Subgroups — C₄×C₂₃×#489 #19169275

Order 5888.

886 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#489	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1993	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1959	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1936	C₂	1	2	yes	maximal
7	C₂₃×#1935	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1836	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂₃×#103	C₂	1	2	yes	maximal, complemented
10	C₄×C₂₃×#102	C₂	1	2	yes	maximal
11	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
15	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#489	C₄	1	8	yes	direct product factor
18	C₂×C₂₃×#103	C₂²	1	2	yes	—
19	C₂×C₂₃×#103	C₂²	1	2	yes	—
20	C₂×C₂₃×#102	C₂²	1	2	yes	—
21	C₂×C₂₃×#102	C₂²	1	2	yes	—
22	C₂×C₂₃×#97	C₂²	1	2	yes	—
23	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
26	C₂₃×#391	—	2	1	no	—
27	C₂₃×#391	C₂²	1	1	yes	characteristic
28	C₂₃×#390	C₂²	1	1	yes	characteristic
29	C₂₃×#389	—	2	1	no	—
30	C₂₃×#389	C₂²	1	2	yes	—
31	C₂₃×#388	—	2	1	no	—
32	C₂₃×#388	—	2	1	no	—
33	C₂₃×#384	C₂²	1	1	yes	characteristic
34	C₂₃×#382	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
36	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
37	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
38	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
39	C₄³×C₂₃	C₂²	1	1	yes	characteristic
40	C₂₃×#340	C₂²	1	2	yes	—
41	C₂₃×#340	C₂²	1	2	yes	—
42	C₂₃×#340	C₂²	1	2	yes	—
43	C₂₃×#340	C₂²	1	2	yes	—
44	C₂²×Q₈×C₂₃	—	2	1	no	—
45	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₂₃	—	2	1	no	—
48	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
50	C₂₃×C₄⋊Q₈	—	2	2	no	—
51	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
52	C₂₃×#124	—	2	2	no	—
53	C₂₃×#124	C₂×C₄	1	4	yes	—
54	C₄×Q₈×C₂₃	—	2	4	no	—
55	C₄×Q₈×C₂₃	—	4	4	no	—
56	C₄×D₄×C₂₃	—	2	4	no	—
57	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
71	C₂×C₄²×C₂₃	—	2	1	no	—
72	C₂×C₄²×C₂₃	—	2	2	no	—
73	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
80	C₂₃×#103	—	2	4	no	—
81	C₂₃×#103	C₂×C₄	1	8	yes	complemented
82	C₂₃×#102	—	2	4	no	—
83	C₂₃×#102	C₂×C₄	1	8	yes	—
84	C₂₃×#97	—	2	4	no	—
85	C₂₃×#97	—	2	2	no	—
86	C₂₃×#97	C₂×C₄	1	4	yes	complemented
87	C₄×C₈×C₂₃	—	2	4	no	—
88	C₂₃×#95	—	2	1	no	—
89	C₂₃×#95	—	2	1	no	—
90	C₂₃×#95	—	2	1	no	—
91	C₂₃×#95	—	2	1	no	—
92	C₂₃×#95	—	2	2	no	—
93	C₂₃×#95	—	2	2	no	—
94	C₂⁴×C₂₃	—	2	1	no	—
95	C₂×Q₈×C₂₃	—	2	4	no	—
96	C₂×Q₈×C₂₃	—	2	1	no	—
97	C₂×Q₈×C₂₃	—	2	2	no	—
98	C₂×Q₈×C₂₃	—	2	2	no	—
99	C₂×Q₈×C₂₃	—	4	2	no	—
100	C₂×Q₈×C₂₃	—	4	2	no	—
101	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
102	C₂×D₄×C₂₃	—	2	2	no	—
103	C₂×D₄×C₂₃	—	2	2	no	—
104	C₂×D₄×C₂₃	—	2	1	no	—
105	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₂₃	—	2	1	no	—
107	C₂²×C₄×C₂₃	—	2	2	no	—
108	C₂²×C₄×C₂₃	—	2	1	no	—
109	C₂²×C₄×C₂₃	—	2	2	no	—
110	C₂²×C₄×C₂₃	—	2	1	no	—
111	C₂²×C₄×C₂₃	—	4	2	no	—
