Subgroups — C₂₃×#61840 #19209466

Order 5888.

1928 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3497	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3386	C₂	1	3	yes	maximal, complemented
4	C₂₃×#3384	C₂	1	3	yes	maximal, complemented
5	Q₈×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#3368	C₂	1	3	yes	maximal, complemented
8	C₂₃×#3366	C₂	1	3	yes	maximal, complemented
9	C₂₃×#3362	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#3358	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#3235	C₂	1	3	yes	maximal, complemented
12	C₂₃×#3228	C₂	1	3	yes	maximal, complemented
13	C₂₃×#3115	C₂	1	3	yes	maximal, complemented
14	C₂₃×#3106	C₂	1	3	yes	maximal, complemented
15	C₂₃×#2984	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×2₋¹⁺⁴	—	2	1	no	—
17	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#557	C₂²	1	3	yes	—
19	C₂₃×#554	—	2	3	no	—
20	C₂₃×#554	C₂²	1	3	yes	—
21	C₂₃×#550	C₂²	1	1	yes	characteristic
22	C₂₃×#550	C₂²	1	3	yes	—
23	D₄×Q₈×C₂₃	—	2	1	no	—
24	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
25	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
26	D₄×Q₈×C₂₃	C₂²	1	3	yes	—
27	C₂₃×#548	—	2	1	no	—
28	C₂₃×#548	—	2	3	no	—
29	C₂₃×#537	—	2	3	no	—
30	C₂₃×#533	—	2	3	no	—
31	C₂₃×#533	C₂²	1	3	yes	—
32	C₂₃×#532	C₂²	1	3	yes	—
33	C₂₃×#532	C₂²	1	3	yes	—
34	C₄×#41×C₂₃	C₂²	1	3	yes	—
35	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
36	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
37	C₂₃×#501	C₂²	1	6	yes	—
38	C₂₃×#500	C₂²	1	3	yes	—
39	C₂₃×#498	C₂²	1	3	yes	—
40	C₂₃×#490	C₂²	1	6	yes	—
41	C₂₃×#486	C₂²	1	3	yes	—
42	C₂₃×#486	C₂²	1	3	yes	—
43	C₂₃×#482	C₂²	1	3	yes	—
44	C₂₃×#482	C₂²	1	6	yes	—
45	C₂₃×#482	C₂²	1	3	yes	—
46	C₂₃×#478	C₂²	1	6	yes	—
47	C₂₃×#478	C₂²	1	6	yes	—
48	C₂₃×#477	C₂²	1	3	yes	—
49	C₂₃×#476	C₂²	1	3	yes	—
50	C₂₃×#476	C₂²	1	3	yes	—
51	C₂₃×#476	C₂²	1	3	yes	—
52	C₂₃×#475	C₂²	1	3	yes	—
53	C₂₃×#474	C₂²	1	3	yes	—
54	C₂₃×#474	C₂²	1	3	yes	—
55	C₂₃×#474	C₂²	1	3	yes	—
56	C₂₃×#460	C₂²	1	3	yes	—
57	C₂₃×#460	C₂²	1	3	yes	—
58	C₂₃×#459	C₂²	1	3	yes	—
59	C₂₃×#459	C₂²	1	3	yes	—
60	C₂₃×#449	C₂²	1	1	yes	characteristic
61	C₂₃×#449	C₂²	1	3	yes	—
62	C₂₃×#446	C₂²	1	3	yes	—
63	C₂₃×#446	C₂²	1	3	yes	—
64	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
65	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
66	C₂₃×#442	C₂²	1	6	yes	—
67	C₂₃×#440	C₂²	1	6	yes	—
68	C₄×SD₁₆×C₂₃	C₂²	1	3	yes	—
69	C₄×SD₁₆×C₂₃	C₂²	1	3	yes	—
70	C₄×D₈×C₂₃	C₂²	1	3	yes	—
71	C₄×D₈×C₂₃	C₂²	1	3	yes	—
72	C₂₃×#432	C₂²	1	3	yes	—
73	C₂₃×#431	C₂²	1	3	yes	—
74	C₂₃×2₋¹⁺⁴	—	4	8	no	—
75	C₂×#41×C₂₃	—	2	3	no	—
76	C₂×#41×C₂₃	—	2	1	no	—
77	C₂×#41×C₂₃	—	2	3	no	—
78	C₂×#41×C₂₃	—	2	3	no	—
79	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
