Subgroups — C₂₃×#8981 #19156607

Order 5888.

664 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#474	C₂	1	1	yes	maximal, characteristic
2	D₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1981	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×#474	—	2	2	no	—
13	C₂₃×#474	C₄	1	4	yes	complemented
14	C₂×C₂₃×#107	—	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	1	no	—
18	C₂×C₂₃×#103	—	2	1	no	—
19	C₂×C₂₃×#102	—	2	1	no	—
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#389	—	2	1	no	—
22	C₂₃×#385	—	2	2	no	—
23	C₂₃×#384	C₂²	1	1	yes	characteristic
24	C₂₃×#380	—	2	1	no	—
25	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#340	—	2	2	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×#107	—	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	2	no	—
66	C₂₃×#103	—	4	2	no	—
67	C₂₃×#102	—	2	4	no	—
68	C₄×C₈×C₂₃	—	2	1	no	—
69	C₄×C₈×C₂₃	—	2	1	no	—
70	C₂₃×#95	—	2	2	no	—
71	C₂₃×#95	—	2	1	no	—
72	C₂⁴×C₂₃	—	2	1	no	—
73	C₂×Q₈×C₂₃	—	2	2	no	—
74	C₂×Q₈×C₂₃	—	4	1	no	—
75	C₂×Q₈×C₂₃	—	4	1	no	—
76	C₂×Q₈×C₂₃	—	4	1	no	—
77	C₂×Q₈×C₂₃	—	4	1	no	—
78	C₂×Q₈×C₂₃	—	4	1	no	—
79	C₂×D₄×C₂₃	—	2	1	no	—
80	C₂×D₄×C₂₃	—	2	1	no	—
81	C₂×D₄×C₂₃	—	2	1	no	—
82	C₂×D₄×C₂₃	C₂²⋊C₄	1	2	yes	—
83	C₂×D₄×C₂₃	SD₁₆	1	2	yes	—
84	C₂×D₄×C₂₃	Q₁₆	1	2	yes	—
85	C₂²×C₄×C₂₃	—	2	1	no	—
86	C₂²×C₄×C₂₃	—	2	1	no	—
87	C₂²×C₄×C₂₃	—	2	2	no	—
88	C₂²×C₄×C₂₃	—	2	2	no	—
89	C₂²×C₄×C₂₃	—	2	1	no	—
90	C₂²×C₄×C₂₃	—	2	2	no	—
91	C₂²×C₄×C₂₃	—	4	4	no	—
92	C₂²×C₄×C₂₃	—	4	1	no	—
93	C₂²×C₄×C₂₃	—	4	1	no	—
94	C₂²×C₄×C₂₃	—	4	1	no	—
95	C₂²×C₄×C₂₃	—	4	1	no	—
96	C₂²×C₄×C₂₃	—	4	4	no	—
97	C₂²×C₄×C₂₃	—	4	1	no	—
98	C₂²×C₄×C₂₃	—	8	2	no	—
99	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₂₃	—	2	2	no	—
103	C₂×C₈×C₂₃	—	2	2	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	C₂×C₈×C₂₃	—	4	1	no	—
109	C₂×C₈×C₂₃	—	4	1	no	—
110	C₂×C₈×C₂₃	—	4	1	no	—
111	C₄⋊C₄×C₂₃	—	2	1	no	—
112	C₄⋊C₄×C₂₃	—	2	1	no	—
113	C₄⋊C₄×C₂₃	—	2	2	no	—
114	C₄⋊C₄×C₂₃	—	2	2	no	—
115	C₄⋊C₄×C₂₃	—	2	1	no	—
116	C₄⋊C₄×C₂₃	—	2	1	no	—
117	C₄⋊C₄×C₂₃	—	4	2	no	—
118	C₄⋊C₄×C₂₃	—	4	4	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	—	4	1	no	—
123	C₄⋊C₄×C₂₃	—	4	1	no	—
124	C₄⋊C₄×C₂₃	—	4	1	no	—
125	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
126	C₄⋊C₄×C₂₃	D₈	1	2	yes	—
127	C₂²⋊C₄×C₂₃	—	2	2	no	—
128	C₂²⋊C₄×C₂₃	—	2	2	no	—
129	C₂²⋊C₄×C₂₃	—	4	4	no	—
130	C₄²×C₂₃	—	4	1	no	—
131	C₄²×C₂₃	—	4	1	no	—
132	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₂₃	#41	1	1	yes	characteristic
135	C₄²×C₂₃	#41	1	1	yes	characteristic
136	#8981	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂³×C₂₃	—	2	2	no	—