112	C₂²×C₄×C₂₃	—	4	2	no	—
113	C₂²×C₄×C₂₃	—	4	2	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	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
121	C₂×C₈×C₂₃	—	2	2	no	—
122	C₂×C₈×C₂₃	—	2	2	no	—
123	C₂×C₈×C₂₃	—	2	2	no	—
124	C₂×C₈×C₂₃	—	2	2	no	—
125	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
126	C₂×C₈×C₂₃	#41	1	4	yes	—
127	C₄⋊C₄×C₂₃	—	2	4	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	2	no	—
132	C₄⋊C₄×C₂₃	—	2	2	no	—
133	C₄⋊C₄×C₂₃	—	4	2	no	—
134	C₄⋊C₄×C₂₃	—	4	2	no	—
135	C₄⋊C₄×C₂₃	—	4	4	no	—
136	C₄⋊C₄×C₂₃	—	4	2	no	—
137	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
138	C₂²⋊C₄×C₂₃	—	2	4	no	—
139	C₂²⋊C₄×C₂₃	—	2	4	no	—
140	C₂²⋊C₄×C₂₃	—	4	2	no	—
141	C₂²⋊C₄×C₂₃	—	4	2	no	—
142	C₂²⋊C₄×C₂₃	—	4	2	no	—
143	C₄²×C₂₃	—	2	2	no	—
144	C₄²×C₂₃	—	2	2	no	—
145	C₄²×C₂₃	—	2	2	no	—
146	C₄²×C₂₃	—	4	1	no	—
147	C₄²×C₂₃	—	4	1	no	—
148	C₄²×C₂₃	—	4	2	no	—
149	C₄²×C₂₃	—	4	4	no	—
150	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
151	C₄²×C₂₃	#41	1	2	yes	—
152	C₄²×C₂₃	#41	1	4	yes	—
153	C₄×#489	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	C₂²×D₄	1	1	yes	characteristic
162	Q₈×C₂₃	—	2	4	no	—
163	Q₈×C₂₃	—	4	8	no	—
164	Q₈×C₂₃	—	4	2	no	—
165	D₄×C₂₃	—	2	4	no	—
166	D₄×C₂₃	—	4	2	no	—
167	C₂×C₄×C₂₃	—	2	1	no	—
168	C₂×C₄×C₂₃	—	2	1	no	—
169	C₂×C₄×C₂₃	—	2	1	no	—
170	C₂×C₄×C₂₃	—	2	1	no	—
171	C₂×C₄×C₂₃	—	2	1	no	—
172	C₂×C₄×C₂₃	—	2	1	no	—
173	C₂×C₄×C₂₃	—	2	2	no	—
174	C₂×C₄×C₂₃	—	2	4	no	—
175	C₂×C₄×C₂₃	—	2	1	no	—
176	C₂×C₄×C₂₃	—	2	1	no	—
177	C₂×C₄×C₂₃	—	2	1	no	—
178	C₂×C₄×C₂₃	—	2	1	no	—
179	C₂×C₄×C₂₃	—	2	1	no	—
180	C₂×C₄×C₂₃	—	2	1	no	—
181	C₂×C₄×C₂₃	—	4	1	no	—
182	C₂×C₄×C₂₃	—	4	1	no	—
183	C₂×C₄×C₂₃	—	4	2	no	—
184	C₂×C₄×C₂₃	—	4	2	no	—
185	C₂×C₄×C₂₃	—	4	1	no	—
186	C₂×C₄×C₂₃	—	4	1	no	—
187	C₂×C₄×C₂₃	—	4	2	no	—
188	C₂×C₄×C₂₃	—	4	2	no	—
189	C₂×C₄×C₂₃	—	4	1	no	—
190	C₂×C₄×C₂₃	—	4	1	no	—
191	C₂×C₄×C₂₃	—	4	1	no	—
192	C₂×C₄×C₂₃	—	4	1	no	—
193	C₂×C₄×C₂₃	—	4	2	no	—
194	C₂×C₄×C₂₃	—	8	4	no	—
195	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
196	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
197	C₂×C₄×C₂₃	#124	1	2	yes	—
198	C₂×C₄×C₂₃	#124	1	2	yes	—
199	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
200	C₂×C₄×C₂₃	#137	1	2	yes	—
201	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
202	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
203	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
204	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
205	C₈×C₂₃	—	2	4	no	—
206	C₈×C₂₃	—	2	4	no	—
207	C₂×#489	C₂×C₂₃	1	1	yes	characteristic
208	C₄×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
209	C₄×#124	C₂×C₂₃	1	1	yes	characteristic
210	#1993	C₂×C₂₃	1	1	yes	characteristic, complemented
211	#1959	C₂×C₂₃	1	1	yes	characteristic
212	#1936	C₂×C₂₃	1	2	yes	—
213	#1935	C₂×C₂₃	1	2	yes	complemented
214	#1836	C₂×C₂₃	1	1	yes	characteristic, complemented
