Subgroups — C₂₃×#8894 #19156520

Order 5888.

602 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1948	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1802	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1296	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₂₃	—	2	1	no	—
9	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
11	C₂×C₂₃×#106	—	2	1	no	—
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#102	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#389	—	2	1	no	—
17	C₂₃×#385	—	2	2	no	—
18	C₂₃×#384	C₂²	1	1	yes	characteristic
19	C₂₃×#384	C₂²	1	1	yes	characteristic
20	C₂₃×#380	—	2	1	no	—
21	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#340	—	2	1	no	—
23	C₂₃×#340	—	2	1	no	—
24	C₂₃×#340	—	2	2	no	—
25	C₂₃×D₄⋊C₈	C₄	1	4	yes	complemented
26	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₂₃	—	2	1	no	—
28	C₂³×C₄×C₂₃	—	2	2	no	—
29	C₂²×C₈×C₂₃	—	2	1	no	—
30	C₂²×C₈×C₂₃	—	2	1	no	—
31	C₂²×C₈×C₂₃	—	2	1	no	—
32	C₄×D₄×C₂₃	—	2	1	no	—
33	C₄×D₄×C₂₃	—	2	1	no	—
34	C₄×D₄×C₂₃	—	4	2	no	—
35	C₄×D₄×C₂₃	—	4	2	no	—
36	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
37	C₄×D₄×C₂₃	Q₈	1	2	yes	—
38	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
45	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
46	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
50	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
51	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
52	C₂×C₄²×C₂₃	—	2	1	no	—
53	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
54	C₂₃×#106	—	4	2	no	—
55	C₂₃×C₄⋊C₈	—	2	1	no	—
56	C₂₃×C₄⋊C₈	—	2	1	no	—
57	C₂₃×C₄⋊C₈	—	2	1	no	—
58	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	complemented
59	C₂₃×#102	—	4	2	no	—
60	C₄×C₈×C₂₃	—	2	1	no	—
61	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	complemented
62	C₂₃×#95	—	2	2	no	—
63	C₂₃×#95	—	2	1	no	—
64	C₂⁴×C₂₃	—	2	1	no	—
65	C₂×D₄×C₂₃	—	2	1	no	—
66	C₂×D₄×C₂₃	—	2	1	no	—
67	C₂×D₄×C₂₃	—	2	1	no	—
68	C₂×D₄×C₂₃	C₄⋊C₄	1	2	yes	—
69	C₂×D₄×C₂₃	SD₁₆	1	2	yes	—
70	C₂×D₄×C₂₃	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₂₃	—	2	1	no	—
72	C₂²×C₄×C₂₃	—	2	1	no	—
73	C₂²×C₄×C₂₃	—	2	2	no	—
74	C₂²×C₄×C₂₃	—	2	2	no	—
75	C₂²×C₄×C₂₃	—	2	1	no	—
76	C₂²×C₄×C₂₃	—	2	2	no	—
77	C₂²×C₄×C₂₃	—	4	4	no	—
78	C₂²×C₄×C₂₃	—	4	1	no	—
79	C₂²×C₄×C₂₃	—	4	1	no	—
80	C₂²×C₄×C₂₃	—	4	1	no	—
81	C₂²×C₄×C₂₃	—	4	1	no	—
82	C₂²×C₄×C₂₃	—	4	4	no	—
83	C₂²×C₄×C₂₃	—	4	1	no	—
84	C₂²×C₄×C₂₃	—	8	2	no	—
85	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₂₃	—	2	2	no	—
89	C₂×C₈×C₂₃	—	2	2	no	—
90	C₂×C₈×C₂₃	—	4	1	no	—
91	C₂×C₈×C₂₃	—	4	1	no	—
92	C₂×C₈×C₂₃	—	4	1	no	—
93	C₂×C₈×C₂₃	—	4	1	no	—
94	C₂×C₈×C₂₃	—	4	1	no	—
95	C₂×C₈×C₂₃	—	4	1	no	—
96	C₂×C₈×C₂₃	—	4	1	no	—
97	C₄⋊C₄×C₂₃	—	2	1	no	—
98	C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₄⋊C₄×C₂₃	—	2	1	no	—
101	C₄⋊C₄×C₂₃	—	4	2	no	—
102	C₄⋊C₄×C₂₃	—	4	4	no	—
103	C₄⋊C₄×C₂₃	—	4	2	no	—
104	C₄⋊C₄×C₂₃	—	4	1	no	—
105	C₄⋊C₄×C₂₃	—	4	1	no	—
106	C₄⋊C₄×C₂₃	—	4	1	no	—
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	C₄⋊C₄×C₂₃	—	4	1	no	—
112	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
113	C₄⋊C₄×C₂₃	D₈	1	2	yes	—
114	C₂²⋊C₄×C₂₃	—	2	2	no	—
115	C₂²⋊C₄×C₂₃	—	2	2	no	—
116	C₂²⋊C₄×C₂₃	—	4	4	no	—
117	C₄²×C₂₃	—	4	1	no	—
118	C₄²×C₂₃	—	4	1	no	—
119	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
121	C₄²×C₂₃	#41	1	1	yes	characteristic
