Subgroups — C₂₃×#8973 #19156599

Order 5888.

664 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#476	C₂	1	1	yes	maximal, characteristic
2	D₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1987	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1956	C₂	1	1	yes	maximal, characteristic, complemented
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×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₂₃	—	2	1	no	—
10	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
12	C₂₃×#476	—	2	2	no	—
13	C₂₃×#476	C₄	1	4	yes	complemented
14	C₂×C₂₃×#106	—	2	1	no	—
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#103	—	2	2	no	—
18	C₂×C₂₃×#102	—	2	1	no	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#389	—	2	1	no	—
21	C₂₃×#385	—	2	2	no	—
22	C₂₃×#384	C₂²	1	1	yes	characteristic
23	C₂₃×#380	—	2	1	no	—
24	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#340	—	2	1	no	—
26	C₂₃×#340	—	2	1	no	—
27	C₂₃×D₄⋊C₈	—	2	2	no	—
28	C₂²×Q₈×C₂₃	—	2	1	no	—
29	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₂₃	—	2	1	no	—
31	C₂³×C₄×C₂₃	—	2	2	no	—
32	C₂²×C₈×C₂₃	—	2	1	no	—
33	C₂²×C₈×C₂₃	—	2	1	no	—
34	C₂²×C₈×C₂₃	—	2	1	no	—
35	C₂₃×C₄⋊Q₈	—	2	1	no	—
36	C₂₃×C₄⋊Q₈	—	2	1	no	—
37	C₂₃×C₄⋊Q₈	—	2	1	no	—
38	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	complemented
39	C₄×D₄×C₂₃	—	2	1	no	—
40	C₄×D₄×C₂₃	—	2	1	no	—
41	C₄×D₄×C₂₃	—	4	2	no	—
42	C₄×D₄×C₂₃	—	4	2	no	—
43	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
44	C₄×D₄×C₂₃	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
46	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
47	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
48	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
49	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
50	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
51	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
52	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
56	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
57	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
58	C₂×C₄²×C₂₃	—	2	1	no	—
59	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
60	C₂₃×#106	—	2	4	no	—
61	C₂₃×C₄⋊C₈	—	2	1	no	—
62	C₂₃×C₄⋊C₈	—	2	1	no	—
63	C₂₃×C₄⋊C₈	—	2	1	no	—
64	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	complemented
65	C₂₃×#103	—	4	4	no	—
66	C₂₃×#102	—	2	4	no	—
67	C₄×C₈×C₂₃	—	2	1	no	—
68	C₄×C₈×C₂₃	—	2	1	no	—
69	C₂₃×#95	—	2	2	no	—
70	C₂₃×#95	—	2	1	no	—
71	C₂⁴×C₂₃	—	2	1	no	—
72	C₂×Q₈×C₂₃	—	2	2	no	—
73	C₂×Q₈×C₂₃	—	4	2	no	—
74	C₂×Q₈×C₂₃	—	4	2	no	—
75	C₂×Q₈×C₂₃	—	4	1	no	—
76	C₂×D₄×C₂₃	—	2	1	no	—
77	C₂×D₄×C₂₃	—	2	1	no	—
78	C₂×D₄×C₂₃	—	2	1	no	—
79	C₂×D₄×C₂₃	C₂²⋊C₄	1	2	yes	—
80	C₂×D₄×C₂₃	SD₁₆	1	2	yes	—
81	C₂×D₄×C₂₃	Q₁₆	1	2	yes	—
82	C₂²×C₄×C₂₃	—	2	1	no	—
83	C₂²×C₄×C₂₃	—	2	1	no	—
84	C₂²×C₄×C₂₃	—	2	2	no	—
85	C₂²×C₄×C₂₃	—	2	2	no	—
86	C₂²×C₄×C₂₃	—	2	1	no	—
