Subgroups — C₂₃×#8449 #19156075

Order 5888.

788 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#556	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1803	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1802	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#556	—	2	2	no	—
9	C₂₃×#556	—	2	2	no	—
10	C₂₃×#556	—	2	2	no	—
11	C₂₃×#556	C₄	1	4	yes	complemented
12	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#126	—	2	2	no	—
14	C₂×C₂₃×#124	—	2	1	no	—
15	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
16	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
17	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#117	—	2	1	no	—
20	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#103	—	2	1	no	—
23	C₂×C₂₃×#102	—	2	1	no	—
24	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#384	C₂²	1	1	yes	characteristic
26	C₂₃×#340	—	2	2	no	—
27	C₂₃×#340	—	2	1	no	—
28	C₂₃×#340	—	2	1	no	—
29	C₂₃×#332	—	2	2	no	—
30	C₂₃×Q₈⋊C₈	—	2	2	no	—
31	C₂₃×D₄⋊C₈	—	2	2	no	—
32	C₂²×Q₈×C₂₃	—	2	1	no	—
33	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
34	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₂₃	—	2	1	no	—
36	C₂²×C₈×C₂₃	—	2	1	no	—
37	C₂²×C₈×C₂₃	—	2	1	no	—
38	C₂²×C₈×C₂₃	—	2	1	no	—
39	C₂₃×C₄⋊Q₈	—	2	1	no	—
40	C₂₃×C₄⋊Q₈	—	2	1	no	—
41	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
42	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
43	C₂₃×#126	—	2	8	no	—
44	C₂₃×#126	—	2	8	no	—
45	C₂₃×#124	—	2	4	no	—
46	C₂₃×#124	—	2	4	no	—
47	C₂₃×C₂²⋊Q₈	—	2	4	no	—
48	C₂₃×C₂²⋊Q₈	—	2	4	no	—
49	C₄×Q₈×C₂₃	—	2	4	no	—
50	C₄×Q₈×C₂₃	—	2	4	no	—
51	C₄×Q₈×C₂₃	—	2	1	no	—
52	C₄×Q₈×C₂₃	—	2	1	no	—
53	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
54	C₄×Q₈×C₂₃	D₄	1	2	yes	—
55	C₄×D₄×C₂₃	—	2	1	no	—
56	C₄×D₄×C₂₃	—	2	1	no	—
57	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
58	C₄×D₄×C₂₃	D₄	1	2	yes	—
59	C₂₃×#117	—	2	4	no	—
60	C₂₃×#117	—	2	4	no	—
61	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
63	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
64	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
65	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
67	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
70	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
73	C₂×C₄²×C₂₃	—	2	2	no	—
74	C₂×C₄²×C₂₃	—	2	1	no	—
75	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
76	C₂₃×C₄⋊C₈	—	2	1	no	—
77	C₂₃×C₄⋊C₈	—	2	1	no	—
78	C₂₃×C₄⋊C₈	—	2	1	no	—
79	C₂₃×C₄⋊C₈	—	2	1	no	—
80	C₂₃×#103	—	4	2	no	—
81	C₂₃×#102	—	4	2	no	—
82	C₄×C₈×C₂₃	—	2	1	no	—
83	C₄×C₈×C₂₃	—	2	1	no	—
84	C₂⁴×C₂₃	—	2	1	no	—
85	C₂×Q₈×C₂₃	—	2	2	no	—
86	C₂×Q₈×C₂₃	—	2	1	no	—
87	C₂×Q₈×C₂₃	—	2	2	no	—
88	C₂×Q₈×C₂₃	—	2	1	no	—
89	C₂×Q₈×C₂₃	—	2	1	no	—
90	C₂×Q₈×C₂₃	—	2	1	no	—
91	C₂×Q₈×C₂₃	—	2	1	no	—
92	C₂×Q₈×C₂₃	—	4	1	no	—
93	C₂×Q₈×C₂₃	—	4	1	no	—
94	C₂×Q₈×C₂₃	—	4	1	no	—
95	C₂×Q₈×C₂₃	—	4	1	no	—
96	C₂×Q₈×C₂₃	C₂²⋊C₄	1	2	yes	—
97	C₂×D₄×C₂₃	—	2	1	no	—
98	C₂×D₄×C₂₃	—	2	1	no	—
99	C₂×D₄×C₂₃	—	2	1	no	—
100	C₂×D₄×C₂₃	—	2	1	no	—
101	C₂×D₄×C₂₃	—	2	1	no	—
