Subgroups — C₂₃×#19392 #19167018

Order 5888.

728 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3226	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3036	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2972	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1681	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1673	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1571	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1540	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
11	C₂²×SD₁₆×C₂₃	—	2	1	no	—
12	C₂₃×#532	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#500	C₂²	1	2	yes	complemented
16	C₂₃×#486	C₂²	1	2	yes	complemented
17	C₂₃×#480	—	2	2	no	—
18	C₂₃×#474	—	2	2	no	—
19	C₂₃×#463	—	2	2	no	—
20	C₂₃×#461	—	2	2	no	—
21	C₂₃×#461	C₂²	1	2	yes	—
22	C₂₃×#461	C₂²	1	4	yes	—
23	C₂₃×#459	C₂²	1	2	yes	—
24	C₈×D₄×C₂₃	—	2	2	no	—
25	C₈×D₄×C₂₃	C₂²	1	4	yes	—
26	C₂₃×#431	C₂²	1	1	yes	characteristic
27	C₂₃×#431	C₂²	1	1	yes	characteristic
28	C₂₃×#423	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂₃×#417	—	2	1	no	—
31	C₂×C₂₃×#103	—	2	1	no	—
32	C₂×C₂₃×C₂²⋊C₈	—	2	2	no	—
33	C₂₃×#335	C₂²	1	4	yes	complemented
34	C₂₃×#332	C₂²	1	2	yes	complemented
35	C₂₃×#332	C₂²	1	4	yes	complemented
36	C₂₃×#331	C₂²	1	2	yes	complemented
37	C₂×#41×C₂₃	—	2	1	no	—
38	C₂²×Q₈×C₂₃	—	2	1	no	—
39	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₂₃	—	2	1	no	—
41	C₂₃×AGL(1,ℤ/8)	—	4	4	no	—
42	C₂×SD₁₆×C₂₃	—	2	4	no	—
43	C₂×SD₁₆×C₂₃	—	2	2	no	—
44	C₂×SD₁₆×C₂₃	—	4	4	no	—
45	C₂×D₈×C₂₃	—	2	2	no	—
46	C₂×M₁₆×C₂₃	—	2	1	no	—
47	C₂²×C₈×C₂₃	—	2	1	no	—
48	C₂²×C₈×C₂₃	—	2	4	no	—
49	C₂₃×C₄⋊Q₈	—	2	1	no	—
50	C₂₃×C₄⋊Q₈	—	2	1	no	—
51	C₂₃×C₄⋊Q₈	—	2	1	no	—
52	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
53	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
54	C₂₃×#127	C₂³	1	1	yes	characteristic
55	C₂₃×#124	—	2	1	no	—
56	C₂₃×#121	—	2	1	no	—
57	C₂₃×#121	—	2	1	no	—
58	C₄×D₄×C₂₃	—	2	1	no	—
59	C₄×D₄×C₂₃	—	2	1	no	—
60	C₄×D₄×C₂₃	—	2	1	no	—
61	C₄×D₄×C₂₃	—	2	1	no	—
62	C₄×D₄×C₂₃	D₄	1	2	yes	—
63	C₄×D₄×C₂₃	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
65	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
68	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
69	C₂₃×#107	—	2	4	no	—
70	C₂₃×C₄⋊C₈	—	2	1	no	—
71	C₂₃×C₄⋊C₈	—	2	1	no	—
72	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
73	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
74	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
75	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
76	C₂₃×#103	—	2	4	no	—
77	C₂₃×#103	—	2	2	no	—
78	C₂₃×#102	—	2	4	no	—
79	C₂₃×#102	—	2	2	no	—
80	C₂₃×C₂²⋊C₈	—	2	1	no	—
81	C₂₃×C₂²⋊C₈	—	2	4	no	—