80	C₂²×Q₈×C₂₃	—	2	3	no	—
81	C₂²×Q₈×C₂₃	—	2	1	no	—
82	C₂²×Q₈×C₂₃	—	2	1	no	—
83	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
84	C₂×SD₁₆×C₂₃	C₂³	1	1	yes	characteristic
85	C₂×SD₁₆×C₂₃	C₂³	1	3	yes	—
86	C₂×D₈×C₂₃	C₂³	1	1	yes	characteristic
87	C₂×D₈×C₂₃	C₂³	1	3	yes	—
88	C₂₃×C₄⋊Q₈	—	2	3	no	—
89	C₂₃×C₄⋊Q₈	—	2	3	no	—
90	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
91	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
92	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
93	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
94	C₂₃×#127	C₂³	1	3	yes	—
95	C₂₃×#127	C₂³	1	3	yes	—
96	C₂₃×#125	—	2	6	no	—
97	C₂₃×#125	C₂³	1	6	yes	—
98	C₂₃×#124	—	2	3	no	—
99	C₂₃×#124	—	2	3	no	—
100	C₂₃×C₂²⋊Q₈	—	2	3	no	—
101	C₂₃×C₂²⋊Q₈	—	2	3	no	—
102	C₂₃×C₂²⋊Q₈	—	2	3	no	—
103	C₂₃×C₂²⋊Q₈	—	2	3	no	—
104	C₂₃×C₂²⋊Q₈	—	2	6	no	—
105	C₂₃×C₂²⋊Q₈	—	2	3	no	—
106	C₂₃×C₂²⋊Q₈	—	2	3	no	—
107	C₂₃×C₂²⋊Q₈	—	2	6	no	—
108	C₂₃×#121	—	2	3	no	—
109	C₂₃×#121	—	2	3	no	—
110	C₂₃×#121	—	2	3	no	—
111	C₂₃×#121	—	2	6	no	—
112	C₂₃×#121	—	2	3	no	—
113	C₂₃×#121	C₂³	1	1	yes	characteristic
114	C₂₃×#121	C₂³	1	3	yes	—
115	C₄×Q₈×C₂₃	—	2	1	no	—
116	C₄×Q₈×C₂₃	—	2	3	no	—
117	C₄×Q₈×C₂₃	—	2	3	no	—
118	C₄×Q₈×C₂₃	—	2	3	no	—
119	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
120	C₄×Q₈×C₂₃	D₄	1	3	yes	—
121	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
122	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
123	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
124	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
125	C₄×D₄×C₂₃	—	2	3	no	—
126	C₄×D₄×C₂₃	—	2	3	no	—
127	C₄×D₄×C₂₃	—	2	3	no	—
128	C₄×D₄×C₂₃	—	2	3	no	—
129	C₄×D₄×C₂₃	C₂³	1	3	yes	—
130	C₄×D₄×C₂₃	C₂³	1	3	yes	—
131	C₄×D₄×C₂₃	C₂³	1	6	yes	—
132	C₄×D₄×C₂₃	C₂³	1	3	yes	—
133	C₄×D₄×C₂₃	C₂³	1	3	yes	—
134	C₂₃×#117	—	2	3	no	—
135	C₂₃×#117	C₂³	1	3	yes	—
136	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
137	C₂×C₄⋊C₄×C₂₃	D₄	1	3	yes	—
138	C₂×C₄²×C₂₃	—	2	3	no	—
139	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
140	C₂₃×#107	C₂³	1	3	yes	—
141	C₂₃×#107	C₂³	1	3	yes	—
142	C₂₃×#107	C₂³	1	6	yes	—
143	C₂₃×#106	C₂³	1	3	yes	—
144	C₂₃×#106	C₂³	1	3	yes	—
145	C₂₃×#106	C₂³	1	6	yes	—
146	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
147	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
148	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
149	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
150	C₂₃×#103	C₂³	1	1	yes	characteristic
151	C₂₃×#103	C₂³	1	3	yes	—
152	C₂₃×#102	C₂³	1	1	yes	characteristic
153	C₂₃×#102	C₂³	1	3	yes	—
154	C₂₃×#102	C₂³	1	3	yes	—
155	C₂₃×#102	C₂³	1	3	yes	—
156	C₂₃×#102	C₂³	1	3	yes	—
157	C₂₃×#102	C₂³	1	3	yes	—
158	C₂₃×#102	C₂³	1	6	yes	—
159	C₂₃×#102	C₂³	1	6	yes	—