87	C₂²×C₄×C₂₃	—	2	2	no	—
88	C₂²×C₄×C₂₃	—	4	4	no	—
89	C₂²×C₄×C₂₃	—	4	1	no	—
90	C₂²×C₄×C₂₃	—	4	1	no	—
91	C₂²×C₄×C₂₃	—	4	1	no	—
92	C₂²×C₄×C₂₃	—	4	1	no	—
93	C₂²×C₄×C₂₃	—	4	4	no	—
94	C₂²×C₄×C₂₃	—	4	1	no	—
95	C₂²×C₄×C₂₃	—	8	2	no	—
96	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₂₃	—	2	2	no	—
100	C₂×C₈×C₂₃	—	2	2	no	—
101	C₂×C₈×C₂₃	—	4	1	no	—
102	C₂×C₈×C₂₃	—	4	1	no	—
103	C₂×C₈×C₂₃	—	4	1	no	—
104	C₂×C₈×C₂₃	—	4	1	no	—
105	C₂×C₈×C₂₃	—	4	1	no	—
106	C₂×C₈×C₂₃	—	4	1	no	—
107	C₂×C₈×C₂₃	—	4	1	no	—
108	C₄⋊C₄×C₂₃	—	2	1	no	—
109	C₄⋊C₄×C₂₃	—	2	1	no	—
110	C₄⋊C₄×C₂₃	—	2	2	no	—
111	C₄⋊C₄×C₂₃	—	2	2	no	—
112	C₄⋊C₄×C₂₃	—	2	1	no	—
113	C₄⋊C₄×C₂₃	—	2	1	no	—
114	C₄⋊C₄×C₂₃	—	4	2	no	—
115	C₄⋊C₄×C₂₃	—	4	4	no	—
116	C₄⋊C₄×C₂₃	—	4	1	no	—
117	C₄⋊C₄×C₂₃	—	4	1	no	—
118	C₄⋊C₄×C₂₃	—	4	1	no	—
119	C₄⋊C₄×C₂₃	—	4	1	no	—
120	C₄⋊C₄×C₂₃	—	4	1	no	—
121	C₄⋊C₄×C₂₃	—	4	1	no	—
122	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
123	C₄⋊C₄×C₂₃	SD₁₆	1	2	yes	—
124	C₂²⋊C₄×C₂₃	—	2	2	no	—
125	C₂²⋊C₄×C₂₃	—	2	2	no	—
126	C₂²⋊C₄×C₂₃	—	4	4	no	—
127	C₄²×C₂₃	—	4	1	no	—
128	C₄²×C₂₃	—	4	1	no	—
129	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₂₃	#41	1	1	yes	characteristic
132	C₄²×C₂₃	#41	1	1	yes	characteristic
133	#8973	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂³×C₂₃	—	2	2	no	—
135	C₂³×C₂₃	—	2	2	no	—
136	C₂³×C₂₃	—	2	2	no	—
137	C₂³×C₂₃	—	4	1	no	—
138	C₂³×C₂₃	—	4	1	no	—
139	C₂³×C₂₃	—	4	1	no	—
140	C₂³×C₂₃	—	4	1	no	—
141	C₂³×C₂₃	#120	1	1	yes	characteristic
142	Q₈×C₂₃	—	4	4	no	—
143	Q₈×C₂₃	—	8	2	no	—
144	D₄×C₂₃	—	2	2	no	—
145	D₄×C₂₃	—	2	2	no	—
146	D₄×C₂₃	—	2	2	no	—
147	D₄×C₂₃	#103	1	4	yes	complemented
148	C₂×C₄×C₂₃	—	2	1	no	—
149	C₂×C₄×C₂₃	—	2	1	no	—
150	C₂×C₄×C₂₃	—	2	2	no	—
151	C₂×C₄×C₂₃	—	2	2	no	—
152	C₂×C₄×C₂₃	—	4	1	no	—
153	C₂×C₄×C₂₃	—	4	1	no	—
154	C₂×C₄×C₂₃	—	4	1	no	—
155	C₂×C₄×C₂₃	—	4	1	no	—
156	C₂×C₄×C₂₃	—	4	2	no	—
157	C₂×C₄×C₂₃	—	4	2	no	—
158	C₂×C₄×C₂₃	—	4	2	no	—
159	C₂×C₄×C₂₃	—	4	2	no	—
160	C₂×C₄×C₂₃	—	4	2	no	—
161	C₂×C₄×C₂₃	—	4	1	no	—
162	C₂×C₄×C₂₃	—	4	2	no	—
163	C₂×C₄×C₂₃	—	4	2	no	—
164	C₂×C₄×C₂₃	—	4	1	no	—
165	C₂×C₄×C₂₃	—	4	2	no	—
166	C₂×C₄×C₂₃	—	4	1	no	—
167	C₂×C₄×C₂₃	—	8	8	no	—
168	C₂×C₄×C₂₃	—	8	1	no	—
169	C₂×C₄×C₂₃	—	8	1	no	—
170	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
172	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
173	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
174	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
175	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
176	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
177	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
178	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
179	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
180	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
183	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
184	C₈×C₂₃	—	4	2	no	—
185	C₈×C₂₃	—	4	2	no	—