82	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
83	C₄×C₈×C₂₃	—	2	1	no	—
84	C₄×C₈×C₂₃	C₂³	1	2	yes	—
85	C₂⁴×C₂₃	—	2	1	no	—
86	#41×C₂₃	—	4	2	no	—
87	#41×C₂₃	—	4	2	no	—
88	C₂×Q₈×C₂₃	—	2	1	no	—
89	C₂×Q₈×C₂₃	—	2	2	no	—
90	C₂×Q₈×C₂₃	—	4	1	no	—
91	C₂×Q₈×C₂₃	—	4	2	no	—
92	C₂×Q₈×C₂₃	—	4	2	no	—
93	C₂×D₄×C₂₃	—	2	1	no	—
94	C₂×D₄×C₂₃	—	2	2	no	—
95	C₂×D₄×C₂₃	—	2	1	no	—
96	C₂×D₄×C₂₃	—	2	1	no	—
97	C₂×D₄×C₂₃	—	2	1	no	—
98	C₂×D₄×C₂₃	—	4	1	no	—
99	C₂×D₄×C₂₃	D₈	1	4	yes	—
100	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₂₃	—	2	1	no	—
102	C₂²×C₄×C₂₃	—	2	2	no	—
103	C₂²×C₄×C₂₃	—	2	2	no	—
104	C₂²×C₄×C₂₃	—	4	1	no	—
105	C₂²×C₄×C₂₃	—	4	1	no	—
106	C₂²×C₄×C₂₃	—	8	2	no	—
107	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
110	SD₁₆×C₂₃	—	4	8	no	—
111	SD₁₆×C₂₃	—	4	4	no	—
112	D₈×C₂₃	—	4	4	no	—
113	M₁₆×C₂₃	—	4	2	no	—
114	C₂×C₈×C₂₃	—	2	2	no	—
115	C₂×C₈×C₂₃	—	2	2	no	—
116	C₂×C₈×C₂₃	—	2	2	no	—
117	C₂×C₈×C₂₃	—	4	2	no	—
118	C₂×C₈×C₂₃	—	8	4	no	—
119	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
120	C₄⋊C₄×C₂₃	—	2	1	no	—
121	C₄⋊C₄×C₂₃	—	2	2	no	—
122	C₄⋊C₄×C₂₃	—	2	1	no	—
123	C₄⋊C₄×C₂₃	—	2	2	no	—
124	C₄⋊C₄×C₂₃	—	2	1	no	—
125	C₄⋊C₄×C₂₃	—	4	1	no	—
126	C₄⋊C₄×C₂₃	—	4	1	no	—
127	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
128	C₂²⋊C₄×C₂₃	—	2	2	no	—
129	C₂²⋊C₄×C₂₃	—	2	2	no	—
130	C₂²⋊C₄×C₂₃	—	4	1	no	—
131	C₂²⋊C₄×C₂₃	—	4	1	no	—
132	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₂₃	#41	1	1	yes	characteristic
135	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
136	#19392	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂³×C₂₃	—	2	2	no	—
138	C₂³×C₂₃	—	2	4	no	—
139	C₂³×C₂₃	—	4	1	no	—
140	C₂³×C₂₃	—	8	1	no	—
141	C₂³×C₂₃	—	8	1	no	—
142	C₂³×C₂₃	#121	1	1	yes	characteristic
143	Q₈×C₂₃	—	4	2	no	—
144	Q₈×C₂₃	—	4	1	no	—
145	Q₈×C₂₃	—	4	2	no	—
146	Q₈×C₂₃	—	4	1	no	—
147	Q₈×C₂₃	—	8	2	no	—
148	D₄×C₂₃	—	2	2	no	—
149	D₄×C₂₃	—	2	4	no	—
150	D₄×C₂₃	—	4	1	no	—
151	D₄×C₂₃	—	4	1	no	—
152	D₄×C₂₃	—	4	2	no	—
153	D₄×C₂₃	—	8	1	no	—
154	D₄×C₂₃	—	8	2	no	—
155	D₄×C₂₃	C₂×D₈	1	4	yes	—
156	C₂×C₄×C₂₃	—	2	2	no	—
157	C₂×C₄×C₂₃	—	2	2	no	—
158	C₂×C₄×C₂₃	—	4	1	no	—
159	C₂×C₄×C₂₃	—	4	4	no	—
160	C₂×C₄×C₂₃	—	4	1	no	—
161	C₂×C₄×C₂₃	—	4	2	no	—
162	C₂×C₄×C₂₃	—	8	2	no	—
163	C₂×C₄×C₂₃	—	8	1	no	—
164	C₂×C₄×C₂₃	—	8	2	no	—
165	C₂×C₄×C₂₃	—	8	2	no	—
166	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
167	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
168	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
169	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
170	C₂×C₄×C₂₃	#135	1	2	yes	—
171	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
174	C₈×C₂₃	—	2	4	no	—
175	C₈×C₂₃	—	4	2	no	—
176	C₈×C₂₃	—	4	2	no	—