160	C₂₃×C₂²⋊C₈	C₂³	1	1	yes	characteristic
161	C₂₃×C₂²⋊C₈	C₂³	1	3	yes	—
162	C₂₃×#97	C₂³	1	6	yes	—
163	C₄×C₈×C₂₃	C₂³	1	3	yes	—
164	C₄×C₈×C₂₃	C₂³	1	3	yes	—
165	#41×C₂₃	—	2	2	no	—
166	#41×C₂₃	—	2	2	no	—
167	#41×C₂₃	—	4	12	no	—
168	#41×C₂₃	—	4	12	no	—
169	#41×C₂₃	—	4	12	no	—
170	#41×C₂₃	—	4	4	no	—
171	C₂×Q₈×C₂₃	—	2	4	no	—
172	C₂×Q₈×C₂₃	—	2	3	no	—
173	C₂×Q₈×C₂₃	—	2	3	no	—
174	C₂×Q₈×C₂₃	—	2	3	no	—
175	C₂×Q₈×C₂₃	—	2	3	no	—
176	C₂×Q₈×C₂₃	—	4	12	no	—
177	C₂×Q₈×C₂₃	—	4	4	no	—
178	C₂×Q₈×C₂₃	—	4	4	no	—
179	C₂×Q₈×C₂₃	C₂×D₄	1	3	yes	—
180	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₂×Q₈×C₂₃	C₂⁴	1	1	yes	characteristic
182	C₂×D₄×C₂₃	—	2	1	no	—
183	C₂×D₄×C₂₃	—	2	3	no	—
184	C₂×D₄×C₂₃	—	2	1	no	—
185	C₂×D₄×C₂₃	—	2	3	no	—
186	C₂×D₄×C₂₃	—	2	3	no	—
187	C₂×D₄×C₂₃	—	2	3	no	—
188	C₂×D₄×C₂₃	—	2	3	no	—
189	C₂×D₄×C₂₃	C₂⁴	1	1	yes	characteristic
190	C₂×D₄×C₂₃	C₂⁴	1	3	yes	—
191	C₂×D₄×C₂₃	C₂⁴	1	1	yes	characteristic
192	C₂²×C₄×C₂₃	—	2	1	no	—
193	C₂²×C₄×C₂₃	—	2	3	no	—
194	C₂²×C₄×C₂₃	—	2	6	no	—
195	C₂²×C₄×C₂₃	—	2	3	no	—
196	C₂²×C₄×C₂₃	—	2	3	no	—
197	C₂²×C₄×C₂₃	—	2	3	no	—
198	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
199	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
200	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
201	SD₁₆×C₂₃	—	2	2	no	—
202	SD₁₆×C₂₃	—	2	6	no	—
203	D₈×C₂₃	—	2	2	no	—
204	D₈×C₂₃	—	2	6	no	—
205	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
206	C₂×C₈×C₂₃	C₂⁴	1	3	yes	—
207	C₂×C₈×C₂₃	C₂⁴	1	3	yes	—
208	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
209	C₄⋊C₄×C₂₃	—	2	6	no	—
210	C₄⋊C₄×C₂₃	—	2	3	no	—
211	C₄⋊C₄×C₂₃	—	2	3	no	—
212	C₄⋊C₄×C₂₃	—	2	3	no	—
213	C₄⋊C₄×C₂₃	—	2	3	no	—
214	C₄⋊C₄×C₂₃	—	2	3	no	—
215	C₄⋊C₄×C₂₃	—	2	3	no	—
216	C₄⋊C₄×C₂₃	—	2	6	no	—
217	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
218	C₄⋊C₄×C₂₃	C₂×D₄	1	6	yes	—
219	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
220	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
221	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
222	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
223	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
224	C₂²⋊C₄×C₂₃	—	2	3	no	—
225	C₂²⋊C₄×C₂₃	—	2	3	no	—
226	C₂²⋊C₄×C₂₃	—	2	3	no	—
227	C₂²⋊C₄×C₂₃	—	2	1	no	—
228	C₂²⋊C₄×C₂₃	—	2	3	no	—
229	C₂²⋊C₄×C₂₃	—	2	6	no	—
230	C₂²⋊C₄×C₂₃	—	2	3	no	—
231	C₂²⋊C₄×C₂₃	—	2	3	no	—
232	C₄²×C₂₃	—	2	3	no	—
233	C₄²×C₂₃	—	2	3	no	—
234	C₄²×C₂₃	—	2	3	no	—
235	C₄²×C₂₃	C₂×D₄	1	3	yes	—
236	C₄²×C₂₃	C₂×D₄	1	3	yes	—
237	C₄²×C₂₃	C₂⁴	1	3	yes	—
238	C₄²×C₂₃	C₂⁴	1	3	yes	—
239	#61840	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₂³×C₂₃	—	2	1	no	—
241	C₂³×C₂₃	—	2	3	no	—
242	C₂³×C₂₃	—	2	1	no	—
243	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
244	Q₈×C₂₃	—	2	6	no	—