102	C₂×D₄×C₂₃	C₂²⋊C₄	1	2	yes	—
103	C₂²×C₄×C₂₃	—	2	1	no	—
104	C₂²×C₄×C₂₃	—	2	1	no	—
105	C₂²×C₄×C₂₃	—	2	1	no	—
106	C₂²×C₄×C₂₃	—	2	2	no	—
107	C₂²×C₄×C₂₃	—	2	2	no	—
108	C₂²×C₄×C₂₃	—	2	2	no	—
109	C₂²×C₄×C₂₃	—	4	1	no	—
110	C₂²×C₄×C₂₃	—	4	1	no	—
111	C₂²×C₄×C₂₃	—	4	1	no	—
112	C₂²×C₄×C₂₃	—	4	1	no	—
113	C₂²×C₄×C₂₃	—	4	1	no	—
114	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₂₃	—	2	2	no	—
118	C₂×C₈×C₂₃	—	4	1	no	—
119	C₂×C₈×C₂₃	—	4	1	no	—
120	C₂×C₈×C₂₃	—	4	1	no	—
121	C₂×C₈×C₂₃	—	4	1	no	—
122	C₂×C₈×C₂₃	—	4	1	no	—
123	C₂×C₈×C₂₃	—	4	1	no	—
124	C₂×C₈×C₂₃	—	4	1	no	—
125	C₂×C₈×C₂₃	—	4	1	no	—
126	C₄⋊C₄×C₂₃	—	2	8	no	—
127	C₄⋊C₄×C₂₃	—	2	2	no	—
128	C₄⋊C₄×C₂₃	—	2	2	no	—
129	C₄⋊C₄×C₂₃	—	2	2	no	—
130	C₄⋊C₄×C₂₃	—	2	2	no	—
131	C₄⋊C₄×C₂₃	—	2	1	no	—
132	C₄⋊C₄×C₂₃	—	2	1	no	—
133	C₄⋊C₄×C₂₃	—	2	2	no	—
134	C₄⋊C₄×C₂₃	—	4	2	no	—
135	C₄⋊C₄×C₂₃	—	4	1	no	—
136	C₄⋊C₄×C₂₃	—	4	1	no	—
137	C₄⋊C₄×C₂₃	—	4	1	no	—
138	C₄⋊C₄×C₂₃	—	4	1	no	—
139	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
140	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
141	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
142	C₂²⋊C₄×C₂₃	—	2	8	no	—
143	C₂²⋊C₄×C₂₃	—	2	8	no	—
144	C₂²⋊C₄×C₂₃	—	2	2	no	—
145	C₂²⋊C₄×C₂₃	—	2	2	no	—
146	C₄²×C₂₃	—	2	8	no	—
147	C₄²×C₂₃	—	2	2	no	—
148	C₄²×C₂₃	—	2	2	no	—
149	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
150	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
153	#8449	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂³×C₂₃	—	2	2	no	—
155	C₂³×C₂₃	—	4	1	no	—
156	C₂³×C₂₃	—	4	1	no	—
157	C₂³×C₂₃	—	4	1	no	—
158	C₂³×C₂₃	—	4	1	no	—
159	C₂³×C₂₃	—	4	1	no	—
160	C₂³×C₂₃	—	4	1	no	—
161	C₂³×C₂₃	#120	1	1	yes	characteristic
162	Q₈×C₂₃	—	2	2	no	—
163	Q₈×C₂₃	—	2	2	no	—
164	Q₈×C₂₃	—	4	2	no	—
165	Q₈×C₂₃	—	4	2	no	—
166	Q₈×C₂₃	—	4	1	no	—
167	Q₈×C₂₃	—	4	2	no	—
168	Q₈×C₂₃	—	4	1	no	—
169	Q₈×C₂₃	—	4	2	no	—
170	D₄×C₂₃	—	2	2	no	—
171	D₄×C₂₃	—	2	2	no	—
172	D₄×C₂₃	—	4	1	no	—
173	D₄×C₂₃	—	4	1	no	—
174	C₂×C₄×C₂₃	—	2	1	no	—
175	C₂×C₄×C₂₃	—	2	1	no	—
176	C₂×C₄×C₂₃	—	2	2	no	—
177	C₂×C₄×C₂₃	—	2	2	no	—
178	C₂×C₄×C₂₃	—	2	2	no	—
179	C₂×C₄×C₂₃	—	2	4	no	—
180	C₂×C₄×C₂₃	—	4	1	no	—
181	C₂×C₄×C₂₃	—	4	1	no	—
182	C₂×C₄×C₂₃	—	4	2	no	—
183	C₂×C₄×C₂₃	—	4	1	no	—
184	C₂×C₄×C₂₃	—	4	1	no	—
185	C₂×C₄×C₂₃	—	4	2	no	—
186	C₂×C₄×C₂₃	—	4	1	no	—
187	C₂×C₄×C₂₃	—	4	1	no	—
188	C₂×C₄×C₂₃	—	4	2	no	—
189	C₂×C₄×C₂₃	—	4	2	no	—
190	C₂×C₄×C₂₃	—	4	2	no	—
191	C₂×C₄×C₂₃	—	4	2	no	—
192	C₂×C₄×C₂₃	—	8	1	no	—
193	C₂×C₄×C₂₃	—	8	1	no	—
194	C₂×C₄×C₂₃	—	8	1	no	—
195	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
199	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
200	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
201	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
202	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
203	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
204	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
205	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
207	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