177	C₈×C₂₃	—	4	2	no	—
178	#3226	C₂×C₂₃	1	1	yes	characteristic, complemented
179	#3036	C₂×C₂₃	1	1	yes	characteristic
180	C₂×#461	C₂×C₂₃	1	1	yes	characteristic, complemented
181	#2972	C₂×C₂₃	1	1	yes	characteristic, complemented
182	#1681	C₂×C₂₃	1	4	yes	complemented
183	#1673	C₂×C₂₃	1	4	yes	complemented
184	#1571	C₂×C₂₃	1	1	yes	characteristic, complemented
185	#1546	C₂×C₂₃	1	1	yes	characteristic, complemented
186	#1540	C₂×C₂₃	1	1	yes	characteristic, complemented
187	C₂²×C₂₃	—	2	1	no	—
188	C₂²×C₂₃	—	2	1	no	—
189	C₂²×C₂₃	—	2	4	no	—
190	C₂²×C₂₃	—	4	2	no	—
191	C₂²×C₂₃	—	4	2	no	—
192	C₂²×C₂₃	—	8	4	no	—
193	C₂²×C₂₃	—	8	1	no	—
194	C₂²×C₂₃	—	8	1	no	—
195	C₂²×C₂₃	—	8	1	no	—
196	C₂²×C₂₃	#466	1	2	yes	—
197	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
198	C₄×C₂₃	—	2	1	no	—
199	C₄×C₂₃	—	2	1	no	—
200	C₄×C₂₃	—	2	1	no	—
201	C₄×C₂₃	—	4	2	no	—
202	C₄×C₂₃	—	8	1	no	—
203	C₄×C₂₃	—	8	2	no	—
204	C₄×C₂₃	#466	1	2	yes	—
205	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
206	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
207	C₄×C₂₃	#577	1	2	yes	—
208	C₂×AGL(1,ℤ/8)	—	2	1	no	—
209	C₂²×SD₁₆	—	2	1	no	—
210	#532	C₂²×C₂₃	1	1	yes	characteristic
211	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
212	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
213	#500	C₂²×C₂₃	1	2	yes	complemented
214	#486	C₂²×C₂₃	1	2	yes	complemented
215	#480	—	2	2	no	—
216	#474	—	2	2	no	—
217	#463	—	2	2	no	—
218	#461	—	2	2	no	—
219	#461	C₂²×C₂₃	1	2	yes	—
220	#461	C₂²×C₂₃	1	4	yes	—
221	#459	C₂²×C₂₃	1	2	yes	—
222	C₈×D₄	—	2	2	no	—
223	C₈×D₄	C₂²×C₂₃	1	4	yes	—
224	#431	C₂²×C₂₃	1	1	yes	characteristic
225	#431	C₂²×C₂₃	1	1	yes	characteristic
226	#423	C₂²×C₂₃	1	1	yes	characteristic
227	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
228	#417	—	2	1	no	—
229	C₂×#103	—	2	1	no	—
230	C₂×C₂²⋊C₈	—	2	2	no	—
231	#335	C₂²×C₂₃	1	4	yes	complemented
232	#332	C₂²×C₂₃	1	2	yes	complemented
233	#332	C₂²×C₂₃	1	4	yes	complemented
234	#331	C₂²×C₂₃	1	2	yes	complemented
235	C₂×C₂₃	—	2	2	no	—
236	C₂×C₂₃	—	4	4	no	—
237	C₂×C₂₃	—	16	1	no	—
238	C₂×C₂₃	#3042	1	1	yes	characteristic
239	C₂×C₂₃	C₂×#466	1	1	yes	characteristic
240	C₂×C₂₃	#3069	1	1	yes	characteristic
241	C₂×#41	—	2	1	no	—
242	C₂²×Q₈	—	2	1	no	—
243	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
244	C₂³×C₄	—	2	1	no	—
245	AGL(1,ℤ/8)	—	4	4	no	—
246	C₂×SD₁₆	—	2	4	no	—
247	C₂×SD₁₆	—	2	2	no	—
248	C₂×SD₁₆	—	4	4	no	—
249	C₂×D₈	—	2	2	no	—
250	C₂×M₁₆	—	2	1	no	—
251	C₂²×C₈	—	2	1	no	—
252	C₂²×C₈	—	2	4	no	—
253	C₄⋊Q₈	—	2	1	no	—
254	C₄⋊Q₈	—	2	1	no	—
255	C₄⋊Q₈	—	2	1	no	—
256	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
257	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
