M₁₆	—	2	2	no	—
387	M₁₆	—	2	4	no	—
388	M₁₆	—	2	2	no	—
389	M₁₆	—	2	2	no	—
390	M₁₆	C₂²×C₄×C₂₃	1	2	yes	—
391	M₁₆	C₂²×C₄×C₂₃	1	4	yes	—
392	M₁₆	#41×C₂₃	1	2	yes	—
393	C₂×C₈	—	2	4	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	4	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
402	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
403	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
404	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
405	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
406	C₂×C₈	#41×C₂₃	1	2	yes	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
411	C₂²⋊C₄	—	2	1	no	—
412	C₂²⋊C₄	—	2	2	no	—
413	C₂²⋊C₄	—	4	2	no	—
414	C₂²⋊C₄	—	4	2	no	—
415	C₄²	—	2	2	no	—
416	C₄²	—	2	1	no	—
417	C₄²	C₂×D₄×C₂₃	1	2	yes	—
418	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
419	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
420	C₂³	—	2	1	no	—
421	C₂³	—	2	4	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	2	no	—
424	C₂³	—	8	2	no	—
425	C₂³	—	8	2	no	—
426	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
427	Q₈	—	2	1	no	—
428	Q₈	C₄×D₄×C₂₃	1	2	yes	—
429	D₄	—	2	2	no	—
430	D₄	—	2	2	no	—
431	D₄	—	2	4	no	—
432	D₄	—	2	4	no	—
433	D₄	—	2	1	no	—
434	D₄	—	4	2	no	—
435	D₄	—	4	2	no	—
436	D₄	C₄×D₄×C₂₃	1	2	yes	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	8	2	no	—
444	C₂×C₄	—	8	4	no	—
445	C₂×C₄	—	8	4	no	—
446	C₂×C₄	—	8	2	no	—
447	C₂×C₄	C₂₃×#121	1	2	yes	—
448	C₂×C₄	C₂₃×#121	1	2	yes	—
449	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
450	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
451	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
452	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
453	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
454	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
455	C₈	—	2	2	no	—
456	C₈	—	2	4	no	—
457	C₈	—	2	1	no	—
458	C₈	—	2	2	no	—
459	C₈	—	2	1	no	—
460	C₈	—	4	2	no	—
461	C₈	C₄×D₄×C₂₃	1	2	yes	—
462	C₈	C₄×D₄×C₂₃	1	2	yes	—
463	C₂²	—	2	2	no	—
464	C₂²	—	2	2	no	—
465	C₂²	—	4	2	no	—
466	C₂²	—	4	4	no	—
467	C₂²	—	8	4	no	—
468	C₂²	—	8	4	no	—
469	C₂²	—	8	2	no	—
470	C₂²	—	8	2	no	—
471	C₂²	C₄×#41×C₂₃	1	1	yes	characteristic
472	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
473	C₂²	C₂₃×#529	1	1	yes	characteristic
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	2	2	no	—
477	C₄	—	4	2	no	—
478	C₄	—	8	2	no	—
479	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
480	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
481	C₄	C₂₃×#525	1	1	yes	characteristic
482	C₄	C₂₃×#532	1	1	yes	characteristic
483	C₂	—	2	1	no	—
484	C₂	—	2	1	no	—
485	C₂	—	4	2	no	—
486	C₂	—	8	4	no	—
487	C₂	C₄×C₂₃×#121	1	1	yes	characteristic
488	C₂	C₂₃×#3068	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#121
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#472	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₂₃×AGL(1,ℤ/8)	C₂	semidirect
3. C₂×C₂₃×#442	C₂	semidirect
4. C₂₃×#2875	C₂	semidirect
5. C₂₃×#1978	C₂	semidirect
6. C₂₃×#1951	C₂	semidirect
7. C₂₃×#1931	C₂	semidirect
8. C₂₃×#1907	C₂	semidirect
9. C₂₃×#1815	C₂	semidirect
10. C₂₃×#1778	C₂	semidirect
11. C₄×C₂₃×#100	C₂	semidirect
18. C₂₃×#472	C₄	semidirect
52. C₂₃×AGL(1,ℤ/8)	C₂×C₄	semidirect
58. C₂×D₈×C₂₃	C₂×C₄	semidirect
62. C₂₃×#131	C₂×C₄	semidirect
87. C₂₃×#108	C₂×C₄	semidirect
99. C₂₃×#100	C₂×C₄	semidirect
175. #21555	C₂₃	direct (N × H)
220. C₄×AGL(1,ℤ/8)	C₂×C₂₃	semidirect
221. C₂×#442	C₂×C₂₃	semidirect
222. #2875	C₂×C₂₃	semidirect
223. #1978	C₂×C₂₃	semidirect
224. #1951	C₂×C₂₃	semidirect
225. #1931	C₂×C₂₃	semidirect
226. #1907	C₂×C₂₃	semidirect
227. #1815	C₂×C₂₃	semidirect
228. #1778	C₂×C₂₃	semidirect
229. C₄×#100	C₂×C₂₃	semidirect
256. #472	C₄×C₂₃	semidirect
296. AGL(1,ℤ/8)	C₂×C₄×C₂₃	semidirect
302. C₂×D₈	C₂×C₄×C₂₃	semidirect
306. #131	C₂×C₄×C₂₃	semidirect
331. #108	C₂×C₄×C₂₃	semidirect
343. #100	C₂×C₄×C₂₃	semidirect
356. C₂₃	#21555	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×#472	2
C₄×C₂₃×AGL(1,ℤ/8)	2
C₂×C₂₃×#442	2
C₂₃×#2875	2
C₂₃×#1978	2
C₂₃×#1951	2
C₂₃×#1931	2
C₂₃×#1907	2
C₂₃×#1815	2
C₂₃×#1778	2
C₄×C₂₃×#100	2
#21555	23

Subgroups — C23×#21555 #19169181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21555 #19169181