208	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
209	C₈×C₂₃	—	4	2	no	—
210	C₈×C₂₃	—	8	1	no	—
211	C₈×C₂₃	—	8	1	no	—
212	C₂×#556	C₂×C₂₃	1	1	yes	characteristic
213	#1990	C₂×C₂₃	1	1	yes	characteristic, complemented
214	#1803	C₂×C₂₃	1	1	yes	characteristic, complemented
215	#1802	C₂×C₂₃	1	1	yes	characteristic
216	C₂×#332	C₂×C₂₃	1	1	yes	characteristic, complemented
217	C₂×Q₈⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
218	C₂×D₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
219	C₂²×C₂₃	—	4	2	no	—
220	C₂²×C₂₃	—	4	2	no	—
221	C₂²×C₂₃	—	4	2	no	—
222	C₂²×C₂₃	—	8	1	no	—
223	C₂²×C₂₃	—	8	1	no	—
224	C₂²×C₂₃	—	8	1	no	—
225	C₂²×C₂₃	—	8	1	no	—
226	C₂²×C₂₃	#379	1	1	yes	characteristic
227	C₂²×C₂₃	#449	1	1	yes	characteristic
228	C₂²×C₂₃	#449	1	1	yes	characteristic
229	C₂²×C₂₃	#452	1	1	yes	characteristic
230	C₂²×C₂₃	#452	1	1	yes	characteristic
231	C₂²×C₂₃	#454	1	1	yes	characteristic
232	C₂²×C₂₃	#456	1	1	yes	characteristic
233	C₄×C₂₃	—	2	1	no	—
234	C₄×C₂₃	—	2	1	no	—
235	C₄×C₂₃	—	4	2	no	—
236	C₄×C₂₃	—	4	2	no	—
237	C₄×C₂₃	—	4	2	no	—
238	C₄×C₂₃	—	4	4	no	—
239	C₄×C₂₃	—	16	1	no	—
240	C₄×C₂₃	#416	1	1	yes	characteristic
241	C₄×C₂₃	#416	1	1	yes	characteristic
242	C₄×C₂₃	#417	1	1	yes	characteristic
243	C₄×C₂₃	#417	1	1	yes	characteristic
244	C₄×C₂₃	#449	1	1	yes	characteristic
245	C₄×C₂₃	#449	1	1	yes	characteristic
246	C₄×C₂₃	#452	1	1	yes	characteristic
247	C₄×C₂₃	#452	1	1	yes	characteristic
248	#556	—	2	2	no	—
249	#556	—	2	2	no	—
250	#556	—	2	2	no	—
251	#556	C₄×C₂₃	1	4	yes	complemented
252	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
253	C₂×#126	—	2	2	no	—
254	C₂×#124	—	2	1	no	—
255	C₂×C₂²⋊Q₈	—	2	1	no	—
256	C₂×C₄×Q₈	—	2	1	no	—
257	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
258	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
259	C₂×#117	—	2	1	no	—
260	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
261	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
262	C₂×#103	—	2	1	no	—
263	C₂×#102	—	2	1	no	—
264	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
265	#384	C₂²×C₂₃	1	1	yes	characteristic
266	#340	—	2	1	no	—
267	#340	—	2	1	no	—
268	#340	—	2	2	no	—
269	#332	—	2	2	no	—
270	Q₈⋊C₈	—	2	2	no	—
271	D₄⋊C₈	—	2	2	no	—
272	C₂×C₂₃	—	8	2	no	—
273	C₂×C₂₃	#1628	1	1	yes	characteristic
274	C₂×C₂₃	#1638	1	1	yes	characteristic
275	C₂×C₂₃	#1650	1	1	yes	characteristic
276	C₂×C₂₃	#1652	1	1	yes	characteristic
277	C₂×C₂₃	#1788	1	1	yes	characteristic
278	C₂×C₂₃	#1788	1	1	yes	characteristic
279	C₂×C₂₃	#1793	1	1	yes	characteristic
280	C₂²×Q₈	—	2	1	no	—
281	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
282	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
283	C₂³×C₄	—	2	1	no	—
284	C₂²×C₈	—	2	1	no	—
285	C₂²×C₈	—	2	1	no	—
286	C₂²×C₈	—	2	1	no	—
287	C₄⋊Q₈	—	2	1	no	—
288	C₄⋊Q₈	—	2	1	no	—
289	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
290	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
291	#126	—	2	8	no	—
292	#126	—	2	8	no	—
293	#124	—	2	4	no	—