215	C₄×#103	C₂×C₂₃	1	2	yes	complemented
216	C₄×#102	C₂×C₂₃	1	2	yes	—
217	C₄×#97	C₂×C₂₃	1	1	yes	characteristic, complemented
218	C₂²×C₂₃	—	4	2	no	—
219	C₂²×C₂₃	—	4	2	no	—
220	C₂²×C₂₃	—	4	2	no	—
221	C₂²×C₂₃	—	8	1	no	—
222	C₂²×C₂₃	—	8	1	no	—
223	C₂²×C₂₃	—	8	1	no	—
224	C₂²×C₂₃	—	8	1	no	—
225	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
226	C₂²×C₂₃	C₂×#124	1	1	yes	characteristic
227	C₂²×C₂₃	#532	1	1	yes	characteristic
228	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
229	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
230	C₂²×C₂₃	#575	1	1	yes	characteristic
231	C₂²×C₂₃	#575	1	1	yes	characteristic
232	C₄×C₂₃	—	2	2	no	—
233	C₄×C₂₃	—	2	2	no	—
234	C₄×C₂₃	—	2	2	no	—
235	C₄×C₂₃	—	4	1	no	—
236	C₄×C₂₃	—	4	1	no	—
237	C₄×C₂₃	—	4	4	no	—
238	C₄×C₂₃	—	4	2	no	—
239	C₄×C₂₃	—	8	1	no	—
240	C₄×C₂₃	—	8	1	no	—
241	C₄×C₂₃	—	8	2	no	—
242	C₄×C₂₃	#489	1	4	yes	direct product factor
243	C₄×C₂₃	C₄×#41	1	2	yes	—
244	C₄×C₂₃	#529	1	2	yes	—
245	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
246	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
247	C₂×C₄×Q₈	—	2	1	no	—
248	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
249	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
250	#489	C₄×C₂₃	1	8	yes	direct product factor
251	C₂×#103	C₂²×C₂₃	1	2	yes	—
252	C₂×#103	C₂²×C₂₃	1	2	yes	—
253	C₂×#102	C₂²×C₂₃	1	2	yes	—
254	C₂×#102	C₂²×C₂₃	1	2	yes	—
255	C₂×#97	C₂²×C₂₃	1	2	yes	—
256	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
257	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
258	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
259	#391	—	2	1	no	—
260	#391	C₂²×C₂₃	1	1	yes	characteristic
261	#390	C₂²×C₂₃	1	1	yes	characteristic
262	#389	—	2	1	no	—
263	#389	C₂²×C₂₃	1	2	yes	—
264	#388	—	2	1	no	—
265	#388	—	2	1	no	—
266	#384	C₂²×C₂₃	1	1	yes	characteristic
267	#382	C₂²×C₂₃	1	1	yes	characteristic
268	C₄×C₄⋊C₄	—	2	1	no	—
269	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
270	C₄×C₂²⋊C₄	—	2	1	no	—
271	C₄×C₂²⋊C₄	—	2	1	no	—
272	C₄³	C₂²×C₂₃	1	1	yes	characteristic
273	#340	C₂²×C₂₃	1	2	yes	—
274	#340	C₂²×C₂₃	1	2	yes	—
275	#340	C₂²×C₂₃	1	2	yes	—
276	#340	C₂²×C₂₃	1	2	yes	—
277	C₂×C₂₃	—	8	2	no	—
278	C₂×C₂₃	C₄×#124	1	1	yes	characteristic
279	C₂×C₂₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
280	C₂×C₂₃	C₄×#137	1	1	yes	characteristic
281	C₂×C₂₃	C₂×#489	1	1	yes	characteristic
282	C₂×C₂₃	#3135	1	1	yes	characteristic
283	C₂×C₂₃	#3171	1	1	yes	characteristic
284	C₂×C₂₃	#3172	1	1	yes	characteristic
285	C₂²×Q₈	—	2	1	no	—
286	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
287	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
288	C₂³×C₄	—	2	1	no	—
289	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
290	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
291	C₄⋊Q₈	—	2	2	no	—
292	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
293	#124	—	2	2	no	—
294	#124	C₂×C₄×C₂₃	1	4	yes	—