186	C₈×C₂₃	—	8	1	no	—
187	C₂×#476	C₂×C₂₃	1	1	yes	characteristic
188	D₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
189	#1987	C₂×C₂₃	1	1	yes	characteristic, complemented
190	#1956	C₂×C₂₃	1	1	yes	characteristic, complemented
191	C₂×D₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
192	#1296	C₂×C₂₃	1	1	yes	characteristic, complemented
193	#1279	C₂×C₂₃	1	1	yes	characteristic, complemented
194	C₂²×C₂₃	—	2	4	no	—
195	C₂²×C₂₃	—	4	2	no	—
196	C₂²×C₂₃	—	4	2	no	—
197	C₂²×C₂₃	—	4	2	no	—
198	C₂²×C₂₃	—	4	2	no	—
199	C₂²×C₂₃	—	4	2	no	—
200	C₂²×C₂₃	—	4	2	no	—
201	C₂²×C₂₃	#386	1	1	yes	characteristic
202	C₂²×C₂₃	#450	1	1	yes	characteristic
203	C₂²×C₂₃	#450	1	1	yes	characteristic
204	C₂²×C₂₃	#451	1	1	yes	characteristic
205	C₂²×C₂₃	#452	1	1	yes	characteristic
206	C₂²×C₂₃	#453	1	1	yes	characteristic
207	C₂²×C₂₃	#454	1	1	yes	characteristic
208	C₄×C₂₃	—	2	1	no	—
209	C₄×C₂₃	—	2	1	no	—
210	C₄×C₂₃	—	4	2	no	—
211	C₄×C₂₃	—	4	2	no	—
212	C₄×C₂₃	—	8	2	no	—
213	C₄×C₂₃	—	8	2	no	—
214	C₄×C₂₃	—	8	1	no	—
215	C₄×C₂₃	C₂×#103	1	1	yes	characteristic
216	C₄×C₂₃	#417	1	1	yes	characteristic
217	C₄×C₂₃	C₄×SD₁₆	1	1	yes	characteristic
218	C₄×C₂₃	#440	1	1	yes	characteristic
219	C₄×C₂₃	#461	1	1	yes	characteristic
220	C₄×C₂₃	#463	1	1	yes	characteristic
221	C₄×C₂₃	#483	1	1	yes	characteristic
222	C₄×C₂₃	#484	1	1	yes	characteristic
223	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
224	C₂×C₄×D₄	—	2	1	no	—
225	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
226	C₂²×C₄⋊C₄	—	2	1	no	—
227	#476	—	2	2	no	—
228	#476	C₄×C₂₃	1	4	yes	complemented
229	C₂×#106	—	2	1	no	—
230	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
231	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
232	C₂×#103	—	2	2	no	—
233	C₂×#102	—	2	1	no	—
234	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
235	#389	—	2	1	no	—
236	#385	—	2	2	no	—
237	#384	C₂²×C₂₃	1	1	yes	characteristic
238	#380	—	2	1	no	—
239	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
240	#340	—	2	1	no	—
241	#340	—	2	1	no	—
242	D₄⋊C₈	—	2	2	no	—
243	C₂×C₂₃	—	4	4	no	—
244	C₂×C₂₃	#1629	1	1	yes	characteristic
245	C₂×C₂₃	#1632	1	1	yes	characteristic
246	C₂×C₂₃	#1635	1	1	yes	characteristic
247	C₂×C₂₃	#1637	1	1	yes	characteristic
248	C₂×C₂₃	#1872	1	1	yes	characteristic
249	C₂×C₂₃	#1877	1	1	yes	characteristic
250	C₂×C₂₃	#1883	1	1	yes	characteristic
251	C₂²×Q₈	—	2	1	no	—
252	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
253	C₂³×C₄	—	2	1	no	—
254	C₂³×C₄	—	2	2	no	—
255	C₂²×C₈	—	2	1	no	—
256	C₂²×C₈	—	2	1	no	—
257	C₂²×C₈	—	2	1	no	—
258	C₄⋊Q₈	—	2	1	no	—
259	C₄⋊Q₈	—	2	1	no	—
260	C₄⋊Q₈	—	2	1	no	—
261	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	complemented
262	C₄×D₄	—	2	1	no	—
263	C₄×D₄	—	2	1	no	—
264	C₄×D₄	—	4	2	no	—
265	C₄×D₄	—	4	2	no	—
266	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