245	Q₈×C₂₃	—	2	1	no	—
246	Q₈×C₂₃	—	4	6	no	—
247	Q₈×C₂₃	—	4	6	no	—
248	Q₈×C₂₃	—	4	6	no	—
249	Q₈×C₂₃	—	8	3	no	—
250	Q₈×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
251	Q₈×C₂₃	2₋¹⁺⁴	1	2	yes	complemented
252	D₄×C₂₃	—	2	4	no	—
253	D₄×C₂₃	—	2	1	no	—
254	D₄×C₂₃	—	2	1	no	—
255	D₄×C₂₃	—	2	3	no	—
256	D₄×C₂₃	—	2	1	no	—
257	D₄×C₂₃	—	2	3	no	—
258	D₄×C₂₃	—	4	6	no	—
259	D₄×C₂₃	—	4	6	no	—
260	D₄×C₂₃	—	4	2	no	—
261	D₄×C₂₃	—	4	6	no	—
262	D₄×C₂₃	—	8	3	no	—
263	D₄×C₂₃	—	8	3	no	—
264	D₄×C₂₃	2₋¹⁺⁴	1	2	yes	complemented
265	C₂×C₄×C₂₃	—	2	4	no	—
266	C₂×C₄×C₂₃	—	2	1	no	—
267	C₂×C₄×C₂₃	—	2	6	no	—
268	C₂×C₄×C₂₃	—	2	1	no	—
269	C₂×C₄×C₂₃	—	2	1	no	—
270	C₂×C₄×C₂₃	—	2	3	no	—
271	C₂×C₄×C₂₃	—	2	3	no	—
272	C₂×C₄×C₂₃	—	2	3	no	—
273	C₂×C₄×C₂₃	—	2	1	no	—
274	C₂×C₄×C₂₃	—	4	3	no	—
275	C₂×C₄×C₂₃	—	4	12	no	—
276	C₂×C₄×C₂₃	—	4	6	no	—
277	C₂×C₄×C₂₃	—	4	6	no	—
278	C₂×C₄×C₂₃	—	4	6	no	—
279	C₂×C₄×C₂₃	—	8	3	no	—
280	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
281	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
282	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
283	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
284	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
285	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
286	C₈×C₂₃	—	2	3	no	—
287	C₈×C₂₃	—	2	1	no	—
288	C₈×C₂₃	—	2	1	no	—
289	C₈×C₂₃	—	2	3	no	—
290	#3497	C₂×C₂₃	1	1	yes	characteristic, complemented
291	Q₈×#41	C₂×C₂₃	1	1	yes	characteristic, complemented
292	#3386	C₂×C₂₃	1	3	yes	complemented
293	#3384	C₂×C₂₃	1	3	yes	complemented
294	Q₈×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
295	Q₈×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
296	#3368	C₂×C₂₃	1	3	yes	complemented
297	#3366	C₂×C₂₃	1	3	yes	complemented
298	#3362	C₂×C₂₃	1	1	yes	characteristic, complemented
299	#3358	C₂×C₂₃	1	1	yes	characteristic, complemented
300	#3235	C₂×C₂₃	1	3	yes	complemented
301	#3228	C₂×C₂₃	1	3	yes	complemented
302	#3115	C₂×C₂₃	1	3	yes	complemented
303	#3106	C₂×C₂₃	1	3	yes	complemented
304	#2984	C₂×C₂₃	1	1	yes	characteristic, complemented
305	C₂²×C₂₃	—	2	1	no	—
306	C₂²×C₂₃	—	2	1	no	—
307	C₂²×C₂₃	—	2	2	no	—
308	C₂²×C₂₃	—	2	1	no	—
309	C₂²×C₂₃	—	4	1	no	—
310	C₂²×C₂₃	—	4	1	no	—
311	C₂²×C₂₃	—	4	1	no	—
312	C₂²×C₂₃	—	4	3	no	—
313	C₂²×C₂₃	—	4	3	no	—
314	C₂²×C₂₃	—	4	1	no	—
315	C₂²×C₂₃	—	4	3	no	—
316	C₂²×C₂₃	—	4	1	no	—
317	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
318	C₄×C₂₃	—	2	3	no	—
319	C₄×C₂₃	—	2	3	no	—
320	C₄×C₂₃	—	2	1	no	—
321	C₄×C₂₃	—	2	2	no	—
322	C₄×C₂₃	—	2	3	no	—
323	C₄×C₂₃	—	4	3	no	—
324	C₄×C₂₃	—	4	3	no	—
325	C₄×C₂₃	—	4	1	no	—
326	C₄×C₂₃	—	4	3	no	—
327	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	6	yes	—
328	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