122	C₄²×C₂₃	#41	1	1	yes	characteristic
123	#8894	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂³×C₂₃	—	2	2	no	—
125	C₂³×C₂₃	—	2	2	no	—
126	C₂³×C₂₃	—	2	2	no	—
127	C₂³×C₂₃	—	4	1	no	—
128	C₂³×C₂₃	—	4	1	no	—
129	C₂³×C₂₃	—	4	1	no	—
130	C₂³×C₂₃	—	4	1	no	—
131	C₂³×C₂₃	#120	1	1	yes	characteristic
132	D₄×C₂₃	—	2	2	no	—
133	D₄×C₂₃	—	2	2	no	—
134	D₄×C₂₃	—	2	2	no	—
135	D₄×C₂₃	#106	1	4	yes	complemented
136	C₂×C₄×C₂₃	—	2	1	no	—
137	C₂×C₄×C₂₃	—	2	1	no	—
138	C₂×C₄×C₂₃	—	2	2	no	—
139	C₂×C₄×C₂₃	—	2	2	no	—
140	C₂×C₄×C₂₃	—	4	1	no	—
141	C₂×C₄×C₂₃	—	4	1	no	—
142	C₂×C₄×C₂₃	—	4	1	no	—
143	C₂×C₄×C₂₃	—	4	1	no	—
144	C₂×C₄×C₂₃	—	4	2	no	—
145	C₂×C₄×C₂₃	—	4	2	no	—
146	C₂×C₄×C₂₃	—	4	2	no	—
147	C₂×C₄×C₂₃	—	4	2	no	—
148	C₂×C₄×C₂₃	—	4	2	no	—
149	C₂×C₄×C₂₃	—	4	1	no	—
150	C₂×C₄×C₂₃	—	4	2	no	—
151	C₂×C₄×C₂₃	—	4	1	no	—
152	C₂×C₄×C₂₃	—	4	1	no	—
153	C₂×C₄×C₂₃	—	4	2	no	—
154	C₂×C₄×C₂₃	—	8	8	no	—
155	C₂×C₄×C₂₃	—	8	1	no	—
156	C₂×C₄×C₂₃	—	8	1	no	—
157	C₂×C₄×C₂₃	—	8	1	no	—
158	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
161	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
163	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
164	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
165	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
166	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
167	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
168	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
171	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
172	C₈×C₂₃	—	4	2	no	—
173	C₈×C₂₃	—	4	2	no	—
174	C₈×C₂₃	—	8	1	no	—
175	D₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
176	#1948	C₂×C₂₃	1	1	yes	characteristic, complemented
177	#1834	C₂×C₂₃	1	1	yes	characteristic, complemented
178	#1802	C₂×C₂₃	1	1	yes	characteristic
179	C₂×D₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
180	#1296	C₂×C₂₃	1	1	yes	characteristic, complemented
181	#1279	C₂×C₂₃	1	1	yes	characteristic, complemented
182	C₂²×C₂₃	—	2	4	no	—
183	C₂²×C₂₃	—	4	2	no	—
184	C₂²×C₂₃	—	4	2	no	—
185	C₂²×C₂₃	—	4	2	no	—
186	C₂²×C₂₃	—	4	2	no	—
187	C₂²×C₂₃	—	4	2	no	—
188	C₂²×C₂₃	—	4	2	no	—
189	C₂²×C₂₃	#385	1	1	yes	characteristic
190	C₂²×C₂₃	#447	1	1	yes	characteristic
191	C₂²×C₂₃	#448	1	1	yes	characteristic
192	C₂²×C₂₃	#450	1	1	yes	characteristic
193	C₂²×C₂₃	#452	1	1	yes	characteristic
194	C₂²×C₂₃	#454	1	1	yes	characteristic
195	C₂²×C₂₃	#455	1	1	yes	characteristic
196	C₄×C₂₃	—	2	1	no	—
197	C₄×C₂₃	—	2	1	no	—
198	C₄×C₂₃	—	4	2	no	—
199	C₄×C₂₃	—	4	2	no	—
200	C₄×C₂₃	—	8	1	no	—
201	C₄×C₂₃	—	8	2	no	—
202	C₄×C₂₃	—	16	1	no	—
203	C₄×C₂₃	C₂×#106	1	1	yes	characteristic
204	C₄×C₂₃	#428	1	1	yes	characteristic
205	C₄×C₂₃	C₄×D₈	1	1	yes	characteristic
206	C₄×C₂₃	#440	1	1	yes	characteristic
207	C₄×C₂₃	#474	1	1	yes	characteristic
208	C₄×C₂₃	#478	1	1	yes	characteristic
209	C₄×C₂₃	#481	1	1	yes	characteristic
210	C₄×C₂₃	#483	1	1	yes	characteristic
211	C₂×C₄×D₄	—	2	1	no	—
212	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
213	C₂²×C₄⋊C₄	—	2	1	no	—
214	C₂×#106	—	2	1	no	—
215	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
216	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
217	C₂×#102	—	2	1	no	—
218	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
219	#389	—	2	1	no	—
220	#385	—	2	2	no	—
221	#384	C₂²×C₂₃	1	1	yes	characteristic
