Subgroups — C₄×C₂₃×#471 #19169178

Order 5888.

914 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#471	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×#137	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1977	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1951	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1931	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1930	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1908	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1907	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#1815	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×C₂₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₂₃×#101	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₂₃×#100	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#137	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
18	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
21	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#471	C₄	1	8	yes	direct product factor
25	C₂₃×#442	—	2	1	no	—
26	C₂₃×#441	—	2	1	no	—
27	C₂₃×#440	—	2	1	no	—
28	C₂₃×#440	—	2	1	no	—
29	C₄×Q₁₆×C₂₃	—	2	1	no	—
30	C₄×SD₁₆×C₂₃	—	2	1	no	—
31	C₄×SD₁₆×C₂₃	—	2	1	no	—
32	C₄×SD₁₆×C₂₃	—	2	1	no	—
33	C₄×SD₁₆×C₂₃	—	2	1	no	—
34	C₄×SD₁₆×C₂₃	—	2	1	no	—
35	C₄×SD₁₆×C₂₃	—	2	1	no	—
36	C₄×D₈×C₂₃	—	2	1	no	—
37	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
39	C₂₃×#428	C₂²	1	1	yes	characteristic
40	C₂₃×#428	C₂²	1	1	yes	characteristic
41	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
42	C₂₃×#419	C₂²	1	1	yes	characteristic
43	C₂₃×#418	C₂²	1	1	yes	characteristic
44	C₂₃×#417	C₂²	1	1	yes	characteristic
45	C₂₃×#417	C₂²	1	1	yes	characteristic
46	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
47	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
48	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
49	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
50	C₂×C₂₃×#100	C₂²	1	1	yes	characteristic
51	C₂×C₂₃×#100	C₂²	1	1	yes	characteristic
52	C₂₃×#405	—	2	1	no	—
53	C₂₃×#405	—	2	1	no	—
54	C₂₃×#405	C₂²	1	1	yes	characteristic
55	C₂₃×#405	C₂²	1	1	yes	characteristic
56	C₄×M₁₆×C₂₃	—	2	1	no	—
57	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
58	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
59	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
60	C₂×C₄×C₈×C₂₃	—	2	1	no	—
61	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
62	C₂₃×#343	C₂²	1	1	yes	characteristic
63	C₂₃×#343	C₂²	1	1	yes	characteristic
64	C₂₃×#343	C₂²	1	1	yes	characteristic
65	C₂₃×#343	C₂²	1	1	yes	characteristic
66	C₂₃×#341	C₂²	1	1	yes	characteristic
67	C₂₃×#341	C₂²	1	1	yes	characteristic
68	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₂₃	—	2	1	no	—
72	C₂₃×#137	—	2	2	no	—
73	C₂₃×#137	C₂×C₄	1	4	yes	complemented
74	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
75	C₂₃×AGL(1,ℤ/8)	C₂×C₄	1	4	yes	—
76	C₂×Q₁₆×C₂₃	—	2	1	no	—
77	C₂×SD₁₆×C₂₃	—	2	1	no	—
78	C₂×SD₁₆×C₂₃	—	2	1	no	—
79	C₂×SD₁₆×C₂₃	—	2	1	no	—
80	C₂×SD₁₆×C₂₃	—	2	1	no	—
81	C₂×SD₁₆×C₂₃	—	2	2	no	—