329	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
330	C₂×2₋¹⁺⁴	—	2	1	no	—
331	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
332	#557	C₂²×C₂₃	1	3	yes	—
333	#554	—	2	3	no	—
334	#554	C₂²×C₂₃	1	3	yes	—
335	#550	C₂²×C₂₃	1	1	yes	characteristic
336	#550	C₂²×C₂₃	1	3	yes	—
337	D₄×Q₈	—	2	1	no	—
338	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
339	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
340	D₄×Q₈	C₂²×C₂₃	1	3	yes	—
341	#548	—	2	1	no	—
342	#548	—	2	3	no	—
343	#537	—	2	3	no	—
344	#533	—	2	3	no	—
345	#533	C₂²×C₂₃	1	3	yes	—
346	#532	C₂²×C₂₃	1	3	yes	—
347	#532	C₂²×C₂₃	1	3	yes	—
348	C₄×#41	C₂²×C₂₃	1	3	yes	—
349	C₂×C₄×Q₈	—	2	1	no	—
350	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
351	#501	C₂²×C₂₃	1	6	yes	—
352	#500	C₂²×C₂₃	1	3	yes	—
353	#498	C₂²×C₂₃	1	3	yes	—
354	#490	C₂²×C₂₃	1	6	yes	—
355	#486	C₂²×C₂₃	1	3	yes	—
356	#486	C₂²×C₂₃	1	3	yes	—
357	#482	C₂²×C₂₃	1	3	yes	—
358	#482	C₂²×C₂₃	1	3	yes	—
359	#482	C₂²×C₂₃	1	6	yes	—
360	#478	C₂²×C₂₃	1	6	yes	—
361	#478	C₂²×C₂₃	1	6	yes	—
362	#477	C₂²×C₂₃	1	3	yes	—
363	#476	C₂²×C₂₃	1	3	yes	—
364	#476	C₂²×C₂₃	1	3	yes	—
365	#476	C₂²×C₂₃	1	3	yes	—
366	#475	C₂²×C₂₃	1	3	yes	—
367	#474	C₂²×C₂₃	1	3	yes	—
368	#474	C₂²×C₂₃	1	3	yes	—
369	#474	C₂²×C₂₃	1	3	yes	—
370	#460	C₂²×C₂₃	1	3	yes	—
371	#460	C₂²×C₂₃	1	3	yes	—
372	#459	C₂²×C₂₃	1	3	yes	—
373	#459	C₂²×C₂₃	1	3	yes	—
374	#449	C₂²×C₂₃	1	1	yes	characteristic
375	#449	C₂²×C₂₃	1	3	yes	—
376	#446	C₂²×C₂₃	1	3	yes	—
377	#446	C₂²×C₂₃	1	3	yes	—
378	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
379	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
380	#442	C₂²×C₂₃	1	6	yes	—
381	#440	C₂²×C₂₃	1	6	yes	—
382	C₄×SD₁₆	C₂²×C₂₃	1	3	yes	—
383	C₄×SD₁₆	C₂²×C₂₃	1	3	yes	—
384	C₄×D₈	C₂²×C₂₃	1	3	yes	—
385	C₄×D₈	C₂²×C₂₃	1	3	yes	—
386	#432	C₂²×C₂₃	1	3	yes	—
387	#431	C₂²×C₂₃	1	3	yes	—
388	C₂×C₂₃	—	4	1	no	—
389	C₂×C₂₃	—	4	2	no	—
390	C₂×C₂₃	—	8	3	no	—
391	C₂×C₂₃	—	8	1	no	—
392	C₂×C₂₃	#3486	1	1	yes	characteristic
393	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
394	C₂×C₂₃	#3584	1	1	yes	characteristic
395	2₋¹⁺⁴	—	4	8	no	—
396	C₂×#41	—	2	3	no	—
397	C₂×#41	—	2	1	no	—
398	C₂×#41	—	2	3	no	—
399	C₂×#41	—	2	3	no	—
400	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
401	C₂²×Q₈	—	2	1	no	—
402	C₂²×Q₈	—	2	3	no	—
403	C₂²×Q₈	—	2	1	no	—
404	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
405	C₂×SD₁₆	C₂³×C₂₃	1	1	yes	characteristic
406	C₂×SD₁₆	C₂³×C₂₃	1	3	yes	—
407	C₂×D₈	C₂³×C₂₃	1	1	yes	characteristic
408	C₂×D₈	C₂³×C₂₃	1	3	yes	—
409	C₄⋊Q₈	—	2	3	no	—
410	C₄⋊Q₈	—	2	3	no	—
411	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
412	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