295	C₄×Q₈	—	2	4	no	—
296	C₄×Q₈	—	4	4	no	—
297	C₄×D₄	—	2	4	no	—
298	C₂×C₄⋊C₄	—	2	1	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	—	2	2	no	—
301	C₂×C₄⋊C₄	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
306	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
307	C₂×C₂²⋊C₄	—	2	1	no	—
308	C₂×C₂²⋊C₄	—	2	1	no	—
309	C₂×C₂²⋊C₄	—	2	1	no	—
310	C₂×C₂²⋊C₄	—	2	1	no	—
311	C₂×C₂²⋊C₄	—	2	1	no	—
312	C₂×C₄²	—	2	2	no	—
313	C₂×C₄²	—	2	1	no	—
314	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
315	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
316	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
317	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
318	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
319	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
320	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
321	#103	—	2	4	no	—
322	#103	C₂×C₄×C₂₃	1	8	yes	complemented
323	#102	—	2	4	no	—
324	#102	C₂×C₄×C₂₃	1	8	yes	—
325	#97	—	2	4	no	—
326	#97	—	2	2	no	—
327	#97	C₂×C₄×C₂₃	1	4	yes	complemented
328	C₄×C₈	—	2	4	no	—
329	#95	—	2	1	no	—
330	#95	—	2	1	no	—
331	#95	—	2	1	no	—
332	#95	—	2	1	no	—
333	#95	—	2	2	no	—
334	#95	—	2	2	no	—
335	C₂₃	C₄×#489	1	1	yes	characteristic, direct product factor, Sylow
336	C₂⁴	—	2	1	no	—
337	C₂×Q₈	—	2	4	no	—
338	C₂×Q₈	—	2	1	no	—
339	C₂×Q₈	—	2	2	no	—
340	C₂×Q₈	—	2	2	no	—
341	C₂×Q₈	—	4	2	no	—
342	C₂×Q₈	—	4	2	no	—
343	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	2	1	no	—
347	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	1	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	4	2	no	—
354	C₂²×C₄	—	4	2	no	—
355	C₂²×C₄	—	4	2	no	—
356	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
357	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
358	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
360	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
361	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
362	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
368	C₂×C₈	#41×C₂₃	1	4	yes	—
369	C₄⋊C₄	—	2	4	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	4	2	no	—
376	C₄⋊C₄	—	4	4	no	—
377	C₄⋊C₄	—	4	2	no	—
378	C₄⋊C₄	—	4	2	no	—
379	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
380	C₂²⋊C₄	—	2	4	no	—
381	C₂²⋊C₄	—	2	4	no	—
382	C₂²⋊C₄	—	4	2	no	—
383	C₂²⋊C₄	—	4	2	no	—
384	C₂²⋊C₄	—	4	2	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	4	1	no	—
389	C₄²	—	4	1	no	—
390	C₄²	—	4	2	no	—
391	C₄²	—	4	4	no	—
392	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
393	C₄²	#41×C₂₃	1	2	yes	—
394	C₄²	#41×C₂₃	1	4	yes	—
395	C₂³	—	2	2	no	—
396	C₂³	—	4	1	no	—
397	C₂³	—	4	1	no	—
398	C₂³	—	4	1	no	—
399	C₂³	—	4	1	no	—
400	C₂³	—	4	1	no	—
401	C₂³	—	4	1	no	—
402	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
403	Q₈	—	2	4	no	—
404	Q₈	—	4	8	no	—
405	Q₈	—	4	2	no	—
406	D₄	—	2	4	no	—