82	C₂×SD₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
83	C₂×D₈×C₂₃	—	2	1	no	—
84	C₂₃×#131	—	2	2	no	—
85	C₂₃×#131	—	2	2	no	—
86	C₂₃×#131	—	2	2	no	—
87	C₂₃×#131	C₂×C₄	1	4	yes	complemented
88	C₂×M₁₆×C₂₃	—	2	1	no	—
89	C₂×M₁₆×C₂₃	—	2	1	no	—
90	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
91	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
92	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
94	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
95	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
96	C₂²×C₈×C₂₃	—	2	1	no	—
97	C₂²×C₈×C₂₃	—	2	1	no	—
98	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
99	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
100	C₄×Q₈×C₂₃	—	2	1	no	—
101	C₄×Q₈×C₂₃	—	2	1	no	—
102	C₄×Q₈×C₂₃	—	2	1	no	—
103	C₄×Q₈×C₂₃	—	2	1	no	—
104	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
105	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
106	C₄×D₄×C₂₃	—	2	1	no	—
107	C₄×D₄×C₂₃	—	2	1	no	—
108	C₄×D₄×C₂₃	—	2	1	no	—
109	C₄×D₄×C₂₃	—	2	1	no	—
110	C₄×D₄×C₂₃	—	2	1	no	—
111	C₄×D₄×C₂₃	—	2	1	no	—
112	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
113	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
114	C₂₃×#117	—	2	1	no	—
115	C₂₃×#117	C₂³	1	1	yes	characteristic
116	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
117	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
118	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
119	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
120	C₂×C₄²×C₂₃	—	2	1	no	—
121	C₂×C₄²×C₂₃	—	2	1	no	—
122	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
123	C₂₃×#108	—	2	2	no	—
124	C₂₃×#108	C₂×C₄	1	4	yes	complemented
125	C₂₃×#107	—	2	1	no	—
126	C₂₃×#107	—	2	1	no	—
127	C₂₃×#106	—	2	1	no	—
128	C₂₃×#106	—	2	1	no	—
129	C₂₃×#106	—	2	1	no	—
130	C₂₃×#106	—	2	1	no	—
131	C₂₃×#103	—	2	1	no	—
132	C₂₃×#103	—	2	1	no	—
133	C₂₃×#103	—	2	1	no	—
134	C₂₃×#103	—	2	1	no	—
135	C₂₃×#103	—	2	1	no	—
136	C₂₃×#103	—	2	1	no	—
137	C₂₃×#102	—	2	1	no	—
138	C₂₃×#102	—	2	1	no	—
139	C₂₃×#102	—	2	1	no	—
140	C₂₃×#102	—	2	1	no	—
141	C₂₃×#102	—	2	1	no	—
142	C₂₃×#102	—	2	1	no	—
143	C₂₃×#101	—	2	2	no	—
144	C₂₃×#101	C₂×C₄	1	4	yes	complemented
145	C₂₃×#100	—	2	2	no	—
146	C₂₃×#100	C₂×C₄	1	4	yes	complemented
147	C₂₃×#97	—	2	1	no	—
148	C₂₃×#97	—	2	1	no	—
149	C₂₃×#97	—	2	1	no	—
150	C₂₃×#97	—	2	1	no	—
151	C₂₃×#97	—	2	1	no	—
152	C₄×C₈×C₂₃	—	2	1	no	—
153	C₄×C₈×C₂₃	—	2	1	no	—
154	C₄×C₈×C₂₃	—	2	1	no	—
155	C₄×C₈×C₂₃	—	2	1	no	—
156	C₄×C₈×C₂₃	—	2	1	no	—
157	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
158	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
159	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
160	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
161	C₂⁴×C₂₃	—	2	1	no	—
162	#41×C₂₃	—	2	1	no	—
163	#41×C₂₃	—	2	1	no	—
164	#41×C₂₃	C₂²×C₄	1	2	yes	—
165	#41×C₂₃	#41	1	2	yes	—
166	C₂×Q₈×C₂₃	—	2	1	no	—
167	C₂×Q₈×C₂₃	—	2	1	no	—
168	C₂×Q₈×C₂₃	—	2	1	no	—
169	C₂×Q₈×C₂₃	—	2	1	no	—
170	C₂×Q₈×C₂₃	—	2	1	no	—
171	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
172	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂×D₄×C₂₃	—	2	1	no	—