138	C₂³×C₂₃	—	2	2	no	—
139	C₂³×C₂₃	—	2	2	no	—
140	C₂³×C₂₃	—	4	1	no	—
141	C₂³×C₂₃	—	4	1	no	—
142	C₂³×C₂₃	—	4	1	no	—
143	C₂³×C₂₃	—	4	1	no	—
144	C₂³×C₂₃	#120	1	1	yes	characteristic
145	Q₈×C₂₃	—	4	2	no	—
146	Q₈×C₂₃	—	4	2	no	—
147	Q₈×C₂₃	—	8	1	no	—
148	Q₈×C₂₃	—	8	1	no	—
149	D₄×C₂₃	—	2	2	no	—
150	D₄×C₂₃	—	2	2	no	—
151	D₄×C₂₃	—	2	2	no	—
152	D₄×C₂₃	#103	1	4	yes	complemented
153	C₂×C₄×C₂₃	—	2	1	no	—
154	C₂×C₄×C₂₃	—	2	1	no	—
155	C₂×C₄×C₂₃	—	2	2	no	—
156	C₂×C₄×C₂₃	—	2	2	no	—
157	C₂×C₄×C₂₃	—	4	1	no	—
158	C₂×C₄×C₂₃	—	4	1	no	—
159	C₂×C₄×C₂₃	—	4	1	no	—
160	C₂×C₄×C₂₃	—	4	1	no	—
161	C₂×C₄×C₂₃	—	4	2	no	—
162	C₂×C₄×C₂₃	—	4	2	no	—
163	C₂×C₄×C₂₃	—	4	2	no	—
164	C₂×C₄×C₂₃	—	4	2	no	—
165	C₂×C₄×C₂₃	—	4	2	no	—
166	C₂×C₄×C₂₃	—	4	2	no	—
167	C₂×C₄×C₂₃	—	4	1	no	—
168	C₂×C₄×C₂₃	—	4	1	no	—
169	C₂×C₄×C₂₃	—	4	2	no	—
170	C₂×C₄×C₂₃	—	4	2	no	—
171	C₂×C₄×C₂₃	—	4	1	no	—
172	C₂×C₄×C₂₃	—	8	8	no	—
173	C₂×C₄×C₂₃	—	8	1	no	—
174	C₂×C₄×C₂₃	—	8	1	no	—
175	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
177	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
178	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
179	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
180	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
181	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
182	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
183	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
184	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
185	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
189	C₈×C₂₃	—	4	2	no	—
190	C₈×C₂₃	—	4	2	no	—
191	C₈×C₂₃	—	8	1	no	—
192	C₂×#474	C₂×C₂₃	1	1	yes	characteristic
193	D₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
194	#1990	C₂×C₂₃	1	1	yes	characteristic, complemented
195	#1981	C₂×C₂₃	1	1	yes	characteristic, complemented
196	C₂×D₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
197	#1296	C₂×C₂₃	1	1	yes	characteristic, complemented
198	#1279	C₂×C₂₃	1	1	yes	characteristic, complemented
199	C₂²×C₂₃	—	2	4	no	—
200	C₂²×C₂₃	—	4	2	no	—
201	C₂²×C₂₃	—	4	2	no	—
202	C₂²×C₂₃	—	4	2	no	—
203	C₂²×C₂₃	—	4	2	no	—
204	C₂²×C₂₃	—	4	2	no	—
205	C₂²×C₂₃	—	4	2	no	—
206	C₂²×C₂₃	#386	1	1	yes	characteristic
207	C₂²×C₂₃	#447	1	1	yes	characteristic
208	C₂²×C₂₃	#449	1	1	yes	characteristic
209	C₂²×C₂₃	#450	1	1	yes	characteristic
210	C₂²×C₂₃	#451	1	1	yes	characteristic
211	C₂²×C₂₃	#455	1	1	yes	characteristic
212	C₂²×C₂₃	#456	1	1	yes	characteristic
213	C₄×C₂₃	—	2	1	no	—
214	C₄×C₂₃	—	2	1	no	—
215	C₄×C₂₃	—	4	2	no	—
216	C₄×C₂₃	—	4	2	no	—
217	C₄×C₂₃	—	8	2	no	—