267	C₄×D₄	D₄×C₂₃	1	2	yes	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	2	2	no	—
271	C₂×C₄⋊C₄	—	2	1	no	—
272	C₂×C₄⋊C₄	—	2	1	no	—
273	C₂×C₄⋊C₄	—	4	2	no	—
274	C₂×C₄⋊C₄	—	4	2	no	—
275	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
276	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
277	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
278	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
279	C₂×C₂²⋊C₄	—	2	1	no	—
280	C₂×C₂²⋊C₄	—	2	2	no	—
281	C₂×C₄²	—	2	1	no	—
282	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
283	#106	—	2	4	no	—
284	C₄⋊C₈	—	2	1	no	—
285	C₄⋊C₈	—	2	1	no	—
286	C₄⋊C₈	—	2	1	no	—
287	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	complemented
288	#103	—	4	4	no	—
289	#102	—	2	4	no	—
290	C₄×C₈	—	2	1	no	—
291	C₄×C₈	—	2	1	no	—
292	#95	—	2	2	no	—
293	#95	—	2	1	no	—
294	C₂₃	#8973	1	1	yes	characteristic, direct product factor, Sylow
295	C₂⁴	—	2	1	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	4	2	no	—
298	C₂×Q₈	—	4	2	no	—
299	C₂×Q₈	—	4	1	no	—
300	C₂×D₄	—	2	1	no	—
301	C₂×D₄	—	2	1	no	—
302	C₂×D₄	—	2	1	no	—
303	C₂×D₄	C₂²⋊C₄×C₂₃	1	2	yes	—
304	C₂×D₄	SD₁₆×C₂₃	1	2	yes	—
305	C₂×D₄	Q₁₆×C₂₃	1	2	yes	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	4	4	no	—
313	C₂²×C₄	—	4	1	no	—
314	C₂²×C₄	—	4	1	no	—
315	C₂²×C₄	—	4	1	no	—
316	C₂²×C₄	—	4	1	no	—
317	C₂²×C₄	—	4	4	no	—
318	C₂²×C₄	—	4	1	no	—
319	C₂²×C₄	—	8	2	no	—
320	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
321	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	4	1	no	—
326	C₂×C₈	—	4	1	no	—
327	C₂×C₈	—	4	1	no	—
328	C₂×C₈	—	4	1	no	—
329	C₂×C₈	—	4	1	no	—
330	C₂×C₈	—	4	1	no	—
331	C₂×C₈	—	4	1	no	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	4	2	no	—
339	C₄⋊C₄	—	4	4	no	—
340	C₄⋊C₄	—	4	1	no	—
341	C₄⋊C₄	—	4	1	no	—
342	C₄⋊C₄	—	4	1	no	—
343	C₄⋊C₄	—	4	1	no	—
344	C₄⋊C₄	—	4	1	no	—
345	C₄⋊C₄	—	4	1	no	—
346	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
347	C₄⋊C₄	SD₁₆×C₂₃	1	2	yes	—
348	C₂²⋊C₄	—	2	2	no	—
349	C₂²⋊C₄	—	2	2	no	—
350	C₂²⋊C₄	—	4	4	no	—
351	C₄²	—	4	1	no	—
352	C₄²	—	4	1	no	—
353	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
354	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
355	C₄²	#41×C₂₃	1	1	yes	characteristic
356	C₄²	#41×C₂₃	1	1	yes	characteristic
357	C₂³	—	2	2	no	—
358	C₂³	—	2	2	no	—
359	C₂³	—	2	2	no	—
360	C₂³	—	4	1	no	—
361	C₂³	—	4	1	no	—
362	C₂³	—	4	1	no	—
363	C₂³	—	4	1	no	—
364	C₂³	C₂₃×#120	1	1	yes	characteristic
365	Q₈	—	4	4	no	—
366	Q₈	—	8	2	no	—
367	D₄	—	2	2	no	—
368	D₄	—	2	2	no	—
369	D₄	—	2	2	no	—
370	D₄	C₂₃×#103	1	4	yes	complemented
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	4	1	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	1	no	—
390	C₂×C₄	—	8	8	no	—
391	C₂×C₄	—	8	1	no	—
392	C₂×C₄	—	8	1	no	—
393	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
394	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