174	C₂×D₄×C₂₃	—	2	1	no	—
175	C₂×D₄×C₂₃	—	2	1	no	—
176	C₂×D₄×C₂₃	—	2	1	no	—
177	C₂×D₄×C₂₃	—	2	1	no	—
178	C₂×D₄×C₂₃	—	2	1	no	—
179	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
180	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₂₃	—	2	1	no	—
182	C₂²×C₄×C₂₃	—	2	1	no	—
183	C₂²×C₄×C₂₃	—	2	1	no	—
184	C₂²×C₄×C₂₃	—	2	1	no	—
185	C₂²×C₄×C₂₃	—	2	1	no	—
186	C₂²×C₄×C₂₃	—	2	1	no	—
187	C₂²×C₄×C₂₃	—	4	1	no	—
188	C₂²×C₄×C₂₃	—	4	1	no	—
189	C₂²×C₄×C₂₃	—	8	2	no	—
190	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
191	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
192	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
193	Q₁₆×C₂₃	—	2	4	no	—
194	SD₁₆×C₂₃	—	2	4	no	—
195	SD₁₆×C₂₃	—	2	4	no	—
196	SD₁₆×C₂₃	—	2	4	no	—
197	SD₁₆×C₂₃	—	2	4	no	—
198	D₈×C₂₃	—	2	4	no	—
199	M₁₆×C₂₃	—	2	4	no	—
200	M₁₆×C₂₃	—	2	1	no	—
201	M₁₆×C₂₃	—	2	1	no	—
202	M₁₆×C₂₃	—	2	1	no	—
203	M₁₆×C₂₃	—	2	1	no	—
204	M₁₆×C₂₃	—	2	4	no	—
205	M₁₆×C₂₃	—	2	1	no	—
206	M₁₆×C₂₃	—	2	1	no	—
207	M₁₆×C₂₃	—	2	1	no	—
208	M₁₆×C₂₃	—	2	1	no	—
209	M₁₆×C₂₃	C₂²×C₄	1	2	yes	—
210	M₁₆×C₂₃	C₂²×C₄	1	2	yes	—
211	M₁₆×C₂₃	C₂²×C₄	1	2	yes	—
212	M₁₆×C₂₃	#41	1	2	yes	—
213	C₂×C₈×C₂₃	—	2	4	no	—
214	C₂×C₈×C₂₃	—	2	1	no	—
215	C₂×C₈×C₂₃	—	2	1	no	—
216	C₂×C₈×C₂₃	—	2	1	no	—
217	C₂×C₈×C₂₃	—	2	1	no	—
218	C₂×C₈×C₂₃	—	2	1	no	—
219	C₂×C₈×C₂₃	—	2	1	no	—
220	C₂×C₈×C₂₃	—	2	4	no	—
221	C₂×C₈×C₂₃	—	2	1	no	—
222	C₂×C₈×C₂₃	—	2	1	no	—
223	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
224	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
225	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
226	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
227	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
228	C₂×C₈×C₂₃	#41	1	2	yes	—
229	C₄⋊C₄×C₂₃	—	2	1	no	—
230	C₄⋊C₄×C₂₃	—	2	1	no	—
231	C₄⋊C₄×C₂₃	—	2	1	no	—
232	C₄⋊C₄×C₂₃	—	2	1	no	—
233	C₄⋊C₄×C₂₃	—	2	1	no	—
234	C₄⋊C₄×C₂₃	—	2	1	no	—
235	C₄⋊C₄×C₂₃	—	4	1	no	—
236	C₄⋊C₄×C₂₃	—	4	1	no	—
237	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
238	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
239	C₂²⋊C₄×C₂₃	—	2	1	no	—
240	C₂²⋊C₄×C₂₃	—	2	1	no	—
241	C₂²⋊C₄×C₂₃	—	2	1	no	—
242	C₂²⋊C₄×C₂₃	—	4	1	no	—
243	C₂²⋊C₄×C₂₃	—	4	1	no	—
244	C₄²×C₂₃	—	2	1	no	—
245	C₄²×C₂₃	—	2	1	no	—
246	C₄²×C₂₃	—	2	1	no	—
247	C₄²×C₂₃	—	4	1	no	—
248	C₄²×C₂₃	—	4	1	no	—
249	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
250	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
251	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
252	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
253	C₄×#471	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
254	C₂³×C₂₃	—	2	1	no	—
255	C₂³×C₂₃	—	2	2	no	—
256	C₂³×C₂₃	—	4	1	no	—
257	C₂³×C₂₃	—	4	1	no	—
258	C₂³×C₂₃	—	8	1	no	—
259	C₂³×C₂₃	—	8	1	no	—
260	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
261	Q₈×C₂₃	—	2	1	no	—
262	Q₈×C₂₃	—	2	2	no	—
263	Q₈×C₂₃	—	2	2	no	—
264	Q₈×C₂₃	—	4	1	no	—
265	Q₈×C₂₃	—	4	1	no	—
266	Q₈×C₂₃	C₄×D₄	1	2	yes	—
267	D₄×C₂₃	—	2	2	no	—
268	D₄×C₂₃	—	2	2	no	—
269	D₄×C₂₃	—	2	2	no	—
270	D₄×C₂₃	—	2	2	no	—
271	D₄×C₂₃	—	2	1	no	—