294	#124	—	2	4	no	—
295	C₂²⋊Q₈	—	2	4	no	—
296	C₂²⋊Q₈	—	2	4	no	—
297	C₄×Q₈	—	2	4	no	—
298	C₄×Q₈	—	2	4	no	—
299	C₄×Q₈	—	2	1	no	—
300	C₄×Q₈	—	2	1	no	—
301	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
302	C₄×Q₈	D₄×C₂₃	1	2	yes	—
303	C₄×D₄	—	2	1	no	—
304	C₄×D₄	—	2	1	no	—
305	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
306	C₄×D₄	D₄×C₂₃	1	2	yes	—
307	#117	—	2	4	no	—
308	#117	—	2	4	no	—
309	C₂×C₄⋊C₄	—	2	2	no	—
310	C₂×C₄⋊C₄	—	2	1	no	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	—	2	1	no	—
313	C₂×C₄⋊C₄	—	2	1	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
316	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
317	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
318	C₂×C₂²⋊C₄	—	2	2	no	—
319	C₂×C₂²⋊C₄	—	2	2	no	—
320	C₂×C₂²⋊C₄	—	2	1	no	—
321	C₂×C₄²	—	2	2	no	—
322	C₂×C₄²	—	2	1	no	—
323	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
324	C₄⋊C₈	—	2	1	no	—
325	C₄⋊C₈	—	2	1	no	—
326	C₄⋊C₈	—	2	1	no	—
327	C₄⋊C₈	—	2	1	no	—
328	#103	—	4	2	no	—
329	#102	—	4	2	no	—
330	C₄×C₈	—	2	1	no	—
331	C₄×C₈	—	2	1	no	—
332	C₂₃	#8449	1	1	yes	characteristic, direct product factor, Sylow
333	C₂⁴	—	2	1	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	1	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	—	2	1	no	—
338	C₂×Q₈	—	2	1	no	—
339	C₂×Q₈	—	2	1	no	—
340	C₂×Q₈	—	2	1	no	—
341	C₂×Q₈	—	4	1	no	—
342	C₂×Q₈	—	4	1	no	—
343	C₂×Q₈	—	4	1	no	—
344	C₂×Q₈	—	4	1	no	—
345	C₂×Q₈	C₂²⋊C₄×C₂₃	1	2	yes	—
346	C₂×D₄	—	2	1	no	—
347	C₂×D₄	—	2	1	no	—
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	—	2	1	no	—
351	C₂×D₄	C₂²⋊C₄×C₂₃	1	2	yes	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	4	1	no	—
359	C₂²×C₄	—	4	1	no	—
360	C₂²×C₄	—	4	1	no	—
361	C₂²×C₄	—	4	1	no	—
362	C₂²×C₄	—	4	1	no	—
363	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
364	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
365	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	4	1	no	—
368	C₂×C₈	—	4	1	no	—
369	C₂×C₈	—	4	1	no	—
370	C₂×C₈	—	4	1	no	—
371	C₂×C₈	—	4	1	no	—
372	C₂×C₈	—	4	1	no	—
373	C₂×C₈	—	4	1	no	—
374	C₂×C₈	—	4	1	no	—
375	C₄⋊C₄	—	2	8	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	4	2	no	—
384	C₄⋊C₄	—	4	1	no	—
385	C₄⋊C₄	—	4	1	no	—
386	C₄⋊C₄	—	4	1	no	—
387	C₄⋊C₄	—	4	1	no	—
388	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
389	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
390	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
391	C₂²⋊C₄	—	2	8	no	—
392	C₂²⋊C₄	—	2	8	no	—
393	C₂²⋊C₄	—	2	2	no	—
394	C₂²⋊C₄	—	2	2	no	—
395	C₄²	—	2	8	no	—
396	C₄²	—	2	2	no	—
397	C₄²	—	2	2	no	—
398	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
399	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
400	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
401	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
402	C₂³	—	2	2	no	—
403	C₂³	—	4	1	no	—
404	C₂³	—	4	1	no	—
405	C₂³	—	4	1	no	—
406	C₂³	—	4	1	no	—
407	C₂³	—	4	1	no	—
408	C₂³	—	4	1	no	—
409	C₂³	C₂₃×#120	1	1	yes	characteristic