413	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
414	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
415	#127	C₂³×C₂₃	1	3	yes	—
416	#127	C₂³×C₂₃	1	3	yes	—
417	#125	—	2	6	no	—
418	#125	C₂³×C₂₃	1	6	yes	—
419	#124	—	2	3	no	—
420	#124	—	2	3	no	—
421	C₂²⋊Q₈	—	2	3	no	—
422	C₂²⋊Q₈	—	2	3	no	—
423	C₂²⋊Q₈	—	2	3	no	—
424	C₂²⋊Q₈	—	2	3	no	—
425	C₂²⋊Q₈	—	2	6	no	—
426	C₂²⋊Q₈	—	2	3	no	—
427	C₂²⋊Q₈	—	2	6	no	—
428	C₂²⋊Q₈	—	2	3	no	—
429	#121	—	2	3	no	—
430	#121	—	2	3	no	—
431	#121	—	2	3	no	—
432	#121	—	2	6	no	—
433	#121	—	2	3	no	—
434	#121	C₂³×C₂₃	1	1	yes	characteristic
435	#121	C₂³×C₂₃	1	3	yes	—
436	C₄×Q₈	—	2	1	no	—
437	C₄×Q₈	—	2	3	no	—
438	C₄×Q₈	—	2	3	no	—
439	C₄×Q₈	—	2	3	no	—
440	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
441	C₄×Q₈	D₄×C₂₃	1	3	yes	—
442	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
443	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
444	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
445	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
446	C₄×D₄	—	2	3	no	—
447	C₄×D₄	—	2	3	no	—
448	C₄×D₄	—	2	3	no	—
449	C₄×D₄	—	2	3	no	—
450	C₄×D₄	C₂³×C₂₃	1	3	yes	—
451	C₄×D₄	C₂³×C₂₃	1	3	yes	—
452	C₄×D₄	C₂³×C₂₃	1	3	yes	—
453	C₄×D₄	C₂³×C₂₃	1	3	yes	—
454	C₄×D₄	C₂³×C₂₃	1	6	yes	—
455	#117	—	2	3	no	—
456	#117	C₂³×C₂₃	1	3	yes	—
457	C₂×C₄⋊C₄	—	2	3	no	—
458	C₂×C₄⋊C₄	D₄×C₂₃	1	3	yes	—
459	C₂×C₄²	—	2	3	no	—
460	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
461	#107	C₂³×C₂₃	1	3	yes	—
462	#107	C₂³×C₂₃	1	3	yes	—
463	#107	C₂³×C₂₃	1	6	yes	—
464	#106	C₂³×C₂₃	1	3	yes	—
465	#106	C₂³×C₂₃	1	3	yes	—
466	#106	C₂³×C₂₃	1	6	yes	—
467	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
468	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
469	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
470	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
471	#103	C₂³×C₂₃	1	1	yes	characteristic
472	#103	C₂³×C₂₃	1	3	yes	—
473	#102	C₂³×C₂₃	1	1	yes	characteristic
474	#102	C₂³×C₂₃	1	3	yes	—
475	#102	C₂³×C₂₃	1	3	yes	—
476	#102	C₂³×C₂₃	1	3	yes	—
477	#102	C₂³×C₂₃	1	3	yes	—
478	#102	C₂³×C₂₃	1	3	yes	—
479	#102	C₂³×C₂₃	1	6	yes	—
480	#102	C₂³×C₂₃	1	6	yes	—
481	C₂²⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
482	C₂²⋊C₈	C₂³×C₂₃	1	3	yes	—
483	#97	C₂³×C₂₃	1	6	yes	—
484	C₄×C₈	C₂³×C₂₃	1	3	yes	—
485	C₄×C₈	C₂³×C₂₃	1	3	yes	—
486	C₂₃	#61840	1	1	yes	characteristic, direct product factor, Sylow
487	#41	—	2	2	no	—
488	#41	—	2	2	no	—
489	#41	—	4	12	no	—
490	#41	—	4	12	no	—
491	#41	—	4	12	no	—
492	#41	—	4	4	no	—
493	C₂×Q₈	—	2	4	no	—
494	C₂×Q₈	—	2	3	no	—
495	C₂×Q₈	—	2	3	no	—
496	C₂×Q₈	—	2	3	no	—
497	C₂×Q₈	—	2	3	no	—
498	C₂×Q₈	—	4	12	no	—
499	C₂×Q₈	—	4	4	no	—
500	C₂×Q₈	—	4	4	no	—
501	C₂×Q₈	C₂×D₄×C₂₃	1	3	yes	—