222	#384	C₂²×C₂₃	1	1	yes	characteristic
223	#380	—	2	1	no	—
224	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
225	#340	—	2	1	no	—
226	#340	—	2	2	no	—
227	#340	—	2	1	no	—
228	D₄⋊C₈	C₄×C₂₃	1	4	yes	complemented
229	C₂×C₂₃	—	4	4	no	—
230	C₂×C₂₃	#1627	1	1	yes	characteristic
231	C₂×C₂₃	#1635	1	1	yes	characteristic
232	C₂×C₂₃	#1639	1	1	yes	characteristic
233	C₂×C₂₃	#1643	1	1	yes	characteristic
234	C₂×C₂₃	#1853	1	1	yes	characteristic
235	C₂×C₂₃	#1860	1	1	yes	characteristic
236	C₂×C₂₃	#1862	1	1	yes	characteristic
237	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
238	C₂³×C₄	—	2	1	no	—
239	C₂³×C₄	—	2	2	no	—
240	C₂²×C₈	—	2	1	no	—
241	C₂²×C₈	—	2	1	no	—
242	C₂²×C₈	—	2	1	no	—
243	C₄×D₄	—	2	1	no	—
244	C₄×D₄	—	2	1	no	—
245	C₄×D₄	—	4	2	no	—
246	C₄×D₄	—	4	2	no	—
247	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
248	C₄×D₄	Q₈×C₂₃	1	2	yes	—
249	C₂×C₄⋊C₄	—	2	2	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	4	2	no	—
256	C₂×C₄⋊C₄	—	4	2	no	—
257	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
258	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
259	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
260	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
261	C₂×C₂²⋊C₄	—	2	1	no	—
262	C₂×C₂²⋊C₄	—	2	2	no	—
263	C₂×C₄²	—	2	1	no	—
264	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
265	#106	—	4	2	no	—
266	C₄⋊C₈	—	2	1	no	—
267	C₄⋊C₈	—	2	1	no	—
268	C₄⋊C₈	—	2	1	no	—
269	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	complemented
270	#102	—	4	2	no	—
271	C₄×C₈	—	2	1	no	—
272	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	complemented
273	#95	—	2	2	no	—
274	#95	—	2	1	no	—
275	C₂₃	#8894	1	1	yes	characteristic, direct product factor, Sylow
276	C₂⁴	—	2	1	no	—
277	C₂×D₄	—	2	1	no	—
278	C₂×D₄	—	2	1	no	—
279	C₂×D₄	—	2	1	no	—
280	C₂×D₄	C₄⋊C₄×C₂₃	1	2	yes	—
281	C₂×D₄	SD₁₆×C₂₃	1	2	yes	—
282	C₂×D₄	SD₁₆×C₂₃	1	2	yes	—
283	C₂²×C₄	—	2	1	no	—
284	C₂²×C₄	—	2	1	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	4	4	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	4	1	no	—
294	C₂²×C₄	—	4	4	no	—
295	C₂²×C₄	—	4	1	no	—
296	C₂²×C₄	—	8	2	no	—
297	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
298	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	4	1	no	—
303	C₂×C₈	—	4	1	no	—
304	C₂×C₈	—	4	1	no	—
305	C₂×C₈	—	4	1	no	—
306	C₂×C₈	—	4	1	no	—
307	C₂×C₈	—	4	1	no	—
308	C₂×C₈	—	4	1	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	4	2	no	—
314	C₄⋊C₄	—	4	4	no	—
315	C₄⋊C₄	—	4	2	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₄⋊C₄	—	4	1	no	—
318	C₄⋊C₄	—	4	1	no	—
319	C₄⋊C₄	—	4	1	no	—
320	C₄⋊C₄	—	4	1	no	—
321	C₄⋊C₄	—	4	1	no	—
322	C₄⋊C₄	—	4	1	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
325	C₄⋊C₄	D₈×C₂₃	1	2	yes	—
326	C₂²⋊C₄	—	2	2	no	—
327	C₂²⋊C₄	—	2	2	no	—
328	C₂²⋊C₄	—	4	4	no	—
329	C₄²	—	4	1	no	—
330	C₄²	—	4	1	no	—
331	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
332	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
333	C₄²	#41×C₂₃	1	1	yes	characteristic
334	C₄²	#41×C₂₃	1	1	yes	characteristic
335	C₂³	—	2	2	no	—
336	C₂³	—	2	2	no	—
337	C₂³	—	2	2	no	—
338	C₂³	—	4	1	no	—
339	C₂³	—	4	1	no	—
340	C₂³	—	4	1	no	—
341	C₂³	—	4	1	no	—
342	C₂³	C₂₃×#120	1	1	yes	characteristic
343	D₄	—	2	2	no	—
344	D₄	—	2	2	no	—
345	D₄	—	2	2	no	—
346	D₄	C₂₃×#106	1	4	yes	complemented