410	Q₈	—	2	2	no	—
411	Q₈	—	2	2	no	—
412	Q₈	—	4	2	no	—
413	Q₈	—	4	2	no	—
414	Q₈	—	4	1	no	—
415	Q₈	—	4	2	no	—
416	Q₈	—	4	1	no	—
417	Q₈	—	4	2	no	—
418	D₄	—	2	2	no	—
419	D₄	—	2	2	no	—
420	D₄	—	4	1	no	—
421	D₄	—	4	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	8	1	no	—
441	C₂×C₄	—	8	1	no	—
442	C₂×C₄	—	8	1	no	—
443	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
444	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
445	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
446	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
447	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
448	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
449	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
450	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
451	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
452	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
453	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
454	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
455	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
456	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
457	C₈	—	4	2	no	—
458	C₈	—	8	1	no	—
459	C₈	—	8	1	no	—
460	C₂²	—	4	2	no	—
461	C₂²	—	4	2	no	—
462	C₂²	—	4	2	no	—
463	C₂²	—	8	1	no	—
464	C₂²	—	8	1	no	—
465	C₂²	—	8	1	no	—
466	C₂²	—	8	1	no	—
467	C₂²	C₂₃×#379	1	1	yes	characteristic
468	C₂²	C₂₃×#449	1	1	yes	characteristic
469	C₂²	C₂₃×#449	1	1	yes	characteristic
470	C₂²	C₂₃×#452	1	1	yes	characteristic
471	C₂²	C₂₃×#452	1	1	yes	characteristic
472	C₂²	C₂₃×#454	1	1	yes	characteristic
473	C₂²	C₂₃×#456	1	1	yes	characteristic
474	C₄	—	2	1	no	—
475	C₄	—	2	1	no	—
476	C₄	—	4	2	no	—
477	C₄	—	4	2	no	—
478	C₄	—	4	4	no	—
479	C₄	—	4	2	no	—
480	C₄	—	16	1	no	—
481	C₄	C₂₃×#416	1	1	yes	characteristic
482	C₄	C₂₃×#416	1	1	yes	characteristic
483	C₄	C₂₃×#417	1	1	yes	characteristic
484	C₄	C₂₃×#417	1	1	yes	characteristic
485	C₄	C₂₃×#449	1	1	yes	characteristic
486	C₄	C₂₃×#449	1	1	yes	characteristic
487	C₄	C₂₃×#452	1	1	yes	characteristic
488	C₄	C₂₃×#452	1	1	yes	characteristic
489	C₂	—	8	2	no	—
490	C₂	C₂₃×#1628	1	1	yes	characteristic
491	C₂	C₂₃×#1638	1	1	yes	characteristic
492	C₂	C₂₃×#1650	1	1	yes	characteristic
493	C₂	C₂₃×#1652	1	1	yes	characteristic
494	C₂	C₂₃×#1788	1	1	yes	characteristic
495	C₂	C₂₃×#1788	1	1	yes	characteristic
496	C₂	C₂₃×#1793	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×#1990	C₂	semidirect
3. C₂₃×#1803	C₂	semidirect
5. C₂×C₂₃×#332	C₂	semidirect
6. C₂×C₂₃×Q₈⋊C₈	C₂	semidirect
11. C₂₃×#556	C₄	semidirect
153. #8449	C₂₃	direct (N × H)
213. #1990	C₂×C₂₃	semidirect
214. #1803	C₂×C₂₃	semidirect
216. C₂×#332	C₂×C₂₃	semidirect
217. C₂×Q₈⋊C₈	C₂×C₂₃	semidirect
251. #556	C₄×C₂₃	semidirect
332. C₂₃	#8449	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×#556	2
C₂₃×#1990	2
C₂₃×#1803	2
C₂₃×#1802	2
C₂×C₂₃×#332	2
C₂×C₂₃×Q₈⋊C₈	2
C₂×C₂₃×D₄⋊C₈	2
#8449	23

Subgroups — C23×#8449 #19156075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8449 #19156075