272	D₄×C₂₃	—	4	1	no	—
273	D₄×C₂₃	—	4	1	no	—
274	D₄×C₂₃	C₄×D₄	1	2	yes	—
275	C₂×C₄×C₂₃	—	2	1	no	—
276	C₂×C₄×C₂₃	—	2	2	no	—
277	C₂×C₄×C₂₃	—	2	2	no	—
278	C₂×C₄×C₂₃	—	2	2	no	—
279	C₂×C₄×C₂₃	—	2	2	no	—
280	C₂×C₄×C₂₃	—	2	1	no	—
281	C₂×C₄×C₂₃	—	2	1	no	—
282	C₂×C₄×C₂₃	—	4	1	no	—
283	C₂×C₄×C₂₃	—	4	2	no	—
284	C₂×C₄×C₂₃	—	4	1	no	—
285	C₂×C₄×C₂₃	—	4	1	no	—
286	C₂×C₄×C₂₃	—	4	2	no	—
287	C₂×C₄×C₂₃	—	4	1	no	—
288	C₂×C₄×C₂₃	—	4	1	no	—
289	C₂×C₄×C₂₃	—	8	2	no	—
290	C₂×C₄×C₂₃	—	8	2	no	—
291	C₂×C₄×C₂₃	—	8	1	no	—
292	C₂×C₄×C₂₃	—	8	1	no	—
293	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
294	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
295	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
296	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
297	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
298	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
300	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
301	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
302	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
303	C₈×C₂₃	—	2	2	no	—
304	C₈×C₂₃	—	2	2	no	—
305	C₈×C₂₃	—	2	2	no	—
306	C₈×C₂₃	—	2	1	no	—
307	C₈×C₂₃	—	2	2	no	—
308	C₈×C₂₃	—	2	1	no	—
309	C₈×C₂₃	—	4	1	no	—
310	C₈×C₂₃	—	4	1	no	—
311	C₈×C₂₃	C₄×D₄	1	2	yes	—
312	C₈×C₂₃	C₄×D₄	1	2	yes	—
313	C₂×#471	C₂×C₂₃	1	1	yes	characteristic
314	C₄×#137	C₂×C₂₃	1	1	yes	characteristic, complemented
315	C₄×AGL(1,ℤ/8)	C₂×C₂₃	1	1	yes	characteristic
316	C₂×C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
317	C₄×#131	C₂×C₂₃	1	1	yes	characteristic, complemented
318	#1977	C₂×C₂₃	1	1	yes	characteristic, complemented
319	#1951	C₂×C₂₃	1	1	yes	characteristic, complemented
320	#1931	C₂×C₂₃	1	1	yes	characteristic, complemented
321	#1930	C₂×C₂₃	1	1	yes	characteristic, complemented
322	#1908	C₂×C₂₃	1	1	yes	characteristic, complemented
323	#1907	C₂×C₂₃	1	1	yes	characteristic
324	#1815	C₂×C₂₃	1	1	yes	characteristic, complemented
325	C₄×#108	C₂×C₂₃	1	1	yes	characteristic, complemented
326	C₄×#101	C₂×C₂₃	1	1	yes	characteristic, complemented
327	C₄×#100	C₂×C₂₃	1	1	yes	characteristic, complemented
328	C₂²×C₂₃	—	2	1	no	—
329	C₂²×C₂₃	—	2	1	no	—
330	C₂²×C₂₃	—	2	2	no	—
331	C₂²×C₂₃	—	4	1	no	—
332	C₂²×C₂₃	—	4	1	no	—
333	C₂²×C₂₃	—	4	2	no	—
334	C₂²×C₂₃	—	8	2	no	—
335	C₂²×C₂₃	—	8	2	no	—
336	C₂²×C₂₃	—	8	1	no	—
337	C₂²×C₂₃	—	8	1	no	—
338	C₂²×C₂₃	C₄×#41	1	1	yes	characteristic
339	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
340	C₂²×C₂₃	#529	1	1	yes	characteristic
341	C₄×C₂₃	—	2	1	no	—
342	C₄×C₂₃	—	2	1	no	—
343	C₄×C₂₃	—	2	1	no	—
344	C₄×C₂₃	—	2	2	no	—
345	C₄×C₂₃	—	4	1	no	—
346	C₄×C₂₃	—	4	1	no	—
347	C₄×C₂₃	—	4	2	no	—
348	C₄×C₂₃	—	8	1	no	—
349	C₄×C₂₃	—	8	1	no	—
350	C₄×C₂₃	#471	1	2	yes	direct product factor
351	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
352	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
353	C₄×C₂₃	#525	1	1	yes	characteristic
354	C₄×C₂₃	#532	1	1	yes	characteristic
355	C₂×#137	C₂²×C₂₃	1	1	yes	characteristic
356	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	1	yes	characteristic