218	C₄×C₂₃	—	8	2	no	—
219	C₄×C₂₃	—	8	1	no	—
220	C₄×C₂₃	C₂×#103	1	1	yes	characteristic
221	C₄×C₂₃	#417	1	1	yes	characteristic
222	C₄×C₂₃	C₄×D₈	1	1	yes	characteristic
223	C₄×C₂₃	#442	1	1	yes	characteristic
224	C₄×C₂₃	#459	1	1	yes	characteristic
225	C₄×C₂₃	#463	1	1	yes	characteristic
226	C₄×C₂₃	#483	1	1	yes	characteristic
227	C₄×C₂₃	#484	1	1	yes	characteristic
228	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
229	C₂×C₄×D₄	—	2	1	no	—
230	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
231	C₂²×C₄⋊C₄	—	2	1	no	—
232	#474	—	2	2	no	—
233	#474	C₄×C₂₃	1	4	yes	complemented
234	C₂×#107	—	2	1	no	—
235	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
236	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
237	C₂×#103	—	2	1	no	—
238	C₂×#103	—	2	1	no	—
239	C₂×#102	—	2	1	no	—
240	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
241	#389	—	2	1	no	—
242	#385	—	2	2	no	—
243	#384	C₂²×C₂₃	1	1	yes	characteristic
244	#380	—	2	1	no	—
245	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
246	#340	—	2	2	no	—
247	D₄⋊C₈	—	2	2	no	—
248	C₂×C₂₃	—	4	4	no	—
249	C₂×C₂₃	#1639	1	1	yes	characteristic
250	C₂×C₂₃	#1644	1	1	yes	characteristic
251	C₂×C₂₃	#1649	1	1	yes	characteristic
252	C₂×C₂₃	#1654	1	1	yes	characteristic
253	C₂×C₂₃	#1872	1	1	yes	characteristic
254	C₂×C₂₃	#1879	1	1	yes	characteristic
255	C₂×C₂₃	#1882	1	1	yes	characteristic
256	C₂²×Q₈	—	2	1	no	—
257	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
258	C₂³×C₄	—	2	1	no	—
259	C₂³×C₄	—	2	2	no	—
260	C₂²×C₈	—	2	1	no	—
261	C₂²×C₈	—	2	1	no	—
262	C₂²×C₈	—	2	1	no	—
263	C₄⋊Q₈	—	2	1	no	—
264	C₄⋊Q₈	—	2	1	no	—
265	C₄⋊Q₈	—	2	1	no	—
266	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	complemented
267	C₄×D₄	—	2	1	no	—
268	C₄×D₄	—	2	1	no	—
269	C₄×D₄	—	4	2	no	—
270	C₄×D₄	—	4	2	no	—
271	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
272	C₄×D₄	D₄×C₂₃	1	2	yes	—
273	C₂×C₄⋊C₄	—	2	1	no	—
274	C₂×C₄⋊C₄	—	2	1	no	—
275	C₂×C₄⋊C₄	—	2	2	no	—
276	C₂×C₄⋊C₄	—	2	1	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	4	2	no	—
279	C₂×C₄⋊C₄	—	4	2	no	—
280	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
281	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
282	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
283	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
284	C₂×C₂²⋊C₄	—	2	1	no	—
285	C₂×C₂²⋊C₄	—	2	2	no	—
286	C₂×C₄²	—	2	1	no	—
287	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
288	#107	—	2	4	no	—
289	C₄⋊C₈	—	2	1	no	—
290	C₄⋊C₈	—	2	1	no	—
291	C₄⋊C₈	—	2	1	no	—
292	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	complemented