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	4	1	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	1	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	8	8	no	—
366	C₂×C₄	—	8	1	no	—
367	C₂×C₄	—	8	1	no	—
368	C₂×C₄	—	8	1	no	—
369	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
370	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
371	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
372	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
373	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
374	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
375	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
376	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
377	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
378	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
379	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
380	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
381	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
382	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
383	C₈	—	4	2	no	—
384	C₈	—	4	2	no	—
385	C₈	—	8	1	no	—
386	C₂²	—	2	4	no	—
387	C₂²	—	4	2	no	—
388	C₂²	—	4	2	no	—
389	C₂²	—	4	2	no	—
390	C₂²	—	4	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	4	2	no	—
393	C₂²	C₂₃×#385	1	1	yes	characteristic
394	C₂²	C₂₃×#447	1	1	yes	characteristic
395	C₂²	C₂₃×#448	1	1	yes	characteristic
396	C₂²	C₂₃×#450	1	1	yes	characteristic
397	C₂²	C₂₃×#452	1	1	yes	characteristic
398	C₂²	C₂₃×#454	1	1	yes	characteristic
399	C₂²	C₂₃×#455	1	1	yes	characteristic
400	C₄	—	2	1	no	—
401	C₄	—	2	1	no	—
402	C₄	—	4	2	no	—
403	C₄	—	4	2	no	—
404	C₄	—	8	1	no	—
405	C₄	—	8	2	no	—
406	C₄	—	16	1	no	—
407	C₄	C₂×C₂₃×#106	1	1	yes	characteristic
408	C₄	C₂₃×#428	1	1	yes	characteristic
409	C₄	C₄×D₈×C₂₃	1	1	yes	characteristic
410	C₄	C₂₃×#440	1	1	yes	characteristic
411	C₄	C₂₃×#474	1	1	yes	characteristic
412	C₄	C₂₃×#478	1	1	yes	characteristic
413	C₄	C₂₃×#481	1	1	yes	characteristic
414	C₄	C₂₃×#483	1	1	yes	characteristic
415	C₂	—	4	4	no	—
416	C₂	C₂₃×#1627	1	1	yes	characteristic
417	C₂	C₂₃×#1635	1	1	yes	characteristic
418	C₂	C₂₃×#1639	1	1	yes	characteristic
419	C₂	C₂₃×#1643	1	1	yes	characteristic
420	C₂	C₂₃×#1853	1	1	yes	characteristic
421	C₂	C₂₃×#1860	1	1	yes	characteristic
422	C₂	C₂₃×#1862	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#120
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂²×D₄×C₂₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1948	C₂	semidirect
3. C₂₃×#1834	C₂	semidirect
6. C₂₃×#1296	C₂	semidirect
7. C₂₃×#1279	C₂	semidirect
25. C₂₃×D₄⋊C₈	C₄	semidirect
58. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
61. C₄×C₈×C₂₃	C₂×C₄	semidirect
123. #8894	C₂₃	direct (N × H)
135. D₄×C₂₃	#106	semidirect
176. #1948	C₂×C₂₃	semidirect
177. #1834	C₂×C₂₃	semidirect
180. #1296	C₂×C₂₃	semidirect
181. #1279	C₂×C₂₃	semidirect
228. D₄⋊C₈	C₄×C₂₃	semidirect
269. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
272. C₄×C₈	C₂×C₄×C₂₃	semidirect
275. C₂₃	#8894	direct (N × H)
346. D₄	C₂₃×#106	semidirect

Maximal subgroups

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

subgroup	index
D₄×C₄⋊C₄×C₂₃	2
C₂₃×#1948	2
C₂₃×#1834	2
C₂₃×#1802	2
C₂×C₂₃×D₄⋊C₈	2
C₂₃×#1296	2
C₂₃×#1279	2
#8894	23

Subgroups — C23×#8894 #19156520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8894 #19156520