357	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
358	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
359	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
360	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
361	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
362	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
363	#471	C₄×C₂₃	1	8	yes	direct product factor
364	#442	—	2	1	no	—
365	#441	—	2	1	no	—
366	#440	—	2	1	no	—
367	#440	—	2	1	no	—
368	C₄×Q₁₆	—	2	1	no	—
369	C₄×SD₁₆	—	2	1	no	—
370	C₄×SD₁₆	—	2	1	no	—
371	C₄×SD₁₆	—	2	1	no	—
372	C₄×SD₁₆	—	2	1	no	—
373	C₄×SD₁₆	—	2	1	no	—
374	C₄×SD₁₆	—	2	1	no	—
375	C₄×D₈	—	2	1	no	—
376	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
377	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
378	#428	C₂²×C₂₃	1	1	yes	characteristic
379	#428	C₂²×C₂₃	1	1	yes	characteristic
380	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
381	#419	C₂²×C₂₃	1	1	yes	characteristic
382	#418	C₂²×C₂₃	1	1	yes	characteristic
383	#417	C₂²×C₂₃	1	1	yes	characteristic
384	#417	C₂²×C₂₃	1	1	yes	characteristic
385	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
386	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
387	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
388	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
389	C₂×#100	C₂²×C₂₃	1	1	yes	characteristic
390	C₂×#100	C₂²×C₂₃	1	1	yes	characteristic
391	#405	—	2	1	no	—
392	#405	—	2	1	no	—
393	#405	C₂²×C₂₃	1	1	yes	characteristic
394	#405	C₂²×C₂₃	1	1	yes	characteristic
395	C₄×M₁₆	—	2	1	no	—
396	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
397	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
398	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
399	C₂×C₄×C₈	—	2	1	no	—
400	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
401	#343	C₂²×C₂₃	1	1	yes	characteristic
402	#343	C₂²×C₂₃	1	1	yes	characteristic
403	#343	C₂²×C₂₃	1	1	yes	characteristic
404	#343	C₂²×C₂₃	1	1	yes	characteristic
405	#341	C₂²×C₂₃	1	1	yes	characteristic
406	#341	C₂²×C₂₃	1	1	yes	characteristic
407	C₂×C₂₃	—	2	1	no	—
408	C₂×C₂₃	—	2	1	no	—
409	C₂×C₂₃	—	4	2	no	—
410	C₂×C₂₃	—	8	2	no	—
411	C₂×C₂₃	C₄×#121	1	1	yes	characteristic
412	C₂×C₂₃	C₂×#471	1	1	yes	characteristic
413	C₂×C₂₃	#3067	1	1	yes	characteristic
414	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
415	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
416	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
417	C₂³×C₄	—	2	1	no	—
418	#137	—	2	2	no	—
419	#137	C₂×C₄×C₂₃	1	4	yes	complemented
420	AGL(1,ℤ/8)	—	2	2	no	—
421	AGL(1,ℤ/8)	C₂×C₄×C₂₃	1	4	yes	—
422	C₂×Q₁₆	—	2	1	no	—
423	C₂×SD₁₆	—	2	1	no	—
424	C₂×SD₁₆	—	2	1	no	—
425	C₂×SD₁₆	—	2	1	no	—
426	C₂×SD₁₆	—	2	1	no	—
427	C₂×SD₁₆	—	2	2	no	—
428	C₂×SD₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
429	C₂×D₈	—	2	1	no	—
430	#131	—	2	2	no	—
431	#131	—	2	2	no	—
432	#131	—	2	2	no	—
433	#131	C₂×C₄×C₂₃	1	4	yes	complemented
434	C₂×M₁₆	—	2	1	no	—
435	C₂×M₁₆	—	2	1	no	—
436	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
437	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