293	#103	—	4	2	no	—
294	#103	—	4	2	no	—
295	#102	—	2	4	no	—
296	C₄×C₈	—	2	1	no	—
297	C₄×C₈	—	2	1	no	—
298	#95	—	2	2	no	—
299	#95	—	2	1	no	—
300	C₂₃	#8981	1	1	yes	characteristic, direct product factor, Sylow
301	C₂⁴	—	2	1	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	—	4	1	no	—
304	C₂×Q₈	—	4	1	no	—
305	C₂×Q₈	—	4	1	no	—
306	C₂×Q₈	—	4	1	no	—
307	C₂×Q₈	—	4	1	no	—
308	C₂×D₄	—	2	1	no	—
309	C₂×D₄	—	2	1	no	—
310	C₂×D₄	—	2	1	no	—
311	C₂×D₄	C₂²⋊C₄×C₂₃	1	2	yes	—
312	C₂×D₄	SD₁₆×C₂₃	1	2	yes	—
313	C₂×D₄	Q₁₆×C₂₃	1	2	yes	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	4	4	no	—
321	C₂²×C₄	—	4	1	no	—
322	C₂²×C₄	—	4	1	no	—
323	C₂²×C₄	—	4	1	no	—
324	C₂²×C₄	—	4	1	no	—
325	C₂²×C₄	—	4	4	no	—
326	C₂²×C₄	—	4	1	no	—
327	C₂²×C₄	—	8	2	no	—
328	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
329	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
330	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	4	1	no	—
334	C₂×C₈	—	4	1	no	—
335	C₂×C₈	—	4	1	no	—
336	C₂×C₈	—	4	1	no	—
337	C₂×C₈	—	4	1	no	—
338	C₂×C₈	—	4	1	no	—
339	C₂×C₈	—	4	1	no	—
340	C₄⋊C₄	—	2	1	no	—
341	C₄⋊C₄	—	2	1	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	4	2	no	—
347	C₄⋊C₄	—	4	4	no	—
348	C₄⋊C₄	—	4	1	no	—
349	C₄⋊C₄	—	4	1	no	—
350	C₄⋊C₄	—	4	1	no	—
351	C₄⋊C₄	—	4	1	no	—
352	C₄⋊C₄	—	4	1	no	—
353	C₄⋊C₄	—	4	1	no	—
354	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
355	C₄⋊C₄	D₈×C₂₃	1	2	yes	—
356	C₂²⋊C₄	—	2	2	no	—
357	C₂²⋊C₄	—	2	2	no	—
358	C₂²⋊C₄	—	4	4	no	—
359	C₄²	—	4	1	no	—
360	C₄²	—	4	1	no	—
361	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
362	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
363	C₄²	#41×C₂₃	1	1	yes	characteristic
364	C₄²	#41×C₂₃	1	1	yes	characteristic
365	C₂³	—	2	2	no	—
366	C₂³	—	2	2	no	—
367	C₂³	—	2	2	no	—
368	C₂³	—	4	1	no	—
369	C₂³	—	4	1	no	—
370	C₂³	—	4	1	no	—
371	C₂³	—	4	1	no	—
372	C₂³	C₂₃×#120	1	1	yes	characteristic
373	Q₈	—	4	2	no	—
374	Q₈	—	4	2	no	—
375	Q₈	—	8	1	no	—
376	Q₈	—	8	1	no	—
377	D₄	—	2	2	no	—
378	D₄	—	2	2	no	—
379	D₄	—	2	2	no	—
380	D₄	C₂₃×#103	1	4	yes	complemented
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	8	8	no	—
401	C₂×C₄	—	8	1	no	—
402	C₂×C₄	—	8	1	no	—
403	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
404	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
405	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
406	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
407	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
408	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