258	#127	C₂³×C₂₃	1	1	yes	characteristic
259	#124	—	2	1	no	—
260	#121	—	2	1	no	—
261	#121	—	2	1	no	—
262	C₄×D₄	—	2	1	no	—
263	C₄×D₄	—	2	1	no	—
264	C₄×D₄	—	2	1	no	—
265	C₄×D₄	—	2	1	no	—
266	C₄×D₄	D₄×C₂₃	1	2	yes	—
267	C₄×D₄	C₂³×C₂₃	1	2	yes	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
271	C₂×C₂²⋊C₄	—	2	1	no	—
272	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
273	#107	—	2	4	no	—
274	C₄⋊C₈	—	2	1	no	—
275	C₄⋊C₈	—	2	1	no	—
276	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
277	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
278	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
279	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
280	#103	—	2	4	no	—
281	#103	—	2	2	no	—
282	#102	—	2	4	no	—
283	#102	—	2	2	no	—
284	C₂²⋊C₈	—	2	1	no	—
285	C₂²⋊C₈	—	2	4	no	—
286	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
287	C₄×C₈	—	2	1	no	—
288	C₄×C₈	C₂³×C₂₃	1	2	yes	—
289	C₂₃	#19392	1	1	yes	characteristic, direct product factor, Sylow
290	C₂⁴	—	2	1	no	—
291	#41	—	4	2	no	—
292	#41	—	4	2	no	—
293	C₂×Q₈	—	2	1	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	4	1	no	—
296	C₂×Q₈	—	4	2	no	—
297	C₂×Q₈	—	4	2	no	—
298	C₂×D₄	—	2	1	no	—
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	2	1	no	—
301	C₂×D₄	—	2	1	no	—
302	C₂×D₄	—	2	1	no	—
303	C₂×D₄	—	4	1	no	—
304	C₂×D₄	D₈×C₂₃	1	4	yes	—
305	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	4	1	no	—
310	C₂²×C₄	—	4	1	no	—
311	C₂²×C₄	—	8	2	no	—
312	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
314	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
315	SD₁₆	—	4	8	no	—
316	SD₁₆	—	4	4	no	—
317	D₈	—	4	4	no	—
318	M₁₆	—	4	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	4	2	no	—
323	C₂×C₈	—	8	4	no	—
324	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	4	1	no	—
331	C₄⋊C₄	—	4	1	no	—
332	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
333	C₂²⋊C₄	—	2	2	no	—
334	C₂²⋊C₄	—	2	2	no	—
335	C₂²⋊C₄	—	4	1	no	—
336	C₂²⋊C₄	—	4	1	no	—
337	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
338	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
339	C₄²	#41×C₂₃	1	1	yes	characteristic
340	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
341	C₂³	—	2	2	no	—
342	C₂³	—	2	4	no	—
343	C₂³	—	4	1	no	—
344	C₂³	—	8	1	no	—
345	C₂³	—	8	1	no	—
346	C₂³	C₂₃×#121	1	1	yes	characteristic
347	Q₈	—	4	2	no	—
348	Q₈	—	4	1	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	4	1	no	—
351	Q₈	—	8	2	no	—
352	D₄	—	2	2	no	—
353	D₄	—	2	4	no	—
354	D₄	—	4	1	no	—
355	D₄	—	4	1	no	—
356	D₄	—	4	2	no	—
357	D₄	—	8	1	no	—
358	D₄	—	8	2	no	—
359	D₄	C₂×D₈×C₂₃	1	4	yes	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	4	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	8	2	no	—