438	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
439	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
440	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
441	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
442	C₂²×C₈	—	2	1	no	—
443	C₂²×C₈	—	2	1	no	—
444	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
445	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
446	C₄×Q₈	—	2	1	no	—
447	C₄×Q₈	—	2	1	no	—
448	C₄×Q₈	—	2	1	no	—
449	C₄×Q₈	—	2	1	no	—
450	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
451	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
452	C₄×D₄	—	2	1	no	—
453	C₄×D₄	—	2	1	no	—
454	C₄×D₄	—	2	1	no	—
455	C₄×D₄	—	2	1	no	—
456	C₄×D₄	—	2	1	no	—
457	C₄×D₄	—	2	1	no	—
458	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
459	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
460	#117	—	2	1	no	—
461	#117	C₂³×C₂₃	1	1	yes	characteristic
462	C₂×C₄⋊C₄	—	2	1	no	—
463	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
464	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
465	C₂×C₂²⋊C₄	—	2	1	no	—
466	C₂×C₄²	—	2	1	no	—
467	C₂×C₄²	—	2	1	no	—
468	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
469	#108	—	2	2	no	—
470	#108	C₂×C₄×C₂₃	1	4	yes	complemented
471	#107	—	2	1	no	—
472	#107	—	2	1	no	—
473	#106	—	2	1	no	—
474	#106	—	2	1	no	—
475	#106	—	2	1	no	—
476	#106	—	2	1	no	—
477	#103	—	2	1	no	—
478	#103	—	2	1	no	—
479	#103	—	2	1	no	—
480	#103	—	2	1	no	—
481	#103	—	2	1	no	—
482	#103	—	2	1	no	—
483	#102	—	2	1	no	—
484	#102	—	2	1	no	—
485	#102	—	2	1	no	—
486	#102	—	2	1	no	—
487	#102	—	2	1	no	—
488	#102	—	2	1	no	—
489	#101	—	2	2	no	—
490	#101	C₂×C₄×C₂₃	1	4	yes	complemented
491	#100	—	2	2	no	—
492	#100	C₂×C₄×C₂₃	1	4	yes	complemented
493	#97	—	2	1	no	—
494	#97	—	2	1	no	—
495	#97	—	2	1	no	—
496	#97	—	2	1	no	—
497	#97	—	2	1	no	—
498	C₄×C₈	—	2	1	no	—
499	C₄×C₈	—	2	1	no	—
500	C₄×C₈	—	2	1	no	—
501	C₄×C₈	—	2	1	no	—
502	C₄×C₈	—	2	1	no	—
503	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
504	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
505	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
506	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
507	C₂₃	C₄×#471	1	1	yes	characteristic, direct product factor, Sylow
508	C₂⁴	—	2	1	no	—
509	#41	—	2	1	no	—
510	#41	—	2	1	no	—
511	#41	C₂²×C₄×C₂₃	1	2	yes	—
512	#41	#41×C₂₃	1	2	yes	—
513	C₂×Q₈	—	2	1	no	—
514	C₂×Q₈	—	2	1	no	—
515	C₂×Q₈	—	2	1	no	—
516	C₂×Q₈	—	2	1	no	—
517	C₂×Q₈	—	2	1	no	—
518	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
519	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
520	C₂×D₄	—	2	1	no	—
521	C₂×D₄	—	2	1	no	—
522	C₂×D₄	—	2	1	no	—
523	C₂×D₄	—	2	1	no	—
524	C₂×D₄	—	2	1	no	—
525	C₂×D₄	—	2	1	no	—
526	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
527	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
528	C₂²×C₄	—	2	1	no	—
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	—	2	1	no	—
531	C₂²×C₄	—	2	1	no	—
532	C₂²×C₄	—	2	1	no	—
533	C₂²×C₄	—	2	1	no	—
534	C₂²×C₄	—	4	1	no	—
535	C₂²×C₄	—	4	1	no	—
536	C₂²×C₄	—	8	2	no	—