409	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
410	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
411	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
412	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
413	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
414	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
415	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
416	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
417	C₈	—	4	2	no	—
418	C₈	—	4	2	no	—
419	C₈	—	8	1	no	—
420	C₂²	—	2	4	no	—
421	C₂²	—	4	2	no	—
422	C₂²	—	4	2	no	—
423	C₂²	—	4	2	no	—
424	C₂²	—	4	2	no	—
425	C₂²	—	4	2	no	—
426	C₂²	—	4	2	no	—
427	C₂²	C₂₃×#386	1	1	yes	characteristic
428	C₂²	C₂₃×#447	1	1	yes	characteristic
429	C₂²	C₂₃×#449	1	1	yes	characteristic
430	C₂²	C₂₃×#450	1	1	yes	characteristic
431	C₂²	C₂₃×#451	1	1	yes	characteristic
432	C₂²	C₂₃×#455	1	1	yes	characteristic
433	C₂²	C₂₃×#456	1	1	yes	characteristic
434	C₄	—	2	1	no	—
435	C₄	—	2	1	no	—
436	C₄	—	4	2	no	—
437	C₄	—	4	2	no	—
438	C₄	—	8	2	no	—
439	C₄	—	8	1	no	—
440	C₄	—	8	2	no	—
441	C₄	C₂×C₂₃×#103	1	1	yes	characteristic
442	C₄	C₂₃×#417	1	1	yes	characteristic
443	C₄	C₄×D₈×C₂₃	1	1	yes	characteristic
444	C₄	C₂₃×#442	1	1	yes	characteristic
445	C₄	C₂₃×#459	1	1	yes	characteristic
446	C₄	C₂₃×#463	1	1	yes	characteristic
447	C₄	C₂₃×#483	1	1	yes	characteristic
448	C₄	C₂₃×#484	1	1	yes	characteristic
449	C₂	—	4	4	no	—
450	C₂	C₂₃×#1639	1	1	yes	characteristic
451	C₂	C₂₃×#1644	1	1	yes	characteristic
452	C₂	C₂₃×#1649	1	1	yes	characteristic
453	C₂	C₂₃×#1654	1	1	yes	characteristic
454	C₂	C₂₃×#1872	1	1	yes	characteristic
455	C₂	C₂₃×#1879	1	1	yes	characteristic
456	C₂	C₂₃×#1882	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×#1990	C₂	semidirect
4. C₂₃×#1981	C₂	semidirect
6. C₂₃×#1296	C₂	semidirect
7. C₂₃×#1279	C₂	semidirect
13. C₂₃×#474	C₄	semidirect
38. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
64. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
136. #8981	C₂₃	direct (N × H)
152. D₄×C₂₃	#103	semidirect
194. #1990	C₂×C₂₃	semidirect
195. #1981	C₂×C₂₃	semidirect
197. #1296	C₂×C₂₃	semidirect
198. #1279	C₂×C₂₃	semidirect
233. #474	C₄×C₂₃	semidirect
266. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
292. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
300. C₂₃	#8981	direct (N × H)
380. 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×#474	2
D₄×C₄⋊C₄×C₂₃	2
C₂₃×#1990	2
C₂₃×#1981	2
C₂×C₂₃×D₄⋊C₈	2
C₂₃×#1296	2
C₂₃×#1279	2
#8981	23

Subgroups — C23×#8981 #19156607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8981 #19156607