395	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
396	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
397	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
398	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
399	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
400	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
401	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
402	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
403	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
404	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
405	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
406	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
407	C₈	—	4	2	no	—
408	C₈	—	4	2	no	—
409	C₈	—	8	1	no	—
410	C₂²	—	2	4	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	4	2	no	—
413	C₂²	—	4	2	no	—
414	C₂²	—	4	2	no	—
415	C₂²	—	4	2	no	—
416	C₂²	—	4	2	no	—
417	C₂²	C₂₃×#386	1	1	yes	characteristic
418	C₂²	C₂₃×#450	1	1	yes	characteristic
419	C₂²	C₂₃×#450	1	1	yes	characteristic
420	C₂²	C₂₃×#451	1	1	yes	characteristic
421	C₂²	C₂₃×#452	1	1	yes	characteristic
422	C₂²	C₂₃×#453	1	1	yes	characteristic
423	C₂²	C₂₃×#454	1	1	yes	characteristic
424	C₄	—	2	1	no	—
425	C₄	—	2	1	no	—
426	C₄	—	4	2	no	—
427	C₄	—	4	2	no	—
428	C₄	—	8	2	no	—
429	C₄	—	8	1	no	—
430	C₄	—	8	2	no	—
431	C₄	C₂×C₂₃×#103	1	1	yes	characteristic
432	C₄	C₂₃×#417	1	1	yes	characteristic
433	C₄	C₄×SD₁₆×C₂₃	1	1	yes	characteristic
434	C₄	C₂₃×#440	1	1	yes	characteristic
435	C₄	C₂₃×#461	1	1	yes	characteristic
436	C₄	C₂₃×#463	1	1	yes	characteristic
437	C₄	C₂₃×#483	1	1	yes	characteristic
438	C₄	C₂₃×#484	1	1	yes	characteristic
439	C₂	—	4	4	no	—
440	C₂	C₂₃×#1629	1	1	yes	characteristic
441	C₂	C₂₃×#1632	1	1	yes	characteristic
442	C₂	C₂₃×#1635	1	1	yes	characteristic
443	C₂	C₂₃×#1637	1	1	yes	characteristic
444	C₂	C₂₃×#1872	1	1	yes	characteristic
445	C₂	C₂₃×#1877	1	1	yes	characteristic
446	C₂	C₂₃×#1883	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
3. C₂₃×#1987	C₂	semidirect
4. C₂₃×#1956	C₂	semidirect
6. C₂₃×#1296	C₂	semidirect
7. C₂₃×#1279	C₂	semidirect
13. C₂₃×#476	C₄	semidirect
38. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
64. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
133. #8973	C₂₃	direct (N × H)
147. D₄×C₂₃	#103	semidirect
189. #1987	C₂×C₂₃	semidirect
190. #1956	C₂×C₂₃	semidirect
192. #1296	C₂×C₂₃	semidirect
193. #1279	C₂×C₂₃	semidirect
228. #476	C₄×C₂₃	semidirect
261. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
287. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
294. C₂₃	#8973	direct (N × H)
370. D₄	C₂₃×#103	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#476	2
D₄×C₄⋊C₄×C₂₃	2
C₂₃×#1987	2
C₂₃×#1956	2
C₂×C₂₃×D₄⋊C₈	2
C₂₃×#1296	2
C₂₃×#1279	2
#8973	23

Subgroups — C23×#8973 #19156599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8973 #19156599