367	C₂×C₄	—	8	1	no	—
368	C₂×C₄	—	8	2	no	—
369	C₂×C₄	—	8	2	no	—
370	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
371	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
372	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
373	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
374	C₂×C₄	C₂₃×#135	1	2	yes	—
375	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
376	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
377	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
378	C₈	—	2	4	no	—
379	C₈	—	4	2	no	—
380	C₈	—	4	2	no	—
381	C₈	—	4	2	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	4	no	—
385	C₂²	—	4	2	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	8	4	no	—
388	C₂²	—	8	1	no	—
389	C₂²	—	8	1	no	—
390	C₂²	—	8	1	no	—
391	C₂²	C₂₃×#466	1	2	yes	—
392	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
393	C₄	—	2	1	no	—
394	C₄	—	2	1	no	—
395	C₄	—	2	1	no	—
396	C₄	—	4	2	no	—
397	C₄	—	8	1	no	—
398	C₄	—	8	2	no	—
399	C₄	C₂₃×#466	1	2	yes	—
400	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
401	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
402	C₄	C₂₃×#577	1	2	yes	—
403	C₂	—	2	2	no	—
404	C₂	—	4	4	no	—
405	C₂	—	16	1	no	—
406	C₂	C₂₃×#3042	1	1	yes	characteristic
407	C₂	C₂×C₂₃×#466	1	1	yes	characteristic
408	C₂	C₂₃×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3036	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ 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
1. C₂₃×#3226	C₂	semidirect
3. C₂×C₂₃×#461	C₂	semidirect
4. C₂₃×#2972	C₂	semidirect
5. C₂₃×#1681	C₂	semidirect
6. C₂₃×#1673	C₂	semidirect
7. C₂₃×#1571	C₂	semidirect
8. C₂₃×#1546	C₂	semidirect
9. C₂₃×#1540	C₂	semidirect
15. C₂₃×#500	C₂²	semidirect
16. C₂₃×#486	C₂²	semidirect
33. C₂₃×#335	C₂²	semidirect
34. C₂₃×#332	C₂²	semidirect
35. C₂₃×#332	C₂²	semidirect
36. C₂₃×#331	C₂²	semidirect
136. #19392	C₂₃	direct (N × H)
178. #3226	C₂×C₂₃	semidirect
180. C₂×#461	C₂×C₂₃	semidirect
181. #2972	C₂×C₂₃	semidirect
182. #1681	C₂×C₂₃	semidirect
183. #1673	C₂×C₂₃	semidirect
184. #1571	C₂×C₂₃	semidirect
185. #1546	C₂×C₂₃	semidirect
186. #1540	C₂×C₂₃	semidirect
213. #500	C₂²×C₂₃	semidirect
214. #486	C₂²×C₂₃	semidirect
231. #335	C₂²×C₂₃	semidirect
232. #332	C₂²×C₂₃	semidirect
233. #332	C₂²×C₂₃	semidirect
234. #331	C₂²×C₂₃	semidirect
289. C₂₃	#19392	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×#3226	2
C₂₃×#3036	2
C₂×C₂₃×#461	2
C₂₃×#2972	2
C₂₃×#1681	2
C₂₃×#1673	2
C₂₃×#1571	2
C₂₃×#1546	2
C₂₃×#1540	2
#19392	23

Subgroups — C23×#19392 #19167018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19392 #19167018