537	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
538	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
539	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
540	Q₁₆	—	2	4	no	—
541	SD₁₆	—	2	4	no	—
542	SD₁₆	—	2	4	no	—
543	SD₁₆	—	2	4	no	—
544	SD₁₆	—	2	4	no	—
545	D₈	—	2	4	no	—
546	M₁₆	—	2	4	no	—
547	M₁₆	—	2	1	no	—
548	M₁₆	—	2	1	no	—
549	M₁₆	—	2	1	no	—
550	M₁₆	—	2	1	no	—
551	M₁₆	—	2	4	no	—
552	M₁₆	—	2	1	no	—
553	M₁₆	—	2	1	no	—
554	M₁₆	—	2	1	no	—
555	M₁₆	—	2	1	no	—
556	M₁₆	C₂²×C₄×C₂₃	1	2	yes	—
557	M₁₆	C₂²×C₄×C₂₃	1	2	yes	—
558	M₁₆	C₂²×C₄×C₂₃	1	2	yes	—
559	M₁₆	#41×C₂₃	1	2	yes	—
560	C₂×C₈	—	2	4	no	—
561	C₂×C₈	—	2	1	no	—
562	C₂×C₈	—	2	1	no	—
563	C₂×C₈	—	2	1	no	—
564	C₂×C₈	—	2	1	no	—
565	C₂×C₈	—	2	1	no	—
566	C₂×C₈	—	2	1	no	—
567	C₂×C₈	—	2	4	no	—
568	C₂×C₈	—	2	1	no	—
569	C₂×C₈	—	2	1	no	—
570	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
571	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
572	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
573	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
574	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
575	C₂×C₈	#41×C₂₃	1	2	yes	—
576	C₄⋊C₄	—	2	1	no	—
577	C₄⋊C₄	—	2	1	no	—
578	C₄⋊C₄	—	2	1	no	—
579	C₄⋊C₄	—	2	1	no	—
580	C₄⋊C₄	—	2	1	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	4	1	no	—
583	C₄⋊C₄	—	4	1	no	—
584	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
585	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
586	C₂²⋊C₄	—	2	1	no	—
587	C₂²⋊C₄	—	2	1	no	—
588	C₂²⋊C₄	—	2	1	no	—
589	C₂²⋊C₄	—	4	1	no	—
590	C₂²⋊C₄	—	4	1	no	—
591	C₄²	—	2	1	no	—
592	C₄²	—	2	1	no	—
593	C₄²	—	2	1	no	—
594	C₄²	—	4	1	no	—
595	C₄²	—	4	1	no	—
596	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
597	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
598	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
599	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
600	C₂³	—	2	1	no	—
601	C₂³	—	2	2	no	—
602	C₂³	—	4	1	no	—
603	C₂³	—	4	1	no	—
604	C₂³	—	8	1	no	—
605	C₂³	—	8	1	no	—
606	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
607	Q₈	—	2	1	no	—
608	Q₈	—	2	2	no	—
609	Q₈	—	2	2	no	—
610	Q₈	—	4	1	no	—
611	Q₈	—	4	1	no	—
612	Q₈	C₄×D₄×C₂₃	1	2	yes	—
613	D₄	—	2	2	no	—
614	D₄	—	2	2	no	—
615	D₄	—	2	2	no	—
616	D₄	—	2	2	no	—
617	D₄	—	2	1	no	—
618	D₄	—	4	1	no	—
619	D₄	—	4	1	no	—
620	D₄	C₄×D₄×C₂₃	1	2	yes	—
621	C₂×C₄	—	2	1	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	1	no	—
628	C₂×C₄	—	4	1	no	—
629	C₂×C₄	—	4	2	no	—
630	C₂×C₄	—	4	1	no	—
631	C₂×C₄	—	4	1	no	—
632	C₂×C₄	—	4	2	no	—
633	C₂×C₄	—	4	1	no	—
634	C₂×C₄	—	4	1	no	—
635	C₂×C₄	—	8	2	no	—
636	C₂×C₄	—	8	2	no	—
637	C₂×C₄	—	8	1	no	—
638	C₂×C₄	—	8	1	no	—
639	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
640	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
641	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
642	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
643	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
644	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
645	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