407	D₄	—	4	2	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	8	4	no	—
436	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
437	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
438	C₂×C₄	C₂₃×#124	1	2	yes	—
439	C₂×C₄	C₂₃×#124	1	2	yes	—
440	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
441	C₂×C₄	C₂₃×#137	1	2	yes	—
442	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
443	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
444	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
445	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
446	C₈	—	2	4	no	—
447	C₈	—	2	4	no	—
448	C₂²	—	4	2	no	—
449	C₂²	—	4	2	no	—
450	C₂²	—	4	2	no	—
451	C₂²	—	8	1	no	—
452	C₂²	—	8	1	no	—
453	C₂²	—	8	1	no	—
454	C₂²	—	8	1	no	—
455	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
456	C₂²	C₂×C₂₃×#124	1	1	yes	characteristic
457	C₂²	C₂₃×#532	1	1	yes	characteristic
458	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
459	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
460	C₂²	C₂₃×#575	1	1	yes	characteristic
461	C₂²	C₂₃×#575	1	1	yes	characteristic
462	C₄	—	2	2	no	—
463	C₄	—	2	2	no	—
464	C₄	—	2	2	no	—
465	C₄	—	4	1	no	—
466	C₄	—	4	1	no	—
467	C₄	—	4	4	no	—
468	C₄	—	4	2	no	—
469	C₄	—	8	2	no	—
470	C₄	—	8	1	no	—
471	C₄	—	8	1	no	—
472	C₄	C₂₃×#489	1	4	yes	direct product factor
473	C₄	C₄×#41×C₂₃	1	2	yes	—
474	C₄	C₂₃×#529	1	2	yes	—
475	C₂	—	8	2	no	—
476	C₂	C₄×C₂₃×#124	1	1	yes	characteristic
477	C₂	C₄×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
478	C₂	C₄×C₂₃×#137	1	1	yes	characteristic
479	C₂	C₂×C₂₃×#489	1	1	yes	characteristic
480	C₂	C₂₃×#3135	1	1	yes	characteristic
481	C₂	C₂₃×#3171	1	1	yes	characteristic
482	C₂	C₂₃×#3172	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂²×D₄
prime radical	C₂²×D₄×C₂₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₂₃×C₄⋊Q₈	C₂	semidirect
4. C₂₃×#1993	C₂	semidirect
7. C₂₃×#1935	C₂	semidirect
8. C₂₃×#1836	C₂	semidirect
9. C₄×C₂₃×#103	C₂	semidirect
11. C₄×C₂₃×#97	C₂	semidirect
17. C₂₃×#489	C₄	direct (N × H)
51. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
81. C₂₃×#103	C₂×C₄	semidirect
86. C₂₃×#97	C₂×C₄	semidirect
153. C₄×#489	C₂₃	direct (N × H)
208. C₄×C₄⋊Q₈	C₂×C₂₃	semidirect
210. #1993	C₂×C₂₃	semidirect
213. #1935	C₂×C₂₃	semidirect
214. #1836	C₂×C₂₃	semidirect
215. C₄×#103	C₂×C₂₃	semidirect
217. C₄×#97	C₂×C₂₃	semidirect
242. C₄×C₂₃	#489	direct (N × H)
250. #489	C₄×C₂₃	direct (N × H)
292. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
322. #103	C₂×C₄×C₂₃	semidirect
327. #97	C₂×C₄×C₂₃	semidirect
335. C₂₃	C₄×#489	direct (N × H)
472. C₄	C₂₃×#489	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×#489	2
C₄×C₂₃×C₄⋊Q₈	2
C₄×C₂₃×#124	2
C₂₃×#1993	2
C₂₃×#1959	2
C₂₃×#1936	2
C₂₃×#1935	2
C₂₃×#1836	2
C₄×C₂₃×#103	2
C₄×C₂₃×#102	2
C₄×C₂₃×#97	2
C₄×#489	23

Subgroups — C4×C23×#489 #19169275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₃×#489 #19169275