502	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
503	C₂×Q₈	C₂⁴×C₂₃	1	1	yes	characteristic
504	C₂×D₄	—	2	1	no	—
505	C₂×D₄	—	2	1	no	—
506	C₂×D₄	—	2	3	no	—
507	C₂×D₄	—	2	3	no	—
508	C₂×D₄	—	2	3	no	—
509	C₂×D₄	—	2	3	no	—
510	C₂×D₄	—	2	3	no	—
511	C₂×D₄	C₂⁴×C₂₃	1	1	yes	characteristic
512	C₂×D₄	C₂⁴×C₂₃	1	3	yes	—
513	C₂×D₄	C₂⁴×C₂₃	1	1	yes	characteristic
514	C₂²×C₄	—	2	1	no	—
515	C₂²×C₄	—	2	3	no	—
516	C₂²×C₄	—	2	6	no	—
517	C₂²×C₄	—	2	3	no	—
518	C₂²×C₄	—	2	3	no	—
519	C₂²×C₄	—	2	3	no	—
520	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
521	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
522	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
523	SD₁₆	—	2	2	no	—
524	SD₁₆	—	2	6	no	—
525	D₈	—	2	2	no	—
526	D₈	—	2	6	no	—
527	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
528	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
529	C₂×C₈	C₂⁴×C₂₃	1	3	yes	—
530	C₂×C₈	C₂⁴×C₂₃	1	3	yes	—
531	C₄⋊C₄	—	2	6	no	—
532	C₄⋊C₄	—	2	3	no	—
533	C₄⋊C₄	—	2	3	no	—
534	C₄⋊C₄	—	2	3	no	—
535	C₄⋊C₄	—	2	3	no	—
536	C₄⋊C₄	—	2	3	no	—
537	C₄⋊C₄	—	2	3	no	—
538	C₄⋊C₄	—	2	6	no	—
539	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
540	C₄⋊C₄	C₂×D₄×C₂₃	1	6	yes	—
541	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
542	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
543	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
544	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
545	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
546	C₂²⋊C₄	—	2	3	no	—
547	C₂²⋊C₄	—	2	3	no	—
548	C₂²⋊C₄	—	2	3	no	—
549	C₂²⋊C₄	—	2	1	no	—
550	C₂²⋊C₄	—	2	3	no	—
551	C₂²⋊C₄	—	2	6	no	—
552	C₂²⋊C₄	—	2	3	no	—
553	C₂²⋊C₄	—	2	3	no	—
554	C₄²	—	2	3	no	—
555	C₄²	—	2	3	no	—
556	C₄²	—	2	3	no	—
557	C₄²	C₂×D₄×C₂₃	1	3	yes	—
558	C₄²	C₂×D₄×C₂₃	1	3	yes	—
559	C₄²	C₂⁴×C₂₃	1	3	yes	—
560	C₄²	C₂⁴×C₂₃	1	3	yes	—
561	C₂³	—	2	1	no	—
562	C₂³	—	2	3	no	—
563	C₂³	—	2	1	no	—
564	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
565	Q₈	—	2	6	no	—
566	Q₈	—	2	1	no	—
567	Q₈	—	4	6	no	—
568	Q₈	—	4	6	no	—
569	Q₈	—	4	6	no	—
570	Q₈	—	8	3	no	—
571	Q₈	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
572	Q₈	C₂₃×2₋¹⁺⁴	1	2	yes	complemented
573	D₄	—	2	4	no	—
574	D₄	—	2	1	no	—
575	D₄	—	2	1	no	—
576	D₄	—	2	3	no	—
577	D₄	—	2	1	no	—
578	D₄	—	2	3	no	—
579	D₄	—	4	6	no	—
580	D₄	—	4	6	no	—
581	D₄	—	4	2	no	—
582	D₄	—	4	6	no	—
583	D₄	—	8	3	no	—
584	D₄	—	8	3	no	—
585	D₄	C₂₃×2₋¹⁺⁴	1	2	yes	complemented
586	C₂×C₄	—	2	4	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	6	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	3	no	—
592	C₂×C₄	—	2	3	no	—
593	C₂×C₄	—	2	3	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	4	3	no	—
596	C₂×C₄	—	4	12	no	—
597	C₂×C₄	—	4	6	no	—
598	C₂×C₄	—	4	6	no	—
599	C₂×C₄	—	4	6	no	—