646	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
647	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
648	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
649	C₈	—	2	2	no	—
650	C₈	—	2	2	no	—
651	C₈	—	2	2	no	—
652	C₈	—	2	1	no	—
653	C₈	—	2	2	no	—
654	C₈	—	2	1	no	—
655	C₈	—	4	1	no	—
656	C₈	—	4	1	no	—
657	C₈	C₄×D₄×C₂₃	1	2	yes	—
658	C₈	C₄×D₄×C₂₃	1	2	yes	—
659	C₂²	—	2	1	no	—
660	C₂²	—	2	1	no	—
661	C₂²	—	2	2	no	—
662	C₂²	—	4	1	no	—
663	C₂²	—	4	1	no	—
664	C₂²	—	4	2	no	—
665	C₂²	—	8	2	no	—
666	C₂²	—	8	2	no	—
667	C₂²	—	8	1	no	—
668	C₂²	—	8	1	no	—
669	C₂²	C₄×#41×C₂₃	1	1	yes	characteristic
670	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
671	C₂²	C₂₃×#529	1	1	yes	characteristic
672	C₄	—	2	1	no	—
673	C₄	—	2	1	no	—
674	C₄	—	2	1	no	—
675	C₄	—	2	2	no	—
676	C₄	—	4	1	no	—
677	C₄	—	4	1	no	—
678	C₄	—	4	2	no	—
679	C₄	—	8	1	no	—
680	C₄	—	8	1	no	—
681	C₄	C₂₃×#471	1	2	yes	direct product factor
682	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
683	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
684	C₄	C₂₃×#525	1	1	yes	characteristic
685	C₄	C₂₃×#532	1	1	yes	characteristic
686	C₂	—	2	1	no	—
687	C₂	—	2	1	no	—
688	C₂	—	4	2	no	—
689	C₂	—	8	2	no	—
690	C₂	C₄×C₂₃×#121	1	1	yes	characteristic
691	C₂	C₂×C₂₃×#471	1	1	yes	characteristic
692	C₂	C₂₃×#3067	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×#121
prime radical	C₂×C₂₃×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₂₃×#137	C₂	semidirect
4. C₂×C₄×SD₁₆×C₂₃	C₂	semidirect
5. C₄×C₂₃×#131	C₂	semidirect
6. C₂₃×#1977	C₂	semidirect
7. C₂₃×#1951	C₂	semidirect
8. C₂₃×#1931	C₂	semidirect
9. C₂₃×#1930	C₂	semidirect
10. C₂₃×#1908	C₂	semidirect
12. C₂₃×#1815	C₂	semidirect
13. C₄×C₂₃×#108	C₂	semidirect
14. C₄×C₂₃×#101	C₂	semidirect
15. C₄×C₂₃×#100	C₂	semidirect
24. C₂₃×#471	C₄	direct (N × H)
73. C₂₃×#137	C₂×C₄	semidirect
82. C₂×SD₁₆×C₂₃	C₂×C₄	semidirect
87. C₂₃×#131	C₂×C₄	semidirect
124. C₂₃×#108	C₂×C₄	semidirect
144. C₂₃×#101	C₂×C₄	semidirect
146. C₂₃×#100	C₂×C₄	semidirect
253. C₄×#471	C₂₃	direct (N × H)
314. C₄×#137	C₂×C₂₃	semidirect
316. C₂×C₄×SD₁₆	C₂×C₂₃	semidirect
317. C₄×#131	C₂×C₂₃	semidirect
318. #1977	C₂×C₂₃	semidirect
319. #1951	C₂×C₂₃	semidirect
320. #1931	C₂×C₂₃	semidirect
321. #1930	C₂×C₂₃	semidirect
322. #1908	C₂×C₂₃	semidirect
324. #1815	C₂×C₂₃	semidirect
325. C₄×#108	C₂×C₂₃	semidirect
326. C₄×#101	C₂×C₂₃	semidirect
327. C₄×#100	C₂×C₂₃	semidirect
350. C₄×C₂₃	#471	direct (N × H)
363. #471	C₄×C₂₃	direct (N × H)
419. #137	C₂×C₄×C₂₃	semidirect
428. C₂×SD₁₆	C₂×C₄×C₂₃	semidirect
433. #131	C₂×C₄×C₂₃	semidirect
470. #108	C₂×C₄×C₂₃	semidirect
490. #101	C₂×C₄×C₂₃	semidirect
492. #100	C₂×C₄×C₂₃	semidirect
507. C₂₃	C₄×#471	direct (N × H)
681. C₄	C₂₃×#471	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×#471	2
C₄×C₂₃×#137	2
C₄×C₂₃×AGL(1,ℤ/8)	2
C₂×C₄×SD₁₆×C₂₃	2
C₄×C₂₃×#131	2
C₂₃×#1977	2
C₂₃×#1951	2
C₂₃×#1931	2
C₂₃×#1930	2
C₂₃×#1908	2
C₂₃×#1907	2
C₂₃×#1815	2
C₄×C₂₃×#108	2
C₄×C₂₃×#101	2
C₄×C₂₃×#100	2
C₄×#471	23

Subgroups — C4×C23×#471 #19169178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₃×#471 #19169178