600	C₂×C₄	—	8	3	no	—
601	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
602	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
603	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
604	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
605	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
606	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
607	C₈	—	2	1	no	—
608	C₈	—	2	3	no	—
609	C₈	—	2	1	no	—
610	C₈	—	2	3	no	—
611	C₂²	—	2	1	no	—
612	C₂²	—	2	2	no	—
613	C₂²	—	2	1	no	—
614	C₂²	—	2	1	no	—
615	C₂²	—	4	1	no	—
616	C₂²	—	4	1	no	—
617	C₂²	—	4	3	no	—
618	C₂²	—	4	1	no	—
619	C₂²	—	4	3	no	—
620	C₂²	—	4	1	no	—
621	C₂²	—	4	1	no	—
622	C₂²	—	4	3	no	—
623	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
624	C₄	—	2	1	no	—
625	C₄	—	2	2	no	—
626	C₄	—	2	3	no	—
627	C₄	—	2	3	no	—
628	C₄	—	2	3	no	—
629	C₄	—	4	3	no	—
630	C₄	—	4	1	no	—
631	C₄	—	4	3	no	—
632	C₄	—	4	3	no	—
633	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	6	yes	—
634	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
635	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
636	C₂	—	4	1	no	—
637	C₂	—	4	2	no	—
638	C₂	—	8	3	no	—
639	C₂	—	8	1	no	—
640	C₂	C₂₃×#3486	1	1	yes	characteristic
641	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
642	C₂	C₂₃×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3497	C₂	semidirect
2. Q₈×#41×C₂₃	C₂	semidirect
3. C₂₃×#3386	C₂	semidirect
4. C₂₃×#3384	C₂	semidirect
5. Q₈×SD₁₆×C₂₃	C₂	semidirect
6. Q₈×D₈×C₂₃	C₂	semidirect
7. C₂₃×#3368	C₂	semidirect
8. C₂₃×#3366	C₂	semidirect
9. C₂₃×#3362	C₂	semidirect
10. C₂₃×#3358	C₂	semidirect
11. C₂₃×#3235	C₂	semidirect
12. C₂₃×#3228	C₂	semidirect
13. C₂₃×#3115	C₂	semidirect
14. C₂₃×#3106	C₂	semidirect
15. C₂₃×#2984	C₂	semidirect
239. #61840	C₂₃	direct (N × H)
251. Q₈×C₂₃	2₋¹⁺⁴	semidirect
264. D₄×C₂₃	2₋¹⁺⁴	semidirect
290. #3497	C₂×C₂₃	semidirect
291. Q₈×#41	C₂×C₂₃	semidirect
292. #3386	C₂×C₂₃	semidirect
293. #3384	C₂×C₂₃	semidirect
294. Q₈×SD₁₆	C₂×C₂₃	semidirect
295. Q₈×D₈	C₂×C₂₃	semidirect
296. #3368	C₂×C₂₃	semidirect
297. #3366	C₂×C₂₃	semidirect
298. #3362	C₂×C₂₃	semidirect
299. #3358	C₂×C₂₃	semidirect
300. #3235	C₂×C₂₃	semidirect
301. #3228	C₂×C₂₃	semidirect
302. #3115	C₂×C₂₃	semidirect
303. #3106	C₂×C₂₃	semidirect
304. #2984	C₂×C₂₃	semidirect
486. C₂₃	#61840	direct (N × H)
572. Q₈	C₂₃×2₋¹⁺⁴	semidirect
585. D₄	C₂₃×2₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3497	2
Q₈×#41×C₂₃	2
C₂₃×#3386	2
C₂₃×#3384	2
Q₈×SD₁₆×C₂₃	2
Q₈×D₈×C₂₃	2
C₂₃×#3368	2
C₂₃×#3366	2
C₂₃×#3362	2
C₂₃×#3358	2
C₂₃×#3235	2
C₂₃×#3228	2
C₂₃×#3115	2
C₂₃×#3106	2
C₂₃×#2984	2
#61840	23

Subgroups — C23×#61840 #19209466

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61840 #19209466